Noncommutative Algebra and Geometry
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Noncommutative Algebra and Geometry

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

Zuhair Nashed University of Central Florida Orlando, Florida

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

Marvin Marcus University of California, Santa Barbara

Walter Schempp Universität Siegen

W. S. Massey Yale University

Mark Teply University of Wisconsin, Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS Recent Titles G. Lumer and L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences J. Cagnol et al., Shape Optimization and Optimal Design J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra G. Chen et al., Control of Nonlinear Distributed Parameter Systems F. Ali Mehmeti et al., Partial Differential Equations on Multistructures D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra Á. Granja et al., Ring Theory and Algebraic Geometry A. K. Katsaras et al., p-adic Functional Analysis R. Salvi, The Navier-Stokes Equations F. U. Coelho and H. A. Merklen, Representations of Algebras S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory G. Lyubeznik, Local Cohomology and Its Applications G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications W. A. Carnielli et al., Paraconsistency A. Benkirane and A. Touzani, Partial Differential Equations A. Illanes et al., Continuum Theory M. Fontana et al., Commutative Ring Theory and Applications D. Mond and M. J. Saia, Real and Complex Singularities V. Ancona and J. Vaillant, Hyperbolic Differential Operators and Related Problems G. R. Goldstein et al., Evolution Equations A. Giambruno et al., Polynomial Identities and Combinatorial Methods A. Facchini et al., Rings, Modules, Algebras, and Abelian Groups J. Bergen et al., Hopf Algebras A. C. Krinik and R. J. Swift, Stochastic Processes and Functional Analysis: A Volume of Recent Advances in Honor of M. M. Rao S. Caenepeel and F. van Oystaeyen, Hopf Algebras in Noncommutative Geometry and Physics J. Cagnol and J.-P. Zolésio, Control and Boundary Analysis S. T. Chapman, Arithmetical Properties of Commutative Rings and Monoids O. Imanuvilov, et al., Control Theory of Partial Differential Equations Corrado De Concini, et al., Noncommutative Algebra and Geometry Alberto Corso, et al., Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects Giuseppe Da Prato and Luciano Tubaro, Stochastic Partial Differential Equations and Applications – VII

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Noncommutative Algebra and Geometry

Edited by

Corrado De Concini University of Rome Rome, Italy

Freddy Van Oystaeyen University of Antwerp/UIA Antwerp, Belgium

Nikolai Vavilov St. Petersburg State University St. Petersburg, Russia

Anatoly Yakovlev St. Petersburg State University St. Petersburg, Russia

Boca Raton London New York

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Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2349-X (Hardcover) International Standard Book Number-13: 978-0-8247-2349-1 (Hardcover) Library of Congress Card Number 2005049748 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Noncommutative algebra and geometry / edited by Corrado De Concini ... [et al.]. p. cm. -- (Lecture notes in pure and applied mathematics ; 243) Includes bibliographical references and index. ISBN 0-8247-2349-X (acid-free paper) 1. Noncommutative algebras--Textbooks. 2. Noncommutative rings--Textbooks. I. De Concini, Corrado. II. Lecture notes in pure and applied mathematics ; v. 243. QA251.4.N657 2005 512'.46--dc22

2005049748

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Introduction The international meeting at St. Petersburg was organized in honor of Prof. Dr. Z. Borevich, but there was no restriction on the topics of the lectures. A proceedings covering all subjects of the meeting would therefore constitute a rather inhomogeneous collection. The present volume, however, is mainly devoted to the contributions related to the ESF workshop organized in the framework of the scientific program “Noncommutative Geometry” of the European Science Foundation and integrated in the Borevich meeting. The topics dealt with here may be classified as noncommutative algebra. The congenial atmosphere at the meeting combined with the city’s preparations for the anniversary festivities provided the perfect setting for a very fruitful meeting. Moreover, the combination of the ESF workshop and the Borevich meeting brought together many participants from East and West (now perhaps old-fashioned terminology) engaging in open discussions, hard work, and the occasional party. Most of this may be blamed on the local organizers, Vavilov and Yakovlev, whom we thank for their great hospitality.

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Contributors Hans-Jochen Bartels Universitat Mannheim Mannheim, Germany

Lucchini, Andrea Dipto di Matematica University Brescia, Italy

Igor Burban Fachbereich Mathematik Kaiserslautern, Germany

Dmitry A. Malinin Belarusian State Pedag. University Minsk, Belarus

Eloisa Detomi Dipto di Matematica Universit Padova, Italy

Janvière Ndirahisha University of Antwerp (UIA) Department of Math and Computer Science Wilrijk, Belgium

Yuriy Drozd Kyiv Taras Shevchenko University Department of Mechanics and Mathematics Kyiv, Ukraine

Toukaiddine Petit University of Antwerp Department of Math and Computer Science Antwerp, Belgium

G. Griffith Elder University of Nebraska/Omaha Department of Mathematics Omaha, Nebraska

Tsetska G. Rashkova University of Rousse Center of Applied Math and Information Rousse, Bulgaria

Eivind Eriksen University of Warwick Institute of Mathematics Coventry, United Kingdom

Wolfgang Rump Universitat Stuttgart Institut f'ur Algebra und Zah Stuttgart, Germany

Michiel Hazewinkel CWI Amsterdam, The Netherlands

Freddy Van Oystaeyen University of Antwerp/UIA Department of Mathematics Antwerp/Wilrijk, Belgium

Lieven Le Bruyn Universiteit Antwerpen Department of Wiskunde and Informatica Antwerpen, Belgium

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Table of Contents Introduction .........................................................................................................................vii Finite Galois Stable Subgroups of GLn ................................................................................1 HANS-JOCHEN BARTELS, DMITRY A. MALININ

Derived Categories for Nodal Rings and Projective Configurations ..............................23 IGOR BURBAN, YURIY DROZD

Crowns in Profinite Groups and Applications ..................................................................47 ELOISA DETOMI, ANDREA LUCCHINI

The Galois Structure of Ambiguous Ideals in Cyclic Extensions of Degree 8................63 G. GRIFFITH ELDER

An Introduction to Noncommutative Deformations of Modules ....................................90 EIVIND ERIKSEN

Symmetric Functions, Noncommutative Symmetric Functions and Quasisymmetric Functions II ...........................................................................................126 MICHIEL HAZEWINKEL

Quotient Grothendieck Representations .........................................................................147 JANVIÈRE NDIRAHISHA, FREDDY VAN OYSTAEYEN

On the Strong Rigidity of Solvable Lie Algebras............................................................162 TOUKAIDDINE PETIT

The Role of a Theorem of Bergman in Investigating Identities in Matrix Algebras with Symplectic Involution ...............................................................................................175 TSETSKA G. RASHKOVA

The Triangular Structure of Ladder Functors ...............................................................184 WOLFGANG RUMP

Non-commutative Algebraic Geometry and Commutative Desingularizations..........203 LIEVEN LE BRUYN

Author Index ......................................................................................................................253

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FINITE GALOIS STABLE SUBGROUPS OF GLn H. -J. BARTELS1 AND D. A. MALININ2

Abstract. Let K/Q be a ﬁnite Galois extension with maximal order OK and Galois group Γ. We consider ﬁnite Γ-stable subgroups G ⊂ GLn (OK ) and prove that they are generated by matrices with coeﬃcients in OKab , Kab the maximal abelian subextension of K over Q. This implies in particular a positive answer to a conjecture of J. Tate on the classiﬁcation of p-divisible groups over Z and answers also a longstanding question of Y. Kitaoka on totally real scalar extensions of positive deﬁnite integral quadratic lattices.

Introduction The starting point of our investigations was the following problem studied by Y. Kitaoka and the ﬁrst named author around 1978 on the behaviour of the automorphism groups of positive deﬁnite quadratic Z-lattices under totally real scalar extensions. There was the Question. If two positive definite quadratic Z-lattices become isomorphic over the ring OK of integers of a totally real field extension K of the rationals Q, are they already isomorphic over Z, the ring of rational integers? Closely connected with this question was the following Conjecture 1. Let K/Q be a finite totally real Galois extension and denote by OK the corresponding ring of integers and let G ⊂ GLn (OK ) be a finite subgroup stable under the operation of the Galois group Γ = Gal(K/Q), then G ⊂ GLn (Z) holds, Z the ring of rational integers. There are several reformulations and generalizations of the above mentioned conjecture. One generalization is the following: Consider an arbitrary not necessarily totally real ﬁnite Galois extension K of the rationals Q and a free Z-module M of rank n n with basis m1 , . . . , mn . The group GLn (OK ) acts in a natural way on OK ⊗ M ∼ = i=1 OK mi . A ﬁnite group G ⊂ GLn (OK ) is said to be of k A-type, if there exists a decomposition M = i=1 Mi such that for every g ∈ G there exists a permutation Π(g) of {1, 2, . . . , k} and roots of unity i (g) such that i (g)gMi = MΠ(g)i for 1 ≤ i ≤ k. The following conjecture generalizes (and would imply) conjecture 1 and would also give a positive answer to the above mentioned question: Conjecture 2. Any finite subgroup of GLn (OK ) stable under the Galois group Γ = Gal(K/Q) is of A-type. For totally real ﬁelds K ± 1 are the only roots of 1 contained in K, and so conjecture 2 reduces to conjecture 1. Partial answers to these questions are given in [2], [3], [4], [8], [9], [10], [14], [16], [17], [19] (compare also the references in mentioned articles).

1991 Mathematics Subject Classification. Primary 20C10, 11R33, 11S23, 11R29.

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In an earlier version of this paper (see [4]) it is shown that conjecture 2 is true in the case of Galois ﬁeld extension K/Q with odd discriminant. Also some partial answers are given in the case of ﬁeld extensions K/Q which are un-ramiﬁed outside 2. The proof of the main part is essentially already contained in the article [17] of the second named author in slightly diﬀerent formulation. While [17] focusses mainly on the proofs of conjecture 1 and contains also some other related results, we observed that the proofs of conjecture 1 can immediately be transfered in order to proof conjecture 2 in the mentioned cases. Using the methods of [2], [3] and discriminant estimations of A. Odlyzko [23] in order to exclude the existence of certain Galois extensions having low ramiﬁcation, the ﬁrst named author proved in an unpublished note eighteen years ago, that conjecture 1 is true in the following cases: i) ii) iii) iv)

Γ = Gal(K/Q) = P SL2 (5) ∼ = A5 the alternating group of order 60, Γ = Gal(K/Q) = P SL2 (7) the simple group of order 168, K/Q is tamely ramiﬁed of degree ≤ 131 K/Q is tamely ramiﬁed of degree ≤ 233 assuming a generalized Riemann hypothesis to be true.

The combination of this approach using discriminant estimations with the far reaching results of [17] and [7] gave us the the following better results: Conjecture 1 is true in the following cases: i) [K : Q] ≤ 960 assuming the generalized Riemann hypothesis for the zeta function of the number ﬁeld K, or if ii) [K : Q] ≤ 480 unconditionally. Conjecture 2 is true if [K : Q] < 288 unconditionally. See [4] for the details. After ﬁnishing the ﬁrst version of our paper [4] we became aware of the recent work [20] of M. Mazur on the same topic. It turned out that in a certain sense the partial results of M. Mazur are complementary to our partial results. Using the the classiﬁcation of ﬁnite ﬂat group schemes over Z annihilated by a prime p for primes p ≤ 17 due to V. A. Abrashkin [1] and J.-M. Fontaine [6] the particular case of ﬁeld extensions K/Q which are unramiﬁed outside 2 follows in full generality from [20]. In this revised version of our paper we restrict therefore ourselves to the case of ramiﬁed primes p = 2. It should be noted that conversely our Main Theorem in combination with the work of M. Mazur has interesting consequences for the classiﬁcation of ﬁnite ﬂat commutative group schemes over Z annihilated by a prime p: It answers a question of J. Tate [28] also for primes p ≥ 17 completing the partial results of Abrashkin [1] and Fontaine [6]. It is interesting to notice that the methods used in the proofs, namely the detailed study of the operation of the higher ramiﬁcation groups of the Galois group on the given Galois stable group G for the ramiﬁed primes in the ﬁeld extension K over Q together with discriminant estimations, in order to eliminate ramiﬁcation with large depth using trivial action of higher ramiﬁcation groups (compare [2] section 1), are similar to the methods used by [1] and [6]. This paper is organized as follows: Section I contains the results and the propositions and lemmata used in the proofs. The proofs themselves are presented in Section II. As far as it is needed the necessary parts of the proofs from [17] are reproduced only slightly changed in this paper for the convenience of the reader. Acknowledgement: The second author is grateful to DAAD for support. Helpful comments from an anonymous referee to an earlier version of this paper are also gratefully acknowledged.

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Notation Q, Qp , Z, Zp , OK denote the ﬁeld of rationals and p-adic rationals, the ring of rational and p-adic rational integers respectively, and the ring of integers of an algebraic number ﬁeld K. to be the intersection of valuation rings of all ramiﬁed prime ideals p ∈ OK We consider OK (if K = Q). T rK/L denotes the trace map from K to L. GLn (R) denotes the general linear group over R. [E : F ] denotes the degree of the ﬁeld extension E/F . Im denotes the unit m × m-matrix, 0n,m and 0m are zero n × m and m × m-matrices, ei,j are square matrices having the only nonzero element 1 in the position (i, j), rankM and detM are rank and determinant of a matrix M . t M denotes a transposed matrix for M, diag(d1 , d2 , . . . , dm ) is a block-diagonal matrix having diagonal components d1 , d2 , . . . , dn . We suppose that K is a Galois extension of the rationals Q. We denote by Γ the Galois group of a normal extension K/F ; if needed we specify K/F as a subscript in ΓK/F . The symbols Γi (p) denote the i-th ramiﬁcation groups of the prime divisor p and Γ0 (p) the inertia group in Γ, ei is the order of Γi (p) for i ≥ 1, while e = e0 is the order of the inertia group. For Γ acting on G and any σ ∈ Γ and g ∈ G we write g σ for the image of g under σ-action. If G is a ﬁnite linear group, F (G) denotes the ﬁeld obtained by adjoining the matrix coeﬃcients of all matrices g ∈ G. Throughout this paper ζm denotes a primitive m-th root of unity. 1. Statement of the main results 1.1. Let E/F be a normal extension of algebraic number ﬁelds, and let ΓE/F = Gal(E/F ) be its Galois group. We consider the problem of integral realizations of ﬁnite subgroups G of the general linear group GLn (E) that are stable under the natural action of ΓE/F on the matrices of the group G. Let OF and OE denote the maximal orders of the number ﬁelds F and E respectively. Let us introduce the class C(F ) of ﬁelds normal over F that are obtained by adjoining to F all coeﬃcients of matrices contained in some ﬁnite ΓE/F -stable group G ⊂ GLn (OE ). In [3] it is shown that if F = Q and the class C(Q) contains some ﬁeld K = Q, then C(Q) will also contain some ﬁeld K1 = Q, K1 ⊂ K such that there exists only one prime p ramiﬁed in K1 . In this paper we use some properties of Galois groups for ﬁelds having restricted ramiﬁcation. In general, the existence of global ﬁelds with a given Galois group and prescribed local properties for ramiﬁcation is a rather subtle question. L. Moret-Bailly proved the existence of extensions of number ﬁelds that have prescribed local structure of ramiﬁcation over a given set of prime divisors and unramiﬁed elsewhere for certain relative extensions [22]. In our case we deal with absolute extensions of the rationals K/Q, and we ﬁx the only ramiﬁed prime p. Let Cp (Q) denote the class of ﬁelds in C(Q) with the unique ramiﬁed prime p. Nilpotent extensions of Q having this property were described by Markshaitis in [18], but there are many examples of extensions in Cp (Q) that are not nilpotent, and also nonsolvable extensions unramiﬁed outside p; for this and also for non-existence theorems compare [27], [7]. Both conjectures 1 and 2 are true for nilpotent extensions K/Q (see [3], [8]), and the proof of this fact uses the special structure of the Galois group of nilpotent extensions unramiﬁed outside a prime p [18]. 1.2. It is well known, that the problem of description of ﬁelds Q(G) can be reduced to the case of commutative groups G of exponent p. Compare Proposition 1 in [17] and section 3 of [19] and [20] chapter 4. The idea of this reduction appears already in [14], [15], [13] and [10] where it was used, in particular, to study conditions for coeﬃcients of the representations of nilpotent groups over integral rings providing their diagonalizability. Hence, if there would be a counterexample to conjecture 1 or conjecture 2, there would exist also an elementary abelian p group G as a counterexample.

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We use also reduction to the case of a GLn (Q)-irreducible group G. Here a matrix group G is reducible in GLn (R) or simply R-reducible (R a ring or a ﬁeld) if there exist h ∈ GLn (R) such that G ∗ , h−1 Gh ⊂ 1 0 G2 , and G is irreducible otherwise. We note that the reduction to the case of an irreducible group G can be done using the following lemma: Lemma 1.2.1. Let E/F be a normal extension of algebraic number fields with Galois group ΓE/F = Gal(E/F ) and let E1 , F1 be rings with quotient fields E and F respectively. If G ⊂ GLn (E1 ) is a finite ΓE /F -stable subgroup which has GLn (F1 )-irreducible components G1 , G2 , . . . , Gr , then F (G) is the composite of the fields F (G1 ), F (G2 ), . . . , F (Gr ). The proof of this Lemma is given at the beginning of section II. 1.3. The essential results of this note can be summarized as follows: Main Theorem. Let K be a finite Galois extension of Q and G be a finite subgroup of GLn (OK ) that is stable under the natural action of the Galois group Γ of the field K. Then G is of A-type and in particular G ⊂ GLn (OKab ) holds, Kab the maximal abelian subextension of K over Q. Let µp denote the multiplicative group scheme over Z of order p and αp the constant group scheme of order p (see [28] and [1]). Due to the results of [1] and [6] in conjunction with [20] one gets immediately the following Corollary 1. If G is a finite flat commutative group scheme over Z annihilated by a prime p, then it is a direct sum of copies of µp , αp and, if p = 2, the nontrivial element in Ext(α2 , µ2 ). We can also express the result of the Main Theorem in the following form: Corollary 2. A finite flat group scheme G over Z satisfies G(Q) = G(Qab ), Q the algebraic closure of Q and Qab the maximal abelian (over Q) subextension of Q. For the proof of the Main Theorem we distinguish essentially two cases and for their treatment we need several results which are recorded in the subsequent sections 1.4 and 1.5. The ﬁrst Proposition 1 gives a criterion for the existence of integral realizations of an abelian matrix group. It shows that the existence of G in question is possible only if certain determinants dk are divisible by the root of the discriminant D of a certain extension of number ﬁelds (for the details see section 1.4 below). In the proof of the Main Theorem in section II we use this for a certain cyclic extension E/F which is tame with respect to a ﬁxed prime ideal (case I). Assume that E/Q is not abelian. Then we can make E/F to be a Kummer extension via adjoining √ appropriate roots of 1. We use the explicit Kummer basis to ﬁnd an index k for which D does not divide dk . The proof of the Main Theorem is divided in to two parts depending on the ramiﬁcation index e = e0 of Q(G). In the ﬁrst part we use Proposition 1. In the second part we use lemma 1.5.2 and the Corollary 1.5.3 of section 1.5.

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We can sketch the scheme of the proof of the Main Theorem:

Let us outline the idea of the proof of the Main Theorem in more detail for the convenience of the reader.

The outline of the proof of the Main Theorem. In virtue of the argument of [3], lemmata 1 and 2 (compare also Theorem 2 in [19]), we can assume that K is unramiﬁed outside a prime p, so we can ﬁx this prime. Since as already remarked in the introduction the particular case of ﬁeld extensions K/Q which are unramiﬁed outside 2 follows in full generality from [20], we can restrict ourself to the case p > 2. We can also assume that G is an abelian group of exponent p, and we can consider G to be irreducible under conjugation in GLn (Q) by Corollary 1.4.1. The proof of the Main Theorem consists of a reduction to special cases, and these special cases are treated with diﬀerent methods. , OL denote the semilocal rings that are obtained by For number ﬁelds E, L be let OE intersection of the valuation rings of all ramiﬁed prime ideals in the rings OE , OL respectively. These semilocal rings are known to be principal ideal domains. Denote G0 = GΓ1 (p) the subgroup of elements in G that are ﬁxed by the ﬁrst ramiﬁcation group Γ1 (p) for some prime divisor p of p. Let e0 be the ramiﬁcation index of Q(G0 ) over Q with respect to p. Then e0 e0 /e1 (= the index of Γ1 (p) in Γ0 (p).) Case I. Assume that e0 does not divide p − 1. In this case we apply Proposition 1 to a certain subgroup G0 ⊂ GΓ1 (p) ⊂ GLn (OE ) for a certain cyclic Kummer extension E/F with a i convenient power basis π , i = 0, . . . , t − 1 and with the explicit action of the generating , namely element σ of order t of the Galois group on the uniformizing element π of OE σ π = πζt , which is convenient for applying Proposition 1 explicitly. Here E and F are the ramiﬁcation ﬁeld and the inertia ﬁeld for some prime divisor p of p adjoined by a primitive t-root of 1, t = e0 . Denote ΓE /F the Galois group of E/F . In case I we determine a ΓE /F -stable subgroup G0 ⊂ G0 which is generated by all conjugates hγ , γ ∈ ΓE /F of some element h ∈ G0 . G0 can not be cyclic provided t = e0 does not divide p − 1, and this is just the case where

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the arguments in case II (see below) can not be applied. So we start the proof of the Main Theorem just from this most diﬃcult case, and apply Proposition 1 to a subgroup G0 ⊂ G. We show that case I is impossible since the conditions of Proposition 1 never hold true for G0 and the extension E/F . In particular, if e0 does not divide p − 1 we have a contradiction with the condition G ⊂ GLn (OE ) which can not hold true since G0 ⊂ GLn (OE ). Case II. Let us suppose that e0 divides p − 1. In this case we can suppose without loss of generality, that K contains a p-th root of unity ζp (see Lemma 2.2.2 below). Using a local argument on the diagonalization of matrices which are congruent to In modulo the prime ideal p (see Corollary 1.5.3 below) a certain subgroup G1 in G is constructed such that K Γ1 (p) (G1 ) is an extension of K Γ1 (p) with ζp ∈ K Γ1 (p) (G1 ), tame ramiﬁcation index p − 1 and K Γ1 (p) (G1 )/K Γ1 (p) is an elementary abelian Kummer extension. In a second step a careful study of the Galois-action of Γ0 (p) on G1 shows that the constructed group G1 can not exist. This gives then the desired contradiction. 1.4. In this section we formulate the mentioned criterion for the existence of an integral realization of an abelian group G with the properties mentioned above. Let E, L be ﬁnite Galois extensions of the number ﬁeld F that are diﬀerent from F with , OL be the semilocal rings Galois groups ΓE/F and ΓL/F respectively. As above let OE that are obtained by intersection of the valuation rings of all ramiﬁed prime ideals in the . Let w1 , w2 , . . . , wt be a basis of OE over OF , and rings OE , OL , and let OF = F ∩ OE let D be the discriminant of this basis. Suppose that some matrix g of prime order p has coeﬃcients in E and all ΓE/F -conjugates g γ , γ ∈ ΓE/F generate a ﬁnite abelian group G of exponent p. Let σ1 = 1, σ2 , . . . , σt denote all automorphisms of the Galois group ΓE/F of the ﬁeld E over F . Assume that L = E(ζ(1) , ζ(2) , . . . , ζ(n) ) where ζ(1) , ζ(2) , . . . , ζ(n) are the eigenvalues of the matrix g, therefore L = E(ζp ), ζp a primitive p-th root of unity. We will reserve the same notations for some extensions of σi to L, and the automorphisms of L/F will be denoted σ1 , σ2 , . . . , σr for some r t. Let E be a numberﬁeld containing F (G) which is obtained by adjoining to F all coeﬃcients of all g ∈ G. For a suitable choice of t elements of ζ(1) , ζ(2) , . . . , ζ(n) say ζ(1) , ζ(2) , . . . , ζ(t) we can prove the following Proposition 1. 1) Let G be generated by all g γ , γ ∈ ΓE/F and irreducible under GLn (F ) conjugation. Then G is conjugate in GLn (F ) to a subgroup of GLn (OE ) if and only if all determinants w1 . . . wk−1 ζ(1) wk+1 · · · wt σ2 σ2 σ2 σ2 w1 · · · wk−1 ζ(2) wk+1 · · · wtσ2 dk = det . .. σ w t · · · w σt ζ σt w σt · · · w σt t 1 k−1 (t) k+1 √ are divisible by D in the ring OL . 2) If any of the three sets of conjugates {g γ , γ ∈ ΓE/F }, {hγ , γ ∈ ΓE/F }, {(gh)γ , γ ∈ ΓE/F } generates G and the corresponding eigenvalues of g and h given in 1) are g g g h h h , ζ(2) , . . . , ζ(t) and ζ(1) , ζ(2) , . . . , ζ(t) respectively, then the eigenvalues for the matrix gh ζ(1) gh g gh g gh h h in 1) can be chosen as products ζ(1) = ζ(1) = ζ(1) ζ(1) , ζ(2) = ζ(2) = ζ(2) ζ(2) , . . . , ζ(t) = ζ(t) = g h ζ(t) . ζ(t)

Note that the conditions of Proposition 1 are always true if E is unramiﬁed over F since = OE in this case. DOE

FINITE GALOIS STABLE SUBGROUPS OF GLn

7

Corollary 1.4.1. If there is an abelian ΓE/F -stable subgroup G ⊂ GLn (OE ) of expoγ nent p generated by g , γ ∈ ΓE/F such that E = F (G) = F , then the GLn (F )-irreducible components Gi ⊂ GLni (E), i = 1, . . . , k of G are conjugate in GLni (F ) to subgroups ) such that E = F (G1 )F (G2 ) . . . F (Gk ). In particular, F (Gi ) = F for some Gi ⊂ GLni (OE indices i.

The following corollary shows that the conditions of Proposition 1 hold true even if G is not irreducible. Corollary 1.4.2. Let E/F be a normal extension of number fields with Galois group ΓE/F . Let G ⊂ GLn (E) be an abelian ΓE/F -stable subgroup of exponent p generated by g and all matrices g γ , γ ∈ ΓE/F , and let E = F (G). Then G is conjugate in GLn (F ) to G ⊂ GLn (OE ) if and only if all eigenvalues of matrices Bi , i = 1, . . . , t are contained in OL , where L = E(ζp ). The latter happens if and only if the criterion of Proposition 1, 1) holds true, i.e. all determinants w1 . . . wk−1 ζ(1) wk+1 · · · wt σ2 σ2 σ2 σ2 σ2 w1 · · · wk−1 ζ(2) wk+1 · · · wt dk = det . .. σ w t · · · w σt ζ σt w σt · · · w σt t 1 k−1 (t) k+1 are divisible by

√ D in the ring OL .

Corollary 1.4.3. Let F = Q. If there is an abelian ΓE/Q -stable subgroup G ⊂ GLn (OE ) of exponent p generated by g γ , γ ∈ ΓE/Q such that E = Q(G) = Q, then the GLn (Q)irreducible components Gi ⊂ GLni (E), i = 1, . . . , k of G are conjugate in GLni (Q) to subgroups Gi ⊂ GLni (OE ) such that E = Q(G1 )Q(G2 ) . . . Q(Gk ). In particular, Q(Gi ) = Q for some indices i. 1.5. For the proof of the Main Theorem (more precisely for the part of the proof dealing with case II) we use a lemma which is a variation on a theme of Minkowski [21] and is – like in the earlier related work [2], [3] - the key ingredient in the proofs of Lemma 1.5.2 and the Main Theorem. For the proof see [11]. Compare also [19], Proposition 1. Lemma 1.5.1. Let J be an ideal in Dedekind ring S of characteristic χ, 0 = J = S, let g be an n × n-matrix of finite order congruent to In (mod J). j

(i) If χ = p > 0, then g p = In for some integer j. If χ = 0, then J contains a prime j number p and g p = In , i ∈ Z. In particular, any finite group of matrices congruent to In (mod J) is a p-group. (ii) Let χ = 0, J = p be a prime ideal having the ramification index e with respect to p, g ≡ In (mod pr ) and mpi−1 (p − 1) ≤ e/r < pi (p − 1), i ≥ 0, m = min{1, i}. Then i g p = In . In particular, any finite group of matrices congruent to In (mod pt ) is trivial if e < t(p − 1). Related to these properties is the following Lemma 1.5.2. Let O be a Dedekind ring in an algebraic number field, and let ζp ∈ O. Let p = pe , e = p − 1. Let G be a finite subgroup of GLn (O) and g ≡ In (mod p) for all g ∈ G. Then G is conjugate in GLn (O) to an abelian group of diagonal matrices of exponent p.

8

H. -J. BARTELS AND D. A. MALININ

Corollary 1.5.3. Let L be an extension of Q and p a prime ideal in the field L(ζp ). Suppose that L is unramified at p and let Op denote the valuation ring of the ramified prime ideal p in L(ζp ). Let Γ denote the Galois group of L(ζp ) over L. If G is a finite Γ-stable subgroup of GLn (Op ) consisting of matrices g, g ≡ In (mod p), then G is conjugate in GLn (L ∩ Op ) to an abelian group of diagonal matrices of exponent p. 2. Proofs 2.1. Proof of Lemma 1.2.1. Let G1 ∗ .. h−1 Gh ⊂ . 0 Gr for h ∈ GLn (F1 ). If there exists g ∈ G such that g γ = g for some automorphism γ of F (G) over F (G1 )F (G2 ) . . . F (Gr ), then g = g γ g −1 = In . The blocks Gi in h−1 Gh are stable under the action of γ, since h ∈ GLn (F1 ) and the elements of F (Gi ) are ﬁxed by γ. Because g1 ∗ h−1 gh = . . . 0 gr and g1 ∗ (h−1 gh)γ = h−1 g γ h = . . . 0 gr are matrices having the same diagonal components, all eigenvalues of the matrix g = g γ g −1 of ﬁnite order are 1 and hence g = In . This contradiction completes the proof of Lemma 1.2.1. Proof of Proposition 1. One proof (namely of the ﬁrst part) is given in the paper [17]. The second part of proposition 1, which is important for the proof of the Main Theorem, follows from the construction given in [17]. But for convenience we give here a proof for the proposition, which is shorter than in [17]. over OF we can write Using the basis w1 , . . . , wt of OE g σj =

t

wi σj Bi

for j = 1, . . . , t

i=1 σ

with semisimple matrices Bi ∈ Mn (F ). Since the matrix W = [wi j ]j,i is nondegenerate, the matrices Bi can be expressed as a linear combination of g σj , i, j = 1, 2, . . . , t: Bi =

t j=1

mij g σj ,

FINITE GALOIS STABLE SUBGROUPS OF GLn

9

where [mij ] = W −1 . Since by assumption the matrices g σj commute pairwise, all matrices Bi also commute with each other. The irreducibility of G implies that the minimal polynomial of Bi is irreducible over F for each i such that Bi is not zero (see [26], page 8, Corollary 3 for then all of them are since they are Galois example). So if one of the eigenvalues of Bi is in OL ∗ ∗ conjugate. Using the dual basis w1 , . . . , wt to w1 , . . . , wt with respect to the traceform one σ can see that the inverse matrix W −1 to W = [wi j ]j,i is of the form W −1 = [wj∗σi ]j,i . In order to prove the claim of the proposition, we need to determine whether or not matrices Bi , i = 1, . . . , t are conjugate in GLn (F ) to matrices Bi ∈ Mn (OF ), since for the generator g of G the equation g = B1 w1 + B2 w2 + · · · + Bt wt , holds with Bi ∈ Mn (F ) and w1 , . . . , wt a basis of OE over OF . In fact each semisimple matrix Bi ∈ Mn (F ) is conjugate in GLn (F ) to a matrix from Mn (OF ) if and only if all its (see Lemma 2.1.1 below). eigenvalues are contained in OL ∗σ Cramer’s rule now implies that wi j = (−1)i+j Wi,j det(W )−1 , where Wi,j is the (i, j)minor of W . Over the splitting ﬁeld L there is a basis which consists of eigenvectors for G. Let u be one such common eigenvector with

g σi u = ti u. σ −1

Then ζ(i) := ti i with eigenvalue

is an eigenvalue of g. It also follows, that u is an eigenvector for Bk

λk =

t j=1

mkj tj =

t j=1

σ

(−1)j+k Wj,k ζ(j)j det(W )−1 .

The cofactor expansion for determinants implies λk = dk /detW and therefore the eigenval iﬀ detW divides dk , which proves the criterion of Proposition 1 and - by ues of Bk are in OL deﬁnition of the eigenvalues ti - also the second statement modulo the proof of the following Lemma 2.1.1. i) Let all eigenvalues λj , j = 1, 2, . . . , k of the semisimple matrices Bi ∈ Mn (F ), i = 1 . . . , t be contained in the ring OL for some field L ⊃ F . Then Bi are conjugate in GLn (F ) simultaneously to matrices that are contained in Mn (OF ). ii) Conversely, if the semisimple matrices Bi are contained in Mn (OF ) and Bi are diag . onalizable over a field L ⊃ F , then their eigenvalues are contained in OL Proof of Lemma 2.1.1. i) By the virtue of [26], chapter 1, sect. 1, corollary 2 we can consider A to be a ﬁeld extending F . Let a1 , a2 , . . . , an be a basis of OA over OF . Then for any B ∈ A . we have B = b1 a1 + · · · + bn an , and the elements bi ∈ F are contained in OF iﬀ B ∈ OA But all coeﬃcients kij of the characteristic polynomials fi (x) = ki0 + ki1 x + · · · + kin xn of the matrices Bi are contained in OL , and kin = 1, so Bi ∈ A are integral over F . It follows that Bi = bi1 a1 + · · · + bin an , and bij ∈ OF . If υ ∈ F n is a non-zero vector in F n , then a1 υ, a2 υ, . . . , an υ is a basis of F n , and Bi aj υ = Σk cijk ak υ, where cijk ∈ OF . It follows that for any i the matrix Ci = [cijk ]k,j belongs to GLn (OF ), and Ci is the matrix of the operator Bi in the basis a1 υ, a2 υ, . . . , an υ of F n . Therefore, Bi is conjugate in GLn (F ) to Ci for any i = 1, . . . , t. ii) Consider the characteristic polynomials fi (x) = ki0 +ki1 x+· · ·+kin xn of the matrices . This completes the Bi . Since kin = 1 and all kij are in OF all roots of f (x) are in OL proof of Lemma 2.1.1.

10

H. -J. BARTELS AND D. A. MALININ

Remark. In the situation of Lemma 2.1.1, i) the F -algebra A = F [B1 , . . . , Bt ] is isomorphic to the field L = F [λ1 , . . . , λk ] where λj , j = 1, 2, . . . , k are all eigenvalues of the matrices Bi , i = 1 . . . , t. Proof of Corollary 1.4.1. If G ⊂ GLn (OE ) is a group of exponent p and g = B1 w1 + over OF , then Bi ∈ Mn (OF ), and it follows B2 w2 + · · · + Bt wt for a basis w1 , . . . , wt of OE . But eigenvalues are from Lemma 2.1.1 that the eigenvalues of Bj are contained in OL preserved under conjugation, so the latter claim is also true for all components Gi . We can apply Proposition 1 to Gi , i = 1, . . . , k. It follows that Gi are conjugate to subgroups ). Now, Lemma 1.2.1 implies E = F (G1 )F (G2 ) . . . F (Gk ). This completes Gi ⊂ GLni (OE the proof of Corollary 1.4.1.

Proof of Corollary 1.4.2. Let G1 ∗ .. C −1 GC = . 0 Gk for C ∈ GLn (F ) and irreducible components Gi ⊂ GLni (E), i = 1, . . . , k. Then for g = B1 w1 + B2 w2 + · · · + Bt wt g1 ∗ C −1 gC = . . . = B1 w1 + B2 w2 + · · · + Bt wt 0 gk holds with Bi = C −1 Bi C. Let us consider the F -algebra A generated by all Bi , i = 1, . . . , t over F . Since A is semisimple, it is completely reducible. It follows that matrices Bi are simultaneously conjugate in GLn (F ) to the block-diagonal form. Therefore, G is conjugate in GLn (F ) to a direct sum of its irreducible components Gi . Since E ⊂ F (Gi ) for all i, and contains all rings OF (Gi ) , we can apply Proposition 1 to each of them. Proposition 1 OE implies that each Gi is conjugate in GLni (F ) to Gi ⊂ GLni (OE ) if and only if all eigenvalues of matrices Bi , i = 1, . . . , t are contained in OLi , where Li = F (Gi )(ζp ) and this happens iﬀ w1 . . . wk−1 ζ(1) wk+1 · · · wt σ2 σ2 σ2 σ2 σ2 w1 · · · wk−1 ζ(2) wk+1 · · · wt dk = det . .. σ w t · · · w σt ζ σt w σt · · · w σt t 1 k−1 (t) k+1 √ are divisible by D in the ring OL . But F (G) = F (G1 )F (G2 ) . . . F (Gk ) by the Lemma in section 1.2, and so L = L1 L2 . . . Lk . This completes the proof of Corollary 1.4.2. Proof of Corollary 1.4.3. The argument of the proof of Corollary 1.4.1 remains true for the rings of integers OE and Z in E and F = Q since Z is a principal ideal domain and OE has a free basis over Z. Therefore, the rest of the proof of Corollary 1.4.3 reproduces the proof and OF respectively. of Corollary 1.4.1 with OE and Z instead of OE

FINITE GALOIS STABLE SUBGROUPS OF GLn

11

2.2. Proof of the Main Theorem. Let us suppose that there exist a counterexample G to the Main Theorem with corresponding Galois extension K/Q, K = Q(G) with Galois group Γ := ΓK/Q . In virtue of Lemmas 1 and 2 in [3] or Theorem 2 in [19] we can assume the ﬁeld K to be unramiﬁed outside the ﬁxed prime p. Since as already remarked above the particular case of ﬁeld extensions K/Q which are unramiﬁed outside 2 follows in full generality from [20], we can restrict our self to the case p > 2. Because of the Proposition in section 1.2 we can also suppose that G is an abelian group of exponent p and we can consider G to be irreducible under conjugation in GLn (Q) by Corollary 1.4.3. Let us assume that G is a counterexample of minimal order of this kind. With the notation of the beginning of this note let Γi (p) ⊂ Γ denote the i-th ramiﬁcation groups of the prime divisor p for i ≥ 1 and Γ0 (p) the inertia group in Γ. Let G0 = GΓ1 (p) denote the subgroup of elements in G that are ﬁxed by the ﬁrst ramiﬁcation group Γ1 (p) for some prime divisor p of p. Let e0 be the ramiﬁcation index of Q(G0 ) over Q with respect to p. Then e0 e0 /e1 (= the index of Γ1 (p) in Γ0 (p).) We distinguish two cases: Case I : e0 does not divide p − 1 and Case II : e0 is a divisor of p − 1. Case I. e0 does not divide p − 1. 1) In this case, where e0 does not divide p − 1, let us ﬁx p and one of its ramiﬁed prime divisors say p. Let E1 and F1 denote the subﬁelds of Γ1 (p)-ﬁxed elements and Γ0 (p)ﬁxed elements of K respectively. We will prove that for p = 2 and a ﬁeld K which has discriminant pj , j ∈ Z, all Γ0 (p)/Γ1 (p)-stable ﬁnite subgroups G of GLn (OE1 ) are already in GLn (OF1 ) for E1 = F1 (GΓ1 (p) ) = F1 (G0 ) ⊂ K Γ1 (p) and F1 = K Γ0 (p) . We can extend the ground ﬁeld F1 by adjoining ζt , t = e0 . Set E = E1 (ζt ) and F = F1 (ζt ). We obtain a cyclic extension E/F such that ζt ∈ F for t = e0 . Since K is unramiﬁed outside p, Q(ζt ) and K have intersection Q and therefore we can identify the Galois group ΓE/F = Gal(E/F ) with the Galois group Gal(E1 /F1 ). With respect to this extension of the corresponding Galois action to E/F we obtain a ΓE/F − stable group G0 ⊂ GLn (OE ). E/F is a tame extension with respect to p, t = e0 is its ramiﬁcation index and p − 1 ≥ 2. We have the following conditions for local e ramiﬁcation: pE0 = (p) = (ζp − 1)p−1 as ideals of the ring OEp (ζp ) , where pE is the e0 prime divisor of p in p-adic completion Ep of E. It is clear that + 1 (p−1) > e0 . 2 Hence p[t/2]+1 does not divide (ζp − 1) as ideals of OE(ζp ) . We can also assume that G is an abelian p-group of exponent p, and E = F because e0 > 1 in the case I. We and OF use the statement of Proposition 1 and its Corollary 1.4.2 for the rings OE t−1 t and a basis 1, π, . . . , π , such that π ∈ F . If ΓE/F , the Galois group of E/F , is generated by an element σ of order t, we can consider the action of ΓE/F on the basis 1, π, . . . , π t−1 in the following way: (π i )σ = π i ζti . Then det W = π t(t−1)/2

(ζtj − ζti ).

1i<jt

Let us consider the determinants of the matrices Wj that are obtained from W by j changing elements of j-th column of W = [(π i )σ ]i,j to appropriate p-roots ζ(1) , ζ(2) , . . . , ζ(t) i of 1 that are the eigenvalues of the matrices g σ , i = 1, 2, . . . , t for some g ∈ G, according to Proposition 1. For simplicity let ζ = ζt , but reserve previous notation for ζp for the rest of this proof. Recall, that G is supposed to be a minimal counterexample to the Main Theorem and that K is unramiﬁed outside p. In the proof of the Case I we pick g ∈ G0 = GΓ1 (p) and a

12

H. -J. BARTELS AND D. A. MALININ

generator σ of the Galois group of E over F ; by our assumption, the order t of σ does not divide p − 1. There is a matrix g ∈ G0 such that matrices g γ , γ ∈ Γ generate G. Indeed, if matrices g γ , γ ∈ Γ generated a proper subgroup G1 of G for any g ∈ G0 , then G1 would be a group of A-type, since G is a minimal counterexample, and the order of e0 would divide p − 1 (because Q(G1 )/Q is unramiﬁed outside p and tamely ramiﬁed at p), contrary to the assumption of the Case I. Let us ﬁx the above G and σ. We need the following auxiliary lemma which speciﬁes the option of g for our proof of the case I: Lemma 2.2.1. Let k be an integer such that 0 < k < p. There is a matrix g ∈ G0 such that matrices g γ , γ ∈ Γ generate G, and the group G is generated by all hγ , γ ∈ Γ, where h := g k g σ . Proof of Lemma 2.2.1. Take a matrix g ∈ G0 such that matrices g γ , γ ∈ Γ generate G. If a group H generated by all hγ , γ ∈ Γ is a proper subgroup of G, it is a group of A-type, and it is ﬁxed elementwise by the commutator subgroup Γ of Γ. Then g σ = g −k h = g l h 2 2 p−1 p−1 for l ≡ −k(modp). We have g σ = g l hl hσ , . . . , g σ = g l h0 = gh0 for some matrix h0 having coeﬃcients ﬁxed by Γ . Since h ∈ G0 , G0 is ﬁxed by Γ1 (p) and K is unramiﬁed p−1 i(p−1) = ζp , and we also have g σ = ghi0 , so outside p, we have h ∈ GLn (Q(ζp )). But ζpσ p(p−1)

= g. The same argument is true for elements g1 , h1 such that for i = p we obtain g σ p(p−1) = g1 . But G0 is g1 = g τ ∈ G0 (τ ∈ Γ) and h1 = g1k g1σ taken instead of g, h. We have g1σ covered by subgroups generated by all elements g1 = g τ since G is generated by elements g1 = g γ , γ ∈ Γ. Therefore, σ p(p−1) acts trivially on G0 . But the order of σ is coprime to p. We conclude that the order of σ divides p − 1, which contradicts the assumption of the Case I. It follows that either the group H or the group H1 generated by all hγ1 , γ ∈ Γ coincides with G. In the latter case we can rename matrix g1 to g. This completes the proof of Lemma 2.2.1. We distinguish the cases of odd and even t, the order of σ. If t is odd, we need a matrix g having at least one eigenvalue θi = ζ(i) = 1 (we use notations of Proposition 1) such that G is generated by all conjugates g γ , γ ∈ Γ. For an even t we have to choose g = g k g σ ζps . The choice of the eigenvalues ζ(i) (see Proposition 1) ensures that the product of the corresponding eigenvalues are in accordance with the product of two matrices h1 , h2 ∈ G (compare the proof of Proposition 1). Now, we intend to replace G0 by a smaller subgroup G0 generated by a single element of G0 which also satisﬁes the conditions of the Case I. G0 is covered by its ΓE/F -stable subgroups Gγ , where Gγ are generated by elements γ σi (ˆ g ) , i = 1, 2, . . . , t for some γ ∈ Γ and any gˆ such that gˆγ ∈ G0 and all gˆτ , τ ∈ Γ, generate G. By deﬁnition, Gγ is generated by the orbit of an element g having the above property. But if h satisﬁes the conditions of the above Lemma, the elements gˆτ , τ ∈ Γ −1 i generate G for gˆ = hγ , so we can assume that Gγ is generated by elements hσ , i = 1, . . . , t for a given γ and some h ∈ G satisfying the conditions of the above Lemma. Since the ramiﬁcation index with respect to p of the composite of the ﬁelds F (Gγ ), γ ∈ Γ, does not divide p − 1, there is γ ∈ Γ such that the ramiﬁcation index e(F (Gγ )/F ) of F (Gγ ) does not divide p − 1. Let us brieﬂy explain this claim. The ﬁeld F (G0 ) is a composite of ﬁelds Ei = F (Gγi ), and F (G0 )/F is a cyclic totally ramiﬁed extension whose Galois group is generated by an element σ of order t equal to the ramiﬁcation index of F (G0 )/F in p. So Ei /F are also cyclic totally ramiﬁed extensions, and their Galois groups are generated by elements σi of orders equal to the ramiﬁcation indices ti of Ei /F . Therefore, if all ti divide p − 1, then the order of σ must also divide p − 1, because σ is a product of pairwise commuting elements of orders ti . This completes the proof of our claim.

FINITE GALOIS STABLE SUBGROUPS OF GLn

13

Let us ﬁx γ and denote G0 = Gγ . The group G0 is not cyclic since the order of σ does not divide p − 1 in the case I. Using Proposition 1 or, alternatively, Corollary 1.4.1 or Corollary 1.4.2 of Proposition 1, we will prove that G0 ⊂ GLn (OF ). Below we use ΓE/F -stability of i G0 in order to apply Proposition 1 to G0 ⊂ G0 generated by all (hγ )σ , i = 1, 2, . . . , t for the ﬁxed γ ∈ Γ. Since E/F is a cyclic Kummer extension, for E = F (G0 ) ⊂ E the extension E /F is also a cyclic Kummer extension, and there are an integer t dividing t, σ ∈ ΓE/F and a basis 1, π, π 2 , . . . , π t−1 such that π t ∈ F, π σ = πζt and the Galois group ΓE /F of E /F is generated by σ. Moreover, both extensions E/F and E /F are totally ramiﬁed in is the ramiﬁcation index of E /F , so we have as earlier the following inequality: p, and t t 2

+ 1 (p − 1) > t, and p[t/2]+1 does not divide (ζp − 1).

Since p is odd and t does not divide p − 1, we can assume that t > 2. We will consider matrices 1 π · · · π j−1 ζ(1) − 1 π j ··· π t−1 1 πζ · · · π j−2 ζ j−2 ζ − 1 π j ζ j · · · π t−1 ζ t−1 (2) , Mj . .. 1 πζ t−1 · · · (π j−2 )σt−1 ζ(t) − 1 (π j )σt−1 · · · (π t−1 )σt−1 j = 2, . . . , t that are obtained from Wj by subtracting ﬁrst column of Wj from j-th column of Wj . For even t we may suppose that only r n − 2 elements from ζ(1) , ζ(2) , . . . , ζ(t ), the eigenvalues of h, are distinct from 1. Indeed, we can choose two elements g1 and g2 of G0 generating a noncyclic subgroup of G0 in such a way that ζpα1 , ζpα2 , . . . and ζpβ1 , ζpβ2 , . . . compose the full set of eigenvalues of g1 and g2 respectively and α1 = α2 . Set k=

−(β1 − β2 ) α1 − α2

and h = ζps · g1k g2

for s = −kα1 − β1 ,

since we are calculating αj , βj and k modulo p we can ﬁnd an integer k with this properties. Then matrix h has two eigenvalues ζ(i) for diﬀerent i, and the group generated by hγ , γ ∈ ΓE (ζp )/F (ΓE (ζp )/F denotes the Galois group of E (ζp )/F ) is abelian of exponent p; we can still apply the criterion of Proposition 1 to the group G0 generated by matrices hγ , γ ∈ ΓE /F . In other words, we can extend the group G0 , if it is needed, by adjoining some scalar matrices and naturally extending Galois action to them, and this does not change ΓE/F -stability of G0 . For convenience we still preserve our previous notation. We can apply our construction to the matrix h = ζps · g0 for some g0 ∈ G0 and if we show that this matrix is not contained in GLn (OE(ζ ), then g0 ∈ GLn (OE ), and this contradiction is exactly the p) aim of our proof of the case 1). Denote Λ = [ζ (i−1)(j−1) ]ti,j=1 . Note that Λ is a symmetric matrix. Let det Wj = det Mj = θj1 (ζ(1) − 1) + θj2 (ζ(2) − 1) + · · · + θjt (ζ(t) − 1), θjk = (−1)j+k π t(t−1)/2−(j−1) ·

where

ζ −(j−1)(k−1) λjk · c = π t(t−1)/2−(j−1) · , t t

for c = detΛ =

(ζ j − ζ i ).

1i<jt

14

H. -J. BARTELS AND D. A. MALININ

and λjk = (−1)k+j ζ −(j−1)(k−1) = λkj . Indeed, denote Λ−1 = [ ζ −(j−1)(i−1)

−(j−1)(i−1)

t

]ti,j=1 , and so

· c. Let us consider the element δ from the (ij)-th cofactor of Wj is (−1)j+i · ζ t −1 Galois group of Q(ζ)/Q such that δ : ζ → ζ , and so δ = 1, δ 2 = 1. δ acts as a complex conjugation on t-th roots of 1. Note that for a t-root η of 1 η δ = η iﬀ η −1 = η or, equivalently, η = ±1. Let us determine some properties of the above elements λij under δ-action. Since the number of rows in Λ that are permuted under δ-action is equal to φ(t), the Euler function, we have cδ = c if φ(t)/2 is even and cδ = −c if φ(t)/2 is odd. Furthermore, δ permutes i-th row and (t + 2 − i)-th row of the matrix Λ for 1 < i < 1 + t/2, and (−1)i+j = (−1)t−i+j = (−1)t (−1)i+j . Therefore, if both t and φ(t)/2 are even, or both t and φ(t)/2 are odd, then λδk,j = λk,t−j+2 = λt−k+2,j for 1 < j < 1 + t/2, otherwise λδk,j = −λk,t−j+2 = −λt−k+2,j . In the general case we can claim that λδk,j = s · λk,t−j+2 = s · λt−k+2,j where s = s(t) = (−1)t+φ(t)/2 = ±1 depends only on t. = [λi,j ]−1 Let t be even, and let Λ1 = [λij ]i,j = [(−1)i+j ζ −(i−1)(j−1) ]i,j . Then Λ−1 1 i,j = [(−1)i+j · ζ

(i−1)(j−1)

t

]i,j , and it follows that cofactors of λij are equal to aij = −1

ζ (i−1)(j−1) , t

and

so all aij ≡ 0(modq), in particular, a1j = t . Let C = [cij ] be a (t − 1) × (t − 1)- matrix obtained via eliminating the ﬁrst row and the ﬁrst column of Λ. Taking an expansion of a1i −1 by 2t -th row of C we obtain: t = ci1 Ai1 +ci2 Ai2 +· · ·+ci,t−1 Ai,t−1 where Aiu are cofactors of the elements ciu in the i-th row of C. It follows that for some m Aim ≡ 0(modq). Now it is possible to ﬁx integers j = 1 and m. We can use matrices g1 = g and g2 = g σ for getting a matrix g whose eigenvalues associated with j-th and m-th blocks are ζ(j) = ζ(m) = 1 (see Proposition 1, 2)) and the above Lemma. For this purpose take the eigenvalues ζpα1 and ζpα2 of g1 and the eigenvalues ‘ζpβ1 and ζpβ2 of g2 associated with j-th and m-th blocks respectively. If ζpα1 = ζpα2 , set g = ζpα1 g, otherwise set g = ζps g1k g2 for s = −kα1 − β1 1 −β2 ) and k = −(β α1 −α2 . Now we can apply Proposition 1 to the group G0 generated by all i

hσ , i = 1, . . . , t for h = g . Let us consider a prime ideal q in the ring of integers O of the ﬁeld Qp (ζp , ζ) such that q divides p. Let us suppose that ζ(l) = 1 and the elements (ζ(t) − 1)λit (ζ(1) − 1)λi1 (ζ(2) − 1)λi2 + + ··· + , i = 1, 2, . . . , t ζ(l) − 1 ζ(l) − 1 ζ(l) − 1 are divisible by (ζ(l) − 1) in the ring O, then the system of congruences x1 λ11 + x2 λ12 + · · · + xt λ1t ≡ 0(mod q) x1 λ21 + x2 λ22 + · · · + xt λ2t ≡ 0(mod q) .. . x1 λt1 + x2 λt2 + · · · + xt λtt ≡ 0(mod q)

(S )

has a nontrivial solution x1 = 1,

x2 =

ζ(2) − 1 , ζ(l) − 1

x3 =

ζ(t) − 1 ζ(3) − 1 , · · · , xt = . ζ(l) − 1 ζ(l) − 1

Let us eliminate the ﬁrst and the (t/2 + 1)-th congruences from system (S), coeﬃcients of which are equal to (λi1 , λi2 , . . . , λit ) = (1, 1, . . . , 1) for i = 1 and (1, −1, 1, −1, . . . , 1, −1),

FINITE GALOIS STABLE SUBGROUPS OF GLn

15

for i = t/2 + 1. We obtained a system containing r = t − 2 congruences in r = t − 2 variables, since two variables xj , xm that correspond to ζ(j) = 1, ζ(k) = 1 do not appear in the system (S). The determinant of the matrix of this system is a r × r-minor N of the matrix [ζ −(i−1)(j−1) ]i,j , and the above choice of j = 1, m (such that ζ(j) = 1, ζ(m) = 1) allows us to assume that det N = 0, since det N = (−1)i+m Aim ≡ 0(modq) as it was proved above. But in this case the system has the unique solution (0, . . . , 0). This contradicts the fact that all xi in question which are diﬀerent from 0 are invertible elements of the ring of integers O of the ﬁeld Qp (ζp , ζ). Therefore, we can claim that

sj =

t

(ζ(k) − 1)λjk ≡ 0(mod(ζ(l) − 1)2 )

k=1

for some j, where summands (ζ(1) − 1)λj1 and (ζ(m) − 1)λjm are equal to 0. Since r = t − 2 and in virtue of the mentioned equality λδk,j = s · λk,t−j+2 = s · λt−k+2,j , where δ 2 = 1, we can consider some j that satisﬁes inequalities 2 + t/2 j t. Let us calculate det Mj : t dj = det Mj = π 1+2+···+(t−1)−(j−1) ( (ζ(i) − 1)λji ) = π(t(t − 1)/2 − (j − 1)) · sj . i=1

We can calculate the determinant detW with respect to the basis 1, π, . . . , π t−1 : det W = π t(t−1)/2

(ζtj − ζti ).

1i<jt

Taking into account that π j−1 does not divide (ζp − 1) for j ≥ 2 + 2t and comparing √ √ determinants D = detW and dj , we obtain that dj · ( D)−1 can not be contained in , L = E(ζp ). By Proposition 1 and its Corollary 1.4.2 this implies that the above matrix OL ) and so G0 ⊂ GLn (OE ). This is a contradiction. g ∈ GLn (OL If t is odd, the same argument is valid, and we can ﬁnd an index j such that (t + 3)/2 ≤ j ≤ n and detWj /detW ∈ Ol . Hence the previous proof remains unchanged if we eliminate the ﬁrst and the (t + 1)/2-th congruences of the above system (S). However, for odd t it would be enough to eliminate only the ﬁrst equation of the system (S). Case II. e0 divides p − 1. Now we can consider the case II. We recall the notation from the beginning of the proof of the Main Theorem. So K = Q(G) is Galois over Q, unramiﬁed outside the prime p, p > 2 and G0 = GΓ1 (p) is the subgroup of elements in G that are ﬁxed by the ﬁrst ramiﬁcation group Γ1 (p) for some prime divisor p of p, and e0 denotes the ramiﬁcation index of Q(G0 ) over Q with respect to p. For case II we suppose, that e0 is a divisor of p − 1. Firstly we need the following Lemma 2.2.2. The only ramified prime in the extension Q(G0 )(ζp )/Q is p, the ramification index e(Q(G0 )(ζp )/Q) of a ramified prime ideal in Q(G0 )(ζp ) lying over p ∩ OQ(G0 ) is p − 1. Proof of lemma 2.2.2. For the calculation of the ramiﬁcation index we consider the corresponding local situation. Therefore, let Qp denote the p-adic numbers and Q(G0 )υ the completion of Q(G0 ) with respect to the valuation υ deﬁned by the prime ideal p ∩ OQ(G0 ) .

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H. -J. BARTELS AND D. A. MALININ

According to the assumptions in case II the ramiﬁcation index e0 of Q(G0 )υ /Qp divides p−1, while the ramiﬁcation of Qp (ζp )/Qp is p−1. The compositum Qp (ζp )·Q(G0 )υ = Q(G0 )υ (ζp ) of these two extensions is tamely ramiﬁed over Qp with a ramiﬁcation index t(p − 1), where the natural number t divides the ramiﬁcation index e0 and therefore divides also p − 1. We claim, that t = 1. For this purpose let Lυ denote the maximal over Qp unramiﬁed extension in Q(G0 )υ (ζp ). Then Q(G0 )υ (ζp )/Lυ is a totally ramiﬁed cyclic Galois extension. Therefore, there is only one subgroup of index t in the Galois group of this cyclic extension. Galois theory give us a uniquely determined subﬁeld of Q(G0 )υ (ζp ) over Lυ with ramiﬁcation index t. But in case t > 1 we would have two such extensions: one is a subﬁeld of Qp (ζp ) · Lυ . This contradiction shows that the ramiﬁcation index of the composite ﬁeld can not exceed p − 1. According to this Lemma 2.2.2 we see that adjoining a p-th root of unity ζp to K and extending the Galois operation to this larger ﬁeld does not inﬂuence the validity of condition II, e0 is still a divisor of p − 1. So we can and do assume ζp ∈ K without loss of generality. As it was already mentioned in the beginning of the proof of the Main Theorem we can assume that G is GLn (Q)-irreducible (using corollary 1.4.3) and that G is a counterexample to the Main Theorem with minimal order. Therefore, also in case II let G ⊂ GLn (OK ) be a group of the minimal order such that the extension Q(G)/Q is not abelian. For the treatment of case II we distinguish two subcases: case II a): Γ1 (p) is trivial, i.e. K is tamely ramified over Q. and case II b): Γ1 (p) is not trivial, i.e. K is wildly ramified over Q. We start with case II a), since we can use an argument of the proof of case I. There we have seen: if the group generated by all g γ , γ ∈ Γ for a g ∈ G is not cyclic, then some element h = ζps g1k g2 has an eigenvalue 1 (for the notation of g1 , g2 see above the proof of case I). We have the following conditions:

e0 + 1 (p − 1) > e0 , 2

and: p[t/2]+1 does not divide (ζp − 1) for t = e0 = p − 1. The argument of the proof of Case I implies that the conditions of Proposition 1 are not satisﬁed for the group generated by all hγ , γ ∈ Γ. Therefore, g γ = g a for all g ∈ G and any γ ∈ Γ0 (p). Moreover, a is the same for all g. Indeed, if g γ = g a and g1γ = g1b , with a = b, then the elements (gg1 )γ , γ ∈ Γ would generate a noncyclic group. So we have g γσ = g σγ −1 for any γ ∈ Γ0 (p), σ ∈ Γ. This implies g γ = g σγσ . If G is generated by all g γ , γ ∈ Γ, this implies the coincidence of all inertia groups Γ0 (p). Since Γ0 (p) is cyclic, it follows that G must be of A-type. Now we consider case II b), where K is wildly ramified. We assumed ζp ∈ K. Since Q(ζp ) is a tame extension of Q, Γ1 (p) operates trivially on the p-th roots of unity ζp , hence K Γ1 (p) contains also ζp . Take now in Corollary 1.5.3L = K Γ0 (p) , then this ﬁeld is unramiﬁed over Q for the prime divisor p of p. Corollary 1.5.3 shows: up to conjugation in GLn (Op ∩ K Γ0 (p)) , where Op is the valuation ring of of K Γ0 (p) (ζp ) at p, the group G0 (p) = {g ∈ G0 , g ≡ In (modp)}

FINITE GALOIS STABLE SUBGROUPS OF GLn

17

consists of diagonal matrices. The group G(p) := {g ∈ G, g ≡ In (mod p)} is a nontrivial p-group and therefore G0 (p) = {In } is not trivial as the subgroup of Γ1 (p)-ﬁxed elements of a nontrivial p-group. G is abelian and therefore in the centralizer of every matrix h ∈ G0 (p). ) holds If in particular h = diag(l1 In1 , . . . , lk Ink ), then g = diag(g1 , . . . , gk ), gi ∈ GLni (OK Γ0 (p) for every g ∈ G and therefore we can split G into GLn (Op ∩K )-irreducible components. ) of G with a In this decomposition we choose an irreducible component G ⊂ GLm (OK suitable natural number m such that G has nontrivial Γ1 (p)-action. Moreover it is worth mentioning, that the described decomposition is stable under the operation of Γ0 (p) (see Corollary 1.5.3), in particular Γ0 (p) operates on the group G . If G0 denotes the subgroup of Γ1 (p)-ﬁxed elements of G , then the group G0 (p) := {g ∈ G0 , g ≡ Im (modp)} consists of scalar matrices. The conditions on the ramiﬁcation of case II are also satisﬁed for G and G0 instead of G and G0 . But now the group G0 (p) is equal to the group µ := {ζIm , ζ p = 1}. Let us now consider the Galois-equivariant homomorphism ψ = ψm : G → GLmp (K) p

given by ψ(g) = g ⊗ . The kernel of ψ is the set of all scalar matrices contained in G . This kernel is not trivial, since G0 (p) Kerψ. Hence we have: There is an exact sequence 1 −→ µ −→ G −→ ψ(G ) −→ 1 of Γ0 (p)-groups. The aim of our proof is the construction of a certain group G1 ⊂ G ⊂ GLm (K) such that: K Γ1 (p) (G1 ) is an extension of K Γ1 (p) with ζp ∈ K Γ1 (p) (G1 ), tame ramiﬁcation index e0 = p − 1 and K Γ1 (p) (G1 )/K Γ1 (p) is an elementary abelian Kummer extension. In a second step a careful study of the Galois-action of Γ0 (p) on G1 will then show that the constructed group G1 can not exist. This gives then the desired contradiction. First step: Construction of G1 . We have H := ψ(G )Γ1 (p) = {Im } since both ψ(G ) and Γ1 (p) are p-groups. For later use we notice, that (i) H is Γ0 (p)- stable, since Γ1 (p) is a normal subgroup of Γ0 (p), and (ii) the action of Γ0 (p) on H is given by the cyclotomic character. (δ)

More precisely, we have for h ∈ H and δ ∈ Γ0 (p)hδ = hχ . Here χ(δ) denotes the unique (δ) integer modulo p such that ζ δ = ζ χ holds for all p-th root of unity ζ and δ ∈ Γ0 (p). This is an immediate consequence of Corollary 1.5.3. Now, if there exist a g ∈ ψ −1 (H) having nontrivial Γ1 (p)-action, then deﬁne G1 as the subgroup of ψ −1 (H) generated by all g δ , δ ∈ Γ0 (p). If such an element g does not exist in ψ −1 (H), we can suppose, that ψ(G ) has nontrivial Γ1 (p)-action (since otherwise

18

H. -J. BARTELS AND D. A. MALININ

g with the needed property would exist). Now consider a suitable irreducible component G of ψ(G ) having non-trivial Γ1 (p)-action and apply the corresponding map ψ to G . For simplicity we call this map ψ also simply ψ. If ψ(G ) is ﬁxed elementwise by Γ1 (p), again we have the needed element g ∈ G with non-trivial Γ1 (p)-action, and we can deﬁne G1 in G correspondingly. Otherwise, we take an irreducible component G ψ(G ) having non-trivial Γ1 (p)-action etc.. Since the order of the groups G , G , G , . . . is becoming smaller and smaller (the kernel of the diﬀerent maps ψ is not trivial), we will have at last G(i) to be ﬁxed by Γ1 (p) with the least possible i, so we have the needed element g ∈ G(i−1) with non-trivial Γ1 (p)-action. Instead of G1 we consider then the subgroup of ψ −1 (ψ(G(i−1) )Γ1 (p) ) generated by all g δ , δ ∈ Γ0 (p). For simplicity let us call these groups again G1 , G and call also the degree of the corresponding linear group again m. step 2: study of the Galois-action of Γ0 (p) on G1 and on K Γ0 (p) (G1 ). For g ∈ G1 and for γ ∈ Γ1 (p) we have ψ(g γ )ψ(g)−1 = ψ(g)γ ψ(g −1 ) = ψ(g)ψ(g)−1 = Im . This implies g γ = gζ for any γ ∈ Γ1 (p) with a suitable p-th root of unity ζ = ζγ . Let σ be an element of Γ0 (p), whose image in Γ0 (p)/Γ1 (p) is a generator of Γ0 (p)/Γ1 (p) and take g ∈ G1 . There are two possibilities: g −1 g σ ∈ GLm (K Γ1 (p) ) or g −1 g σ is not ﬁxed by the ramiﬁcation group Γ1 (p). In the ﬁrst of these two cases we claim that g σ = gζσ for a suitable p-th root of unity ζσ . Let us prove this and show how to get the desired contradiction in that case. For this purpose notice that d := g −1 g σ ≡ Im (mod p) and therefore using Corollary 1.5.3 we can diagonalize this matrix d over GLm (Op ∩ K Γ0 (p) ). But since G is irreducibel over GLm (Op ∩ K Γ0 (p) ) it follows, that d = ζσ Im , for a suitable root of unity ζσ . Now we have g σ = gζσ and at the same time g γ = gζγ for any γ ∈ Γ1 (p). Since Γ1 (p) k operates trivially on the p-th roots of unity ζ we obtain: g σ = g γ , for some integer k and therefore the two Galois automorphisms σ and γ k coincides on K Γ0 (p) (G1 ) since g is any generator of G1 . This gives the contradiction in the case, where g −1 g σ ∈ GLm (K Γ1 (p) ). In the alternative case g0 := g −1 g σ is not ﬁxed by the ramiﬁcation group Γ1 (p). Now ˜ ⊆ G generated by all elements g δ , δ ∈ Γ0 (p). Since for any δ ∈ Γ0 (p) consider the group G 1 0 we have χ(δ)

ψ(g0 δ ) = ψ(g0 )δ = ψ(g0 )χ(δ) = ψ(g0

),

(δ)

it follows that g0δ = g0χ ζδ with suitable p-th roots of unity ζδ depending on the Galois ˜ is generated by g0 and ζp Im and the order of G ˜ is p2 . automorphism δ. Therefore the group G Γ0 (p) ˜ Γ0 ˜ ˜ (G), which is Galois over K (p) by deﬁnition of G. We study the Deﬁne K := K ˜ (like on K Γ0 (p) (G ) in the ﬁrst case). For this purpose we denote by Galois-action on K 1 Γ0 (p) and Γ1 (p) the corresponding inertia respectively ramiﬁcation groups of the extenΓ {1} since the Γ1 (p)-action on G ˜ ˜ is not trivial. We then sion K/K 0 (p). We have Γ1 (p) = claim ﬁrstly, that p is the highest p-power dividing the order of Γ 0 (p). The Galois group

Γ0 (p) ˜ ˜ (considered as a is contained in the group of linear automorphism of G Γ 0 (p) of K/K 2-dimensional vector space over the ﬁeld Fp of p elements), so its order divides the order of GL2 (Fp ), which equals to (p2 − 1)(p2 − p). This implies that p2 does not divide the 1 (p) ˜ ˜ Γ is cyclic of order p, as claimed above. order of Γ 0 (p), so the Galois group of K/K

FINITE GALOIS STABLE SUBGROUPS OF GLn

19

√ 1 (p) p 1 (p) 1 (p) 1 (p) ˜ =K ˜ Γ ˜ Γ ˜ Γ ˜ Γ Hence K ( u) with u ∈ K . Now σ(K )=K since Γ 1 (p) is a normal √ √ σ (p) (p) Γ Γ p p ˜ 1 ( u) = K ˜ 1 ( u ), and one concludes: subgroup of Γ0 (p). Therefore K √ √ p 1 (p) ˜ Γ uσ ( p u)−1 ∈ K K Γ1 (p) . Since g0−1 g0γ = ζγ Im for all γ ∈ Γ1 (p) we have g0 =

√ p

ug1 with g1 ∈ K Γ1 (p) . It follows that

g0−1 g0σ ∈ GLm (K Γ1 (p) ) and we can apply Corollary 1.5.3 to this element. Like in the ﬁrst of the considered two cases with g0 instead of g we can conclude that g0σ = g0 ζσ for a suitable p-th root of unity ζσ . The contradiction follows then analogously to the ﬁrst case (see above). 2.3. Proof of Lemma 1.5.2. It is a generalization of the well known argument proposed by Minkowski [21]. The outline of our proof is given in [13]. It is easy to prove that G is abelian of exponent p. Let Op be the valuation ring of p and π a prime element. Let g1 = In + πB1 , g2 = In + πB2 for some g1 , g2 ∈ G. Then gi−1 ≡ In − πBi (mod π 2 ), i = 1, 2 and h = g1 g2 g1−1 g2−1 ≡ In (mod π 2 ). It follows from Lemma 1.5.1, (ii) that h = In , and the same Lemma 1.5.1, (ii) shows that g p = In for any g ∈ G. First of all, G is conjugate over Op to a group of triangular matrices, since G is abelian and Op is a local ring, see [5] Theorem (73.9) and the remarks in [5] on page 493. On the other hand, we can describe explicitely the matrix M such that M −1 gM = diag(λ1 , λ2 , . . . , λn ) is a diagonal matrix for a triangular matrix g of order p which is congruent to In (mod p). Indeed, let g ∈ G and ζ(1) It1 P21 . . . Pk1 0 ζ(2) It2 . . . Pk2 g= . .. , .. .. . . 0 ··· ζ(k) Itk and let It1 0 . . . 0 It2 · · · S= . . .. .. 0 ...

A1 A2 .. . It k

for t1 + t2 + · · · + tk = n and t1 ≤ t2 ≤ · · · ≤ tk , ζ(i) , i = 1, 2, . . . , k are appropriate p-roots of 1. We consider ζ(1) It1 ∗ . . . Mk1 0 ζ(2) It2 . . . Mk2 S −1 gS = . .. , .. .. . . 0 ··· ζ(k) It k

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H. -J. BARTELS AND D. A. MALININ

and we ﬁnd the system of conditions for providing Mki = 0ti ,tk , the zero ti × tk -matrix. We have the following system of conditions: −1 1 1 1 ζ(1) (1 − ζ(k) ζ(1) )A1 + P2 A2 + · · · + Pk−1 Ak−1 + Pk = 0t1 ,tk . . . −1 k−2 k−2 = 0tk−2 ,tk ζ(k−2) Ak−2 (1 − ζ(k) ζ(k−2) ) + Pk−1 Ak−1 + Pk −1 k−1 ζ = 0tk−1 ,tk . (k−1) Ak−1 (1 − ζ(k) ζ(k−1) ) + Pk The condition g ≡ In (mod p) implies Pij ≡ 0tj ti (mod p), and we can ﬁnd Ai , 1 ≤ i ≤ k−1 sequentially using the results of previous steps: Ak−1 = −

Ak−2 = −

Ak−3 = −

Pkk−1 , −1 ζ(k−1) (1 − ζ(k) ζ(k−1) ) k−2 (Pkk−2 + Pk−1 Ak−1 ) −2 ζ(k−2) (1 − ζ(k) ζ(k−2) )

,

k−3 k−3 (Pkk−3 + Pk−1 Ak−1 + Pk−2 Ak−2 ) −1 ζ(k−3) (1 − ζ(k) ζ(k−3) )

,

and so on. Now, using induction on the degree n we can ﬁnd a matrix M that transforms g to a diagonal form as required. Since G is an abelian group of exponent p this allows to prove our claim locally over the ring Op . We use statement (81.20) in [5] for proving our result globally for the given Dedekind ring (compare for this also the proof of (81.20) and (75.27) in [5]). Remark. Another proof of the fact that G is elementary abelian can be found in [29], sect. 4 and [30], p. 187. Proof of Corollary 1.5.3. We can assume that for some matrix g ∈ G and a generator σ of Γ the condition g σ = g α , 1 < α < p, is fulﬁlled. Indeed, by Lemma 1.5.2 G is an abelian group of exponent p, so it can be considered as an Fp Γ - module over the ﬁeld Fp of p elements. Since Γ is a cyclic group of order p − 1 generated by an element σ this element determines an automorphism of G and all its eigenvalues are contained in Fp . In fact, its matrix is diagonalizable over Fp because the order of σ is prime to p. Hence we can take g ∈ G to be an eigenvector of this automorphism and so g σ = g α , 1 < α < p since not all eigenvalues are 1. Now Lemma 1.5.2 provides the existence of a matrix M ∈ GLn (Op ) such that M −1 GM is a group of diagonal matrices. We shall show that α coincides with the integer β, ζpσ = ζpβ , 1 < β < p. Let us suppose that M −1 gM = h = diag(λ1 In1 , λ2 In2 , . . . , λm Inm ), λj ∈ L(ζp ), then hσ = hβ and (M σ )−1 g σ M σ = hβ . Since M −1 g α M = hα and g σ = g α , it is obvious that (M σ )−1 M hα M −1 M σ = hβ . As Γ coincides with the inertia group of the ideal p and M ∈ GLn (Op ), it follows that M σ ≡ M (mod p). Therefore, the congruence M −1 M σ ≡ In (mod p) is valid and conjugation by

FINITE GALOIS STABLE SUBGROUPS OF GLn

21

matrix M −1 M σ maps diagonal elements of hα to diagonal elements of hβ . But if α = β, then the matrix M −1 M σ must have at least one diagonal element dii = 0, which is impossible. We proved our claim, and α = β. We obtained also that M −1 M σ = λ = diag(d1 , d2 , . . . , dm ) for some nj × nj -matrices dj . Let us introduce the following matrix: M1 =

1 (M σ1 + M σ2 + · · · + M σp−1 ), p−1

M1 = [mij ],

mij ∈ Op ,

σ1 , σ2 , . . . , σp−1 are all elements of Γ. It is clear, that M1 ≡ M (mod p) and det M1 ≡ detM (mod p). It follows that M1 ∈ GLn (Op ). Furthermore, M1 is stable under elementwise Γ-action, so all mij are Γ-stable and mij ∈ L. Hence M1 ∈ GLn (L). Since M σ = M λ, it follows that M1−1 GM1 is contained in the group of diagonal matrices, as it was claimed.

References [1] V.A. Abrashkin, Galois moduli of period p group schemes over a ring of Witt vectors, Math. USSR Izvestiya 31 (1988), 1–46. [2] H.-J. Bartels, Zur Galois Kohomologie deﬁniter arithmetischer Gruppen, J. reine angew. Math. 298 (1978), 89–97. [3] H.-J. Bartels, Y. Kitaoka, Endliche arithmetische Untergruppen der GLn , J. reine angew. Math. 313 (1980), 151–156. [4] H.-J. Bartels, D.A. Malinin, Finite Galois stable subgroups of GLn , Manuskripte der Forschergruppe Arithmetik, see http://www.math.uni-mannheim.de/∼ fga/preprint5.htm Nr.3 (2000), 21 pages. [5] C. W. Curtis, I. Reiner, Representation theory of ﬁnite groups and associative algebras, Interscience, New York, 1962. [6] J.-M. Fontaine, Il n’y a pas de vari´ et´e ab´elienne sur Z, Invent. math. 81 (1985), 515–538. [7] D. Harbater, Galois Groups with Prescribed Ramiﬁcation, Contemporary Mathematics 174 (1994), 35–60. [8] Y. Kitaoka and H. Suzuki, Finite arithmetic subgroups of GLn , IV, Nagoya Math. J. 142 (1996), 183–188. [9] D.A. Malinin., On integral representations stable under Galois action., Preprint MSLU N 5, 27p. (1997). [10] D. A. Malinin, Integral representations of ﬁnite groups with Galois action, Dokl. Russ. Acad. Nauk 349 (1996), 303–305. (Russian) [11] D.A. Malinin, Integral representations of p-groups of given nilpotency class over local ﬁelds, Algebra i analiz 10 (1998), N 1, 58–67 (Russian); English translation in St. Petersburg Math. J. v. 10, N 1, 45–52. [12] D.A. Malinin, On integral representations of ﬁnite p-groups over local ﬁelds, Dokl. Akad. Nauk USSR 309 (1989), 1060–1063 (Russian); English transl. in Sov. Math. Dokl. v.40 (1990), N 3, 619–622. [13] D. A. Malinin, On integral representations of ﬁnite nilpotent groups, Vestnik Beloruss. State Univ. Ser. 1 (1993), N 1, 27–29. (Russian) [14] D. A. Malinin, On realization ﬁelds of integral matrix groups, Vesti Beloruss. Pedag. Univ. 2 (1994), 101–104. (Belarusian) [15] D. A. Malinin, Isometries of positive deﬁnite quadratic lattices, ISLC Math. Coll. Works Lie Lobachevsky Colloquium. Tartu (1992), 21–22. [16] D.A. Malinin, Arithmetic properties of ﬁnite groups with coeﬃcients in Dedekind rings, Dissertation, Saint-Petersburg State University, St. Petersburg, 1993, 164 pages. [17] D. A. Malinin, Galois stability for integral representations of ﬁnite groups, Algebra i Analiz 12 (2000), 106–145 (Russian); English translation in St. Petersburg Math. J. v. 12, N 3.

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[18] G. N. Markshaitis, On p-extensions with one critical number, Izvestija Akad. Nauk USSR 27 (1963), 463–466. (Russian) [19] M. Mazur, Finite Arithmetic Subgroups of GLn , Journal of Number Theory 75 (1999), 109–119. [20] M. Mazur, Finite Arithmetic Subgroups of GLN . The Normalizer of a Group in the Unit Group of its Group Ring and the Isomorphism Problem., Dissertation, Department of Mathematics, Chicago, Illinois, 1999, 112 pages. ¨ ¨ [21] H. Minkowski, Uber den arithmetischen Begriﬀ der Aquivalenz und u ¨ber die endlichen Gruppen linearer ganzzahliger Substitutionen, J. reine angew. Math. 100 (1887), 449–458. [22] L. Moret-Bailly, Extensions de corps globaux a ramiﬁcation et groupe de Galois donnes, C.R. Acad. Sci. Paris, Serie 1 311 (1990), 273–276. [23] A. Odlyzko, Discriminant bounds, unpublished Tables from November 29 (1976), see http://www.research.att.com/∼ amo/unpublished/discr.bound.table. [24] I. Schur, Elementarer Beweis eines Satzes von L. Stickelberger, Math. Z. 29 (1929), 464–465. [25] J.-P. Serre, Corps locaux, Hermann, Paris, 1962. [26] D.A. Suprunenko, R.I. Tyshkevich, Commutative Matrices, Academic Press, New York and London, 1968. [27] J. Tate, The Non-Existence of Certain Galois Extensions of Q Unramiﬁed Outside 2, Contemporary Mathematics 174 (1994), 153–156. [28] J. Tate, p-Divisible Groups (1967), in: Conf. Local Fields (Dreibergen), Springer Verlag, Berlin and New York, 158–183. [29] A. Weiss, Rigidity of p-adic p-torsion, Annals of Math. 127 (1988), 317–322. [30] A. Weiss, Torsion in integral group rings, J. f¨ ur die Reine und angew. Math. 145 (1991), 175–187. 1 Fakulta ¨t fu ¨r Mathematik und Informatik, Universita ¨t Mannheim, Seminar-geba ¨ude A5, D-68131 Mannheim, Germany E-mail address: [email protected] 2 Belarusian State Pedag. University, Sovetskaya str. 18, 220050 Minsk, Belarus E-mail address: [email protected]

DERIVED CATEGORIES FOR NODAL RINGS AND PROJECTIVE CONFIGURATIONS IGOR BURBAN AND YURIY DROZD

Contents Introduction 1. Backstr¨om rings 2. Nodal rings 3. Examples 3.1. Simple node 3.2. Dihedral algebra 3.3. Gelfand problem 4. Projective conﬁgurations 5. Conﬁgurations of type A and A˜ 6. Application: Cohen–Macaulay modules over surface singularities References

23 24 25 29 29 32 33 36 37 43 45

Introduction This paper is devoted to recent results on explicit calculations in derived categories of modules and coherent sheaves. The idea of this approach is actually not new and was eﬀectively used in several questions of module theory (cf. e.g. [10, 12, 13, 7]). Nevertheless it was somewhat unexpected and successful that the same technique could be applied to derived categories, at least in the case of rings and curves with “simple singularities.” We present here two cases: nodal rings and conﬁgurations of projective lines of types A and ˜ when these calculations can be carried out up to a result, which can be presented in A, more or less distinct form, though it involves rather intricate combinatorics of a special sort of matrix problems, namely “bunches of semi-chains” [4] (or, equivalently, “clans” [8]). In Sections 1 and 4 we give a general construction of “categories of triples,” which are a connecting link between derived categories and matrix problems, while in Sections 2 ˜ Section 3 and 5 this construction is applied to nodal rings and conﬁgurations of types A. contains examples of calculations for concrete rings and Section 5 also presents those for nodal cubic. We tried to choose typical examples, which allow to better understand the general procedure of passing from combinatorial data to complexes. Section 6 contains an application to Cohen–Macaulay modules over surface singularities, which was in fact the origin of investigations of vector bundles over projective curves in [13]. More detailed exposition of these results can be found in [5, 6, 14].

2000 Mathematics Subject Classification. 16E05, 16D90. It is a survey of a research supported by the CRDF Award UM 2-2094 and by the DFG Schwerpunkt “Globale Methoden in der komplexen Geometrie”.

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IGOR BURBAN AND YURIY DROZD

¨ m rings 1. Backstro We consider a class of rings, which generalizes in a certain way local rings of ordinary multiple points of algebraic curves. Following the terminology used in the representations theory of orders, we call them Backstr¨ om rings. Since in the ﬁrst three sections we are investigating a local situation, all rings there are supposed to be semi-perfect [3] and noetherian. We denote by A-mod the category of ﬁnitely generated A-modules and by D(A) the derived category D− (A-mod) of right bounded complexes over A-mod. As usually, it can be identiﬁed with the homotopy category K − (A-pro) of (right bounded) complexes of (ﬁnitely generated) projective A-modules. Moreover, since A is semi-perfect, each complex from K − (A-pro) is homotopic to a minimal one, i.e. to such a complex C• = (Cn , dn ) that Im dn ⊆ rad Cn−1 for all n. If C• and C• are two minimal complexes, they are isomorphic in D(A) if and only if they are isomorphic as complexes; moreover, any morphism C• → C• in D(A) can be presented by a morphism of complexes, and f is an isomorphism if and only if the latter one is. Definition 1.1. A ring A is called a Backstr¨ om ring if there is a hereditary ring H ⊇ A (also semi-perfect and noetherian) and a (two-sided) H-ideal I ⊂ A such that both R = H/I and S = A/I are semi-simple. For Backstr¨om rings there is a convenient approach to the study of derived categories. Recall that for a hereditary ring H every object C• from D(H) is isomorphic to the direct sum of its homologies. Especially, any indecomposable object from D(H) is isomorphic to α a shift N [n] for some H-module N , or, the same, to a “short” complex 0 → P −→ P → 0, where P and P are projective modules and α is a monomorphism with Im α ⊆ rad P (maybe P = 0). Thus it is natural to study the category D(A) using this information about D(H) and the functor T : D(A) → D(H) mapping C• to H ⊗A C• .1 Consider a new category T = T (A) (the category of triples) deﬁned as follows: • Objects of T are triples (A• , B• , ι), where – A• ∈ D(H); – B• ∈ D(S); – ι is a morphism B• → R ⊗H A• from D(S) such that the induced morphism ιR : R ⊗S B• → R ⊗H A• is an isomorphism in D(R). • A morphism from a triple (A• , B• , ι) to a triple (A• , B• , ι ) is a pair (Φ, φ), where – Φ : A• → A• is a morphism from D(H); – φ : B• → B• is a morphism from D(S); – the diagram ι

B• −−−−→ R ⊗H A• 1⊗Φ φ ι

(1.1)

B• −−−−→ R ⊗H A• commutes in D(S). One can deﬁne a functor F : D(A) → T (A) setting F(C• ) = (H ⊗A C• , S ⊗A C• , ι), where ι : S ⊗A C• → R ⊗H (H ⊗A C• ) R ⊗A C• is induced by the embedding S → R. The values of F on morphisms are deﬁned in an obvious way. 1 Of course, we mean here the left derived functor of ⊗, but when we consider complexes of projective modules, it restricts indeed to the usual tensor product.

DERIVED CATEGORIES FOR NODAL RINGS

25

Theorem 1.2. The functor F is a full representation equivalence, i.e. it is • dense, i.e. every object from T is isomorphic to an object of the form F(C• ); • full, i.e. each morphism F(C• ) → F(C• ) is of the form F(γ) for some γ : C• → C• ; • conservative, i.e. F(γ) is an isomorphism if and only if so is γ; As a consequence, F maps non-isomorphic objects to non-isomorphic and indecomposable to indecomposable. Note that in general F is not faithful : it is possible that F(γ) = 0 though γ = 0 (cf. Example 3.1.3 below). Sketch of the proof. Consider any triple T = (A• , B• , ι). We may suppose that A• is a minimal complex from K − (A-pro), while B• is a complex with zero diﬀerential (since S is semi-simple) and the morphism ι is a usual morphism of complexes. Note that R ⊗H A• is also a complex with zero diﬀerential. We have an exact sequence of complexes 0 −→ IA• −→ A• −→ R ⊗H A• −→ 0. Together with the morphism ι : B• → R ⊗H A• it gives rise to a commutative diagram in the category of complexes Com− (A-mod) 0 −−−−→ IA• −−−−→ A• −−−−→ R ⊗H A• −−−−→ 0 ι α 0 −−−−→ IA• −−−−→ C• −−−−→

B•

−−−−→ 0,

where C• is the preimage in A• of Im ι. The lower row is also an exact sequence of complexes and α is an embedding. Moreover, since ιR is an isomorphism, IA• = IC• . It implies that C• consists of projective A-modules and H ⊗A C• A• , wherefrom T FC• . Let now (Φ, φ) : FC• → FC• . We suppose again that both C• and C• are minimal, while Φ : H⊗A C• → H⊗A C• and φ : S⊗A C• → S⊗A C• are morphisms of complexes. Then the diagram (1.1) is commutative in the category of complexes, so Φ(C• ) ⊆ C• and Φ induces a morphism γ : C• → C• . It is evident from the construction that F(γ) = (Φ, φ). Moreover, if (Φ, φ) is an isomorphism, so are Φ and φ (since our complexes are minimal). Therefore Φ(C• ) = C• , i.e. Im γ = C• . But ker γ = ker Φ ∩ C• = 0, thus γ is an isomorphism too. 2. Nodal rings We apply these considerations to the class of rings ﬁrst considered in [10], where the second author has shown that they are unique pure noetherian rings such that the classiﬁcation of their modules of ﬁnite length is tame (all others being wild). Definition 2.1. A ring A (semi-perfect and noetherian) is called a nodal ring if it is pure noetherian, i.e. has no minimal ideals, and there is a hereditary ring H ⊇ A, which is semi-perfect and pure notherian such that 1) rad A = rad H; we denote this common radical by R. 2) lengthA (H ⊗A U ) ≤ 2 for every simple left A-module U and lengthA (V ⊗A H) ≤ 2 for every simple right A-module V . Note that condition 2 must be imposed both on left and on right modules.

26

IGOR BURBAN AND YURIY DROZD

It is known that such a hereditary ring H is Morita equivalent to a direct product of rings H(D, n), where D is a discrete valuation ring (maybe non-commutative) and H(D, n) is the subring of Mat(n, D) consisting of all matrices (aij ) with non-invertible entries aij for i < j. Especially, H and A are semi-prime (i.e. without nilpotent ideals) Example 2.2. 1. The ﬁrst example of a nodal ring is the completion of the local ring of a simple node (or a simple double point) of an algebraic curve over a ﬁeld k. It is isomorphic to A = k[[x, y]]/(xy) and can be embedded into H = k[[x1 ]] × k[[x2 ]] as the subring of pairs (f, g) such that f (0) = g(0): x maps to (x1 , 0) and y to (0, x2 ). Evidently this embedding satisﬁes conditions of Deﬁnition 2.1. 2. The dihedral algebra A = k x, y /(x2 , y 2 ) is another example of a nodal ring. In this case H = H(k[[t]], 2) and the embedding A → H is given by the rule x →

0t , 00

00 y → . 10

3. The “Gelfand problem” is that of classiﬁcation of diagrams with relations x+

2

y+

1 x−

3 y−

x+ x− = y+ y− .

If we consider the case when x+ x− is nilpotent (the main part of the problem), such diagrams are just modules over the ring A, which is the subring of Mat(3, k[[t]]) consisting of all matrices (aij ) with a12 (0) = a13 (0) = a23 (0) = a32 (0) = 0. The arrows of the diagram correspond to the following matrices: x+ → te12 ,

x− → e21 ,

y+ → te13 ,

y− → e31 ,

where eij are matrix units. It is also a nodal ring with H being the subring of Mat(3, k[[t]]) consisting of all matrices (aij ) with a12 (0) = a13 (0) = 0 (it is Morita equivalent to H(k[[t]], 2)). 4. The classiﬁcation of quadratic functors, which play an important role in algebraic topology, reduces to the study of modules over the ring A, which is the subring of Z22 × Mat(2, Z2 ) consisting of all triples c 2c a, b, 1 2 c3 c4

with a ≡ c1 (mod 2) and b ≡ c4 (mod 2),

where Z2 is the ring of p-adic integers [11]. It is again a nodal ring: one can take for H the ring of all triples as above, but without congruence conditions; then H = Z22 × H(Z2 , 2). Certainly, a nodal ring is always Backstr¨ om, so Theorem 1.2 can be applied. Moreover, in nodal case the resulting problem belongs to a well-known type. For the sake of simplicity, we consider now the situation, when A is a D-algebra ﬁnitely generated as D-module, where D is a discrete valuation ring with algebraically closed residue ﬁeld k. We denote by U1 , U2 , . . . , Us indecomposable non-isomorphic projective (left) modules over A

DERIVED CATEGORIES FOR NODAL RINGS

27

and by V1 , V2 , . . . , Vr those over H. Condition 2 from Deﬁnition 2.1 implies that there are three possibilities: 1) H ⊗A Ui Vj for some j and Vj does not occur as a direct summand in H ⊗A Uk for k = i; 2) H ⊗A Ui Vj ⊕ Vj (j = j ) and neither Vj nor Vj occur in H ⊗A Uk for k = i; 3) there are exactly two indices i = i such that H ⊗A Ui H ⊗A Ui Vj and Vj does not occur in H ⊗A Uk for k ∈ / { i, i }. We denote by Hj the indecomposable projective H-module such that Hj /RHj Vj . Since H is a semi-perfect hereditary order, any indecomposable complex from D(H) is φ

isomorphic either to 0 → Hk −→ Hj → 0 or to 0 → Hj → 0 (it follows, for instance, from [9]). Moreover, the former complex is completely deﬁned by either j or k and the length l = lengthH Coker φ. We shall denote it both by C(j, −l, n) and by C(k, l, n + 1), while the latter complex will be denoted by C(j, ∞, n), where n denotes the place of Hj ˜ the set (Z \ { 0 }) ∪ { ∞ } and consider the (so the place of Hk is n + 1). We denote by Z ˜ which coincides with the usual ordering separately on positive integers ordering ≤ on Z, and on negative integers, but l < ∞ < −l for any l ∈ N. Note that for each j the submodules of Hj form a chain with respect to inclusion. It immediately implies the following result. Lemma 2.3. There is a homomorphism C(j, l, n) → C(j, l , n), which is an isomorphism ˜ Otherwise the n-th component of any on the n-th components, if and only if l ≤ l in Z. homomorphism C(j, l, n) → C(j, l , n) is zero modulo R. ˜ , so the ˜ to the set Ej,n = C(j, l, n) | l ∈ Z We transfer the ordering from Z latter becomes a chain with respect to this ordering. We also denote by Fj,n the set { (i, j, n) | Vj is a direct summand of H ⊗A Ui }. It has at most two elements. We always consider Fj,n with trivial ordering. Then a triple (A• , B• , ι) from the category T (A) is given by homomorphisms φijn jln : di,j,n Ui → rj,l,n Vj , where (i, j, n) ∈ Fjn , the left Ui comes from Bn and the right Vj comes from direct summands rj,l,n C(j, l, n) of A• . Note that if both C(j, −l, n) and C(k, l, n + 1) correspond to the same complex (then we write ijn C(j, −l, n) ∼ C(k, l, n + 1)), we have rj,−l,n = rk,l,n+1 . We present φijn jln by its matrix Mjln . Then Lemma 2.3 implies the following ijn ijn and Njln describe isomorphic triples Proposition 2.4. Two sets of matrices Mjln if and only if one of them can be transformed to the other by a sequence of the following “elementary transformations”: ijn ijn 1) For any given values of i, n, simultaneously Mjln

→ Mjln S for all j, l such that (ijn) ∈ Fj,n , where S is an invertible matrix of appropriate size. ijn ijn

→ S Mjln for all (i, j, n) ∈ Fjn 2) For any given values of j, l, n, simultaneously Mjln

i,k,n−sgn l i,k,n−sgn l and Mk,−l,n−sgn l → S Mk,−l,n−sgn l for all (i, k, n − sgn l) ∈ Fk,n−sgn l , where S is an invertible matrix of appropriate size and C(j, l, n) ∼ C(k, −l, n − sgn l). If l = ∞, it ijn ijn just means Mj∞n

→ SMj∞n . ijn ijn ijn

→ Mjln + RMjl 3) For any given values of j, l < l, n, simultaneously Mjln n for all (i, j, n) ∈ Fj,n , where R is an arbitrary matrix of appropriate size. Note that, unlike

28

IGOR BURBAN AND YURIY DROZD i,k,n−sgn l the preceding transformation, this one does not touch the matrices Mk,−l,n−sgn l such that C(j, l, n) ∼ C(k, −l, n − sgn l).

This sequence must contain ﬁnitely many transformations for every ﬁxed values of j and n. Therefore we obtain representations of the bunch of semi-chains Ejn , Fjn in the sense of [4], so we can deduce from this paper a description of indecomposables in D(A). We arrange it in terms of strings and bands, often used in representation theory. Definition 2.5. 1. We deﬁne the alphabet X as the set j,n (Ej,n ∪ { (j, n) }). We deﬁne symmetric relations ∼ and − on X by the following exhaustive rules: (a) C(j, l, n) − (j, n) for all l ∈ Z; (b) C(j, −l, n) ∼ C(k, l, n + 1) deﬁned as above; (c) (j, n) ∼ (k, n) (k = j) if Vj ⊕ Vk H ⊗A Ui for some i; (d) (j, n) ∼ (j, n) if Vj H ⊗A Ui H ⊗A Ui for some i = i. 2. We deﬁne an X-word as a sequence w = x1 r1 x2 r2 x3 . . . rm−1 xm , where xk ∈ X, rk ∈ { −, ∼ } such that (a) xk rk xk+1 in X for 1 ≤ k < m; (b) rk = rk+1 for 1 ≤ k < m − 1. We call x1 and xm the ends of the word w. 3. We call an X-word w full if (a) r1 = rm−1 = − (b) x1 ∼ y for each y = x1 ; (c) xm ∼ z for each z = xm . Condition (a) reﬂects the fact that ιR must be an isomorphism, while conditions (b,c) come from generalities on bunches of semi-chains [4]. 4. A word w is called symmetric, if w = w∗ , where w∗ = xm rm−1 xm−1 . . . r1 x1 (the inverse word ), and quasisymmetric, if there is a shorter word v such that w = v ∼ v ∗ ∼ · · · ∼ v ∗ ∼ v. 5. We call the end x1 (xm ) of a word w special if x1 ∼ x1 and r1 = − (respectively, xm ∼ xm and rm−1 = −). We call a word w (a) usual if it has no special ends; (b) special if it has exactly one special end; (c) bispecial if it has two special ends. Note that a special word is never symmetric, a quasisymmetric word is always bispecial, and a bispecial word is always full. 6. We deﬁne a cycle as a word w such that r1 = rm−1 =∼ and xm − x1 . Such a cycle is called non-periodic if it cannot be presented in the form v − v − · · · − v for a shorter cycle v. For a cycle w we set rm = −, xqm+k = xk and rqm+k = rk for any q, k ∈ Z. 7. A (k-th) shift of a cycle w, where k is an even integer, is the cycle w[k] = xk+1 rk+1 xk+2 . . . rk−1 xk . A cycle w is called symmetric if w[k] = w∗ for some k. 8. We also consider inﬁnite words of the sorts w = x1 r1 x2 r2 . . . (with one end) and w = . . . x0 r0 x1 r1 x2 r2 . . . (with no ends) with restrictions (a) every pair (j, n) occurs in this sequence only ﬁnitely many times; (b) there is an n0 such that no pair (j, n) with n < n0 occurs. We extend to such inﬁnite words all above notions in the obvious manner. Definition 2.6 (String and band data). 1. String data are deﬁned as follows: (a) a usual string datum is a full usual non-symmetric X-word w; (b) a special string datum is a pair (w, δ), where w is a full special word and δ ∈ { 0, 1 };

DERIVED CATEGORIES FOR NODAL RINGS

29

(c) a bispecial string datum is a quadruple (w, m, δ1 , δ2 ), where w is a bispecial word that is neither symmetric nor quasisymmetric, m ∈ N and δ1 , δ2 ∈ { 0, 1 }. 2. A band datum is a triple (w, m, λ), where w is a non-periodic cycle, m ∈ N and λ ∈ k∗ ; if w is symmetric, we also suppose that λ = 1. The results of [4, 8] imply Theorem 2.7. Every string or band datum d deﬁnes an indecomposable object C• (d) from D(A), so that 1) Every indecomposable object from D(A) is isomorphic to C• (d) for some d. 2) The only isomorphisms between these complexes are the following: (a) C(w) C(w∗ ); (b) C(w, m, δ1 , δ2 ) C(w∗ , m, δ2 , δ1 ); (c) C(w, m, λ) C(w[k] , m, λ) C(w∗ [k] , m, 1/λ) if k ≡ 0 (mod 4); (d) C(w∗ , m, λ) C(w[k] , m, 1/λ) C(w∗ [k] , m, λ) if k ≡ 2 (mod 4). 3) Every object from D(A) uniquely decomposes into a direct sum of indecomposable objects. The construction of complexes C• (d) is rather complicated, especially in the case, when there are pairs (j, n) with (j, n) ∼ (j, n) (e.g. special ends are involved). So we only show several examples arising from simple node, dihedral algebra and Gelfand problem.

3. Examples 3.1. Simple node. In this case there is only one indecomposable projective A-module (A itself) and two indecomposable projective H-modules H1 , H2 corresponding to the ﬁrst and the second direct factors of the ring H. We have H ⊗A A H H1 ⊕ H2 . So the ∼-relation is given by: 1) (1, n) ∼ (2, n); 2) C(j, l, n) ∼ C(j, −l, n − sgn l) for any l ∈ Z \ { 0 }. Therefore there are no special ends at all. Moreover, any end of a full string must be of the form C(j, ∞, n). Note that the homomorphism in the complex corresponding to C(j, −l, n) and C(j, l, n + 1) (l ∈ N) is just multiplication by xlj . Consider several examples of strings and bands.

Example 3.1.

1. Let w be the cycle

C(2, 1, 1) ∼ C(2, −1, 0) − (2, 0) ∼ (1, 0) − C(1, −2, 0) ∼ C(1, 2, 1)− − (1, 1) ∼ (2, 1) − C(2, 4, 1) ∼ C(2, −4, 0) − (2, 0) ∼ (1, 0)− − C(1, −1, 0) ∼ C(1, 1, 1) − (1, 1) ∼ (2, 1) − C(2, −3, 1) ∼ C(2, 3, 2)− − (2, 2) ∼ (1, 2) − C(1, 2, 2) ∼ C(1, −2, 1) − (1, 1) ∼ (2, 1)

30

IGOR BURBAN AND YURIY DROZD

Then the band complex C• (w, 1, λ) is obtained from the complex of H-modules x2

H2

H2

x21

H1

H1

x42

H2

H2

λ x1

H1 x32

H2

x21

H1

H1

H2

H1

by gluing along the dashed lines (they present the ∼ relations (1, n) ∼ (2, n)). All glueings are trivial, except the last one marked with ‘λ’; the latter must be twisted by λ. It gives the A-complex y

A λx2

y4

A

A

A

x2

A

(3.1)

x

y3

A Here each column presents direct summands of a non-zero component Cn (in our case n = 2, 1, 0) and the arrows show the non-zero components of the diﬀerential. According to the embedding A → H, we have to replace x1 by x and x2 by y. Gathering all data, we can rewrite this complex as

λx2 y 0 2 4 x y 0 0 x y3 A −−−−−→ A ⊕ A ⊕ A −−−−−−→ A ⊕ A ,

though the form (3.1) seems more expressive, so we use it further. If m > 1, one only has to replace A by mA, each element a ∈ A by aE, where E is the identity matrix,

DERIVED CATEGORIES FOR NODAL RINGS

31

and λa by aJm (λ), where Jm (λ) is the Jordan m × m cell with eigenvalue λ. So we obtain the complex 2 x Jm (λ) 0 3

yE

0 y4 E 0 xE y E mA −−−−−−−−−→ mA ⊕ mA ⊕ mA −−−−−−−−−→ mA ⊕ mA . 2 x E

2. Let w be the word

C(1, ∞, 1) − (1, 1) ∼ (2, 1) − C(2, 2, 1) ∼ C(2, −2, 0) − (2, 0) ∼ ∼ (1, 0) − C(1, −3, 0) ∼ C(1, 3, 1) − (1, 1) ∼ (2, 1) − C(2, −1, 1) ∼ ∼ C(2, 1, 2) − (2, 2) ∼ (1, 2) − C(1, 1, 2) ∼ C(1, −1, 1) − (1, 1) ∼ ∼ (2, 1) − C(2, 2, 1) ∼ C(2, −2, 0) − (2, 0) ∼ (1, 0) − C(1, ∞, 0)

Then the string complex C• (w) is

A

A

y

y2

A

x3

A

x

A

y2

A

Note that for string complexes (which are always usual in this case) there are no multiplicities m and all glueings are trivial. a 3. Set a = x + y. Then the factor A/aA is represented by the complex A −→ A, which is the band complex C• (w, 1, 1), where w = C(1, 1, 1) ∼ C(1, −1, 0) − (1, 0) ∼ (2, 0)− − C(2, −1, 0) ∼ C(2, 1, 1) − (2, 1) ∼ (1, 1).

Consider the morphism of this complex to A[1] given on the 1-component by multix plication A −→ A. It is non-zero in D(A), but the corresponding morphism of triples a is (Φ, 0), where Φ arises from the morphism of the complex H −→ H to H[1] given by multiplication with x1 . But Φ is homotopic to 0: x1 = e1 a, where e1 = (1, 0) ∈ H, thus (Φ, 0) = 0 in the category of triples.

32

IGOR BURBAN AND YURIY DROZD

4. The string complex C• (l, 0), where w is the word C(1, ∞, 0) − (1, 0) ∼ (2, 0) − C(2, −1, 0) ∼ C(2, 1, 1) − (2, 1) ∼ ∼ (1, 1) − C(1, −2, 1) ∼ C(1, 1, 2) − (1, 2) ∼ (2, 2) − C(2, −1, 2) ∼ ∼ C(2, 1, 3) − (2, 3) ∼ (1, 3) − C(1, −2, 3) ∼ C(1, 2, 4) − · · · , is x2

x2

y

y

. . . A −→ A −→ A −→ A −→ A −→ 0. Its homologies are not left bounded, so it does not belong to Db (A-mod). 3.2. Dihedral algebra. This case is very similar to the preceding one. Again there is only one indecomposable projective A-module (A itself) and two indecomposable projective Hmodules H1 , H2 corresponding to the ﬁrst and the second columns of matrices from the ring H, and we have H ⊗A A H H1 ⊕ H2 . The main diﬀerence is that now the unique maximal submodule of Hj is isomorphic to Hk , where k = j. So the ∼-relation is given by: 1) (1, n) ∼ (2, n); 2) C(j, l, n) ∼ C(j, −l, n−sgn l) if l ∈ Z\{ 0 } is even, and C(j, l, n) ∼ C(j , −l, n−sgn l), where j = j, if l ∈ Z \ { 0 } is odd. Again there are no special ends. The embeddings Hk → Hj are given by right multiplications with the following elements from H: H1 → H1 − by tr e11 r

H1 → H2 − by t e12 r

H2 → H1 − by t e21 r

H2 → H2 − by t e22

(colength 2r), (colength 2r − 1), (colength 2r + 1), (colength 2r).

When gluing H-complexes into A-complexes we have to replace them respectively tr e11 − by (xy)r , tr e22 − by (yx)r , tr e12 − by (xy)r−1 x, tr e21 − by (yx)r y. The glueings are quite analogous to those for simple node, so we only present the results, without further comments. Example 3.2.

1. Consider the band datum (w, 1, λ), where

w = C(1, −2, 0) ∼ C(1, 2, 1) − (1, 1) ∼ (2, 1) − C(2, −5, 1) ∼ ∼ C(1, 5, 2) − (1, 2) ∼ (2, 2) − C(2, 4, 2) ∼ C(2, −4, 1) − (2, 1) ∼ ∼ (1, 1) − C(1, 3, 1) ∼ C(2, −3, 0) − (2, 0) ∼ (1, 0).

DERIVED CATEGORIES FOR NODAL RINGS

33

The corresponding complex C• (w, m, λ) is mA 2

(yx)2 E

mA

xyxJm (λ)

(xy) xE

mA

xyE

mA

2. Let w be the word C(2, ∞, 0) − (2, 0) ∼ (1, 0) − C(1, −1, 0) ∼ C(2, 1, 1) − (2, 1) ∼ (1, 1) − C(1, 3, 1) ∼ ∼ C(2, −3, 0) − (2, 0) ∼ (1, 0) − C(1, −3, 0) ∼ C(2, 3, 1) − (2, 1) ∼ (1, 1) − C(1, ∞, 1). Then the string complex C• (w) is A

e21

A

2

t e12

A

te21

A

3. The factor A/R is described by the inﬁnite string complex C• (w) ...

e21

A

te12

A

e21

A.

te12

...

te12

A

e21

A

The corresponding word w is

· · · − C(2, 1, 2) ∼ C(1, −1, 1) − (1, 1) ∼ (2, 1)− − C(2, 1, 1) ∼ C(1, −1, 0) − (1, 0) ∼ (2, 0) − C(2, −1, 0) ∼ ∼ C(1, 1, 1) − (1, 1) ∼ (2, 1) − C(2, −1, 1) ∼ C(1, 1, 2) − · · ·

3.3. Gelfand problem. In this case there are 2 indecomposable projective H-modules H1 (the ﬁrst column) and H2 (both the second and the third columns). There are 3 indecomposable A-projectives Ai (i = 1, 2, 3); Ai correspond to the i-th column of A. We have H ⊗A A1 H1 and H ⊗A A2 H ⊗A A3 H2 . So the relation ∼ is given by: 1) (2, n) ∼ (2, n); 2) C(j, l, n) ∼ C(j, −l, n − sgn l) if l is even; 3) C(j, l, n) ∼ C(j , −l, n − sgn l) (j = j) if l is odd. So a special end is always (2, n).

34

IGOR BURBAN AND YURIY DROZD

Example 3.3.

1. Consider the special word w:

(2, 0) − C(2, −2, 0) ∼ C(2, 2, 1) − (2, 1) ∼ (2, 1) − C(2, −4, 1) ∼ ∼ C(2, 4, 2) − (2, 2) ∼ (2, 2) − C(2, 2, 2) ∼ C(2, −2, 1)− − (2, 1) ∼ (2, 1) − C(2, −1, 1) ∼ C(1, 1, 2) − (1, 2) The complex C• (w, 0) is obtained by gluing from the complex of H-modules H2

H2

4

H2

H2

2

H2

H1

1

H2

2

H2

Here the numbers inside arrows show the colengths of the corresponding images. We mark dashed lines deﬁning glueings with arrows going from the bigger complex (with respect to the ordering in Ej,n ) to the smaller one. When we construct the corresponding complex of A-modules, we replace each H2 by A2 and A3 starting with A2 (since δ = 0; if δ = 1 we start from A3 ). Each next choice is arbitrary with the only requirement that every dashed line must touch both A2 and A3 . (Diﬀerent choices lead to isomorphic complexes: one can see it from the pictures below.) All horizontal mappings must be duplicated by slanting ones, carried along the dashed arrow from the starting point or opposite the dashed arrow with the opposite sign from the ending point (the latter procedure will be marked by ‘−’ near the duplicated arrow). So we get the A-complex −

A2

4

A3

2

A2

2

4

A3

2

A2

2 1

2

−

A2

2

A1

1

A3

All mappings are uniquely deﬁned by the colengths in the H-complex, so we just mark them with ‘l.’

DERIVED CATEGORIES FOR NODAL RINGS

35

2. Let w be the bispecial word (2, 2) − C(2, 2, 2) ∼ C(2, −2, 1) − (2, 1) ∼ (2, 1) − C(2, 2, 1) ∼ ∼ C(2, −2, 0) − (2, 0) ∼ (2, 0) − C(2, −4, 0) ∼ C(2, 4, 1)− − (2, 1) ∼ (2, 1) − C(2, 6, 1) ∼ C(2, −6, 0) − (2, 0) The complex C• (w, m, 1, 0) is the following one: aA3 ⊕ bA2

M1

mA3

−M1

2

−

mA2

2

2

mA3

2

mA3

−

4

−

mA2

4

mA2

aA2 ⊕ bA3

M2

where a = [(m + 1)/2], b = [m/2], so a + b = m. (The change of δ1 , δ2 transpose A2 and A3 at the ends.) All arrows are just αl E, where αl is deﬁned by the colength l, except of the “end” matrices Mi . To calculate the latter, write αl E for one of them (say, M1 ) and αl J for anothher one (say, M2 ), where J is the Jordan m × m cell with eigenvalue 1, then put the odd rows or columns into the ﬁrst part of Mi and the even ones to its second part. In our example we get

1 0 M1 = α2 0 0 0

0 0 1 0 0

0 0 0 0 1

0 1 0 0 0

0 0 0 1 0

,

M2 = α6

1 0 0 0 0

1 0 0 1 0

0 1 0 1 0

0 1 0 0 1

0 0 1 . 0 1

(We use columns for M1 and rows for M2 since the left end is the source and the right end is the sink of the corresponding mapping.) 3. The band complex C• (w, 1, λ), where w is the cycle (2, 1) ∼ (2, 1) − C(2, −2, 1) ∼ C(2, 2, 2) − (2, 2) ∼ (2, 2)− − C(2, 4, 2) ∼ C(2, −4, 1) − (2, 1) ∼ (2, 1) − C(2, 6, 1) ∼ ∼ C(2, −6, 0) − (2, 0) ∼ (2, 0) − C(2, −4, 0) ∼ C(2, 4, 1) is

36

IGOR BURBAN AND YURIY DROZD 2

mA2

mA2

2 4

mA3

4λ

mA2

6

24 −

2

−

mA3

−

6

mA3 −

4λ

4λ

mA2 −

4λ

mA3

Superscript ‘λ’ denotes that the corresponding mapping must be twisted by Jm (λ). 4. The projective resolution of the simple A-module U1 is −

A2

1

A1

1

1

A1

1

A3

It coincides with the usual string complex C• (w), where w is (1, 0) − C(1, −1, 0) ∼ C(2, 1, 1) − (2, 1) ∼ (2, 1) − C(2, −1, 1) ∼ C(1, 1, 2) − (1, 2). The projective resolution of U2 (U3 ) is A1 → A2 (respectively A1 → A3 ), which is the special string complex C• (w, 0) (respectively C• (w, 1)), where w = (2, 0) − C(2, −1, 0) ∼ C(1, 1, 1) − (1, 1). Note that gl.dim A = 2.

4. Projective configurations We can “globalize” the results of the preceding sections. The simplest way is to consider the so called projective conﬁgurations, which are a sort of global analogues of Backstr¨om rings. Definition 4.1. Let X be a projective curve over k, which we suppose reduced, but possibly ˜ → X its normalization; then X ˜ is a disjoint union of reducible. We denote by π : X ˜ are rational smooth curves. We call X a projective conﬁguration if all components of X curves (i.e. of genus 0) and all singular points p of X are ordinary. Thelatter means that m m if π −1 (p) = { y1 , y2 , . . . , ym }, the image of OX,p in i=1 OX,y ˜ i contains i=1 mi , where mi is the maximal ideal of OX,y ˜ i.

DERIVED CATEGORIES FOR NODAL RINGS

37

We denote by S = { p1 , p2 , . . . , ps } the set of singular points of X and by S˜ = ˜ We also put O = OX , O ˜ = O ˜ and denote by J the { y1 , y2 , . . . , yr } its preimage in X. X ˜ in O, i.e. the maximal sheaf of π∗ O-ideals ˜ conductor of O contained in O. Set S = O/J ˜ ˜ −1 J . Both these sheaves have 0-dimensional support S, so we O/π and R = π∗ O/J may (and shall) identify them with the algebras of their global sections. In the case of s projective conﬁgurations both these algebras are semi-simple, namely S = i=1 k(pi ) and r R = i=1 k(yi ). Let D(X) = D− (Coh X) be the right bounded derived category of coherent sheaves over X. As X is a projective variety, it can be identiﬁed with the category of fractions K − (VB X)[Q−1 ], where K − (VB X) is the category of right bounded complexes of vector bundles (or, the same, locally free coherent sheaves) over X modulo homotopy and Q is the set of quasi-isomorphisms in K − (VB X). So we always present objects from D(X) ˜ as complexes of vector bundles. We denote by T : D(X) → D(X) ˜ the and from D(X) ∗ left derived functor Lπ . Again if C• is a complex of vector bundles, T C• coincides with π ∗ C• . Just as in Section 1, we deﬁne the category of triples T = T (X) as follows: • Objects of T are triples (A• , B• , ι), where ˜ – A• ∈ D(X); – B• ∈ D(S); – ι is a morphism B• → R ⊗O˜ A• from D(S) such that the induced morphism ιR : R ⊗S B• → R ⊗O˜ A• is an isomorphism in D(R). • A morphism from a triple (A• , B• , ι) to a triple (A• , B• , ι ) is a pair (Φ, φ), where ˜ – Φ : A• → A• is a morphism from D(X); – φ : B• → B• is a morphism from D(S); – the diagram ι

B• −−−−→ R ⊗O˜ A• 1⊗Φ φ ι

(4.1)

B• −−−−→ R ⊗O˜ A• commutes in D(S). We deﬁne a functor F : D(X) → T (X) setting F(C• ) = (π ∗ C• , S ⊗O C• , ι), where ι : S ⊗O C• → R ⊗O˜ (π ∗ C• ) R ⊗O C• is induced by the embedding S → R. Just as in Section 1 the following theorem holds (with almost the same proof, see [6]). Theorem 4.2. The functor F is a representation equivalence, i.e. it is dense and conservative. Remark. We do not now whether it is full, though it seems to be true.

5. Configurations of type A and A˜ As it was shown in [13], even classiﬁcation of vector bundles is wild for almost all projective curves. Among singular curves the only exceptions are projective conﬁgurations of ˜ These curves only have ordinary double points (so no three components type A and A.

38

IGOR BURBAN AND YURIY DROZD

have a common point). Moreover, in A case irreducible components X1 , X2 , . . . , Xs and singular points p1 , p2 , . . . , ps−1 can be so arranged that pi ∈ Xi ∩ Xi+1 , while in A˜ case the components X1 , X2 , . . . , Xs and the singular points p1 , p2 , . . . , ps can be so arranged that pi ∈ Xi ∩ Xi+1 for i < s and ps ∈ Xs ∩ X1 . Note that in A case s > 1, while in A˜ case s = 1 is possible: then there is one component with one ordinary double point (a nodal plane cubic). These projective conﬁgurations are global analogues of nodal rings, and the calculations according Theorem 4.2 are quite similar to those of Section 2. We present here the A˜ case and add remarks explaining which changes should be done for A case. s If s > 1, the normalization of X is just a disjoint union i=1 Xi ; for uniformity, we ˜ if s = 1. We also denote Xqs+i = Xi . Note that Xi P1 for all i. write X1 = X ˜ we suppose that p ∈ Xi correEvery singular point pi has two preimages pi , pi in X; i 1 sponds to the point ∞ ∈ P and pi ∈ Xi+1 corresponds to the point 0 ∈ P1 . Recall that any indecomposable vector bundle over P1 is isomorphic to OP1 (d) for some d ∈ Z. ˜ is isomorphic either to 0 → Oi (d) → 0 So every indecomposable complex from D(X) or to 0 → Oi (−lx) → Oi → 0, where Oi = OXi , d ∈ Z, l ∈ N and x ∈ Xi . The latter complex corresponds to the indecomposable sky-scraper sheaf of length l and support { x }. We denote this complex by C(x, −l, n) and by C(x, l, n + 1). The complex 0 → Oi (d) → is denoted by C(pi , dω, n) and by C(pi−1 , dω, n). As before, n is the unique place, where the complex has non-zero homologies. We deﬁne the symmetric relation ∼ for these symbols setting C(x, −l, n) ∼ C(x, l, n + 1) and C(pi , dω, n) ∼ C(pi−1 , dω, n). Let Zω = (Z ⊕ { 0 }) ∪ Zω, where Zω = { dω | d ∈ Z }. We introduce an ordering on Zω , which is natural on N, on −N and on Zω, but l < dω < −l for each l ∈ N, d ∈ Z. Then an analogue of Lemma 2.3 can be easily veriﬁed. Lemma 5.1. There is a morphism of complexes C(x, z, n) → C(x, z , n) such that its nth component induces a non-zero mapping on Cn (x) if and only if z ≤ z in Zω . We introduce the ordered sets Ex,n = { C(x, z, n) | z ∈ Zω } with the ordering inherited from Zω , We also put Fx,n = { (x, n) } and (pi , n) ∼ (pi−1 , n) for all i, n. Lemma 5.1 shows that the category of triples T (X) can be again described in terms of the bunch of chains { Ex,n , Fx,n }. Thus we can describe indecomposable objects in terms of strings and bands just as for nodal rings. We leave the corresponding deﬁnitions to the reader; they are quite analogous to those from Section 2. If we consider a conﬁguration of type A, we have to exclude the points ps , ps and the corresponding symbols C(ps , z, n), C(ps , z, n), (ps , n), (ps , n). Thus in this case C(ps−1 , dω, n) and C(p1 , dω, n) are not in ∼ relation with any symbol. It makes possible ﬁnite or oneside inﬁnite full strings, while in A˜ case only two-side inﬁnite strings are full. Note that an inﬁnite word must contain a ﬁnite set of symbols (x, n) with any ﬁxed n; moreover there must be n0 such that n ≥ n0 for all entries (x, n) that occur in this word. If x ∈ / S and z ∈ / Zω, the complex C(x, z, n) vanishes after tensoring by R, so gives no essential input into the category of triples. It gives rise to the n-th shift of a sky-scraper sheaf with support at the regular point x. Therefore in the following examples we only consider complexes C(x, z, n) with x ∈ S. Moreover, we conﬁne most examples to the case s = 1 (so X is a nodal cubic). If s > 1, one must distribute vector bundles in the pictures ˜ below among the components of X.

DERIVED CATEGORIES FOR NODAL RINGS

39

Example 5.2. 1. First of all, even a classiﬁcation of vector bundles is non-trivial in A˜ case. They correspond to bands concentrated at 0 place, i.e. such that the underlying cycle w is of the form

(ps , 0) ∼ (ps , 0) − C(ps , d1 ω, 0) ∼ C(p1 , d1 ω, 0)− − (p1 , 0) ∼ (p1 , 0) − C(p1 , d2 ω, 0) ∼ C(p2 , d2 ω, 0)− − (p2 , 0) ∼ (p2 , 0) − C(p2 , d3 ω, 0) ∼ · · · ∼ C(ps , drs ω, 0) (obviously, its length must be a multiple of s, and we can start from any place pk , pk ). Then C• (w, m, λ) is actually a vector bundle, which can be schematically described as ˜ the following gluing of vector bundles over X.

•

•

•

d1

• λ

d2

•

d3

•

.. . •

drs

•

Here horizontal lines symbolize line bundles over Xi of the superscripted degrees, their left (right) ends are basic elements of these bundles at the point ∞ (respectively 0), and the dashed lines show which of them must be glued. One must take m copies of each vector bundle from this picture and make all glueings trivial, except one going from the uppermost right point to the lowermost left one (marked by ‘λ’), where the gluing must be performed using the Jordan m × m cell with eigenvalue λ. In other words, if e1 , e2 , . . . , em and f1 , f2 , . . . , fm are bases of the corresponding spaces, one has to identify f1 with λe1 and fk with λek + ek−1 if k > 1. We denote this vector bundle rover X by V(d, m, λ), where d = (d1 , d2 , . . . , drs ); it is of rank mr and of degree m i=1 di . If r = s = 1, this picture becomes

•

d

λ •

If they are V((d1 , d2 , . . . , ds ), 1, λ) (of degree rs = m = 1, we obtain all line bundles: s ∗ d ). Thus the Picard group is Z × k . i i=1

40

IGOR BURBAN AND YURIY DROZD

In A case there are no bands concentrated at 0 place, but there are ﬁnite strings of this sort:

C(p1 , d1 ω, 0) − (p1 , 0) ∼ (p1 , 0) − C(p1 , d2 ω, 0) ∼ ∼ C(p2 , d2 , 0) − (p2 , 0) ∼ (p2 , 0) − C(p2 , d3 , 0) ∼ · · · ∼ C(ps−1 , ds−1 ω, 0) − (ps−1 , 0) ∼ (ps−1 , 0) − C(ps−1 , ds ω, 0)

So vector bundles over such conﬁgurations are in one-to-one correspondence with integral vectors (d1 , d2 , . . . , ds ); in particular, all of them are line bundles and the Picard group is Zs . In the picture above one has to set r = 1 and to omit the last gluing (marked with ‘λ’). 2. From now on s = 1, so we write p instead of p1 . Let w be the cycle

(p , 1) ∼ (p , 1) − C(p , −2, 1) ∼ C(p , 2, 2) − (p , 2) ∼ (p , 2)− − C(p , 3ω, 2) ∼ C(p , 3ω, 2) − (p , 2) ∼ (p , 2) − C(p , 3, 2) ∼ ∼ C(p , −3, 1) − (p , 1) ∼ (p , 1) − C(p , 1, 1) ∼ C(p , −1, 0)− − (p , 0) ∼ (p , 0) − C(p , −2, 0) ∼ C(p , 2, 1).

Then the band complex C• (w, m, λ) can be pictured as follows:

◦

•

2

•

◦

λ •

◦

3

•

•

3

◦

•

•

◦

1

•

◦

◦

•

2

◦

•

˜ Bullets and circles correspond to Again horizontal lines describe vector bundles over X. the points ∞ and 0; circles show those points, where the corresponding complex gives no input into R⊗O˜ A• . Horizontal arrows show morphisms in A• ; the numbers l inside give the lengths of factors. Dashed and dotted lines describe glueings. Dashed lines (between bullets) correspond to mandatory glueings arising from relations (p , n) ∼

DERIVED CATEGORIES FOR NODAL RINGS

41

(p , n) in the word w, while dotted lines (between circles) can be drawn arbitrarily; the only conditions are that each circle must be an end of a dotted line and the dotted lines between circles sitting at the same level must be parallel (in our picture they are between the 1st and 3rd levels and between the 4th and 5th levels). The degrees of line bundles in complexes C(x, z, n) with z ∈ N ∪ (−N) (they are described by the levels containing 2 lines) can be chosen as d − l and d with arbitrary d (we set d = 0), otherwise (in the second row) they are superscripted over the line. Thus the resulting complex is V((−2, 3, −3), m, 1) −→ V((0, 0, −1, −2), m, λ) −→ V((0, 0), m, 1) (we do not precise mappings, but they can be easily restored). 3. If s = 1, the sky-scraper sheaf k(p) is described by the complex ···

◦

•

···

•

◦

···

◦

•

···

•

◦

◦

•

◦

•

1

•

◦

•

◦

1

•

◦

1

◦

•

◦

•

1

◦

•

•

◦

•

◦

1

1

which is the string complex corresponding to the word . . . C(p , −1, 2) − (p , 2) ∼ (p , 2) − C(p , 1, 2) ∼ C(p , −1, 1)− − (p , 1) ∼ (p , 1) − C(p , 1, 1) ∼ C(p , −1, 0) − (p , 0) ∼ ∼ (p , 0) − C(p , −1, 0) ∼ C(p , 1, 1) − (p , 1) ∼ (p , 1)− − C(p , −1, 1) ∼ C(p , 1, 2) − (p , 2) ∼ (p , 2) − C(p , −1, 2) . . . 4. The band complex C(w, m, λ) , where w is the cycle (p , 0) ∼ (p , 0) − C(p , −3ω, 0) ∼ C(p , −3ω, 0)− − (p , 0) ∼ (p , 0) − C(p , 0ω, 0) ∼ C(p , 0ω, 0) − (p , 0) ∼ ∼ (p , 0) − C(p , −1, 0) ∼ C(p , 1, 1) − (p , 1) ∼ (p , 1)− − C(p , 2, 1) ∼ C(p , −2, 0) − (p , 0) ∼ (p , 0) − C(p , −4, 0) ∼ ∼ C(p , 4, 1) − (p , 1) ∼ (p , 1) − C(p , 5, 1) ∼ C(p , −5, 0)− − (p , 0) ∼ (p , 0) − C(p , 0ω, 0) ∼ C(p , 0ω, 0) describes the complex

42

IGOR BURBAN AND YURIY DROZD

•

-3

•

0

•

λ

•

◦

•

1

◦

•

•

◦

2

•

◦

◦

•

4

◦

•

•

◦

5

•

◦

•

0

•

or V((0, 0), m, 1) ⊕ V((0, 0), m, 1) −→ V((−3, 0, 1, 2, 4, 5, 0), m, λ). Its homologies are zero except the place 0, so it correspond to a coherent sheaf. One can see that this sheaf is a “mixed” one (neither torsion free nor sky-scraper). Note that this time we could trace dotted lines another way, joining the ﬁrst free end with the last one and the second with the third. •

-3

•

0

•

λ

•

◦

•

1

◦

•

•

◦

2

•

◦

◦

•

4

◦

•

•

◦

5

•

◦

•

0

•

It gives an isomorphic object in D(X) V((0, 0, 0, 0), m, 1) −→ V((−3, 0, 1, 5, 0), m, λ) ⊕ V((2, 4), m, 1).

DERIVED CATEGORIES FOR NODAL RINGS

43

Remark. In [6] we used another encoding of strings and bands for projective conﬁgurations, which is equivalent, but uses more speciﬁcs of the situation. In this paper we prefer to use a uniform encoding, which is the same both for nodal rings and for projective conﬁgurations.

6. Application: Cohen–Macaulay modules over surface singularities The results on vector bundles over projective conﬁgurations can be applied to study Cohen–Macaulay modules over normal surface singularities. Recall some related notions. Let A be a noetherian local complete domain of Krull dimension 2, which is normal (i.e. integrally closed in its ﬁeld of fractions), X = Spec A and o be the unique closed points of X (corresponding to the maximal ideal m of A). We call A or X a normal surface singularity. A resolution of this singularity is a morphism of schemes π : Y → X such that • Y is smooth; • π is projective and birational; ˘ = X \ { o }. • the restriction of π onto Y˘ = Y \ π −1 (o) is an isomorphism Y˘ → X We denote by E = π −1 (o)red and call it the exceptional curve of the resolution. It is indeed a projective curve. Let E1 , E2 , . . . , Es be its sirreducible components. We call eﬀective cycles non-zero divisors on Y of the form Z = i=1 ki Ei with ki ≥ 0 and consider such a cycle as a projective curve (non-reduced if some ki > 1), namely the subscheme of Y deﬁned by the sheaf of ideals OY (−Z). Obviously Zred = ki >0 Ei . In [17] C. Kahn established a one-to-one correspondence between Cohen–Macaulay modules over A and some vector bundles over a special eﬀective cycle Z, called a reduction cycle. We shall not present here his result in full generality, but only in the case, when the singularity is minimally elliptic, which means, by deﬁnition, that A is Gorenstein and dimk H1 (Y, OY ) = 1 [19]. We also suppose that the resolution π : Y → X is minimal, i.e. cannot be factored through any other non-isomorphic resolution. Then Kahn’s result can be stated as follows Theorem 6.1 ([17]). Let A be a minimally elliptic surface singularity and Z be the fundamental cycle of its minimal resolution, i.e. the smallest eﬀective cycle such that (Z.Ei ) ≤ 0 for all i. There is one-to-one correspondence between Cohen–Macaulay modules over A and vector bundles F over Z such that F G ⊕ nOZ , where 1) G is generically spanned, i.e. global sections from Γ(E, G) generate G everywhere, except maybe ﬁnitely many closed points; 2) H1 (E, G) = 0; 3) n ≥ dimk H0 (E, G(Z)). Especially, indecomposable Cohen–Macaulay A-modules correspond to vector bundles F G ⊕ nOZ , where either G = 0, n = 1 or G is indecomposable, satisﬁes the above conditions (a,b) and n = dimk H0 (E, G(Z)). (The vector bundle OZ corresponds to the regular A-module, i.e. A itself.) Kahn himself deduced from this theorem and the results of Atiyah [1] a description of Cohen–Macaulay modules over simple elliptic singularities, i.e. such that E is an elliptic curve (smooth curve of genus 1). Using the results of Section 5, one can obtain an analogous ˜ description for cusp singularities, i.e. such that E is a projective conﬁguration of type A. Brieﬂy, one gets the following theorem (for more details see [14]).

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Theorem 6.2. There is a one-to-one correspondence between indecomposable Cohen– Macaulay modules over a cusp singularity A, except the regular module A, and vector bundles V(d, m, λ), where d = (d1 , d2 , . . . , drs ) satisﬁes the following conditions2 : • d > 0, i.e. di ≥ 0 for all i and d = (0, 0, . . . , 0); • no shift of d, i.e. a sequence (dk+1 , . . . , drs , d1 , . . . , dk ), contains a subsequence (0, 1, 1, . . . , 1, 0), in particular (0, 0); • no shift of d is of the form (0, 1, 1, . . . , 1). Moreover, from Theorem 6.1 and the results of [13] one gets the following Theorem 6.3 ([14]). If a minimally elliptic singularity A is neither simple elliptic nor cusp, it is Cohen–Macaulay wild, i.e. the classiﬁcation of Cohen–Macaulay A-modules includes the classiﬁcation of representations of all ﬁnitely generated k-algebras. As a consequence of Theorem 6.2 and the Kn¨orrer periodicity theorem [18, 20], one also obtains a description of Cohen–Macaulay modules over hypersurface singularities of type Tpqr , i.e. factor-rings k[[x1 , x2 , . . . , xn ]]/(xp1 + xq2 + xr3 + λx1 x2 x3 + Q)

(n ≥ 3, 1/p + 1/q + 1/r ≤ 1),

where Q is a non-degenerate quadratic form of x4 , . . . , xn , and over curve singularities of type Tpq , i.e. factor-rings k[[x, y]]/(xp + y q + λx2 y 2 )

(1/p + 1/q ≤ 1/2).

The latter ﬁlls up a ﬂaw in the result of [12], where one has only proved that the curve singularities of type Tpq are Cohen–Macaulay tame, but got no explicit description of modules. Recall that a normal surface singularity A is Cohen–Macaulay ﬁnite, i.e. has only a ﬁnite number of non-isomorphic indecomposable Cohen–Macaulay modules, if and only if it is a quotient singularity, i.e. A k[[x, y]]G , where G is a ﬁnite group of automorphisms [2, 15]. Just in the same way one can show that all singularities of the form A = BG , where B is either simple elliptic or cusp, are Cohen–Macaulay tame, and obtain a description of Cohen–Macaulay modules in this case. We call such singularities elliptic-quotient. There is an evidence that all other singularities are Cohen–Macaulay wild, so Table 1 completely describes Cohen–Macaulay types of isolated singularities (we mark by ‘?’ the places, where the result is still a conjecture).

2 There was a mistake in the preprint [14], where we claimed that d > 0 is enough for V(d, m, λ) to satisfy Kahn’s conditions. It has been improved in the final version. We are thankful to Igor Burban who has noticed this mistake.

DERIVED CATEGORIES FOR NODAL RINGS

45

Table 1. Cohen–Macaulay types of singularities

CM type

curves

surfaces

hypersurfaces

ﬁnite

dominate A-D-E

quotient

simple (A-D-E)

tame

dominate Tpq

elliptic-quotient (only ?)

Tpqr (only ?)

wild

all other

all other ?

all other ?

References [1] M. Atiyah. Vector bundles over an elliptic curve. Proc. London Math. Soc. 7 (1957), 414–452. [2] M. Auslander. Rational singularities and almost split sequences. Trans. Amer. Math. Soc. 293 (1986), 511–531. [3] H. Bass. Finitistic dimension and a homological generalization of semi-primary rings. Trans. Amer. Math. Soc. 95 (1960), 466–488. [4] V. M. Bondarenko. Representations of bundles of semi-chained sets and their applications. Algebra i Analiz 3, No. 5 (1991), 38–61 (English translation: St. Petersburg Math. J. 3 (1992), 973–996). [5] I. I. Burban and Y. A. Drozd. Derived categories of nodal rings. J. Algebra 272 (2004), 46–94. [6] I. I. Burban and Y. A. Drozd. Coherent sheaves on rational curves with simple double points and transversal intersections. Duke Math. J. 121 (2004), 189–229. [7] I. I. Burban, Y. A. Drozd and G.-M. Greuel. Vector bundles on singular projective curves. Applications of Algebraic Geometry to Coding Theory, Physics and Computation. Kluwer Academic Publishers, 2001, 1–15. [8] W. Crawley-Boevey. Functorial filtrations, II. Clans and the Gelfand problem. J. London Math. Soc. 1 (1989), 9–30. [9] Y. A. Drozd. Modules over hereditary orders. Mat. Zametki 29 (1981), 813–816. [10] Y. A. Drozd. Finite modules over pure Noetherian algebras. Trudy Mat. Inst. Steklov Acad. Nauk USSR 183 (1990), 56–68. (English translation: Proc. Steklov Inst. of Math. 183 (1991), 97–108.) [11] Y. A. Drozd. Finitely generated quadratic modules. Manuscripta matem. 104 (2001), 239–256. [12] Y. A. Drozd and G.-M. Greuel. Cohen–Macaulay module type. Compositio Math. 89 (1993), 315–338. [13] Y. A. Drozd and G.-M. Greuel. Tame and wild projective curves and classification of vector bundles. J. Algebra 246 (2001), 1–54. [14] Y. A. Drozd, G.-M. Greuel and I. V. Kashuba. On Cohen–Macaulay modules on surface singularities. Preprint MPI 00–76. Max–Plank–Institut f´ ur Mathematik, Bonn, 2000 (to appear in Moscow Math. J.). ´ [15] H. Esnault. Reflexive modules on quotient surface singularities. J. Reine Angew. Math. 362 (1985), 63–71. [16] R. Hartshorn. Algebraic Geometry. Springer–Verlag, New York, 1977.

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[17] C. Kahn. Reflexive modules on minimally elliptic singularities. Math. Ann. 285 (1989), 141–160. [18] H. Kn¨ orrer. Cohen–Macaulay modules on hypersurface singularities. I. Invent. Math. 88 (1987), 153–164. [19] H. Laufer. On minimally elliptic singularities. Am. J. Math. 99 (1975), 1257–1295. [20] Y. Yoshino. Cohen–Macaulay Modules over Cohen–Macaulay Rings. Cambridge University Press, 1990. Kyiv Taras Shevchenko University, University of Kaiserslautern and Institute of Mathematics of the National Academy of Sciences of Ukraine E-mail address: [email protected] E-mail address: [email protected]

CROWNS IN PROFINITE GROUPS AND APPLICATIONS ELOISA DETOMI AND ANDREA LUCCHINI

In [6] Gasch¨ utz introduced the notion of crown associated with a complemented chief factor H/K of a ﬁnite soluble group G; the crown is a certain normal factor of G, which collects all complemented chief factors of G which are G-isomorphic to H/K. He employed this notion in the construction of a characteristic conjugacy class of subgroups, the prefrattini subgroups. Later this notion has been generalized to all ﬁnite groups (see for example [10] and [8]): it has been deﬁned the crown associated with a non-Frattini chief factor of an arbitrary ﬁnite group. In [4] the notion of crown have been applied to study some properties of the probabilistic zeta function of a ﬁnite group. Let we recall how this function is deﬁned. For a ﬁnite group G and a non-negative integer t let ProbG (t) be the probability that t random elements generate G. In [7] Hall proved that

ProbG (t) =

H≤G

µ(H) |G : H|t

where µ is the M¨ obius function of the subgroup lattice of G. Hence ProbG (t) can be exhibited as a ﬁnite Dirichlet series n∈N an n−t with an ∈ Z and an = 0 unless n divides |G|. So, in view of Hall’s formula, we can speak of ProbG (s) for an arbitrary complex number s. The function ProbG (s) is the multiplicative inverse of a zeta function for G, as described by Mann [11] and Boston [1]. What is shown in [4] is that the properties of the crowns of a ﬁnite group G can be used to study the factors of ProbG (s) in the ring of ﬁnite Dirichlet series with integer coeﬃcients. In the present paper we revise the notion of crown in the contest of proﬁnite groups. We prove that it is possible to extend the deﬁnitions and the results known in the ﬁnite case, to arbitrary proﬁnite groups. Moreover, when G is a ﬁnitely generated proﬁnite group, it is possible to associate to G an inﬁnite formal Dirichlet series, generalizing the deﬁnition given in the ﬁnite case: we apply the crowns to study some properties of this series.

1. G-equivalence and crowns Recall that a proﬁnite group is a compact Hausdorﬀ topological group whose open subgroups form a base for the neighborhoods of the identity; these groups are exactly those obtained as inverse limits of ﬁnite groups. In this paper we are mainly interested in proﬁnite groups, so, unless stated otherwise, “groups” means proﬁnite groups, “subgroups” means closed subgroups and the homomorphisms are assumed to be continuous. Recall that a (closed) subgroup is open if and only if it has ﬁnite index in G. In [8] an equivalence relation among irreducible G-groups is described in the particular case when G is a ﬁnite group; we generalize this notion to the case when G is a proﬁnite group.

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Definition 1. Let G be a proﬁnite group and let A and B be two ﬁnite irreducible G-groups. We say that they are G-equivalent and put A ∼G B, if there are two continuous isomorphisms φ : A → B and Φ : AG → BG such that the following diagram commutes: 1 −−−−→ A −−−−→ φ

AG −−−−→ Φ

G −−−−→ 1

1 −−−−→ B −−−−→ BG −−−−→ G −−−−→ 1 It is immediate that this is an equivalence relation. Notice that if φ : A → B is a G-isomorphism then (ag)Φ = aφ g, a ∈ A, g ∈ G, deﬁnes an isomorphism Φ : AG → BG which makes the above diagram commutative. That is, two G-isomorphic G-groups are G-equivalent. Conversely, if A and B are abelian and G-equivalent then A and B are also G-isomorphic. Indeed for any g ∈ G there exists bg ∈ B with g Φ = bg g, so for any a ∈ A Φ we have (ag )φ = (ag )Φ = (aΦ )g = (aφ )bg g = (aφ )g . But for nonabelian G-groups the G-equivalence is strictly weaker than G-isomorphism; for example the two minimal normal subgroups of G = Alt(5)2 are G-equivalent without being G-isomorphic. Now assume that A and B are ﬁnite irreducible G-groups and consider C = CG (A) ∩ CG (B). Since the actions of G on A and B are assumed to be continuous, CG (A) and CG (B) are open subgroups of G, so in particular C is an open normal subgroup of G and G/C is a ﬁnite group. The following lemma reduces the study of our equivalence relation to the case when G is a ﬁnite group. Lemma 2. A and B are G-equivalent if and only if they are G/C-equivalent. Proof. The statement is obvious when A and B are abelian, since in that case G-equivalent is the same as G-isomorphic. So we may assume that A and B are nonabelian. First assume A ∼G B. For any c ∈ C there exists bc ∈ B with cΦ = bc c. If a ∈ A, we Φ have aφ = (ac )φ = (ac )Φ = (aΦ )c = (aφ )bc c = (aφ )bc , hence bc ∈ Z(Aφ ) = Z(B) = 1. This proves that cφ = c for any c ∈ C. But then it is well deﬁned an isomorphism Ψ : AG/C → BG/C by the position (agC)Ψ = (ag)Φ C which makes commutative the following diagram: 1 −−−−→ A −−−−→ AG/C −−−−→ G/C −−−−→ 1 φ Ψ 1 −−−−→ B −−−−→ BG/C −−−−→ G/C −−−−→ 1. Hence A ∼G/C B. Now assume that A ∼G/C B and let Ψ : AG/C → BG/C be an isomorphism which makes commutative the diagram: 1 −−−−→ A −−−−→ AG/C −−−−→ G/C −−−−→ 1 φ Ψ 1 −−−−→ B −−−−→ BG/C −−−−→ G/C −−−−→ 1.

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49

For any g ∈ G, there exists bg ∈ B such that (gC)Ψ = bg gC. Deﬁne Φ : AG → BG by setting aΦ = aφ if a ∈ A, g Φ = bg g if g ∈ G; it is easy to check that Φ is well deﬁned and the following diagram is commutative: 1 −−−−→ A −−−−→ φ

AG −−−−→ Φ

G −−−−→ 1

1 −−−−→ B −−−−→ BG −−−−→ G −−−−→ 1. Hence A ∼G B. We will say that a section H/K is a chief factor of G if H and K are closed normal subgroups of G with K < H and for any closed normal subgroup X of G with K ≤ X ≤ H either X = K or X = H. Notice that if H/K is a chief factor of G, then there exists an open normal subgroup of N of G with H/K ∼ =G HN/KN ; indeed H (as well as K) is the intersection of all the open normal subgroups that contain it and so, as H = K, we get HN = KN for at least one open normal subgroup N of G. This implies that a chief factor H/K is ﬁnite and that the action of G on H/K is continuous and irreducible. Our ﬁrst aim is to study the G-equivalence relation between the chief factors of G. Recall that a ﬁnite group L is said to be primitive if it has a maximal subgroup with trivial normal core. The socle soc(L) of a primitive group L can be either an abelian minimal normal subgroup (I), or a nonabelian minimal normal subgroup (II), or the product of two nonabelian minimal normal subgroups (III); we say respectively that G is primitive of type I, II, III and in the ﬁrst two cases we say that L is monolithic. As in the case of ﬁnite groups (see [5], [8]) the G-equivalence relation on chief factors of G is strictly related to the primitive epimorphic images of G. We have: Lemma 3. Let G be a proﬁnite group. Two chief factors are G-equivalent as G-groups if and only if they are G-isomorphic either between them or to the two minimal normal subgroups of a ﬁnite primitive epimorphic image of type III of G. Proof. This is true if G is ﬁnite (see [8]) and Lemma 2 allows us to reduce the proof to the ﬁnite case. A chief factor H/K is called Frattini factor if H/K ≤ Frat(G/K). Notice that if H/K is a Frattini factor, then HN/KN is Frattini for every normal closed subgroup N of G. In particular, by considering a ﬁnite image of G, we get that a Frattini chief factor is abelian. Now we are ready to give two crucial deﬁnitions. Let A be a ﬁnite irreducible G-group. We set IG (A) = {g ∈ G | g induces an inner automorphism in A}. Notice that IG (A) contains CG (A), so it is an open normal subgroup of G. Next let XG (A) be the set of open normal subgroups N of G with the properties that N ≤ IG (A), IG (A)/N ∼G A and IG (A)/N is non-Frattini. We deﬁne RG (A) =

N ∈XG (A)

N

50

ELOISA DETOMI AND ANDREA LUCCHINI

if the set XG (A) is nonempty, otherwise we set RG (A) = IG (A). The quotient group IG (A)/RG (A) is called the A-crown of G or the crown of G associated with A. Note that two G-equivalent G-groups, A and B, deﬁne the same crown; indeed IG (A) = IG (B) and so RG (A) = RG (B). Moreover, since RG (A) and IG (A) are closed normal subgroups of G, the quotient groups G/RG (A) and IG (A)/RG (A) are proﬁnite groups; if XG (A) = ∅, then the family of subgroups N/RG (A) where N is an intersection of ﬁnitely many subgroups in XG (A) is a fundamental system of open neighborhoods of the identity in both G/RG (A) and IG (A)/RG (A). We want to study the structure of G/RG (A). First note that RG (A) = IG (A) if and only if A is equivalent to a non-Frattini chief factor of G; so we restrict our attention to this case. Let ρ : G → Aut(A) be deﬁned by g → g ρ , where g ρ : a → ag for all a ∈ A. The monolithic primitive group associated with A is deﬁned as Gρ A ∼ = (G/CG (A))A LG (A) = Gρ ∼ = G/CG (A)

if A is abelian, otherwise.

Observe that LG (A) is a ﬁnite primitive group of type I or II, and soc(LG (A)) ∼ = A. Note that two G-equivalent G-groups may have diﬀerent centralizers in G, but their associated monolithic primitive groups are isomorphic. To simplify our notation we identify A with soc(LG (A)) and we set I = IG (A), R = RG (A), L = LG (A), and X = XG (A). Moreover let Y be the set of normal subgroups of G obtained as intersection of ﬁnitely many subgroups in X ; we remark that, if X = ∅, then G/R is the inverse limit of the family of ﬁnite groups G/N for N ∈ Y (as well as I/R is the inverse limit of the family I/N for N ∈ Y). We want to describe the structure of G/N when N ∈ Y. To do that we recall a deﬁnition: Definition 4. (see [3]) Let now L be a monolithic primitive group and let A be its unique minimal normal subgroup. For each positive integer k, let Lk be the k-fold product of L. The crown-based power of L of size k is the subgroup Lk of Lk deﬁned by Lk = {(l1 , . . . , lk ) ∈ Lk | l1 ≡ · · · ≡ lk mod A}. Clearly, soc(Lk ) = Ak , Lk / soc(Lk ) ∼ = L/A and the quotient group of Lk over any minimal normal subgroup is isomorphic to Lk−1 , for k > 1. Moreover any normal subgroup of Lk either contains or is contained in soc(Lk ). The utility of the previous deﬁnition in our study of the group G/R is explained by the next lemma (see Proposition 9 in [4]): Lemma 5. If Y ∈ Y then G/Y ∼ = Lk where k is the smallest cardinality of a subset {N1 , . . . , Nk } of X with Y = N1 ∩ · · · ∩ Nk . Moreover I/Y = soc(G/Y ) and any chief factor H/K of G with Y ≤ K < H ≤ I is non-Frattini and G-equivalent to A. Corollary 6. If N is a closed normal subgroup of G and R ≤ N then either I ≤ N or N ≤ I. Moreover if N is open and R ≤ N < I then N ∈ Y. Proof. As N is closed and {Y /R}Y ∈Y is a fundamental system of open neighborhoods of the identity in G/R, we have N = Y ∈Y N Y . Now N Y /Y is a normal subgroup of the ﬁnite group G/Y which is isomorphic to Lk for an integer k by the previous lemma. It follows that I/Y = soc(G/Y ) and also either N Y ≤ I or N Y > I. In the ﬁrst case we conclude N ≤ I. Otherwise, N Y > I for every Y ∈ Y and thus N = Y ∈Y N Y ≥ I.

CROWNS IN PROFINITE GROUPS AND APPLICATIONS

51

For any set Ω, the cartesian product LΩ , endowed with the product topology, is a proﬁnite group. Let now consider the subgroup LΩ = {(lω )ω∈Ω ∈ LΩ | lω1 ≡ lω2 mod A for any ω1 , ω2 ∈ Ω}. LΩ is a closed subgroup of LΩ , so it can be viewed as a proﬁnite group; indeed, LΩ is the inverse limit of the family of ﬁnite groups LI , where I is a ﬁnite subset of Ω. Let now D be the set of subsets ∆ of Hom(G, L) satisfying: (1) for any φ ∈ ∆, ker φ ∈ X ; (2) for any ﬁnite subset I = {φ1 , . . . , φk } of ∆, the position g φI = (g φ1 , . . . , g φk ), deﬁnes an homomorphism φI : G → Lk ; in particular g φ1 ≡ g φ2 mod A for any g ∈ G and any φ1 , φ2 ∈ ∆; (3) for any ﬁnite subset I of ∆, the homomorphism φI is surjective. This deﬁnition implies that if ∆ ∈ D, then the functions φI , where I is a ﬁnite subset of ∆, are compatible surjections from G to the inverse system {LI }; thus the corresponding induced mapping of proﬁnite groups Φ : G → L∆ is onto. Moreover, ker φI ∈ Y and so ker Φ = φ∈∆ ker φ is an intersection of elements of X . We may order the elements of D by inclusion. By Zorn’s lemma, D has at least one maximal element. Lemma 7. If ∆ is a maximal element of D then φ∈∆ ker φ = R. Proof. For any φ ∈ ∆ let Nφ = ker φ ∈ X . Suppose that S = φ∈∆ Nφ = R. Then there exists N ∈ X with S ≤ N. Moreover there is an epimorphism α : G → L with ¯ ker α = N. Fix φ¯ ∈ ∆; the map G/(Nφ¯ ∩ N ) → L2 deﬁned by g(Nφ¯ ∩ N ) → (g φ , g α ) is injective; by Lemma 5, G/(Nφ¯ ∩ N ) ∼ = L2 , hence there exists β ∈ Aut(L) such that ¯ β −1 g α β ≡ g φ mod soc(L) for any g ∈ G. Let γ : G → L be deﬁned by g γ = β −1 g α β. ¯ = ∆ ∪ {γ}. We claim that ∆ ¯ ∈ D. The only thing that remains to prove is that Now let ∆ for any ﬁnite subset I = {φ1 , . . . , φk } of ∆, the homomorphism φ¯I : G → Lk+1 deﬁned by g → (g φ1 , . . . , g φk , g γ ) is surjective. By Lemma 5 and the fact that φI is surjective, either φ¯I is surjective or G/(Nφ1 ∩ · · · ∩ Nφk ) ∼ = G/(Nφ1 ∩ · · · ∩ Nφk ∩ N ) ∼ = Lk . But in the latter case S ≤ Nφ1 ∩ · · · ∩ Nφk ≤ N, a contradiction. Let w0 (G) denote the local weight of the proﬁnite group G, i.e. the smallest cardinality of a fundamental system of open neighborhoods of 1 in G. Theorem 8. G/R is homeomorphic to LΩ , for a suitable choice of the set Ω. If X is inﬁnite, then |Ω| = |X |. Proof. By Lemma 7, G/R is homeomorphic to LΩ , where Ω is a maximal element of D. Since a base of neighborhoods of 1 in G/R is given by the subgroups N/R, for N ∈ Y, if X is inﬁnite, then |X | = |Y| = w0 (G/R). On the other hand, w0 (G/R) = w0 (LΩ ) is the cardinality of the set of the ﬁnite subsets of Ω, which is precisely the cardinality of Ω whenever Ω is inﬁnite. In [4] it is proved that if G is a ﬁnite group, then the cardinality of the set Ω which appears in the previous theorem coincides with the number of non-Frattini factors G-equivalent to A in any chief series of G. We want to prove that a similar result holds for arbitrary proﬁnite groups.

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First we recall that any proﬁnite group G has a chain of closed normal subgroups Gµ = 1 ≤ · · · ≤ Gλ ≤ · · · ≤ G0 = G indexed by the ordinals λ ≤ µ such that • Gλ /Gλ+1 is a chief factor of G, for each λ < µ; • if λ is a limit ordinal then Gλ = ν<λ Gν . Such a chain will be called chief series of G. Note that |µ| is an invariant of G, since, if G is inﬁnite, then |µ| = w0 (G). Lemma 9. Let H/K be a chief factor of G. If R ≤ K < H ≤ I then H/K is non-Frattini and G-equivalent to A. Proof. Since Y induces a fundamental systemof open neighborhoods of the identity in I/R, we get that K = N ∈Y KN and H = N ∈Y HN . Thus there exists N ∈ Y such that KN = HN and so H/K ∼ =G HN/KN . By Lemma 5 HN/KN is non-Frattini and G-equivalent to A, and thus the same holds for H/K. Lemma 10. Let H/K be a chief factor of G. Then H/K is non-Frattini and G-equivalent to A if and only if RH/RK = 1 and RH ≤ I. Proof. If RK = RH ≤ I, then, by Lemma 9, RH/RK is non-Frattini and G-equivalent to A; hence also H/K satisﬁes these properties. Now assume that H/K is a non-Frattini chief factor G-equivalent to A. As H/K ≤ Frat(G/K), there exists a closed maximal subgroup, say M, which contains K but not H. Let N be the normal core of M in G; since H and K are normal in G, from K ≤ M and H ≤ M we deduce K ≤ N and H ≤ N. In particular HN/N is a minimal normal subgroup of the primitive group G/N and it is G-isomorphic to H/K, hence G-equivalent to A. Note that, by the deﬁnition of I = IG (A), I/N is the socle of the primitive group G/N . Then either I/N = HN/N and N ∈ X or G/N is primitive of type III and, by Lemma 3, N ∈ Y. In particular R ≤ N = N K < N H ≤ I and so RK = RH ≤ I. Theorem 11. Let {Gλ }λ≤µ be a chief series of G and let Θ be the set of factors Gλ /Gλ+1 which are non-Frattini and G-equivalent to A. The cardinality δG (A) of Θ does not depend on the choice of the chief series. Moreover if G/R ∼ = LΩ then |Ω| = δG (A). Proof. We obtain a chain of closed normal subgroups {Hλ }λ≤µ with H0 = G and Hµ = R by deﬁning Hλ = RGλ . For any λ ≤ µ, either Hλ = Hλ+1 or Hλ /Hλ+1 is a chief factor of G/R. Moreover the set of non trivial factors Hλ /Hλ+1 coincides with the set of factors of a chief series of G/R. By Corollary 6 either Hλ ≤ I or Hλ ≥ I. Let ν be the smallest ordinal with Hν ≤ I. Now Lemma 10 implies that if λ < ν, Gλ /Gλ+1 cannot be non-Frattini and G-equivalent to A; moreover, if λ ≥ ν, then Hλ /Hλ+1 = 1 if and only if Gλ /Gλ+1 is non-Frattini and G-equivalent to A. So, {Hλ /Hλ+1 | Hλ /Hλ+1 = 1} is a chief series of G/R which passes through I/R and has the property that the elements of Θ are in bijective correspondence with the non trivial factors Hλ /Hλ+1 contained in I/R; in particular |Θ| does not depend on the choice of the series. Finally, since G/R ∼ = AΩ , we conclude that |Θ| = |Ω|. = LΩ implies I/R ∼

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Theorem 12. If G is ﬁnitely generated then δG (A) is ﬁnite for every ﬁnite irreducible G-group A. Proof. If X ∈ XG (A), then by deﬁnition I/X ∼ = A and so |G : X| = |G : I| · |A| = n for an integer n. As G is ﬁnitely generated, the number of subgroups of index n is ﬁnite; thus |XG (A)| is ﬁnite and consequently R has ﬁnite index in G. Therefore G/R ∼ = LΩ for a ﬁnite set Ω and the result follows from the previous theorem. 2. Factors of the Dirichlet series PG (s) Let G be a ﬁnitely generated proﬁnite group, that is a proﬁnite group topologically generated by a ﬁnite number of elements. Recall that for every integer n, the number of open subgroups of index n in G is ﬁnite. So we can deﬁne a formal Dirichlet series PG (s) as follows: αn with αn := µ(H) PG (s) := s n n∈N

|G:H|=n

where µ(H) denotes the M¨ obius function of the lattice of open subgroups of G, deﬁned by µ (K) = 0 unless H = G, in which case the sum is 1. H≤K G When G is ﬁnite, PG (s) is actually the series ProbG (s) deﬁned in the introduction; in particular, when t is an integer, PG (t) is the probability that t random elements generate G. Given a normal subgroup N of G we deﬁne a formal Dirichlet series PG,N (s) as follows: PG,N (s) :=

bn ns

with

n∈N

bn :=

µ(H).

|G:H|=n HN =G

Notice that PG (s) = PG,G (s). Moreover, N admits a proper supplement in G if and only if it is not contained in the Frattini subgroup of G; it follows easily that PG,N (s) = 1 if and only if N ≤ Frat(G). Let A(s) = n an /ns and B(s) = n bn /ns be two formal Dirichlet series. We denotes by A(s) ∗ B(s) the convolution product of A(s) and B(s), i.e. the Dirichlet series n cn /n with cn = d|n ad bn/d . Theorem 13. If G is a ﬁnitely generated proﬁnite group and N is a closed normal subgroup of G then PG (s) = PG/N (s) ∗ PG,N (s). Proof. The result is already known when G is a ﬁnite group (see for example [2] section 2.2), so we need an argument to reduce to the ﬁnite case. Let n ∈ N; we have to prove that the coeﬃcients of 1/ns in PG (s) and PG/N (s) ∗ PG,N (s) are equal, that is: |G:H|=n

µ(H) =

d|n

N ≤H1 ≤G |G:H1 |=d

µ(H1 )

H2 N =G |G:H2 |=n/d

µ(H2 )

(2.1)

Let Xn be the intersection of the open subgroups of G with index at most n; as G is ﬁnitely generated, Xn has ﬁnite index in G. Thus PG/Xn (s) = PG/N Xn (s) ∗ PG/Xn ,N Xn /Xn (s).

(2.2)

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Now (2.1) follows from (2.2) since the terms in (2.1) are equal to the coeﬃcients of 1/ns in the two series in (2.2); indeed if |G : H| ≤ n then Xn ≤ H and µ(H) = µ(H/Xn ). If G is a ﬁnite group, by taking a chief series Σ : 1 = Nl < . . . N1 < N0 = G and iterating the above formula, we obtain an expression of PG (s) as a product indexed by the non-Frattini chief factors in the series: PG (s) =

PG/Ni+1 ,Ni /Ni+1 (s).

(2.3)

Ni /Ni+1 ≤Frat(G/Ni+1 )

In [4] it is proved that if G is a ﬁnite group, the factors in (2.3) are independent of the choice of the series Σ. Moreover it is also described how the factors in (2.3) look like. Let L be a ﬁnite monolithic primitive group, and let A be its socle. We deﬁne PL,1 (s) = PL,A (s), (1 + q + · · · + q i−2 )γ PL,i (s) = PL,A (s) − , |A|s

(2.4)

where γ = |CAut A (L/A)| and q = | EndL A| if A is abelian, q = 1 otherwise. In [4] it is proved that if G is a ﬁnite group, then the factors of PG (s) corresponding to the nonFrattini factors in a chief series are all of the kind PL,i (s), for suitable choices of L and i. In particular: Theorem 14. ([4], Theorem 18) Let G be a ﬁnite group. Then

PG (s) =

A

PLA ,i (s)

(2.5)

1≤i≤δG (A)

where A runs over the set of irreducible G-groups G-equivalent to a non-Frattini chief factor of G, and LA is the monolithic primitive group associated with A. Moreover, the factorization of PG (s) corresponding to the non-Frattini factors in a chief series Σ of G is precisely (2.5), independently of the choice of Σ. We want to generalize this result to the case when G is a ﬁnitely generated proﬁnite group. First of all, we get the following result: Theorem 15. Let G be a ﬁnitely generated proﬁnite group and let A = H/K be a nonFrattini chief factor of G. If R = RG (A), we have: PG/K,H/K (s) = PG/RK,RH/RK (s) = P˜L ,k (s) A

where LA is the monolithic primitive group associated with A and k = δG/RK (A).

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Proof. Let Sn = {X ≤ = G, |G : X| = n and µ(X) = 0}; notice G | K ≤ sX ≤ G, XH α /n with α = that PG/K,H/K (s) = n n n∈N X∈Sn µ(X). But µ(X) = 0 only if X is an intersection of closed maximal subgroups of G. Moreover in the proof of Lemma 10 we have seen that if M is a closed maximal subgroup of G with K ≤ M and H ≤ M then R ≤ M ; hence RK ≤ X for any X ∈ Sn . This implies immediately that PG/K,H/K (s) = PG/RK,RH/RK (s). Now, by Theorems 12 and 11, we get that k = δG/RK (A) is ﬁnite and G/RK ∼ = Lk . Moreover RH/RK is a minimal normal subgroup of G/RK and is equivalent to A. Therefore, by [4] Theorem 17, we conclude that PG/RK,RH/RK (s) = P˜L,k (s). Let now G be an inﬁnite ﬁnitely generated proﬁnite group. In that case ω0 (G) = ℵ0 and G has a chief series of length ℵ0 Σ : G = G0 ≥ · · · ≥ Gi ≥ · · · ≥ Gℵ0 = 1;

(2.6)

to each chief factor Gi /Gi+1 it is associated the ﬁnite Dirichlet series Pi (s) = PG/Gi+1 ,Gi /Gi+1 (s).

(2.7)

Now, by Theorem 13, for any i ∈ N we get PG/Gi+1 (s) = P0 (s) ∗ P1 (s) ∗ · · · ∗ Pi (s). So, one is tempted to say that PG (s) is the product of the inﬁnite factors {Pi (s)}i∈N . Unfortunately, since the formal series PG (s) is not necessarily convergent, PG (s) is not a function and a sentence like “PG (s) = limi→∞ P0 (s) . . . Pi (s)” has no meaning if we think to a convergence of complex functions. However we can prove that the formal Dirichlet series PG (s) is uniquely determined as an “inﬁnite convolution” of the factors {Pi (s)}i∈N and that the set of these factors is independent of the choice of the chief series Σ. To do that we give some deﬁnitions. αωn /ns , with Let P = {Pω (s)}ω∈Ω be a family of ﬁnite Dirichlet series, let say Pω (s) = the property that αω1 = 1, for every ω. We say that the family P is suitable for convolution if for every m > 1 the set Ωm = {ω ∈ Ω | αωn = 0 for some 1 < n ≤ m} is ﬁnite. If P has this properties, then, for any m > 1, ∗ (s) = Pm

Pω (s)

ω∈Ωm ∗ (s) = is a well-deﬁned and ﬁnite Dirichlet series, say Pm (inﬁnite) convolution product of P to be

P ∗ (s) =

n∈N

γn /ns

s n cn,m /n .

Then we deﬁne the

where γ1 = 1 and γn = cn,n if n > 1.

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Note that if P is suitable for convolution and ∆ ⊆ Ω, then the family Q = {Pω (s)}ω∈∆ is again suitable for convolution and the following holds: Lemma 16. Let Q∗ (s) = n δn /ns and let m > 1. If Ωm ⊆ ∆, then γn = δn for any n ≤ m. ∗ For example, whenever n ≤ m, the ﬁrst n terms of Pn∗ (s) and Pm (s) are equal. ˜ Now let ΩG be the set of pairs (A, i) where A runs over a set of representatives for the ˜G equivalence classes of ﬁnite irreducible G-groups and 1 ≤ i ≤ δG (A). If ω = (A, i) ∈ Ω ˜ deﬁne Pω (s) = PLA ,i (s) as in (2.4); ﬁnally let

PG = {Pω (s)}ω∈Ω˜ G .

(2.8)

Given a chief series Σ of G, let ΩΣ be the set of non-Frattini chief factors in this series and let PΣ = {PG/K,H/K (s)}H/K∈ΩΣ . Theorem 17. Let G be a ﬁnitely generated proﬁnite group and let Σ be a chief series of G. ˜ G such that if H/K ∈ ΩΣ then PG/K,H/K (s) = Pφ(H/K) (s). There is a bijection φ : ΩΣ → Ω ∗ The two families PΣ and PG are suitable for convolution and PG (s) = PΣ∗ (s) = PG (s). Proof. Let A be a ﬁnite irreducible G-group with δG (A) = 0. By Theorem 11 there are exactly δ = δG (A) non-Frattini factors H1,A /K1,A , . . . , Hδ,A /Kδ,A in the chief series Σ with Hi,A /Ki,A ∼G A, for 1 ≤ i ≤ δ. We may assume Kδ,A < Hδ,A < · · · < Ki,A < Hi,A < · · · < K1,A < H1,A . ˜ G . Moreover, by Theorem 15, The map Hi,A /Ki,A → (A, i) induces a bijection φ : ΩΣ → Ω ˜ PHi,A /Ki,A (s) = PLA ,i (s) = Pφ(Hi,A /Ki,A ) (s). This proves the ﬁrst part of the statement and ∗ (s) for every chief series Σ. that PΣ∗ (s) = PG ¯ such that To complete the proof it now suﬃces to prove that there exists a chief series Σ ∗ PΣ¯ is suitable for convolution and that PG (s) = PΣ¯ (s). For any integer n deﬁne Xn to be the intersection of the open subgroups H of G with |G : H| ≤ n. Since G is ﬁnitely generated, Xn is an open normal subgroup of G. Moreover n Xn = 1, hence we may produce a chief ¯ by reﬁning the series {Xn }n∈N . series Σ Now ﬁx an integer m. Let H/K ∈ ΩΣ¯ and PG/K,H/K (s) = n βn /ns . If βn = 0 for some 1 = n ≤ m, then there exists an open subgroup Y /K of G/K with G = HY and |G : Y | = n; this implies Xn ≤ K, otherwise H ≤ Xn and, as Xn ≤ Y , we get G = HY = Y, a contradiction. Thus Xm ≤ Xn ≤ K. As G/Xm is ﬁnite, there are only ﬁnitely many factors H/K ∈ ΩΣ¯ with Xm ≤ K. This proves that the family PΣ¯ is suitable for convolution. Moreover, if Qm is the subfamily of PΣ¯ indexed by the factors H/K ∈ ΩΣ¯ satisfying Xm ≤ K, then Lemma 16 gives that the coeﬃcients bm and cm in the two series Q∗m (s) =

are equal.

bn , ns n

PΣ∗¯ (s) =

cn ns n

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On theother hand Q∗m (s) = PG/Xm (s); thus, by deﬁnition of Xm , the coeﬃcient am of PG (s) = m am /ms is am =

µ(X) =

|G:X|=m

µ(X/Xm ) = bm

|G/Xm :X/Xm |=m

and we conclude that am = cm . Since this holds for every integer m, the theorem is proved.

3. The probabilistic zeta function Since a proﬁnite group G has a natural compact topology, it has also a Haar measure, which is determined uniquely by the algebraic structure of G. We normalize this measure so that G has measure 1, and is thus a probability space. This allows us to deﬁne, for any positive integer t, ProbG (t) as the measure of the subset {(g1 , . . . , gt ) ∈ Gt | g1 , . . . , gt topologically generate G}. If G is ﬁnite, the function φG (t) = Prob(G)|G|t is the Eulerian function of G, which gives the number of ordered t-uples (g1 , . . . , gt ) generating G. This function was introduced and studied by P. Hall [7], who proved in particular φG (t) =

µ(H)|H|t .

(3.1)

H≤G

This implies that if G is ﬁnite, then the Dirichlet series PG (s) deﬁned in the previous section is a complex function with the property that, for any integer t, PG (t) = ProbG (t) = φG (t)/|G|t ; the complex function ζG (s) = 1/PG (s) is called the probabilistic zeta function associated to the ﬁnite group G. In [11] Mann proposed the problem of looking for a complex function interpolating the values {ProbG (t) | t ∈ N}. Of course in order to discuss this question one must focus his attention on ﬁnitely generated proﬁnite groups with the property that ProbG (t) > 0 for some t ∈ N (otherwise the interpolating function we are looking for is just the zero function); the groups with this property are called positively ﬁnitely generated (PFG). It is worth mentioning that a ﬁnitely generated proﬁnite group is not necessarily PFG. For example Kantor and Lubotzky [9] proved that the free proﬁnite group of rank d is not PFG if d ≥ 2. The conjecture proposed by Mann in [11] is the following: to each PFG group G there corresponds naturally a “zeta function” ζG (s) which is an analytic function deﬁned in some right half plane of the complex numbers, such that ζG (t) = ProbG (t)−1 , for all ˆ denotes the proﬁnite completion of a cyclic suﬃciently large integers t. For example, if Z inﬁnite group, then

ProbZˆ (t) =

µ(n) n

nt

=

1 −1 = ζ(t)−1 t n n

where ζ is the Riemann zeta function. Hence in this case ζ(s) is the function we are looking for.

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Before discussing Mann’s conjecture, we want to prove that it is possible to decide whether a ﬁnitely generated proﬁnite group G is PFG from the knowledge of the family PG of ﬁnite Dirichlet series deﬁned in (2.8). First recall: Proposition 18 (Mann [11] Theorem 1). If G is a ﬁnitely generated proﬁnite group and t is an integer, then ProbG (t) = inf N ProbG/N (t), where N varies over all open normal subgroups of G. In particular, if Σ : G = G0 < · · · < Gℵ0 = 1 is a chief series of G then ProbG (t) = inf ProbG/Gn (t) = lim PG/Gn (t), n∈N

n→∞

(3.2)

since {Gn }n∈N is a base of neighborhoods of 1 in G, ProbG/Gn (t) ≥ ProbG/Gn+1 (t) and PG/Gn (t) = ProbG/Gn (t). This suggests that given an integer t there may be a relation between ProbG (t) and the inﬁnite product of the numbers Pω (t), where Pω (s) ∈ PG as deﬁned in (2.8). Theorem 19. A ﬁnitely generated proﬁnite group G is PFG if and only if the inﬁnite prod uct ω∈Ω˜ G Pω (t) is absolutely convergent for some positive integer t; in that case ProbG (t) = ˜ G Pω (t) for suﬃciently large positive integers. ω∈Ω Proof. If G is ﬁnite, the result follows from Theorem 14. So, let G be inﬁnite and let Σ : G = G0 > . . . > Gℵ0 = 1 be a chief series of G. Let us denote the Dirichlet series PG/Gi+1 ,Gi /Gi+1 (s) by Pi (s); notice that Pi (s) = 1 whenever Gi /Gi+1 is a Frattini factor. Now G is PFG if and only if for some integer t we have 0 = ProbG (t) = lim PG/Gn (t) = lim P0 (t) . . . Pn−1 (t). n→∞

n→∞

(3.3)

Since 0 < Pi (t) < 1 for any i ∈ N, the condition lim n→∞ P0 (t) . . . Pn−1 (t) = 0 is equivalent to the absolute convergence of the inﬁnite product n∈N Pn (t). As the value of an absolutely convergent product does not change if the factors are reordered, and the Frattini 17 we deduce factors do not inﬂuence the product, from Theorem that the inﬁnite product n∈N Pn (t) is absolutely convergent if and only if ω∈Ω˜ G Pω (t) is absolutely convergent. The previous theorem says that if G is PFG then the inﬁnite product ω∈Ω˜ GPω (t) is absolutely convergent for any suﬃciently large integer t. Unfortunately from this result no information can be obtained about the behaviour of the product ω∈Ω˜ G Pω (s) when s is a complex number. Mann [11] proved that if G is prosoluble then ω∈Ω˜ G Pω (s) is absolutely convergent in some right half plane of the complex plane. We conjecture that this holds for an arbitrary PFG group G. This would give the possibility of deﬁning the probabilistic zeta function of G as the multiplicative inverse of the inﬁnite product ω∈Ω˜ G Pω (s). 4. Recognizing PFG groups Mann and Shalev proved that PFG groups can be characterized by the behaviour of the function mn (G) which is deﬁned as the number of closed maximal subgroups of G with index n.

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Theorem 20 (Mann and Shalev [12] Theorem 4). A ﬁnitely generated proﬁnite group G is PFG if and only if G has polynomial maximal subgroup growth, i.e. there exists a constant c such that for all n, the number mn (G) is at most nc . This criterion can be translated in term of “multiplicity” of the chief factors of G. We need ˜ ﬁrst some deﬁnition. For a ﬁnite primitive group L let λ(L) be the minimum of the index |L : X| where X runs over the set of core-free maximal subgroups of L. Let M be the set of closed maximal subgroups of G; as M ∈ M is open, its normal core MG has ﬁnite index in G and G/MG is a primitive group. We deﬁne K = {N G | N = MG for some M ∈ M}, ˜ κn (G) = |{N ∈ K | λ(G/N ) = n}|. It was announced by Pyber that, using the classiﬁcation of the ﬁnite simple groups, the following result can be proved: Theorem 21 (Pyber). There exists a constant b such that for every ﬁnite group G and every n ≥ 2, G has at most nb core-free maximal subgroups of index n. In fact, b = 2 will do. Using this result we can deduce easily: Lemma 22. κn (G) ≤ mn (G) ≤ n2 m≤n κm (G). Proof. The ﬁrst inequality is trivial. We prove the second one. Let M ∈ Mn ; since ˜ ) ≤ n, the normal subgroup M can be chosen in at most λ(G/M G G m≤n κm (G) diﬀerent 2 ways. Given N = MG , by Theorem 21, there are at most n core-free maximal subgroups of index n containing N. Combining Theorem 20 and Lemma 22 we obtain: Corollary 23. Let G be a ﬁnitely generated proﬁnite group. The following are equivalent: (1) G is PFG; (2) there exists a constant c1 such that mn (G) ≤ nc1 for all n ∈ N; (3) there exists a constant c2 such that κn (G) ≤ nc2 for all n ∈ N. Now we study κn (G) using the G-equivalent relation among ﬁnite irreducible G-groups described in the ﬁrst section. In the following G will be a ﬁnitely generated proﬁnite group. Let N be an element of K; the quotient group G/N is a ﬁnite primitive group and its minimal normal subgroups are all equivalent to the same irreducible G-group, say A; indeed either G/N is monolithic or G/N is primitive of type III and, by Lemma 3, its two minimal normal subgroups are G-equivalent. By deﬁnition IG (A)/N is the socle of G/N and so either N ∈ XG (A) or N is the intersection of two diﬀerent elements of XG (A); anyway, RG (A) ≤ N and G/N ∼ = Li where L = LG (A) is the monolithic primitive group associated with A and i = 1, 2. When G/N ∼ = L2 , A is nonabelian and any faithful primitive representation of G/N has degree |A|. Given an irreducible G-group A, we deﬁne KA as the subset of K containing those normal subgroups N with the property that the minimal normal subgroups of G/N are equivalent to A. It is clear that Lemma 24. The set K is the disjoint union of the subsets KA , where A runs over the set of ﬁnite irreducible G-groups, up to equivalence.

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˜ Now deﬁne λ(A) = λ(L), where L = LG (A). Notice that when A is abelian λ(A) = |A| ˜ and λ(G/N ) = |A| for every N ∈ XG (A), since in this case G/N ∼ = L (see Lemma 5); 1 2 ∪ KA therefore, for n = |A|, Kn ⊇ KA = XG (A). When A is nonabelian we get KA = KA 1 2 1 ∼ ∼ with KA = {N ∈ KA | G/N = L} and KA = {N ∈ KA | G/N = L2 }. Notice that KA = 1 2 ˜ ) = λ(A) for every N ∈ KA ; also, KA is the set of the intersections XG (A) and so λ(G/N 2 ˜ . Set γA = of two diﬀerent normal subgroups of XG (A) and λ(G/N ) = |A| for N ∈ KA |CAut A (L/A)| and qA = | EndL A| if A is abelian, qA = 1 otherwise. Since, by Theorems 11 and 12, IG (A)/RG (A) ∼ = AδG (A) and δG (A) is ﬁnite, it can be easily proved that δ (A)−1

Lemma 25. Let n = λ(A). If A is abelian, then |Kn ∩ KA | = 1 + qA + · · · + qAG 1 2 is non abelian, then |Kn ∩ KA | = δG (A) and |Kn ∩ KA | = δG (A)(δG (A) − 1)/2.

; if A

Now deﬁne: κab n =

δ (A)−1

1 + qA + · · · + qAG

;

A =1,|A|=n

κ1n =

δG (A);

A =A,λ(A)=n

κ2n =

A =A,|A|=n

δG (A) . 2

1 2 ab 1 2 By Lemma 25 κn (G) = κab n + κn + κn . So G is PFG if and only if κn , κn , κn are polynomially bounded. Now let αn (G) be the number of ﬁnite abelian irreducible G-groups A, with |A| = n and δG (A) > 0.

Lemma 26. κab n is polynomially bounded if and only if αn (G) is polynomially bounded. Proof. Obviously αn (G) ≤ κab n . We have to prove that if αn (G) is polynomially bounded then the same is true for κab n . Assume that G can be generated by r elements and let A be a ﬁnite abelian irreducible G-group with |A| = n and δG (A) = 0. By [4] Theorem 18, P˜L,δG (A) (r) > 0, where L = LG (A). On the other hand δ (A)−2

(1 + qA + · · · + qAG P˜L,δG (A) (r) = PL,A (r) − |A|r

)γA

.

In particular δ (A)−2

(1 + qA + · · · + qAG |A|r

)γA

< PL,A (r) ≤ 1,

hence δ (A)−2

1 + qA + · · · + qAG

≤

|A|r ≤ |A|r = nr γA

and δ (A)−1

1 + qA + · · · + qAG

δ (A)−2

≤ (1 + qA )(1 + qA + · · · + qAG

r+1 since qA ≤ n. It follows that κab . n ≤ 2αn (G)n

) ≤ (1 + qA )nr = 2nr+1

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61

Now consider the number ξn (G) of ﬁnite nonabelian irreducible G-groups A, with |A| = n and δG (A) ≥ 2. Lemma 27. κ2n (G) is polynomially bounded if and only if ξn (G) is polynomially bounded. Proof. Obviously ξn (G) ≤ κ2n (G). We have to prove that if ξn (G) is polynomially bounded then the same is true for κ2n (G). Assume that G can be generated by r elements and let A be a ﬁnite nonabelian irreducible G-group with |A| = n and δG (A) > 1. By [4] Theorem 18, P˜L,δG (A) (r) > 0, where L = LG (A). On the other hand (δG (A) − 1)γA P˜L,δG (A) (r) = PL,A (r) − . |A|r In particular (δG (A) − 1)γA < PL,A (r) ≤ 1, |A|r hence δG (A) − 1 ≤

|A|r ≤ |A|r = nr . γA

It follows that κ2n (G) ≤ ξn (G)(n + 1)2r . Lemma 28. If κ1n (G) is polynomially bounded then ξn (G) and κ2n (G) are polynomially bounded. Proof. If A is nonabelian and |A| = n, then λ(A) ≤ n; so ξn (G) ≤

m≤n

κ1m (G) ≤ nκ1n (G).

So we conclude: Theorem 29. G is PFG if and only if αn (G) and κ1n (G) are polynomially bounded. We conclude with two question: Question 1. Does there exist a ﬁnitely generated proﬁnite group G such that αn (G) is not polynomially bounded? Question 2. Let βn (G) be the number of nonabelian irreducible G-groups A, with λ(A) = n and δG (A) = 0. Does there exist a ﬁnitely generated proﬁnite group G such that βn (G) is not polynomially bounded? We conjecture that both these questions have a negative answer. This would imply: Conjecture.A ﬁnitely generated proﬁnite group G is PFG if and only if ρG (n) = max{δG (A) | A nonabelian, λ(A) = n} is polynomially bounded.

(4.1)

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ELOISA DETOMI AND ANDREA LUCCHINI

References [1] N. Boston, ‘A probabilistic generalization of the Riemann zeta functions’, Analytic Number Theory 1 (1996), 155–162. [2] K. S. Brown, ‘The coset poset and probabilistic zeta function of a finite group’, J. Algebra 225 (2000), 989–1012. [3] F. Dalla Volta and A. Lucchini, ‘Finite groups that need more generators than any proper quotient’, J. Austral. Math. Soc., Ser. A 64 (1998), 82–91. [4] E. Detomi and A. Lucchini, ‘Crowns and factorization of the probabilistic zeta function of a finite group’, J. Algebra to appear. [5] P. F¨ orster, ‘Chief factors, crowns, and the generalized Jordan-Holder Theorem’, Comm. Algebra 16 (1988), 1627–1638. [6] W. Gasch¨ utz, ‘Praefrattinigruppen’, Arch. Math. 13 (1962), 418–426. [7] P. Hall, ‘The Eulerian functions of a group’, Quart. J. Math. 7 (1936), 134–151. [8] P. Jim´enez-Seral and J. Lafuente, ‘On complemented nonabelian chief factors of a finite group’, Israel J. Math. 106 (1998), 177–188. [9] W. M. Kantor and A. Lubotzky, ‘The probability of generating a finite classical group’, Geom. Ded. 36 (1990), 67–87. [10] J. Lafuente, ‘Crowns and centralizers of chief factors of finite groups’, Comm. Algebra 13 (1985), 657–668. [11] A. Mann, ‘Positively finitely generated groups’, Forum Math. 8 No. 4 (1996), 429–459. [12] A. Mann and A. Shalev, ‘Simple groups, maximal subgroups and probabilistic aspects of profinite groups’, Israel J. Math. 96 (1996), 449–46 8.

A. Lucchini Dipartimento di Matematica Universit`a di Brescia Via Valotti, 9 25133 Brescia, Italy [email protected] E. Detomi Dipartimento di Matematica Universit`a di Brescia Via Valotti 9 25133 Brescia, Italy [email protected] Current address: Dipartimento di Matematica Universit`a di Padova via Belzoni, 7 35131 Padova, Italy [email protected]

THE GALOIS STRUCTURE OF AMBIGUOUS IDEALS IN CYCLIC EXTENSIONS OF DEGREE 8 G. GRIFFITH ELDER

Abstract. In cyclic, degree 8 extensions of algebraic number ﬁelds N/K, ambiguous ideals in N are canonical Z[C8 ]-modules. Their Z[C8 ]-structure is determined here. It is described in terms of indecomposable modules and determined by ramiﬁcation invariants. Although inﬁnitely many indecomposable Z[C8 ]-modules are available (classiﬁcation by Yakovlev), only 23 appear.

1. Introduction We are concerned with the interrelationship between two basic objects in algebraic number theory: the ring of integers and the Galois group. In particular, we seek to understand the eﬀect of the Galois group upon the ring of integers. At the same time, we are also interested in the Galois action upon other fractional ideals. So that the action may be similar, we restrict ourselves to ambiguous ideals – those that are mapped to themselves by the Galois group. The setting for our investigation is the family of C8 -extensions. This choice is guided by by a result of E. Noether as well as results in Integral Representation Theory. Noether’s Normal Integral Basis Theorem. A ﬁnite Galois extension of number ﬁelds N/K is said to be at most tamely ramiﬁed (TAME) if the factorization of each prime ideal PK (of OK ) in ON results in exponents (degrees of ramiﬁcation) that are relatively prime to the ideal PK . A normal integral basis (NIB) is said to exist if there is an element α ∈ ON (in the ring of integers of N ) whose conjugates, {σα : σ ∈ Gal(N/K)}, provide a basis for ON over OK (the integers in K). Noether proved NIB ⇒ TAME; moreover, for local number ﬁelds NIB ⇔ TAME, tying the Galois module structure of the ring of integers to the arithmetic of the extension [?]. This is a nice eﬀect – NIB means that the integers are isomorphic to the group ring, OK [Gal(N/K)]. It is similar to the eﬀect of the Galois group on the ﬁeld itself (i.e. Normal Basis Theorem). The impact of her result is two-fold: (1) We are encouraged to localize. (2) We are directed away from tamely ramiﬁed extensions – toward wildly ramiﬁed extensions and p-groups (See [?]). Integral Representation Theory (Restricted to p-groups G). Classiﬁcation of Modules. The number of indecomposable modules over a group ring Z[G] is, in general, inﬁnite. Only Z[Cp ] and Z[Cp2 ] are of ﬁnite type. Still, among those of inﬁnite type, there are two whose classiﬁcations are somehow manageable. These are the ones of so– called tame type [?]: Z[C2 ×C2 ] (classiﬁcation by L. A. Nazarova [?]) and Z[C8 ] (classiﬁcation by A. V. Yakovlev [?]). Unique Decomposition. The Krull–Schmidt–Azumaya Theorem does not, in general, hold: although a module over a group ring will decompose into indecomposable modules, this

Date: October 6, 2002. 2000 Mathematics Subject Classiﬁcation. Primary 11S23; Secondary 20C10. Key words and phrases. Galois Module Structure, Wild Ramiﬁcation, Integral Representation.

64

G. GRIFFITH ELDER

decomposition may not be unique. Fortunately, it does hold for a few group rings, including Z[C2 × C2 ] and Z[C8 ] [?]. Topic. Let G = Gal(N/K). We are led to ask the following natural question: What is the Z[G]-module structure of ambiguous ideals when • the number theory is ‘bad’ (wild ramiﬁcation), while • the representation theory is ‘good’ (tame type, K–S–A)? In other words: What is the Z[G]-module structure of ambiguous ideals in wildly ramiﬁed C2 × C2 and C8 number ﬁeld extensions? Previous work solved this for C2 × C2 extensions [?], [?]. So our focus here is on C8 -extensions. (Note: This question has already been addressed for those group rings with ‘very good’ representation theory, those of ﬁnite type. See [?] and [?].) As with C2 × C2 -extensions, the Z[G]-module structure of ambiguous ideals in C8 extensions is completely determined by the structure at its 2-adic completion – our global question reduces to a collection of local ones. We leave it to the reader to ﬁll in the details. (One may follow [?, §2] using [?].) 1.1. Local Question, Answer. Let K0 be a ﬁnite extension of the 2-adic numbers Q2 and let Kn be a wildly ramiﬁed, cyclic, degree 2n extension of K0 with G = Gal(Kn /K0 ). The maximal ideal Pn in Kn is unique (therefore ambiguous). So every fractional ideal Pin is ambiguous. We ask: What is the Z2 [G]-module structure of Pin for n = 1, 2, 3? (Z2 denotes the 2-adic integers.) The answered is given by the following theorem and the description of the modules Ms (i, b1 , . . . , bs ). Let T denote the maximal unramiﬁed extension of Q2 in K0 . Following [?, Ch IV], let G = G−1 ⊇ G0 ⊇ G1 ⊇ · · · denote the ramiﬁcation ﬁltration. Use subscripts to denote ﬁeld of reference, so Ok denotes the ring of integers of k. Theorem 1.1. Let Kn /K0 be a cyclic extension of degree 2n and let k ⊆ K0 be an unramiﬁed extension of Q2 . Suppose that |G1 | ≤ 8 (i.e. s = 1, 2 or 3) and let b1 , . . . , bs be the break numbers in the ramiﬁcation ﬁltration of G1 . If Ms (i, b1 , . . . , bs ) is the Z2 [G1 ]-module deﬁned below, then Pin ∼ = Ok [G] ⊗Z2 [G1 ] Ms (i, b1 , . . . , bs )[T :k] as left Ok [G]-modules. 1.1.1. Ms (i, b1 , · · · , bs ). Indecomposable modules are listed in Appendix A and e0 denotes the absolute ramiﬁcation index of K0 . Following [?] and [?], M1 (i, b1 ) = (R0 ⊕ R1 )(i+b1 )/2−i/2 ⊕ Z2 [G1 ]e0 −((i+b1 )/2−i/2) if b2 + 2b1 > 4e0 , HbA ⊕ G cA ⊕ LdA a M2 (i, b1 , b2 ) = I ⊕ bB cB dB H ⊕M ⊕L if b2 + 2b1 < 4e0 .

(1.1) (1.2)

where a = (i+b2 )/4− (i+2b1 )/4, bA = e0 + i/4− (i+b2 )/4, bB = (i+b2 +2b1 )/4−

(i + b2 )/4, cA = (i + b2 + 2b1 )/4 − e0 − i/4, cB = e0 + i/4 − (i + b2 + 2b1 )/4, dA = e0 + (i + 2b1 )/4 − (i + b2 + 2b1 )/4, dB = (i + 2b1 )/4 − i/4. The description of M3 (i, b1 , b2 , b3 ) is given by Tables 1 and 2. Note the eight columns in each table. There are eight cases. Each module that appears in M3 (i, b1 , b2 , b3 ), except for R3 , is listed in the appropriate column of Table 1. The multiplicity of the module is appears in the corresponding spot in Table 2. The multiplicity of R3 follows the tables.

H2 H1,2 M1 L L3 I I2

d + e0 − d¯ a ¯ − d − 2e0 a + e0 − a ¯ c¯ − a c + e0 − c¯ ¯b − c − e0 b − ¯b + e0

Table 2. A ¯ d−b

H2 M M1 L L3 I I2

Table 1. A B H H D H H1 L H1 G H1 H1,2 G4 L L3 L2 I2

a ¯ − d¯ − e0 d + 2e0 − a ¯ a − d − e0 c¯ − a c + e0 − c¯ ¯b − c − e0 b − ¯b + e0

B ¯ d−b

H1 H1,2 G4 L L3 I I2

C H H1 L

F I1 H1 L H1 G H1 G G4 G3 L3 L2 I2

d¯ − w a − d¯ d + e0 − a c¯ − d − e0 z¯ + b1 − c¯ ¯b − c − e0 b − ¯b + e0

C a ¯ − b − e0 ¯ w + e0 − a

H1 G G4 G3 L3 I I2

E I1 H1 L H1 G G4 G3 G2 G1 L1

H I1

D a ¯ − b − e0 y¯ + m − a ¯ d − y¯ + e0 d¯ − d − m a − d¯ w ¯−m−a c¯ − d − e0 z¯ + b1 − c¯ c + e0 − ¯b b−c

H1 G G4 G3 G2 L2 L1

G I1

a−w d¯ − a c¯ − d¯ d + e0 − c¯ y−d ¯b − c − e0 a ¯ − ¯b

E b + e0 − a ¯ w−b

F b + e0 − a ¯ y¯ + m − e0 − b c¯ − y¯ a + e0 − c¯ − m d¯ − a y¯ − d¯ d + e0 − c¯ y + e0 − d c + e0 − ¯b a ¯ − c − e0

a−b d¯ − a c¯ − d¯ ¯b − c¯ d + e0 − ¯b d + e0 − ¯b c + e0 − a ¯

G b−c

a−b d¯ − a c¯ − d¯ ¯b − c¯ a ¯ − ¯b d + e0 − a ¯ c−d

H b−c

GALOIS STRUCTURE 65

66

Cases. A. B. C. D. E. F. G. H.

G. GRIFFITH ELDER

4e0 − 4b1 /3 < b2 (including Stable Ramiﬁcation, b1 ≥ e0 ). 4e0 − 2b1 < b2 < 4e0 − 4b1 /3 4e0 − 4b1 < b2 < 4e0 − 2b1 and b2 > (4e0 + 4b1 )/3 4e0 − 4b1 < b2 < 4e0 − 2b1 and b2 < (4e0 + 4b1 )/3 b2 < 4e0 − 4b1 and b3 > 8e0 + 4b1 − 2b2 b2 < 4e0 − 4b1 and 8e0 + 4b1 − 2b2 < b3 < 8e0 + 4b1 − 2b2 8e0 − 4b1 − 2b2 < b3 < 8e0 − 2b2 b3 < 8e0 − 4b1 − 2b2

A graphic representation of these cases appears in §3.2. Constants used in Table 2. a := (i − 2b2 )/8, a ¯ := (i + b3 − 2b2 )/8, b := (i − 2b2 − 4b1 )/8, ¯b := (i+b3 −2b2 −4b1 )/8, c := (i−4b2 )/8, c¯ := (i+b3 −4b2 )/8, d := (i−4b2 −4b1 )/8, ¯ := (i + b3 − 2b2 − 2b1 )/8, d¯ := (i + b3 − 4b2 − 4b1 )/8, w := (i − 2b2 − 2b1 )/8, w y := (i − 4b2 − 2b1 )/8, y¯ := (i + b3 − 4b2 − 2b1 )/8, z¯ := (i + b3 − 4b2 − 6b1 )/8, m := (b2 − b1 )/2. ¯ − (a + b + c + d) − 3e0 )f . The multiplicity of R3 . In Cases A and B it is ((¯ a + ¯b + c¯ + d) ¯ − (a + b + d) − 2e0 − (¯ In Case C, it is ((¯ a + ¯b + c¯ + d) z + b1 ))f . While in Case D it is ¯ − (a + b) − e0 + m − (w ¯ + z¯ + b1 ))f . In Case E, it is (¯b + d¯ − a − y − e0 )f . ((¯ a + ¯b + c¯ + d) In Case F it is ((¯b + d¯ + c¯) − (a + y + y¯) − e0 )f . Finally, in Cases G and H, the number of R3 that appear is (d¯ − a)f . 1.2. Discussion. Cyclic p-Extensions. The Galois module structure of the ring of integers in fully and wildly ramiﬁed, cyclic, local extensions of degree pn was studied in [?] and more recently in [?]. Both of these papers required a lower bound on the ﬁrst ramiﬁcation number b1 . In particular, [?] restricted b1 to about half of its possible values, under so-called strong ramiﬁcation. In this paper, by focusing on p = 2 we are able to remove this restriction. Our work sheds light (1) on strong ramiﬁcation and (2) on the structures that are possible outside of it. (1) Strong ramiﬁcation for p = 2 means b1 > e0 , a small part of Case A. The structure under strong ramiﬁcation given by [?, Thm 5.3], when restricted to p = 2, remains valid throughout Case A. What then should Case A be, for odd p? (2) Suppose that ‘nice’ refers to the structure under strong ramiﬁcation, indeed under Case A. Does the structure remain relatively ‘nice’ beyond Case A? This depends upon a precise deﬁnition. Let an indecomposable module be nice if it is made up of distinct irreducible modules. Note only nice modules appear in Case A. But then, as we leave Case A, the structure turns nasty immediately. At least one of H1,2 , H1 L and H1 G appears in every Case B through F . Induced Structure. The subﬁeld of Kn ﬁxed by the ﬁrst ramiﬁcation group G1 is tame over the base ﬁeld K0 . Miyata generalized Noether’s Theorem proving that each ideal is relatively projective over G1 [?]. In other words, the ideals are direct summands of modules that have been induced from G1 to G [?, §10]. We ﬁnd, in our situation, that ideals are relatively free over G1 . See [?, Thm 2] for a more general, related result. Extension of Ground Ring. When studying the structure of ideals in an extension Kn /K0 over a group ring, one must choose a ring of coeﬃcients. Does one study ‘ﬁne’ structure – over O0 [G] where the coeﬃcients are integers in K0 . Does one study ‘coarse’ structure – over Z2 [G]. We study a canonical intermediate structure – over OT [G] where the coeﬃcients belong to the Witt ring of the residue class ﬁeld. We determine this structure

GALOIS STRUCTURE

67

by listing generators and relations. Interestingly, the coeﬃcients in these relations always belong Z2 [G]. Therefore OT [G]-structure results, by extension of the ground ring, from Z2 [G]-structure [?, §30B]. Realizable Modules. Let SG denote the set of realizable indecomposable Z2 [G]-modules: Those indecomposable Z2 [G]-modules that appear in the decomposition of some ambiguous ideal in an extension N/K with Gal(N/K) ∼ = G. Chinburg asked whether SG could be inﬁnite. In [?], since SC2 ×C2 is inﬁnite, the answer was found to be yes. We determine here that although the set of indecomposable Z2 [C8 ]-modules is inﬁnite, SC8 is ﬁnite. The sequence |SC2n |, n = 0, 1, 2, . . . begins 1, 3, 7, 23 . . . 1.3. Organization of Paper. Preliminary results are presented in §2, main results in §3. There are two appendixes. Appendix A lists all necessary indecomposable modules. Appendix B lists bases for our ideals. Preliminary Results: In §2.1 we handle the special case when a ramiﬁcation break number is even. In §2.2, we present a strategy for handling odd ramiﬁcation numbers. To motivate our work in §3, we implement this strategy for |G1 | = 2 and 4, in §2.2.1 and §2.2.3 respectively. We conclude, in §2.3, with a reduction to totally ramiﬁed extensions. Main Results: We begin in §3.1 with a brief outline and discussion. Then, we catalog ramiﬁcation numbers and prove some technical lemmas in §3.2. All this sets the stage for our work in §3.3 determining the Galois structure of ideals in fully, though unstably, ramiﬁed C8 -extensions. This is our primary focus. Our work in §3.4 on stably ramiﬁed extensions is essentially contained in [?]. 2. Preliminary Results We continue to use the notation of §1.1. Let K0 be a ﬁnite extension of Q2 and Kn /K0 be a cyclic extension of degree 2n . Let σ generate G = Gal(Kn /K0 ) and use subscripts to i distinguish among subﬁelds. So Ki denotes the ﬁxed ﬁeld of σ 2 , Oi denotes the ring of integers of Ki and Pi denotes the maximal ideal of Oi . Let vi be the additive valuation in Ki , πi its prime element, so that vi (πi ) = 1. Let Tri,j denote the trace from Ki down to Kj . Recall the ramiﬁcation ﬁltration of G. Note G−1 = G0 if and only if Kn /K0 is fully ramiﬁed. Also since G is a 2-group and [G1 : G0 ] is odd, G0 = G1 . Furthermore since G is cyclic and Gi /Gi−1 is elementary abelian for i > 1, there are s = log2 |G1 | breaks in the ﬁltration of G1 [?, p 67]. Let b1 < b2 < · · · < bs denote these break numbers. (The break numbers of G may include −1 as well.) It is a standard exercise to show that b1 , . . . , bs are all either odd or even [?, Ex 3, p 71]. When they are even, we are in an extreme case, called maximal ramiﬁcation. The general case, when they are odd, will be our primary concern. 2.1. Even Ramiﬁcation Numbers. If b1 , . . . , bs are even, we use idempotent elements of the group algebra, Q2 [G], and Ullom’s generalization of Noether’s result [?, Thm 1], to determine the structure of each ideal. In doing so, we rely upon two observations: (1) Idempotent elements in Q2 [G] that map an ideal into itself, decompose the ideal. (2) Modules over a principal ideal domain are free. We illustrate this process in one case, leaving other cases to the reader. Suppose |G| = 8 and |G1 | = 4. So K3 /K0 is only partially ramiﬁed and s = 2. From [?, IV §2 Ex 3], b1 = 2e0 and b2 = 4e0 . Using [?, V §3], one ﬁnds that (1/2)(σ 4 + 1)Pi3 ⊆ Pi3 . As a result, the i/2 idempotent (σ 4 + 1)/2 decomposes the ideal Pi3 ∼ ⊕ M2 with (σ 4 + 1)M2 = 0. = P2

68

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i/2 i/2 i/2 Meanwhile (1/2)(σ 2 + 1)P2 ⊆ P2 . So P2 decomposes as well. This yields Pi3 ∼ = i/4 ⊕ M1 ⊕ M2 with (σ 4 + 1)M2 = 0 and (σ 2 + 1)M1 = 0. Each Mi may be viewed as a P1 i i module over OTK [σ]/(σ 2 +1), a principal ideal domain. So Mi is free over OTK [σ]/(σ 2 +1). i/4 Ullom’s result provides a normal integral basis for P1 . Counting OT -ranks, we ﬁnd that

OT [σ] Pi3 ∼ = 2 (σ − 1)

e0

⊕

OT [σ] (σ 2 + 1)

e0

e

⊕

OT [σ] 0 . (σ 4 + 1)

2.2. Odd Ramiﬁcation Numbers. Henceforth the ramiﬁcation numbers will be odd. In this context we will use the following technical result (with Ki /Ki−1 ). Lemma 2.1. Let k be a ﬁnite extension of Q2 and K/k be a ramiﬁed quadratic extension. Let ek be the absolute ramiﬁcation index of k. Assume that σ generates the Galois group and that the ramiﬁcation number, b < 2ek , is odd. Then (1) vK ((σ ± 1)α) = vK (α) + b for vK (α) odd; (2) if τ ∈ k, there is a ρ ∈ K such that (σ + 1)ρ = τ and vK (ρ) = vK (τ ) − b; (3) if vK (α) is even and (σ + 1)α = 0, there is a θ ∈ K such that α = (σ − 1)θ and vK (θ) = vK (α) − b. Proof. These may be shown using [?, V §3], as in [?, Lem 3.12–14]. Our strategy is based upon the following observations: (1) Under wild ramiﬁcation, Galois action ‘shifts/increases’ valuation (Lemma 2.1(1)). So an element may be used to ‘construct’ other elements with distinct valuation. (2) Elements with distinct valuation may be used to construct bases. If the valuation map vn : Kn → Z is one–to–one on a subset A ⊆ Kn , while vn (A) is onto {i, i+1, . . . , i+vn (2)−1}; then A is a basis for Pin over the integers in the maximal unramiﬁed subﬁeld of Kn . If Kn /K0 is fully ramiﬁed, this subﬁeld is T . The strategy is illustrated below. It is: Use Galois Action to Create Bases. 2.2.1. First Ramiﬁcation Group of Order Two. Suppose that |G1 | = 2. To use Observation (1), we pick α ∈ Kn an element with vn (α) = b1 (e.g. α = πnb1 ). Let αm := α · π0m . Since n−1 vn (π0 ) = 2, vn (αm ) = b1 + 2m. Use Lemma 2.1 with Kn /Kn−1 . So vn ((σ 2 + 1)αm ) = n−1 2b1 + 2m. Since b1 is odd, the valuations of αm and (σ 2 + 1)αm have opposite parity. The valuations for all m lie in one–to–one correspondence with Z. Select those with valuation in {i, . . . , i + vn (2) − 1}. Replace π0e0 by 2 whenever possible. The result is n−1

+ 1)αm : (i − b1 )/2 ≤ m ≤ e0 + i/2 − b1 − 1 B := {αm , (σ 2 n−1 ∪ (σ 2 + 1)αm , 2αm : i/2 − b1 ≤ m ≤ (i − b1 )/2 − 1 .

(2.1)

Since Kn−1 /K0 is unramiﬁed, there is a root of unity ζ with Kn−1 = K0 (ζ). The maximal unramiﬁed extension Q2 in Kn is T (ζ). By Observation (2), B is a basis for Pin over OT (ζ) . n−1 + 1)αm = OT (ζ) · αm + OT (ζ) · σαm yields the group Note that OT (ζ) · αm + OT (ζ) · (σ 2 n−1 n−1 + 1)αm + OT (ζ) · 2αm = OT (ζ) · (σ 2 + 1)αm + OT (ζ) · ring, OT (ζ) [G1 ], while OT (ζ) · (σ 2 n−1 2 − 1)αm yields the maximal order of OT (ζ) [G1 ]. Restricting coeﬃcients and counting (σ leads to the Ok [G1 ]-module structure of Pin , and to M1 (i, b1 ) as in (1.1).

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69

Next, we extend B to a basis upon which the action of the whole group can be followed. Since Kn−1 /K0 is unramiﬁed, there is a normal ﬁeld basis for Pjn−1 /Pj+1 n−1 over O0 /P0 (for each j). Of course, [O0 /P0 : OT /PT ] = 1. So Pjn−1 /Pj+1 has a normal ﬁeld basis n−1 b1 over OT /PT . For j = b1 , this means that there is an element µ ∈ Pn−1 and basis n−1 µ, σµ, . . . , σ 2 −1 µ. Using Lemma 2.1(2), there is an α ∈ Kn with vn (α) = b1 such that n−1 n−1 (σ 2 + 1)α = µ. Then α, σα, . . . , σ 2 −1 α is a normal ﬁeld basis for Pbn1 /Pbn1 +1 over n−1 1 1 +1 OT /PT . Since {σ j (σ 2 + 1)α : j = 0, . . . , 2n−1 − 1} is a basis for Pbn−1 /Pbn−1 , it is 2b1 2b1 +1 also a basis for Pn /Pn over OT /PT . This together with the fact that {σ j α : j = n−1 0, . . . , 2n−1 − 1} is a basis for Pbn1 /Pbn1 +1 over OT /PT leads to ∪2j=0 −1 σ j B being a basis for Pin over OT , and Pin ∼ = OT [G] ⊗Z2 [G1 ] M1 (i, b1 ) as OT [G]-modules. 2.2.2. An Application of Nakayama’s Lemma. In the previous section we were able to follow the Galois action from one basis element to another explicitly. This level of detail becomes overwhelming as we generalize to |G1 | = 4, 8. Fortunately, Nakayama’s Lemma allows us to push some of these details into the background. Lemma 2.2. Let A be a Ok [C2n ]-module (torsion-free over Ok ) where C2n = σ and k denotes an unramiﬁed extension of Q2 . Let H denote the subgroup of order 2, H A the submodule ﬁxed by H, and TrH A the image under the trace. Then TrH A/ H is free over Ok/2Ok . Suppose that B ⊆ A such that TrH B is a (σ − 1)TrH A + 2A basis for TrH A/ (σ − 1)TrH A + 2AH then B can be extended to a Ok [C2n ]/ TrH -basis of A/AH . Proof. Since A/AH is a module over the principal ideal domain Ok [C2n ]/ TrH , it is a free. So C := A/AH ∼ = (Ok [C2n ]/ TrH ) for some exponent a. Now Ok [C2n ]/ TrH is a local ring with maximal ideal σ − 1 dividing 2. Therefore by Nakayama’s Lemma any collection of elements in A that serves as a Ok /2Ok -basis for C/(σ − 1)C will serve as an Ok [C2n ]/ TrH -basis for C. This leaves us to show that B can be extended to a H Ok /2Ok -basis for the − 1)C = A/(A + (σ − 1)A). But since TrH B is vector space C/(σ H a basis for TrH A/ (σ − 1)TrH A + 2A , the elements of B are linearly independent in A/(AH + (σ − 1)A) and therefore span a subspace. 2.2.3. First Ramiﬁcation Group of Order Four. Let |G1 | = 4. This case is important because it illustrates the utility of Lemma 2.2. (Recall that §2.2.1 and §2.2.3 are included in this paper to motivate considerations in §3.) Step 1: Collect |G1 | elements whose valuations are a complete set of residues modulo |G1 |. We begin with the elements used to determine the structure of ideals in Kn−1 (from §2.2.1), n−2 namely αm and (˜ σ + 1)αm ∈ Kn−1 (replacing n by n − 1, expressing σ 2 as σ ˜ ). Note that the ﬁrst ramiﬁcation number of Kn /Kn−2 is the (only) ramiﬁcation number of Kn−1 /Kn−2 (use [?, pg 64 Cor] or switch to upper ramiﬁcation numbers [?, IV §3]). So vn (αm ) = 2vn−1 (αm ) = 2b1 + 4m and vn ((˜ σ + 1)αm ) = 4b1 + 4m. We have two elements of even valuation. To get elements with odd valuation, we apply Lemma 2.1(2). For each X ∈ Kn−1 , Lemma 2.1(2) gives us a preimage X ∈ Kn (under the trace Trn,n−1 ), a preimage that satisﬁes vn (X) = 2vn−1 (X) − b2 . So Trn,n−1 X = (˜ σ 2 + 1)X = X. The integers vn (αm ), vn ((˜ σ + 1)αm ), vn (αm ) = 2b1 − b2 + 4m, vn ((˜ σ + 1)αm ) = 4b1 − b2 + 4m are a complete set of residues modulo 4. Step 2: Collect elements with valuation in {i, i+1, . . . , i+vn (2)−1}. To organize this process we use Wyman’s catalog of ramiﬁcation numbers [?]. If b1 ≥ e0 , the second ramiﬁcation

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number is uniquely determined, b2 = b1 + 2e0 . If b1 < e0 , then either b2 = 3b1 , b2 = 4e0 − b1 , or b2 = b1 + 4t for some t with b1 < 2t < 2e0 − b1 [?, Thm 32]. In any case, we have the bound, 2b1 < b2 .

(2.2)

σ + 1)αm+ke0 , αm+ke0 , Now for a given m, list the inﬁnitely many elements, αm+ke0 , (˜ (˜ σ + 1)αm+ke0 , in terms of increasing valuation. Replace αm+ke0 by 2k αm and drop the subscripts m. So for b2 > 4e0 − 2b1 , beginning at α, we have: 1

2

3

2

· · · −→ α −→ 1/2(˜ σ + 1)α −→ (˜ σ + 1)α −→ α −→ 2α −→ · · · x

Each increase in valuation, denoted by −→, is justiﬁed as follows: For x = 1, the justiﬁcation depends upon the case either b2 > 4e0 − 2b1 or b2 < 4e0 − 2b1 . For x = 2, it is b2 < 4e0 . For x = 3, it is (2.2). If b2 < 4e0 − 2b1 , the list is as follows: 4

3

4

1

σ + 1)α −→ α −→ (˜ σ + 1)α −→ 2α −→ · · · · · · −→ α −→ (˜ Note x = 4 is justiﬁed by b1 > 0. Now collect those elements with valuation in {i, . . . , i + vn (2) − 1}. This will provide us with an OT (ζ) -basis for Pin . Begin with the smallest m such that i ≤ vn (αm ). Note then σ + 1)αm ) < i + vn (2). Associated with this particular m are four elements in that vn (2(˜ {i, . . . , i + vn (2) − 1}. They are listed in the ﬁrst row of the table below. Consider this interval to be a ‘window’. As we increase m, new elements appear (e.g. 2X) – appearance coincides with disappearance (namely of X). Four elements are in ‘view’ always. There are four ‘views’ (four sets). We list the ‘views’ as rows under the appropriate heading. D: The OT (ζ) -basis for Pin . A:

b2 < 4e0 − 2b1 (˜ σ + 1)α

B:

(1)

α

(2)

(˜ σ + 1)α

(3)

α

(˜ σ + 1)α

α

(˜ σ + 1)α

(4)

1/2(˜ σ + 1)α

α

(˜ σ + 1)α

α

2α

2(˜ σ + 1)α

(˜ σ + 1)α

α

b2 > 4e0 − 2b1 2α

(˜ σ + 1)α

2α

(˜ σ + 1)α α

1/2(˜ σ + 1)α α

α

2α

2(˜ σ + 1)α (˜ σ + 1)α

(˜ σ + 1)α

1/2(˜ σ + 1)α

α

(˜ σ + 1)α

2α α

Should we need to determine the subscripts (associated with a particular ‘view’), we can easily do so: For example the four elements listed in A(1) and B(1), appear for m with σ + 1)αm ) ≤ i + 4e0 − 1. In other words, (i − 2b1 )/4 ≤ m ≤ i ≤ vn (αm ) and vn (2(˜

(i + b2 )/4 − b1 − 1. i/2

Step 3: Identify a basis for the quotient module Pin /Pn−1 , and determine the precise image i/2 of each basis element under the trace Trn,n−1 (in terms of the basis for Pn−1 ). Observe i/2 that Pin /Pn−1 is, in a natural way, free over the principal ideal domain OT (ζ) [G]/ ˜ σ 2 + 1. We begin by identifying those elements listed in D, the OT (ζ) -basis from Step 2, that can σ 2 + 1-basis. Take D and partition it into two sets. Let D contain serve as a OT (ζ) [G]/ ˜ those elements X with a bar. Let D0 contain those elements X without a bar. So D is an

GALOIS STRUCTURE i/2

71 i/2

OT (ζ) -basis for Pin /Pn−1 , and D0 is an OT (ζ) -basis for Pn−1 . If we knew which elements i/2 from D provide us with OT (ζ) [G]/ ˜ σ 2 + 1-basis for Pin /Pn−1 we would be done, as it is easy to express the image (under the trace Trn,n−1 ) of each element in D in terms of elements of D0 (there is a one–to–one correspondence). Before we proceed further, note the following. We may assume without loss of generality that for X ∈ D, Trn,n−1 X = 0 if and only if X appears together with X (for the same subscript m) in D. Clearly if X and X appear together, then Trn,n−1 X = X = 0. However when 2X and X appear together, after a change of basis, we may assume that Trn,n−1 2X = 0. The reason for this is as follows: We can change an element of D by adding i/2 an element from D0 and still have a OT (ζ) -basis for Pin /Pn−1 . So whenever 2X and X appear together, replace 2X with 2X − X. Note Trn,n−1 (2X − X) = 0. If we perform this change throughout our basis, but relabel 2X − X as 2X, then we may continue to use the lists, A(1)–A(4) and B(1)–B(4), but assume that Trn,n−1 2X = 0 if 2X appears together with X. i/2 Our next step will be to provide an OT (ζ) [G]/ ˜ σ 2 +1-basis for Pin /Pn−1 . Consider those rows with an X such that Trn,n−1 X = 0 (namely A(2), A(3), A(4), B(2), B(4)). Let S ⊆ D denote the set of left–most X associated with those rows. So, for example, if b2 + 2b1 < 4e0 , then S is made up of the (˜ σ + 1)αm from A(2), and the αm from A(3) and A(4). Verify that i/2 Trn,n−1 S is a OT (ζ) /2OT (ζ) -basis for Trn,n−1 Pin /((˜ σ − 1)Trn,n−1 Pin + 2Pn−1 ) (observe i/2

that Trn,n−1 S generates Trn,n−1 Pin /2Pn−1 over OT (ζ) /2OT (ζ) [G]). Now use Lemma 2.2 i/2

i/2

to extend S to S , an OT (ζ) [G]/ ˜ σ 2 + 1-basis for Pin /Pn−1 . Since Pin /Pn−1 has rank e0 2 over OT (ζ) [G]/ ˜ σ + 1, we have |S | = e0 . σ 2 +1-basis, S , possesses two important properties. First, it contains S. This OT (ζ) [G]/ ˜ Second, without loss of generality we may assume that the elements in S − S are killed by the trace Trn,n−1 . These two properties are shared with another set: The set of all left–most X (an X for every value of m). Clearly the set of all left–most X contains S. Moreover, by an earlier assumption, the compliment of S in the set of all left–most X is mapped to zero under the trace. And so, because the sets have the same cardinality (namely e0 ), we can identify them. Without loss of generality, assume that S is the set of all left–most X. This allows us to use the lists, A(1)–A(4) and B(1)–B(4), in the ‘book-keeping’ necessary for determining the Galois module structure below. i/2 At this point, we know that Pin /Pn−1 is free over OT (ζ) [G]/ ˜ σ 2 + 1. Indeed, S (the i/2 set of all left–most X) provides us a OT (ζ) [G]/ ˜ σ 2 + 1-basis for Pin /Pn−1 . Of course, i/2 the OT (ζ) [G]-structure of Pn−1 is known from §2.2.1 (and can be read oﬀ of D0 ). So a ˜ 2 + 1 in terms of D0 will determine the Galois module description of the image of S under σ structure. See [?, §8]. The Result: For each m associated with A(1) or B(1) we decompose oﬀ an OT (ζ) [G1 ]-summand of OT (ζ) ⊗Z2 I, for A(2) or B(2) we get an OT (ζ) ⊗Z2 H, for A(3) we ﬁnd the group ring, OT (ζ) [G1 ] ∼ = OT (ζ) ⊗Z2 G. But, for B(3) we decompose oﬀ the maximal order of OT (ζ) [G1 ], OT (ζ) ⊗Z2 M. For A(4) and B(4) there is OT (ζ) ⊗Z2 L. All this and counting determines the OT (ζ) [G1 ]-module structure of Pin from which the Ok [G1 ]-module structure can be inferred. It also determines the module M2 (i, b1 , b2 ). To determine the OT [G]-module structure (from which the Ok [G]-module structure can be inferred), we need to take our OT (ζ) -bases for Pin and create OT -bases. 2.3. Partially Ramiﬁed Extensions. Let Ti denote the maximal unramiﬁed extension of Q2 contained in Ki . So T (ζ) of the previous section can be expressed at Tn , while T = T0 . Recall the steps in §2.2.1. We ﬁrst determined a OTn -basis B for Pin , one upon which the

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action of G1 could be explicitly followed. Then noting that we can identify G/G1 with the Galois group for Tn /T0 , we extended B to an OT0 -basis for Pin . This time the action of every element in the Galois group G could be followed. What were the important ingredients in this process? It was important that the elements of B lay in one–to–one correspondence, via valuation, with the integers i, . . . , i + vn (2) − 1. Using this fact and the fact that for had a normal ﬁeld basis over OT0 /PT0 , we were able to make an OT each t, Ptn /Pt+1 n basis for Pin , namely B = ∪σi ∈G/G1 σ i B. At that point we were done. The OT [G]-structure could simply be read oﬀ of this basis. This is not the case when |G1 | = 4. Nor is it the case when |G1 | = 8. At this point we still need to change our basis and use Nakayama’s Lemma, if only to determine OT [G1 ]-structure. We leave it to the reader to check that this process of basis change ‘commutes’ with the process of extending our OTn -basis to an OT0 basis. Simply follow the argument using elements of the form σ t αm , σ t (˜ σ + 1)αm , . . . with σ + 1)αm , . . .. As a consequence, the t = 0, . . . 2[G:G1 ]−1 instead of elements of the form αm , (˜ problem of determining the OT [G]-module structure reduces to the problem of determining the OTn [G1 ]-module structure. 3. Fully Ramified Cyclic Extensions of Degree Eight We consider fully ramiﬁed extensions K3 /K0 with odd ramiﬁcation numbers. 3.1. Outline. Our discussion here is focused on the unstably ramiﬁed case, b1 < e0 . (The stably ramiﬁed case will be addressed separately in §3.4.) Recall Step 1 of §2.2.3 (in reference to K2 /K0 ). But ﬁrst note that the ﬁrst two ramiﬁcation numbers of K3 /K0 are the (only) two ramiﬁcation numbers of K2 /K0 [?, pg 64 Cor]. We began with two elements, namely α, (σ +1)α in the subﬁeld K1 . (The Galois relationship between them was explicit.) Then we created α, (σ + 1)α ∈ K2 , preimages under the trace Tr2,1 . In this section, we will start with these four elements from K2 and use Lemma 2.1(2) to ﬁnd further preimages: of α, (σ + 1)α, α, (σ + 1)α under Tr3,2 . To avoid confusion (confusion resulting from additional bars denoting a preimage under Tr3,2 ), we relabel. Let α := α and let ρ := (σ + 1)α. So the four elements in K2 are labeled α, (σ 2 +1)α, ρ, (σ +1)(σ 2 +1)α (instead of α, α, (σ + 1)α, (σ +1)α respectively). The eight resulting elements (four from K2 along with their preimages) lie in one–to–one correspondence with the residues modulo 8. We would have accomplished all that was accomplished in Step 1 from §2.2.3 if we knew the Galois relationships among α, (σ 2 +1)α, ρ, (σ+1)(σ 2 +1)α explicitly. We need an explicit relationship between α and ρ. This is accomplished in §3.2.2 through a list of technical results – generalizations of Lemma 2.1. Note that ρ is an ‘approximation’ to (σ + 1)α – they have the same image under the trace Tr2,1 . Our results describe their diﬀerence, the ‘error’ in this ‘approximation’. As a prerequisite for the technical results of §3.2.2, and in preparation for the analog of Step 2 from §2.2.3 we use a result of Fontaine to provide a catalog of ramiﬁcation numbers in §3.2.1. We are then ready for Step 2: First we order the eight elements (that we inherit from Step 1) in terms of increasing valuation. This is accomplished in §3.3. There are eight orderings – eight cases. The result is eight diﬀerent bases, listed as A – H (as opposed to just two in D from §2.2.3). For the convenience of the reader, they are listed in Appendix B. We are now ready for the analog of Step 3 from §2.2.3. We are ready to determine those i/2 elements in each OT -basis that serve as an OT [G]/ Tr3,2 -basis, S, for Pi3 /P2 . We will i/2 then be able to describe the image, Tr3,2 S, in terms of our OT -basis for P2 (or more to i/2 the point, explicitly in terms of OT [G]-generators for P2 ). To do all this we will need, as in §2.2.3, to perform certain basis changes. The processes are similar, but there are a

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73

few very important diﬀerences. For the convenience of the reader, the results of this step are summarized in §3.3.1. The steps are then spelled out in §3.3.2 – §3.3.5. The structure of M3 (i, b1 , b2 , b3 ) (given in Tables 1 and 2) can then be read oﬀ of the bases in Appendix B. Note however, that we still need to determine the structure under b1 ≥ e0 (part of Case A). This situation is addressed in § 3.3.4. 3.2. Preliminary Results. We catalog the ramiﬁcation triples and generalize Lemma 2.1, describing the diﬀerence ρ − (σ + 1)α. 3.2.1. Ramiﬁcation Triples. There is stability and instability. Theorem 3.1 ([?, Prop 4.3]). Stability: b1 ≥ e0 ⇒ b2 = b1 + 2e0 ,

and

b1 + b2 ≥ 2e0 ⇒ b3 = b2 + 4e0 .

Instability: b1 < e0 ⇒ 3b1 ≤ b2 ≤ 4e0 − b1 ,

b1 + b2 < 2e0 ⇒ 3b2 + 2b1 ≤ b3 ≤ 8e0 − b2 − 2b1 .

In particular, when b1 < e0 , either b2 = 3b1 , b2 = 4e0 − b1 , or b2 = b1 + 4t for b1 < 2t < 2e0 − b1 , while if b1 + b2 < 2e0 , then either b2 = 3b2 + 2b1 , b2 = 8e0 − b2 − 2b1 , or b3 = 8s − b2 + 2b1 for b2 < 2s < 2e0 − b1 . Plot these ramiﬁcation triples (b1 , b2 , b3 ) in 3 , and project this plot to the ﬁrst two coordinates, (x, y, z) → (x, y, 0), thus creating Figure 1 (next page). This projection is partly a line: for b1 ≥ e0 , each point (b1 , b2 ) is restricted to b2 = b1 + 2e0 . It is partly a triangular region: for b1 < e0 , each point (b1 , b2 ) is bound between the lines b2 = 3b1 and b2 = 4e0 − b1 . The signiﬁcance of the regions A, B, C, . . . will be explained later. Note that for points, (b1 , b2 ), above the line b2 = −b1 + 2e0 , the plot of the (b1 , b2 , b3 ) in 3 will be a plane – b3 is uniquely determined. In Figure 2 we have plotted a slice, at a particular value of b1 , through our plot of ramiﬁcation triples in 3 . Part of this slice is a line – when b3 is uniquely determined. Thus the line from (2e0 − b1 , 6e0 − b1 ) to (4e0 − b1 , 8e0 − b1 ). Indeed, as drawn, Figure 2 implicitly assumes that the slice was taken at b1 for b1 < e0 /2. Otherwise there would be no triangular region. Observe that in Figure 1, the lines b2 = 2e0 − b1 and b2 = 3b1 intersect at b1 = e0 /2. If b1 ≥ e0 /2, the third ramiﬁcation number is uniquely determined by b2 . The triangular region bound by the lines b2 = 3b1 , b3 = 3b2 + 2b1 and b3 = 8e0 − b2 − 2b1 exists only for b1 < e0 /2. Because the ramiﬁcation numbers are odd, the triangular part of Figure 1 can be partitioned as follows: A. B. C. D. E. F.

4e0 − 4b1 /3 < b2 4e0 − 2b1 < b2 < 4e0 − 4b1 /3 4e0 − 4b1 < b2 < 4e0 − 2b1 and b2 > (4e0 + 4b1 )/3 4e0 − 4b1 < b2 < 4e0 − 2b1 and b2 < (4e0 + 4b1 )/3 2e0 − b1 ≤ b2 < 4e0 − 4b1 and b2 > (4e0 + 4b1 )/3 2e0 − b1 ≤ b2 < 4e0 − 4b1 and b2 < (4e0 + 4b1 )/3

74

G. GRIFFITH ELDER

Assuming that b1 < e0 /2, there is a triangular region in Figure 2. This can be partitioned into the following cases: E. F. G. H.

8e0 + 4b1 − 2b2 < b3 8e0 − 2b2 < b3 < 8e0 + 4b1 − 2b2 8e0 − 4b1 − 2b2 < b3 < 8e0 − 2b2 b3 < 8e0 − 4b1 − 2b2

Note that if b1 > 8e0 /17, region G is empty; if b1 > 8e0 /21, region H is empty; if b1 > 8e0 /28, region E is empty. So as drawn, we have assumed that b1 < 2e0 /7. If however the slice were taken for a value 8e0 /17 < b1 < 8e0 /16, note that the triangular region would consist of only one case, namely F . The relationship between E, F and E, F will be explained in §3.3. 3.2.2. Technical Lemmas. The diﬀerence ρ − (σ + 1)α depends upon ramiﬁcation. Unstable Ramiﬁcation. Assume that b1 < e0 . These results may be thought of as consequences of indirect ‘routes’ from α to ρ. For example, we may begin with α ∈ K2 , create (σ 2 + 1)α, then (σ + 1)(σ 2 + 1)α and let ρ be the inverse image of (σ + 1)(σ 2 + 1)α under Tr2,1 . This results in an expression for the ρ − (σ + 1)α. Lemma 3.2. If b2 ≡ b1 mod 4 (equivalently 3b1 < b2 < 4e0 − b1 ), let t = (b2 − b1 )/4. There are elements αm ∈ K2 with v2 (αm ) = b2 + 4m, such that ρm = (σ + 1)αm + (σ 2 ± 1)αm−t has valuation v2 (ρm ) = b2 + 2b1 + 4m. The ‘+’ or ‘−’ depends on our needs. Proof. Let α ∈ K2 with valuation, v2 (α) ≡ b2 mod 4. Using Lemma 2.1, v2 ((σ + 1)α) = v2 (α) + b1 , v2 ((σ 2 + 1)α) = v2 (α) + b2 . Since (σ 2 + 1)α ∈ K1 and v1 ((σ 2 + 1)α) = (v2 (α) + b2 )/2 ≡ b2 mod 2, v1 ((σ + 1)(σ 2 + 1)α) = (v2 (α) + b2 )/2 + b1 . Using Lemma 2.1(2), there is a ρ ∈ K2 with v2 (ρ) = v2 (α) + 2b1 such that (σ 2 + 1)ρ = (σ + 1)(σ 2 + 1)α. Since (σ 2 +1) [ρ − (σ + 1)α] = 0. Using Lem 2.1(3), there is a θ ∈ K2 with v2 (θ) = (v2 (α)−b2 )+b1 and ρ = (σ +1)α+(σ 2 −1)θ. Since b1 < e0 , v2 (2θ) > v2 (ρ). We may replace ρ by ρ := ρ+2θ (they have the same valuation), and get ρ = (σ + 1)α + (σ 2 + 1)θ. Once αm is chosen, we let αm−t := θ. Lemma 3.3. If b2 ≡ −b1 mod 4 (equivalently b2 = 3b1 or b2 = 4e0 −b1 ), let s := (b2 +b1 )/4. There are elements αm ∈ K2 with v2 (αm ) = b2 + 4m, such that ρm = (σ + 1)αm + (σ + 1)(σ 2 + 1)αm−s has valuation, v2 (ρm ) = 2b2 − b1 + 4m. Note if b2 = 3b1 , v2 (ρm ) = b2 + 2b1 + 4m.

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75

Proof. There is a τ ∈ K0 with v0 (τ ) = (b2 − b1 )/2. Using Lemma 2.1(2), let ρ ∈ K2 with v2 (ρ) = b2 − 2b1 such that (σ 2 + 1)ρ = τ . Clearly (σ 2 + 1) · (σ − 1)ρ = 0, so there is a θ ∈ K2 with v2 (θ) = −b1 such that (σ − 1)ρ = (σ 2 − 1)θ. Since (σ − 1) · [ρ − (σ + 1)θ] = 0, τ := ρ − (σ + 1)θ is a unit in K0 . Let ρ = ρ/τ and θ = θ/τ , so 1 = ρ − (σ + 1)θ . Now let β = (σ + 1)(σ 2 + 1)θ . Clearly (σ 2 + 1)θ ∈ K1 and v1 ((σ 2 + 1)θ ) = (b2 − b1 )/2 is odd. Therefore v2 (β) = b2 + b1 . Replacing 1 with the expression, (σ + 1)(σ 2 + 1)(θ /β), yields ρ = (σ + 1)θ + (σ + 1)(σ 2 + 1)(θ /β).

(3.1)

By choosing τ ∈ K0 with other valuations, the result follows. Unfortunately, if b2 = 4e0 − b1 then s = e0 (valuation can not distinguish between αm /2 and αm−s ). To avoid this confusion, we include the following. Lemma 3.4. Let b2 = 4e0 − b1 . There are αm ∈ K2 with v2 (αm ) = b2 + 4m, so 1 1 ρm := (σ + 1)αm − (σ + 1)(σ 2 + 1)αm + (σ + 1)(σ 2 + 1)αm+e0 −b1 2 2 has valuation, v2 (ρm ) = 2b2 − b1 + 4m. Proof. From (3.1) we have ρ = (σ + 1)θ + (σ + 1)(σ 2 + 1)(θ /β). Apply (σ 2 + 1)/β to both sides. So (σ 2 + 1)(ρ /β) = 1 + 2/β. Since v2 ((σ 2 + 1)ρ ) = 8e0 − 4b1 and v2 (β) = 4e0 , then v0 (1 + 2/β) = e0 − b1 . Replace θ /β with (1/2) · [−θ + θ (1 + 2/β)], and distribute (σ + 1)(σ 2 + 1). Remark 3.5. Note (σ − 1)ρm = (σ 2 − 1)αm and (σ 2 + 1)ρm = (σ + 1)(σ 2 + 1)αm+e0 −b1 , using Lemma 3.4. Apparently, ρm is ‘torn’ between αm and αm+e0 −b1 . We chose to emphasize ρm ’s tie to αm . If we relabel ρm−e0 +b1 as ρm (keep the αm the same), Lemma 3.4 reads 1 1 ρm := (σ + 1)αm−e0 +b1 − (σ + 1)(σ 2 + 1)αm−e0 +b1 + (σ + 1)(σ 2 + 1)αm 2 2 has valuation, v2 (ρm ) = b2 + 2b1 + 4m – thus tying ρm to (1/2)(σ + 1)(σ 2 + 1)αm . This valuation of ρm is as in Lemmas 3.2 and 3.3 (for b2 = 3b1 ). Stably Ramiﬁed Extensions. Assume that b1 ≥ e0 . The results may be seen as direct routes from α to ρ. We create ρ immediately from (σ +1)α ∈ K2 . For discussion and generalization, see [?]. Lemma 3.6. Let b1 > e0 . For every odd integer, a, there are elements α, ρ ∈ K2 with v2 (α) = a, v2 (ρ) = a + (b2 − b1 ). such that (σ + 1)α − ρ = µ ∈ K1 , with v2 (µ) = v2 (α) + b1 . Furthermore µ ∈ K0 for v2 (µ) = v2 (α) + b1 ≡ 0 mod 4. Proof. Since v2 ((σ + 1)α) = v2 (α) + b1 is even, we may express (σ + 1)α as a sum µ + ρ with µ ∈ K1 , ρ ∈ K2 , v2 (µ) = v2 (α) + b1 and odd v2 (ρ). Apply (σ − 1). So (σ 2 − 1)α = (σ − 1)µ + (σ − 1)ρ. Since b2 = b1 + 2e0 < 3b1 , v2 ((σ 2 − 1)α) = v2 (α) + b2 < v2 (α) + 3b1 ≤ v2 ((σ − 1)µ). So v2 ((σ 2 − 1)α) = v2 ((σ − 1)ρ) and v2 (ρ) = v2 (α) + (b2 − b1 ). If v2 (µ) ≡ 0 mod 4, we may

76

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choose α so that µ ∈ K0 . Pick a µ∗ ∈ K0 with v2 (µ∗ ) = v2 (µ). Relabel α as α0 . Choose αi ∈ K i = µi + ρi . Clearly 2∞with v2 (αi ) = v2 (α0 ) + 2i. As before, generate ∞ µi and ρi with α ∞ µ∗ = i=0 ai µi for some units ai ∈ K0 . Let α∗ = i=0 ai αi and ρ∗ = i=0 ai ρi . Lemma 3.7. Let b1 = e0 be odd. For every odd integer, a, there are elements α, ρ ∈ K2 with v2 (α) = a, v2 (ρ) = a + (b2 − b1 ) such that (σ − 1)α − ρ = µ1 ∈ K1 if a ≡ e0 mod 4, (σ + 1)α − ρ = µ0 ∈ K0 if a ≡ 3e0 mod 4. with v2 (µi ) = v2 (α) + b1 . Proof. Let τ ∈ K0 be a unit. From Lemma 2.1(2), there is a ρ ∈ K2 with v2 (ρ) = −b2 and (σ 2 + 1)ρ = τ . So (σ 2 + 1) · (σ − 1)ρ = 0. Use Lemma 2.1(3) to ﬁnd θ ∈ K2 with v2 (θ) = b1 − 2b2 and (σ 2 − 1)θ = (σ − 1)ρ. For a ≡ e0 mod 4, we may assume that α = ρπ0m for some m. Let µ1 = (σ 2 + 1)θπ0m ∈ K1 and ρ = −2θπ0m ∈ K2 . The statement follows. For a ≡ 3e0 mod 4, (σ 2 − 1)θ = (σ − 1)ρ can be interpreted to mean that ρ − (σ + 1)θ ∈ K0 . Multiplying by an appropriate power of π0 , we let α = θπ0m , µ0 = −(ρ − (σ + 1)θ)π0m ∈ K0 and ρ = ρπ0m ∈ K2 . 3.3. The Galois module structure under unstable ramiﬁcation. Assume b1 < e0 . First we determine the OT -bases in Appendix B. From Lemmas 3.2, 3.3, 3.4 we have αm , ρm , (σ 2 +1)αm , (σ +1)(σ 2 +1)αm ∈ K2 , with valuations (measured in v2 ) for every residue class modulo 4. Recall v2 (αm ) = b2 +4m, v2 ((σ 2 +1)αm ) = 2b2 +4m, v2 ((σ+1)(σ 2 +1)αm ) = 2b2 + 2b1 +4m and v2 (ρm ) = 8e0 −3b1 +4m if b2 = 4e0 −b1 , otherwise v2 (ρm ) = b2 +2b1 +4m. Using Lemma 2.1(2) we determine elements αm , ρm , (σ 2 + 1)αm , (σ + 1)(σ 2 + 1)αm ∈ K3 , with (σ 4 + 1)X = X and v3 (X) = 2v2 (X) − b3 . These eight elements have valuations (measured in v3 ) in one–to–one correspondence with the residue classes modulo 8. By varying m, it is possible to choose those with valuation i ≤ v3 (x) < 8e0 + i. To organize this process, we list these elements in terms of increasing valuation. There are eight orderings – eight cases. In each case X (or X), an increase in valuation is denoted by an arrow, −→, and justiﬁed by an inequality assigned a number. Numbers above an arrow apply to X. Numbers below the arrow apply to X. As we see below, the ordering of the elements in E is the same as in E (also in F as in F ). This explains the use of similar notation. A.

1

2

1

1

1

ρ −→ 2ρ −→ (σ 2 + 1)α −→ 2(σ 2 + 1)α −→ 2α −→ 6

1

4

4α −→ (σ + 1)(σ 2 + 1)α −→ 2(σ + 1)(σ 2 + 1)α −→ 2ρ In Case A, the valuation of ρm depends upon whether or not b2 = 4e0 − b1 . If b2 = 4e0 − b1 , 0 < b1 justiﬁes 2 while b1 < 2e0 justiﬁes 4. All other increases, including 2 and 4 for b2 = 4e0 − b1 , are justiﬁed by the inequalities listed below. In Cases B through H, there is only one valuation of ρm . 1

2

1

4

5

B. ρ −→ 2ρ −→ (σ 2 + 1)α −→ 2(σ 2 + 1)α −→ 2α −→ 6

5

4

(σ + 1)(σ 2 + 1)α −→ 4α −→ 2(σ + 1)(σ 2 + 1)α −→ 2ρ

GALOIS STRUCTURE

C.

1

2

1

77

7

5

7

5

ρ −→ 2ρ −→ (σ 2 + 1)α −→ 2(σ 2 + 1)α −→ (σ + 1)(σ 2 + 1)α −→ 5

7

7

2α −→ 2(σ + 1)(σ 2 + 1)α −→ 4α −→ 2ρ D.

2

9

9

ρ −→ (σ 2 + 1)α −→ 2ρ −→ 2(σ 2 + 1)α −→ (σ + 1)(σ 2 + 1)α −→ 5

7

7

2α −→ 2(σ + 1)(σ 2 + 1)α −→ 4α −→ 2ρ E = E.

7

8

2

8

7

11

8

13

8

11

ρ −→ 2α −→ 2ρ −→ (σ 2 + 1)α −→ (σ + 1)(σ 2 + 1)α −→ 8

7

8

8

14

8

2(σ 2 + 1)α −→ 2(σ + 1)(σ 2 + 1)α −→ 2α −→ 2ρ 7

10

2

12

13

7

10

F = F . ρ −→ 2α −→ (σ 2 + 1)α −→ 2ρ −→ (σ + 1)(σ 2 + 1)α −→ 11

8

12

11

7

8

14

8

2(σ 2 + 1)α −→ 2(σ + 1)(σ 2 + 1)α −→ 2α −→ 2ρ 8

G.

9

12

15

12

9

8

8

9

8

ρ −→ (σ 2 + 1)α −→ 2α −→ (σ + 1)(σ 2 + 1)α −→ 2ρ −→ 2(σ 2 + 1)α −→ 14

8

2(σ + 1)(σ 2 + 1)α −→ 2α −→ 2ρ 9

15

8

H. ρ −→ (σ 2 + 1)α −→ (σ + 1)(σ 2 + 1)α −→ 2α −→ 2ρ −→ 2(σ 2 + 1)α −→ 14

8

2(σ + 1)(σ 2 + 1)α −→ 2α −→ 2ρ Numbered Inequalities: (1) b1 < 2e0 , b2 < 4e0 , b3 < 8e0 . (2) 3b2 > 4e0 + 4b1 . (2 ) 3b2 < 4e0 +4b1 . (3) 4e0 −4b1 < 3b2 (true for A–F since b2 ≥ 2e0 −b1 ). (4) 2b2 < b3 . (5) 4e0 −2b1 < b2 . (5 ) 4e0 − 2b1 > b2 . (6) 4e0 − 4b1 /3 < b2 . (6 ) 4e0 − 4b1 /3 > b2 . (7) 4e0 − 4b1 < b2 . (7 ) 4e0 − 4b1 > b2 . (8) b1 > 0. (9) b2 > 2b1 . (10) b2 > 4e0 /3 (true for A–F , since b2 ≥ 3e0 /2). (11) Since b2 > 2b1 and b3 ≤ 8e0 − 2b1 − b2 , b3 < 8e0 − 4b1 . (12) 8e0 − 2b2 < b3 . (12 ) 8e0 − 2b2 > b3 . (13) 8e0 + 4b1 − 2b2 < b3 . (13 ) 8e0 + 4b1 − 2b2 > b3 . (14) Since b2 > 2b1 and b3 ≥ 3b2 + 2b1 , b3 > 2b1 + 4b2 . (15) 8e0 − 4b1 − 2b2 < b3 . (15 ) 8e0 − 4b1 − 2b2 > b3 . We leave it to the reader to verify Appendix B. 3.3.1. Summary: Results of Basis Changes and Nakayama’s Lemma. Basis Changes. Except in four rows, C(2), D(2), E(2), F (2),

(3.2)

we ﬁnd we may change the OT -bases in Appendix B so that the Galois action upon each basis is as if ρ and ρ had been everywhere replaced by (σ + 1)α and (σ + 1)α. In the four exceptional cases there are nontrivial Galois relationships among the basis elements. This is explained in §3.3.5. Nakayama’s Lemma. We ﬁnd, without loss of generality, that the set S of ‘left–most’ elements X (as in S of §2.2.3) from each basis in Appendix B will serve as a OT [G]/ Tr3,2 i/2 basis for Pi3 /P2 , except that S contains both (σ + 1)(σ 2 + 1)α and 2α in B(3), C(3), D(3). At this point, the reader can skip the veriﬁcation of these assertions, ignore Cases C through F , replace ρ with (σ + 1)α, and lift the Galois module structure oﬀ of the bases listed in Appendix B. See [?, §8] The result of the readers eﬀort will be the statement of our main result in every case except those associated with (3.2).

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3.3.2. Trivial Diﬀerence. The elements αm , ρm (or ρm , 2αm ) from each basis in Appendix B i/2 i/4 i/4 provide a OT -basis for P2 /P1 . We can change ρm by an element in P1 and still i/4 have a OT -basis. So when ρm −(σ +1)αm ∈ P1 , the diﬀerence between ρm and (σ +1)αm is trivial. i/4 is equivSince v2 ((σ + 1)α) = v2 (ρ − (σ + 1)α), checking ρm − (σ + 1)αm ∈ P1 alent to checking v3 ((σ + 1)α) ≥ i. In Case A, because b2 + b1 ≤ 4e0 we ﬁnd that v3 ((1/2) · (σ + 1)(σ 2 + 1)αm ) ≤ v3 ((σ + 1)αm ). Therefore, in A(3) through A(8), we may replace ρm by (σ + 1)αm . We refrain from doing so in A(8) as it may hamper our ability to determine the eﬀect of Tr3,2 on ρ. We will return to this issue in §3.3.4. In Case B, because b2 > 4e0 − 2b1 we ﬁnd v3 (2α) < v3 ((σ + 1)α). We may replace ρ in B(3) through B(8). For similar reasons, we refrain in B(8). In Cases C and D, b3 > 2b2 +2b1 (since b3 = b2 +4e0 and b2 < 4e0 − 2b1 ). As a consequence, v3 ((σ + 1)(σ 2 + 1)α) < v3 ((σ + 1)α). We may replace ρ in C(3) through C(8), and in D(3) through D(6). In Cases E through H, we clearly have v3 (α) < v3 ((σ + 1)α). We may replace ρ in E(1) or E(3) – E(8), F (1) or F (3) – F (8), G(1) or G(3) – G(8), H(1) or H(3) – H(8). We replace ρ everywhere that we may, except that we refrain for A(8), B(8), C(8), D(7), D(8), E(8), F (7), F (8), G(6), G(7), H(6).

(3.3)

Now we consider the diﬀerence between ρ and (σ + 1)α and replace ρ with (σ + 1)α (2ρ with (σ + 1)2α) in E(1), F (1), G(1), G(8), H(1), H(7), H(8).

(3.4)

Since (σ 4 + 1) · [ρ − (σ + 1)α] = 0, we may use Lem 2.1(2) and ﬁnd an element ω ∈ K3 with v3 (ω) = 2b2 + b1 − 2b3 so that (σ 4 − 1)ω = ρ − (σ + 1)α. As long as b3 < 8e0 − 3b1 , which holds in Cases E through H, we have v3 (ρ) = v3 (ρ + 2ω). On the basis of valuation, we may replace ρ with ρ + 2ω and still have a basis (i.e. Observation (2)). Now since (ρ + 2ω) − (σ + 1)α = (σ 4 + 1)ω ∈ K2 , we may replace (ρ + 2ω) with (σ + 1)α and still have a basis. All we need is v3 ((σ + 1)α) ≥ i. But this clearly holds since v3 (α) ≥ i. i/2

3.3.3. Nakayama’s Lemma and an OT [G]/ Tr3,2 -basis for Pi3 /P2 . The collection of i/2 X in our bases provide an OT -basis for Pi3 /P2 . As in §2.2.3, whenever X and (1/2)· X appear in the same row, we may replace X with X − (1/2) · X and still have a OT -basis. Since Tr3,2 (X − (1/2) · X) = 0, we relabel and assume, without loss of generality, that for these X’s, Tr3,2 X = 0. Let T=0 denote this set (trace zero). Let T=0 denote the set of X’s with X in the same row. For each such X ∈ T=0 , Tr3,2 X ≡ 0 mod 2. This is the set i/2 of trace not zero. Note that Tr3,2 T=0 is an OT /2OT -basis for Tr3,2 Pi3 /2P2 . Following §2.2.3, we select from T=0 a set S (notation as in §2.2.3) such that Tr3,2 S is a OT /2OT -basis i/2 for Tr3,2 Pi3 /((σ − 1)Tr3,2 Pi3 + 2P2 ). It turns out that just as in §2.2.3, S is the set of left-most X for which X appears in the same row, except that S contains both X’s in T=0 from B(3), C(3), D(3). Note that σ acts trivially (modulo 2) upon (σ + 1)(σ 2 + 1)α and 2α in B(3), C(3) and D(3). These elements are linearly independent over OT /2OT [G]. Since both contribute to i/2 the OT /2OT -basis for Tr3,2 Pi3 /2P2 , both (σ + 1)(σ 2 + 1)α and 2α are in S. When a row contributes exactly one X to T=0 , the phrase ‘left–most’ is unnecessary. Indeed σ acts trivially (modulo 2) on the lone X = Tr3,2 X, and since X is needed for the OT /2OT -basis for i/2 Tr3,2 Pi3 /2P2 , X must appear in S. Note this is the only situation to consider in Case A.

GALOIS STRUCTURE

79

In the other cases, we need to show that each X, corresponding to the left–most X of T=0 , generates over OT /2OT [G] all other elements in the same row (in Tr3,2 T=0 ). This is easy to see for rows E(1), E(5), F (1), F (5), G(1), G(5), G(8) and H(1), H(5), H(7), H(8). More work is required for rows D(7), F (7), G(6), G(7), H(6). Note that ρ − (σ + 1)αm = (σ 2 + 1)αm−t or (σ + 1)(σ 2 + 1)αm−s depending upon b2 > 3b1 or b2 = 3b1 , respectively. If i/2 ρ−(σ+1)αm = (σ+1)(σ 2 +1)αm−s , then (σ−1)ρ = (σ 2 +1)α−2α ≡ (σ 2 +1)α mod 2P2 . 2 2 So ρ generates (σ + 1)α. If ρ − (σ + 1)αm = (σ + 1)αm−t the analysis is a little more i/2 involved. Note (σ − 1)ρ − (σ 2 + 1)α ≡ (σ − 1)(σ 2 + 1)αm−t mod 2P2 . For m associated with D(7), F (7), G(6), G(7), H(6), check that m − t lies in D(3), F (4), G(4), H(4) or later. In any case (σ + 1)(σ 2 + 1)αm−t ∈ Pi3 . So ρm and another X, namely (σ + 1)(σ 2 + 1)αm−t , combine together to generate (σ 2 + 1)αm . i/2 Apply Lemma 2.2 and extend S to an OT [G]/ Tr3,2 -basis for Pi3 /P2 . Except in Cases B, C, D (where a row contributes more than one element), we may assume that this basis is the set of left–most elements X, one from each row. 3.3.4. Essentially Trivial Diﬀerence. In §3.3.2 we did not replace ρ by (σ + 1)α in rows A(1), A(2), B(1), B(2), C(1), C(2), D(1), D(2), E(2), F (2), G(2), H(2). It was not clear i/4 that the diﬀerence ρ − (σ + 1)α lay in P1 . Neither did we replace ρ by (σ + 1)α in the rows listed in (3.3). In this section we remedy this situation. We show, except in four cases, C(2), D(2), E(2), F (2), we may change our basis so that the Galois action is as if ρ had been replaced by (σ + 1)α (ρ by (σ + 1)α). We begin with Case A, explaining why the diﬀerence between ρ and (σ+1)α is essentially trivial and then determine the Galois module structure (to illustrate the process). Consider A(1), A(2) and A(8). Recall there are three expressions for ρm corresponding to 3b1 < b2 < 4e0 − b1 , b2 = 3b1 , and b2 = 4e0 − b1 . Suppose 3b1 < b2 < 4e0 − b1 , and ρm = (σ + 1)αm + (σ 2 − 1)αm−t . Consider ρm in A(8). Since b1 + b2 < 4e0 , v3 (ρm ) ≤ v3 (2αm−t ). So for m in A(8), m − t is in A(4) or later. In any case, (σ − 1)αm−t = (σ + 1)αm−t − 2αm−t ∈ Pi3 and (1/2)(σ−1)(σ 2 +1)αm−t = (1/2)(σ+1)(σ 2 +1)αm−t −(σ 2 +1)αm−t ∈ Pi3 (i.e. these elements are available). We replace αm with x = αm + (σ − 1)αm−t − (1/2)(σ − 1)(σ 2 + 1)αm−t . Note (σ + 1)x = ρ and (σ 2 + 1)x = (σ 2 + 1)αm . The Galois action on x and ρm is the same as the Galois action on αm and (σ + 1)αm . It is as if ρm had been replaced by (σ + 1)αm and ρm by (σ + 1)αm . Now consider A(1) and A(2), ρm = (σ + 1)αm + (1/2)(σ 2 − 1)αm−t+e0 . Since v2 (ρm ) < v2 ((1/2)(σ + 1)(σ 2 + 1)αm−t+e0 ), for m in A(1) or A(2), m − t + e0 lies in A(3) or later. In any case, (1/2)(σ − 1)(σ 2 − 1)αm−t+e0 is available. So in A(1) and A(2), we replace 2αm by 2αm − (1/2)(σ − 1)(σ 2 − 1)αm−t+e0 . The eﬀect of this replacement on the Galois action is, again, the same as if we replaced ρm by (σ + 1)αm . Now suppose b2 = 3b1 and ρm = (σ + 1)αm + (σ + 1)(σ 2 + 1)αm−s . Note s = b1 . Starting with the smallest m such that i ≤ v3 (ρm ) we replace αm by αm + (1/2)(σ + 1)αm+e0 −b1 so long as m + e0 − b1 is associated with A(8). If i ≤ v3 (ρm−b1 ), we replace αm by αm + (σ 2 + 1)αm−b1 . In any case, we can systematically replace αm by x = αm + (1/2)(σ 2 + 1)αm+e0 −b1 or αm + (σ 2 + 1)αm−b1 (1/2)(σ 2 + 1)αm by (1/2)(σ 2 + 1)x and (1/2)(σ +1)(σ 2 +1)αm by (1/2)(σ +1)(σ 2 +1)x. The Galois action after this change of basis is as if ρm = (σ + 1)αm and ρm = (σ + 1)αm . Consider A(1) and A(2). Note (σ − 1)ρm = (σ −1)·(σ +1)αm . Moreover, for m associated with these two cases, (σ +1)(σ 2 +1)αm+e0 −b1 and (σ 2 + 1)αm+e0 −b1 are available elsewhere in our basis. So we replace (σ 2 + 1)αm by (σ 2 + 1)(αm + αm+e0 −b1 ) and (σ + 1)(σ 2 + 1)αm by (σ + 1)(σ 2 + 1)(αm + αm+e0 −b1 ). Note for m associated with A(2), m + e0 − b1 is associated with A(3) or later. We achieve the desired eﬀect by replacing (σ + 1)(σ 2 + 1)αm with (σ + 1)(σ 2 + 1)αm + (σ + 1)(σ 2 + 1)αm+e0 −b1 .

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This leaves b2 = 4e0 −b1 . Because this case is more complicated (recall Remark 3.5: ρm is ‘torn’ between αm and αm+e0 −b1 ), we ﬁrst determine the Galois module structure for b2 < 4e0 −b1 . Each m in A(1) results in an OT ⊗Z2 (R3 ⊕H); m in A(2) in an OT ⊗Z2 H2 ; m in A(3) in an OT ⊗Z2 (R3 ⊕ M); m in A(4) in an OT ⊗Z2 M1 ; m in A(5) in an OT ⊗Z2 (R3 ⊕ L); m in A(6) in an OT ⊗Z2 L3 ; m in A(7) in an OT ⊗Z2 (R3 ⊕ I); m in A(8) in an OT ⊗Z2 I2 . Counting the number of m associated with each A(j) yields the ﬁrst column of Table 2. Now consider b2 = 4e0 − b1 . Because v2 (ρm ) = 2b2 − b1 + 4m, the number of m associated with A(1) and A(7) are diﬀerent. The number for A(7) is e0 − b1 too low, while A(1) is e0 − b1 too high. We seem to be missing e0 − b1 of OT ⊗Z2 I and have e0 − b1 too many of OT ⊗Z2 H. Let us look at this more carefully. Note ρm in A(8) maps (via Tr3,2 ) to ρm

(1/2)(σ + 1)(σ 2 + 1)αm+e0 −b1 = (σ + 1)(αm − (1/2)(σ 2 + 1)αm ) + (σ + 1)(σ 2 + 1)αm−b1

So ρm maps into the OT -module spanned by αm − (1/2)(σ 2 + 1)αm and (σ + 1)(αm − (1/2)(σ 2 +1)αm ) along with either (1/2)(σ 2 +1)αm+e0 −b1 and (1/2)(σ +1)(σ 2 +1)αm+e0 −b1 or (σ 2 + 1)αm−b1 and (σ + 1)(σ 2 + 1)αm−b1 . In any case, the elements (1/2)(σ 2 + 1)αm and (1/2)(σ + 1)(σ 2 + 1)αm for (i + b3 − 4b2 + 2b1 )/8 ≤ m ≤ (i + b3 − 4b2 + 2b1 )/8 + e0 − b1 − 1 are not associated with a ρm in A(8). The ρm in A(1) map to (σ 2 − 1)αm (under (σ − 1)) and so (σ +1)(σ 2 +1)αm+e0 −b1 (under (σ 2 +1)) yielding a H, unless m+e0 −b1 is associated with A(2). In fact, there are e0 − b1 ρm that map into A(2) under (σ 2 + 1). For each m in A(2) we have (σ 4 + 1)(σ + 1)(σ 2 + 1)αm = (σ 2 + 1)ρm−e0 +b1 = (σ + 1)(σ 2 + 1)αm , yielding a copy of H2 . But for the last e0 − b1 elements ρm in A(2), namely those m such that m + e0 − b1 is in A(3) we may replace ρm by ρm − (1/2)(σ + 1)(σ 2 + 1)αm+e0 −b1 . For each of these m we have the OT [G]-submodule spanned by ρm − (1/2)(σ + 1)(σ 2 + 1)αm+e0 −b1 and (σ 2 − 1)αm . These e0 − b1 together with the elements left out of a module in A(8) yield a e0 − b1 copies of I, precisely making up the counts. Cases B – H: In the remaining cases, we only have two situations: b2 = 3b1 and 3b1 < b2 < 4e0 − b1 . Consider 3b1 < b2 < 4e0 − b1 ﬁrst, and ρm = (σ + 1)αm + (σ 2 ± 1)αm−t where we may choose between ± as we like. We are concerned with the image of the trace, Tr3,2 , in particular Tr3,2 ρm = (σ + 1)αm + (σ 2 + 1)αm−t , for ρm appearing in B(8), C(8), D(7), D(8), E(8), F (7), F (8), G(6), G(7), and H(6). Note if (σ 2 + 1)αm−t ∈ Pi3 , we may replace ρm with ρm − (σ 2 + 1)αm−t . So if (σ 2 + 1)αm−t appears in B(6), C(6), D(6), E(5), F (5), G(5), H(5) or later we may replace ρm with (σ + 1)αm and ρm with ρm − (σ 2 + 1)αm−t . The later replacement exhibits the same Galois action as a replacement of ρm by (σ + 1)αm . Without loss of generality we will call it a replacement of ρm by (σ + 1)αm . Since b2 ≤ 4e0 − b1 , v3 (2αm ) ≥ v3 (ρm ). What happens when (σ 2 + 1)αm−t appears in B(3) – B(5), C(3) – C(5), D(3) – D(5), E(4), F (6), G(4), H(6)? In this case (σ − 1)αm−t = (σ + 1)αm−t − 2αm−t ∈ Pi3 . In B(8), C(8), D(7), D(8), E(8), F (8), G(6), G(7), H(6), we replace αm with αm + (σ − 1)αm−t , and (σ 2 + 1)αm with (σ 2 + 1)αm + (σ − 1)(σ 2 + 1)αm−t . Note ρm = (σ + 1) · [αm + (σ − 1)αm−t ]. The Galois action upon these basis elements: Tr3,2 ρm = ρm = (σ + 1) · [αm + (σ − 1)αm−t ], (σ 2 + 1) · [αm + (σ − 1)αm−t ] = (σ 2 + 1) αm + (σ − 1)(σ 2 + 1)αm−t , and (σ + 1) · [(σ 2 + 1)αm + (σ − 1)(σ 2 + 1)αm−t ] = (σ 2 + 1) ρm = (σ + 1)(σ 2 + 1)αm , is similar to the Galois action upon: (σ + 1)αm , αm , (σ + 1)αm , (σ 2 + 1)αm , (σ + 1)(σ 2 + 1)αm . We may assume (σ + 1)αm and (σ + 1)αm appear instead of ρm and ρm . Now consider the appearance of ρ in B(1), B(2), C(1), D(1), G(2), H(2). Suppose ρm = (σ+1)αm +(1/2)·(σ 2 −1)αm+e0 −t . One may check v3 (ρm ) ≤ v3 ((1/2)(σ+1)(σ 2 +1)αm+e0 −t

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and v3 (2ρm+e0 −t ) ≤ v3 (4αm ). So (1/2)(σ+1)(σ 2 +1)αm+e0 −t appears in B(4) – B(7), C(6) – C(8) or D(6) – D(8). Note in these sets of elements, ρm+e0 −t has already been replaced by (σ + 1)αm+e0 −t . Importantly, (1/2)(σ − 1)(σ 2 + 1)αm+e0 −t along with (σ − 1)αm+e0 −t are available to us. We replace 2αm with 2αm −(1/2)(σ −1)(σ 2 +1)αm+e0 −t +(σ −1)αm+e0 −t = 2αm − (1/2)(σ − 1)(σ 2 − 1)αm+e0 −t in B(1), B(2), C(1) and D(1). The eﬀect of this change of basis is the same as if we replaced ρm by (σ + 1)αm . Now consider G(2) and H(2). Again ρm = (σ+1)αm +(1/2)(σ 2 −1)αm+e0 −t . In G and H, b3 ≤ 8e0 − 2b2 . As a result, v3 (ρm ) ≤ v3 ((σ − 1)αm+e0 −t ). Note we refer to (σ − 1)αm+e0 −t and not (σ − 1)αm+e0 −t . The valuation of the ﬁrst is b1 more than the valuation of the second. As one may check v3 (ρm ) ≤ v3 ((1/2)(σ +1)(σ 2 +1)αm+e0 −t ), so (1/2)(σ +1)(σ 2 +1) αm+e0 −t appears in G(7), G(8) or H(8). If (σ + 1)(σ 2 + 1)α appeared in G(1) or H(1), (σ 2 + 1)αm−t would be available and so we would replace ρm with ρm − (σ 2 + 1)αm−t . If (1/2)(σ + 1)(σ 2 + 1)αm+e0 −t appears in G(7), then we may assume (σ − 1)αm+e0 −t appears there instead of ρm+e0 −t , because v3 ((σ 2 + 1)αm+e0 −2t ) = v3 ((σ − 1)αm+e0 −2t ) ≥ i, and we would have replaced ρm+e0 −t previously in our discussion with ρm+e0 −t −(σ 2 + 1)αm+e0 −2t . We may now replace 2αm with 2αm − (σ − 1)αm+e0 −t . We replace (σ 2 + 1)αm with (σ 2 + 1) αm + (1/2)(σ − 1)(σ 2 )αm+e0 −t . We may assume without loss of generality that (σ + 1)αm appears in G(2) and H(2) instead of ρm . Now we work with Cases B through H under the assumption b2 = 3b1 . So ρm = (σ + 1) · [αm + (σ 2 + 1)αm−b1 ]. First note if (σ + 1)(σ 2 + 1)αm−b1 appears in B(2), C(3), D(3), E(4), F (4), G(4), H(4), or later we may replace ρm in B(8), C(8), D(7), D(8), E(7), F (7), G(5), G(6), H(5) with ρm − (σ + 1)(σ 2 + 1)αm . Suppose (σ 2 + 1)αm−b1 appears elsewhere. In B, these elements can appear in B(1), B(2), or as (1/2)·(σ 2 +1)αm+e0 −b1 elsewhere in B(8). In cases C through H, since b1 < 4e0 /5, v3 (ρm ) ≤ v3 (ρm−b1 ). So (σ 2 +1)αm−b1 appears in C(1), C(2), D(1), D(2), E(2), E(3), F (2), F (3), G(2), G(3), H(2), H(3). In these cases, we may either replace αm with αm +(1/2)·(σ 2 +1)αm+e0 −b1 or αm +(σ 2 +1)αm+−b1 . If for example, we replace αm with αm + (σ 2 + 1)αm−b1 , (σ 2 + 1)αm with (σ 2 + 1)αm + 2(σ 2 + 1)αm−b1 , and (σ + 1)(σ 2 + 1)αm with (σ + 1)(σ 2 + 1)αm + 2(σ + 1)(σ 2 + 1)αm−b1 , then the Galois action on this new basis is the same as if (σ + 1)αm and (σ + 1)αm appear instead of ρm and ρm . We now concern ourselves with B(1), B(2), C(1), D(1), G(2) and H(2). Check v3 ((σ 2 + 1) αm+e0 −b1 ) ≥ v3 (ρm ). We replace (σ 2 +1)αm with (σ 2 +1)αm +(σ 2 +1)αm+e0 −b1 , and (σ + 1)(σ 2 + 1)αm with (σ + 1)(σ 2 + 1)αm + (σ + 1)(σ 2 + 1)αm+e0 −b1 . In B(2), v3 ((σ + 1)(σ 2 + 1)αm+e0 −b1 ) ≥ v3 ((σ + 1)(σ 2 + 1)αm ), we replace (σ + 1)(σ 2 + 1)αm with (σ + 1)(σ 2 + 1)αm + (σ + 1)(σ 2 + 1)αm+e0 −b1 . All this has the same eﬀect upon the Galois action as a replacement of ρm by (σ + 1)αm . 3.3.5. Non-Trivial Diﬀerence. We consider ρ in C(2), D(2), E(2), F (2). First consider the case b2 = 3b1 where ρm = (σ + 1)αm + (1/2)(σ + 1)(σ 2 + 1)αm+e0 −b1 . Note C and E do not intersect the line b2 = 3b1 . We focus on D(2), F (2). In D with b2 = 3b1 , we have b1 < 4e0 /5. So v3 (2αm ) ≤ v3 (αm+e0 −b1 ). Since v3 (2(σ + 1)(σ 2 + 1)αm ) ≤ v3 ((σ + 1)(σ 2 + 1)αm+e0 −b1 ), for m associated with D(2), (σ + 1)(σ 2 + 1)αm+e0 −b1 appears in D(4), or (1/2)(σ + 1)(σ 2 + 1)αm+e0 −b1 appears in D(5) or later. If (1/2)(σ + 1)(σ 2 + 1) αm+e0 −b1 is available, we may replace ρm with (σ + 1)αm . The Galois action when m is in D(2) and m + e0 − b1 is in D(4) is our primary concern. But ﬁrst consider F (or F ) with b2 = 3b1 . Note then b3 ≤ 8e0 + 2b2 − 8b1 . So v3 (ρm ) ≤ v3 ((σ + 1)(σ 2 + 1)αm+e0 −b1 ). Since b3 ≤ 8e0 +2b2 −8b1 , v3 (αm ) ≤ v3 (2(σ 2 + 1)αm+e0 −b1 ). So for m associated with F (2), (σ+1) (σ 2 + 1)αm+e0 −b1 appears in F (4), or in F (5) or later. If m + e0 − b1 is associated with F (5) or later, we have (σ 2 + 1)αm+e0 −b1 available. We replace 2αm with 2αm +(σ 2 + 1)αm+e0 −b1 .

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We replace (σ 2 + 1)αm and (σ + 1)(σ 2 + 1)αm with (σ 2 + 1)αm + (σ 2 + 1)αm+e0 −b1 and (σ + 1)(σ 2 + 1)αm + (σ + 1)(σ 2 + 1)αm+e0 −b1 . The eﬀect of these changes upon the Galois action is the same as the replacement of ρm by (σ + 1)αm . This leaves the situation when m belongs to D(2), F (2) while m + e0 − b1 belongs to D(4), F (4). In both of these cases, we replace (σ + 1)(σ 2 + 1)αm+e0 −b1 with (σ + 1)(σ 2 + 1)αm+e0 −b1 + (σ + 1)2αm − ρm . This new basis element has trace, Tr3,2 , zero. For each such pair (m, m + e0 − t) we get a copy of H1 G ⊕ R3 . Let us now turn to the case where 3b1 < b2 < 4e0 −b1 and ρm = (σ +1)αm +(1/2)(σ 2 +1) αm+e0 −t . Consider cases C and E. Because v3 (2α) ≤ v3 ((σ 2 + 1)α), if m appears in C(2), then m + e0 − t appears in C(6) or later. Since v3 ((σ 2 + 1)αm+e0 −t > v3 (2(σ + 1)(σ 2 + 1)α), not every m + e0 − t is in C(6) when m is in C(2). Since v3 (ρ) ≤ v3 ((1/2)(σ 2 + 1)α), if m appears in E(2), then m+e0 −t appears in E(6) or later. Since v3 ((σ 2 +1)αm+e0 −t > v3 (2α), some m + e0 − t spill over into C(7). Consequently, whenever a pair (m, m + e0 − t) has m in C(2), E(2) while m + e0 − t is in C(6), E(6) we get a copy of H1 L ⊕ R3 . Consider cases D and F (including F ). Consider D ﬁrst. Since v3 (2αm ) < v3 ((σ 2 + 1)αm+e0 −t ), for m in D(2), m + e0 − t lands in D(6) or later. Note since v3 (2αm ) > v3 (ρm+e0 −t ), some m + e0 − t land in D(6). Since v3 ((σ 2 + 1)αm+e0 −t ) > v3 (2(σ 2 + 1)αm ), the collection of m + e0 − t overlap into D(8). When m + e0 − t is in D(8), the element (1/2)(σ 2 + 1)αm+e0 −t is available and we replace ρm by ρm − (1/2)(σ 2 + 1)αm+e0 −t = (σ + 1)αm . For each pair (m, m + e0 − t) such that m is associated with D(2) and m + e0 − t is associated with D(6), we get a copy of H1 L ⊕ R3 . What we are principally concerned with is what happens when for m in D(2), m + e0 − t is in D(7). In this case, because ρm+e0 −t = (σ + 1)αm+e0 −t + (σ 2 + 1)αm+e0 −2t , there is some new interaction to consider. Suppose m is in D(2), while m + e0 − t is in D(7). Since v3 (ρm+e0 −t ) ≤ v3 (αm+e0 −2t ) and v3 (2(σ + 1)(σ 2 + 1)αm ) ≤ v3 ((σ + 1)(σ 2 + 1)αm+e0 −2t ), for m in D(2) and m + e0 − t in D(7), we ﬁnd m + e0 − 2t is associated with D(4), or D(5) or later. Consider m in D(2), m + e0 − t in D(7), and m + e0 − 2t in D(4). Perform change of basis: Replace 2αm with 2αm + 2αm+e0 −t − 2αm+e0 −t , ρm with ρm − αm+e0 −t , (σ 2 + 1)αm with (σ 2 + 1) αm +(σ 2 +1)αm+e0 −2t +1/2(σ−1)(σ 2 +1)αm+e0 −t , and (σ+1)(σ 2 +1)αm with (σ+1)(σ 2 +1) αm + (σ + 1)(σ 2 + 1)αm+e0 −2t . The eﬀect of these base changes upon the Galois action is the same as if we were to replace ρm with (σ + 1)αm − (1/2)(σ + 1)(σ 2 + 1)αm+e0 −2t . Notice the similarity between this expression and the expression for ρm used when b2 = 3b1 . Consequently, this scenario results in copies of H1 G ⊕ R3 . (Note if b2 = 3b1 , then 2t = b1 .) In the alternative situation, when m is in D(2), m+e0 −t in D(7), and m+e0 −2t is in D(5) or later, we perform the same basis changes. Except, since the element (1/2)(σ + 1)(σ 2 + 1) αm+e0 −2t is available, we replace ρm with ρm − αm+e0 −t + (1/2)(σ + 1)(σ 2 + 1)αm+e0 −2t . The eﬀect of this alternative basis change upon the Galois action is the same as a simple replacement of ρm with (σ + 1)αm . We now turn our attention to Cases F and F . Since 0 < 2b1 , v3 (ρm ) < v3 ((1/2)(σ +1)(σ 2 +1)αm+e0 −t . So for m associated with F (2), m+e0 −t is associated with F (6) or later. We leave it to the reader to check that m + e0 − t lands in F (6) or F (7). If m + e0 − t is associated with F (7), then m + e0 − 2t lands in F (4) or F (5). In any case, all this is analogous to D. 3.4. The Galois module structure under stable ramiﬁcation. For p = 2, stable ramiﬁcation b1 ≥ e0 is nearly strong ramiﬁcation b1 > (1/2) · pe0 /(p − 1), (the conditions diﬀer only when e0 is odd – K0 tame over Q2 ). In [?], the structure of the ring of integers was determined under strong ramiﬁcation for any prime p. We revisit that argument extending it to ambiguous ideals and the case b1 = e0 .

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e0 i/2 i/4 ∼ i/2 i/4 Following §2.1, P2 /P1 = OT [σ]/ σ 2 + 1 . So e0 elements generate P2 /P1 over OT [G]. Use Lemmas 3.6, 3.7 to select elements, α, with odd valuation a such that

i/2 ≤ a ≤ i/2 + 2e0 − 1. Each of these e0 elements gives rise (via the action of (σ ± 1)) to another element, ρ in K2 , with odd valuation, a + (b2 − b1 ) = a + 2e0 . These α along i/4 with their Galois translates, ρ ≡ (σ ± 1)α mod P1 , have valuations in one–to–one correspondence (via v2 ) with the odd integers in i/2, . . . , 4e0 + i/2 − 1, and as a result i/4 i/2 serve as a OT -basis for P2 /P1 . The α provide a OT [G]/ Tr2,1 -basis. i/2 i/4 i/4 to be compatible with our OT -basis for P1 We need this basis for P2 /P1 i/2 (as determined as in §2.2.1), as well as our OT [G]/ Tr3,2 -basis for Pi3 /P2 . First we i/4 i/4 consists of pairs: either ((σ + consider compatibility with P1 . The OT -basis for P1 1)η, η) or ((σ+1)η, 2η) ∈ K0 ×K1 where v1 (η) is odd. Because of Lemma 2.1 each coordinate uniquely determines the other. Now consider pairs where the valuation v3 of both elements is bound between i and 8e0 + i − 1. For example, pairs of the form ((σ + 1)η, η) appear for i/4 ≤ v1 (η) ≤ 2e0 + i/4 − b1 − 1, while pairs of the form ((σ + 1)η, 2η) appear for

i/4 − b1 ≤ v1 (η) ≤ i/4 − 1. The coordinates of all pairs provides us with an OT basis i/4 for P1 . Each α with v2 ((σ 2 + 1)α) ≤ 4e0 + i/2 − 1 determines (via (σ 2 + 1)α ∈ K1 ) i/4 a pair of elements in the OT -basis for P1 . If v1 ((σ 2 + 1)α) is odd, then α determines a pair of the form ((σ + 1)η, 2η). If even, it determines a pair of the form ((σ + 1)η, η). In general for α with v2 ((σ 2 + 1)α) ≥ 4e0 + i/2, v2 (1/2(σ 2 + 1)α) ≥ i/2. So 1/2(σ 2 + 1)α is available and we may replace α in by α − 1/2(σ 2 + 1)α and still have a basis. Note 2 2 (σ +1) α − 1/2(σ + 1)α = 0. So we can assume, without loss of generality, (σ 2 +1)α = 0. This posses no complication, unless (σ ± 1)α = µ + ρ with ρ in the image of Tr3,2 Pi3 . In other words, v2 (ρ) ≥ (b3 + i + 1)/2. (Note for α with v2 ((σ 2 + 1)α) ≤ 4e0 + i/2 − 1 and (σ ± 1)α = µ + ρ, we have v2 (ρ) < (b3 + i + 1)/2.) For these α (actually α − 1/2(σ 2 + 1)α), µ (actually µ − (σ ± 1)1/2(σ 2 + 1)α) will determine a pair ((σ + 1)η, 2η) or ((σ + 1)η, η) i/4 in our OT -basis for P1 . We need simply to show µ and µ − (σ ± 1)1/2(σ 2 + 1)α have the same properties. We leave it to the reader to do this (use Lemma 3.6 and 3.7 to show that the valuations are the same, that µ − (σ ± 1)1/2(σ 2 + 1)α ∈ K0 if and only if µ ∈ K0 ). The only issue that remains is whether there can be any conﬂict between a pair of basis i/4 determined directly, via (σ 2 + 1)α, and a pair determined indirectly via elements for P1 µ = (σ ± 1)α − ρ. Note any element in the image of the trace, Tr2,1 , has valuation that is larger than the valuation of every µ ∈ K1 that arises from the expression for a Galois translate ρ = (σ ± 1)α − µ. i/2 We select our OT [G]-basis for Pi3 /P2 now. There is one element X in our OT -basis i/2 for P2 for each valuation v2 in (i + b3 + 1)/2, . . . , 4e0 + i/2 − 1.

(3.5)

The reader may check for v2 (X) even, X = (σ 2 + 1)α for some α in our OT [G]-basis for i/4 i/2 P2 /P1 . For v2 (X) odd, since i/2+(b2 −b1 ) < (i+b3 +1)/2, X = ρ = (σ ±1)α−µ also for some α. Use Lem 2.1 to create elements X ∈ Pi3 such that Tr3,2 X = X and v3 (X) = v3 (X) − b3 . Note the elements (σ 2 + 1)α and µ (from each case) have expressions i/4 in terms of our OT -basis for P1 . These expressions depend solely upon the valuations 2 of (σ + 1)α and µ. i/2 Before we move on to our result, we should say something about our basis for Pi3 /P2 . i/2 Since OT [σ]/ σ 4 + 1 is a principal ideal domain, Pi3 /P2 is free over OT [σ]/ σ 4 + 1 of

84

G. GRIFFITH ELDER

rank e0 . Given elements of K2 with valuation v2 listed in (3.5) we may use Lem 2.1(2) to ﬁnd elements, ρ ∈ Pi3 , whose images under the trace, Tr3,2 , lie one–to–one correspondence (via valuation) with (3.5). Refer to this set of elements in Pi3 as S. One can check b1 + (i + i/2 b3 + 1)/2 > 4e0 + i/2. Therefore (σ − 1)Tr3,2 Pi3 ⊆ 2P2 . Since Tr3,2 S is an OT -basis i/2 i/2 for Tr3,2 Pi3 ⊆ 2P2 and σ acts trivially upon Tr3,2 Pi3 ⊆ 2P2 we may use Lemma 2.2 i/2 and extend S to an OT [G]/ σ 4 + 1-basis for Pi3 /P2 . At this point we may put the preceding discussion together with our work in §2.2.3 (that i/2 determines the structure of P2 ) and determine the Galois module structure of Pi3 . We i/2 need to express the image of S under the trace, Tr3,2 , in terms of our OT [G]-basis for P2 . This is the same as a determination of the expression (in terms of Galois generators of i/2q P2 ) for each valuation in (3.5). First note under stable ramiﬁcation, b2 > 4e0 −2b1 so the i/2 structure of P2 is determined by the basis listed as Case B in §2.2.3. However it is more convenient for us to use the basis listed as Case A in Appendix B. To translate between the two bases, note in the elements α, (σ + 1)α, α, (σ + 1)α from §2.2.3 are referred to as α, ρ, (σ 2 +1)α, (σ+1)(σ 2 +1)α in §3.1 and then in Appendix B. So row B(1) in §2.2.3 corresponds with a pair of rows A(7) and A(8) in Appendix B. Moreover B(2) corresponds to rows A(1) and A(2), B(3) corresponds to A(3) and A(4), and B(4) corresponds to A(5) and A(6). There are four types of expression with valuation listed in (3.5). If the valuation a satisﬁes a − (b2 − 2b1 ) ≡ 0 mod 4 then a is the valuation of a Galois translate ρ where the diﬀerence i/4 between (σ ± 1)α and ρ is an element (σ + 1)µ ∈ K0 where µ is in the basis for P1 . Note each such a corresponds with the appearance of I2 in the OT [G] decomposition of Pi3 . Counting such a one ﬁnds the same count as in A(8). Note therefore A(7) counts the number of I that are not mapped to under the trace, Tr3,2 , from Pi3 . Each valuation a satisfying a ≡ 0 mod 4 is the valuation of (σ 2 + 1)α = (σ + 1)µ for some i/2 i/4 i/8 α in the basis for P2 and µ in the basis for P1 P0 . So each such a, corresponds with the appearance of an H2 . A count of such a equals the count in A(2). Note A(1) counts the number of H not interacted with. Each valuation a satisfying a − (b2 − 2b1 ) ≡ 2 mod 4 is the valuation of a Galois translate ρ where the diﬀerence between (σ ± 1)α and ρ is an element i/4 i/8 2µ ∈ P1 where (σ + 1)µ is in the basis for P0 . Each such a, therefore corresponds with the appearance of an M1 . The count of such a is the same as the count for A(4). The number of M that appear in Pi3 is the same as the count for A(3). Finally each valuation a satisfying a ≡ 2 mod 4 is the valuation of (σ 2 + 1)α = 2µ for some α in the basis for i/2 i/8 P2 . Also (σ + 1)µ is in the basis for P0 , so each such a, therefore corresponds with the appearance of an L3 . The count of such a is the same as the count for A(6). Again, A(5) counts the number of L in Pi3 . Note the structure of Pi3 under stable ramiﬁcation is consistent with the structure of Pi3 under unstable ramiﬁcation so long as b2 > 4e0 − 4b1 /3. Appendix A. The Modules In this section we introduce twenty–three indecomposable Z2 [C8 ]-modules. It is left to the interested reader to translate our notation into Yakovlev’s [?]. Irreducibles: Four of the Z2 [C8 ]-modules are irreducible: R0 , R1 , R2 , and R3 where Rn := Z2 [ζ2n ], ζ2n denotes a primitive 2n root of unity, and σ the generator of C8 acts via multiplication by ζ2n . The other nineteen modules are ‘compounds’. They are organized according to ﬁxed part – those ﬁxed by σ 2 are listed ﬁrst, followed by those ﬁxed by σ 4 , etc.

GALOIS STRUCTURE

85

Z2 [C2 ]-modules: Besides the two irreducibles R0 , R1 , the group ring Z2 [σ]/ σ 2 is the only other indecomposable module that is ﬁxed by σ 2 . Notation for ‘compounds’: The group ring, Z2 [σ]/ σ 2 , is made up of two irreducibles. To make the relationships between irreducibles and their ‘compounds’ explicit, we will use diagrams like R1 → 1 ∈ R0 (instead of Z2 [σ]/ σ 2 ). These diagrams are to be interpreted as follows: The number of Z2 [σ]-generators is the number of irreducible modules that appear in the diagram. For example, R1 → 1 ∈ R0 means two generators. Let us call them c and d. (Think: c generates R1 while d generates R0 .) Relations determine the module. If there is no ‘arrow’ leaving an irreducible Ri , then the trace Φ2i (σ) maps the generator to zero. So Φ20 (σ)d = 0. Note Φ2i (x) denotes the cyclotomic polynomial and x8 − 1 = Φ20 (x) · Φ21 (x) · Φ22 (x) · Φ23 (x). If there is an ‘arrow’ leaving an irreducible Ri (pointing to an element), then the trace Φ2i (σ) maps the generator to that element. In this case Φ21 (σ)c = 1 · d. Z2 [C4 ]-modules: There are three indecomposable modules ﬁxed by σ 4 (yet not ﬁxed by σ 2 ). Notation for two other decomposable modules is included as it will be needed to describe certain modules later (those not ﬁxed by σ 4 ). For three (of these ﬁve), the submodule ﬁxed by σ 2 is the group ring Z2 [σ]/ σ 2 (note how their diagams include R1 → 1 ∈ R0 ): (G) : R2 → 1 ∈ R1 → 1 ∈ R0 ,

R2 (H) : , R1 → 1 ∈ R0

(I) : R2 ⊕ (R1 → 1 ∈ R0 ).

Denote the three generators by b, c, d. (Think: generating R2 , R1 , R0 , respectively.) Recall (σ − 1)d = 0 while (σ + 1)c = d. In G, we have Φ22 (σ)b = 1 · c. So G is the group ring Z2 [σ]/ σ 4 . In H, we have Φ22 (σ)b = 1 · d. While in I, Φ22 (σ)b = 0. For two (of these ﬁve), the submodule ﬁxed by σ 2 is the maximal order of Z2 [σ]/ σ 2 (note how R1 ⊕ R0 appears): (L) : R2 →

1 ∈ R1 ⊕ , 1 ∈ R0

(M) : R2 ⊕ R1 ⊕ R0 .

Denote the three generators by b, c, d where (σ − 1)d = 0 and (σ + 1)c = 0. In L, we have Φ22 (σ)b = 1 · c + 1 · d. In M, we have Φ22 (σ)b = 0. So M is the maximal order of Z2 [σ]/ σ 4 . Z2 [C8 ]-modules: The remaining ﬁfteen indecomposable modules can now be listed. They are collected according to submodule ﬁxed by σ 4 . Fixed part G. R3

R2 → 1 ∈ R1 → 1 ∈ R0 R3 (G2 ) : R3 → λ ∈ R2 → 1 ∈ R1 → 1 ∈ R0 (G4 ) : R2 → 1 ∈ R1 → 1 ∈ R0 (G1 ) : R3 → 1 ∈ R2 → 1 ∈ R1 → 1 ∈ R0 (G3 ) :

Call the generators a, b, c, d, where the Z2 [σ]-relations among b, c, d are as in G. In G1 , we have Φ23 (σ)a = 1 · b. So G1 is the group ring Z2 [σ]. In G2 , we have Φ23 (σ)a = λ · b where λ = σ − 1. In G3 , Φ23 (σ)a = 1 · c. In G4 , Φ23 (σ)a = 1 · d.

86

G. GRIFFITH ELDER

Fixed part H. (H1 ) : R3 →

R3 λ ∈ R2 ⊕ (H2 ) : R2 → 1 ∈ R0 1 ∈ R1 → 1 ∈ R0 R1

Call the generators a, b, c, d, where the Z2 [σ]-relationships among b, c, d are as in H. In H1 , Φ23 (σ)a = λ · 1 · b + 1 · c. In H2 , Φ23 (σ)a = d. Fixed part I. (I1 ) : R3 →

1 ∈ R2 1 ∈ R2 ⊕ (I2 ) : R3 → ⊕ 1 ∈ R1 → 1 ∈ R0 R1 → 1 ∈ R0

Each module is generated by a, b, c, d, where the Z2 [σ]-relationships among b, c, d are as in I. In I1 , Φ23 (σ)a = 1 · b + 1 · c. In I2 , Φ23 (σ)a = 1 · b + 1 · d. Fixed part L or M. R3 1 ∈ R1 1 ∈ R1 (L3 ) : (L1 ) : R3 → 1 ∈ R2 → ⊕ 1 ∈ R0 R2 → ⊕ 1 ∈ R0 λ ∈ R2 ⊕ 1 ∈ R1 (L2 ) : R3 → λ ∈ R2 → ⊕ (M1 ) : R3 → 1 ∈ R1 1 ∈ R0 ⊕ 1 ∈ R0

The generators are a, b, c, d, where the Z2 [σ]-relationships among b, c, d are as in L or M respectively. In L1 , Φ23 (σ)a = b. In L2 , Φ23 (σ)a = λ · b. In L3 , Φ23 (σ)a = 1 · c + 1 · d. In M1 , Φ23 (σ)a = 1 · b + 1 · c + 1 · d. Hybrids of H1 . The next three modules result from the linking of an H1 with either another R3 , or with a G, or with a L.

(H1,2 ) :

R3 →

R3

1 ∈ R1 → 1 ∈ R0 ⊕ λ ∈ R2

This module is generated by a1 , a2 , b, c, d with the Z2 [σ]-relationships among b, c, d as in H, while Φ23 (σ)a1 = λ · b + 1 · c and Φ23 (σ)a2 = d. If Φ23 (σ)a1 = 0, H2 would decompose oﬀ. If Φ23 (σ)a2 = 0, H1 would decompose oﬀ. It is a mixture of H1 and H2 , hence the name. (H1 G) :

R3 →

1 ∈ R1 → 1 ∈ R0 ⊕ λ ∈ R2 → ⊕

R2 →

1 ∈ R1 → 1 ∈ R0

GALOIS STRUCTURE

87

This module is generated by a1 , b1 , c1 , d1 and b2 , c2 , d2 . The Z2 [σ]-relationships among b2 , c2 , d2 are as in G. The Z2 [σ]-relationships among a1 , c1 , d1 are as in H1 with (σ 2 + 1)b1 = 1 · d1 + 1 · d 2 . R3 → (H1 L) :

1 ∈ R1 → 1 ∈ R0 ⊕ λ ∈ R2 → ⊕ 1 ∈ R0 R2 → ⊕ 1 ∈ R1

This module is generated by a1 , b1 , c1 , d1 and b2 , c2 , d2 . The Z2 [σ]-relationships among b2 , c2 , d2 are as in L. The Z2 [σ]-relationships among a1 , c1 , d1 are as in H1 with (σ 2 + 1)b1 = 1 · d1 + (1 · c2 + 1 · d2 ).

Appendix B. The Bases by Case, A through H From §3.4, we inherit sequences of elements ordered in terms of increasing valuation (for Case A, we have . . . ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α, 4α, (σ + 1)(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2ρ, . . .). Following §2.2.3, we are interested in those elements ‘in view’ (i.e. with valuation in i, i + 1, . . . , i + v3 (2) − 1). As we vary m the ‘view’ changes. Indeed, for each case, there are eight views (eight sets). They are listed below. Recall from §2.2.3 it is easy to determine the subscripts m associated with a particular ‘view’. For example, the elements in A(2) appear for i ≤ v3 (σ + 1)(σ 2 + 1)α and v3 (σ + 1)(σ 2 + 1)α ≤ 8e0 + i − 1. In other words, (i + b3 − 4b1 − 4b2 )/8 ≤ m ≤ (i + 8e0 − 4b1 − 4b2 )/8 − 1.

Case A (1)

2

ρ, 2ρ, (σ + 1)α,

2(σ 2

+ 1)α, 2α, 4α, (σ + 1)(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α

(σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α, 4α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α, 4α (3) 2 1 (4) 2α, (σ + 1)(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α 2 1 (5) α, 2α, (σ + 1)(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α 2 1 (6) (σ 2 + 1)α, α, 2α, (σ + 1)(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α 2 1 1 2 (σ + 1)α, (σ 2 + 1)α, α, 2α, (σ + 1)(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ (7) 2 2 1 1 (8) ρ, (σ 2 + 1)α, (σ 2 + 1)α, α, 2α, (σ + 1)(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, ρ 2 2 (2)

88

G. GRIFFITH ELDER Case B 2

+ 1)α, 2α, (σ + 1)(σ 2 + 1)α, 4α, 2(σ + 1)(σ 2 + 1)α

(1)

ρ, 2ρ, (σ + 1)α,

(2)

(σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, 4α

(3)

2α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α 2 1 α, (σ + 1)(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α 2 1 (σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α 2 1 1 2 (σ + 1)α, (σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ 2 2 1 2 1 2 ρ, (σ + 1)α, (σ + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, ρ 2 2

(4) (5) (6) (7) (8)

2(σ 2

Case C (1)

ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2(σ + 1)(σ 2 + 1)α, 4α

(2)

2α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2(σ + 1)(σ 2 + 1)α

(3)

(σ + 1)(σ 2 + 1)α, 2α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α

(4)

α, (σ + 1)(σ 2 + 1)α, 2α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α 2 1 (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ, 2ρ, (σ 2 + 1)α 2 1 2 1 (σ + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ, 2ρ 2 2 1 2 1 ρ, (σ + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ 2 2

(5) (6) (7) (8)

Case D (1)

ρ, (σ + 1)α, 2ρ, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2(σ + 1)(σ 2 + 1)α, 4α

(2)

2α, ρ, (σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2(σ + 1)(σ 2 + 1)α

(3)

(σ + 1)(σ 2 + 1)α, 2α, ρ, (σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α

(4)

α, (σ + 1)(σ 2 + 1)α, 2α, ρ, (σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ, (σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α 2 1 (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ, (σ 2 + 1)α, 2ρ 2 1 ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ, (σ 2 + 1)α 2 1 2 1 (σ + 1)α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ 2 2

(5) (6) (7) (8)

2

Case E 2

(1)

2α, 2ρ, (σ + 1)α, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α, 2ρ

(2)

ρ, 2α, 2ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α

(3)

α, ρ, 2α, 2ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α

GALOIS STRUCTURE (4)

(σ + 1)(σ 2 + 1)α, α, ρ, 2α, 2ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α

(5)

(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α, 2ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α, 2ρ, (σ 2 + 1)α 2 1 1 2 (σ + 1)α, (σ + 1)(σ 2 + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α, 2ρ 2 2 1 2 1 ρ, (σ + 1)α, (σ + 1)(σ 2 + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α 2 2 Case F

(6) (7) (8)

(1)

2α, (σ 2 + 1)α, 2ρ, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α, 2ρ

(2)

ρ, 2α, (σ 2 + 1)α, 2ρ, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α

(3)

α, ρ, 2α, (σ 2 + 1)α, 2ρ, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α

(4)

(σ + 1)(σ 2 + 1)α, α, ρ, 2α, (σ 2 + 1)α, 2ρ, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α

(5)

(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α, (σ 2 + 1)α, 2ρ, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α, (σ 2 + 1)α, 2ρ 2 1 ρ, (σ + 1)(σ 2 + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α, (σ 2 + 1)α 2 1 2 1 (σ + 1)α, ρ, (σ + 1)(σ 2 + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α 2 2 Case G

(6) (7) (8)

(1)

(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α, 2ρ

(2)

ρ, (σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α

(3)

α, ρ, (σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α

(4)

(σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α

(5)

(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, 2ρ

(6)

ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, 2α 2 1 α, (σ + 1)(σ 2 + 1)α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α 2 Case H

(7) (8)

(1)

(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2ρ, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α, 2ρ

(2)

ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2ρ, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α

(3)

α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2ρ, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α

(4)

(σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2ρ, 2(σ 2 + 1)α

(5)

(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2ρ

(6)

ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α

(7)

α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α 2

(8)

89

Department of Mathematics, University of Nebraska at Omaha, Omaha, Nebraska 68132-0243 E-mail address: [email protected]

AN INTRODUCTION TO NONCOMMUTATIVE DEFORMATIONS OF MODULES EIVIND ERIKSEN

Abstract. Let k be an algebraically closed (commutative) field, let A be an associative k-algebra, and let M = {M1 , . . . , Mp } be a finite family of left A-modules. We study the simultaneous formal deformations of this family, described by the noncommutative deformation functor DefM : ap → Sets introduced in Laudal [8]. In particular, we prove that this deformation functor has a pro-representing hull, and describe how to calculate this hull using the cohomology groups Extn A (Mi , Mj ) and their matric Massey products.

Introduction In this paper, I shall give an elementary introduction to the noncommutative deformation theory for modules, due to Laudal. This theory, which generalizes the classical deformation theory for modules, was introduced by Laudal in [8]. Earlier versions of this material appeared in the preprints Laudal [3], [4], [5], [6], [7]. This noncommutative deformation theory has several applications: In the paper Laudal [8], Laudal used it to construct algebras with a prescribed set of simple modules, and also to study the moduli space of iterated extensions of modules. In the preprint Laudal [7], he also showed that this theory is a useful tool in the study of algebras, and in establishing a noncommutative algebraic geometry. These applications are an important part of the motivation for the noncommutative deformation theory. But we shall not go into the details of these applications in this elementary introduction. Instead, we refer to the papers and preprints of Laudal mentioned above for applications and further developments of the theory. Throughout this paper, we shall ﬁx the following notations: Let k be an algebraically closed (commutative) ﬁeld, let A be an associative k-algebra, and let M = {M1 , . . . , Mp } be a ﬁnite family of left A-modules. Notice that this notation diﬀers from Laudal’s: While Laudal considers families of right modules in all his paper, I consider families of left modules. Of course, the diﬀerence is only in the appearance — the resulting theories are obviously equivalent. We shall present a noncommutative deformation functor DefM : ap → Sets, which describes the simultaneous formal deformations of the family M of left A-modules. Furthermore, we shall prove that this deformation functor has a pro-representable hull (H, ξ) when the family M satisfy a certain ﬁniteness condition. We shall also describe a method for ﬁnding the pro-representable hull explicitly. In section 1, we describe the category ap . It is a full sub-category of the category Ap of p-pointed k-algebras. The objects of Ap are the k-algebras R equipped with k-algebra homomorphisms k p → R → k p , such that the composition k p → k p is the identify. For any

This research has been supported by a Marie Curie Fellowship of the European Community programme “Improving Human Research Potential and the Socio-economic Knowledge Base” under contract number HPMF-CT-2000-01099.

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91

such object, R = (Rij ) is a k-algebra of p × p matrices. The radical of this object is the ideal I(R) = ker(R → k p ) ⊆ R. The category ap is the full sub-category of Ap consisting of objects such that R is Artinian and complete in the I(R)-adic topology. In section 2, we describe the noncommutative deformation functor associated to the family M of left A-modules, DefM : ap → Sets It is constructed in the following way: Let R be an object of ap , and consider the vector space MR = (Mi ⊗k Rij ), equipped with the natural right R-module structure induced by the multiplication in R. A deformation of M to R consists of the following data: • A left A-module structure on MR making MR a left A ⊗k Rop -module, • Isomorphisms ηi : MR ⊗R ki → Mi of left A-modules for 1 ≤ i ≤ p. The set of equivalence classes of such deformations is denoted DefM (R), and this deﬁnes the covariant functor DefM . Notice that the fact that MR ∼ = (Mi ⊗k Rij ) as right R-modules replaces the ﬂatness condition in classical deformation theory. If p = 1 and R is commutative, the above condition is of course equivalent to the ﬂatness condition, so the noncommutative deformation functor generalizes the classical one. In section 3, we look at noncommutative deformations from the point of view of resolutions. Let R be any object of ap . An M-free module over R is a left A ⊗k Rop -module F of the form F = (Li ⊗ Rij ), where L1 , . . . , Lp are free left A-modules. M-free complexes and M-free resolutions are deﬁned similarly. Let us ﬁx a free resolution of Mi the form 0 ← Mi ← L0,i ← · · · ← Lm,i ← · · · for 1 ≤ i ≤ p. We prove that there is a bijective correspondence between deformations of M to R and complexes of M-free modules over R of the form (L0,i ⊗k Rij ) ← · · · ← (Lm,i ⊗k Rij ) ← · · · In fact, each such complex of M-free modules is an M-free resolution of the corresponding deformation MR of M to R. In section 4, we recall some general facts about pointed functors and their representability. In section 5, we consider the special case of the noncommutative deformation functor DefM . From this point in the text, we assume that the family M satisfy the ﬁniteness condition (FC)

dimk ExtnA (Mi , Mj ) is ﬁnite for 1 ≤ i, j ≤ p, n = 1, 2.

When this condition holds, we deﬁne T1 , T2 to be the formal matrix rings (in the sense of section 1) given by the families of k-vector spaces Vij = ExtnA (Mj , Mi )∗ for n = 1, 2.

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Assuming condition (FC), we show the following theorem of Laudal, which generalizes the corresponding theorem for the classical deformation functor: ˆ T2 k p Theorem 0.1. There exists an obstruction morphism o : T2 → T1 , such that H = T1 ⊗ is a pro-representable hull for the noncommutative deformation functor DefM : ap → Sets. In the rest of the paper, we show how to construct the hull H explicitly, which can be accomplished by using matric Massey products. In section 6, we introduce the immediately deﬁned matric Massey products. In section 7, we deﬁne the matric Massey products in general, and show that the hull H of the noncommutative deformation functor DefM is determined by the vector spaces ExtnA (Mi , Mj ) for n = 1, 2 and 1 ≤ i, j ≤ p and their matric Massey products. We also describe a general method for calculating the hull H in concrete terms. In appendix A, we describe the Yoneda and Hochschild representations of the cohomology groups ExtnA (Mi , Mj ). In this paper, we have chosen to express the matric Massey products using the Yoneda representation and M-free resolutions. It is also possible to express the matric Massey products using the Hochschild representation, see for instance Laudal [8]. 1. Categories of pointed algebras Let p be a ﬁxed natural number, and consider the ring k p . This ring has a natural kalgebra structure given by the map α → (α, . . . , α) for α ∈ k. Let pri : k p → k p be the i’th projection, and consider the ideal ki = pri (k p ) ⊆ k p as a k p -module for 1 ≤ i ≤ p. Clearly, k p is an Artinian k-algebra and {k1 , . . . , kp } is the full set of isomorphism classes of simple k p -modules, each of them of dimension 1 over k. This simple example will serve as a model for the p-pointed algebras that we shall consider in this section. A p-pointed k-algebra is a triple (R, f, g), where R is an associative ring and f : k p → R, g : R → k p are ring homomorphisms such that g ◦ f = id. A morphism u : (R, f, g) → (R , f , g ) of p-pointed k-algebras is a ring homomorphism u : R → R such that the natural diagrams commute (that is, such that u ◦ f = f and g ◦ u = g). We shall denote the category of p-pointed k-algebras by Ap . Notice that if (R, f, g) is an object of Ap , then f is injective and g is surjective, and we shall identify k p with its image in R. We often write R for the object (R, f, g) to simplify notation. Let (R, f, g) be an object in Ap . We deﬁne the radical of R to be I(R) = ker(g), which is an ideal in R. Furthermore, we denote by J(R) the Jacobson radical of R J(R) = {x ∈ R : xM = 0 for all simple left R-modules M }, which is also an ideal in R. We shall write I, J for the radicals I(R), J(R) when there is no danger of confusion. Notice that the Jacobson radical J depends only on the ring R, while the radical I depends on the structural morphism g as well. For all objects R in Ap , we have an inclusion J(R) ⊆ I(R): We have J(k p ) = 0 since p k is semi-simple, and g(J(R)) ⊆ J(k p ) = 0 since g : R → k p is a surjection. In general, we know that R and R/J(R) have the same simple left modules. So if we consider ki as a left R-module via the morphism g : R → k p for 1 ≤ i ≤ p, we see that {k1 , . . . , kp } is contained in the set of isomorphism classes of simple left R-modules, and the equality J(R) = I(R) holds if and only if {k1 , . . . , kp } is the full set of isomorphism classes of simple left R-modules. Equivalently, the equality I(R) = J(R) holds if and only if there are exactly p isomorphism classes of simple left R-modules.

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It is therefore clear that the equality I(R) = J(R) does not hold in general: It is easy to ﬁnd examples where R has ‘too many’ simple modules. For instance, consider R = k[x]/(x − x2 ) with the natural k-algebra structure f : k → R and let g : R → k be given by x → 0. Then R is an object of A1 , but J(R) = I(R) because R has two non-isomorphic simple left R-modules (given by x → 0 and x → 1). Let ei be the idempotent (0, 0, . . . , 1, . . . , 0) ∈ k p for 1 ≤ i ≤ p. Notice that ei ej = 0 if i = j, and that e1 + · · · + ep = 1. For any object R in Ap , we identify {e1 , . . . , ep } with idempotents in R via the inclusion k p → R. Denote by Rij the k-linear sub-space ei Rej ⊆ R. We immediately see, using the properties of the idempotents, that the following relations hold for 1 ≤ i, j, l, m ≤ p: (1) Rij Rlm ⊆ δjl Rim , (2) Rij ∩ Rlm = 0 if (i, j) = (l, m), (3) Rij = R. In particular, we have that R = ⊕Rij , so every element r ∈ R may be written in matrix form r = (rij ) with rij ∈ Rij for 1 ≤ i, j ≤ p. Furthermore, elements of R multiply as matrices when we write them in this form. It is therefore reasonable to call an object R in Ap a matrix ring, and to write it R = (Rij ). Notice that Rii is an associative ring (with identity ei ), and that Rij is a (unitary) Rii − Rjj bimodule for 1 ≤ i, j ≤ p. For any ideal K ⊆ R, we see that ei Kej = K ∩ Rij , and we shall denote this k-linear subspace Kij for 1 ≤ i, j ≤ p. Since K = ⊕ Kij , we write K = (Kij ). Let R be an object of Ap , so R = (Rij ) is a matrix ring in the above sense. The following standard result gives useful information on when R is an Artinian or Noetherian ring: Proposition 1.1. Let R = (Rij ) be an object in Ap . Then R is Noetherian (Artinian) if and only if the following conditions hold: i) Rii is Noetherian (Artinian) for 1 ≤ i ≤ p, ii) Rij is a Noetherian (Artinian) left Rii -module and a Noetherian (Artinian) right Rjj module for 1 ≤ i = j ≤ p. We recall that a ﬁnitely generated, associative k-algebra is not necessarily Noetherian. That is, Hilbert’s basis theorem does not hold for associative rings. For a counter-example, let R = k{x1 , . . . , xn } be the free associative k-algebra on n generators. It is well-known that R is Noetherian only if n = 1. However, we know from the Hopkins-Levitzki theorem that an associative Artinian ring is Noetherian. A k-algebra R of ﬁnite dimension as vector space over k is Artinian. This is clear, since every one-sided ideal is a vector space over k of ﬁnite dimension. We have a converse statement under the following conditions: Lemma 1.2. Let R be an object of Ap . If R is Artinian and I(R) is nilpotent, then R has finite dimension as a vector space over k. Proof. We write I = I(R). Since R is Artinian and therefore Noetherian, I m is ﬁnitely generated as a left R-module for all m. Consequently, I m /I m+1 is a ﬁnitely generated R/I-module for all m, and hence has ﬁnite k-dimension. But I n = 0 for some n, so I m has ﬁnite kdimension for all m ≥ 0. In particular, R has ﬁnite dimension as a vector space over k. We deﬁne the category ap to be the full sub-category of Ap consisting of objects R in Ap such that R is Artinian and I(R) = J(R). The condition I(R) = J(R) might equivalently be replaced by the condition that I(R) is a nilpotent ideal, since the Jacobson radical is the

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largest nilpotent ideal in an Artinian ring. So by lemma 1.2, all objects R in ap have ﬁnite k-dimension. Since R is Artinian, the condition that I(R) is nilpotent is also equivalent to ∩ I(R)n = 0. Finally, there is a geometric interpretation of the condition I(R) = J(R): By the comment earlier in this section, I(R) = J(R) if and only if {k1 , . . . , kp } is the full set of isomorphism classes of simple left R-modules (or equivalently, that the number of such isomorphism classes is exactly p). Lemma 1.3. Let R be an associative ring. Then there exists morphisms f : k p → R and g : R → k p making (R, f, g) an object of ap if and only if R is an Artinian k-algebra with exactly p isomorphism classes of simple left R-modules, each of them of dimension 1 over k. Proof. One implication follows from the comments above. For the other, assume that R is Artinian with the prescribed isomorphism classes of simple left R-modules. This deﬁnes a morphism g : R → k p . Clearly, I = ker(g) = J(R) by the comments above. So it is enough to lift the idempotents {e1 , . . . , ep } of k p to idempotents {r1 , . . . , rp } in R such that r1 + · · · + rp = 1 and ri rj = 0 when i = j. But R is Artinian and therefore I = J(R) is nilpotent, so this is clearly possible. Let R be an object in Ap with radical I = I(R). Then the I-adic ﬁltration deﬁnes a topology on R compatible with the ring operations, and we shall always consider R a topological ring in this way. We say that the topology on R is Hausdorﬀ (or separated) if and only if ∩I n = 0. ˆ of R and a canonical morphism For all objects R in Ap , there is an I-adic completion R ˆ is deﬁned by the projective limit ˆ in Ap . The I-adic completion R R→R ˆ = lim R/I n , R ←

ˆ is the natural one induced by this projective limit. Notice that and the morphism R → R the kernel of this morphism is ∩I n . We say that R is complete (or separated complete) if ˆ is an isomorphism in Ap . In particular, this implies that the the natural morphism R → R morphism is injective, so R is Hausdorﬀ (or separated). This gives a new characterization of the category ap : Lemma 1.4. The category ap is the full sub-category of Ap consisting of objects such that R is Artinian and I-adic complete. We deﬁne the pro-category ˆ ap of ap to be the full sub-category of Ap consisting of objects such that R is complete and R/I(R)n belongs to ap for all n ≥ 1. It is clear that we have an inclusion of (full) sub-catgories ap ⊆ ˆ ap . Let R be an object in ˆ ap with radical I = I(R). To ﬁx notation, we write grn (R) = I n /I n+1 for n ≥ 0 (with I 0 = R). We also write gr R = ⊕ grn (R), this is the graded ring associated to the I-adic ﬁltration of R. The tangent space of R is deﬁned to be the k-linear space dual to gr1 (R), tR = Homk (I/I 2 , k) = (I/I 2 )∗ , which is clearly of ﬁnite dimension over k. In particular, we have (tR )∗ ∼ = I/I 2 . Let u : R → S be a morphism in ˆ ap . As usual, we consider R and S with the I-adic ﬁltrations, where I is I(R) and I(S) respectively. Since u preserves these ﬁltrations, it induces a morphism of graded rings gr(u) : gr R → gr S. This morphism is homogeneous of degree 0, so u also induces morphisms of k-vector spaces grn (u) : grn (R) → grn (S) for all

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n ≥ 0. In particular, we have a morphism of k-vector spaces gr1 (u) : gr1 (R) → gr1 (S), and a dual morphism tu : tS → tR . Proposition 1.5. Let u : R → S be a morphism in ˆ ap . Then u is a surjection if and only if gr1 (u) is a surjection. Furthermore, u is injective if gr(u) is injective. Proof. If u is surjective, then clearly gr1 (u) is also surjective. To prove the other implication, let us consider the map gr(u) : gr(R) → gr(S). Since gr S is generated by the elements in gr1 S as an algebra, it follows that if gr1 (u) is surjective, then gr(u) is also surjective. From Bourbaki [1], chapter III, §2, no. 8, corollary 1 and 2, we have that u is surjective (injective) if gr(u) is surjective (injective), and the result follows. Let n be any natural number. We deﬁne the category ap (n) to be the full sub-category of ap consisting of objects R in ap such that I(R)n = 0. Notice that ap (n) ⊆ ap (n + 1) for all n ≥ 1. Furthermore, each object R in ap belongs to a sub-category ap (n) for some integer n. Let u : R → S be a morphism in ap , and denote by K = ker(u) the kernel of u. We say that u is a small morphism if we have I(R) · K = K · I(R) = 0. We prove the following important fact about small surjections: Lemma 1.6. Let u : R → S be a surjection in ap . Then u can be factored into a finite number of small surjections. Proof. Let I = I(R), then I n K = 0 for some n ≥ 0. Consider the surjection uq : R/I q K → R/I q−1 K for 1 ≤ q ≤ n. Clearly I(R/I q K) ker(uq ) = 0 for all q. Moreover, u1 ◦ · · · ◦ un = u when u1 : R/IK → R/K is considered as a morphism onto S ∼ = R/K. It is therefore enough to prove the lemma for a surjection u : R → S with IK = 0. In this situation, KI n = 0 for some n ≥ 0. Now consider the surjection vq : R/KI q → R/KI q−1 for 1 ≤ q ≤ n. Clearly, vq is a small surjection for all q. Moreover, u = v1 ◦ · · · ◦ vn when v1 : R/KI → R/K is considered as a morphism onto S ∼ = R/K. It follows that u can be factorized in a ﬁnite number of small surjections in ap . We conclude this section with an important family of examples: Let Vij be a ﬁnite dimensional k-vector space for 1 ≤ i, j ≤ p, with dimk Vij = dij . Let furthermore {rij (l) : 1 ≤ l ≤ dij } be a basis of Vij for 1 ≤ i, j ≤ p (or simply {rij } if dij = 1). We deﬁne the free matrix ring R = R({Vij }) deﬁned by the vector spaces Vij in the following way: We say that a monomial in R of type (i, j) and degree n is an expression of the form ri0 i1 (l1 )ri1 i2 (l2 ) . . . rin−1 in (ln ) with i0 = i, in = j. To these, we add the monomials ei for 1 ≤ i ≤ p, which we consider to be of type (i, i) and degree 0. We deﬁne R to be the k-linear space generated by all monomials in R, with the obvious multiplication: If M is a monomial of type (i, j), and M is a monomial of type (l, m), then M M = 0 if j = l, and M M is the monomial obtained by juxtapositioning M and M (possibly after having erased unnecessary ei ’s) if j = l. We see that (R, f, g) is an object of the category Ap , where f, g are the obvious maps k p → R → k p . In fact, Rij is the k-linear subspace generated by monomials in R of type (i, j), and the ideal I = I(R) is the k-linear subspace generated by all monomials of positive degree. ˆ = R({V ˆ We denote by R ij }) the completion of R = R({Vij }), and call this the formal ˆ ij is an inﬁnite matrix ring deﬁned by the vector spaces Vij . Explicitly, every element in R k-linear sum of monomials in R of type (i, j). Let I = I(R), then we have that Rn = R/I n ∼ =

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ˆ R) ˆ n belongs to ap for n ≥ 1: Clearly, Rn has ﬁnite dimension as k-vector space, so Rn R/I( ˆ clearly is complete, it is Artinian, and I(Rn ) = I/I n , so the radical is nilpotent. Since R ˆ belongs to ˆ follows that R ap . ˆ is Noetherian in Notice that neither the free matrix ring R nor the formal matrix ring R general. For a counter-example, it is enough to consider the case when p = 1 and d11 = 2, or the case when p = 2 and d11 = d12 = d21 = 1, d22 = 0. In the ﬁrst case, R ∼ = k{x, y}, which we know is not Noetherian. In the second case, we have that R11 = k{r11 , r12 r21 } ∼ = k{x, y}, which again is not Noetherian. So by proposition 1.1, R is not Noetherian in this case either. ˆ is not Noetherian in any of the two cases. A similar argument shows that R 2. Noncommutative deformations of modules We recall that k is an algebraically closed (commutative) ﬁeld, A is an associative k-algebra, and M = {M1 , . . . , Mp } is a ﬁnite family of left A-modules. In this section, we shall deﬁne the noncommutative deformation functor DefM : ap → Sets describing the simultaneous formal deformations of the family M. Let R be an object of ap . A lifting of the family M of left A-modules to R is a left A ⊗k Rop -module MR , together with isomorphisms ηi : MR ⊗R ki → Mi of left A-modules for 1 ≤ i ≤ p, such that MR ∼ = (Mi ⊗k Rij ) as right R-modules. We remark that a left A⊗k Rop -module is the same as an A-R bimodule such that the left and right k-vector space structures coincide. Furthermore, the notation (Mi ⊗k Rij ) refers to the k-vector space (Mi ⊗k Rij ) = ⊕ (Mi ⊗k Rij ) i,j

with the natural right R-module structure coming from the multiplication in R. The condition that MR ∼ = (Mi ⊗k Rij ) as right R-modules generalizes the ﬂatness condition in commutative deformation theory. Let MR , MR be two liftings of M to R. We say that these two liftings are equivalent if there exists an isomorphism τ : MR → MR of left A ⊗k Rop -modules such that the natural diagrams commute (that is, such that ηi ◦ (τ ⊗R ki ) = ηi for 1 ≤ i ≤ p). We let DefM (R) denote the set of equivalence classes of liftings of M to R, and we refer to these equivalence classes as deformations of M to R. We shall often denote a deformation represented by (MR , ηi ) by MR to simplify notation. Let u : R → S be a morphism in ap , and let MR be a lifting of M to R, representing an element in DefM (R). We deﬁne MS = MR ⊗R S, which has a natural structure as a left A ⊗k S op -module. Since u is a morphism in ap , we have natural isomorphisms of left A-modules (MR ⊗R S) ⊗S ki ∼ = MR ⊗R ki , inducing isomorphisms of left A-modules ρi : MS ⊗S ki → Mi via ηi for 1 ≤ i ≤ p. A straight-forward calculation shows that MS together with the isomorphisms ρi for 1 ≤ i ≤ p constitutes a lifting of M to S, and furthermore that the equivalence class of this lifting is independent upon the representative of the equivalence class of MR . Hence, we obtain a map DefM (u) : DefM (R) → DefM (S), and we see that DefM : ap → Sets is a covariant functor.

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Let R = (Rij ) be an object in ap . We shall describe how one, in principle, could attempt to calculate DefM (R) explicitly: We may assume that every element of DefM (R) is represented by a lifting MR , such that MR = (Mi ⊗k Rij ) considered as a right R-module. In order to describe this lifting completely, it is enough to describe the left action of A on MR . Furthermore, it is enough to describe this action on elements of the form mi ⊗ ei with mi ∈ Mi , since we have a(mi ⊗ rij ) = (a(mi ⊗ ei ))rij for all a ∈ A, mi ∈ Mi , rij ∈ Rij. For a ﬁxed a ∈ A, mi ∈ Mi , assume that a(mi ⊗ ei ) = (mj ⊗ rjl ) with mj ∈ Mj , rjl ∈ Rjl . Then multiplication by ei on the right gives the equality a(mi ⊗ ei ) = (mj ⊗ rji ), j

and the isomorphism ηi gives a further restriction on the left action of A, expressed by the formula (1)

a(mi ⊗ ei ) = (ami ) ⊗ ei +

mj ⊗ rji ,

j where a ∈ A, mi ∈ Mi , mj ∈ Mj , rji ∈ I(R)ji . Consequently, the set DefM (R) consists of all possible choices of left A-actions on elements of the form mi ⊗ ei , fulﬁlling condition (1) and the associativity condition, up to equivalence. Let R be any object in ap . Then the formula a(mi ⊗ ei ) = (ami ) ⊗ ei for a ∈ A, mi ∈ Mi deﬁnes a left A-module structure on (Mi ⊗ Rij ) compatible with the right R-module structure. Hence, there exists a trivial lifting MR to R for all objects R in ap , and DefM (R) is non-empty. Notice that in the case R = k p , we have I = I(R) = 0, so this trivial lifting is the only one possible. Consequently, we have DefM (k p ) = {∗}, where ∗ denotes the equivalence class of the trivial lifting. Let u : R → S be a morphism in ap , and let MS ∈ DefM (S) be a given deformation. We say that a deformation MR ∈ DefM (R) is a lifting of MS or is lying over MS if DefM (u)(MR ) = MS . Given any object R in ap and a deformation MR ∈ DefM (R), we see that MR is a lifting of the trivial deformation ∗ in DefM (k p ) in the above sense via the structural morphism g : R → k p . Hence, our notation is consistent. For another example, consider the test algebras R(α, β) for 1 ≤ α, β ≤ p, constructed in the following way: Let R be the free matrix algebra deﬁned by the k-vector spaces Vij with dimensions dα,β = 1 and dij = 0 when (i, j) = (α, β). We deﬁne R(α, β) = R/I(R)2 , which is an object in ap (2) by construction. We know that any lifting of M to R(α, β) is deﬁned by a left A-action

a(mβ ⊗ eβ ) = (amβ ) ⊗ eβ + ψ(a)(mβ ) ⊗ εα,β for all a ∈ A, mβ ∈ Mβ , where ψ : A × Mβ → Mα is a k-bilinear map and εα,β is the class of rα,β . Clearly, we must have a(mi ⊗ ei ) = (ami ) ⊗ ei for all a ∈ A, mi ∈ Mi when i = β. Moreover, ψ deﬁnes an associative A-module structure if and only if ψ ∈ Derk (A, Homk (Mβ , Mα )). In this case, we shall denote the corresponding lifting by M(ψ) ∈ DefM (R(α, β)). Given two derivations ψ, ψ , we see that M(ψ) and M(ψ ) are equivalent liftings if and only if there is a φ ∈ Homk (Mβ , Mα ) such that (ψ − ψ )(a)(mβ ) = aφ(mβ ) − φ(amβ )

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for all a ∈ A, mβ ∈ Mβ . Lemma 2.1. There is a bijective correspondence DefM (R(α, β)) ∼ = Ext1A (Mβ , Mα ) for 1 ≤ α, β ≤ p. Proof. From the deﬁnition of Hochschild cohomology (see appendix A), we see that ψ → M(ψ) induces a bijective correspondence between HH1 (A, Homk (Mβ , Mα )) and DefM (R(α, β)). Moreover, HH1 (A, Homk (Mβ , Mα )) ∼ = Ext1A (Mβ , Mα ) by proposition A.3.

3. M-free resolutions and noncommutative deformations We recall that k is an algebraically closed (commutative) ﬁeld, A is an associative kalgebra, and M = {M1 , . . . , Mp } is a ﬁnite family of left A-modules. In this section, we shall deﬁne M-free resolutions and relate them to noncommutative deformations of modules. In particular, we shall show that M-free resolutions are useful computational tools in order to study the deformation functor DefM . Let R be any object of ap . An M-free module over R is a left A ⊗k Rop -module F of the form F = (Li ⊗k Rij ), where L1 , . . . , Lp are free left A-modules, and the left A-module structure on F is the trivial one. In other words, F is the trivial lifting of a family {L1 , . . . , Lp } of free left A-modules to R. Although an M-free module over R is not free considered as a left A ⊗k Rop -module, it behaves as a free module when interpreted as a module of matrices in the correct way: Lemma 3.1. Let u : R → S be a surjection in ap , and consider a left A ⊗k Rop -module MR = (Mi ⊗k Rij ) and a left A ⊗k S op -module MS = (Mi ⊗k Sij ) such that the natural map v : MR → MS induced by u is left A-linear. If F S is any M-free module over S given by the free left A-modules L1 , . . . , Lp and fS : F S → MS is any left A ⊗k S op -linear map, then there exists a left A ⊗k Rop -linear map fR : F R → MR making the diagram MR

fR

v

MS

FR (id ⊗u)

fS

FS

commutative, where F R is the M-free module over R given by the free left A-modules L1 , . . . , Lp . Proof. Clearly, the map fS is determined by its values on Li ⊗ ei , and therefore by the corresponding left A-linear maps Lj → ⊕(Mi ⊗k Sij ). Since each left A-module Lj is projective, we can lift these maps to left A-linear maps Lj → ⊕(Mi ⊗k Rij ), and these maps determine fR . Let R be any object of ap , and let MR = (Mi ⊗k Rij ) ∈ DefM (R) be a lifting of M to R. An M-free resolution of MR is an exact sequence of left A ⊗k Rop -linear maps R 0 ← MR ← F0R ← F1R ← · · · ← Fm ← ···

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R R where Fm is an M-free module over R for m ≥ 0. So we have Fm = (Lm,i ⊗k Rij ) where Lm,i are free left A-modules for 1 ≤ i ≤ p, m ≥ 0. We shall denote the diﬀerentials by R R dR m : Fm+1 → Fm for m ≥ 0. We ﬁx a k-linear basis {rij (l) : 1 ≤ l ≤ dimk Rij } of Rij for 1 ≤ i, j ≤ p such that ei is contained in the basis of Rii for 1 ≤ i ≤ p. Consider the diﬀerential dR m in the M-free uniquely in the form resolution of MR above. Clearly, we can write dR m

dR m =

(2)

α(rij (l))m ⊗ rij (l)

i,j,l

for all m ≥ 0, where α(rij (l))m : Lm+1,j → Lm,i is a homomorphism of left A-modules for 1 ≤ i, j ≤ p, 1 ≤ l ≤ dimk Rij . In particular, the M-free resolution of MR deﬁnes a family of 1-cochains α(rij (l)) ∈ Hom1 (L∗j , L∗i ), indexed by a k-linear basis for R. From now on, we ﬁx a free resolution (L∗i , d∗i ) of Mi considered as left A-module for 1 ≤ i ≤ p. These free resolutions correspond to an M-free resolution (F∗ , d∗ ) of the trivial p deformation (Mi ⊗k (k p )ij ) ∈ Def M (k ). In fact, the M-free resolution (F∗ , d∗ ) is given by p Fm = (Lm,i ⊗k (k )ij ) and dm = dm,i ⊗ ei for m ≥ 0. We have therefore ﬁxed an M-free resolution (F∗ , d∗ ) of the trivial lifting of M to k p . R Let R be any object of ap . We say that a complex (F∗R , dR ∗ ) of M-free modules Fm = (Lm,i ⊗k Rij ) over R is a lifting of the complex (F∗ , d∗ ) if the following diagram commutes F0R

dR 0

v0

F0

F1R

dR 1

v1 d0

F1

F2R

...

v2 d1

F2

...

R where vm : Fm → Fm are the natural maps induced by R → k p .

Lemma 3.2. Let R be any object of ap , and let (F∗R , dR ∗ ) be a lifting of the complex (F∗ , d∗ ). Then we have: (1) H m (F∗R , dR ∗ ) = 0 for all m ≥ 0, (2) H 0 (F∗R , dR ∗ ) is a lifting of the family M to R. Proof. Clearly, the lemma holds for R = k p . We shall consider a small surjection u : R → S in ap and liftings of complexes (F∗U , dU ∗ ) of (F∗ , d∗ ) to U for U = R, S such that the following diagram commutes: F0R

dR 0

v0

F0S

F1R

dR 1

v1

dS 0

F1S

F2R

...

v2

dS 1

F2S

...

In this situation, we shall prove that if the conclusion of the lemma holds for S, it holds for R as well. This is clearly enough to prove the lemma. Let K = ker(u), then we clearly have ker(vm ) = (Fm,i ⊗k Kij ) with the trivial left K , then (F∗K , dK A-action for all m ≥ 0. We denote this kernel by Fm ∗ ) is a complex of

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R left A ⊗k Rop -modules, where dK ∗ is the restriction of d∗ . Moreover, it is clear that vm is 0 U U surjective for m ≥ 0. Deﬁne MU = H (F∗ , d∗ ) for U = R, S, let v : MR → MS be the induced map, and denote the kernel by MK = ker(v). Then clearly v is surjective, and we have the following commutative diagram of complexes:

0

0

0

MK

ρK

R

MR

ρ

MS

0

ρS

F1K

0 dK 1

i1

F0R

dR 0

v0

v

0

dK 0

i0

i

0

F0K

0

0

dS 0

...

i2

F1R

dR 1

v1

F0S

F2K

F2R

...

v2

F1S

0

dS 1

F2S

...

0

Clearly all columns are exact, so the diagram gives a short exact sequence of complexes. By assumption, the bottom row is exact and MS = (Mi ⊗k Sij ) is a lifting of M to S. Let us ﬁrst show that H m (F∗K , dK ∗ ) = 0 for m ≥ 1: This follows since the complex is a lifting of (F∗ , d∗ ) and because I(R)K = 0 (since u : R → S is small). The long exact sequence of cohomologies of the complexes above now implies that H m (F∗R , dR ∗ ) = 0 for all m ≥ 1 and that we have a short exact sequence 0 → H 0 (F∗K , dK ∗ ) → MR → MS = (Mi ⊗k Sij ) → 0, of left A-modules, so in particular MK ∼ = H 0 (F∗K , dK ∗ ). But since I(R)K = 0, it follows 0 K K ∼ 0 that H (F∗ , d∗ ) = (H (L∗,i , d∗,i ) ⊗k Kij ) = (Mi ⊗k Kij ) with the trivial left A-module structure. It follows that MR ∼ = (Mi ⊗k Rij ) considered as a k-vector space, and therefore MR is a lifting of M to R. Lemma 3.3. Let R be any object of ap , and let MR be a lifting of M to R. Then there exists an M-free resolution of MR which lifts the complex (F∗ , d∗ ) to R. Proof. Clearly, the lemma holds for R = k p . We shall consider a small surjection u : R → S in ap , deformations MU ∈ DefM (U ) for U = R, S such that MR lifts MS to R, and an M-free resolution (F∗S , dS∗ ) of MS which lifts the complex (F∗ , d∗ ) to S. In this situation, we shall prove that there exists an M-free resolution (F∗R , dR ∗ ) of MR compatible with the M-free resolution of MS . This is clearly enough to prove the lemma. R K = (Lm,i ⊗k Rij ) for all m ≥ 0. Moreover, we write Fm = (Lm,i ⊗k Kij ) for all Let Fm m ≥ 0, where K = ker(u). To complete the proof, we have to ﬁnd the diﬀerentials dR m for m ≥ 0 and the augmentation map ρR : By lemma 3.1, we can ﬁnd a homomorphism ρR : F0R → MR lifting ρS . Denote by ρK : F0K → MK its restriction, where MK = ker(MR → MS ). Since u is small, ρK is surjective, and this implies that the induced map ker(ρR ) → R R S ker(ρS ) is surjective. By lemma 3.1, we can ﬁnd a homomorphism dR 0 : F1 → F0 lifting d0 R R K R K K such that ρ d0 = 0. Let d0 be the restriction of d0 , then clearly ker(ρ ) = Im(d0 ) since

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u is small. An easy induction argument shows that we can construct a complex (F∗R , dR ∗) lifting the complex (F∗S , dS∗ ) in such a way that the restriction (F∗K , dK ∗ ) is a resolution of MK . By the proof of lemma 3.2, it follows that H m (F∗R , dR ∗ ) = 0 for m ≥ 1 and that there is an exact sequence 0 → MK → H 0 (F∗R , dR ∗ ) → MS → 0. R R This implies that MR = H 0 (F∗R , dR ∗ ), and (F∗ , d∗ ) is the required M-free resolution of MR compatible with the given M-free resolution of MS .

Proposition 3.4. Let u : R → S be a surjection in ap , and consider a deformation MS ∈ DefM (S) and any M-free resolution (F∗S , dS∗ ) of MS which lifts the complex (F∗ , d∗ ) to S. There is a bijective correspondence between the set of liftings {MR ∈ DefM (R) : DefM (u)(MR ) = MS } S S and the set of M-free complexes (F∗R , dR ∗ ) which lift the resolution (F∗ , d∗ ) to R, up to equivalence.

Proof. For a small surjection, this follows from lemma 3.2 and lemma 3.3. But any surjection in ap is a composition of small surjections. Let R be any object in ap . In section 2, we described how to, in principle, calculate DefM (R) by considering the possible left A-module structures on the right R-module (Mi ⊗k Rij ). The M-free resolutions give us another way of viewing deformations in DefM (R): By proposition 3.4, we can view DefM (R) as the set of liftings of the complex (F∗ , d∗ ) to R, up to equivalence. Using equation 2, each lifting of complexes corresponds to a family of 1-cochains α(rij (l)) ∈ Hom1 (L∗j , L∗i ), parametrized by a k-basis for R. We leave it as an exercise for the reader to use this approach to calculate DefM (R) in the case R = Rα,β — this will give a new proof of lemma 2.1 via the Yoneda representation of Ext1A (Mβ , Mα ).

4. Pro-representing hulls of pointed functors We say that a covariant functor F : ap → Sets is pointed if F(k p ) = {∗}. In this section, we shall consider pointed functors deﬁned on the category ap , and study their representability. Of course, the motivation for this is the fact that DefM is such a pointed functor. Let R be any object of ˆ ap , and consider the functor hR : ap → Sets given by hR (S) = Mor(R, S) for all objects S in ap . The notation Mor(R, S) denotes the set of morphisms from R to S in the pro-category ˆ ap . Then hR is clearly a pointed functor deﬁned on ap . We say that a pointed functor F : ap → Sets is representable is F is isomorphic to hR for some object R in ap , and pro-representable if F is isomorphic to hR for some object R in ˆ ap . However, it is well-known that deformation functors seldom are representable or even pro-representable. So a weaker notion is required, and we shall deﬁne the notion of a pro-representing hull of a pointed functor on ap . We start by introducing some notation: ˆ :ˆ ap → Sets deﬁned Any pointed functor F : ap → Sets has an extension to a functor F on the pro-category ˆ ap . This extension is deﬁned by the formula ˆ F(R) = lim F(R/I n ) ←

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for any object R in ˆ ap with I = I(R). Clearly, any pointed functor F : ap → Sets also has a restriction to the sub-category ap (n) ⊆ ap for all n ≥ 1. We shall denote this restriction by Fn : ap (n) → Sets. Lemma 4.1. Let R be an object in ˆ ap , and let F : ap → Sets be a pointed functor. Then ˆ there is a natural isomorphism of sets α : F(R) → Mor(hR , F). ˆ Proof. Let ξ ∈ F(R), then ξ = (ξn ) with ξn ∈ F(R/I n ) for all n ≥ 1. For any object S in ap , we construct a map of sets α(ξ)S : Mor(R, S) → F(S): Let u : R → S be a morphism in ˆ ap , then u(I(R)) ⊆ I(S), and I(S) is nilpotent since S is in ar , so there exists n ≥ 1 such that u factorizes through un : R/I(R)n → S. We deﬁne α(ξ)S (u) = F(un )(ξn ), and a straight-forward calculation shows that this expression is independent upon the choice of n, and gives rise to a natural transformation of functors. Conversely, let φ : hR → F be a natural transformation of functors on ap . Then we deﬁne ξn ∈ F(R/I(R)n ) to be ξn = φR/I(R)n (R → R/I(R)n ), where R → R/I(R)n is the natural morphism. Again, a ˆ straight-forward calculation shows that ξ = (ξn ) deﬁnes an element in F(R), and that this map of sets deﬁnes an inverse to α. There is also a version of lemma 4.1 for the category ap (n): For an object R in ap (n), and a pointed functor F : ap (n) → Sets, there is a natural isomorphism of sets αn : F(R) → Mor(hR , F). The construction of this isomorphism is similar to the construction in lemma 4.1. We recall that a morphism φ : F → G of pointed functors F, G : ap → Sets is smooth if the following condition holds: For all surjective morphisms u : R → S in ap , the natural map of sets (3)

F(R) → F(S) × G(R), G(S)

given by x → (F(u)(x), φR (x)) for all x ∈ F(R), is a surjection. Clearly, it is enough to check this for small surjections in ap . Also notice that any morphism φ : F → G of functors ˆ → G ˆ of functors on ˆ naturally extends to a morphism φˆ : F ap , and if φ is a smooth ˆ ˆ ˆ morphism, then φR : F(R) → G(R) is surjective for all objects R in ˆ ap . Similarly, we say that a morphism φ : F → G of functors F, G : ap (n) → Sets on ap (n) is smooth if the map of sets (3) is surjective for all surjective morphisms u : R → S in ap (n). Clearly, a morphism φ : F → G of functors on ap is smooth if and only if the restriction φn : Fn → Gn is smooth for all n ≥ 1. Let F be a pointed functor on ap . A pro-couple for F is a pair (R, ξ), where R is an ˆ A morphism u : (R, ξ) → (R , ξ ) of pro-couples is a morphism object in ˆ ap and ξ ∈ F(R). ˆ ap such that F(u)(ξ) = ξ . If (R, ξ) is a pro-couple for F such that R is also u : R → R in ˆ an object of ap , then it is called a couple for F. We say that a pro-couple (R, ξ) pro-represents F if α(ξ) : hR → F is an isomorphism of functors on ap . If (R, ξ) pro-represents F and (R, ξ) is also a couple for F, then we say that (R, ξ) represents F. It is clear that if the couple (R, ξ) represents F, then (R, ξ) is unique up to a unique isomorphism of couples. Similarly, let F be a pointed functor on ap (n). A couple for F is a pair (R, ξ), where R is an object of ap (n) and ξ ∈ F(R). We say that the couple (R, ξ) represents F if and only if αn (ξ) is an isomorphism of functors deﬁned on ap (n). It is clear that if this is the case, the couple (R, ξ) is unique up to a unique isomorphism of couples.

NONCOMMUTATIVE DEFORMATIONS OF MODULES

103

Let F be a functor on ap , and let (R, ξ) be a pro-couple for F. For all n ≥ 1, let (Rn , ξn ) be given by Rn = R/I(R)n and ξn = F(un )(ξ), where un : R → Rn is the natural surjection. Then (Rn , ξn ) is a couple for the restriction Fn : ap (n) → Sets of F for all n ≥ 1. Notice that αn (ξn ) is the restriction of the morphism α(ξ) to ap (n) for all n ≥ 1. Consequently, (R, ξ) pro-represents F if and only if (Rn , ξn ) represents Fn for all n ≥ 1. In particular, it follows that if (R, ξ) pro-represents F, then (R, ξ) is unique up to a unique isomorphism of pro-couples. Let F : ap → Sets be a pointed functor on ap . A pro-representing hull of F is a pro-couple (R, ξ) of F such that the following conditions hold: (1) α(ξ) : hR → F is a smooth morphism of functors on ap (2) α2 (ξ2 ) : hR2 → F2 is an isomorphism of functors on ap (2) To simplify notation, we sometimes call the pro-representing hull (R, ξ) a hull of F. Proposition 4.2. Let F : ap → Sets be a pointed functor on ap , and assume that (R, ξ), (R , ξ ) are pro-representing hulls of F. Then there exists an isomorphism of procouples u : (R, ξ) → (R , ξ ). Proof. Let φ = α(ξ), φ = α(ξ ). Since φ, φ are smooth morphisms, we have that φR and φR are surjective. So we can ﬁnd morphisms u : (R, ξ) → (R , ξ ) and v : (R , ξ ) → (R, ξ) of pro-couples of F. The restriction to ap (2) gives us morphisms u2 : (R2 , ξ2 ) → (R2 , ξ2 ) and v2 : (R2 , ξ2 ) → (R2 , ξ2 ). But both (R2 , ξ2 ) and (R2 , ξ2 ) represent F2 , so u2 and v2 are inverses. In particular, gr1 (u2 ) and gr1 (v2 ) are inverses, and (v ◦ u)2 = v2 ◦ u2 = id. From the proof of proposition 1.5, we see that gr(v ◦ u) is surjective. This means that grn (v ◦ u) is a surjective endomorphims of a ﬁnite dimensional k-vector space for all n ≥ 1, so gr(v ◦ u) is an isomorphism. By proposition 1.5, v ◦ u is an isomorphism as well, and the same holds for u ◦ v by a symmetric argument. It follows that u and v are isomorphisms. So if there exists a pro-representing hull of a pointed functor F, we know that it is unique, and we shall denote it by (H, ξ). Notice that (H, ξ) is only unique up to non-canonical isomorphism. By abuse of language, we shall sometimes omit ξ from the notation, and say that H is the hull of F. 5. Hulls of noncommutative deformation functors We recall that k is an algebraically closed (commutative) ﬁeld, A is an associative k-algebra, and M = {M1 , . . . , Mp } is a ﬁnite family of left A-modules. In this section, we prove that if the family M satisfy the ﬁniteness condition (FC), then there exists a hull H = H(M) of the noncommutative deformation functor DefM . The proof follows Laudal [8], and the essential point is the following obstruction calculus: Proposition 5.1. Let u : R → S be a small surjective morphism in ap with kernel K = ker(u), and let MS ∈ DefM (S) be a deformation. Then there exists a canonical obstruction o(u, MS ) ∈ (Ext2A (Mj , Mi ) ⊗k Kij ), such that o(u, MS ) = 0 if and only if there exists a deformation MR ∈ DefM (R) lifting MS . If this is the case, the set of deformations in DefM (R) lifting MS is a torsor under the k-vector space (Ext1A (Mj , Mi ) ⊗k Kij ).

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Proof. We recall from section 2 that up to equivalence, we may assume that MS has the following form: MS = (Mi ⊗k Sij ) with right S-module structure given by the multiplication in S, and with left A-module structure given by k-linear homomorphisms ai : Mi → ⊕(Mj ⊗k Sji ) for all a ∈ A. Via the natural projections, the map ai gives rise to k-linear maps aji : Mi → Mj ⊗k Sji for a ∈ A, 1 ≤ i, j ≤ p. Since u is surjective, we may choose k-linear maps L(a)ji : Mi → Mj ⊗k Rji such that (id ⊗ u) ◦ L(a)ji = aji for a ∈ A, 1 ≤ i, j ≤ p. Let L(a) = (L(a)ij ) ∈ (Homk (Mj , Mi ⊗k Rij )), this deﬁnes a k-linear left action of A on MR = (Mi ⊗k Rij ), lifting the left A-module structure on MS . We let Q = (Homk (Mj , Mi ⊗k Rij )), and remark that this is an associative k-algebra in a natural way: We compose the k-linear morphisms in Q by using the multiplication in R. For a, b ∈ A, consider the expression L(ab)−L(a)L(b) ∈ Q . By the associativity of the left A-module structure on MS , we see that L(ab)−L(a)L(b) ∈ Q, where Q = (Homk (Mj , Mi ⊗k Kij )) ⊆ Q . Furthermore, we notice that Q ⊆ Q is an ideal, and Q has a natural structure as an A-A bimodule via L, since K 2 = 0. We deﬁne ψ ∈ Homk (A ⊗k A, Q) to be given by ψ(a, b) = L(ab) − L(a)L(b) for all a, b ∈ A. A straight-forward calculation shows that ψ is a 2-cocycle in HC∗ (A, Q), so ψ gives rise to an element o(u, MS ) ∈ HH2 (A, Q) — see appendix A for the deﬁnition of the Hochschild complex and its cohomology. Since K 2 = 0, it follows that if L is another k-linear lifting of the left A-module structure on MS , then the A-A bimodule structures of Q given by L and L coincide. Therefore, HH∗ (A, Q) is independent upon the choice of L, and a straight-forward calculation shows that the same holds for the obstruction o(u, MS ). We remark that there exists a deformation MR ∈ DefM (R) lifting MS if and only if there exists some k-linear lifting L : A → Q of the left A-module structure of MS such that L (ab) = L (a)L (b) for all a, b ∈ A. Let τ = L − L, then τ : A → Q is a k-linear map, and a straight-forward calculation shows that L (ab) = L (a)L (b) if and only if the relation L(ab) − L(a)L(b) = L(a)τ (b) − τ (ab) + τ (a)L(b) + τ (a)τ (b) holds. Since K 2 = 0, the last term vanishes. The fact that the above relation holds for all a, b ∈ A is therefore equivalent to the fact that o(u, MS ) = 0 in HH2 (A, Q). So we have established that there exists a canonical obstruction o(u, MS ) ∈ HH2 (A, Q) such that o(u, MS ) = 0 if and only if there is a lifting of MS to R. Assume that L : A → Q is such that L(ab) = L(a)L(b) for all a, b ∈ A, that is, such that it deﬁnes a deformation MR lying over MS . For any other k-linear lifting L : A → Q of the left A-module structure on MS , we may consider the diﬀerence τ = L − L : A → Q. A straight-forward calculation shows that τ is a 1-cocycle in HC∗ (A, Q) if and only if L (ab) = L (a)L (b) for all a, b ∈ A, that is, if and only if L deﬁnes a left A-module structure on MR . Furthermore, we have that L and L give rise to equivalent deformations if and only if τ is a 1-coboundary: It is clear that any equivalence between the left A-module structures of MR = (Mi ⊗k Rij ) given by L and L has the form id + ψ, where ψ ∈ Q. Furthermore, the map id + ψ : MR → MR (with the left A-module structure from L and L respectively) is a left A-module homomorphism if and only if L(r)(id + ψ) = (id + ψ)L (r) holds for all a ∈ A, and this last condition is equivalent with the fact that τ = d(ψ), so that τ is a 1-coboundary. If τ is a 1-boundary in HC∗ (A, Q), it is also clear that id + ψ deﬁnes an equivalence between the two deformations given by L and L . Therefore, the set of deformations MR lying over MS is a torsor under the k-vector space HH1 (A, Q).

NONCOMMUTATIVE DEFORMATIONS OF MODULES

105

To end the proof, we have to show that there are isomorphisms of k-vector spaces HHn (A, Q) ∼ = (ExtnA (Mj , Mi ) ⊗k Kij ) for n = 1, 2: Since L(a) is a lifting to MR of the left multiplication of a on MS (satisfying equation 1), L(a) satisﬁes equation 1 as well. That is, we have L(a)ji (mi ) − δij (ami ) ⊗ ei ∈ Mj ⊗k Iji for all a ∈ A, mi ∈ Mi , 1 ≤ i, j ≤ p. Since K 2 = 0, this means that the A-A bimodule structure of Q deﬁned via L coincides with the following natural one: Since Mi , Mj ⊗k Kji are left A-modules, we have that Qij = Homk (Mj , Mi ⊗k Kij ) and Q = ⊕Qij has natural A-A bimodule structures. Clearly, we have HHn (A, Q) ∼ = ⊕ HHn (A, Qij ) = (HHn (A, Qij )). i,j

∼ Extn (Mj , Mi ⊗k Kij ) for By appendix A, proposition A.3, we have that HHn (A, Qij ) = A n n n ≥ 0. Moreover, ExtA (Mj , Mi ⊗k Kij ) ∼ = ExtA (Mj , Mi ) ⊗k Kij since Kij is a k-vector space of ﬁnite dimension. This completes the proof of the proposition. We remark that it is easy to ﬁnd an alternative proof of proposition 5.1 using resolutions and the Yoneda representation of ExtnA (Mi , Mj ). This is straight-forward, but makes essential use of proposition 3.4. Also notice that the obstruction calculus is functorial in the following sense: Let u : R → S and u : R → S be two small surjections in ap , and write K = ker(u) and K = ker(u ). Assume that v : R → R and w : S → S are morphisms such that u ◦ v = w ◦ u. Then v(K) ⊆ K , and the map v induces a k-linear map of obstruction spaces ). (Ext2A (Mj , Mi ) ⊗k Kij ) → (Ext2A (Mj , Mi ) ⊗k Kij

If MS is a deformation of M to S and MS = DefM (w)(MS ) is the corresponding deformation to S , then this map of obstruction spaces maps o(u, MS ) to o(u , MS ). This follows from the proof of proposition 5.1. Let us start the construction of the pro-representing hull (H, ξ) of DefM , using the obstruction calculus for DefM given above. From now on, we shall assume that the family M satisfy the ﬁniteness condition (FC)

dimk ExtnA (Mi , Mj )is ﬁnite for 1 ≤ i, j ≤ p, n = 1, 2.

We ﬁx the following notation: Let {xij (l) : 1 ≤ l ≤ dij } be a basis for Ext1A (Mj , Mi )∗ and let {yij (l) : 1 ≤ l ≤ rij } be a basis for Ext2A (Mj , Mi )∗ for 1 ≤ i, j ≤ p, with dij = dimk Ext1A (Mj , Mi ) and rij = dimk Ext2A (Mj , Mi ). Moreover, we consider the formal ˆ matrix rings in ˆ ap corresponding to these vector spaces, and denote them by T1 = R 1 2 2 ∗ ∗ ˆ ({ExtA (Mj , Mi ) }) and T = R({ExtA (Mj , Mi ) }). First, let us show that DefM restricted to ap (2) is representable: We deﬁne H2 to be the object H2 = T12 = T1 /I(T1 )2 in ap (2). For all objects R in ap (2), we get Mor(H2 , R) ∼ = (Homk (Ext1A (Mj , Mi )∗ , I(R)ij )) ∼ = (Ext1A (Mj , Mi ) ⊗k I(R)ij ), and 1 ∼ DefM (R) = (ExtA (Mj , Mi ) ⊗k I(R)ij ) by proposition 5.1 applied to the small surjection R → k p . The isomorphisms we obtain in this way are compatible, so they induce an isomorphism φ2 : hH2 → DefM of functors on ap (2). From the version of lemma 4.1 for the category ap (2), we see that there is a unique deformation ξ2 ∈ DefM (H2 ) such that α2 (ξ2 ) = φ2 . By deﬁnition, (H2 , ξ2 ) represents the deformation functor DefM restricted to ap (2).

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Let us also give an explicit description of the deformation ξ2 : We have H2 = T12 , so let us denote by ij (l) the image of xij (l) in H2 for 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij . In this notation, ξ2 is represented by the right H2 -module (Mi ⊗k (H2 )ij ), with left A-module structure deﬁned by a(mj ⊗ ej ) = amj ⊗ ej +

l ψij (a)(mj ) ⊗ ij (l)

i,l l for all a ∈ A, mj ∈ Mj , 1 ≤ j ≤ p, where ψij ∈ Derk (A, Homk (Mj , Mi )) is a representative 1 ∗ of xij (l) ∈ ExtA (Mj , Mi ) via Hochschild cohomology. There is also an alternative description of ξ2 using M-free resolutions and the Yoneda representation of Ext1A (Mi , Mj ): Let α(ij (l)) ∈ Hom1 (L∗j , L∗i ) be a 1-cocycle representing xij (l)∗ ∈ Ext1A (Mj , Mi ) for 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij . Then by construction, the formula 2 dH m =

dm,i ⊗ ei +

i

α(ij (l))m ⊗ ij (l)

i,j,l

deﬁnes a diﬀerential which lifts the complex (F∗ , d∗ ) to H2 . By proposition 3.4, the lifted complex is in fact an M-free resolution of some deformation of M to H2 , and this deformation is ξ2 ∈ DefM (H2 ). Theorem 5.2. Assume that dimk ExtnA (Mi , Mj ) is finite for 1 ≤ i, j ≤ p, n = 1, 2. Then ˆ T2 k p is a pro-representing ap such that H(M) = T1 ⊗ there exists a morphism o : T2 → T1 in ˆ hull for DefM . Proof. For simplicity, let us write I for the ideal I = I(T1 ), and for all n ≥ 1, let us write T1n for the quotient T1n = T1 /I n , and tn : T1n+1 → T1n for the natural morphism. From the paragraphs preceding this theorem, we know that (H2 , ξ2 ) represents DefM restricted to ap (2). Let o2 : T2 → T12 be the trivial morphism given by o2 (I(T2 )) = 0 and let a2 = I 2 , then H2 = T1 /a2 ∼ = T12 ⊗T2 k p . Using o2 and ξ2 as a starting point, we shall construct on and ξn for n ≥ 3 by an inductive process. So let n ≥ 2, and assume that the morphism on : T2 → T1n and the deformation ξn ∈ DefM (Hn ) is given, with Hn = T1n ⊗T2 k p . We shall also assume that tn−1 ◦ on = on−1 and that ξn is a lifting of ξn−1 . Let us now construct the morphism on+1 : T2 → T1n+1 : We let an be the ideal in 1 Tn generated by on (I(T2 )). Then an = an /I n for an ideal an ⊆ T1 with I n ⊆ an , and Hn ∼ = T1 /an . Let bn = Ian + an I, then we obtain the following commutative diagram: T2

T1n+1

T1 /bn

T1n

Hn = T1 /an ,

on

Observe that T1 /bn → T1 /an is a small surjection. So by proposition 5.1, there is an obstruction on+1 = o(T1 /bn → Hn , ξn ) for lifting ξn to T1 /bn , and we have on+1 ∈ (Ext2A (Mj , Mi ) ⊗k (an /bn )ij ) ∼ = (Homk (gr1 (T2 )ij , (an /bn )ij )).

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Consequently, we obtain a morphism on+1 : T2 → T1 /bn . Let an+1 be the ideal in T1 /bn generated by on+1 (I(T2 )). Then an+1 = an+1 /bn for an ideal an+1 ⊆ T1 with bn ⊆ an+1 ⊆ an . We deﬁne Hn+1 = T1 /an+1 and obtain the following commutative diagram: on+1

T2

T1n+1

T1 /bn

T1n

Hn = T1 /an

Hn+1 = T1 /an+1

on

By the choice of an+1 , the obstruction for lifting ξn to Hn+1 is zero. We can therefore ﬁnd a lifting ξn+1 ∈ DefM (Hn+1 ) of ξn to Hn+1 . The next step of the construction is to ﬁnd a morphism on+1 : T2 → T1n+1 which commutes with on+1 and on : We know that tn−1 ◦ on = on−1 , which means that an−1 = I n−1 + an . For simplicity, let us write O(K) = (Homk (gr1 (T2 )ij , Kij )) for any ideal K ⊆ T1 . Consider the following commutative diagram of k-vector spaces, in which the columns are exact: 0

0

O(bn /I n+1 )

O(an /I n+1 )

jn

O(bn−1 /I n )

kn

O(an−1 /I n )

rn+1

O(an /bn )

0

rn ln

O(an−1 /bn−1 )

0

We may consider consider on as an element in O(an−1 /I n ), since an ⊆ an−1 . On the other hand, on+1 ∈ O(an /bn ). Let on = rn (on ), then the natural map T1 /bn → T1 /bn−1 maps the obstruction on+1 to the obstruction on by the second remark following proposition 5.1. This implies that on+1 commutes with on , so ln (on+1 ) = on = rn (on ). But we have on (I(T2 )) ⊆ an , so we can ﬁnd an element on+1 ∈ O(an /I n+1 ) such that kn (on+1 ) = on . Since an−1 = an + I n−1 , jn is surjective. Elementary diagram chasing using the snake lemma implies that we can ﬁnd on+1 ∈ O(an /I n+1 ) such that rn+1 (on+1 ) = on+1 and kn (on+1 ) = on . It follows that the obstruction on+1 deﬁnes a morphism on+1 : T2 → T1n+1 compatible with on such that T1n+1 ⊗T2 k p ∼ = Hn+1 . By induction, it follows that we can ﬁnd a morphism on : T2 → T1n and a deformation ξn ∈ DefM (Hn ), with Hn = T1n ⊗T2 k p , for all n ≥ 1. From the construction, we see that tn−1 ◦ on = on−1 for all n ≥ 2, so we obtain a morphism o : T2 → T1 by the universal property of the projective limit. Moreover, the induced morphisms hn : Hn+1 → Hn are such that ξn+1 ∈ DefM (Hn+1 ) is a lifting of ξn ∈ DefM (Hn ) to Hn+1 . Notice that I(Hn )n = 0

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and that Hn /I(Hn )n−1 ∼ = Hn−1 for all n ≥ 2. It follows that H/I(H)n = Hn for all n ≥ 1, so H is an object of the pro-category ˆ ap . Let ξ = (ξn ), then clearly ξ ∈ DefM (H), so (H, ξ) is a pro-couple for DefM . It remains to show that (H, ξ) is a pro-representable hull for DefM . It is clearly enough to show that (Hn , ξn ) is a pro-representing hull for DefM restricted to ap (n) for all n ≥ 3. So let φn = αn (ξn ) be the morphism of functors on ap (n) corresponding to ξn . We shall prove that φn is a smooth morphism. So let u : R → S be a small surjection in ap (n), and assume that MR ∈ DefM (R) and v ∈ Mor(Hn , S) are given such that DefM (u)(MR ) = DefM (v)(ξn ) = MS . Let us consider the following commutative diagram: T1 /bn

T1

Hn+1

R u

Hn

v

S

Let v : T1 → R be any morphism making the diagram commutative. Then v (an ) ⊆ K, where K = ker(u), so v (bn ) = 0. But the induced map T1 /bn → R maps the obstruction on+1 to o(u, MS ), and we know that o(u, MS ) = 0. So we have v (an+1 ) = 0, and v induces a morphism v : Hn+1 → R making the diagram commutative. Since v (I(Hn+1 )n ) = 0, we may consider v a map from Hn+1 /I(Hn+1 )n ∼ = Hn . So we have constructed a map v ∈ Mor(Hn , R) such that u ◦ v = v. Let MR = DefM (v )(ξn ), then MR is a lifting of MS to R. By proposition 5.1, the diﬀerence between MR and MR is given by an element d ∈ (Ext1A (Mj , Mi ) ⊗k Kij ) = (Homk (gr1 (T1 )ij , Kij )). Let v : T1 → R be the morphism given by v (xij (l)) = v (xij (l))+d(xij (l)) for 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij . Since an+1 ⊆ I(T1 )2 , we have v (an+1 ) ⊆ v (an+1 ) + I(R)K + KI(R) + K 2 . But u is small, so v (an+1 ) = 0 and v induces a morphism v : Hn → R. Clearly, u ◦ v = u ◦ v = v, and DefM (v )(ξn ) = MR by construction. It follows that φn is smooth for all n ≥ 3. We remark that the conclusion of the theorem still holds if we relax the ﬁniteness condition (FC). If we only assume that dimk Ext1A (Mi , Mj )is ﬁnite for 1 ≤ i, j ≤ p, then the object T2 is in Ap , but not necessarily in ˆ ap . However, the rest of the proof is still valid as stated, so the ﬁniteness condition on Ext2A (Mi , Mj ) is clearly not essential. In general, it is possible to generalize theorem 5.2 to the case when ExtnA (Mi , Mj ) has countable dimension as a vector space over k for 1 ≤ i, j ≤ p, n = 1, 2, see Laudal [2]. However, we shall always assume (FC) in this paper.

NONCOMMUTATIVE DEFORMATIONS OF MODULES

109

Assume that M satisfy (FC). If Ext2A (Mi , Mj ) = 0 for 1 ≤ i, j ≤ p, we say that the deformation functor DefM is unobstructed. For instance, DefM is unobstructed for any ﬁnite family M of left A-modules satisfying (FC) if A is left hereditary (that is, the left global homological dimension of A is at most 1). If DefM is unobstructed, H = T1 is the hull of DefM . In general, DefM can be obstructed, and there is no simple formula for the hull H of DefM if this is the case. However, there exists an algorithm for calculating the hull H using matric Massey products. In the next sections, we shall introduce the matric Massey products and explain how the hull can be calculated when M satisfy (FC). 6. Immediately defined matric Massey products We recall that k is an algebraically closed (commutative) ﬁeld, A is an associative k-algebra, and M = {M1 , . . . , Mp } is a ﬁnite family of left A-modules. From now on, we also assume that the family M satisfy the ﬁniteness condition (FC). In this section, we shall deﬁne the immediately deﬁned matric Massey products and their deﬁning system, and show how to calculate these products using matrices. Let us ﬁx a monomial X ∈ I(T1 ) of type (i, j) and degree n ≥ 2. Then we can write X uniquely in the form X = xi0 i1 (l1 )xi1 i2 (l2 ) . . . xin−1 in (ln ), where (i0 , in ) = (i, j). Let X be another monomial in T1 . We shall say that X divides X if there exist monomials X(l), X(r) ∈ T1 such that X = X(l)X X(r), and write X | X if this is the case. Consider the set of monomials {X ∈ I(T1 ) : X | X}, and denote by J(X) the ideal in 1 T generated by these monomials. We deﬁne R(X) = T1 /J(X) and S(X) = R(X)/(X) = T1 /(J(X), X). Then the natural map π(X) : R(X) → S(X) is a small surjection in ap , and it has a 1-dimensional kernel which is generated by the monomial X. We write I(X) = I(S(X)) and S(X)n = S(X)/I(X)n for all n ≥ 1. Let us consider the set B(X) = {(i, j, l) : 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij , xij (l) | X}, and denote by vij (l) the image of xij (l) in S(X)2 for all (i, j, l) ∈ B(X). Then the set {vij (l) : (i, j, l) ∈ B(X)} is a natural k-basis for I(X)/I(X)2 . Assume that a morphism φ(X) : H → S(X) is given, and denote the composition of φ(X) with the natural morphism S(X) → S(X)2 by φ(X)2 : H → S(X)2 . This morphism can be written uniquely in the form φ(X)2 =

αij (l) ⊗ vij (l),

(i,j,l)∈B(X)

where αij (l) ∈ Ext1A (Mj , Mi ) for all (i, j, l) ∈ B(X).

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EIVIND ERIKSEN

Conversely, consider a family {αij (l) ∈ Ext1A (Mj , Mi ) : (i, j, l) ∈ B(X)} of extensions indexed by B(X), corresponding to a morphism φ(X)2 : H → S(X)2 given by φ(X)2 = αij (l) ⊗ vij (l). If there exists a lifting of φ(X)2 to a morphism φ(X) : H → S(X), we say that the matric Massey product α; X = αi0 ,i1 (l1 ), αi1 ,i2 (l2 ), . . . , αin−1 ,in (ln ) is deﬁned, and that φ(X) is a defining system for this matric Massey product. If this is the case, we denote the deformation induced by the deﬁning system φ(X) by MX ∈ DefM (S(X)), and by proposition 5.1, the obstruction for lifting MX to R(X) is an element o(π(X), MX ) ∈ (Ext2A (Mj , Mi ) ⊗k K(X)ij ) ∼ = Ext2A (Mj , Mi ), where K(X) = ker(π(X)) ∼ = kX. In general, this element depends upon the deformation MX , and therefore on the deﬁning system φ(X). We deﬁne the value of the matric Massey product to be α; X = αi0 ,i1 (l1 ), αi1 ,i2 (l2 ), . . . , αin−1 ,in (ln ) = o(π(X), MX ). Consequently, the value of the matric Massey product α; X will in general depend upon the chosen deﬁning system. Let us ﬁx the monomial X. Then the matric Massey product α → α; X is a not everywhere deﬁned k-linear map Ext1A (Mi1 , Mi0 ) ⊗k · · · ⊗k Ext1A (Min , Min−1 )

Ext2A (Min , Mi0 ).

In fact, this map is deﬁned for α if and only if the morphism φ(X)2 : H → S(X)2 corresponding to α can be lifted to a morphism φ(X) : H → S(X). Moreover, even when this map is deﬁned for α, it is not necessarily uniquely deﬁned: In general, its value α; X depends upon the chosen lifting φ(X), the deﬁning system. The matric Massey products α; X deﬁned above are called the immediately defined matric Massey products. We remark that if X is a monomial of degree n = 2, then the situation is much simpler: We have S(X) = S(X)2 , so the matric Massey product α; X is uniquely deﬁned for any family of extensions {αij (l) : (i, j, l) ∈ B(X)}. In fact, the matric Massey product is just the usual cup product in this case. Let us ﬁx a monomial X ∈ I(T1 ) of degree n ≥ 2. Then there exists a natural family of extensions indexed by B(X) given by αij (l) = xij (l)∗ , {xij (l)∗ ∈ Ext1A (Mj , Mi ) : (i, j, l) ∈ B(X)}. The matric Massey products of these extensions are the ones that we shall use for the construction of the hull H of DefM in the next section. We therefore introduce the notation x∗ ; X = xi0 i1 (l1 )∗ , xi1 i2 (l2 )∗ , . . . , xin−1 in (ln )∗ for their immediately deﬁned matric Massey products.

NONCOMMUTATIVE DEFORMATIONS OF MODULES

111

The matric Massey products are called matric because these products (and their deﬁning systems) can be described completely in terms of linear algebra and matrices. We shall end this section by giving such a description. Let {αij (l) ∈ Ext1A (Mj , Mi ) : (i, j, l) ∈ B(X)} be a family of extensions indexed by B(X), and consider the corresponding matric Massey product α; X = αi0 i1 (l1 ), αi1 i2 (l2 ), . . . , αin−1 in (ln ).

(4)

We assume that there exists a deﬁning system φ(X) : H → S(X) for this matric Massey product. Then φ(X) induces a deformation MX ∈ DefM (S(X)). We notice that the matric Massey product (4) only depends upon this deformation. By abuse of language, we shall therefore let the notion defining system refer to the deformation MX as well as the morphism φ(X) : H → S(X) which induces MX . We know that any deformation MX ∈ DefM (S(X)) can be described by a complex which lifts (F∗ , d∗ ) to S(X). Such a complex is given by diﬀerentials of the form dS(X) : (Lm+1,i ⊗k S(X)ij ) → (Lm,i ⊗k S(X)ij ). m We write v(X ) for the image of X in S(X) whenever X is a monomial in T1 , and deﬁne B(X) = {X ∈ I(T1 ) : X is a monomial such that X | X} ∪ {e1 , . . . , ep }. Then the set {v(X ) : X ∈ B(X)} is a natural k-basis for S(X), and B(X) contains {xij (l) : (i, j, l) ∈ B(X)} and {e1 , . . . , ep } as subsets. Let us write B(X)ij = B(X) ∩ S(X)ij for 1 ≤ i, j ≤ p. With this notation, the above diﬀerentials have the form dS(X) = m

1≤i≤p

dm,i ⊗ ei +

α(X )m ⊗ v(X ),

X ∈B(X)

where α(X ) ∈ Hom1A (L∗j , L∗i ) is a 1-cochain whenever X ∈ B(X )ij . S(X) Let dm be arbitrary maps between M-free modules over S(X) deﬁned by a family of 1-cochains {α(X ) : X ∈ B(X)} as above. These maps lifts the complex (F∗ , d∗ ) if α(ei ) = d∗i for 1 ≤ i ≤ p. Moreover, these maps are diﬀerentials if and only if the following condition holds: For all monomials Z ∈ B(X) and for all integers m ≥ 0, we have (5)

X ,X ∈B(X) X X =Z

α(X )m ◦ α(X )m+1 =

α(X )m+1 α(X )m = 0.

X ,X ∈B(X) X X =Z

In the ﬁrst sum, the symbol ◦ denotes composition of maps. We recall that each of the maps involved can be considered as right multiplication by a matrix. In the second summation, we identify the maps with such matrices, and re-write the composition of maps as multiplication of the corresponding matrices. Assume that these conditions hold. Then the family {α(X ) : X ∈ B(X)} of 1-cochains deﬁnes a lifting of complexes of (L∗ , d∗ ) to S(X) given by the diﬀerentials dS(X) as above,

112

EIVIND ERIKSEN

and this lifting corresponds to a deformation MX ∈ DefM (S(X)). The deformation MX is a deﬁning system for the matric Massey product (4) if and only if α(X ) is a 1-cocycle which represents αij (l) ∈ Ext1A (Mj , Mi ) whenever X = xij (l) for some (i, j, l) ∈ B(X). In this case, we shall refer to the family of 1-cochains {α(X ) : X ∈ B(X)} as a defining system for the matric Massey product (4). Finally, assume that the family of 1-cochains {α(X ) : X ∈ B(X)} is a deﬁning system of the matric Massey product (4). Then the value of this matric Massey product is given by

α; Xm =

(6)

α(X )m+1 α(X )m

X ,X ∈B(X) X X =X

for all m ≥ 0, where the multiplication denotes matrix multiplication of the corresponding matrices. Proposition 6.1. Let {αij (l) ∈ Ext1A (Mj , Mi ) : (i, j, l) ∈ B(X)} be a family of extensions. A defining system for the matric Massey product α; X = αi0 i1 (l1 ), . . . , αin−1 in (ln ) corresponds to a family {α(X ) ∈ Hom1A (L∗j , L∗i ) : 1 ≤ i, j ≤ p, X ∈ B(X)ij } of 1-cochains satisfying the following conditions: • α(ei ) = d∗i for 1 ≤ i ≤ p, • α(X ) is a 1-cocycle representing αij (l) whenever X = xij (l) for some (i, j, l) ∈ B(X), • For all Z ∈ B(X) and for all m ≥ 0, we have

α(X )m+1 α(X )m = 0.

X ,X ∈B(X) X X =Z

Moreover, given such a family of 1-cochains, the matric Massey product α; X is represented by the 2-cocyle given by α; Xm =

α(X )m+1 α(X )m

X ,X ∈B(X) X X =X

for all m ≥ 0. Hence we have described the immediately deﬁned matric Massey products and their deﬁning systems in terms of linear algebra and matrices, as we set out to do. We remark that the description given in proposition 6.1 is extremely useful for doing concrete calculations with matric Massey products, and even for implementing such computations on computers. It also justiﬁes the name matric.

NONCOMMUTATIVE DEFORMATIONS OF MODULES

113

7. Calculating hulls using matric Massey products We recall that k is an algebraically closed (commutative) ﬁeld, A is an associative k-algebra, and M = {M1 , . . . , Mp } is a ﬁnite family of left A-modules. We also assume that the family M satisfy the ﬁniteness condition (FC). In this section, we show how to calculate the hull H of the deformation functor DefM using matric Massey products. By theorem 5.2, there exists an obstruction morphism o : T2 → T1 in ˆ ap such that ˆ T2 k p is a hull for the deformation functor DefM . We shall write I = I(T1 ) and H = T1 ⊗ 2 by fij (l) = o(yij (l)) for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . Then fij (l) is a formal power series in Iij 1 construction. Let us deﬁne a ⊆ T to be the ideal generated by {fij (l) : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij }. Then a ⊆ I 2 , and we have ˆ T2 k p ∼ H = T1 ⊗ = T1 /a. We shall use the matric Massey products from section 6 to calculate the coeﬃcients of the power series fij (l). Clearly, this is suﬃcient to determine the hull H. Let us ﬁx an integer N ≥ 2 such that a ⊆ I N . This is always possible, since a ⊆ I 2 . So fij (l) ∈ I N for all fij (l), and we can write fij (l) in the form fij (l) =

alij (X) · X +

|X|=N

alij (X) · X

|X|>N

for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij , with alij (X) ∈ k for all monomials X ∈ I N . As usual, we use the notation |X| to denote the degree of the monomial X. Let 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij and let n ≥ N . Then we agree to write fij (l)n for the truncated power series n alij (X) · X. fij (l)n = |X|=N

Moreover, let an+1 = I n+1 + (f n ) for all n ≥ N , where (f n ) ⊆ T1 is the ideal generated by {fij (l)n : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij }, and let an = I n for 2 ≤ n ≤ N . We write Hn = H/I(H)n as usual, then Hn = T1 /an for all n ≥ 2, in accordance with the notation in the proof of theorem 5.2. Recall that H2 = T12 and that ξ2 ∈ DefM (H2 ) denotes the universal deformation with the property that the couple (H2 , ξ2 ) represents DefM restricted to ap (2). We have assumed that a ⊆ I N , and this means that there exists a lifting of ξ2 to HN = T1 /aN = T1N . Let us proceed to ﬁnd such a lifting MN ∈ DefM (HN ) explicitly. We choose to describe the deformation MN in terms of M-free resolutions. Let us deﬁne B(N − 1) to be the set of all monomials in T1 of degree at most N − 1. Then {X : X ∈ B(N − 1)} is a monomial basis of HN , and any M-free resolution of MN can be described by a family {α(X) : X ∈ B(N − 1)} of 1-cochains satisfying the following conditions: • α(ei ) = d∗i for 1 ≤ i ≤ p, • α(xij (l)) is a 1-cocycle representing xij (l)∗ for 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij , • For all Z ∈ B(N − 1) and for all m ≥ 0, we have X ,X ∈B(N −1) X X =Z

α(X )m+1 α(X )m = 0.

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EIVIND ERIKSEN

We know that a family of 1-cochains with the above properties exists, since we can ﬁnd a lifting MN of ξ2 to HN and this deformation must have some M-free resolution. So we choose one such family {α(X) : X ∈ B(N −1)} and ﬁx this choice. This means that we have ﬁxed a deformation MN ∈ DefM (HN ) with an M-free resolution given by the corresponding diﬀerentials. So (HN , MN ) is a pro-representing hull for DefM restricted to ap (N ). Lemma 7.1. Let π : R → S be any small surjection in ap , let φ : H → S be any morphism, and denote by Mφ ∈ DefM (S) the deformation induced by φ. Then we can lift φ to a morphism φ : T1 → R making the diagram φ

T1

R π

H

φ

S

commutative, and the obstruction o(π, Mφ ) for lifting Mφ to R is given by o(π, Mφ ) =

yij (l)∗ ⊗ φ(fij (l)).

i,j,l

Proof. By construction and functoriality, the obstruction o(π, Mφ ) is given as the restriction of the composition φ ◦ o to the k-linear subspace (Ext2A (Mj , Mi )∗ ) ⊆ T2 . Since {yij (l)} is a k-linear basis for this subspace, we get the desired expression for the obstruction. Let us deﬁne bN ⊆ T1 to be the ideal bN = IaN + aN I = I N +1 , and consider the natural map rN : RN → HN , where RN = T1 /bN = T1N +1 . By construction, rN is a small surjection in ap , and the natural surjection φN : T1 → RN makes the diagram o

T2

T1

φN

RN rN

H

φN

HN

commutative. Let B (N ) be the set of all monomials in T1 of degree N . Since ker(rN ) = I N /I N +1 , we see that {X : X ∈ B (N )} is a monomial basis for ker(rN ). Moreover, let B (N ) = B (N ) ∪ B(N − 1). Then clearly {X : X ∈ B (N )} is a monomial basis for RN . Since rN is a small surjection, there is an obstruction o(rN , MN ) for lifting MN to RN , and we see from lemma 7.1 that this obstruction can be expressed as o(rN , MN ) = yij (l)∗ ⊗ φN (fij (l)) i,j,l

=

i,j,l

=

yij (l)∗ ⊗ f ij (l)

yij (l)∗ ⊗ (alij (X) · X),

i,j,l X∈B (N )

where f ij (l) and X denote the images of fij (l) and X in RN .

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115

We say that the family D(N ) = {α(X) : X ∈ B(N − 1)} of 1-cochains is a defining system for the matric Massey products of order N , x∗ ; X for X ∈ B (N ). Let X ∈ B (N ) be any monomial of type (i, j). We deﬁne the matric Massey product x∗ ; X to be the coeﬃcient of X in the obstruction o(rN , MN ) above. Then we immediately see that this matric Massey product has value ∗

x ; X =

rij

alij (X) · yij (l)∗ .

l=1

In other words, the coeﬃcient of X in the power series fij (l) is given by the matric Massey product x∗ ; X above as alij (X) = yij (l)(x∗ ; X) for 1 ≤ l ≤ rij . We notice that the matric Massey products of order N deﬁned above are immediately defined. In other words, they can be expressed in terms of the matric Massey products of section 6. In fact, the deﬁning system D(N ) induces a deﬁning system {α(X ) : X | X, X = X} in the sense of section 6, and the value of the corresponding matric Massey product x∗ ; X is exactly the coeﬃcient of X in the obstruction o(rN , MN ). On the other hand, we can calculate the obstruction o(rN , MN ) using the deﬁning system D(N ), and therefore also the coeﬃcient of X in this obstruction for each X ∈ B (N ). A straight-forward calculation show that this coeﬃcient is given by the 2-cocycle y(X) deﬁned by α(X )m+1 α(X )m y(X)m = X ,X ∈B(N −1) X X =X

for all m ≥ 0. This means that the matric Massey product x∗ ; X is represented by y(X), so we can easily calculate all matric Massey products of order N using the deﬁning system D(N ). This determines the truncated power series fij (l)N , since we have fij (l)N =

alij (X) · X =

X∈B (N )

yij (l)(x∗ ; X) · X

X∈B (N )

for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . Let hN : HN +1 → HN be the natural map. Then ker(hN ) = I N /aN +1 , so we can ﬁnd a subset B(N ) ⊆ B (N ) of monomials in T1 of degree N such that {X : X ∈ B(N )} is a monomial basis for ker(hN ). Let B(N ) = B(N ) ∪ B(N − 1), then clearly {X : X ∈ B(N )} is a monomial basis for HN +1 . So for each monomial X ∈ T1 with |X| ≤ N , we have a unique relation in HN +1 of the form X=

X ∈B(N )

β(X, X ) X ,

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with β(X, X ) ∈ k for all X ∈ B(N ). Since we have o(hN , MN ) = 0, we deduce that

x∗ ; X β(X, X ) = 0

|X|=N

for all X ∈ B(N ). Notice that β(X, X ) = 0 if the monomials X and X do not have the same type. Therefore, it makes sense to consider the 1-cocycle

β(X, X ) y(X),

|X|=N

and by the relation above, this is a 1-coboundary. It follows that we can ﬁnd a 1-cochain α(X ) such that d α(X ) = − β(X, X ) y(X), |X|=N

and we ﬁx such a choice. Consider the family {α(X) : X ∈ B(N )}. This deﬁnes an M-free complex over HN +1 if and only if we have

β(X, Z)

|X|=N

α(X ) α(X ) = 0

X ,X ∈B(N ) X X =X

for all Z ∈ B(N ). By the deﬁnition of α(X ) when X ∈ B(N ), this condition holds, and we denote by MN +1 ∈ DefM (HN +1 ) the deformation with the complex deﬁned by {α(X) : X ∈ B(N )} as M-free resolution. It is clear from the construction that MN +1 is a lifting of MN , so (HN +1 , MN +1 ) is a pro-representing hull for DefM restricted to ap (N + 1). Let bN +1 ⊆ T1 be the ideal bN +1 = IaN +1 + aN +1 I = I N +2 + I(f N )+(f N )I, and consider the natural map rN +1 : RN +1 → HN +1 , where RN +1 = T1 /bN +1 . By construction, rN +1 is a small surjection in ap , and it is clear that the natural morphism φN +1 : T1 → RN +1 makes the diagram T2

o

T1

φN +1

RN +1 rN +1

H

φN +1

φN

HN +1 hN

HN commutative. We see that ker(rN +1 ) = aN +1 /bN +1 , which we can re-write in the following way: ker(rN +1 ) = (I N +1 + (f N ))/(I N +2 + I(f N ) + (f N )I) = (f N )/(I(f N ) + (f N )I) ⊕ I N +1 /(I N +2 + I(f N ) + (f N )I)

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Let us write c(N + 1) = I N +1 /(I N +2 + I(f N ) + (f N )I). Then c(N + 1) ⊆ ker(rN +1 ) is an ideal, and we can clearly ﬁnd a set B (N + 1) of monomials in T1 of degree N + 1 such that {X : X ∈ B (N + 1)} is a monomial basis for cN +1 . Let us choose B (N + 1) such that for every X ∈ B (N + 1), there is a monomial X ∈ B(N ) such that X | X, this is clearly possible. We let B (N + 1) = B (N + 1) ∪ B(N ), then {X : X ∈ B (N + 1)} ∪ {fij (l)N : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij } is a basis for RN +1 . So for each monomial X ∈ T1 with |X| ≤ N + 1, we have a unique relation in RN +1 of the form X=

β (X, X )X +

β (X, i, j, l)f ij (l)N ,

i,j,l

X ∈B (N +1)

with β (X, X ), β (X, i, j, l) ∈ k for all X ∈ B (N + 1), 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . Since rN +1 is a small surjection, there is an obstruction o(rN +1 , MN +1 ) for lifting MN +1 to RN +1 , and we see from lemma 7.1 that this obstruction can be expressed as o(rN +1 , MN +1 ) =

yij (l)∗ ⊗ φN +1 (fij (l))

i,j,l

=

yij (l)∗ ⊗ f ij (l)

i,j,l

=

yij (l)∗ ⊗ (f ij (l)N +

i,j,l

alij (X) · X),

X∈B (N +1)

where f ij (l), f ij (l)N and X denote the images of fij (l), fij (l)N and X in RN +1 . We say that the family D(N + 1) = {α(X) : X ∈ B(N )} is a defining system for the matric Massey products of order N + 1, x∗ ; X for X ∈ B (N + 1) Let X ∈ B (N + 1) be any monomial of type (i, j). We deﬁne the matric Massey product x∗ ; X to be the coeﬃcient of X in the obstruction o(rN +1 , MN +1 ) above. Then we immediately see that this matric Massey product has value ∗

x ; X =

rij

alij (X) · yij (l)∗ .

l=1

In other words, the coeﬃcient of X in the power series fij (l) is given by the matric Massey product x∗ ; X above as alij (X) = yij (l)(x∗ ; X) for 1 ≤ l ≤ rij . On the other hand, we can calculate the obstruction o(rN +1 , MN +1 ) using the deﬁning system D(N + 1), and therefore also the coeﬃcient of X in this obstruction for each

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X ∈ B (N + 1). A straight-forward calculation show that this coeﬃcient is given by the 2-cocycle y(X) deﬁned by y(X)m =

β (Z, X)

|Z|≤N +1

α(X )m+1 α(X )m

X ,X ∈B(N ) X X =Z

for all m ≥ 0. This means that the matric Massey product x∗ ; X is represented by y(X), so we can easily calculate all matric Massey products of order N + 1 using the deﬁning system D(N + 1). By the construction in the proof of theorem 5.2, we have that HN +2 is the quotient of RN +1 by the ideal generated by the obstruction o(rN +1 , MN +1 ). On the other hand, we know that HN +2 = T1 /(I N +2 + (f N +1 ). This implies that for all monomials X ∈ B (N + 1) of degree N + 1, the coeﬃcient alij (X) = 0 for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . In other words, the truncated power series fij (l)N +1 is determined by the matric Massey products of order N + 1 above, since we have fij (l)N +1 = fij (l)N +

alij (X) · X

X∈B (N +1)

= fij (l)N +

yij (l)(x∗ ; X) · X

X∈B (N +1)

for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . Let hN +1 : HN +2 → HN +1 be the natural map, and consider its kernel. By deﬁnition, we have ker(hN +1 ) = aN +1 /aN +2 = ((f N ) + I N +1 )/((f N +1 ) + I N +2 ), so we can clearly ﬁnd a subset B(N + 1) ⊆ B (N + 1) of monomials of degree N + 1 such that {X : X ∈ B(N + 1)} ∪ {f ij (l)N : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij } is a basis for ker(hN +1 ). Let B(N + 1) = B(N + 1) ∪ B(N ), then clearly {X : X ∈ B(N + 1)} ∪ {f ij (l)N : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij } is a monomial basis for HN +2 . So for each monomial X ∈ T1 with |X| ≤ N + 1, we have a unique relation in HN +2 of the form X=

β(X, X ) X +

X ∈B(N +1)

β(X, i, j, l) fij (l)N ,

i,j,l

with β(X, X ), β(X, i, j, l) ∈ k for all X ∈ B(N + 1), 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . Since we have o(hN +1 , MN +1 ) = 0, we deduce that |X|≤N +1

x∗ ; X β(X, X ) = 0

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for all X ∈ B(N + 1). Notice that β(X, X ) = 0 if the monomials X and X do not have the same type. Therefore, it makes sense to consider the 1-cocycle

β(X, X ) y(X),

|X|≤N +1

and by the relation above, this is a 1-coboundary. It follows that we can ﬁnd a 1-cochain α(X ) such that dα(X ) = − β(X, X ) y(X), |X|≤N +1

and we ﬁx such a choice. Consider the family {α(X) : X ∈ B(N + 1)}. This deﬁnes an M-free complex over HN +2 if and only if we have |X|≤N +1

β(X, Z)

α(X ) α(X ) = 0

X ,X ∈B(N +1) X X =X

for all Z ∈ B(N + 1). By the deﬁnition of α(X ) when X ∈ B(N + 1), this condition holds, and we denote by MN +2 ∈ DefM (HN +2 ) the deformation with the complex deﬁned by {α(X) : X ∈ B(N + 1)} as M-free resolution. It is clear from the construction that MN +2 is a lifting of MN +1 , so (HN +2 , MN +2 ) is a pro-representing hull for DefM restricted to ap (N + 2). It is clear that we can continue in this way. For every k ≥ 1, we can calculate the coeﬃcients in the truncated power series fij (l)N +k , and therefore ﬁnd HN +k+1 . At the same time, we ﬁnd the deﬁning systems {α(X) : X ∈ B(N + k)} necessary to calculate the matric Massey products of order N + k + 1, and these deﬁning systems completely determine the deformation MN +k+1 . We have described how to do this in the case k = 1, and the general case is similar. We conclude that the method that we have described above can be used to calculate the pro-representing hull (Hn , Mn ) for the deformation functor DefM restricted to ap (n) for any n ≥ N . We can therefore, in principle, ﬁnd the hull H = lim Hn ←

of DefM , and also the corresponding versal family deﬁned over H, ξ = M = lim Mn . ←

It follows that the pro-representing hull (H, ξ) of the deformation functor DefM can be calculated using matric Massey products. 8. An example Let k be an algebraically closed ﬁeld of characteristic 0, and let A = A2 (k) be the second Weyl algebra over k. We shall think of A as the ring of diﬀerential operators in the plane deﬁned over k with coordinates x and y. Thus, we can write A = k[x, y]∂x, ∂y, where

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∂x = ∂/∂x and ∂y = ∂/∂y. In other words, A is the k-algebra generated by x, y, ∂x, ∂y with relations [∂x, x] = [∂y, y] = 1. Let us consider the family of left A-modules M = {M1 , M2 , M3 , M4 }, where Mi = A/Ii for 1 ≤ i ≤ 4 and Ii ⊆ A are left ideals given by I1 = A(∂x, ∂y)

I2 = A(∂x, y)

I3 = A(x, ∂y)

I4 = A(x, y)

We immediately notice that the left A-modules in the family M have the following free resolutions:

∂x ∂y ∂y −∂x 2 0 ←M1 ← A ←−−−− A ←−−−−−−− A ← 0 ∂x y y −∂x 0 ←M2 ← A ←−−−− A2 ←−−−−−− A ← 0

x ∂y ∂y −x 2 0 ←M3 ← A ←−−−− A ←−−−−−− A ← 0

x y y −x 2 0 ←M4 ← A ←−−− A ←−−−−− A ← 0

We consider the elements of the free A-modules An as row vectors, and the maps in the free resolutions above as right multiplication of these row vectors by the given matrices. Notice that for 1 ≤ i ≤ 4, the free A-module Lm,i in the free resolution of Mi does not depend upon i. We shall therefore write Lm = Lm,i for all m ≥ 0, 1 ≤ i ≤ 4. It is known that M is a family of simple holonomic left A-modules, so this family satisfy the ﬁniteness condition (FC). Therefore, there exists a pro-representing hull (H, ξ) for the deformation functor DefM : a4 → Sets by theorem 5.2. We shall use the methods from section 7 to construct this hull explicitly. Let us start by calculating ExtnA (Mi , Mj ) for n = 1, 2, 1 ≤ i, j ≤ 4. We need both the dimensions and k-linear bases for these vector spaces, where each basis vector is represented by a cocycle in the corresponding Yoneda complex. The calculations are straight-forward, so we only state the results here: 1 if i = 1 or i = 4 and j = 2 or j = 3, or 1 dimk ExtA (Mi , Mj ) = if i = 2 or i = 3 and j = 1 or j = 4, 0 otherwise dimk Ext2A (Mi , Mj )

1 if (i, j) = (1, 4), (2, 3), (3, 2), (4, 1), = 0 otherwise

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We denote the basis vectors of Ext1A (Mj , Mi ) by x∗ij since there is at most one for each pair of indices (i, j). From the dimensions listed above, we see that we have the following basis vectors: x∗12 , x∗13 , x∗21 , x∗24 , x∗31 , x∗34 , x∗42 , x∗43 We choose a Yoneda representative for each vector x∗ij in this list, and we denote this representative by α(xij ). From the free resolutions above, we see that we can write each of these representatives in the form α(X) = {α(X)0 , α(X)1 }, where α(X)0 : L1 → L0 is right multiplication by a matrix ( ab ) with entries a, b ∈ A, and α(X)1 : L2 → L1 is right multiplication by a matrix ( c d ) with entries c, d ∈ A for each monomial X = xij . We ﬁnd the following representatives: α(x12 ) = α(x21 ) = α(x34 ) = α(x43 ) = {( 01 ) , ( 1 0 )} α(x13 ) = α(x31 ) = α(x24 ) = α(x42 ) = {( 10 ) , ( 0 −1 )} ∗ Similarly, we denote the basis vectors of Ext2A (Mj , Mi ) by yij since there is at most one for each pair of indices (i, j). From the dimensions listed above, we see that we have the following basis vectors: ∗ ∗ ∗ ∗ y14 , y23 , y32 , y41 ∗ We choose a Yoneda representative for each vector yij in this list, and we denote this representative α(yij ). From the free resolutions above, we see that we can write each of these representatives in the form

α(Y ) = {α(Y )0 }, where α(Y )0 : L2 → L0 is given by right multiplication of an element a ∈ A for each monomial Y = yij . We ﬁnd the following representatives: α(y14 ) = α(y23 ) = α(x32 ) = α(x41 ) = {( 1 )} This completes the calculations of ExtnA (Mi , Mj ) for n = 1, 2 and 1 ≤ i, j ≤ 4. We know that these calculations determine the hull at the tangent level, (H2 , ξ2 ). The next step is to ﬁnd the the hull H and the versal family ξ, and we shall employ the notations and methods of section 7 to accomplish this. Let N = 2, we know that this choice is always possible. As usual, we let T1 be the formal matrix algebra generated by the monomials xij in the above list, and let I = I(T1 ) be its radical. Furthermore, denote 2 n for (i, j) = (1, 4), (2, 3), (3, 2), (4, 1), and by fij the corresponding by fij = o(yij ) ∈ Iij truncated power series for each n ≥ N . First, we have to ﬁnd a deﬁning system {α(X) : |X| < 2} for the matric Massey products x∗ ; X when X is any monomial of degree 2 in T1 . This is easily done: The 1-cocycle α(ei ) is the free resolution of Mi for 1 ≤ i ≤ 4, and the 1-cocycle α(X) was chosen above for each monomial X = xij of degree 1.

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Let us calculate the matric Massey products of order 2: Using the deﬁning system given above, we ﬁnd that the cocycles y(X) representing the matric Massey products x∗ ; X are given by −1 if X = x12 x24 , x21 x13 , x34 x42 , x43 x31 , y(X)0 = 1 if X = x13 x34 , x24 x43 , x31 x12 , x42 x21 , 0 otherwise for all monomials X of degree 2 in T1 . This means that the corresponding matric Massey products are given by x12 , x24 = −y14

x13 , x34 = y14

x21 , x13 = −y23

x24 , x43 = y23

x31 , x12 = y32

x34 , x42 = −y32

x42 , x21 = y41

x43 , x31 = −y41 ,

and all other matric Massey products of order 2 are zero. This translates to the following 2 : truncated power series fij 2 = x13 x34 − x12 x24 f14 2 f23 = x24 x43 − x21 x13 2 f32 = x31 x12 − x34 x42 2 f41 = x42 x21 − x43 x31 2 2 2 2 By the general theory, we therefore have H3 = T1 /(f14 , f23 , f32 , f41 ) + I 3 . We know that we can ﬁnd a lifting ξ3 of ξ2 to H3 , and that (H3 , ξ3 ) is a pro-representing hull of DefM restricted to a4 (3). In order to ﬁnd ξ3 , we let B(2) = {X : |X| = 2} \ {x13 x34 , x24 x43 , x31 x12 , x42 x21 }. We also let B(2) = B(2) ∪ B(1), where B(1) = {X : |X| ≤ 1}. Then {X : X ∈ B(2)} is a monomial basis for H3 . We observe that if we choose α(X) = 0 for all X ∈ B(2), the family {α(X) : X ∈ B(2)} deﬁnes an M-free complex over H3 . In other words, this family completely deﬁnes the deformation ξ3 ∈ DefM (H3 ) lifting ξ2 . Clearly, we could continue in this way. But after the last computations, it is tempting 2 for (i, j) = (1, 4), (2, 3), (3, 2), (4, 1). Let us check if this is the case: to think that fij = fij 1 2 2 2 2 , f23 , f32 , f41 ), and choose a monomial basis B of T containing B(2). We put T = T /(f14 Furthermore, we let α(X) be as before when X ∈ B(2) and let α(X) = 0 for all monomials X ∈ B of degree at least 3. This choice corresponds to maps dT0 , dT1 of M-free modules over T , and a computation shows that 2 2 2 2 + f23 + f32 + f41 )) = 0. dT0 ◦ dT1 = (1 ⊗ (f14

So the family {α(X) : X ∈ B} deﬁnes an M-free complex over T , and therefore a deformation ξ ∈ DefM (T ) lifting ξ3 . This proves that H = T , or in other words, that H = T1 /(x13 x34 − x12 x24 , x24 x43 − x21 x13 , x31 x12 − x34 x42 , x42 x21 − x43 x31 ) 2 for all i, j. Moreover, the family is a pro-representing hull of DefM . In particular, fij = fij {α(X) : X ∈ B} deﬁnes the versal family ξ ∈ DefM (H).

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123

Appendix A. Yoneda and Hochschild representations Let k be an algebraically closed (commutative) ﬁeld, let A be an associative k-algebra, and let M, N be left A-modules. In this appendix, we recall several diﬀerent descriptions of the k-vector space ExtnA (M, N ) for n ≥ 0. In particular, we show how to realize this cohomology group using the Yoneda and Hochschild complexes. A.1. The Yoneda representation. Fix free resolutions (L∗ , d∗ ) of M and (L∗ , d∗ ) of N . We shall write di : Li+1 → Li and di : Li+1 → Li for the diﬀerentials, and denote the augmentation morphisms by ρ : L0 → M and ρ : L0 → N . For all integers n ≥ 0, the cohomology group ExtnA (M, N ) is deﬁned to be the n’th cohomology group of the complex HomA (L∗ , N ), ExtnA (M, N ) = H n (HomA (L∗ , N )). Notice that in general, this Abelian group does not have a left A-module structure, but only a left C(A)-module structure, where C(A) is the centre of A. In particular, if A is commutative, then ExtnA (M, N ) has the structure of an A-module, and if A is a k-algebra, then ExtnA (M, N ) has the structure of a k-vector space. We denote by Hom∗ (L∗ , L∗ ) the Yoneda complex given by the given free resolutions. This complex is deﬁned in the following way: For each integer n ≥ 0, let Homn (L∗ , L∗ ) be the left A-module Homn (L∗ , L∗ ) = i HomA (Li+n , Li ). Moreover, let the diﬀerential dn : Homn (L∗ , L∗ ) → Homn+1 (L∗ , L∗ ) for n ≥ 0 be the A-linear map given by the formula dn (φ)i = φi dn+i + (−1)n+1 di φi+1 for all i ≥ 0, where we write φ = (φi ) with φi ∈ HomA (Li+n , Li ) for all i ≥ 0. It is easy to check that this map is a well-deﬁned diﬀerential, so the Yoneda complex is a complex of Abelian groups. We shall write H n (Hom(L∗ , L∗ )) for the cohomology groups of the Yoneda complex. Since the diﬀerential d = dn is left C(A)-linear, these cohomology groups have a natural structure as left C(A)-modules. Lemma A.1. For all integers n ≥ 0, there is a canonical isomorphism of left C(A)-modules H n (Hom(L∗ , L∗ )) ∼ = ExtnA (M, N ). Proof. There is a natural map fn : Homn (L∗ , L∗ ) → HomA (Ln , N ), given by f (φ) = ρ φ0 , where φ = (φi ) ∈ Homn (L∗ , L∗ ). It is easy to see that these maps are compatible with the diﬀerentials, and a small calculation show that fn induces an isomorphism on cohomology H n (Hom(L∗ , L∗ )) → ExtnA (M, N ) for all integers n ≥ 0. A.2. Definition of Hochschild cohomology. Let Q be an A-A bimodule. We deﬁne the Hochschild complex of A with values in Q in the following way: Let HCn (A, Q) = Homk (⊗nk A, Q) for all n ≥ 0. So any ψ ∈ HCn (A, Q) corresponds to a k-multilinear map from n copies of A into Q, and we shall therefore write ψ(a1 , . . . , an ) in place of ψ(a1 ⊗ · · · ⊗

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EIVIND ERIKSEN

an ) for ψ ∈ HCn (A, Q), a1 , . . . , an ∈ A. Moreover, let dn : HCn (A, Q) → HCn+1 (A, Q) for n ≥ 0 be the k-linear map given by the formula (7)

dn (ψ)(a0 , . . . , an ) = a0 ψ(a1 , . . . , an ) +

n

(−1)i ψ(a0 , . . . , ai−1 ai , . . . , an )

i=1

+(−1)n+1 ψ(a0 , . . . , an−1 )an for all ψ ∈ HCn (A, Q), a0 , . . . , an ∈ A. Lemma A.2. HC∗ (A, Q) is a complex of k-vector spaces. Proof. Let ψ ∈ HCn (A, Q). Then ψ = dn (ψ) is a sum of n + 1 summands, and we denote these by ψ0 , . . . , ψn , in the order they appear in formula 7. We let ψ = dn+1 ψ = dn+1 dn ψ. for Each dn+1 ψi for 0 ≤ i ≤ n is a sum of n + 2 summands, and we denote these by ψij 0 ≤ j ≤ n + 1 in the order they appear in formula 7. A straight-forward calculation shows + ψj,i+1 = 0 for all indices i, j with 0 ≤ j ≤ n + 2, j ≤ i ≤ n + 1. Since that we have ψi,j ψ = ψij , it follows that ψ = 0 in HCn+2 (A, Q). Consequently, HC∗ (A, Q) is a complex of k-vector spaces. We deﬁne the Hochschild cohomology of A with values in Q to be the cohomology of the Hochschild complex HC∗ (A, Q), so we have HHn (A, Q) = H n (HC∗ (A, Q)) = ker(dn )/ Im(dn−1 ) for all n ≥ 0. In particular, the cohomology groups HHn (A, Q) have a natural structure as k-vector spaces. Let ψ ∈ HC1 (A, Q), then ψ is a 1-cocycle if and only if ψ(ab) = aψ(b) + ψ(a)b for all a, b ∈ A. So we have ker(d1 ) = Derk (A, Q). We say that a derivation ψ ∈ Derk (A, Q) is trivial if there is an element q ∈ Q such that ψ is of the form ψ(a) = aq − qa for all a ∈ A. Clearly, the set of trivial derivations is the image Im(d0 ). So HH1 (A, Q) ∼ = Derk (A, Q)/T where T is the trivial derivations of A into Q. A.3. The Hochschild representation. We remark that Q = Homk (M, N ) is an A-A bimodule in a natural way: For any a ∈ A, let La : M → M denote left multiplication on M by a, and La : N → N left multiplication on N by a. The bimodule structure is given by aφ = La φ, φa = φLa for a ∈ A, φ ∈ Homk (M, N ). We shall consider the Hochschild cohomology of A with values in Q = Homk (M, N ). By deﬁnition, we have that HH0 (A, Q) = HomA (M, N ) when Q = Homk (M, N ). So we have a natural isomorphism of k-vector spaces Ext0A (M, N ) ∼ = HH0 (A, Q). Notice that since n k ⊆ C(A), ExtA (M, N ) has a natural k-vector space structure for all n ≥ 0. It is possible to extend the above isomorphism to the higher cohomology groups: Proposition A.3. For all integers n ≥ 0, there is an isomorphism of k-vector spaces σn : ExtnA (M, N ) → HHn (A, Homk (M, N )). Proof. From Weibel [9], lemma 9.1.9, there is an isomorphism of k-vector spaces between HHn (A, Homk (M, N )) and ExtnA/k (M, N ) for n ≥ 0. But since k is a commutative ﬁeld, there is a canonical isomorphism between ExtnA/k (M, N ) and ExtnA (M, N ), see theorem 8.7.10 in Weibel [9].

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We shall give an explicit identiﬁcation of k-vector spaces between Ext1A (M, N ) and HH1 (A, Homk (M, N )): Let (L∗ , d∗ ) be a free resolution of M , with augmentation morphism ρ : L0 → M , and let τ : M → L0 be a k-linear section of ρ. For any 1-cocycle φ ∈ HomA (L1 , N ), let ψ = ψ(φ) ∈ Derk (A, Homk (M, N )) be the following derivation: For any a ∈ A, m ∈ M , let x = x(a, m) ∈ L1 be such that d0 (x) = aτ (m) − τ (am). Notice that such an x exists, and is uniquely deﬁned modulo the image Im d1 . We deﬁne ψ by the equation ψ(a)(m) = φ(x) with x = x(a, m). Since φ is a cocycle, ψ is a well-deﬁned homomorphism in Homk (A, Homk (M, N )), and a straight-forward calculation shows that ψ is a derivation. Lemma A.4. Assume that Ext1A (M, N ) is a finite dimensional k-vector space. Then the assignment φ → ψ(φ) defined in the above paragraph induces an isomorphism σ1 : Ext1A (M, N ) → HH1 (A, Homk (M, N )). Proof. Assume that φ is a co-boundary, so φ = d0 (φ ), where φ ∈ HomA (L0 , N ). Then ψ = d0 (φ ), where ψ = φ τ ∈ Homk (M, N ), so φ is a trivial derivation. Consequently, the assignment induces a well-deﬁned map of k-linear spaces. This map is furthermore injective: Assume that ψ is a trivial derivation, so ψ = d0 (ψ ), where ψ ∈ Homk (M, N ). Then, we can construct an A-linear map φ ∈ HomA (L0 , N ) in the following way: Choose a basis for L0 , and for each basis vector y ∈ L0 , choose y ∈ L1 such that d0 (y ) = y − ψ ρ(y). Then we deﬁne φ (y) = ψ ρ(y) + φ(y ) for each basis vector y ∈ L0 . We obtain a morphism φ ∈ HomA (L0 , N ) by A-linear extension, and d0 (φ ) = φ, so φ is a co-boundary. To show that σ1 is an isomorphism as well, it is enough to notice that dimk Ext1A (M, N ) = dimk HH1 (A, Homk (M, N )) by proposition A.3, since Ext1A (M, N ) has ﬁnite k-dimension. The identiﬁcation σn : ExtnA (M, N ) → HHn (A, Homk (M, N )) for n ≥ 2 can be constructed in a similar way. References ´ ements de math´ [1] N. Bourbaki, Alg`ebre commutative, El´ ematique, Masson, 1985. [2] Olav Arnfinn Laudal, Formal moduli of algebraic structures, Lecture notes in mathematics, no. 754, Springer-Verlag, 1979. [3] ——, A generalized burnside theorem, Preprint Series no. 42, University of Olso, 1995. [4] ——, Noncommutative deformations of modules, Preprint Series no. 2, University of Oslo, 1995. [5] ——, Noncommutative algebraic geometry, Preprint Series no. 28, University of Olso, 1996. [6] ——, Noncommutative algebraic geometry II, Preprint Series no. 12, University of Olso, 1998. [7] ——, Noncommutative algebraic geometry, Preprint Series no. 115, Max Planck Institute of Mathematics, 2000. [8] ——, Noncommutative deformations of modules, Homology, Homotopy and Applications 4 (2002), no. 2, 357–396. [9] Charles A. Weibel, An introduction to homological algebra, Cambridge studies in advanced mathematics, no. 38, Cambridge University Press, 1994. Institute of Mathematics, University of Warwick, Coventry CV4 7AL, UK E-mail address: [email protected]

SYMMETRIC FUNCTIONS, NONCOMMUTATIVE SYMMETRIC FUNCTIONS AND QUASISYMMETRIC FUNCTIONS II by MICHIEL HAZEWINKEL CWI, POBox 94079, 1090GB Amsterdam, The Netherlands

Abstract. Like its precursor this paper is concerned with the Hopf algebra of noncommutative symmetric functions and its graded dual, the Hopf algebra of quasisymmetric functions. It complements and extends the previous paper but is also selfcontained. Here we concentrate on explicit descriptions (constructions) of a basis of the Lie algebra of primitives of NSymm and an explicit free polynomial basis of QSymm. As before everything is done over the integers. As applications the matter of the existence of suitable analogues of Frobenius and Verschiebung morphisms is discussed. MSCS: 16W30, 05E05, 05E10, 20C30, 14L05 Key words and key phrases: symmetric function, quasisymmetric function, noncommutative symmetric function, Hopf algebra, divided power sequence, endomorphism of Hopf algebras, automorphism of Hopf algebras, Frobenius operation, Verschiebung operation, Adams operator, power sum, Newton primitive, Solomon descent algebra, cofree coalgebra, free algebra, dual Hopf algebra, lambda-ring, Leibniz Hopf algebra, Lie Hopf algebra, Lie polynomial, formal group, primitive of a Hopf algebra shuﬄe algebra, overlapping shuﬄe algebra.

1. Introduction As said before, [24], the symmetric functions are an exceedingly fascinating object of study; they are best studied from the Hopf algebraic point of view (in my opinion), although they carry quite a good deal more important structures, indeed so much that whole books do not suﬃce, but see [26, 27, 31, 33, 34]. The ﬁrst of the two generalizations to be discussed is the Hopf algebra, NSymm, of noncommutative symmetric functions (over the integers). As an algebra, more precisely a ring, this is simply the free associative ring over the integers, Z, in countably many indeterminates N Symm = ZZ1 , Z2 , . . .

(1.1)

and the coalgebra structure is given by the comultiplication determined by µ : Zn →

Zi ⊗ Zj ,

where

Z0 = 1

(1.2)

i+j=n

and i and j are in N ∪ {0} = {0, 1, 2, · · · }. The augmentation is given by ε(Zn ) = 0,

n = 1, 2, 3, . . .

(1.3)

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127

(and, of course ε(Z0 ) = ε(1) = 1). The Hopf algebra NSymm is a noncommutative covering generalization of the Hopf algebra of symmetric functions, Symm = Z[z1 , z2 , . . .]

(1.4)

where the zn are seen as either the elementary symmetric functions en or the complete symetric functions hn . The interpretation of the zn as the hn seems to work out somewhat nicer, for instance in obtaining the standard inner product autoduality of Symm in terms of the natural duality between NSymm and QSymm, the Hopf algebra of quasisymmetric functions, see [24], section 6. QSymm will be described and discussed later in this paper. The projection is given by N Symm −→ Symm,

Zn → Zn

(1.5)

and is a morphism of Hopf algebras. The systematic investigation of NSymm as a noncommutative generalization of Symm was started in [14] and continued in a whole slew of subsequent papers, e.g. [7, 8, 9, 20, 21, 22, 23, 25, 28, 29, 30, 32, 46]. It is amazing how much of the theory of Symm has natural noncommutative analogues. This includes Newton primitives, Schur functions, representation theoretic interpretations, determinental formulas now involving the quasideterminants of Gel’fand - Retakh, [12, 13]), Capelli and Sylvester identities, and much more. And, not rarely, the noncommutative versions are more elegant than their commutative counterparts. Note, however, that in most of these papers the noncommutative symmetric functions are studied over a ﬁxed ﬁeld K of characteristic zero and not over the integers (or a ﬁeld of positive characteristic). This makes quite a diﬀerence, see section 3 below. The papers [19, 20, 21, 22, 23] focuss on the case over the integers, as does the present paper. It should be stressed that NSymm attracts a lot of attention not only as a natural generalization of Symm. It turns up spontaneously. For instance in terms of representations of the Hecke algebras at zero, [8, 24, 30, 46] and as the direct sum of the Solomon descent algebras of the symmetric groups, [1, 10, 14, 35, 43, 44] and [39], Ch. 9. Moreover there are e.g. applications to noncommutative continued fractions, Pad´e approximants, and a variety of interrelations with quantum groups and quantum enveloping algebras, [2, 14, 29, 37]. Further, the duals, the quasisymmetric functions, ﬁrst turned up (under that name) in the theory of plane partitions and counting permutations with given descent sets, [15, 16, 45]. Actually, QSymm, precisely as the graded dual of NSymm, goes back at least to 1972 in the theory of noncommutative formal groups, [5]. See [20] for an outline of the role played by QSymm in that context. An application of NSymm to chromatic polynomials is in [11]. Given a Hopf algebra H, with multiplication m and comultiplication µ, a primitive in H is an element P of H such that µ(P ) = 1 ⊗ P + P ⊗ 1

(1.6)

The primitives of a Hopf algebra form a Lie algebra under the commutator product [P1 , P2 ] = P1 P2 − P2 P1

(1.7)

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MICHIEL HAZEWINKEL

which is denoted Prim(H). For any Hopf algebra there is strong interest ina description of its Lie algebra of primitives. For instance because of the Milnor - Moore theorem, [36], that says that a graded connected cocommutative Hopf algebra over a ﬁeld of characteristic zero is isomorphic to to the universal enveloping algebra of its Lie algebra of primitives. Also, far from unrelated, let Q(H) = I(H)/I(H)2 be the module of indecomposables of a graded Hopf algebra H. Here I(H) is the augmentation ideal of H. Then there is an induced duality between Q(H) and Prim(H ∗ ), and there is the (classical) Leray theorem that says that for a connected commutative graded Hopf algebra H over a characteristic zero ﬁeld any section of I(H) −→ Q(H) induces an isomorphism of the free commutative algebra over Q(A) to H. This last theorem now has been considerably generalized to the setting of operads, see [38], and the references quoted there. The ﬁrst main topic that is treated in some detail (but without proofs) in this survey is an explicit and algorithmic description of a basis over the integers of Prim(NSymm). A divided power sequence in a Hopf algebra H is a sequence of elements d = (d(0) = 1, d(1), d(2), . . .)

(1.8)

such that for all n µH (d(n)) =

d(i) ⊗ d(j)

i, j ∈ {1, 2, 3, . . .}

(1.9)

i+j=n

Note that d(1) is a primitive. Is is sometimes useful to write a DPS (divided power sequence) as a power series in a counting variable t: d(t) = 1 + d(1)t + d(2)t2 + d(3)t3 + · · ·

(1.10)

That makes it easier to talk about the inverse of a DPS (inverse power series), the product of two DPS’s (multiplication of power series) and shifted DPS’s: d(t) → d(tn ), all operations that give new DPS’s from old ones. When written in the form (1.10) a DPS is often called a curve. It turns out that each primitive of Prim(NSymm) can be extended to a divided power sequence. This is important because it implies that as a coalgebra NSymm is the cocommutative cofree graded coalgebra over the module Prim(NSymm). Now let QSymm be the graded dual Hopf algebra (over the integers) of NSymm. For an explicit description of QSymm, the Hopf algebra of quasisymmetric functions, see below in section 2. A most important question concerning QSymm is whether it is free polynomial as a commutative algebra. This has been an important issue since 1972, since it is crucial for the development of certain parts of the theory of noncommutative formal groups, [5, 6, 17]. The matter was ﬁnally settled in 1999, [21], in the positive sense that it is indeed free. A second proof follows from the cofreeness of NSymm. However, both these proofs fail to produce explicit generators. This has now also been taken care of, [23], and is the second main topic that will be discussed in some detail below. One most interesting and important aspect of the structure of Symm is the presence of two families of Hopf algebra morphisms that are called Frobenius and Verschiebung morphisms. They satisfy a large number of beautiful relations. The third main topic of this survey is to what extent these can be lifted to NSymm, respectively, extended to QSymm.

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There are both positive and negative results. However, the matter has not yet been quite completely settled. This paper is an expanded write-up of two talks that I gave on the subject: in Krasnoyarsk in August 2002 at the occasion of the International Conference “Algebra and its applications” in honour of the 70-th anniversary of V P Shunkov and the 65-th anniversary of V M Busarkin, and at the Z. Borewicz memorial conference in Skt Petersburg in September 2002.

2. The Hopf algebra QSymm of quasisymmetric functions Above, in the introduction, the graded Hopf algebra NSymm of noncommutative symmetric functions was deﬁned. The grading is deﬁned by wt(Zn ) = n

(2.1)

and, more generally, if α = [a1 , a2 , . . . , am ] is a nonempty word over the positive integers N = {1, 2, . . .}, let Zα be the noncomutative monomial Zα = Za1 Za2 · · · Zam

(2.2)

wt(Zα ) = wt(α) = a1 + · · · + am

(2.3)

then

Let Z[ ] = 1, where [ ] is the empty word, then the Zα , α ∈ N∗ , the monoid of words over N form a basis of NSymm (as a graded Abelian group). The empty word, and also Z[ ] = 1, has weight zero. As a free Abelian graded group QSymm, the graded dual of NSymm can be taken to be the free Abelian group with as basis N∗ , the words over the set of natural numbers. The duality is then < Zα , β >= δαβ

(2.4)

The duality induced comultiplication is easy to describe. It is ‘cut’:

[a1 , a2 , . . . , am ] →

m

[a1 , . . . , ai ] ⊗ [ai+1 , . . . , am ]

(2.5)

i=0

where of course [a1 , . . . , ai ] = [ ] = 1 if i = 0 and [ai+1 , . . . am ] = [ ] = 1 if i = m. The duality induced multiplication is more diﬃcult to describe. It is the socalled ‘overlapping shuﬄe multiplication’ which can be described as follows. Let α = [a1 , a2 , . . . , am ] and β = [b1 , b2 , . . . , bn ] be two compositions or words. Take a ‘sofar empty’ word with n + m − r slots where r is an integer between 0 and min{m, n}, 0 ≤ r ≤ min{m, n}. Choose m of the available n + m − r slots and place in it the natural numbers from α in their original order; choose r of the now ﬁlled places; together with the remaining n+m−r−m = n−r places these form n slots; in these place the entries from β in their orginal order; ﬁnally, for those slots which have two entries, add them. The product of

130

MICHIEL HAZEWINKEL

the two words α and β is the sum (with multiplicities) of all words that can be so obtained. So, for instance, [a, b] ×osh [c, d] = [a, b, c, d] + [a, c, b, d] + [a, c, d, b] + [c, a, b, d] + [c, a, d, b] + [c, d, a, b]+ + [a + c, b, d] + [a + c, d, b] + [c, a + d, b] + [a, b + c, d] + [a, c, b + d]+ (2.6) + [c, a, b + d] + [a + c, b + d] and [1] ×osh [1] ×osh [1] = 6[1, 1, 1] + 3[1, 2] + 3[2, 1] + [3]. There is a concrete realization of QSymm much like the standard realization of Symm as the ring of symmetric functions in inﬁnitely many indeterminates x1 , x2 , . . . . See [34], Chapter 1 for some detail on how to work with inﬁnitely many indeterminates in this context. Let X be a ﬁnite or inﬁnite set (of commuting variables) and consider the ring of polynomials, R[X], and the ring of power series, R[[X]], over a commutative ring R with unit element in the commuting variables from X. A polynomial or power series f (X) ∈ R[[X]] is called symmetric if for any two ﬁnite sequences of indeterminates x1 , x2 , . . . , xn and y1 , y2 , . . . , yn from X and any sequence of exponents i1 , i2 , . . . , in ∈ N, the coeﬃcients in f (X) of xi11 xi22 . . . xinn and y1i1 y2i2 . . . ynin are the same. The quasi-symmetric formal power series are a generalization introduced by Gessel, [15], in connection with the combinatorics of plane partitions. This time one takes a totally ordered set of indeterminates, e.g. V = {v1 , v2 , . . .}, with the ordering that of the natural numbers, and the condition is that the coeﬃcients of xi11 xi22 . . . xinn and y1i1 y2i2 . . . ynin are equal for all totally ordered sets of indeterminates x1 < x2 < · · · < xn and y1 < y2 < · · · < yn . Thus, for example, x1 x22 + x2 x23 + x1 x23

(2.7)

is a quasi-symmetric polynomial in three variables that is not symmetric. Products and sums of quasi-symmetric polynomials and power series are again quasisymmetric (obviously), and thus one has, for example, the ring of quasi-symmetric power series QSymm∧ in countably many commuting variables over the integers and its subring QSymm

(2.8)

of quasi-symmetric polynomials in ﬁnite of countably many indeterminates, which are the quasi-symmetric power series of bounded degree. The notation is justiﬁed. The quasisymmetric functions in {x1 , x2 , . . .} in this sense are a concrete realization of the quasisymmetric functions as introduced above as the graded dual of NSymm. In detail, given a word α = [a1 , a2 , . . . , am ] over N, also called a composition in this context, consider the quasi-monomial function Mα =

i1 <···

xai11 xai22 . . . xaimm

(2.9)

deﬁned by α. It is now an easy exercise to verify that as power series in the {x1 , x2 , . . .} the Mα satisfy Mα Mβ = Mα×osh β where ×osh is the overlapping shuﬄe product of words just deﬁned above (2.6).

(2.10)

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131

3. NSymm and QSymm over a field of characteristic zero Let K be a ﬁeld of characteristic zero, in particular the ﬁeld of rational numbers Q. The Hopf algebras of noncommutative symmetric functions and quasisymmetric functions over K are denoted N SymmK = N Symm ⊗Z K,

QSymmK = QSymm ⊗Z K

(3.1)

As remarked in the introduction, working over a ﬁeld of characteristic zero tends to simplify things considerably. Not that then everything becomes clear and easy and straightforward. Very far from it; witness the many papers quoted in the introduction. However, it is certainly true, that for the three groups of questions which form the main topic of this paper: primitives of noncommutative symmetric functions, freeness of the algebra of quasisymmetric functions, existence of Frobenius and Verschiebung morphisms, things either reduce to known things, or become fairly straightforward. At least up to fairly explicitly given isomorphisms. And in that connection it is well to reﬂect that knowing something well up to an isomorphism can be not all the same as really controlling things. To start with, consider another Hopf algebra over the integers, the Lie Hopf algebra U = ZU1 , U2 , . . .,

µ(Un ) = 1 ⊗ Un + Un ⊗ 1,

ε(Un ) = 0,

n = 1, 2, . . .

(3.2)

That is, as an algebra U is the free algebra over the integers in the noncommuting variables Un and the coalgebra structure is given by requiring that the two right hand formulas of (3.2) are algebra homomorphisms. The primitives of U are called Lie polynomials and they form the free Lie algebra over Z in the alphabet {U1 , U2 , · · · }. Also the universal enveloping algebra of Prim(U) is U. Now, over the rationals (and hence over any ﬁeld of characteristic zero or ring containing Q) N Symm1 and U are isomorphic. One particularly beautiful isomorphism ϕ N SymmQ −→UQ is given by setting 1 + Z1 t + Z2 t2 + Z3 t3 + · · · = exp(U1 t + U2 t2 + U3 t3 + · · · ) =

∞

(U1 t + U2 t2 + U3 t3 + · · · )i

(3.3)

i=0

where t is a (counting) variable commuting with everything in sight. Equating equal powers of t on the left and right hand sides of (3.3) gives Zn =

i1 + · · · ik = n ij ∈ N

Ui1 Ui2 · · · Uik k!

(3.4)

For a proof that the algebra isomorphism deﬁned by setting ϕ(Zn ) equal to the right hand side of (3.4) deﬁnes an isomorphism of Hopf algebras see [18]. Thus, up to the isomorphism ϕ, the matter of describing Prim(NSymm), comes down to describing Prim(U), or, equivalently writing down some explicit bases for the free Lie algebra 1 The

Hopf algebra NSymm is sometimes called the Leibniz Hopf algebra.

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generated by a countable set of indeterminates {U1 , U2 , . . .}. The matter of constructing bases for free Lie algebras has had a lot af attention. There are many more or less diﬀerent bases such as Hall bases, Shirshov bases, Lyndon bases, see [3, 39, 41, 42, 47]. Here, partly because most of the concepts will be needed below anyway, is a description of the Lyndon basis (also called Chen - Fox - Lyndon basis). Consider the free monoid N* of words on the alphabet N (or any other totally ordered alphabet for that matter). The lexicographic order (also called dictionary order, or alphabetical order) on N* is deﬁned as follows. If α = [a1 , a2 , . . . , am ] and β = [b1 , b2 , . . . , bn ] are two words of length m and n respectively, lg(α) = m, lg(β) = n,

α >lex

∃k ≤ min{m, n} such that a1 = b1 , . . . , ak−1 = bk−1 and ak > bk β⇔ or lg(α) = m > lg(β) = n and a1 = b1 , . . . , an = bn

(3.5)

The empty word is smaller than any other word. This deﬁnes a total order. Of course, if one accepts the dictum that anything is larger than nothing, the second clause of (3.5) is superﬂuous. The proper tails (suﬃxes) of the word α = [a1 , a2 , . . . , am ] are the words [ai , ai+1 , . . . am ], i = 2, 3, . . . , m. Words of length 1 or 0 have no proper tails. The preﬁx corresponding to a tail α = [ai , ai+1 , . . . am ] is α = [a1 , . . . , ai−1 ] so that α = α ∗ α where * denotes concatenation of words. A word is Lyndon iﬀ it is lexicographically smaller than each of its proper tails. For instance [4], [1, 3, 2], [1, 2, 1, 3] are Lyndon and [1, 2, 1] and [2, 1, 3] are not Lyndon. For each Lyndon word α of length > 1 consider the lexicographically smallest proper tail α of α. Let α be the corresponding preﬁx to α . Then α and α are both Lyndon and α = α ∗ α is called the canonical factorization of α. A basis of the free Lie algebra on {U1 , U2 , . . .}, i.e. a basis of Prim(U) ⊂ U, is now obtained as follows. For each word α = [a1 , a2 , . . . , am ] let Uα = Ua1 Ua2 . . . Uam be the corresponding monomial. Now, by recursion in length, deﬁne for a word of length 1 Q[i] = Ui

(3.6)

and for α Lyndon and of length lg(α) ≥ 2 let α = α ∗ α be its canonical factorization and set Qα = [Qα , Qα ]

(3.7)

then the {Qα : α Lyndon} form a basis of Prim(U) ⊂ U. For a proof see e.g. [39], p. 105ﬀ. The next topic to be taken up is the matter of the freeness of QSymm over the rationals. The graded dual of U is the socalled shuﬄe algebra. As a free module over Z it has the words over N as a basis and the product is the shuﬄe product which is like the overlapping shuﬄe product except that the overlap terms, i.e. those which involve additions of entries are left out. Thus for example [a, b] ×sh [c, d] = [a, b, c, d] + [a, c, b, d] + [a, c, d, b] + [c, a, b, d] + [c, a, d, b] + [c, d, a, b] (compare (2.6) above). It is well known that the shuﬄe algebra is free polynomial with as generators (for example) the Lyndon words. See, for example, [39]. p. 111 for a proof. Thus via the

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133

isomorphism ϕ, or rather its graded dual, it follows that QSymmQ is a free commutative algebra. But the description of the generators is rather involved and they do not look very nice. Actually the situation is rather better and a modiﬁcation of the proof of the freeness of the shuﬄe algebra (using a diﬀerent ordering on words) gives that in fact QSymmQ is commutative free polynomial on the Lyndon words. The ordering to be used is the wll-ordering. The acronym stands for weight ﬁrst, than length, than lexicographic. See [20] for details. The third main topic of this survey is the existence of Frobenius and Verschiebung type Hopf algebra endomorphisms of NSymm and QSymm which lift, respectively extend, those on Symm. Again, over the rationals, this is a relatively straightforward matter. Though there are some unanswered questions. Recall the situation for Symm, see [17, 24] for more details. On Symm there are two families of Hopf algebra endomorphims, called Frobenius and Verschiebung morphisms, denoted fn , vn , n ∈ N, which among others have the following beautiful properties: (i) f1 = v1 = id (ii) fn is homogeneous of degree n, i.e. fn (Symmk ) ⊂ Symmnk Here, for any graded Hopf algebra, H, Hn is the homogeneous part of of weight n of H. (iii) vn is homogenous of degree n−1 , i.e. vn (Symmk ) ⊂ Symn−1 k if n divides k, and vn (Symmk ) = 0 if n does not divide k. (iv) fn fm = fnm for all n, m ∈ N (v) vn vm = vnm for all n, m ∈ N (vi) fn vm = vm fn provided n and m are relatively prime, gcd(m, n) = 1 (vii) vn fn = n, where n is the n-fold convolution of the identiy. Now there is the natural projection N Symm −→ Symm, Zn → hn

(3.8)

and the natural (graded dual) inclusion Symm ⊂ QSymm

(3.9)

obtained by regarding a symmetric function as a special kind of quasisymmetric function. The question is whether there are lifts, respectively extensions, on NSymm, respectively QSymm, which also have the properties (i) - (vii). Retaining property (vii) can be ruled out immediately for trivial reasons. The simple fact is that n on either Qsymm or NSymm simply is not a Hopf algebra endomorphism. So it is natural to concentrate on the other six properties. And then the answer over the rationals is yes. But, as will be stated below, the answer over the integers is no. But there are interesting substitutes. Let pn = xn1 + xn2 + xn3 . . .

(3.10)

denote the power sums in Symm. They are related to the complete symmetric functions by the recursion relation nhn = pn + pn−1 h1 + pn−2 h2 + · · · p1 hn−1

(3.11)

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MICHIEL HAZEWINKEL

The Frobenius and Verschiebung morphisms on Symm are characterized by

fn pk = pnk ,

npk/n if n divides k, vn p k = 0 if n does not divide k.

(3.12)

On the polynomial generators hn this characterization of vn works out as hk/n if n divides k, vn h k = 0 otherwise.

(3.13)

Deﬁne the (noncommutative) Newton primitives in NSymm by Pn (Z) =

(−1)k+1 rk Zr1 Zr2 . . . Zrk ,

ri ∈ N = {1, 2, . . .}

(3.14)

r1 +···rk =n

or, equivalently, by the recursion relation nZn = Pn (Z) + Z1 Pn−1 (Z) + Z2 Pn−2 (Z) + · · · + Zn−1 P1 (Z)

(3.15)

Note that under the projection Zn → hn by (3.15) and (3.11) Pn (Z) goes to pn . It is easily proved by induction, using (3.15), or directly from (3.14), that the Pn (Z) are primitives of NSymm, and it is also easy to see from (3.15) that over the rationals NSymm is the free associative algebra generated by the Pn (Z). Thus over the rationals the Lie algebra of primitives of NSymm is simply the free Lie algebra generated by the Pn (Z), giving a second description of Prim(N SymmQ ). There are obvious candidate lifts of the vn on Symm to Hopf algebra endomorphisms on NSymm., viz Zk/n if k is divisible by n vn (Zk ) = (3.16) 0 otherwise By (3.14) or (3.15) this implies nPk/n if n divides k vn (Pk ) = 0 otherwise

(3.17)

Now on N SymmQ deﬁne the Frobenius morphisms as the algebra morphisms given by fn (Pk (Z)) = Pnk (Z)

(3.18)

It is now easily checked that the vn and fn as deﬁned by (3.16) and (3.18) are Hopf algebra endomorphisms of N SymmQ , that they satisfy (the analogues on N SymmQ of) properties (i)-(vi) and that they descend to the usual Frobenius and Verschiebung morphisms on Symm.

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A priori, the fn as deﬁned by (3.18) are only deﬁned over the rationals and indeed nontrivial denominators show up almost immediately. For instance f2 (Z1 ) = 2Z2 − Z12 f2 (Z2 ) = 2Z4 − 32 Z1 Z3 − 12 Z3 Z1 + Z22 + 12 Z1 Z2 Z1 + 12 Z12 Z2

(3.19)

On Symm a certain amount of coeﬃcient magic sees to it that all coeﬃcients become integral. But of course over Symm there are much better deﬁnitions of the Frobenius morphisms that immediately show that they are deﬁned over the integers, see [24] or [17], §17. As we shall see later, over the integers there are even no algebra endomorphisms fn of NSymm that lift the fn on Symm such that together with the vn as deﬁned by (3.16) they satisfy (i)-(vi). Note there is nothing unique about this solution (3.18) of the Frobenius-Verschiebung lifting problem over the rationals. For instance one could work instead with the seond set of Newton primitives deﬁned by Pn (Z) =

(−1)k+1 r1 Zr1 Zr2 . . . Zrk ,

ri ∈ N = {1, 2, . . .}

(3.20)

r1 +···rk =n

and satisfying the recursion relation nZn = Pn (Z) + Pn−1 (Z)Z1 + Pn−2 (Z)Z2 + · · · + P1 (Z)Zn−1

(3.20)

4. The primitives of NSymm Above, some primitives of NSymm were already written down and they generate a free graded Lie algebra contained in Prim(NSymm). Denote this Lie algebra by FrLie(P ) and its homogeneous part of weight n by FrLie(P )n . The Lie algebra Prim(NSymm) is also graded of course. Let Prim(N Symm)n ⊂ N Symmn be the homogeneous part of degree n of Prim(NSymm). Both Prim(N Symm)n and FrLie(P )n are free Abelian groups of rank βn , the number of weight n Lyndon words2 . The index of FrLie(P )n ⊂ Prim(N Symm)n as a function of n measures how large FrLie(P ) is in Prim(NSymm). As it turns out FrLie(P ) is only a tiny part. Indeed, the value of the index alluded to is

Index of FrLie(P )n in Prim(N Symm)n =

α∈LYN,

wt(α)=n

k(α) g(α)

(4.1)

where for a word α = [a1 , a2 , . . . , am ] over the natural numbers g(α) is the gcd (greatest common divisor) of its entries a1 , a2 , . . . , am and k(α) is the product of its entries. Thus the values of (4.1) for the ﬁrst six n are 1, 1, 2, 6, 576, 69120. Thus taking iterated commutators of the known Newton primitives is not nearly good enough. One can see this coming very quickly. Indeed [P1 , P2 ] = 2(Z1 Z2 − Z2 Z1 ). It also follows that Prim(NSymm) is not a free Lie algebra over the integers. Rather it tries to be something like a free divided power Lie algebra (though I do not know what such a thing might be). 2 The

numbers βn are given by the identity (1 − t)−1 (1 − 2t) = Witt, [48].

∞

n=1 (1

− tn )βn which goes back to

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MICHIEL HAZEWINKEL

Instead of taking commutators of primitives it turns out to be a good idea to work with whole DPS’s (divided power sequences, see (1.8)). Accordingly, the next thing to be described are techniques for producing new divided power sequences from known ones. There are two more techniques for this (besides the ones mentioned in the introduction, which do not suﬃce) coming from two socalled isobaric decomposition theorems. For the ﬁrst isobaric decomposition theorem consider the Hopf algebra

2N Symm = ZX1 , Y1 , X2 , Y2 , · · · , µ(Xn ) =

Xi ⊗ Xj ,

µ(Yn ) =

i+j=n

Yi ⊗ Yj (4.2)

i+j=n

and the two natural curves X(s) = 1 + X1 s + X2 s2 + · · · ,

Y (t) = 1 + Y1 t + Y2 t2 + · · ·

(4.3)

and consider the commutator product X(s)−1 Y (t)−1 X(s)Y (t)

(4.4)

On the set of pairs of nonnegative integers consider the ordering (u, v) <wl (u , v )

⇔

u + v < u + v or (u + v = u + v and u < u )

(4.5)

(Here the index wl on <wl is supposed to be a mnemonic for weight ﬁrst, then lexicographic.) 4.6 Theorem. (ﬁrst bi-isobaric decomposition theorem, Shay [40], Ditters). There are ‘higher commutators’ (or perhaps better ‘corrected commutators’) Lu,v (X, Y ) ∈ ZX, Y , (u, v) ∈ N × N

(4.7)

such that X(s)−1 Y (t)−1 X(s)Y (t) =

→

(1 + La,b (X, Y )sa tb + L2a,2b (X, Y )s2a t2b + · · · )

(4.8)

gcd(a,b)=1

where the product is an ordered product for the ordering <wl just introduced, (4.5). Moreover (i) Lu,v (X, Y ) = [Xu , Yv ] + (terms of length ≥ 3) (ii) Lu,v (X, Y ) is homogeneous of weight u in X and of weight v in Y . (iii) For gcd(a, b) = 1, 1 + La,b (X, Y )sa tb + L2a,2b (X, Y )s2a t2b + · · · is a 2-curve.

(4.9) (4.10)

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137

Here a 2-curve is a two dimensional version of a curve. A power series in two variables with constant term 1 d(i, j)si tj (4.11) d(s, t) = i,j

is a 2-curve iﬀ

µ(d(m, n)) =

d(m1 , n1 ) ⊗ d(m2 , n2 )

(4.12)

m1 +m2 =m n1 +n2 =n

This is not at all diﬃcult to prove. The only thing needed is to observe that pure powers of s or t do not occur on the LHS of (4.8) and that each pair of nonnegative integers (u, v) occurs just once in one of the factors on the right of (4.8). That gives the decomposition. The fact that the factors are two curves then follows easily with induction from the observation that the LHS of (4.8) is a 2-curve. Also (4.8) implies an explicit recursion formula for the Lu,v (X, Y ). 4.13. Theorem. (Second bi-isobaric decomposition theorem, Hazewinkel [22]). There are unique homogeneous noncommutative polynomials Nu,v (Z) ∈ N Symm such that

Z(s)−1 Z(t)−1 Z(s + t) =

→

(1 + Na,b (Z)sa tb + N2a,2b (Z)s2a t2b + · · · ).

(4.14)

a,b∈N gcd(a,b)=1

Moreover

u+v Zu+v + (terms of length ≥ 2) u (ii) Nu,v (Z) is homogeneous of weight u + v (iii) For each a, b ∈ N2 , gcd(a, b) = 1, 1 + Na,b (Z)sa tb + N2a,2b (Z)s2a t2b + · · · is a 2-curve. (iv) For each n ≥ 2, N1,n−1 (Z) = Pn (Z) (i) Nu,v (Z) =

(4.15) (4.16) (4.17) (4.18)

Again, the decomposition is not at all diﬃcult to prove and the ﬁnal observation results directly from the recursion formula implied by (4.14) compared to the recursion formula (3.15) for the Pn (Z). There is now a suﬃciency of tools to describe a basis of Prim(NSymm) and more. The essential fact is that if d1 , d2 are two divided power sequences in some Hopf algebra H, than so is Da,b (d1 , d2 ) = (1, La,b (d1 , d2 ), L2a,2b (d1 , d2 ), . . .)

(4.19)

where, as the notation suggests, Lu,v (d1 , d2 ) is obtained from the Lu,v (X, Y ) of theorem 4.6 by substituting d1 (k) for Xk and d2 (l) for Yl . This follows immediately from the fact that for gcd(a, b) = 1 1 + La,b (X, Y )sa tb + L2a,2b (X, Y )s2a t2b + · · ·

138

MICHIEL HAZEWINKEL

is a 2-curve. Similarly, if d is a curve in any Hopf algebra H Na,b (d) = 1 + Na,b (d)t + N2a,2b (d)t2 + · · ·

(4.20)

is another curve.3 Let LYN denote the set of Lyndon words over the natural numbers N = {1, 2, 3, . . .}. Then to each α = [a1 , . . . , am ] ∈ LYN there are associated three things (i) A number g(α) = gcd{a1 , . . . , am } (ii) A divided power sequence dα (iii) A primitive Pα The items (ii) and (iii) are deﬁned recursively as follows. If lg(α) = m is 1, d[n] = (1, Z1 , Z2 , . . .) and P[n] = Pn (d[n] ) = Pn (Z). If lg(α) ≥ 2, let α = α ∗ α be the canonical factorization of α (see just above (3.6)). Then dα = (1.Lg(α )/g(α),g(α )/g(α) (dα , dα ), L2g(α )/g(α),2g(α )/g(α) (dα , dα ), . . .) = Dg(α )/g(α),g(α )/g(α) (dα , dα )

(4.21)

and Pα = Pg(α) (dα )

(4.22)

Note that the divided power sequences associated to [a1 , . . . , am ] and [ra1 , . . . , ram ], r ∈ N are the same. 4.23. Theorem. The Pα , α ∈ LYN form a basis over the integers of Prim(NSymm). Each of the Pα is the ﬁrst term of a DPS. The second property of theorem 4.23 guarantees that NSymm is the cofree cocommutative graded coalgebra over the graded module Prim(NSymm), see the appendix of [22] for a proof of that. It follows that the graded dual QSymm is a free commutative algebra, and implicitly speciﬁes a set of generators for QSymm. However, this does not give a convenient description of such a set of generators. The original proof of theorem 4.23, [22], is rather long and intricate. Fortunately there is now a much shorter proof, which will be discussed in the next section. The basis Pα , α ∈ LYN of Prim(NSymm) has a number of nice properties, particularly with respect to the Verschiebung endomorphisms vn of NSymm, see (3.16). Consider the following ordering on words: α <wll β if and only if (wt(α) < wt(β) or (wt(α) = wt(β) and lg(α) < lgβ)) or (wt(α) = wt(β) and lg(α) = lgβ) and α

3 The operations on curves defined by (4.19) and (4.20) are functorial. Thus assigning to a Hopf algebra H its group of curves gives a group valued functor with many operations on it, viz 2-ary operations La,b for every pair a, b with gcd(a, b) = 1, and also unary operations Na,b for any such pair. In addition there are the operations which come from the Verschiebung Hopf algebra endomorphisms on NSymm (see section 6) which commute with the La,b and Na,b . And then there are also Frobenius type operations (see also section 6). It is almost totally uninvestigated whether some subset of all this structure can be used for classification purposes, say, of noncommutative formal groups.

SYMM, NSYMM AND QSYMM FUNCTIONS

139

4.24. Theorem. For each Lyndon word α = [a1 , . . . , am ] (i) Pα = g(α)Zα + (wll larger terms)

(ii)

(4.25)

nP[n−1 a1 ,··· ,n−1 am ] if n divides g(α) vn (Pα ) = 0 otherwise.

(4.26)

The ﬁrst property of theorem 4.24 results directly from the construction. The second Nra,rb (Z)sra trb property comes from the fact that the curves Lra,rb (X, Y )sra trb and of the two isobaric decomposition theorems 4.6 and 4.13 are particularly nice curves in that Ln−1 u,n−1 v (X, Y ) if n divides u, v, vn (Lu,v (X, Y )) = 0 otherwise. Nn−1 u,n−1 v (Z) if n divides u, v vn (Nu,v (Z)) = 0 otherwise.

(4.27)

(4.28)

where vn is deﬁned on 2NSymm just like (3.16) for both the X’s and the Y ’s. 5. Free polynomial generators for QSymm over the integers As in section 2 above, consider Symm and QSymm as symmetric functions in an inﬁnity of ideterminates Symm = Z[e1 , e2 , . . .] ⊂ QSymm ⊂ Z[x1 , x2 , . . .]

(5.1)

where the ei are the elementary symmetric functions in the xj . There is a well known λ-ring structure on Z[x1 , x2 , . . .] given by xj λi (xj ) = 0

if i = 1 , if i ≥ 2

j = 1, 2, . . .

(5.2)

For information on λ-rings see [27]. The asscociated Adams operators, determined by the formula ∞ d (−1)n fn (a)tn (5.3) t log λt (a) = dt n=1 where λt (a) = 1 +

∞

λn (a)tn

(5.4)

n=1

are the ring endomorphisms fn : xj → xnj

(5.5)

140

MICHIEL HAZEWINKEL

There are well-known determinantal relations between the λn and the fn as follows 0 ··· 0 .. f2 (a) f1 (a) 2 . . . . . . .. .. n!λn (a) = det .. .. . . 0 fn−1 (a) fn−2 (a) · · · f1 (a) n − 1 fn (a) fn−1 (a) · · · f2 (a) f1 (a)

(5.6)

··· 0 .. .. . λ1 (a) 1 . λ2 (a) . . . . fn (a) = det .. .. .. .. 0 (n − 1)λn−1 (a) λn−2 (a) · · · λ1 (a) 1 λn−1 (a) · · · λ2 (a) λ1 (a) nλn (a)

(5.7)

f1 (a)

λ1 (a)

1

1

0

(which come from the Newton relations between the elementary symmetric functions and the power sum symmetric functions). It follows that the subrings Symm and QSymm are stable under the λn and fn because λn (QSymm ⊗z Q) ⊂ QSymm ⊗z Q by (1.6) and (QSymm ⊗z Q) ∩ Z[x1 , x2 , . . .] = QSymm, and similarly for Symm. It follows that QSymm and Symm have induced λ-ring structures. On Symm this is of course the standard one. It follows immediately from (5.5) that fn ([a1 , a2 , . . . , am ]) = [na1 , na2 , . . . , nam ]

(5.8)

when α = [a1 , a2 , . . . , am ] is seen as a monomial quasisymmetric function as in the realization of QSymm described in section 2. It is more customary to denote the Adams operations (power operations) associated to a λ-ring structure by Ψ‘s. However, in the present case they coincide with the Frobenius endomorphisms on Symm ([17], section E.2, p. 144ﬀ), and their natural extensions to QSymm; so it seems natural to use f ’s instead in this case. For any λ − ring R there is an associated mapping Symm × R −→ R, (ϕ, a) → ϕ(λ1 (a), λ2 (a), . . . , λn (a), . . .)

(5.9)

I.e. write ϕ ∈ Symm as a polynomial in the elementary symmetric functions e1 , e2 , . . . and then substitute λi (a) for ei , i = 1, 2, . . .. For a ﬁxed a ∈ R this is obviously a homomorphism of rings Symm −→ R. We shall often simply write ϕ(a) for ϕ(λ1 (a), λ2 (a), . . . , λn (a), . . .). Another way to see (5.9) is to observe that for ﬁxed a ∈ R(ϕ, a) → ϕ(λ1 (a), λ2 (a), . . .) = ϕ(a) is the unique homomorphism of λ-rings that takes e1 into a. (Symm is the free λ-ring on one generator, see also [27].) Note that en (α) = λn (α),

pn (α) = fn (α) = [na1 , na2 , . . . nam ]

(5.10)

The ﬁrst formula of (5.10) is by deﬁnition and the second follows from (5.7) because the relations between the en and pn are precisely the same as between the λn (a) and the fn (a).

SYMM, NSYMM AND QSYMM FUNCTIONS

141

Now let P ∈ N Symm be a primitive. Then, by duality P, αβ = µ(P ), α ⊗ β = 1 ⊗ P + P ⊗ 1, α ⊗ β = 0

(5.11)

for any words of length ≥ 1. Using the Newton relations pn (α) = pn−1 (α)e1 (α) − pn−2 (α)e2 (α) + · · · (−1)n−2 p1 (α)en−1 (α) + (−1)n−1 nen (α) (5.12) it follows that for any primitive in NSymm P, pn (α) = ±nP, en (α)

(5.13)

Now for any α ∈ LYN , α = [a1 , a2 , . . . am ] let αred = [g(α)−1 a1 , g(α)−1 a2 , . . . , g(α)−1 am ] and deﬁne Eα = eg(α) (αred )

(5.14)

5.15. Theorem. The Eα , α ∈ LYN form a free polynomial basis of QSymm over the integers For a rather simple direct proof of this (based on Chen - Fox - Lyndon factorization, [4]), see [23]. Sometimes it is useful to relabel the Eα , α ∈ LYN a bit. Let eLYN be the set of elementary Lyndon words, i.e those Lyndon words α for which g(α) = 1. Then the en (α), α ∈ eLYN, n = 1, 2, 3, . . .

(5.16)

are a sometimes convenient relabeling of the free polynomial basis Eα , α ∈ LYN . Note that for a ﬁxed α ∈ eLYN , the en (α) generate a subalgebra of QSymm that is isomorphic to Symm. Now, for all weights n, consider the matrices of integers (Pα , Eβ )α,β∈LYNn

(5.17)

where LYNn is the set of Lyndon words of weight n and the columns and rows of (5.17) are ordered by increasing wll-order. Claim: this matrix is diagonal with entries ±1 on the main diagonal. The triangularity follows from theorem 4.24 (ii). As to the diagonal part: g(α)Pα , Eα = g(α)Pα , λg(α) (αred ) by deﬁnition = ±Pα , pg(α) (αred ) by (5.13) = ±Pα , α by (5.10) = ±g(α) by (4.25) Using that there are precisely βn = #LY Nn Pα ‘s and Eα ‘s with α of weight n, the invertibility (over the integers) of the matrix (5.17) immediately implies both that the Pα are a basis of Prim(NSymm) and that the Eα are a free polynomial basis for QSymm over the integers

142

MICHIEL HAZEWINKEL

6. Frobenius and Verschiebung on NSymm and QSymm To ﬁx notations let fnSymm and vnSymm be the classical Frobenius and Verschiebung Hopf algebra endomorphisms over the integers of Symm, characterized by fnSymm (pk ) = pnk ,

npk/n if n divides k vnSymm (pk ) = 0 otherwise

(6.1)

Now NSymm comes with a canonical projection N Symm −→ Symm, Zn → hn and there is the inclusion Symm ⊂ QSymm (see section 2). The question arises whether ther are lifts fnNSymm , vnNSymm on NSymm and extensions fnQSymm , vnQSymm on QSymm that satisfy respectively (i) (ii) (iii) (iv) (v) (vi)

?Symm ?Symm = fnm fn?Symm fm fn is homogeneous of degree n, i.e. fn (?Symmk ) ⊂? Symmnk f1 = v1 = id fm vn = vn fm if (n, m) = 1 ?Symm ?Symm = vnm vn?Symm vm vn is homogeneous of degree n−1 , ?Symmn−1 k if n divides k i.e. vn (? Symmk ) ⊂ 0 otherwise.

(6.2)

Here ‘?’ can be ‘N’, or ‘Q’. Now there exist a natural lifts of the vnSymm to NSymm given by the Hopf algebra endomorphisms Zk/n if n divides k NSymm vn (Zk ) = (6.3) 0 otherwise and there exist natural extension of the Frobenius morphisms on Symm to QSymm ⊃ Symm given by the Hopf algebra endomorphisms fnQSymm ([a1 , . . . , am ]) = [na1 , . . . , nam ]

(6.4)

which, moreover, have the Frobenius-like property fpQSymm (α) = αp

mod p

(6.5)

for each prime number p. These two families are so natural and beautiful that nothing better can be expected and in the following these are ﬁxed as the Verschiebung morphisms on NSymm and Frobenius morphisms on QSymm. They are also dual to each other. The question to be examined in this section is whether there are supplementary families of morphisms fn on NSymm, respectively vn on QSymm, such that (6.2) holds. The ﬁrst result is negative

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6.6. Theorem. There are no (Verschiebung-like) coalgebra endomorphisms vn of QSymm that extend the vn on Symm, such that parts (iii)-(vi) of (6.2) hold. Dually there are no (Frobenius-like) algebra homomorphisms of NSymm that lift the fn on Symm such that parts (i)-(iv) of (6.2) hold. It is the coalgebra morphism property which makes it diﬃcult for (6.2) (iii)-(vi) to hold. It is not particularly diﬃcult to ﬁnd algebra endomorphisms of QSymm that do the job. For instance deﬁne the vn on QSymm as the algebra endomorphisms given on the generators (5.16) by ek/n (α) if n divides k vn (ek (α)) = (6.6) 0 otherwise It then follows from (5.12) that npk/n (α) if n divides k vn (pk (α)) = 0 otherwise

(6.7)

and all the properties (6.2) follow. The last topic I would like to discuss is whether there are Hopf algebra endomorphisms of QSymm (and dually, NSymm) such that some weaker versions of (6.2) hold. To this end we ﬁrst discuss a ﬁltration by Hopf subalgebras of QSymm. Deﬁne

Gi (QSymm) =

Zα ⊂ QSymm

(6.8)

α, lg(α)≤i

the free subgroup spanned by all α of length ≤ i, and let Fi (QSymm) = Z[en (α) : lg(α) ≤ i]

(6.9)

the subalgebra spanned by those generators en (α), α ∈ eLYN , lg(α) ≤ i of length less or equal to i. Note that this does not mean that the elements of Fi (QSymm) are bounded in length. For instance F1 (QSymm) = Symm ⊂ QSymm contains the elements [1, 1, . . . , 1] = en = en ([1])

(6.10)

n

for any n. 6.11. Theorem. Gi (QSymm) ⊂ Fi (QSymm) This is a consequence of the proof of the free generation theorem 5.15. Moreover, Fi (QSymm) ⊗z Q = Z[pn (α) : lg(α) ≤ i] Fi (QSymm) = Z[pn (α) : lg(α) ≤ i] ∩ QSymm

(6.11)

It follows from (6.11) and (6.12) that the Fi (QSymm) are not only subalgebras but sub pn (α ) ⊗ pn (α )).4 Hopf algebras (because µ(pn (α)) = α ∗α =α

4I

believe the corresponding Hopf ideals in NSymm to be the iterated commutator ideals.

144

MICHIEL HAZEWINKEL

Now consider a coalgebra endomorphism of QSymm. Because of the commutative cofreeness of QSymm as a coalgebra over the module tZ[t] for the projection QSymm −→ tZ[t],

[ ] → 0, [n] → tn ,

α → 0 if lg(α) ≥ 2

∞ or, equivalently, because of the freeness of NSymm over its submodule i=1 ZZi , a homogeneous coalgebra morphism of degree n−1 of QSymm is necessarily given by an expression of the form vϕ (α) =

ϕ(α1 ) · · · ϕ(αr )[n−1 wt(α1 ), . . . , n−1 wt(αr )]

(6.12)

α1 ∗···∗αr =α

for some morphism of Abelian groups ϕ : QSymm −→ Z. The endomorphism vϕ is a Hopf algebra endomorphism iﬀ ϕ is a morphism of algebras. 6.13. Proposition. vϕ (Fi (QSymm)) ⊂ Fi (QSymm) One particularly interesting family of ϕ’s is the family of ring morphisms given by τn (en ([1]) = τn (en ) = (−1)n−1 τn (ek (α)) = 0 for k = n or lg(α) ≥ 2(α ∈ eLYN )

(6.14)

Let vn be the Verschiebung type Hopf algebra endomorphism deﬁned by τn according to formula (6.12). Then 6.15. Theorem. nm [n−1 a1 , . . . n−1 am ] mod(length m − 1) if n|ai ∀i (i) vn ([a1 , . . . , am ]) ≡ 0 mod(length m − 1) otherwise (ii) vn extends vnSymm on Symm = F1 (QSymm) ⊂ QSymm (iii) vp vq = vq vp on F2 (QSymm) (iv) vn fn (α) = nlg(α) α mod(Flg(α)−1 (QSymm)) And of course there is a corresponding dual theorem concerning Frobenius type endomorphisms of NSymm. This seems about the best one can do. One unsatisfactory aspect of theorem 6.15 is that there are also other families vn that work. To conclude I would like to conjecture a stronger version of theorem 6.6, viz that there is no family fn of algebra endomorphisms of NSymm over the integers that satisﬁes (6.2) (i)-(iii) and such that these fn descend to the fnSymm on Symm. References [1] F Bergeron, A Garsia, C Reutenauer, Homomorphisms between Solomon descent algebras, J. of Algebra 150 (1992), 503–519. [2] F Bergeron, D Krob, Acyclic complexes related to noncommutative symmetric functions, J. of algebraic combinatorics 6 (1997), 103–117. [3] N Bourbaki, Groupes et alg`ebres de Lie. Chapitres 2 et 3 (Alg`ebres de Lie libres; Groupes de Lie), Hermann, 1972. [4] K T Chen, R H Fox, R C Lyndon, Free diﬀerential calculus IV, Ann. Math. 68 (1958), 81–95. [5] E J Ditters, Curves and formal (co) groups, Inv. Math. 17 (1972), 1–20.

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[6] E J Ditters, Groupes formels, Lecture notes, Universit´e de Paris XI, Orsay, 1974. Chapitre II, §5, p. 29 [7] G´erard Duchamp, Alexander Klyachko, Daniel Krob, Jean-Yves Thibon, Noncommutative symmetric functions III: deformations of Cauchy and convolution algebras, Preprint, Universit´e la Marne-la-Vall´ee, 1996. [8] G Duchamp, D Krob, B Leclerc, J-Y Thibon, Fonctions quasi-symm´ etriques, fonctions symm´ etriques noncommutatives, et alg`ebres de Hecke ` a q = 0, CR Acad. Sci. Paris 322(1996), 107–112. [9] G Duchamp, D Krob, E A Vassilieva, Zassenhaus Lie idempotents, q-bracketing, and a new exponential/logarithmic correspondence, J algebraic combinatorics 12:3(2000), 251–278. [10] A M Garsia, C Reutenauer, A decomposition of Solomon’s descent algebra, Adv. in Mathematics 77(1989), 189–262. [11] David Gebhard, Noncommutative symmetric functions and the chromatic polynomial, Preprint, Dept. Math., Michigan State University, 1994. [12] I M Gel’fand, V S Retakh, Determinants of matrices over noncommutative rings, Funct. Anal. and Appl. 25(1991), 91–102. [13] I M Gel’fand, V S Retakh, A theory of noncommutative determinants and characteristic functions of graphs, Funct. Anal. and Appl. 26 (1992), 1–20. [14] Israel M Gel’fand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir S Retakh, JeanYves-Thibon, Noncommutative symmetric functions, Adv. Math. 112 (1995), 218–348. [15] Ira M Gessel, Multipartite P-partitions and inner product of skew Schur functions. In: 1984, 289–301. [16] Ira M Gessel, Christophe Reutenauer, Counting permutations with given cycle-structure and descent set, J Combinatorial Theory, Series A :64 (1993), 189–215. [17] Michiel Hazewinkel, Formal groups and applications, Acad. Press, 1978. [18] Michiel Hazewinkel, The Leibniz-Hopf algebra and Lyndon words, preprint, CWI, Amsterdam, 1996. [19] Michiel Hazewinkel, Generalized overlapping shuﬄe algebras. In: S M Aseev and S A Vakhrameev (ed.), Proceedings Pontryagin memorial conference, Moscow 1998, VINITI, 2000, 193–222. [20] Michiel Hazewinkel, Quasisymmetric functions. In: D Krob, A A Mikhalev and A V Mikhalev (ed.), Formal series and algebraic combinatorics. Proc of the 12-th international conference, Moscow, June 2000, Springer, 2000, 30–44. [21] Michiel Hazewinkel, The algebra of quasi-symmetric functions is free over the integers, Advances in Mathematics 164 (2001), 283–300. [22] Michiel Hazewinkel, The primitives of the Hopf algebra of noncommutative symmetric functions over the integers, Preprint, CWI, 2001. [23] Michiel Hazewinkel, Explicit polynomial generators for the ring of quasisymmetric functions over the integers, submitted C.R. Acad. Sci. Paris (2002), [24] Michiel Hazewinkel, Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions, Acta Appl. Math. 75 (2003), 55–83. [25] Florent Hivert, Hecke algebras, diﬀerence operators and quasi symmetic functions, Adv. Math. 155 (2000), 181–238. [26] P Hoﬀman, Exponential maps and λ-rings, J. pure and appl. Algebra 27 (1983), 131–162. [27] D Knutson, λ-Rings and the representation theory of the symmetric group, Springer, 1973. [28] D Krob, B Leclerc, J-Y Thibon, Noncommutative symmetric functions II: transformations of alphabeths, Int. J. Algebra and Computation 7:2 (1997), 181–264. [29] Daniel Krob, Jean-Yves Thibon, Noncommutative symmetric functions IV: quantum linear groups and Hecke algebras at q = 0 , J. of algebraic combinatorics 6 (1997), 339–376. [30] D Krob, J-Y Thibon, Noncommutative symmetric functions V: a degenerate version of Uq (glN ), Int. J. Algebra and Computation 9:3&4 (1997), 405–430. [31] A Lascoux, Polynˆ omes symm´ etriques, foncteurs de Schur et Grassmanniens, Th`ese, Univ. de Paris VII, 1977.

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QUOTIENT GROTHENDIECK REPRESENTATIONS J.NDIRAHISHA AND F.VAN OYSTAEYEN Dept. of Mathematics and Computer Science, University of Antwerp (UIA) B-2610 Wilrijk, Belgium @-mail: [email protected]

Abstract. We generalize the concept of a Grothendieck representation to quotient Grothendieck representations; this allows the construction of a noncommutative projective geometry for geometrically graded rings that need not be positively graded. Mathematics Subject Classifications (2000): 16 B 50; 14 A 15; 14 A 22. Key words: (Quotient) Grothendieck representations; topological nerve; spectral representation and scheme

Introduction A Grothendieck representation of an arbitrary category B has been introduced in [9], with an aim to explain the appearance of canonical noncommutative topologies in noncommutative geometry. This point of view also provided a natural framework for a theory of schemes for noncommutative algebras, including a general categorical version of Serre’s global sections theorem. Now the noncommutative theory of P roj may also be ﬁt into the context of a Grothendieck representation but if one wants to relate the projective to the corresponding “aﬃne” theory, two Grothendieck representations of the same category have to be compared. In this note we introduce the quotient of a Grothendieck representation with respect to a so-called topological nerve with respect to B. We introduce some restricting condition, i.e. a spectral representation, allowing to retranslate certain properties holding in the representing Grothendieck categories into properties related to a notion of spectrum in B. The main result in Section 1, states that the quotient Grothendieck representation of a spectral representation is again a spectral representation. In Section 2, we study an application to noncommutative algebra (or geometry). As an example of the general construction in Section 1, we provide the construction and basic noncommutative scheme theory for the scheme P roj of an arbitrary ZZ-graded ring (noncommutative and not necessarily positively graded). The main result following from these considerations is that P roj may indeed be viewed as a gluing of aﬃne noncommutative schemes.

1. Quotient Grothendieck Representations Let B be a general category allowing products and coproducts. To R in B we associate a Grothendieck category Rep(R), to f : S → R in B we associate an exact functor F : Rep(R) → Rep(S) commuting with products and coproducts (we sometimes

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write F = f 0 ) such that: (IR )0 = IRep(R) for every R ∈ B, (f ◦ g)0 = g 0 ◦ f 0 for g: T → S, f : S → R in B. A Grothendieck categorical representation (G. C.-representation) is a contravariant functor Rep : B → R where R is the class consisting of the objects Rep(R), R ∈ B with HomR (Rep(R), Rep(S)) = HomB (S, R)0 . The functor Rep associates to f : S → R in B the exact functor f 0 : Rep(R) → Rep(S) given before. We say that Rep: B → R measures B if for every R ∈ B there exists an object R R in Rep(R) such that to any f : S → R in B there is an associated morphism f ∗ : S S → Rep(f )(R R) in Rep(S), we put Rep(f )(R R) = S R, such that 1∗S is the identity of S S in Rep(S) and for g: T → S, f : S → R in B, (f ◦ g)∗ = Rep(g)(f ∗ ) ◦ g ∗ : T T →T S →T R, where TR = Rep(g)Rep(f )(R R). For detail on torsion theory (i.e localization theory, quotient categories, Serre localizing subcategories, . . .) in a Grothendieck category, we refer to [2], [12]. For an arbitrary Grothendieck category G, the set of all torsion theories of G will be denoted by T ors(G). It is known that T ors(G) is a complete distributive lattice. We use notation as in [9], e.g we write κ, τ, γ, . . ., for torsion theories of G and Tκ , Tτ , Tγ , . . . resp., for the classes of torsion objects in G. A partial order, ≤, is deﬁned on T ors(G) by putting σ ≤ τ if and only if Tσ ⊂ Tτ . An M ∈ G is in Tσ∧τ if and only if M is both in Tσ and Tτ ; on the other hand Tσ∨τ is the torsion class generated by Tσ and Tτ . Recall that torsion classes in G are exactly those that are closed under taking: subobjects, images, ﬁnite products, extensions. For τ ∈ T ors(G), Tτ : G → G is the torsion functor corresponding to τ , i.e for M ∈ G, Tτ (M ) is the largest object of Tτ contained in M . Any subfunctor T of the identity functor in G is of type Tτ (-) if and only if it is left exact and idempotent in the sense that T (M/T (M )) = 0. As in [9] we abbreviate T orsRep to T op. To τ ∈ T ors(G) we associate the Serre quotient category (G, τ ) together with the canonical functors: iτ : (G, τ ) → G aτ : G → (G, τ ), the reflector for τ , with aτ iτ the identity of (G, τ ) and iτ aτ = Qτ the localization functor G → G associated to τ . To any exact functor F : Rep(R) → Rep(S) that commutes with direct sums (coproducts) there corresponds a functor F 0 : T op(S) → T op(R), deﬁning F 0 γ by letting its torsion class consist of those objects X in Rep(R) such that F (X) ∈ Tγ . In case F derives from a morphism f : S → R in B, we write F 0 = f˜. For any γ ∈ Top(A), A ∈ B, we put gen(γ) = {τ, τ ≥ γ}; for U ⊂ T op(A):

gen(∧U ) ⊃ ∪ gen(τ ) τ ∈U

gen(∨U ) = ∩ gen(τ ) τ ∈U

Hence, the subsets of the type gen(τ ) for τ in Top(A) deﬁne a topology on T op(A). For an exact functor F : Rep(R) → Rep(S) as above, F 0 (γ) ≤ F 0 (τ ) if γ ≤ τ and F 0 (∧U ) = ∧F 0 (U ) for any U ⊂ T op(S). Moreover, (F 0 )−1 (gen(τ )) = gen(ξτ ) for τ ∈ T op(R), where ξτ is such that Tξτ is the torsion class generated by F (Tτ ). Note that F 0 is not necessarily a lattice morphism (F 0 does not necessarily respect ∨) but it is a continuous map in the gen-topologies. In case γ ∈ T op(A) is perfect (Qγ is exact and commutes with arbitrary direct sums), then a0γ deﬁnes a homeomorphism (in the gen topologies): T ors(Rep(A), γ) gen(γ) ( see also Lemma 1.4 in [9]).

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The foregoing expresses that open sets of type gen(τ ) with perfect τ , behave as aﬃne open sets in scheme theory, i.e. viewed as topological spaces with the induced topologies they are homeomorphic to the topological space of a localized object. In the sequel, we only consider G.C.-representations that are faithful in the sense that for any S −→ R in B, Rep(f )(X) = 0 if and only if X = 0 (we write 0 for the zero object f

in the category considered). 1.1. Definition. A faithful G.C.-representation that measures B is said to be spectral if for every A ∈ B, γ ∈ T op(A) and τ ∈ gen(γ) we are given the following: A(γ) and fγ : A → A(γ) in B such that the morphism in Rep(A), fγ∗ :A A → Rep(fγ )(A(γ) A(γ)), is exactly the localization morphism A A → Qγ (A); a morphism fτγ : A(γ) → A(τ ) in B fitting into a commutative triangle:

Moreover, if ξ0 (A) stands for the trivial element of T op(A) (i.e. the one having only the zeroobject for its torsion class) then A(ξ0 (A)) = A and the identity of A is the corresponding morphism in B. 1.2. Proposition. Suppose Rep is spectral and γ ∈ T op(A) for A ∈ B, τ ∈ gen(γ). Write τ ), τ˜ for f˜γ (τ ) ∈ T op(A(γ)) and fτ˜ for the corresponding morphism in B, fτ˜ : A(γ) → A(γ)(˜ existing because of the spectral property of Rep (applied to A(γ) ∈ B). With notation as in Definition 1.1. we have: i) Rep(fτγ )(A(τ ) A(τ )) = Qτ˜ (A(γ) A(γ)). ii) Rep(fγ )(Qτ˜ (A(γ) A(γ))) = Qτ (A A). Proof. Since Rep is spectral there exists an A(γ)(˜ τ ) in B together with a morphism τ ) with corresponding morphism fτ˜∗ in Rep(A(γ)), fτ˜∗ : A(γ) A(γ) → A(γ) −→ A(γ)(˜ fτ˜

Rep(fτ˜ )(A(γ)(˜τ ) A(γ)(˜ τ )), which is nothing but the localization homomorphism A(γ) A(γ) → Qτ˜ (A(γ) A(γ)), in Rep(A(γ)). On the other hand we may look at Rep(fγ )(fτ˜∗ ): Qγ (A A) → Rep(fγ )(Qτ˜ (A(γ) A(γ))), where the latter is in fact equal to: Rep(fγ )Rep(fτ˜ )(A(γ)(˜τ ) A(γ)(˜ τ )) = Rep(fτ˜ fγ )(A(γ)(˜τ ) A(γ)(˜ τ )). Looking at A −→ A(γ) −→ A(γ)(˜ τ ), we obtain: fγ

fτ˜

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Now A(τ ) A(τ ) in Rep(A(τ )) is such that the localization morphism → Qτ (A) is exactly given by fτ∗ , fτ∗ : A A → Rep(fτ )(A(τ ) A(τ )). Look at the object Mτ (γ) in Rep(A(γ)) deﬁned by Mτ (γ) = Rep(fτγ )(A(τ ) A(τ )) and fτ∗γ : A(γ) A(γ) → Mτ (γ) the morphism in Rep(A(γ)) given by the fact that Rep is measuring. We have τ Mτ (γ)) τ˜(Mτ (γ)) ⊂ Mτ (γ) in Rep(A(γ)). By the deﬁnition of τ˜ we have that Rep(fγ )(˜ τ Mτ (γ)) ⊂ is τ -torsion in Rep(A), moreover the exactness of Rep(fγ ) yields that Rep(fγ )(˜ Rep(fγ )(Mτ (γ)) = Rep(fτ )(A(τ ) A(τ )), the latter being τ -torsionfree because it is equal to Qτ (A A). Consequently Rep(fγ )(˜ τ Mτ (γ)) = 0 and τ˜Mτ (γ) = 0 then follows from faithfulness τ (A(γ) A(γ)). We have of Rep. Hence fτ∗γ :A(γ) A(γ) → Mτ (γ) factorizes over Bτ (γ)=A(γ) A(γ)/˜ a sequence in Rep(A):A A −→ Qγ (A A) −−−−−−−− → Qτ (A A). So it is clear that in Rep(A), ∗ ∗ AA

fγ

Rep(fγ )(fτγ )

Rep(fγ )(Mτ (γ)/Bτ (γ)) is τ -torsion because the cokernel of Rep(fγ )(fτ∗γ ) is τ -torsion, therefore Mτ (γ)/Bτ (γ) is τ˜-torsion in Rep(A(γ)). It follows that Mτ (γ) ⊂ Qτ˜ (A(γ) A(γ)) in Rep(A(γ)). By deﬁnition Qτ˜ (A(γ) A(γ)) is τ˜-torsion over Bτ (γ) in Rep(A(γ)), hence Rep(fγ )(Qτ˜ (A(γ) A(γ))) is contained in Qτ (A A) as it is τ -torsion over A A/τ (A A). Since Mτ (γ) ⊂ Qτ˜ (A(γ) A(γ)) the functor Rep(fγ ) takes the value Qτ (A) for both objects and again from exactness and faithfulness of Rep(fγ ) it follows that: Mτ (γ) = Qτ˜ (A(γ) A(γ)). This proves i) and() yields that the morphisms Rep(fγ )(fτ˜∗ )fγ∗ and Rep(fγ )(fτ∗γ )fγ∗ are the same. Now ii) is obtained from i) by applying Rep(fγ ) to both members, so it follows again from the exact faithfulness of Rep(fγ ) that i) holds to. 1.3. Corollary. With conventions and notation as before, consider δ ≤ τ and γ ≤ τ with τ˜1 ∈ T op(A(δ)) and τ˜2 ∈ T op(A(γ)). We obtain the following diagram of morphisms in B:

with objects: τ1 ) in Rep(A(δ)(˜ τ1 )) M1 =A(δ)(˜τ1 ) A(δ)(˜ τ2 ) in Rep(A(γ)(˜ τ2 )) M2 =A(δ)(˜τ2 ) A(γ)(˜ M =A(τ ) A(τ ) in Rep(A(τ )) such that: i) Rep(fτ˜1 )(M1 ) = Qτ˜ (A(δ) A(δ)) = Rep(fτδ )(M ) ii) Rep(fτ˜2 )(M2 ) = Qτ˜ (A(γ) A(γ)) = Rep(fτγ )(M ) iii) Rep(fδ )Rep(fτ˜1 )(M1 ) = Qτ (A A) = Rep(fγ )Rep(fτ˜2 )(M2 ) = Rep(fτ )(A(τ ) A(τ )). This may be compared to the classical fact for localization in R-mod for a ring R, Qτ (R) = Qτ˜ (Qγ (R)) for τ ≥ γ, where Qτ˜ is a localization in Qγ (R)-mod. In our abstract setting Rep(A(γ)) replaces Qγ (R)-mod and we have traced the basic properties of a

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G.C.-representation (i.e. spectral) necessary to generalize the foregoing classical fact to the more general situation. τ2 ) as in the corollary are a priori not related so not necesNote that A(δ)(˜ τ1 ) and A(γ)(˜ sarily isomorphic in B. Up to introducing restrictive conditions on B and Rep, e.g. in terms of properties making A A a generator for Rep(A) etc. . ., one could study this further, however we do not aim to push category theoretical generality to the limit, contenting ourselves with the observation that the example(s) considered later will have an even better behaviour. Let us write T op(A)0 for the opposite lattice of T op(A). It would be natural to consider the functor A P: T op(A)0 → Rep(A), A P(τ ) = Qτ (A A), as a structural presheaf (in fact: sheaf) of A A with values in Rep(A). The spectral property of Rep allows to “realize” this structure sheaf inside B, in some sense, by considering P: T op(A)0 → B, deﬁned by P(τ ) = A(τ ) with structure morphism fτ : A → A(τ ). We have P(ξ0 (A)) = A with IA : A → A as the corresponding morphism. Moreover for γ ≤ τ we take fτγ : A(γ) → A(τ ) as the restriction morphism from γ to τ (note that in T op(A)0 the partial order is reversed when viewing γ and τ in T op(A) as open sets). For P to be a presheaf we need an extra property of Rep, as follows. A spectral Rep is said to be schematic if in addition to the properties in Deﬁnition 1.1. we have for every triple γ ≤ τ ≤ δ in T op(A) for any A ∈ B, a commutative diagram:

1.4. Corollary. If Rep is schematic then P: T op(A)0 → B is a presheaf of B-objects over the lattice T op(A), for every A ∈ B. Proof. The composition property of “sections” follows from fδγ = fδτ fτδ

(in the proof of the foregoing proposition one may start from a triangle

then use the same arguments for the triangles

and

and

to retract information

further to the Rep(A)-level). If f : S → R is a morphism in B then we have deﬁned F 0 = f˜ : T op(S) → T op(R),

and we observed that f˜ is a continuous map with respect to the gen-topologies on T op(S)and T op(R), (see remarks before Deﬁnition 1.1).

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A topological nerve η associates to each A ∈ B a ηA ∈ Top(A) such that for any f : A → B in B we have ηB ≤ f˜(ηA ) in T op(B). The fact that the latter domination relation is assumed for every f : A → B makes the topological nerve rather narrowly deﬁned; in practical situations the choice of ηA in each T op(A) follows from a radical-type of construction. A perfect example will be given by the torsion theory associated to the positive part A+ of a positively graded K-algebra or ring, where B is then the category of positively graded K-algebras, resp. rings. We adopt the following notation: iA : (Rep(A), ηA ) → Rep(A), aA : Rep(A) → (Rep(A), ηA ), QA : QηA = iA aA . To a G.C.-representation Rep and a topological nerve η we associate (Rep, η) such that for A ∈ B we have (Rep, η)(A) = (Rep(A), ηA ) and for any morphism f : A → B in B we consider the functor aA F iB : (Rep(B), ηB ) → (Rep(A), ηA ) where F = f 0 . For g: B → C in B we obtain that the composition (aA F iB )(aB GiC ): (Rep(C), ηC ) → Rep(A, ηA ) corresponds to g ◦ f , so we have to relate this to aA F GiC . This will follow from the following proposition. 1.5. Proposition. With notation as before, for any f : A → B in B we have aA F iB aB (M ) = aA F (M ) for every M ∈ Rep(B). Proof. Clearly QB (M )/M is ηB -torsion in Rep(B), hence it is η˜A -torsion because ηB ≤ η˜A and η˜A is deﬁned by f as usual, η˜A = f˜(ηA ). By deﬁnition of η˜A (and using exactness of F ) we obtain that F iB aB (M )/F (M ) is ηA -torsion in Rep(A). The exact sequence 0 → ηB (M ) → M → iB aB (M ) in Rep(B) yields an exact sequence in Rep(A): 0 → F ηB (M ) → F M → F iB aB (M ) → T → 0,

()

where T is ηA -torsion in Rep(A) in view of the foregoing. Clearly ηB (M ) is η˜A -torsion, hence F ηB (M ) is ηA -torsion and it follows that QA (F (M )) = QA (F iB aB (M )), or also aA F (M ) = aA F iB aB (M ). We introduce a notion slightly more general than a G.C.-representation, say a generalized G.C.-representation by weakening the deﬁnition of a G.C.-representation such that for f : S → R in B, F : Rep(R) → Rep(S) is only assumed to commute with ﬁnite products. 1.6. Corollary. With notation as before: aA F iB aB GiC = aA F GiC corresponds to A −→ f

B −→ C. Hence, associating f 2 = aA F iB to f : A → B, defines a generalized G.C.g

representation (Rep, η) as before (observe in particular that aA IM iA is IM by definition of iA and aA , for all M ∈ (Rep(A), ηA ). Proof. The ﬁrst statement is clear from the Proposition 1.5, so we only have to verify that aA F iB is exact and commutes with ﬁnite products (and coproducts i.e direct sums). This is obvious, except perhaps the exactness needs a word of explanation, since iB is not an exact functor in general. Exactness in the quotient categories may be interpreted as exactness up to torsion so the presence of aA in aA F iB (and the exactness of F ) makes aA F iB : (Rep(B), ηB ) → (Rep(A), ηA )

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exact. Explicitely, if 0 → M → M → M → 0 is exact in (Rep(B), ηB ) then: 0 → iB M → iB M → iB M → TB → 0 is exact in Rep(B) where TB is ηB -torsion in Rep(B). Applying the exact functor F yields an exact sequence in Rep(A): 0 → F iB (M ) → F iB (M ) → F iB (M ) → F (TB ) → 0. Since TB is ηB -torsion it is η˜A -torsion and thus F (TB ) is ηA -torsion in Rep(A). Applying the exact functor aA : Rep(A) → (Rep(A), ηA ) we obtain an exact sequence in (Rep(A), ηA ): 0 → aA F iB (M ) → aA F iB (M ) → aA F iB (M ) → 0.

The representation (Rep, η) associated to a topological nerve η is called the quotient generalized G.C.-representation of Rep. 1.7. Observation. If η 1 and η 2 are two topological nerves then η 1 ∧η 2 deﬁned by associating 1 2 ∧ ηA to A ∈ B is again a topological nerve but this does not hold for ∨, see the remarks ηA preceding Deﬁnition 1.1. (i.e. F 0 (∧U ) = ∧F 0 (U ) for any U ⊂ T op(S)). 1.8. Theorem. Let Rep and η be as before, then we have: i) If Rep is measuring B then so is (Rep, η). ii) If Rep is weakly spectral, i.e same definition as spectral but without assuming the faithfulness, then so is (Rep, η).

Proof. i) Since Rep is measuring, there are A A ∈ Rep(A) for A ∈ B and for f : S → R in B, a corresponding f ∗ : S S →S R = Rep(f )(R R) such that IS∗ = IS S and (f ◦g)∗ = Rep(g)(f ∗ )◦g ∗ for g: T → S in B. Now take aA (A A) ∈ (Rep(A), ηA ) for A ∈ B. To f : A → B in B we associate f 2 = aA f 0 iB as in the Corollary 1.6. and then f : aA (A A) → f 2 (aB (B B)). In view of Proposition 1.5. we now have f 2 (aB (B B)) = aA f 0 (B B) = aA (A B) and for f we just take aA (f ∗ ), the localized map of f ∗ . Obviously (IA ) = IaA (A) and the composition rule follows as in Corollary 1.6. ii) For every γ ∈ T op(A), A ∈ B, we have A(γ) ∈ B and fγ : A → A(γ) in B such that fγ∗ : 0 A A → fγ (A(γ) A(γ)) is exactly the localization morphism A A → Qγ (A A) on Rep(A). Now if we consider some γ ∈ T ors(Rep(A), ηA ) then this γ corresponds to a γ ∈ T orsRep(A) = T op(A) such that γ ≥ ηA . (Since aA is exact and commutes with ﬁnite products, we may

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take γ to be given by its torsion class consisting of M ∈ Rep(A) such that aA (M ) is γ-torsion in (Rep(A), ηA ). We obtain aA (fγ∗ ) = fγ as follows: fγ : aA (A A) aA fγ0 iA(γ) aA(γ) (A(γ) A(γ)) || aA fγ0 (A(γ) A(γ)) || aA (Qγ (A A))

(Proposition 1.5) (choice of A(γ))

Now since γ ≥ ηA we have QA Qγ (A A) = Qγ (A A) = Qγ QA (A A) (this is also a special case of compatibility of localization functors cf. [13]) = Qγ QA (A A). So we arrive at aA (Qγ (A A)) = aγ QA (A A)) = aγ (A A), and fγ is the localization map corresponding to γ. 1.9. Remarks. 1. The whole theory of G.C.-representations including the results of [9] extends to the case of generalized G.C.-representations. Since the case where the functors F are “restrictions of scalars” functors over rings is very interesting we have made the deﬁnition of G.C.-representation as we did. 2. Given f : A → B then aA f 0 iB (M ) = 0 if and only if iB (M ) is η˜A -torsion. Since ηB ≤ η˜A this means that M is η A -torsion, η A induced on (Rep(B), ηB ) by η˜A . Thus even if Rep is faithful, (Rep, η) need not be. However since f 2 (M ) = 0 exactly if M is η˜A -torsion, we see for τ ≥ γ in T ors(Rep(A), ηA ), corresponding to τ ≥ γ ≥ ηA in T op(A), that f˜γ (τ ) ≥ f˜γ (γ) ≥ f˜γ (ηA ) ≥ ηA(γ) in notation of Proposition 1.2; in particular we see that a γ˜ = f˜γ (γ)-torsion object is certainly a τ˜ = f˜γ (τ )torsion object. This allows to obtain an equivalent of Proposition 1.2 for (Rep, η). 1.10. Proposition. Let Rep be a spectral G.C.-representation then (Rep, η) is a weakly spectral generalized G.C.-representation such that: i) Qη˜A (fτ2γ (A(τ ) A(τ ))) = Qτ˜ (A(γ) A(γ)) ii) fγ2 (aτ˜ (A(γ) A(γ)) = Qτ (A A) = Qτ˜ (QA (A A)) Proof. Follow the lines of proof of the Proposition 1.2 when deﬁning Mτ (γ) as fτ2γ (A(τ ) A(τ )), and replacing fγ0 = Rep(fγ ) by fγ2 , fτ∗ by fτ , etc. . ., we obtain that f 2 (Mτ (γ)) is ηA torsionfree by deﬁnition, hence Mτ (γ) is η˜A -torsionfree. Thus fγ2 (˜ τ Mτ (γ)) = 0 leads to τ˜Mτ (γ) being η˜A -torsion, contradicting the foregoing unless τ˜Mτ (γ) = 0. As in the proof of Proposition 1.2, one then arrives at fγ2 (Qτ˜ (A(γ) A(γ))/Mτ (γ)) = 0, hence Qτ˜ (A(γ) A(γ)) is η˜A -torsion over Mτ (γ) and this yields i). We know already (the weakly spectral property) that fγ2 fτ2γ (A(τ ) A(τ )) = Qτ˜ (QA (A A)) hence fγ2 Mτ (γ) is τ˜-closed so fγ2 (Qτ˜ (A(γ) A(γ))) = fγ2 Mτ (γ) follows, or ii) follows too. 1.11. Example. Let B be the category of positively ZZ-graded rings with graded morphisms of degree zero for the morphisms. For A ∈ B we let Rep(A) be the category of graded (left) A-modules and graded A-linear morphisms. Full detail on the theory of graded rings and modules may be found in [7], [8]. We shall write Gr(A) = A − gr for the category as deﬁned above. To a graded ring morphism f : A → B in B we associate the restriction of scalars functor F = f 0 : B-gr → A-gr. It is easily seen that we have obtained a G.C.-representation

QUOTIENT GROTHENDIECK REPRESENTATIONS

155

Gr with Gr(A) = A-gr for A ∈ B. In T ors(A-gr) we may consider the graded torsion theory κA (+) by taking for its torsion class the graded A-modules such that every element may be annihilated by some power An+ , n ∈ IN , of the ideal A+ = A1 ⊕ . . . ⊕ An ⊕ . . .. For any f : A → B in B we have that f (A+ ) ⊂ B+ it follows easily that κB (+) ≤ f˜(κA (+)); consequently we have deﬁned a topological nerve κ(+) for Gr. The quotient generalized G.C.-representation (Gr, κ(+)) is denoted by P roj. It is not hard to see that Gr is measuring, just take A A = A viewed as a left graded A-module, and also spectral (for a graded (rigid) τ in T ors(Gr(A)) we have a graded ring of quotients Qgτ (A) and a graded morphism of rings jτg : A → Qgτ (A), then we look at Qgτ (A) Qgτ (A) i.e Qgτ (A) as a left module over itself). Detail on rigid graded localization may be found in [7]; a ﬁrst attempt at a projective scheme theory already appeared in [15], using results of [17]. The following section is devoted to a generalization of the foregoing example, to a more general useful P roj.

Let R =

⊕ Rn

n∈Z Z

2. Proj of a ZZ-graded ring be a ZZ-graded ring. Then δ = R−n Rn is an ideal of R0 and n=0

I = δ ⊕ ( ⊕ Rn ) is an ideal of R that is obviously graded and we have: I0 = δ. An ideal I n=0

of R0 is said to be invariant if for all n ∈ ZZ, Rn IR−n ⊂ I. Obviously, δ is an invariant ideal. In fact, one easily veriﬁes that an ideal I of R0 is invariant if and only if I ⊕ ( ⊕ Rn ) n=0

is an ideal of R. Observe that every ideal I of R0 that contains δ is necessarily invariant! Indeed, for all n ∈ ZZ we have that: Rn IR−n ⊂ Rn R−n ⊂ δ ⊂ I. On R0 -mod we may deﬁne a torsion theory deﬁned by the Gabriel ﬁlter L(δ), L(δ) = {L left ideal of R0 , L contains an ideal I of R0 such that δ ⊂ rad(I)}. Such torsion theories were called (symmetric) radical torsion theories in [16]. An arbitrary -mod, say σ is given by its ﬁlter L(σ) and is said to be invariant if torsion theory σ on R0 Rn IR−n ∈ L(σ). I ∈ L(σ) entails that n=0

In this section we assume that R0 is a (left) Noetherian ring. 2.1. Observation. L(δ) is invariant. Proof. Pick I ∈ L(δ) and consider I =

n=0

Rn IR−n . Look at rad(I) and let m ∈ IN be

such that: rad(I)m ⊂ I. m ) ⊂ Rn rad(I)m R−n it follows that P ⊂I for some prime ideal ¿From (Rn rad(I)R−n Rn rad(I)R−n and consequently: P ⊃ Rn δR−n . In particular P of R0 , yields P ⊃ n=0

n=0

P ⊃ Rn R−n Rn R−n , hence P ⊃ Rn R−n , and this holds for all n = 0 in ZZ. Therefore we arrive at P ⊃ δ. Hence rad(I ) ⊃ δ, or I ∈ L(δ). An ideal I of R0 is said to be strongly invariant whenever Rn I = IRn for all n ∈ ZZ. For example, if R is a centralizing extension of R0 then every ideal I of R0 is strongly invariant. Clearly a strongly invariant ideal is invariant in general. The properties of interest in this section will all follow in case R is a centralizing extension of R0 ; in fact, an acceptable generalization of projective scheme theory for a positively

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graded algebra A with A0 = k, a ﬁeld in the centre of A would already be obtained by looking at ZZ-graded rings with A0 ⊂ Z(A). Nevertheless it is somewhat remarkable that a much less restrictive condition will suﬃce. We say that the ZZ-graded ring R is 0-normal if for all a ∈ ZZ we have: R−a (Ra R−a ) ⊂ (Ra R−a )R−a . 2.2. Proposition. Let R be a 0-normal graded ring with R0 being (left) Noetherian. If δ = R0 then there exists a d ∈ ZZ such that Rd R−d = R0 = R−d Rd . Proof. Fix an a = 0 in ZZ and look at (Ra R−a )n for n ∈ IN . Assuming that n ≥ 1, we obtain: (Ra R−a )n = (Ra R−a )(Ra R−a )(Ra R−a ) . . . ⊂ Ra (Ra R−a )R−a (Ra R−a ) . . . . 2 The latter equals Ra2 R−a (Ra R−a ) . . . and repeated application of the passing Ra R−a over some R−a , to the left, we arrive at n (Ra R−a )n ⊂ Ran R−a ⊂ Rna R−na .

Consequently Ra R−a ⊂ rad(Rna R−na ), for any n ∈ IN . Now observe that: (R−a Ra )(R−a Ra ) = R−a (Ra R−a )Ra ⊂ Ra R−a R−a Ra and the latter is contained in Ra R−a since this is an ideal of R0 . Consequently R−a Ra ⊂ rad(Ra R−a ). Now if δ = R0 the 1 ∈ δ and thus there exist ﬁnitely many a1 , a2 , . . . , am ∈ ZZ such that 1 ∈ Ra1 R−a1 + Ra2 R−a2 . . . + Ram R−am

()

Since we have observed that R−a Ra ⊂ rad(Ra R−a ) for all a ∈ ZZ, it follows that we may interchange ai and −ai in the expression () and obtain 1 ∈ rad(Ra1 R−a1 + · · · + Ram R−am ), where we now have ai ∈ IN for i = 1, . . . , m (up to renaming). Since Ra R−a ⊂ rad(Rna R−na ) for any n ∈ IN we obtain that: rad(Ra1 R−a1 + · · · + Ram R−am ) ⊂ rad(Rd R−d ) where d ∈ IN is the lowest common multiple of a1 , . . . , am ∈ IN . It follows then that R0 = rad(Rd R−d ) or R0 = Rd R−d . Now look at R−d Rd and calculate: R−d Rd = R−d (Rd R−d )Rd = R−d (Rd R−d Rd ) ⊂ R−d (R−d Rd Rd )

QUOTIENT GROTHENDIECK REPRESENTATIONS

157

the latter following from 0-mormality applied to −d. ¿From R−d Rd ⊂ R−d R−d Rd Rd , we obtain: Rd (R−d Rd )R−d ⊂ Rd (R−d R−d Rd Rd )R−d , but applying Rd R−d = R0 then leads to: R0 ⊂ R−d Rd , hence R0 = R−d Rd , as well.

For d ∈ IN we let R(d) be the ZZ-graded ring deﬁned by R(d)n = Rnd (the dth Veronese subring of R). A ZZ-graded ring R is said to be d-strongly graded if R(d) is strongly graded (i.e R(d)1 R(d)−1 = R(d)−1 R(d)1 = R(d)0 ). 2.3. Corollary. Let R be a 0-normal (left) Noetherian ZZ-graded ring. If δ = R0 then R is d-strongly graded for some d ∈ IN . 2.4. Proposition. Let R be a left Noetherian ZZ-graded 0-normal ring and consider a perfect rigid torsion theory τ on R-gr, say with graded filter Lg (τ ). If δ = 0 and Rδ ∈ Lg (τ ) then the graded localization Qgτ (R) is a d-strongly graded ring for some d ∈ IN . Proof. Since R is left Noetherian, R0 is left Noetherian too. Hence rad(δ) is ﬁnitely generated as a left ideal and so we obtain that: rad(δ) = rad(Ra1 R−a1 + · · · + Ram R−am ), with a1 , . . . , am ∈ IN , using arguments formally similar to those used in the proof of Proposition 2.2. Then we also obtain that rad(δ) = rad(Rd R−d ) for some d ∈ IN , in fact we may take for d the lowest common multiple of a1 , . . . , am ∈ IN . Since τ is perfect and Rδ ∈ Lg (τ ), we obtain that Sδ = S, where we put S = Qgτ (R). Looking at the parts of degree zero, we obtain S0 δ = S0 , thus S0 rad(Rd R−d ) = S0 . Then for any m ∈ IN we also have that S0 rad(Rd R−d )

m

= S0

and because for some m0 ∈ IN , (rad(Rd R−d ))

m0

⊂ Rd R−d ,

it also follows that S0 Rd R−d = S0 . ¿From the obvious inclusions Sd ⊃ Rd , S−d ⊃ R−d , it then follows that Sd S−d = S0 .

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Applying the foregoing argument to R−d Rd , noting that rad(Rd R−d ) = rad(R−d Rd ), we arrive at Sd S−d = S−d Sd = S0 .

The equivalent of Proposition 2.4 in case δ = 0 and assuming I ∈ Lg (τ ) does hold when S is 0-normal graded; however at this moment we do not know whether the 0-normality of S follows from the 0-normality of R. 2.5. Lemma. In case R is a domain, δ = 0 if and only if R is positively or negatively graded, i.e resp. R = R≥0 or R = R≤0 . n Proof. Rn = 0 for some n > 0, then for any −m < 0 we have R−m Rnm = 0 because m n R−nm Rnm = 0 as δ = 0. Now Rn = 0 and thus R−m and R−m are zero.

For homogeneous Ore sets T in R, putting T (d) = T ∩ R(d), one easily veriﬁes that T (d) is an Ore set in R(d); indeed, for t ∈ T (d) and r ∈ R(d) we ﬁnd t ∈ T, r ∈ R, such that t r = r t hence (t )d r = (t )d−1 r t, with (t )d ∈ T (d) and (t )d−1 r ∈ R(d). However, not every Ore set of R(d) is of the form T (d). Such problems may be circumvented by developing a “weighted” space theory generalizing the commutative case but we do not go into that here. We content ourselves to pointing out an interesting case, allowing the noncommutative scheme theory and an interpretation in terms of quotient Grothendieck representations as in Section 1. The ZZ-graded ring R is said to be geometrically graded if R is a Noetherian, R0 is central in R (hence certainly 0-normalizing) and R is generated over R0 by R1 ∪ R−1 as a ring. 2.6. Proposition. Consider a geometrically graded ring R and a perfect rigid torsion theory τ on R-gr given by its graded filter Lg (τ ). 1. If δ = 0 and Rδ ∈ Lg (τ ) then S = Qgτ (R) is strongly graded. 2. If δ = 0 and I ∈ Lg (τ ) then S0 = S−1 S1 = S−n Sn for n ≥ 0. In case τ corresponds to an Ore set T of R that is homogeneous and not contained in R0 , then S is strongly graded. Proof. An arbitrary r ∈ R can be written as a sum of monomials of type r0 x1 x2 . . . xn where r0 ∈ R0 and each xi is either in R1 or in R−1 . In case xi ∈ R1 and xi+1 ∈ R−1 , or conversely, then xi xi+1 ∈ R0 and therefore it is central in R. Consequently, such a monomial e e or in R−1 R1d for suitable e and d in IN . In fact, if r ∈ Rn with n ≥ 0, then is in R1d R−1 we see in the same way that r ∈ R1n by putting factors in degree zero at the beginning −m of the monomials in the expression of r as above; for m ≤ 0 we ﬁnd that Rm = R−1 . n n Summarizing, for n ≥ 0 we have Rn = R1 , R−n = R−1 . 1. When δ = 0 then we establish for some d ∈ IN that S0 Rd R−d = S0 R−d Rd = S0 , just as in the proof of Proposition 2.4. However: d Rd R−d = R1d R−1 = (R1 R−1 )d

QUOTIENT GROTHENDIECK REPRESENTATIONS

159

d

follows from foregoing remarks. Then it follows from S0 (R1 R−1 ) = S0 that S0 (R1 R−1 ) = S0 . Similarly we arrive at S0 (R−1 R1 ) = S0 from S0 R−d Rd = S0 . Clearly we then obtain that S1 S−1 = S0 = S−1 S1 . 2. In this situation δ = 0 if and only if R1 R−1 = R−1 R1 = 0 or Rn Rm = 0 for n > 0, m < 0. Consequently δ = 0 if and only if R is either positively or negatively graded depending whether R1 = 0, or resp. R−1 = 0. Let us treat the positively graded case (the negatively graded case may be treated in a similar way, note however that the statement in the proposition should then be given as S1 S−1 = S0 ). So R = ⊕ Rn , I = ⊕ Rn . The assumption I ∈ Lg (τ ) and τ being a perfect n≥0

n>0

torsion theory, leads to S = SI, hence by looking at the parts of degree zero: S0 =

n>0

S−n In =

S−n Rn .

n>0

Look at a typical element s−n rn with s−n ∈ S−n , rn ∈ Rn . For some I ∈ Lg (τ ) we have: Ip s−n rn ⊂ Rp−n Rn ⊂ Rp ,

()

because Ip s−n ⊂ Rp−n for all p. Thus, for any p > 0; S−p Ip s−n rn ⊂ S−p Rp = S−p R1p−1 R1 ⊂ S−1 R1 ⊂ S−1 S1 . Now s−n

S−p Ip = (SI)0 = S0 because I ∈ Lg (τ ), this leads to S0 s−n rn ⊂ S−1 S1 where ∈ S−n , rn ∈ Rn as well as n were arbitrary. ¿From S0 = S−n Rn it thus follows p>0

n>0

that S0 = S−1 S1 . Note that in () we may indeed use that I0 = 0 because if I ∈ Lg (τ ) then I ∩ I ∈ Lg (τ ) has (I ∩ I)0 = 0 and we may replace I by the smaller I ∩ I in (). Note also that from the foregoing information it does not follow that S1 S−1 = S0 ! However if τ is associated to a (left)Ore Set, T say, then the strongly graded condition does follow. Indeed, for y ∈ S0 look at ytm for some tm ∈ T ∩ Rm , m > 0. Since tm −1 is invertible in S and t−1 m ∈ S−m we may consider (ytm )tm = y ∈ Sm S−m . ¿From m S−1 S1 = S0 we may derive Sm = S1 , indeed Sm = Sm S0 = Sm S−1 S1 yields Sm = Sm−1 S1 and by repetition of the argument (Sm−1 = Sm−2 S1 etc. . .) we obtain Sm = S1m . Now y ∈ S1m S−m = S1 (S1m−1 S−m ) ⊂ S1 S−1 . Thus S0 = S1 S−1 . To a ZZ-graded ring R we associate a torsion theory on R-gr, denoted by κR , deﬁned by taking for its graded ﬁlter Lg (κR ) the graded ﬁlter generated by Rδ and I. Note that the ideal I = δ ⊕ ( ⊕ Rn ) is automatically in Lg (κR ) because it contains Rδ, in case n=0

δ = 0. We can now deﬁne schematically graded rings by looking at the class of graded ring R such that there is a ﬁnite number of homogeneous Ore sets T1 , . . . , Tm such that κR = κT1 ∧ . . . ∧ κTm (if so desired one may weaken the deﬁnition to κTi that are only perfect rigid torsion theories).

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We entend to use the κR in deﬁning a topological nerve as mentioned after the proof of Corollary 1.4 and used in Proposition 1.5 and Theorem 1.8. For that we need the following easy lemma. 2.7. Lemma. Let R and S be ZZ-graded rings with either δR = 0 and δS = 0 or else δR = δS = 0. If f : R → S is a morphism of graded rings then κS ≤ f˜(κR ). Proof. Since f (Rn ) ⊂ Sn for every n ∈ ZZ, it is clear that f (δR ) ⊂ δS and also that f (IR ) ⊂ IS . Now L ∈ L(f˜(κR )) means that S/L is a κR -torsion as an R-module, i.e L contains some I ∈ L(κR ). The statement now follows easily. Consider the category B, consisting of ZZ-graded rings R with δR = 0 and graded ring morphisms. The association of R-gr to R deﬁnes a Grotendieck representation. The foregoing lemma entails that κR deﬁnes a nerve and therefore also a quotient representation with respect to the nerve κR . Applying the methods of [15] we obtain a satisfactory theory for P rojR which is deﬁned by the noncommutative topology on (R-gr, κR ) (see Section 1) and the corresponding sheaf theory. 2.8. Conclusion. If R is geometrically graded and schematic then P rojR deﬁned on (Rgr, κR ) satisﬁes all the properties valid in the positively graded case, in particular the schematic property in combination with Proposition 2.6 yields the existence of an aﬃne covering (in the sense of [15], [17]), moreover the proof of Serre’s global section theorem given for P rojR of a positively graded ring carries over this situation too. All this follows from a trivial modiﬁcation of the proofs given in the positively graded case, taking into account the results included in this section, so we omit this repetition here.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13] [14]

P. Gabriel, Des cat´egories ab´eliennes, Bull. Soc. Math. France 90, (1962), 323–448. J. Golan, Localization of Noncommutative Rings, M. Dekker, New York, 1975. O. Goldman, Rings and Modules of Quotients, J. of Algebra 13, (1969), 10–47. L. Le Bruyn, M. Van den Bergh and F. Van Oystaeyen, Graded Orders, Birkhauser Verlag, Basel 1988. H. Li and F. Van Oystaeyen, Zariskian Filtrations, K-Monogr. Math. 2, Kluwer Acad. Publ. Dordrect 1996. S. Maclane, Categories for the Working Mathematician, Springer-Verlag, New York, 1974. C. Nˇ astˇ asescu, F. Van Oystaeyen, Graded Ring Theory, Library of Math. 28, North-Holland, Amsterdam 1982. C. Nˇ astˇ asescu, F. Van Oystaeyen, Dimensions of Ring Theory, D. Reidel Publ. Co., 1987. J. Ndirahisha, F. Van Oystaeyen, Grothendieck Representations of Categories and Canonical Noncommutative Topologies, J.of K- Theory, to appear. R. Sallam, F. Van Oystaeyen, A microstructure Sheaf and Quantum Sections over a Projective Scheme, J. Algebra 158(1), 1993, 201–225. J.P. Serre, Faisceaux Alg´ebriques Coh´ erents, Ann. Math. 61, 1955, 197–278. B. Stenstr¨ om, An Introduction to Methods of Ring Theory, Die Grundlehren der Mathematischen Wissenschaften, Vol. 217, Springer, Berlin, 1975. F. Van Oystaeyen, Compatibility of Kernel Functors and Localization Functors, Bull. Soc. Math. Belg., XVIII, 1976, 131–137 F. Van Oystaeyen, Prime Spectra in Noncommutative Algebra, Lect. Notes in Math. 444, Springer-Verlag, Berlin 1978.

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[15] F. Van Oystaeyen, Algebraic Geometry for Associative Algebras, Pure and Applied Mathematics, Vol. 232, M. Dekker, New York, 2000. [16] F. Van Oystaeyen, A. Verschoren, Non-commutative Algebraic Geometry, LNM 887, Springer-Verlag, Berlin 1981. [17] F. Van Oystaeyen, L. Willaert, Grothendieck Topology, Coherent Sheaves and Serre’s Theorem for Schematic Algebras, J. Pure Applied Algebra 104, 1995, 109–122.

ON THE STRONG RIGIDITY OF SOLVABLE LIE ALGEBRAS TOUKAIDDINE. PETIT1 Departement Wiskunde en Informatica, Universiteit Antwerpen, B-2020 (Belgium)

Abstract. We call a ﬁnite-dimensional complex Lie algebra g strongly rigid if its universal enveloping algebra Ug is rigid as an associative algebra, i.e. every formal associative deformation is equivalent to the trivial deformation. The aim of this paper is to study the strong rigidity properties of solvable Lie algebras. First, we show that a strongly rigid Lie algebra has to be rigid as Lie algebra, this restricts the research to rigid Lie algebras. In addition the second scalar cohomology group has to vanish. Therefore the nilpotent Lie algebras of dimension greater or equal than two are not strongly rigid and the torus’s dimension of strongly rigid solvable Lie algebra has to be one. Moreover, the Kontsevitch’s theory of deformation quantization helps to see that every polynomial deformation of the linear Poisson structure on g∗ which induces a nonzero cohomology class of g leads to a nontrivial deformation of Ug. Since the rigidity is intimately related to cohomology, the cohomology groups are characterized. At last, we classify the n-dimensional strongly rigid solvable Lie algebras where n ≤ 6 and give some remarks on linearizability of their corresponding Poisson structure.

1. Introduction The deformation of of rings and algebras was introduced by M. Gerstenhaber in 1964 ([12]). He gave a tool to deform algebraic structure based on formal power series. The interest on deformation has grown with the development of quantum groups related to quantum mechanics ([2]). Examples of quantum groups may be obtained as Hopf algebra deformation of enveloping algebra of Lie algebra. A formal deformation of an associative (resp. Lie) algebra (A, µ) is an associative (resp. Lie) algebra A[[t]] with a multiplication µt deﬁned by µt (p, q) = µ(p, q) + tµ1 (p, q) + t2 µ2 (p, q) + · · ·

(1.1)

where p, q ∈ A The algebra is said rigid if every formal deformation is isomorphic to a trivial deformation. The rigidity theorem of Gerstenhaber [12] (resp. of Nijenhuis-Richardson [20]) insure that if the 2nd Hochschild cohomological group H2H (A, A) (resp. Chevalley-Eilenberg H2CE (g, g)) of an associative algebra A (resp. a Lie algebra g) vanishes then the algebra (rep. Lie algebra) is rigid. Therefore the semisimple associative (resp. Lie) algebras are rigid because their second cohomology groups are trivial ([14]). The rigidity of n-dimensional complex rigid Lie algebras was studied by R.Carles, Y.Diakit´e, M.Goze and J.M. Ancochea-Bermudez. Carles and Diakit´e established the classiﬁcation for n ≤ 7 ([6],[4]), and Ancochea with Goze did the classiﬁcation for solvable 1 Author supported by the Scientific Programme NOG of the European Science Foundation, e-mail:[email protected]

ON THE STRONG RIGIDITY OF SOLVABLE LIE ALGEBRAS

163

Lie algebras for n = 8 and some classes ([10], [1]). The classiﬁcation of associative rigid algebras are known up to n ≤ 6 (see [19]). In this paper we are interested in the deformation and rigidity of enveloping algebras associated to solvable Lie algebras. We have introduced in ([3]) the notion of strong rigidity of a Lie algebra. A Lie algebra is said strongly rigid if its enveloping algebra is rigid as an associative algebra. The paper is organized as follows. In Section 2 we summarize the deﬁnitons and recall some important and useful results, namely the Cartan-Eilenberg theorem and HochschildSerre factorization theorem. In Section 3 we introduce the strong rigidity of a Lie algebra and give some properties. We show that a strongly rigid Lie algebra has to be rigid as a Lie algebra. In addition, the scalar second cohomology group has to vanish. Therefore, it permits to construct some classes of non strongly rigid Lie algebras. As an example of a strongly rigid Lie algebra, we consider the 2-dimensional non abelian Lie algebra. We show by a direct calculation that the second Hochschild cohomology group of its enveloping algebra with values in the algebra is trivial. Thus this Lie algebra is strongly rigid. Section 4 is devoted to the deformation of the enveloping algebra via the Poisson structures. We recall the result of [3] that every nontrivial polynomial deformation of the linear Poisson structure associated the the Lie algebra induces a nontrivial deformation of the enveloping algebra. In Section 5 we characterize the cohomology groups HnCE (g, Ug)). At last, we apply the previous results to classify the strongly rigid Lie algebras in small dimensions and deduce some remarks on the linearization of Poisson structures. 2. Preliminaries 1. Let g be a ﬁnite dimensional decomposable solvable Lie algebra, i.e g = t ⊕ n where n is the nilradical and t is an exterior torus of derivations in Malcev’s sense; that is t is an abelian subalgebra of g such that adX is semisimple for all X ∈ t. This class of solvable Lie algebra contains the rigid Lie algebras ([4]). 2. Let K be a commutative ring and g be a Lie algebra over K. Recall that a (left) g-representation of g is a K-module M and a K-homomorphism g ⊗ M → M x ⊗ a → xa

(2.1)

such that x(ya) − y(xa) = [x, y]a. To each Lie algebra g, we associate an associative K-algebra Ug such that every (left) g-representation may be viewed as (left) Ugrepresentation and vice-versa. The algebra Ug is constructed as follows Let T g be a tensor algebra of K-module g , T g = T 0 ⊕ T 1 ⊕ · · · ⊕ T n ⊕ · · · where T n = g ⊗ g ⊗ · · · ⊗ g (n times). In particular T 0 = K1 and T 1 = g. The multiplication in T g is the tensor product. Every K-linear map g ⊗ M → M has a unique extension to a map T g ⊗ M → M. The g-module is a g-representation if and only if the elements of T g of the form x ⊗ y − y ⊗ x − [x, y] where x, y ∈ g annihilate M. Consequently, we are led to introduce the two-sided ideal I generated by the elements x ⊗ y − y ⊗ x − [x, y] where x, y ∈ g. We deﬁne the enveloping algebra Ug of g as T g/I. Thus, g-representations and the Ug-modules may be identiﬁed. Recall that every bimodule M is a g-module by (x, m) → xm − mx, denoted by Ma . Assume that g is a free Lie algebra. Let {xi } be a ﬁxed basis of g and yi be the image of xi by the K-homomorphism i : g → Ug. We set yI = yi1 · · · yip with I a ﬁnite

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sequence of indices i1 , . . . , ip and yI = 1 if I = ∅. The Poincar´e-Birkhoﬀ-Witt theorem insures that the enveloping algebra Ug is generated by the elements yI corresponding to the increasing sequences I. We denote by SV the symmetric algebra over a K-module V . If Q ∈ K, then there exists a canonical bijection between Sg and Ug which is a g-module isomorphism between Sg and Uga ( [9, pp.78–79] ) 3. Unless otherwise stated, K denotes an algebraically closed ﬁeld of characteristic 0. Let K[[t]] be the power series ring with coeﬃcients in K. For a K-vector space E we denote by E[[t]] the K[[t]]-module of the power series with coeﬃcients in E. Let (A, µ0 ) be an associative (resp Lie) K-algebra, then (A[[t]], µ0 ) is an associative (resp. Lie) K[[t]]-algebra. (a) A formal deformation of an associative (resp. Lie algebra) A is an associative (resp. Lie) K[[t]]-algebra (A[[t]], µt ) such that µt = µ0 + tµ1 + t2 µ2 + · · · + tn µn + · · · , where µn ∈ HomK (A ⊗K A, A). (resp. µn ∈ HomK (A ∧K A, A)). (b) Two deformations (A[[t]], µt ) and (A[[t]], µt ) are said equivalent if there exists a formal isomorphism ϕt = ϕ0 + ϕ1 t + · · · + ϕn tn + · · · , with ϕ0 = IdA (Identity map on A) and ϕn ∈ End(A) such that µt (a, b) = ϕ−1 t (µt (ϕt (a), ϕt (b)) ∀a, b ∈ A.

(c) A deformation of A is said trivial if it is equivalent to (A[[t]], µ0 ). (d) An associative (resp. Lie) algebra A is said rigid if every deformation of A is trivial. 4. The deformation theory is related to Hochschild cohomology in the case of associative algebra and Chevalley-Eilenberg cohomology in the case of Lie algebra. We denote by HnH (A, M) the n-th Hochschild cohomology group of an associative algebra A with values in the bimodule M and by HnCE (g, M) the n-th Chevalley-Eilenberg cohomology group of a Lie algebra A with values in a g-module M. The second Hochschild cohomology group of an associative algebra (resp. Chevalley-Eilenberg cohomology group of a Lie algebra ) with values in the algebra may be interpreted as the group of inﬁnitesimal deformations. It follows that if this group is trivial then the algebra is rigid. The third cohomology group corresponds to the obstructions to extend a deformation of order n to a deformation of order n + 1 ([12],[13] and [20]). 5. The following classical theorem due to H.Cartan et S.Eilenberg, ([7, pp.277]) gives a link between the Hochschild cohomology of an enveloping algebra with values in an Ug-bimodule M (in particular M = Ug) and the Chevalley-Eilenberg cohomology of the Lie algebra with values in the same module. Theorem 2.1. Let g be a ﬁnite dimensional Lie algebra over K. Then HnH (Ug, M) HnCE (g, Ma ) ∀n ∈ N

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In particular, if Q ⊂ K HnH (Ug, Ug) HnCE (g, Uga ) HnCE (g, Sg) ∀n ∈ N 6. The Hochschild-Serre theorem [17] gives the following factorization of the ChevalleyEilenberg cohomology groups in the case of a decomposable solvable Lie algebra. Theorem 2.2. Let g = n⊕t be a ﬁnite dimensional solvable Lie algebra over K, where n is the largest nilpotent ideal of g and t the supplementary subalgebra of n, reductive in g, such that the t-module induced on Ua g is semisimple, then for all positive integers p, we have Hp CE (g, U a (g))

i+j=p

t

HiCE (t, K) ⊗ HjCE (n, Ua g) .

where HjCE (n, Ua g)t denotes the subspace of the t-invariant elements.

3. Strongly rigid Lie algebras and properties We recall here the notion of strong rigid Lie algebra introduced in [3]. Definition 3.1. A Lie algebra g is said strongly rigid if its enveloping algebra Ug is rigid as an associative algebra. The semisimple Lie algebras give examples of strongly rigid Lie algebras. In fact, the Whitehead lemmas induce that the ﬁrst and second cohomology groups of a Lie algebra g with values in every ﬁnite dimensional K-module vanish. Therefore these Lie algebras are rigid as Lie algebra. Using the ﬁltration of Sg and the Cartan-Eilenberg theorem we obtain 2 HH (Ug,Ug) = 0. Therefore, the enveloping algebra of a semisimple Lie algebra is rigid. 3.1. The rigidity of the Lie algebra. Theorem 3.1. If g is a ﬁnite dimensional strongly rigid Lie algebra over K, then g is rigid as a Lie algebra. Proof. We suppose that the enveloping algebra Ug of g is rigid, but not theLie algebra g. ∞ Then there exists a nontrivial formal deformation (g[[t]], µt ) of g with µt = n=0 µn tn and the cohomology class of µ1 is nontrivial in H2CE (g, g). Since g is ﬁnite dimensional, then the K[[t]]-module g[[t]] is isomorphic to the free module g ⊗K K[[t]]. Let yI := yi1 · · · yik be the generators of the PBW basis of Ug, let yI := yi1 • · · · • yik be the generators of PBW basis of U g[[t]] over K[[t]] and that • is the multiplication in U g[[t]] . The map Φ : Ug ⊗K K[[t]] → U g[[t]] deﬁned by Φ(yI ) := yI is a K[[t]]-module isomorphism. Let on the module Ug⊗K K[[t]] πt : Ug⊗K K[[t]] × Ug⊗K K[[t]] → Ug⊗KK[[t]] the multiplication induced by • and Φ, i.e. πt (a, b) := Φ−1 Φ(a) • Φ(b) . The restriction of πt to elements of Ug × Ug deﬁned a K-bilinear map Ug × Ug → Ug ⊗K K[[t]] ⊂ Ug[[t]] which we denote also ∞ by πt , i.e. πt (u, v) = n=0 tn πn (u, v) for all u, v ∈ Ug where πn ∈ HomK (Ug ⊗ Ug, Ug). The K-bilinear map πt deﬁned naturally a K[[t]]-bilinear associative multiplication over the

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K[[t]]-module Ug [[t]] (which contains Ug ⊗K K[[t]] as a dense submodule with the t-adique topology) : ∞ ∞ ∞ s s πt t us , t vs := tr πs (us , vs ) s

r=0

s =0

s,s ,s ≥0

s+s +s =r

In particular, the map π0 deﬁned an associative multiplication over the vector space Ug, and (Ug [[t]], πt ) is a formal associative deformation of (Ug, π0 ). For a ﬁnite increasing sequence I, J we have π0 (yI , yJ ) = Φ−1 (yI • yJ )|t=0 . By ordering the product yI • yJ we obtain that π0 is the multiplication of Ug and (Ug [[t]], πt ) is a formal deformation of Ug. It follows that π1 is a Hochschild 2-cocycle of Ug, and the restriction of π1 to X, Y ∈ g satiﬁes µ1 (X, Y ) = π1 (X, Y ) − π1 (Y, X)

∀X, Y ∈ g.

(3.1)

because the Lie algebra (g[[t]], µt ) is a Lie subalgebra of U(g[[t]]) which may considered as an associative subalgebra ∞of (Ug [[t]], πt ). The rigidity of Ug implies that there exists a formal isomorphism ϕt = r=0 ϕr tr , where ϕ0 = IdU g and ϕn ∈ HomK (Ug, Ug) such that ϕt (πt (u, v)) = πt (ϕt (u), ϕt (v)),

∀u, v ∈ Ug,

which is equivalent to ∞ r=0

tr

ϕa (πb (u, v)) =

∞ r=0

a,b≥0

a+b=r

tr

πa (ϕb (u), ϕc (v))

∀u, v ∈ Ug.

(3.2)

a,b,c≥0

a+b+c=n

If r = 1, the relation becomes π1 (u, v) = (δH ϕ1 )(u, v)

∀u, v ∈ Ug

(3.3)

where δH is a Hochschild cobord operator (see [16]) with respect the multiplication π0 of the enveloping algebra. Then the formulae (3.1) and (3.3) imply µ1 (X, Y ) = (δH ϕ1 )(X, Y ) − (δH ϕ1 )(Y, X) = Xϕ1 (Y ) − ϕ(XY ) + ϕ(X)Y − Y ϕ1 (X) + ϕ(Y X) − ϕ(Y )X = (δCE ϕ1 )(X, Y ) ∀X, Y ∈ g

(3.4)

where δCE is the Chevalley-Eilenberg cobord operator. (see [7]). Therefore the class of µ1 in H2CE (g, g) is trivial. contradiction. This result show that the class of strongly rigid Lie algebras is contained in the class of rigid Lie algebras.

ON THE STRONG RIGIDITY OF SOLVABLE LIE ALGEBRAS

167

3.2. Second scalar cohomology group. In this Section we give a necessary condition on the scalar Chevalley-Eilenberg cohomology group for the strong rigidity of a Lie algebra. Let ω ∈ Z2CE (g, K) be a scalar 2-cocycle of the Lie algebra g. Let gω = g ⊕ Kc be a central extension of g with ω such that the new bracket [ , ] is deﬁned as usually by [X + ac, Y + bc] := [X, Y ] + ω(X, Y )c ∀X, Y ∈ g; a, b ∈ K.

(3.5)

Theorem 3.2. Let g be a ﬁnite dimensional Lie algebra over K such that the second scalar cohomology group H2CE (g, K) is diﬀerent from 0, then g is not strongly rigid. Proof. Let ω ∈ Z2CE (g, K) be a 2-cocycle with a nonzero class and let gtω [[t]] be the onedimensional central extension of the Lie algebra g[[t]] = g ⊗K K[[t]] over K = K[[t]] (see (3.5)). The multiplication of the enveloping algebra U(gtω [[t]]) of gtω [[t]] is denoted by •. Let consider the two-sided ideal I := (1 − c ) • U(gtω [[t]]) = U(gtω [[t]]) • (1 − c ) (where c denote the image of c in U gtω [[t]] ) and the quotient algebra Utω g := U(gtω [[t]])/I. Let e1 , . . . , en be the K-basis of g. Then c, e1 , . . . , en is a K[[t]]-basis of gtω [[t]]. Let y1 , . . . , yn be the images of the basis vectors in Ug and c , y1 , . . . , yn be the images of the basis vectors in U gtω [[t]] . Let yI := yi1 . . . yik in Ug over K be the generators of the PBW basis. The elements c •i0 • yI (where i0 ∈ N and c •i0 : = 1) form a basis of U gtω [[t]] over K[[t]]. (the Lie algebra is a free module over a commutative ring, see [7], p.271). In the quotient algebra Utω g, the element c •i0 is identiﬁed to 1. We denote the multiplication in Utω g by · and by the canonical projection, the images of y1 , . . . , yn by y1 , . . . , yn the elements yI give yI := yi1 · . . . · yin . It follows that the elements yI form a basis of the quotient algebra Utω g. As in the proof of the previous theorem 3.1, the map Φ : Ug ⊗K K[[t]] → U gtω [[t]] given by yI → yI deﬁnes an isomorphism of free K[[t]]-modules. In a similar way we show that the multiplicationinduced on Ug ⊗K K[[t]] by the multiplication · of Utω g and Φ deﬁne a ∞ sequence of πt = r=0 πr tr , where πr ∈ HomK (Ug ⊗ Ug, Ug) with the following properties: 1. πt deﬁnes a formal associative deformation of (Ug, π0 ), 2. π0 is the usual multiplication of the enveloping algebra Ug of g. Therefore, π1 is a Hochschild 2-cocycle of Ug, and for all X, Y ∈ g ⊂ Ug we have the relation: ω(X, Y )1 = π1 (X, Y ) − π1 (Y, X) because the Lie algebra gtω [[t]] is injected in the quotient algebra Utω g, then in Ug ⊗K K[[t]] ⊂ Ug [[t]]. Suppose that Ug is rigid, then the deformation πt is trivial. Therefore there exists a Hochschild 1-cocycle ϕ1 ∈ C1H (Ug, Ug) such that π1 = δH (ϕ1 ). It follows ∀X, Y ∈ g: ω(X, Y )1 = π1 (X, Y ) − π1 (Y, X) = δH (ϕ1 )(X, Y ) − δH (ϕ1 )(Y, X) = δCE (ϕ1 )(X, Y ). Then ω is a Chevalley-Eilenberg cobord and its class is trivial in H2CE (g, K), contradiction. 3.3. Examples of non strongly rigid Lie algebras. The previous theorems permit to show that some classes of solvable Lie algebras are not strongly rigid. Corollary 3.1. The following Lie algebras are not strongly rigid : 1. Every n-dimensional nilpotent Lie algebra g with n greater or equal than 2. 2. Every Lie algebra g = t ⊕ n where the dimension of the torus t is greater or equar than 2. Proof. The ﬁrst assertion is a consequence of a classical result of Dixmier concerning the nilpotent Lie algebras ([8]): H2CE (g, K) = 0 if dim(g ≥ 2). For the second, we have H2CE (t, K) = {0} for an abelian Lie subalgebra.

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3.4. Example of a strongly rigid Lie algebra. In this section we prove that the 2-dimensional non abelian solvable Lie algebra is strongly rigid. We denote by r2 the solvable Lie algebra generated by X, Y such that [X, Y ] = Y . Lemma 3.1. 1. ∀n, m ∈ N : Y X n = (X − 1)n Y , [X, Y m ] = mY m and ∀n ∈ N, ∀m ∈ ∗ N (m − 1)X n Y m = [X, X n Y m ] − X n Y m . 2. There exists a polynomial Pn+1 (X) in X of degree n + 1 such that : n+1 k Pn+2−k (X), if n ≥ 1. (a) P1 (X) = X and Pn+1 (X) = X n+1 + k=2 (−1)k Cn+2 n (b) (n + 1)X Y = [Pn+1 (X), Y ]. Proof. The ﬁrst assertion may easily be proved by induction. Let us prove the property (2) by induction on n. It is true for n = 0, because [P1 (X), Y ] = [X, Y ] = Y . Assume that it is true until n. We have (a): [X n+2 , Y ] = X n+2 Y − Y X n+2 = X n+2 Y − (X − 1)n+2 Y following (1) =X

n+2

Y −

n+2

k (−1)k Cn+2 X n+2−k Y

k=0

= (n + 2)X n+1 Y −

n+2

k (−1)k Cn+2 X n+2−k Y

k=2

Applying the induction hypothesis on n + 2 − k with k ≥ 2, we obtain X n+2−k Y = [Pn+3−k (X), Y ], (the degree of Pn+3−k (X) = n + 3 − k ≤ n + 1). Then (b) becomes : n+2 k Pn+3−k (X), Y ] = [Pn+2 (X), Y ] (n + 2)X n+1 Y = [X n+2 + k=2 (−1)k Cn+2 In the following we show by a direct calculation, for the Lie algebra r2 , that the second Hochschild cohomology group of its enveloping algebra with values in the algebra is trivial. Thus this Lie algebra is strongly rigid. Proposition 3.1. Let r2 be the 2-dimensional non abelian Lie algebra. We have H2H (Ur2 , Ur2 ) H2CE (r2 , Ur2 ) = 0 Thus, the Lie algebra r2 is strongly rigid. Proof. By Cartan-Eilenberg theorem we have H2H (Ur2 , Ur2 ) H2CE (r2 , Ur2 ). We will show that ∀Φ ∈ Z2CE (r2 , Ur2 ) ∃f ∈ C1CH (r2 , Ur2 ) s.t. δCE (f ) = Φ

(∗)

Let {X n Y m : n, m ∈ N} be the Poincar´e-Birkhoﬀ-Witt basis of Ur2 . Let Φ be an element of Z2CE (r2 , Ur2 ). It is deﬁned by Φ(X, Y ) =: u =: n,m∈N un,m X n Y m where un,m ∈ K

ON THE STRONG RIGIDITY OF SOLVABLE LIE ALGEBRAS

169

are nonzero for a ﬁnite number of n, m. Let f be an element of C1CH (r 2 , Ur2 ). It is deﬁned by two elements f (X) =: v =: n,m∈N vn,m X n Y m and f (Y ) = w = n,m∈N wn,m X n Y m where vn,m , wn,m ∈ K are nonzero for a ﬁnite number of n, m. Then ∀u =

un,m X n Y m ∈ Ur2 ∃v, w ∈ Ur2 tels que u = [X, w] − w + [v, Y ] (∗∗)

n,m∈N

We study two cases Case 1: m = 1. un,m We set wn,m = m−1 , then vn,m = 0 if m = 1 and v, w satisfy (∗∗) by lemma (3.1,(2)). Case 2: m = 1. 1 We set vn,m = n+1 un,1 Pn+1 (X) then wn,m = 0 if m = 1 and v, w satisfy (∗∗) by lemma (3.1,(1)). We conclude that the relation (∗∗) is satisﬁed. Therefore H2CE (r2 , Ur2 ) = 0, and the Lie algebra r2 is strongly rigid.

4. Deformation of enveloping algebras by quantification In this section, we recall a result of [3] which said that a nontrivial polynomial deformation of the linear Poisson structure associated to the Lie algebra induces a nontrivial deformation of the enveloping algebra. We recall ﬁrst the Poisson structure. We refer to Vaisman’s book ([21]) for a complete description. 1. A Poisson algebra is a commutative associative algebra A over K with a bilinear map {, } : A × A → A satisfying for f, g, h ∈ A (1) {f, g} = −{g, f } (2) {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0 (Jacobi identity ) (3) {h, f g} = {h, f }g + f {h, g} ( Leibniz relation) We denote by (A, ·, {, }) such an algebra. A manifold M is called a Poisson manifold if the algebra of functions, C∞ (M ), has a Poisson structure. A Poisson structure is determined by a skew-symmetric bilinear form on T ∗ M . In other word there exists a tensor ﬁeld P∈ Γ(M, Λ2 T M ) (with T M the ﬁbre bundle of M ) such that {f, g} = P (df, dg) = i,j P ij ∂i f ∂j g, where ∂i denotes the partial derivative with respect to the local coordinate xi . The tensor ﬁeld P is called the Poisson bivector of (M, { , }). A Poisson structure on M is given by a bivector P ∈ Γ(M, Λ2 T M ) satisfying h

P ih ∂h P jk + P jh ∂h P ki + P kh ∂h P ij = 0

i = Γ(M, Λi T M ) be the space of all skew symmetric tensor ﬁelds of rank 2. Let Tpoly 0 i i on a manifold M , Tpoly = C ∞ (M ) and Tpoly = (⊕n≥0 Tpoly , ∧) the algebra of multivectors on M .

A bivecteur P ∈ T2poly deﬁned a Poisson structure if and only if the SchoutenNijenhuis bracket [P, P ]s = 0. The operator δP := [P, −]s determines the so-called Poisson cohomology.

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3. Let g be a ﬁnite dimensional Lie algebra over K and g∗ its algebraic dual. The symmetric algebra Sg is identiﬁed to the algebra of polynomial functions on g∗ . The Lie algebra structure of g induces a linear Poisson structure on g∗ . {f, g}(x) = x([df (x), dg(x)]) with f, g ∈ Sg and x ∈ g∗ . Let (ei )i=1...n be a basis of g, (ei )i=1...n the dual basis and n x = i=1 xi ei ∈ g∗ , {f, g} = P0 (df, dg) with P0 the bivector deﬁned by P0 =

1 ij k P0 ∂i ∧ ∂j o` u P0ij (x) = Cij xk 2 i,j

(4.1)

k

k where Cij are the structure constants of g. Therefore, the Poisson algebra structure on S(g). 4. In the theory of deformation by quantiﬁcation ([2]), one associates to the Poisson structure a formal deformation of the associative commutative algebra C ∞ (M ), called star product, see e.g. [21] for the deﬁnition. The existence of a star product for every Poisson structure was established by Kontsewitsh see [18]. Using this result we have proved in [3] the following theorem

Theorem 4.1. Let g be a ﬁnite dimensional Lie algebra over K. Let P0 be the bivector deﬁning the linear Poisson structure on g∗ . Assume that it exists a sequence (Pn )n∈N of polynomial bivectors (Pn ∈ Sg ⊗ ∧2 g∗ ) such that i+j=n [Pi , Pj ]s = 0 for n ∈ N and P1 is not cohomologeous to 0. Then Pt = n≥0 tn Pn is a nontrivial deformation of the Poisson structure P0 and it induces a nontrivial deformation of the enveloping algebra Ug. Therefore, the Lie algebra g is not strongly rigid.

5. Some cohomological properties Let g = t ⊕ n be a ﬁnite-dimensional decomposable solvable Lie algebra, where n is the largest nilpotent ideal and t is an exterior torus of derivations of g such that the center of t is t (this condition holds for rigid Lie algebras ([4])). The group HH (Ug, Ug) is isomorphic to HCE (g, Uga ) using the Cartan-Eilenberg theorem 2.1 where the enveloping algebra Ug is considered as Ug-bimodule and Ua g is considered as an adjoint g-module with X.u := [X, u] := Xu − uX, where X ∈ g, u ∈ Ug [9]. Let Un be the two-sided ideal of Ug generated by n and Z(Ug) be the center of the enveloping algebra. We denote by Ugt (resp. Unt , Utt ), the t-invariant elements of Ug (resp. Un , Ut). The group HCE (g, Uga ) may be deduced from the t-invariant cohomology group HCE (n, Un )t under some assumptions on the torus t (over g). The g-adjoint module Uga is an inductive limit of adjoint sub-g-modules (Uk g)k≥0 where (Uk g)k≥0 is the canonical ﬁltration Ug ([9]). In order to simplify the notation, we denote next the adjoint module Uga by Ug. By the Hochschild-Serre factorization theorem 2.2 we obtain : Proposition 5.1. Let g = t ⊕ n be a decomposable solvable Lie algebra. Then H1CE (g, Ug) t∗ ⊗ Z(Ug) ⊕ H1CE (n, Ug)t H2CE (g, Ug) (∧2 t∗ ) ⊗ Z(Ug) ⊕ t∗ ⊗ H1CE (n, Ug)t ⊕ H2CE (n, Ug)t

(5.1)

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171

The diﬀerent t-modules deduced canonically from the t action on Ug are locally semisimple. The exact sequence of t-modules : 0 → Un → Ug → Ug/n → 0

(5.2)

implies a cohomological exact sequence, which corresponds if we restrict to t-invariant groups the exact sequence : 0 → H0CE (n, Un )t → H0CE (n, Ug)t → H0CE (n, Ug/n)t → H1CE (n, Un )t p+1 t t → · · · → HpCE (n, Ug/n)t → Hp+1 CE (n, Un ) → HCE (n, Ug) → t → Hp+1 CE (n, Ug/n) → · · ·

(5.3)

Assume that HpCE (n, Ug/n)t = 0 for p ≥ 1, and (Un )t = 0. This conditions holds if it exists an element X of t such that the eigenvalues of adX|n are positive in an ordered subﬁeld of K. Recently, M. Goze and E. Remm showed that H2CE (g, C) = 0 if and only if λ = 0 is an eigenvalue ([11]). The sequence (5.3) implies the exact sequence 0 → Ugt → Utt → H1CE (n, Un )t → H1CE (n, Ug)t → 0

(5.4)

and the isomorphisms : p+1 t t Hp+1 CE (n, Un ) HCE (n, Ug) for all p ≥ 1

(5.5)

Then, the Chevalley-Eilenberg cohomology groups of g with values in Ug become: H1CE (g, Ug) t∗ ⊗ Z(Ug) ⊕ H1CE (n, Un )t /(Ut/Z(Ug)).

(5.6)

H2CE (g, Ug) ∧2 t∗ ⊗ Z(Ug) ⊕ t∗ ⊗ H1CE (n, Un )t /(Ut /Z(Ug)) ⊕ H2CE (n, Un )t

(5.7)

HnCE (g, Ug) ∧n t∗ ⊗ Z(Ug) + ∧n−1 t∗ ⊗ H1CE (n, Un )t /(Ut )/Z(Ug) ∧i t∗ ⊗ HjCE (n, Un )t + i+j=n; j≥2

∀n ≥ 2

(5.8)

Now, we characterize the center. Suppose that there exists X0 ∈ t such that the eigenvalues of adX|n are positive in an ordered subﬁeld of K and let Y0 , . . . , Yr be a basis of n and X0 , . . . , Xs be a basis of t. Suppose that the action of X0 on an element ...js j0 u = aji00...i X0 . . . Xsjs Y0i0 . . . Yrir in Ug vanishes. If X0 Yk = λk Yk then 0 = X0 u = r ...js j0 (λ1 i1 + · · · + λr ir )aji00...i X0 . . . Xsjs Y0i0 . . . Yrir . Since λk > 0 then the center of Ug is K. r One can see that Theorem 5.1. Let g = t ⊕ n be a decomposable solvable Lie algebra. We suppose that there exists an element X of t such that the eigenvalues of adX|n are positive in an ordered subﬁeld of K.

172

TOUKAIDDINE. PETIT

Then, we have the following properties: Z(Ug) = K

(5.9)

H1CE (g, Ug) t∗ ⊕ H2CE (g, Ug) ∧2 t∗

(Der(n, Un ) /Ut+ ) ⊕ (Der(n, Un )t /Ut+ ) t

(5.10) ⊗t

∗

⊕

H2CE (n, Un )t

(5.11)

Where Der(n, Un )t denote the t-invariant exterior derivations and Ut+ = Ut /Z(Ug) Since the center is nontrivial, we ﬁnd again, the following necessary condition : Corollary 5.1. If the solvable Lie algebra g = t⊕n is strongly rigid with trivial H2H (Ug, Ug) then dim(t) ≤ 1. Since Ug and Sg are isomorphic as g-module. We can replace in the previous cohomological characterization the algebra Ug by Sg and the two-sided ideal Un by Sn . 6. The classification of strongly rigid solvable Lie algebras in low dimensions Let K be the complex ﬁeld. The classiﬁcation of n-dimensional rigid Lie algebras is known until n ≤ 8 [10]. For 2-dimensional Lie algebras, there is one isomorphism class, namely the Lie algebra r2 which is strongly rigid (see proposition 3.1) In dimension 3, there is no solvable rigid Lie algebras. In dimension 4, there is only one rigid Lie algebras, r2 + r2 . Since the torus is 2-dimensional, then according to corollary (3.1) this algebra is not strongly rigid. In dimension 5, there is only one rigid class with 2-dimensional torus. There is no strongly rigid Lie algebra. In dimension 6, there is 3 isomorphism classes of 6-dimensional rigid solvable Lie algebras. Only one has a one-dimensional torus. Let us consider this Lie algebra, it is denoted in [10] by t1 ⊕ n5,6 . Setting the basis {X0 , X1 , X2 , X3 , X4 , X5 } the Lie algebra is deﬁned by [X0 , Xi ] = iXi

i = 1, . . . , 5

(6.1)

[X1 , Xi ] = Xi+1 [X2 , X3 ] = X5

i = 2, 3, 4

(6.2) (6.3)

The other bracket are equal to 0 or deduced by skew-symmetry from the previous one. In the following we give a nontrivial deformation of the linear Poisson structure associated to the Lie algebra t1 ⊕ n5,6 . Proposition 6.1. Let P0 be the Poisson structure associated to the Lie algebra t1 ⊕ n5,6 and P1 ∈ Sg ⊗ ∧2 g∗ deﬁned by (α, β, γ ∈ C3 \ {(0, 0, 0)}): P1 = βX22

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∧ + γ(−X2 X3 ∧ + X2 X5 ∧ ) + αX1 X5 ∧ ∂X1 ∂X3 ∂X1 ∂X4 ∂X3 ∂X4 ∂X2 ∂X4

Then [P1 , P1 ]s = 0 = [P0 , P1 ]s and the cohomology class of P1 is not 0. Thus, g is not strongly rigid.

ON THE STRONG RIGIDITY OF SOLVABLE LIE ALGEBRAS

173

Proof. Straightforward computation. Theorem 4.1 implies that the Lie algebra is not strongly rigid. Theorem 6.1. There is only one n-dimensional solvable strongly rigid Lie algebra for n ≤ 6, namely the 2-dimensional Lie algebra r2 . Given a Poisson structure, if there exists a formal isomorphism such that this Poissons structure is isomorphic to its linear part then one says that this Poisson structure is linearizable. This problem was formulated ﬁrst by A.Weinstein (based on considerations by Sophus Lie) ([22]). Using the theorem 4.1, we may deduce : Proposition 6.2. Every Poisson structure which is a deformation of linear Poisson structure of n-dimensional strong rigid solvable Lie algebra is linearizable. It follows that every Poisson structure which is a deformation of linear Poisson structure of n-dimensional solvable Lie algebra, with 3 ≤ n ≤ 6, is linearizable. The Poisson structure P0 + P1 (deﬁned in proposition 6.1) is not linearizable.

References ´dez, M. Goze, algebras de Lie rigides dont le nilradical est filiforme. [1] J. M. Ancochea Bermu Notes aux C.R.A.Sc.Paris, 312 (1991), 21–24. [2] F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz, D. Sternheimer Deformation theory and quantization I/ II, Ann. Phys. 111 (1978), 61–110, 111–151. [3] M. Bordemann, A. Makhlouf, T. Petit, D´eformation par quantification et rigidit´ e des alg`ebres enveloppantes, Journal of Algebra (to appear). [4] R. Carles, Sur la structure des alg`ebres de Lie rigides, Ann. Inst. Fourier 34 (1984), 65–82. [5] R. Carles, Weight systems for complex Lie algebras. Preprint Universit´e de Poitiers, 96 (1996). ´: Sur les vari´et´es d’alg`ebres de Lie de dimension 7. J. of Algebra. 91, [6] R. Carles, Y. Diakite 53–63 (1984). [7] H. Cartan, S. Eilenberg Homological algebra. Princeton University Press (1946). [8] J. Dixmier Cohomologie des alg` ebres de Lie nilpotentes, Acta Sci. Math. 16, Nos.3–4 (1955), 246–250. [9] J. Dixmier, alg`ebres enveloppantes, Gauthier-Villars, Paris, (1974). enveloping algebras, GSM AMS, (1996). ´dez, On the classification of Rigid Lie algebras. J. Algebra, [10] M. Goze, J. M. Ancochea Bermu 245 (2001), 68–91. [11] M. Goze, E. Remm, valued deformation of Lie algebra. Preprint (2002). [12] M. Gerstenhaber, On the deformation of rings and algebras II, Ann. of Math., 79 (1964), pp.59–103. [13] M. Gerstenhaber, The cohomology structure of an associative ring. Ann.of Math. 78, 2, 267–288 (1963). [14] M. Gerstenhaber, S. D. Shack: Relative Hochschild cohomology, rigid algebras, and the Bockstein, J. of Pure and Appl. Alg. 43, 53–74 (1986). [15] P. J. Hilton, U. Stammbach, A Course in Homological Algebra, Springer, New York/Berlin, 1996. [16] G. Hochschild On the cohomology groups of an associative algebra. Ann. Math. 46 (1945), 58–87. [17] G. Hochschild, J-P. Serre, Cohomology of Lie algebras, Ann. Math. 57 (1953), 72–144. [18] M. Kontsevitch Deformation quantization of Poisson manifolds, arXiv:q-alg/9709040, 1997.

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[19] A. Makhlouf, M. Goze, Classification of rigid associative algebras in low dimensions, in: Lois d’algebras et vari´ et´es alg´ebriques Hermann, Collection travaux en cours 50 (1996). [20] A. Nijenhuis, R. W. Richardson, Cohomology and deformations in graded Lie Algebras, Bull. Amer. Math. Soc. 72, 1, (1966). [21] I. Vaisman, Lectures on the geometry of Poisson manifolds, Birkh¨ auser (1994). [22] A. Weinstein, The local structure of Poisson manifold, J. of diﬀ geometry. 18, 3, (1983).

THE ROLE OF A THEOREM OF BERGMAN IN INVESTIGATING IDENTITIES IN MATRIX ALGEBRAS WITH SYMPLECTIC INVOLUTION∗ TSETSKA GRIGOROVA RASHKOVA University of Rousse ”A.Kanchev” E-mail: [email protected]

The talk is a survey on a series of results considering as applications of a theorem of Bergman [1], which is connected with investigating a class of identities for matrix algebras. These applications are based both on the essential use of an analogue of the stated theorem concerning matrix algebras with symplectic involution and its elegant proof via graph theory as a method for proving other results as well related to the same theorem. We recall that in the matrix algebra over a ﬁeld Kmof characteristics zero M2n (K, ∗) the symplectic involution ∗ is deﬁned by

AB CD

∗ =

Dt −B t −C t At

,

where A, B, C, D are n × n matrices and t is the usual transpose. For an algebra R with involution ∗ we have (R, ∗) = R+ ⊕ R− , where R+ = {r ∈ R | ∗ r = r} and R+ = {r ∈ R | r∗ = −r}. Let KX be the free associative algebra. We call f (x1 , . . . , xn ) ∈ KX a ∗-polynomial identity for (R, ∗) in symmetric variables if f (r1+ , . . . , rn+ ) = 0 for all r1+ , . . . , rn+ ∈ R+ . Analogously f (x1 , . . . , xs ) ∈ KX is a ∗-polynomial identity for (R, ∗) in skew-symmetric variables if f (r1− , . . . , rs− ) = 0 for all r1− , . . . , rs− ∈ R− . Some of the investigations concerning such identities are based on the classical P.I. theory as every identity in symmetric (or skew-symmetric) variables for M2n (K, ∗) is an ordinary identity for Mn (K). Constructive results however in the symplectic case need stronger tools. They take into account the following considerations. The algebra R+ is a Jordan algebra with respect to the multiplication r1+ ◦ r2+ = r1+ r2+ + + + r2 r1 ; r1+ , r2+ ∈ R+ and the identities in symmetric variables are weak polynomial identities for the pair (R, R+ ). Similarly, the algebra R− is a Lie algebra with respect to the new multiplication [r1− , r2− ] = − − r1 r2 − r2− r1− ; r1− , r2− ∈ R− and the identities in skew-symmetric variables for (R, ∗) are weak polynomial identities for the pair (R, R− ). For polynomials in symmetric variables the Cayley-Hamilton theorem gives an identity 2 in two variables for M2n (K, ∗) of degree n +3n . For n = 3 this identity appears to be of 2 minimal degree [5]. A partial linearization of it gives rise to a Bergman type identity, namely

∗ Partially

supported by Grant MM1106/2001 of the Bulgarian Foundation for Scientific Research.

176

TSETSKA GRIGOROVA RASHKOVA

a homogeneous (of degree k) and multilinear in y1 , . . . , yn polynomial f (x, y1 , . . . , yn ) from the free associative algebra Kx, y1 , . . . , yn which can be written as (1)

f (x, y1 , . . . , yn ) =

v(gi )(x, yi1 , . . . , yin ),

i=(i1 ,...,in )∈Sym(n)

where gi ∈ K[t1 , . . . , tn+1 ] are homogeneous (of degree k − n) polynomials in commuting variables gi (t1 , . . . , tn+1 ) =

p

n+1 αp tp11 . . . tn+1

and (2)

v(gi ) = v(gi )(x, yi1 , . . . , yin ) =

αp xp1 yi1 . . . xpn yin xpn+1 .

For polynomials of type (1) A. Giambruno and A. Valenti [2] gave a lower bound of their degree as identities in skew-symmetric variables for M2n (K, ∗). For any n they constructed a special multilinear polynomial of degree 4n − 1 and found a polynomial of minimal degree for M4 (K, ∗). It leads to the existence of an identity of minimal degree 7 of type (1) for the considered algebra. In [3] on its base a full description of the Bergman type identities in skew-symmetric variables for M4 (K, ∗) was made. In the survey we deﬁne polynomials of type (1) of minimal degree for M6 (K, ∗). Two diﬀerent classes of Bergman type identities in skew-symmetric variables are given for n = 3. One of the class is related to the existence of central polynomials in skew-symmetric variables described in [4]. A polynomial c(x1 , . . . , xm ) ∈ KX is central in skew-symmetric variables for the algebra − − − (R, ∗) if it is non-zero in R− and [c(r1− , . . . , rm ), rm+1 ] = 0 for any r1− , . . . , rm+1 ∈ R− . The existence of Bergman type identities in the general case is discussed in the talk as well. In the sequel we use the following notation: g2n,0 =

(t2p − t2q )(t1 − tn+1 ).

1≤p

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

Zuhair Nashed University of Central Florida Orlando, Florida

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

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S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

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W. S. Massey Yale University

Mark Teply University of Wisconsin, Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS Recent Titles G. Lumer and L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences J. Cagnol et al., Shape Optimization and Optimal Design J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra G. Chen et al., Control of Nonlinear Distributed Parameter Systems F. Ali Mehmeti et al., Partial Differential Equations on Multistructures D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra Á. Granja et al., Ring Theory and Algebraic Geometry A. K. Katsaras et al., p-adic Functional Analysis R. Salvi, The Navier-Stokes Equations F. U. Coelho and H. A. Merklen, Representations of Algebras S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory G. Lyubeznik, Local Cohomology and Its Applications G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications W. A. Carnielli et al., Paraconsistency A. Benkirane and A. Touzani, Partial Differential Equations A. Illanes et al., Continuum Theory M. Fontana et al., Commutative Ring Theory and Applications D. Mond and M. J. Saia, Real and Complex Singularities V. Ancona and J. Vaillant, Hyperbolic Differential Operators and Related Problems G. R. Goldstein et al., Evolution Equations A. Giambruno et al., Polynomial Identities and Combinatorial Methods A. Facchini et al., Rings, Modules, Algebras, and Abelian Groups J. Bergen et al., Hopf Algebras A. C. Krinik and R. J. Swift, Stochastic Processes and Functional Analysis: A Volume of Recent Advances in Honor of M. M. Rao S. Caenepeel and F. van Oystaeyen, Hopf Algebras in Noncommutative Geometry and Physics J. Cagnol and J.-P. Zolésio, Control and Boundary Analysis S. T. Chapman, Arithmetical Properties of Commutative Rings and Monoids O. Imanuvilov, et al., Control Theory of Partial Differential Equations Corrado De Concini, et al., Noncommutative Algebra and Geometry Alberto Corso, et al., Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects Giuseppe Da Prato and Luciano Tubaro, Stochastic Partial Differential Equations and Applications – VII

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Noncommutative Algebra and Geometry

Edited by

Corrado De Concini University of Rome Rome, Italy

Freddy Van Oystaeyen University of Antwerp/UIA Antwerp, Belgium

Nikolai Vavilov St. Petersburg State University St. Petersburg, Russia

Anatoly Yakovlev St. Petersburg State University St. Petersburg, Russia

Boca Raton London New York

DK3043_Discl.fm Page 1 Wednesday, July 20, 2005 9:59 AM

Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2349-X (Hardcover) International Standard Book Number-13: 978-0-8247-2349-1 (Hardcover) Library of Congress Card Number 2005049748 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Noncommutative algebra and geometry / edited by Corrado De Concini ... [et al.]. p. cm. -- (Lecture notes in pure and applied mathematics ; 243) Includes bibliographical references and index. ISBN 0-8247-2349-X (acid-free paper) 1. Noncommutative algebras--Textbooks. 2. Noncommutative rings--Textbooks. I. De Concini, Corrado. II. Lecture notes in pure and applied mathematics ; v. 243. QA251.4.N657 2005 512'.46--dc22

2005049748

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group is the Academic Division of T&F Informa plc.

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Introduction The international meeting at St. Petersburg was organized in honor of Prof. Dr. Z. Borevich, but there was no restriction on the topics of the lectures. A proceedings covering all subjects of the meeting would therefore constitute a rather inhomogeneous collection. The present volume, however, is mainly devoted to the contributions related to the ESF workshop organized in the framework of the scientific program “Noncommutative Geometry” of the European Science Foundation and integrated in the Borevich meeting. The topics dealt with here may be classified as noncommutative algebra. The congenial atmosphere at the meeting combined with the city’s preparations for the anniversary festivities provided the perfect setting for a very fruitful meeting. Moreover, the combination of the ESF workshop and the Borevich meeting brought together many participants from East and West (now perhaps old-fashioned terminology) engaging in open discussions, hard work, and the occasional party. Most of this may be blamed on the local organizers, Vavilov and Yakovlev, whom we thank for their great hospitality.

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Contributors Hans-Jochen Bartels Universitat Mannheim Mannheim, Germany

Lucchini, Andrea Dipto di Matematica University Brescia, Italy

Igor Burban Fachbereich Mathematik Kaiserslautern, Germany

Dmitry A. Malinin Belarusian State Pedag. University Minsk, Belarus

Eloisa Detomi Dipto di Matematica Universit Padova, Italy

Janvière Ndirahisha University of Antwerp (UIA) Department of Math and Computer Science Wilrijk, Belgium

Yuriy Drozd Kyiv Taras Shevchenko University Department of Mechanics and Mathematics Kyiv, Ukraine

Toukaiddine Petit University of Antwerp Department of Math and Computer Science Antwerp, Belgium

G. Griffith Elder University of Nebraska/Omaha Department of Mathematics Omaha, Nebraska

Tsetska G. Rashkova University of Rousse Center of Applied Math and Information Rousse, Bulgaria

Eivind Eriksen University of Warwick Institute of Mathematics Coventry, United Kingdom

Wolfgang Rump Universitat Stuttgart Institut f'ur Algebra und Zah Stuttgart, Germany

Michiel Hazewinkel CWI Amsterdam, The Netherlands

Freddy Van Oystaeyen University of Antwerp/UIA Department of Mathematics Antwerp/Wilrijk, Belgium

Lieven Le Bruyn Universiteit Antwerpen Department of Wiskunde and Informatica Antwerpen, Belgium

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Table of Contents Introduction .........................................................................................................................vii Finite Galois Stable Subgroups of GLn ................................................................................1 HANS-JOCHEN BARTELS, DMITRY A. MALININ

Derived Categories for Nodal Rings and Projective Configurations ..............................23 IGOR BURBAN, YURIY DROZD

Crowns in Profinite Groups and Applications ..................................................................47 ELOISA DETOMI, ANDREA LUCCHINI

The Galois Structure of Ambiguous Ideals in Cyclic Extensions of Degree 8................63 G. GRIFFITH ELDER

An Introduction to Noncommutative Deformations of Modules ....................................90 EIVIND ERIKSEN

Symmetric Functions, Noncommutative Symmetric Functions and Quasisymmetric Functions II ...........................................................................................126 MICHIEL HAZEWINKEL

Quotient Grothendieck Representations .........................................................................147 JANVIÈRE NDIRAHISHA, FREDDY VAN OYSTAEYEN

On the Strong Rigidity of Solvable Lie Algebras............................................................162 TOUKAIDDINE PETIT

The Role of a Theorem of Bergman in Investigating Identities in Matrix Algebras with Symplectic Involution ...............................................................................................175 TSETSKA G. RASHKOVA

The Triangular Structure of Ladder Functors ...............................................................184 WOLFGANG RUMP

Non-commutative Algebraic Geometry and Commutative Desingularizations..........203 LIEVEN LE BRUYN

Author Index ......................................................................................................................253

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FINITE GALOIS STABLE SUBGROUPS OF GLn H. -J. BARTELS1 AND D. A. MALININ2

Abstract. Let K/Q be a ﬁnite Galois extension with maximal order OK and Galois group Γ. We consider ﬁnite Γ-stable subgroups G ⊂ GLn (OK ) and prove that they are generated by matrices with coeﬃcients in OKab , Kab the maximal abelian subextension of K over Q. This implies in particular a positive answer to a conjecture of J. Tate on the classiﬁcation of p-divisible groups over Z and answers also a longstanding question of Y. Kitaoka on totally real scalar extensions of positive deﬁnite integral quadratic lattices.

Introduction The starting point of our investigations was the following problem studied by Y. Kitaoka and the ﬁrst named author around 1978 on the behaviour of the automorphism groups of positive deﬁnite quadratic Z-lattices under totally real scalar extensions. There was the Question. If two positive definite quadratic Z-lattices become isomorphic over the ring OK of integers of a totally real field extension K of the rationals Q, are they already isomorphic over Z, the ring of rational integers? Closely connected with this question was the following Conjecture 1. Let K/Q be a finite totally real Galois extension and denote by OK the corresponding ring of integers and let G ⊂ GLn (OK ) be a finite subgroup stable under the operation of the Galois group Γ = Gal(K/Q), then G ⊂ GLn (Z) holds, Z the ring of rational integers. There are several reformulations and generalizations of the above mentioned conjecture. One generalization is the following: Consider an arbitrary not necessarily totally real ﬁnite Galois extension K of the rationals Q and a free Z-module M of rank n n with basis m1 , . . . , mn . The group GLn (OK ) acts in a natural way on OK ⊗ M ∼ = i=1 OK mi . A ﬁnite group G ⊂ GLn (OK ) is said to be of k A-type, if there exists a decomposition M = i=1 Mi such that for every g ∈ G there exists a permutation Π(g) of {1, 2, . . . , k} and roots of unity i (g) such that i (g)gMi = MΠ(g)i for 1 ≤ i ≤ k. The following conjecture generalizes (and would imply) conjecture 1 and would also give a positive answer to the above mentioned question: Conjecture 2. Any finite subgroup of GLn (OK ) stable under the Galois group Γ = Gal(K/Q) is of A-type. For totally real ﬁelds K ± 1 are the only roots of 1 contained in K, and so conjecture 2 reduces to conjecture 1. Partial answers to these questions are given in [2], [3], [4], [8], [9], [10], [14], [16], [17], [19] (compare also the references in mentioned articles).

1991 Mathematics Subject Classification. Primary 20C10, 11R33, 11S23, 11R29.

2

H. -J. BARTELS AND D. A. MALININ

In an earlier version of this paper (see [4]) it is shown that conjecture 2 is true in the case of Galois ﬁeld extension K/Q with odd discriminant. Also some partial answers are given in the case of ﬁeld extensions K/Q which are un-ramiﬁed outside 2. The proof of the main part is essentially already contained in the article [17] of the second named author in slightly diﬀerent formulation. While [17] focusses mainly on the proofs of conjecture 1 and contains also some other related results, we observed that the proofs of conjecture 1 can immediately be transfered in order to proof conjecture 2 in the mentioned cases. Using the methods of [2], [3] and discriminant estimations of A. Odlyzko [23] in order to exclude the existence of certain Galois extensions having low ramiﬁcation, the ﬁrst named author proved in an unpublished note eighteen years ago, that conjecture 1 is true in the following cases: i) ii) iii) iv)

Γ = Gal(K/Q) = P SL2 (5) ∼ = A5 the alternating group of order 60, Γ = Gal(K/Q) = P SL2 (7) the simple group of order 168, K/Q is tamely ramiﬁed of degree ≤ 131 K/Q is tamely ramiﬁed of degree ≤ 233 assuming a generalized Riemann hypothesis to be true.

The combination of this approach using discriminant estimations with the far reaching results of [17] and [7] gave us the the following better results: Conjecture 1 is true in the following cases: i) [K : Q] ≤ 960 assuming the generalized Riemann hypothesis for the zeta function of the number ﬁeld K, or if ii) [K : Q] ≤ 480 unconditionally. Conjecture 2 is true if [K : Q] < 288 unconditionally. See [4] for the details. After ﬁnishing the ﬁrst version of our paper [4] we became aware of the recent work [20] of M. Mazur on the same topic. It turned out that in a certain sense the partial results of M. Mazur are complementary to our partial results. Using the the classiﬁcation of ﬁnite ﬂat group schemes over Z annihilated by a prime p for primes p ≤ 17 due to V. A. Abrashkin [1] and J.-M. Fontaine [6] the particular case of ﬁeld extensions K/Q which are unramiﬁed outside 2 follows in full generality from [20]. In this revised version of our paper we restrict therefore ourselves to the case of ramiﬁed primes p = 2. It should be noted that conversely our Main Theorem in combination with the work of M. Mazur has interesting consequences for the classiﬁcation of ﬁnite ﬂat commutative group schemes over Z annihilated by a prime p: It answers a question of J. Tate [28] also for primes p ≥ 17 completing the partial results of Abrashkin [1] and Fontaine [6]. It is interesting to notice that the methods used in the proofs, namely the detailed study of the operation of the higher ramiﬁcation groups of the Galois group on the given Galois stable group G for the ramiﬁed primes in the ﬁeld extension K over Q together with discriminant estimations, in order to eliminate ramiﬁcation with large depth using trivial action of higher ramiﬁcation groups (compare [2] section 1), are similar to the methods used by [1] and [6]. This paper is organized as follows: Section I contains the results and the propositions and lemmata used in the proofs. The proofs themselves are presented in Section II. As far as it is needed the necessary parts of the proofs from [17] are reproduced only slightly changed in this paper for the convenience of the reader. Acknowledgement: The second author is grateful to DAAD for support. Helpful comments from an anonymous referee to an earlier version of this paper are also gratefully acknowledged.

FINITE GALOIS STABLE SUBGROUPS OF GLn

3

Notation Q, Qp , Z, Zp , OK denote the ﬁeld of rationals and p-adic rationals, the ring of rational and p-adic rational integers respectively, and the ring of integers of an algebraic number ﬁeld K. to be the intersection of valuation rings of all ramiﬁed prime ideals p ∈ OK We consider OK (if K = Q). T rK/L denotes the trace map from K to L. GLn (R) denotes the general linear group over R. [E : F ] denotes the degree of the ﬁeld extension E/F . Im denotes the unit m × m-matrix, 0n,m and 0m are zero n × m and m × m-matrices, ei,j are square matrices having the only nonzero element 1 in the position (i, j), rankM and detM are rank and determinant of a matrix M . t M denotes a transposed matrix for M, diag(d1 , d2 , . . . , dm ) is a block-diagonal matrix having diagonal components d1 , d2 , . . . , dn . We suppose that K is a Galois extension of the rationals Q. We denote by Γ the Galois group of a normal extension K/F ; if needed we specify K/F as a subscript in ΓK/F . The symbols Γi (p) denote the i-th ramiﬁcation groups of the prime divisor p and Γ0 (p) the inertia group in Γ, ei is the order of Γi (p) for i ≥ 1, while e = e0 is the order of the inertia group. For Γ acting on G and any σ ∈ Γ and g ∈ G we write g σ for the image of g under σ-action. If G is a ﬁnite linear group, F (G) denotes the ﬁeld obtained by adjoining the matrix coeﬃcients of all matrices g ∈ G. Throughout this paper ζm denotes a primitive m-th root of unity. 1. Statement of the main results 1.1. Let E/F be a normal extension of algebraic number ﬁelds, and let ΓE/F = Gal(E/F ) be its Galois group. We consider the problem of integral realizations of ﬁnite subgroups G of the general linear group GLn (E) that are stable under the natural action of ΓE/F on the matrices of the group G. Let OF and OE denote the maximal orders of the number ﬁelds F and E respectively. Let us introduce the class C(F ) of ﬁelds normal over F that are obtained by adjoining to F all coeﬃcients of matrices contained in some ﬁnite ΓE/F -stable group G ⊂ GLn (OE ). In [3] it is shown that if F = Q and the class C(Q) contains some ﬁeld K = Q, then C(Q) will also contain some ﬁeld K1 = Q, K1 ⊂ K such that there exists only one prime p ramiﬁed in K1 . In this paper we use some properties of Galois groups for ﬁelds having restricted ramiﬁcation. In general, the existence of global ﬁelds with a given Galois group and prescribed local properties for ramiﬁcation is a rather subtle question. L. Moret-Bailly proved the existence of extensions of number ﬁelds that have prescribed local structure of ramiﬁcation over a given set of prime divisors and unramiﬁed elsewhere for certain relative extensions [22]. In our case we deal with absolute extensions of the rationals K/Q, and we ﬁx the only ramiﬁed prime p. Let Cp (Q) denote the class of ﬁelds in C(Q) with the unique ramiﬁed prime p. Nilpotent extensions of Q having this property were described by Markshaitis in [18], but there are many examples of extensions in Cp (Q) that are not nilpotent, and also nonsolvable extensions unramiﬁed outside p; for this and also for non-existence theorems compare [27], [7]. Both conjectures 1 and 2 are true for nilpotent extensions K/Q (see [3], [8]), and the proof of this fact uses the special structure of the Galois group of nilpotent extensions unramiﬁed outside a prime p [18]. 1.2. It is well known, that the problem of description of ﬁelds Q(G) can be reduced to the case of commutative groups G of exponent p. Compare Proposition 1 in [17] and section 3 of [19] and [20] chapter 4. The idea of this reduction appears already in [14], [15], [13] and [10] where it was used, in particular, to study conditions for coeﬃcients of the representations of nilpotent groups over integral rings providing their diagonalizability. Hence, if there would be a counterexample to conjecture 1 or conjecture 2, there would exist also an elementary abelian p group G as a counterexample.

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H. -J. BARTELS AND D. A. MALININ

We use also reduction to the case of a GLn (Q)-irreducible group G. Here a matrix group G is reducible in GLn (R) or simply R-reducible (R a ring or a ﬁeld) if there exist h ∈ GLn (R) such that G ∗ , h−1 Gh ⊂ 1 0 G2 , and G is irreducible otherwise. We note that the reduction to the case of an irreducible group G can be done using the following lemma: Lemma 1.2.1. Let E/F be a normal extension of algebraic number fields with Galois group ΓE/F = Gal(E/F ) and let E1 , F1 be rings with quotient fields E and F respectively. If G ⊂ GLn (E1 ) is a finite ΓE /F -stable subgroup which has GLn (F1 )-irreducible components G1 , G2 , . . . , Gr , then F (G) is the composite of the fields F (G1 ), F (G2 ), . . . , F (Gr ). The proof of this Lemma is given at the beginning of section II. 1.3. The essential results of this note can be summarized as follows: Main Theorem. Let K be a finite Galois extension of Q and G be a finite subgroup of GLn (OK ) that is stable under the natural action of the Galois group Γ of the field K. Then G is of A-type and in particular G ⊂ GLn (OKab ) holds, Kab the maximal abelian subextension of K over Q. Let µp denote the multiplicative group scheme over Z of order p and αp the constant group scheme of order p (see [28] and [1]). Due to the results of [1] and [6] in conjunction with [20] one gets immediately the following Corollary 1. If G is a finite flat commutative group scheme over Z annihilated by a prime p, then it is a direct sum of copies of µp , αp and, if p = 2, the nontrivial element in Ext(α2 , µ2 ). We can also express the result of the Main Theorem in the following form: Corollary 2. A finite flat group scheme G over Z satisfies G(Q) = G(Qab ), Q the algebraic closure of Q and Qab the maximal abelian (over Q) subextension of Q. For the proof of the Main Theorem we distinguish essentially two cases and for their treatment we need several results which are recorded in the subsequent sections 1.4 and 1.5. The ﬁrst Proposition 1 gives a criterion for the existence of integral realizations of an abelian matrix group. It shows that the existence of G in question is possible only if certain determinants dk are divisible by the root of the discriminant D of a certain extension of number ﬁelds (for the details see section 1.4 below). In the proof of the Main Theorem in section II we use this for a certain cyclic extension E/F which is tame with respect to a ﬁxed prime ideal (case I). Assume that E/Q is not abelian. Then we can make E/F to be a Kummer extension via adjoining √ appropriate roots of 1. We use the explicit Kummer basis to ﬁnd an index k for which D does not divide dk . The proof of the Main Theorem is divided in to two parts depending on the ramiﬁcation index e = e0 of Q(G). In the ﬁrst part we use Proposition 1. In the second part we use lemma 1.5.2 and the Corollary 1.5.3 of section 1.5.

FINITE GALOIS STABLE SUBGROUPS OF GLn

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We can sketch the scheme of the proof of the Main Theorem:

Let us outline the idea of the proof of the Main Theorem in more detail for the convenience of the reader.

The outline of the proof of the Main Theorem. In virtue of the argument of [3], lemmata 1 and 2 (compare also Theorem 2 in [19]), we can assume that K is unramiﬁed outside a prime p, so we can ﬁx this prime. Since as already remarked in the introduction the particular case of ﬁeld extensions K/Q which are unramiﬁed outside 2 follows in full generality from [20], we can restrict ourself to the case p > 2. We can also assume that G is an abelian group of exponent p, and we can consider G to be irreducible under conjugation in GLn (Q) by Corollary 1.4.1. The proof of the Main Theorem consists of a reduction to special cases, and these special cases are treated with diﬀerent methods. , OL denote the semilocal rings that are obtained by For number ﬁelds E, L be let OE intersection of the valuation rings of all ramiﬁed prime ideals in the rings OE , OL respectively. These semilocal rings are known to be principal ideal domains. Denote G0 = GΓ1 (p) the subgroup of elements in G that are ﬁxed by the ﬁrst ramiﬁcation group Γ1 (p) for some prime divisor p of p. Let e0 be the ramiﬁcation index of Q(G0 ) over Q with respect to p. Then e0 e0 /e1 (= the index of Γ1 (p) in Γ0 (p).) Case I. Assume that e0 does not divide p − 1. In this case we apply Proposition 1 to a certain subgroup G0 ⊂ GΓ1 (p) ⊂ GLn (OE ) for a certain cyclic Kummer extension E/F with a i convenient power basis π , i = 0, . . . , t − 1 and with the explicit action of the generating , namely element σ of order t of the Galois group on the uniformizing element π of OE σ π = πζt , which is convenient for applying Proposition 1 explicitly. Here E and F are the ramiﬁcation ﬁeld and the inertia ﬁeld for some prime divisor p of p adjoined by a primitive t-root of 1, t = e0 . Denote ΓE /F the Galois group of E/F . In case I we determine a ΓE /F -stable subgroup G0 ⊂ G0 which is generated by all conjugates hγ , γ ∈ ΓE /F of some element h ∈ G0 . G0 can not be cyclic provided t = e0 does not divide p − 1, and this is just the case where

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H. -J. BARTELS AND D. A. MALININ

the arguments in case II (see below) can not be applied. So we start the proof of the Main Theorem just from this most diﬃcult case, and apply Proposition 1 to a subgroup G0 ⊂ G. We show that case I is impossible since the conditions of Proposition 1 never hold true for G0 and the extension E/F . In particular, if e0 does not divide p − 1 we have a contradiction with the condition G ⊂ GLn (OE ) which can not hold true since G0 ⊂ GLn (OE ). Case II. Let us suppose that e0 divides p − 1. In this case we can suppose without loss of generality, that K contains a p-th root of unity ζp (see Lemma 2.2.2 below). Using a local argument on the diagonalization of matrices which are congruent to In modulo the prime ideal p (see Corollary 1.5.3 below) a certain subgroup G1 in G is constructed such that K Γ1 (p) (G1 ) is an extension of K Γ1 (p) with ζp ∈ K Γ1 (p) (G1 ), tame ramiﬁcation index p − 1 and K Γ1 (p) (G1 )/K Γ1 (p) is an elementary abelian Kummer extension. In a second step a careful study of the Galois-action of Γ0 (p) on G1 shows that the constructed group G1 can not exist. This gives then the desired contradiction. 1.4. In this section we formulate the mentioned criterion for the existence of an integral realization of an abelian group G with the properties mentioned above. Let E, L be ﬁnite Galois extensions of the number ﬁeld F that are diﬀerent from F with , OL be the semilocal rings Galois groups ΓE/F and ΓL/F respectively. As above let OE that are obtained by intersection of the valuation rings of all ramiﬁed prime ideals in the . Let w1 , w2 , . . . , wt be a basis of OE over OF , and rings OE , OL , and let OF = F ∩ OE let D be the discriminant of this basis. Suppose that some matrix g of prime order p has coeﬃcients in E and all ΓE/F -conjugates g γ , γ ∈ ΓE/F generate a ﬁnite abelian group G of exponent p. Let σ1 = 1, σ2 , . . . , σt denote all automorphisms of the Galois group ΓE/F of the ﬁeld E over F . Assume that L = E(ζ(1) , ζ(2) , . . . , ζ(n) ) where ζ(1) , ζ(2) , . . . , ζ(n) are the eigenvalues of the matrix g, therefore L = E(ζp ), ζp a primitive p-th root of unity. We will reserve the same notations for some extensions of σi to L, and the automorphisms of L/F will be denoted σ1 , σ2 , . . . , σr for some r t. Let E be a numberﬁeld containing F (G) which is obtained by adjoining to F all coeﬃcients of all g ∈ G. For a suitable choice of t elements of ζ(1) , ζ(2) , . . . , ζ(n) say ζ(1) , ζ(2) , . . . , ζ(t) we can prove the following Proposition 1. 1) Let G be generated by all g γ , γ ∈ ΓE/F and irreducible under GLn (F ) conjugation. Then G is conjugate in GLn (F ) to a subgroup of GLn (OE ) if and only if all determinants w1 . . . wk−1 ζ(1) wk+1 · · · wt σ2 σ2 σ2 σ2 w1 · · · wk−1 ζ(2) wk+1 · · · wtσ2 dk = det . .. σ w t · · · w σt ζ σt w σt · · · w σt t 1 k−1 (t) k+1 √ are divisible by D in the ring OL . 2) If any of the three sets of conjugates {g γ , γ ∈ ΓE/F }, {hγ , γ ∈ ΓE/F }, {(gh)γ , γ ∈ ΓE/F } generates G and the corresponding eigenvalues of g and h given in 1) are g g g h h h , ζ(2) , . . . , ζ(t) and ζ(1) , ζ(2) , . . . , ζ(t) respectively, then the eigenvalues for the matrix gh ζ(1) gh g gh g gh h h in 1) can be chosen as products ζ(1) = ζ(1) = ζ(1) ζ(1) , ζ(2) = ζ(2) = ζ(2) ζ(2) , . . . , ζ(t) = ζ(t) = g h ζ(t) . ζ(t)

Note that the conditions of Proposition 1 are always true if E is unramiﬁed over F since = OE in this case. DOE

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Corollary 1.4.1. If there is an abelian ΓE/F -stable subgroup G ⊂ GLn (OE ) of expoγ nent p generated by g , γ ∈ ΓE/F such that E = F (G) = F , then the GLn (F )-irreducible components Gi ⊂ GLni (E), i = 1, . . . , k of G are conjugate in GLni (F ) to subgroups ) such that E = F (G1 )F (G2 ) . . . F (Gk ). In particular, F (Gi ) = F for some Gi ⊂ GLni (OE indices i.

The following corollary shows that the conditions of Proposition 1 hold true even if G is not irreducible. Corollary 1.4.2. Let E/F be a normal extension of number fields with Galois group ΓE/F . Let G ⊂ GLn (E) be an abelian ΓE/F -stable subgroup of exponent p generated by g and all matrices g γ , γ ∈ ΓE/F , and let E = F (G). Then G is conjugate in GLn (F ) to G ⊂ GLn (OE ) if and only if all eigenvalues of matrices Bi , i = 1, . . . , t are contained in OL , where L = E(ζp ). The latter happens if and only if the criterion of Proposition 1, 1) holds true, i.e. all determinants w1 . . . wk−1 ζ(1) wk+1 · · · wt σ2 σ2 σ2 σ2 σ2 w1 · · · wk−1 ζ(2) wk+1 · · · wt dk = det . .. σ w t · · · w σt ζ σt w σt · · · w σt t 1 k−1 (t) k+1 are divisible by

√ D in the ring OL .

Corollary 1.4.3. Let F = Q. If there is an abelian ΓE/Q -stable subgroup G ⊂ GLn (OE ) of exponent p generated by g γ , γ ∈ ΓE/Q such that E = Q(G) = Q, then the GLn (Q)irreducible components Gi ⊂ GLni (E), i = 1, . . . , k of G are conjugate in GLni (Q) to subgroups Gi ⊂ GLni (OE ) such that E = Q(G1 )Q(G2 ) . . . Q(Gk ). In particular, Q(Gi ) = Q for some indices i. 1.5. For the proof of the Main Theorem (more precisely for the part of the proof dealing with case II) we use a lemma which is a variation on a theme of Minkowski [21] and is – like in the earlier related work [2], [3] - the key ingredient in the proofs of Lemma 1.5.2 and the Main Theorem. For the proof see [11]. Compare also [19], Proposition 1. Lemma 1.5.1. Let J be an ideal in Dedekind ring S of characteristic χ, 0 = J = S, let g be an n × n-matrix of finite order congruent to In (mod J). j

(i) If χ = p > 0, then g p = In for some integer j. If χ = 0, then J contains a prime j number p and g p = In , i ∈ Z. In particular, any finite group of matrices congruent to In (mod J) is a p-group. (ii) Let χ = 0, J = p be a prime ideal having the ramification index e with respect to p, g ≡ In (mod pr ) and mpi−1 (p − 1) ≤ e/r < pi (p − 1), i ≥ 0, m = min{1, i}. Then i g p = In . In particular, any finite group of matrices congruent to In (mod pt ) is trivial if e < t(p − 1). Related to these properties is the following Lemma 1.5.2. Let O be a Dedekind ring in an algebraic number field, and let ζp ∈ O. Let p = pe , e = p − 1. Let G be a finite subgroup of GLn (O) and g ≡ In (mod p) for all g ∈ G. Then G is conjugate in GLn (O) to an abelian group of diagonal matrices of exponent p.

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Corollary 1.5.3. Let L be an extension of Q and p a prime ideal in the field L(ζp ). Suppose that L is unramified at p and let Op denote the valuation ring of the ramified prime ideal p in L(ζp ). Let Γ denote the Galois group of L(ζp ) over L. If G is a finite Γ-stable subgroup of GLn (Op ) consisting of matrices g, g ≡ In (mod p), then G is conjugate in GLn (L ∩ Op ) to an abelian group of diagonal matrices of exponent p. 2. Proofs 2.1. Proof of Lemma 1.2.1. Let G1 ∗ .. h−1 Gh ⊂ . 0 Gr for h ∈ GLn (F1 ). If there exists g ∈ G such that g γ = g for some automorphism γ of F (G) over F (G1 )F (G2 ) . . . F (Gr ), then g = g γ g −1 = In . The blocks Gi in h−1 Gh are stable under the action of γ, since h ∈ GLn (F1 ) and the elements of F (Gi ) are ﬁxed by γ. Because g1 ∗ h−1 gh = . . . 0 gr and g1 ∗ (h−1 gh)γ = h−1 g γ h = . . . 0 gr are matrices having the same diagonal components, all eigenvalues of the matrix g = g γ g −1 of ﬁnite order are 1 and hence g = In . This contradiction completes the proof of Lemma 1.2.1. Proof of Proposition 1. One proof (namely of the ﬁrst part) is given in the paper [17]. The second part of proposition 1, which is important for the proof of the Main Theorem, follows from the construction given in [17]. But for convenience we give here a proof for the proposition, which is shorter than in [17]. over OF we can write Using the basis w1 , . . . , wt of OE g σj =

t

wi σj Bi

for j = 1, . . . , t

i=1 σ

with semisimple matrices Bi ∈ Mn (F ). Since the matrix W = [wi j ]j,i is nondegenerate, the matrices Bi can be expressed as a linear combination of g σj , i, j = 1, 2, . . . , t: Bi =

t j=1

mij g σj ,

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where [mij ] = W −1 . Since by assumption the matrices g σj commute pairwise, all matrices Bi also commute with each other. The irreducibility of G implies that the minimal polynomial of Bi is irreducible over F for each i such that Bi is not zero (see [26], page 8, Corollary 3 for then all of them are since they are Galois example). So if one of the eigenvalues of Bi is in OL ∗ ∗ conjugate. Using the dual basis w1 , . . . , wt to w1 , . . . , wt with respect to the traceform one σ can see that the inverse matrix W −1 to W = [wi j ]j,i is of the form W −1 = [wj∗σi ]j,i . In order to prove the claim of the proposition, we need to determine whether or not matrices Bi , i = 1, . . . , t are conjugate in GLn (F ) to matrices Bi ∈ Mn (OF ), since for the generator g of G the equation g = B1 w1 + B2 w2 + · · · + Bt wt , holds with Bi ∈ Mn (F ) and w1 , . . . , wt a basis of OE over OF . In fact each semisimple matrix Bi ∈ Mn (F ) is conjugate in GLn (F ) to a matrix from Mn (OF ) if and only if all its (see Lemma 2.1.1 below). eigenvalues are contained in OL ∗σ Cramer’s rule now implies that wi j = (−1)i+j Wi,j det(W )−1 , where Wi,j is the (i, j)minor of W . Over the splitting ﬁeld L there is a basis which consists of eigenvectors for G. Let u be one such common eigenvector with

g σi u = ti u. σ −1

Then ζ(i) := ti i with eigenvalue

is an eigenvalue of g. It also follows, that u is an eigenvector for Bk

λk =

t j=1

mkj tj =

t j=1

σ

(−1)j+k Wj,k ζ(j)j det(W )−1 .

The cofactor expansion for determinants implies λk = dk /detW and therefore the eigenval iﬀ detW divides dk , which proves the criterion of Proposition 1 and - by ues of Bk are in OL deﬁnition of the eigenvalues ti - also the second statement modulo the proof of the following Lemma 2.1.1. i) Let all eigenvalues λj , j = 1, 2, . . . , k of the semisimple matrices Bi ∈ Mn (F ), i = 1 . . . , t be contained in the ring OL for some field L ⊃ F . Then Bi are conjugate in GLn (F ) simultaneously to matrices that are contained in Mn (OF ). ii) Conversely, if the semisimple matrices Bi are contained in Mn (OF ) and Bi are diag . onalizable over a field L ⊃ F , then their eigenvalues are contained in OL Proof of Lemma 2.1.1. i) By the virtue of [26], chapter 1, sect. 1, corollary 2 we can consider A to be a ﬁeld extending F . Let a1 , a2 , . . . , an be a basis of OA over OF . Then for any B ∈ A . we have B = b1 a1 + · · · + bn an , and the elements bi ∈ F are contained in OF iﬀ B ∈ OA But all coeﬃcients kij of the characteristic polynomials fi (x) = ki0 + ki1 x + · · · + kin xn of the matrices Bi are contained in OL , and kin = 1, so Bi ∈ A are integral over F . It follows that Bi = bi1 a1 + · · · + bin an , and bij ∈ OF . If υ ∈ F n is a non-zero vector in F n , then a1 υ, a2 υ, . . . , an υ is a basis of F n , and Bi aj υ = Σk cijk ak υ, where cijk ∈ OF . It follows that for any i the matrix Ci = [cijk ]k,j belongs to GLn (OF ), and Ci is the matrix of the operator Bi in the basis a1 υ, a2 υ, . . . , an υ of F n . Therefore, Bi is conjugate in GLn (F ) to Ci for any i = 1, . . . , t. ii) Consider the characteristic polynomials fi (x) = ki0 +ki1 x+· · ·+kin xn of the matrices . This completes the Bi . Since kin = 1 and all kij are in OF all roots of f (x) are in OL proof of Lemma 2.1.1.

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Remark. In the situation of Lemma 2.1.1, i) the F -algebra A = F [B1 , . . . , Bt ] is isomorphic to the field L = F [λ1 , . . . , λk ] where λj , j = 1, 2, . . . , k are all eigenvalues of the matrices Bi , i = 1 . . . , t. Proof of Corollary 1.4.1. If G ⊂ GLn (OE ) is a group of exponent p and g = B1 w1 + over OF , then Bi ∈ Mn (OF ), and it follows B2 w2 + · · · + Bt wt for a basis w1 , . . . , wt of OE . But eigenvalues are from Lemma 2.1.1 that the eigenvalues of Bj are contained in OL preserved under conjugation, so the latter claim is also true for all components Gi . We can apply Proposition 1 to Gi , i = 1, . . . , k. It follows that Gi are conjugate to subgroups ). Now, Lemma 1.2.1 implies E = F (G1 )F (G2 ) . . . F (Gk ). This completes Gi ⊂ GLni (OE the proof of Corollary 1.4.1.

Proof of Corollary 1.4.2. Let G1 ∗ .. C −1 GC = . 0 Gk for C ∈ GLn (F ) and irreducible components Gi ⊂ GLni (E), i = 1, . . . , k. Then for g = B1 w1 + B2 w2 + · · · + Bt wt g1 ∗ C −1 gC = . . . = B1 w1 + B2 w2 + · · · + Bt wt 0 gk holds with Bi = C −1 Bi C. Let us consider the F -algebra A generated by all Bi , i = 1, . . . , t over F . Since A is semisimple, it is completely reducible. It follows that matrices Bi are simultaneously conjugate in GLn (F ) to the block-diagonal form. Therefore, G is conjugate in GLn (F ) to a direct sum of its irreducible components Gi . Since E ⊂ F (Gi ) for all i, and contains all rings OF (Gi ) , we can apply Proposition 1 to each of them. Proposition 1 OE implies that each Gi is conjugate in GLni (F ) to Gi ⊂ GLni (OE ) if and only if all eigenvalues of matrices Bi , i = 1, . . . , t are contained in OLi , where Li = F (Gi )(ζp ) and this happens iﬀ w1 . . . wk−1 ζ(1) wk+1 · · · wt σ2 σ2 σ2 σ2 σ2 w1 · · · wk−1 ζ(2) wk+1 · · · wt dk = det . .. σ w t · · · w σt ζ σt w σt · · · w σt t 1 k−1 (t) k+1 √ are divisible by D in the ring OL . But F (G) = F (G1 )F (G2 ) . . . F (Gk ) by the Lemma in section 1.2, and so L = L1 L2 . . . Lk . This completes the proof of Corollary 1.4.2. Proof of Corollary 1.4.3. The argument of the proof of Corollary 1.4.1 remains true for the rings of integers OE and Z in E and F = Q since Z is a principal ideal domain and OE has a free basis over Z. Therefore, the rest of the proof of Corollary 1.4.3 reproduces the proof and OF respectively. of Corollary 1.4.1 with OE and Z instead of OE

FINITE GALOIS STABLE SUBGROUPS OF GLn

11

2.2. Proof of the Main Theorem. Let us suppose that there exist a counterexample G to the Main Theorem with corresponding Galois extension K/Q, K = Q(G) with Galois group Γ := ΓK/Q . In virtue of Lemmas 1 and 2 in [3] or Theorem 2 in [19] we can assume the ﬁeld K to be unramiﬁed outside the ﬁxed prime p. Since as already remarked above the particular case of ﬁeld extensions K/Q which are unramiﬁed outside 2 follows in full generality from [20], we can restrict our self to the case p > 2. Because of the Proposition in section 1.2 we can also suppose that G is an abelian group of exponent p and we can consider G to be irreducible under conjugation in GLn (Q) by Corollary 1.4.3. Let us assume that G is a counterexample of minimal order of this kind. With the notation of the beginning of this note let Γi (p) ⊂ Γ denote the i-th ramiﬁcation groups of the prime divisor p for i ≥ 1 and Γ0 (p) the inertia group in Γ. Let G0 = GΓ1 (p) denote the subgroup of elements in G that are ﬁxed by the ﬁrst ramiﬁcation group Γ1 (p) for some prime divisor p of p. Let e0 be the ramiﬁcation index of Q(G0 ) over Q with respect to p. Then e0 e0 /e1 (= the index of Γ1 (p) in Γ0 (p).) We distinguish two cases: Case I : e0 does not divide p − 1 and Case II : e0 is a divisor of p − 1. Case I. e0 does not divide p − 1. 1) In this case, where e0 does not divide p − 1, let us ﬁx p and one of its ramiﬁed prime divisors say p. Let E1 and F1 denote the subﬁelds of Γ1 (p)-ﬁxed elements and Γ0 (p)ﬁxed elements of K respectively. We will prove that for p = 2 and a ﬁeld K which has discriminant pj , j ∈ Z, all Γ0 (p)/Γ1 (p)-stable ﬁnite subgroups G of GLn (OE1 ) are already in GLn (OF1 ) for E1 = F1 (GΓ1 (p) ) = F1 (G0 ) ⊂ K Γ1 (p) and F1 = K Γ0 (p) . We can extend the ground ﬁeld F1 by adjoining ζt , t = e0 . Set E = E1 (ζt ) and F = F1 (ζt ). We obtain a cyclic extension E/F such that ζt ∈ F for t = e0 . Since K is unramiﬁed outside p, Q(ζt ) and K have intersection Q and therefore we can identify the Galois group ΓE/F = Gal(E/F ) with the Galois group Gal(E1 /F1 ). With respect to this extension of the corresponding Galois action to E/F we obtain a ΓE/F − stable group G0 ⊂ GLn (OE ). E/F is a tame extension with respect to p, t = e0 is its ramiﬁcation index and p − 1 ≥ 2. We have the following conditions for local e ramiﬁcation: pE0 = (p) = (ζp − 1)p−1 as ideals of the ring OEp (ζp ) , where pE is the e0 prime divisor of p in p-adic completion Ep of E. It is clear that + 1 (p−1) > e0 . 2 Hence p[t/2]+1 does not divide (ζp − 1) as ideals of OE(ζp ) . We can also assume that G is an abelian p-group of exponent p, and E = F because e0 > 1 in the case I. We and OF use the statement of Proposition 1 and its Corollary 1.4.2 for the rings OE t−1 t and a basis 1, π, . . . , π , such that π ∈ F . If ΓE/F , the Galois group of E/F , is generated by an element σ of order t, we can consider the action of ΓE/F on the basis 1, π, . . . , π t−1 in the following way: (π i )σ = π i ζti . Then det W = π t(t−1)/2

(ζtj − ζti ).

1i<jt

Let us consider the determinants of the matrices Wj that are obtained from W by j changing elements of j-th column of W = [(π i )σ ]i,j to appropriate p-roots ζ(1) , ζ(2) , . . . , ζ(t) i of 1 that are the eigenvalues of the matrices g σ , i = 1, 2, . . . , t for some g ∈ G, according to Proposition 1. For simplicity let ζ = ζt , but reserve previous notation for ζp for the rest of this proof. Recall, that G is supposed to be a minimal counterexample to the Main Theorem and that K is unramiﬁed outside p. In the proof of the Case I we pick g ∈ G0 = GΓ1 (p) and a

12

H. -J. BARTELS AND D. A. MALININ

generator σ of the Galois group of E over F ; by our assumption, the order t of σ does not divide p − 1. There is a matrix g ∈ G0 such that matrices g γ , γ ∈ Γ generate G. Indeed, if matrices g γ , γ ∈ Γ generated a proper subgroup G1 of G for any g ∈ G0 , then G1 would be a group of A-type, since G is a minimal counterexample, and the order of e0 would divide p − 1 (because Q(G1 )/Q is unramiﬁed outside p and tamely ramiﬁed at p), contrary to the assumption of the Case I. Let us ﬁx the above G and σ. We need the following auxiliary lemma which speciﬁes the option of g for our proof of the case I: Lemma 2.2.1. Let k be an integer such that 0 < k < p. There is a matrix g ∈ G0 such that matrices g γ , γ ∈ Γ generate G, and the group G is generated by all hγ , γ ∈ Γ, where h := g k g σ . Proof of Lemma 2.2.1. Take a matrix g ∈ G0 such that matrices g γ , γ ∈ Γ generate G. If a group H generated by all hγ , γ ∈ Γ is a proper subgroup of G, it is a group of A-type, and it is ﬁxed elementwise by the commutator subgroup Γ of Γ. Then g σ = g −k h = g l h 2 2 p−1 p−1 for l ≡ −k(modp). We have g σ = g l hl hσ , . . . , g σ = g l h0 = gh0 for some matrix h0 having coeﬃcients ﬁxed by Γ . Since h ∈ G0 , G0 is ﬁxed by Γ1 (p) and K is unramiﬁed p−1 i(p−1) = ζp , and we also have g σ = ghi0 , so outside p, we have h ∈ GLn (Q(ζp )). But ζpσ p(p−1)

= g. The same argument is true for elements g1 , h1 such that for i = p we obtain g σ p(p−1) = g1 . But G0 is g1 = g τ ∈ G0 (τ ∈ Γ) and h1 = g1k g1σ taken instead of g, h. We have g1σ covered by subgroups generated by all elements g1 = g τ since G is generated by elements g1 = g γ , γ ∈ Γ. Therefore, σ p(p−1) acts trivially on G0 . But the order of σ is coprime to p. We conclude that the order of σ divides p − 1, which contradicts the assumption of the Case I. It follows that either the group H or the group H1 generated by all hγ1 , γ ∈ Γ coincides with G. In the latter case we can rename matrix g1 to g. This completes the proof of Lemma 2.2.1. We distinguish the cases of odd and even t, the order of σ. If t is odd, we need a matrix g having at least one eigenvalue θi = ζ(i) = 1 (we use notations of Proposition 1) such that G is generated by all conjugates g γ , γ ∈ Γ. For an even t we have to choose g = g k g σ ζps . The choice of the eigenvalues ζ(i) (see Proposition 1) ensures that the product of the corresponding eigenvalues are in accordance with the product of two matrices h1 , h2 ∈ G (compare the proof of Proposition 1). Now, we intend to replace G0 by a smaller subgroup G0 generated by a single element of G0 which also satisﬁes the conditions of the Case I. G0 is covered by its ΓE/F -stable subgroups Gγ , where Gγ are generated by elements γ σi (ˆ g ) , i = 1, 2, . . . , t for some γ ∈ Γ and any gˆ such that gˆγ ∈ G0 and all gˆτ , τ ∈ Γ, generate G. By deﬁnition, Gγ is generated by the orbit of an element g having the above property. But if h satisﬁes the conditions of the above Lemma, the elements gˆτ , τ ∈ Γ −1 i generate G for gˆ = hγ , so we can assume that Gγ is generated by elements hσ , i = 1, . . . , t for a given γ and some h ∈ G satisfying the conditions of the above Lemma. Since the ramiﬁcation index with respect to p of the composite of the ﬁelds F (Gγ ), γ ∈ Γ, does not divide p − 1, there is γ ∈ Γ such that the ramiﬁcation index e(F (Gγ )/F ) of F (Gγ ) does not divide p − 1. Let us brieﬂy explain this claim. The ﬁeld F (G0 ) is a composite of ﬁelds Ei = F (Gγi ), and F (G0 )/F is a cyclic totally ramiﬁed extension whose Galois group is generated by an element σ of order t equal to the ramiﬁcation index of F (G0 )/F in p. So Ei /F are also cyclic totally ramiﬁed extensions, and their Galois groups are generated by elements σi of orders equal to the ramiﬁcation indices ti of Ei /F . Therefore, if all ti divide p − 1, then the order of σ must also divide p − 1, because σ is a product of pairwise commuting elements of orders ti . This completes the proof of our claim.

FINITE GALOIS STABLE SUBGROUPS OF GLn

13

Let us ﬁx γ and denote G0 = Gγ . The group G0 is not cyclic since the order of σ does not divide p − 1 in the case I. Using Proposition 1 or, alternatively, Corollary 1.4.1 or Corollary 1.4.2 of Proposition 1, we will prove that G0 ⊂ GLn (OF ). Below we use ΓE/F -stability of i G0 in order to apply Proposition 1 to G0 ⊂ G0 generated by all (hγ )σ , i = 1, 2, . . . , t for the ﬁxed γ ∈ Γ. Since E/F is a cyclic Kummer extension, for E = F (G0 ) ⊂ E the extension E /F is also a cyclic Kummer extension, and there are an integer t dividing t, σ ∈ ΓE/F and a basis 1, π, π 2 , . . . , π t−1 such that π t ∈ F, π σ = πζt and the Galois group ΓE /F of E /F is generated by σ. Moreover, both extensions E/F and E /F are totally ramiﬁed in is the ramiﬁcation index of E /F , so we have as earlier the following inequality: p, and t t 2

+ 1 (p − 1) > t, and p[t/2]+1 does not divide (ζp − 1).

Since p is odd and t does not divide p − 1, we can assume that t > 2. We will consider matrices 1 π · · · π j−1 ζ(1) − 1 π j ··· π t−1 1 πζ · · · π j−2 ζ j−2 ζ − 1 π j ζ j · · · π t−1 ζ t−1 (2) , Mj . .. 1 πζ t−1 · · · (π j−2 )σt−1 ζ(t) − 1 (π j )σt−1 · · · (π t−1 )σt−1 j = 2, . . . , t that are obtained from Wj by subtracting ﬁrst column of Wj from j-th column of Wj . For even t we may suppose that only r n − 2 elements from ζ(1) , ζ(2) , . . . , ζ(t ), the eigenvalues of h, are distinct from 1. Indeed, we can choose two elements g1 and g2 of G0 generating a noncyclic subgroup of G0 in such a way that ζpα1 , ζpα2 , . . . and ζpβ1 , ζpβ2 , . . . compose the full set of eigenvalues of g1 and g2 respectively and α1 = α2 . Set k=

−(β1 − β2 ) α1 − α2

and h = ζps · g1k g2

for s = −kα1 − β1 ,

since we are calculating αj , βj and k modulo p we can ﬁnd an integer k with this properties. Then matrix h has two eigenvalues ζ(i) for diﬀerent i, and the group generated by hγ , γ ∈ ΓE (ζp )/F (ΓE (ζp )/F denotes the Galois group of E (ζp )/F ) is abelian of exponent p; we can still apply the criterion of Proposition 1 to the group G0 generated by matrices hγ , γ ∈ ΓE /F . In other words, we can extend the group G0 , if it is needed, by adjoining some scalar matrices and naturally extending Galois action to them, and this does not change ΓE/F -stability of G0 . For convenience we still preserve our previous notation. We can apply our construction to the matrix h = ζps · g0 for some g0 ∈ G0 and if we show that this matrix is not contained in GLn (OE(ζ ), then g0 ∈ GLn (OE ), and this contradiction is exactly the p) aim of our proof of the case 1). Denote Λ = [ζ (i−1)(j−1) ]ti,j=1 . Note that Λ is a symmetric matrix. Let det Wj = det Mj = θj1 (ζ(1) − 1) + θj2 (ζ(2) − 1) + · · · + θjt (ζ(t) − 1), θjk = (−1)j+k π t(t−1)/2−(j−1) ·

where

ζ −(j−1)(k−1) λjk · c = π t(t−1)/2−(j−1) · , t t

for c = detΛ =

(ζ j − ζ i ).

1i<jt

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H. -J. BARTELS AND D. A. MALININ

and λjk = (−1)k+j ζ −(j−1)(k−1) = λkj . Indeed, denote Λ−1 = [ ζ −(j−1)(i−1)

−(j−1)(i−1)

t

]ti,j=1 , and so

· c. Let us consider the element δ from the (ij)-th cofactor of Wj is (−1)j+i · ζ t −1 Galois group of Q(ζ)/Q such that δ : ζ → ζ , and so δ = 1, δ 2 = 1. δ acts as a complex conjugation on t-th roots of 1. Note that for a t-root η of 1 η δ = η iﬀ η −1 = η or, equivalently, η = ±1. Let us determine some properties of the above elements λij under δ-action. Since the number of rows in Λ that are permuted under δ-action is equal to φ(t), the Euler function, we have cδ = c if φ(t)/2 is even and cδ = −c if φ(t)/2 is odd. Furthermore, δ permutes i-th row and (t + 2 − i)-th row of the matrix Λ for 1 < i < 1 + t/2, and (−1)i+j = (−1)t−i+j = (−1)t (−1)i+j . Therefore, if both t and φ(t)/2 are even, or both t and φ(t)/2 are odd, then λδk,j = λk,t−j+2 = λt−k+2,j for 1 < j < 1 + t/2, otherwise λδk,j = −λk,t−j+2 = −λt−k+2,j . In the general case we can claim that λδk,j = s · λk,t−j+2 = s · λt−k+2,j where s = s(t) = (−1)t+φ(t)/2 = ±1 depends only on t. = [λi,j ]−1 Let t be even, and let Λ1 = [λij ]i,j = [(−1)i+j ζ −(i−1)(j−1) ]i,j . Then Λ−1 1 i,j = [(−1)i+j · ζ

(i−1)(j−1)

t

]i,j , and it follows that cofactors of λij are equal to aij = −1

ζ (i−1)(j−1) , t

and

so all aij ≡ 0(modq), in particular, a1j = t . Let C = [cij ] be a (t − 1) × (t − 1)- matrix obtained via eliminating the ﬁrst row and the ﬁrst column of Λ. Taking an expansion of a1i −1 by 2t -th row of C we obtain: t = ci1 Ai1 +ci2 Ai2 +· · ·+ci,t−1 Ai,t−1 where Aiu are cofactors of the elements ciu in the i-th row of C. It follows that for some m Aim ≡ 0(modq). Now it is possible to ﬁx integers j = 1 and m. We can use matrices g1 = g and g2 = g σ for getting a matrix g whose eigenvalues associated with j-th and m-th blocks are ζ(j) = ζ(m) = 1 (see Proposition 1, 2)) and the above Lemma. For this purpose take the eigenvalues ζpα1 and ζpα2 of g1 and the eigenvalues ‘ζpβ1 and ζpβ2 of g2 associated with j-th and m-th blocks respectively. If ζpα1 = ζpα2 , set g = ζpα1 g, otherwise set g = ζps g1k g2 for s = −kα1 − β1 1 −β2 ) and k = −(β α1 −α2 . Now we can apply Proposition 1 to the group G0 generated by all i

hσ , i = 1, . . . , t for h = g . Let us consider a prime ideal q in the ring of integers O of the ﬁeld Qp (ζp , ζ) such that q divides p. Let us suppose that ζ(l) = 1 and the elements (ζ(t) − 1)λit (ζ(1) − 1)λi1 (ζ(2) − 1)λi2 + + ··· + , i = 1, 2, . . . , t ζ(l) − 1 ζ(l) − 1 ζ(l) − 1 are divisible by (ζ(l) − 1) in the ring O, then the system of congruences x1 λ11 + x2 λ12 + · · · + xt λ1t ≡ 0(mod q) x1 λ21 + x2 λ22 + · · · + xt λ2t ≡ 0(mod q) .. . x1 λt1 + x2 λt2 + · · · + xt λtt ≡ 0(mod q)

(S )

has a nontrivial solution x1 = 1,

x2 =

ζ(2) − 1 , ζ(l) − 1

x3 =

ζ(t) − 1 ζ(3) − 1 , · · · , xt = . ζ(l) − 1 ζ(l) − 1

Let us eliminate the ﬁrst and the (t/2 + 1)-th congruences from system (S), coeﬃcients of which are equal to (λi1 , λi2 , . . . , λit ) = (1, 1, . . . , 1) for i = 1 and (1, −1, 1, −1, . . . , 1, −1),

FINITE GALOIS STABLE SUBGROUPS OF GLn

15

for i = t/2 + 1. We obtained a system containing r = t − 2 congruences in r = t − 2 variables, since two variables xj , xm that correspond to ζ(j) = 1, ζ(k) = 1 do not appear in the system (S). The determinant of the matrix of this system is a r × r-minor N of the matrix [ζ −(i−1)(j−1) ]i,j , and the above choice of j = 1, m (such that ζ(j) = 1, ζ(m) = 1) allows us to assume that det N = 0, since det N = (−1)i+m Aim ≡ 0(modq) as it was proved above. But in this case the system has the unique solution (0, . . . , 0). This contradicts the fact that all xi in question which are diﬀerent from 0 are invertible elements of the ring of integers O of the ﬁeld Qp (ζp , ζ). Therefore, we can claim that

sj =

t

(ζ(k) − 1)λjk ≡ 0(mod(ζ(l) − 1)2 )

k=1

for some j, where summands (ζ(1) − 1)λj1 and (ζ(m) − 1)λjm are equal to 0. Since r = t − 2 and in virtue of the mentioned equality λδk,j = s · λk,t−j+2 = s · λt−k+2,j , where δ 2 = 1, we can consider some j that satisﬁes inequalities 2 + t/2 j t. Let us calculate det Mj : t dj = det Mj = π 1+2+···+(t−1)−(j−1) ( (ζ(i) − 1)λji ) = π(t(t − 1)/2 − (j − 1)) · sj . i=1

We can calculate the determinant detW with respect to the basis 1, π, . . . , π t−1 : det W = π t(t−1)/2

(ζtj − ζti ).

1i<jt

Taking into account that π j−1 does not divide (ζp − 1) for j ≥ 2 + 2t and comparing √ √ determinants D = detW and dj , we obtain that dj · ( D)−1 can not be contained in , L = E(ζp ). By Proposition 1 and its Corollary 1.4.2 this implies that the above matrix OL ) and so G0 ⊂ GLn (OE ). This is a contradiction. g ∈ GLn (OL If t is odd, the same argument is valid, and we can ﬁnd an index j such that (t + 3)/2 ≤ j ≤ n and detWj /detW ∈ Ol . Hence the previous proof remains unchanged if we eliminate the ﬁrst and the (t + 1)/2-th congruences of the above system (S). However, for odd t it would be enough to eliminate only the ﬁrst equation of the system (S). Case II. e0 divides p − 1. Now we can consider the case II. We recall the notation from the beginning of the proof of the Main Theorem. So K = Q(G) is Galois over Q, unramiﬁed outside the prime p, p > 2 and G0 = GΓ1 (p) is the subgroup of elements in G that are ﬁxed by the ﬁrst ramiﬁcation group Γ1 (p) for some prime divisor p of p, and e0 denotes the ramiﬁcation index of Q(G0 ) over Q with respect to p. For case II we suppose, that e0 is a divisor of p − 1. Firstly we need the following Lemma 2.2.2. The only ramified prime in the extension Q(G0 )(ζp )/Q is p, the ramification index e(Q(G0 )(ζp )/Q) of a ramified prime ideal in Q(G0 )(ζp ) lying over p ∩ OQ(G0 ) is p − 1. Proof of lemma 2.2.2. For the calculation of the ramiﬁcation index we consider the corresponding local situation. Therefore, let Qp denote the p-adic numbers and Q(G0 )υ the completion of Q(G0 ) with respect to the valuation υ deﬁned by the prime ideal p ∩ OQ(G0 ) .

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According to the assumptions in case II the ramiﬁcation index e0 of Q(G0 )υ /Qp divides p−1, while the ramiﬁcation of Qp (ζp )/Qp is p−1. The compositum Qp (ζp )·Q(G0 )υ = Q(G0 )υ (ζp ) of these two extensions is tamely ramiﬁed over Qp with a ramiﬁcation index t(p − 1), where the natural number t divides the ramiﬁcation index e0 and therefore divides also p − 1. We claim, that t = 1. For this purpose let Lυ denote the maximal over Qp unramiﬁed extension in Q(G0 )υ (ζp ). Then Q(G0 )υ (ζp )/Lυ is a totally ramiﬁed cyclic Galois extension. Therefore, there is only one subgroup of index t in the Galois group of this cyclic extension. Galois theory give us a uniquely determined subﬁeld of Q(G0 )υ (ζp ) over Lυ with ramiﬁcation index t. But in case t > 1 we would have two such extensions: one is a subﬁeld of Qp (ζp ) · Lυ . This contradiction shows that the ramiﬁcation index of the composite ﬁeld can not exceed p − 1. According to this Lemma 2.2.2 we see that adjoining a p-th root of unity ζp to K and extending the Galois operation to this larger ﬁeld does not inﬂuence the validity of condition II, e0 is still a divisor of p − 1. So we can and do assume ζp ∈ K without loss of generality. As it was already mentioned in the beginning of the proof of the Main Theorem we can assume that G is GLn (Q)-irreducible (using corollary 1.4.3) and that G is a counterexample to the Main Theorem with minimal order. Therefore, also in case II let G ⊂ GLn (OK ) be a group of the minimal order such that the extension Q(G)/Q is not abelian. For the treatment of case II we distinguish two subcases: case II a): Γ1 (p) is trivial, i.e. K is tamely ramified over Q. and case II b): Γ1 (p) is not trivial, i.e. K is wildly ramified over Q. We start with case II a), since we can use an argument of the proof of case I. There we have seen: if the group generated by all g γ , γ ∈ Γ for a g ∈ G is not cyclic, then some element h = ζps g1k g2 has an eigenvalue 1 (for the notation of g1 , g2 see above the proof of case I). We have the following conditions:

e0 + 1 (p − 1) > e0 , 2

and: p[t/2]+1 does not divide (ζp − 1) for t = e0 = p − 1. The argument of the proof of Case I implies that the conditions of Proposition 1 are not satisﬁed for the group generated by all hγ , γ ∈ Γ. Therefore, g γ = g a for all g ∈ G and any γ ∈ Γ0 (p). Moreover, a is the same for all g. Indeed, if g γ = g a and g1γ = g1b , with a = b, then the elements (gg1 )γ , γ ∈ Γ would generate a noncyclic group. So we have g γσ = g σγ −1 for any γ ∈ Γ0 (p), σ ∈ Γ. This implies g γ = g σγσ . If G is generated by all g γ , γ ∈ Γ, this implies the coincidence of all inertia groups Γ0 (p). Since Γ0 (p) is cyclic, it follows that G must be of A-type. Now we consider case II b), where K is wildly ramified. We assumed ζp ∈ K. Since Q(ζp ) is a tame extension of Q, Γ1 (p) operates trivially on the p-th roots of unity ζp , hence K Γ1 (p) contains also ζp . Take now in Corollary 1.5.3L = K Γ0 (p) , then this ﬁeld is unramiﬁed over Q for the prime divisor p of p. Corollary 1.5.3 shows: up to conjugation in GLn (Op ∩ K Γ0 (p)) , where Op is the valuation ring of of K Γ0 (p) (ζp ) at p, the group G0 (p) = {g ∈ G0 , g ≡ In (modp)}

FINITE GALOIS STABLE SUBGROUPS OF GLn

17

consists of diagonal matrices. The group G(p) := {g ∈ G, g ≡ In (mod p)} is a nontrivial p-group and therefore G0 (p) = {In } is not trivial as the subgroup of Γ1 (p)-ﬁxed elements of a nontrivial p-group. G is abelian and therefore in the centralizer of every matrix h ∈ G0 (p). ) holds If in particular h = diag(l1 In1 , . . . , lk Ink ), then g = diag(g1 , . . . , gk ), gi ∈ GLni (OK Γ0 (p) for every g ∈ G and therefore we can split G into GLn (Op ∩K )-irreducible components. ) of G with a In this decomposition we choose an irreducible component G ⊂ GLm (OK suitable natural number m such that G has nontrivial Γ1 (p)-action. Moreover it is worth mentioning, that the described decomposition is stable under the operation of Γ0 (p) (see Corollary 1.5.3), in particular Γ0 (p) operates on the group G . If G0 denotes the subgroup of Γ1 (p)-ﬁxed elements of G , then the group G0 (p) := {g ∈ G0 , g ≡ Im (modp)} consists of scalar matrices. The conditions on the ramiﬁcation of case II are also satisﬁed for G and G0 instead of G and G0 . But now the group G0 (p) is equal to the group µ := {ζIm , ζ p = 1}. Let us now consider the Galois-equivariant homomorphism ψ = ψm : G → GLmp (K) p

given by ψ(g) = g ⊗ . The kernel of ψ is the set of all scalar matrices contained in G . This kernel is not trivial, since G0 (p) Kerψ. Hence we have: There is an exact sequence 1 −→ µ −→ G −→ ψ(G ) −→ 1 of Γ0 (p)-groups. The aim of our proof is the construction of a certain group G1 ⊂ G ⊂ GLm (K) such that: K Γ1 (p) (G1 ) is an extension of K Γ1 (p) with ζp ∈ K Γ1 (p) (G1 ), tame ramiﬁcation index e0 = p − 1 and K Γ1 (p) (G1 )/K Γ1 (p) is an elementary abelian Kummer extension. In a second step a careful study of the Galois-action of Γ0 (p) on G1 will then show that the constructed group G1 can not exist. This gives then the desired contradiction. First step: Construction of G1 . We have H := ψ(G )Γ1 (p) = {Im } since both ψ(G ) and Γ1 (p) are p-groups. For later use we notice, that (i) H is Γ0 (p)- stable, since Γ1 (p) is a normal subgroup of Γ0 (p), and (ii) the action of Γ0 (p) on H is given by the cyclotomic character. (δ)

More precisely, we have for h ∈ H and δ ∈ Γ0 (p)hδ = hχ . Here χ(δ) denotes the unique (δ) integer modulo p such that ζ δ = ζ χ holds for all p-th root of unity ζ and δ ∈ Γ0 (p). This is an immediate consequence of Corollary 1.5.3. Now, if there exist a g ∈ ψ −1 (H) having nontrivial Γ1 (p)-action, then deﬁne G1 as the subgroup of ψ −1 (H) generated by all g δ , δ ∈ Γ0 (p). If such an element g does not exist in ψ −1 (H), we can suppose, that ψ(G ) has nontrivial Γ1 (p)-action (since otherwise

18

H. -J. BARTELS AND D. A. MALININ

g with the needed property would exist). Now consider a suitable irreducible component G of ψ(G ) having non-trivial Γ1 (p)-action and apply the corresponding map ψ to G . For simplicity we call this map ψ also simply ψ. If ψ(G ) is ﬁxed elementwise by Γ1 (p), again we have the needed element g ∈ G with non-trivial Γ1 (p)-action, and we can deﬁne G1 in G correspondingly. Otherwise, we take an irreducible component G ψ(G ) having non-trivial Γ1 (p)-action etc.. Since the order of the groups G , G , G , . . . is becoming smaller and smaller (the kernel of the diﬀerent maps ψ is not trivial), we will have at last G(i) to be ﬁxed by Γ1 (p) with the least possible i, so we have the needed element g ∈ G(i−1) with non-trivial Γ1 (p)-action. Instead of G1 we consider then the subgroup of ψ −1 (ψ(G(i−1) )Γ1 (p) ) generated by all g δ , δ ∈ Γ0 (p). For simplicity let us call these groups again G1 , G and call also the degree of the corresponding linear group again m. step 2: study of the Galois-action of Γ0 (p) on G1 and on K Γ0 (p) (G1 ). For g ∈ G1 and for γ ∈ Γ1 (p) we have ψ(g γ )ψ(g)−1 = ψ(g)γ ψ(g −1 ) = ψ(g)ψ(g)−1 = Im . This implies g γ = gζ for any γ ∈ Γ1 (p) with a suitable p-th root of unity ζ = ζγ . Let σ be an element of Γ0 (p), whose image in Γ0 (p)/Γ1 (p) is a generator of Γ0 (p)/Γ1 (p) and take g ∈ G1 . There are two possibilities: g −1 g σ ∈ GLm (K Γ1 (p) ) or g −1 g σ is not ﬁxed by the ramiﬁcation group Γ1 (p). In the ﬁrst of these two cases we claim that g σ = gζσ for a suitable p-th root of unity ζσ . Let us prove this and show how to get the desired contradiction in that case. For this purpose notice that d := g −1 g σ ≡ Im (mod p) and therefore using Corollary 1.5.3 we can diagonalize this matrix d over GLm (Op ∩ K Γ0 (p) ). But since G is irreducibel over GLm (Op ∩ K Γ0 (p) ) it follows, that d = ζσ Im , for a suitable root of unity ζσ . Now we have g σ = gζσ and at the same time g γ = gζγ for any γ ∈ Γ1 (p). Since Γ1 (p) k operates trivially on the p-th roots of unity ζ we obtain: g σ = g γ , for some integer k and therefore the two Galois automorphisms σ and γ k coincides on K Γ0 (p) (G1 ) since g is any generator of G1 . This gives the contradiction in the case, where g −1 g σ ∈ GLm (K Γ1 (p) ). In the alternative case g0 := g −1 g σ is not ﬁxed by the ramiﬁcation group Γ1 (p). Now ˜ ⊆ G generated by all elements g δ , δ ∈ Γ0 (p). Since for any δ ∈ Γ0 (p) consider the group G 1 0 we have χ(δ)

ψ(g0 δ ) = ψ(g0 )δ = ψ(g0 )χ(δ) = ψ(g0

),

(δ)

it follows that g0δ = g0χ ζδ with suitable p-th roots of unity ζδ depending on the Galois ˜ is generated by g0 and ζp Im and the order of G ˜ is p2 . automorphism δ. Therefore the group G Γ0 (p) ˜ Γ0 ˜ ˜ (G), which is Galois over K (p) by deﬁnition of G. We study the Deﬁne K := K ˜ (like on K Γ0 (p) (G ) in the ﬁrst case). For this purpose we denote by Galois-action on K 1 Γ0 (p) and Γ1 (p) the corresponding inertia respectively ramiﬁcation groups of the extenΓ {1} since the Γ1 (p)-action on G ˜ ˜ is not trivial. We then sion K/K 0 (p). We have Γ1 (p) = claim ﬁrstly, that p is the highest p-power dividing the order of Γ 0 (p). The Galois group

Γ0 (p) ˜ ˜ (considered as a is contained in the group of linear automorphism of G Γ 0 (p) of K/K 2-dimensional vector space over the ﬁeld Fp of p elements), so its order divides the order of GL2 (Fp ), which equals to (p2 − 1)(p2 − p). This implies that p2 does not divide the 1 (p) ˜ ˜ Γ is cyclic of order p, as claimed above. order of Γ 0 (p), so the Galois group of K/K

FINITE GALOIS STABLE SUBGROUPS OF GLn

19

√ 1 (p) p 1 (p) 1 (p) 1 (p) ˜ =K ˜ Γ ˜ Γ ˜ Γ ˜ Γ Hence K ( u) with u ∈ K . Now σ(K )=K since Γ 1 (p) is a normal √ √ σ (p) (p) Γ Γ p p ˜ 1 ( u) = K ˜ 1 ( u ), and one concludes: subgroup of Γ0 (p). Therefore K √ √ p 1 (p) ˜ Γ uσ ( p u)−1 ∈ K K Γ1 (p) . Since g0−1 g0γ = ζγ Im for all γ ∈ Γ1 (p) we have g0 =

√ p

ug1 with g1 ∈ K Γ1 (p) . It follows that

g0−1 g0σ ∈ GLm (K Γ1 (p) ) and we can apply Corollary 1.5.3 to this element. Like in the ﬁrst of the considered two cases with g0 instead of g we can conclude that g0σ = g0 ζσ for a suitable p-th root of unity ζσ . The contradiction follows then analogously to the ﬁrst case (see above). 2.3. Proof of Lemma 1.5.2. It is a generalization of the well known argument proposed by Minkowski [21]. The outline of our proof is given in [13]. It is easy to prove that G is abelian of exponent p. Let Op be the valuation ring of p and π a prime element. Let g1 = In + πB1 , g2 = In + πB2 for some g1 , g2 ∈ G. Then gi−1 ≡ In − πBi (mod π 2 ), i = 1, 2 and h = g1 g2 g1−1 g2−1 ≡ In (mod π 2 ). It follows from Lemma 1.5.1, (ii) that h = In , and the same Lemma 1.5.1, (ii) shows that g p = In for any g ∈ G. First of all, G is conjugate over Op to a group of triangular matrices, since G is abelian and Op is a local ring, see [5] Theorem (73.9) and the remarks in [5] on page 493. On the other hand, we can describe explicitely the matrix M such that M −1 gM = diag(λ1 , λ2 , . . . , λn ) is a diagonal matrix for a triangular matrix g of order p which is congruent to In (mod p). Indeed, let g ∈ G and ζ(1) It1 P21 . . . Pk1 0 ζ(2) It2 . . . Pk2 g= . .. , .. .. . . 0 ··· ζ(k) Itk and let It1 0 . . . 0 It2 · · · S= . . .. .. 0 ...

A1 A2 .. . It k

for t1 + t2 + · · · + tk = n and t1 ≤ t2 ≤ · · · ≤ tk , ζ(i) , i = 1, 2, . . . , k are appropriate p-roots of 1. We consider ζ(1) It1 ∗ . . . Mk1 0 ζ(2) It2 . . . Mk2 S −1 gS = . .. , .. .. . . 0 ··· ζ(k) It k

20

H. -J. BARTELS AND D. A. MALININ

and we ﬁnd the system of conditions for providing Mki = 0ti ,tk , the zero ti × tk -matrix. We have the following system of conditions: −1 1 1 1 ζ(1) (1 − ζ(k) ζ(1) )A1 + P2 A2 + · · · + Pk−1 Ak−1 + Pk = 0t1 ,tk . . . −1 k−2 k−2 = 0tk−2 ,tk ζ(k−2) Ak−2 (1 − ζ(k) ζ(k−2) ) + Pk−1 Ak−1 + Pk −1 k−1 ζ = 0tk−1 ,tk . (k−1) Ak−1 (1 − ζ(k) ζ(k−1) ) + Pk The condition g ≡ In (mod p) implies Pij ≡ 0tj ti (mod p), and we can ﬁnd Ai , 1 ≤ i ≤ k−1 sequentially using the results of previous steps: Ak−1 = −

Ak−2 = −

Ak−3 = −

Pkk−1 , −1 ζ(k−1) (1 − ζ(k) ζ(k−1) ) k−2 (Pkk−2 + Pk−1 Ak−1 ) −2 ζ(k−2) (1 − ζ(k) ζ(k−2) )

,

k−3 k−3 (Pkk−3 + Pk−1 Ak−1 + Pk−2 Ak−2 ) −1 ζ(k−3) (1 − ζ(k) ζ(k−3) )

,

and so on. Now, using induction on the degree n we can ﬁnd a matrix M that transforms g to a diagonal form as required. Since G is an abelian group of exponent p this allows to prove our claim locally over the ring Op . We use statement (81.20) in [5] for proving our result globally for the given Dedekind ring (compare for this also the proof of (81.20) and (75.27) in [5]). Remark. Another proof of the fact that G is elementary abelian can be found in [29], sect. 4 and [30], p. 187. Proof of Corollary 1.5.3. We can assume that for some matrix g ∈ G and a generator σ of Γ the condition g σ = g α , 1 < α < p, is fulﬁlled. Indeed, by Lemma 1.5.2 G is an abelian group of exponent p, so it can be considered as an Fp Γ - module over the ﬁeld Fp of p elements. Since Γ is a cyclic group of order p − 1 generated by an element σ this element determines an automorphism of G and all its eigenvalues are contained in Fp . In fact, its matrix is diagonalizable over Fp because the order of σ is prime to p. Hence we can take g ∈ G to be an eigenvector of this automorphism and so g σ = g α , 1 < α < p since not all eigenvalues are 1. Now Lemma 1.5.2 provides the existence of a matrix M ∈ GLn (Op ) such that M −1 GM is a group of diagonal matrices. We shall show that α coincides with the integer β, ζpσ = ζpβ , 1 < β < p. Let us suppose that M −1 gM = h = diag(λ1 In1 , λ2 In2 , . . . , λm Inm ), λj ∈ L(ζp ), then hσ = hβ and (M σ )−1 g σ M σ = hβ . Since M −1 g α M = hα and g σ = g α , it is obvious that (M σ )−1 M hα M −1 M σ = hβ . As Γ coincides with the inertia group of the ideal p and M ∈ GLn (Op ), it follows that M σ ≡ M (mod p). Therefore, the congruence M −1 M σ ≡ In (mod p) is valid and conjugation by

FINITE GALOIS STABLE SUBGROUPS OF GLn

21

matrix M −1 M σ maps diagonal elements of hα to diagonal elements of hβ . But if α = β, then the matrix M −1 M σ must have at least one diagonal element dii = 0, which is impossible. We proved our claim, and α = β. We obtained also that M −1 M σ = λ = diag(d1 , d2 , . . . , dm ) for some nj × nj -matrices dj . Let us introduce the following matrix: M1 =

1 (M σ1 + M σ2 + · · · + M σp−1 ), p−1

M1 = [mij ],

mij ∈ Op ,

σ1 , σ2 , . . . , σp−1 are all elements of Γ. It is clear, that M1 ≡ M (mod p) and det M1 ≡ detM (mod p). It follows that M1 ∈ GLn (Op ). Furthermore, M1 is stable under elementwise Γ-action, so all mij are Γ-stable and mij ∈ L. Hence M1 ∈ GLn (L). Since M σ = M λ, it follows that M1−1 GM1 is contained in the group of diagonal matrices, as it was claimed.

References [1] V.A. Abrashkin, Galois moduli of period p group schemes over a ring of Witt vectors, Math. USSR Izvestiya 31 (1988), 1–46. [2] H.-J. Bartels, Zur Galois Kohomologie deﬁniter arithmetischer Gruppen, J. reine angew. Math. 298 (1978), 89–97. [3] H.-J. Bartels, Y. Kitaoka, Endliche arithmetische Untergruppen der GLn , J. reine angew. Math. 313 (1980), 151–156. [4] H.-J. Bartels, D.A. Malinin, Finite Galois stable subgroups of GLn , Manuskripte der Forschergruppe Arithmetik, see http://www.math.uni-mannheim.de/∼ fga/preprint5.htm Nr.3 (2000), 21 pages. [5] C. W. Curtis, I. Reiner, Representation theory of ﬁnite groups and associative algebras, Interscience, New York, 1962. [6] J.-M. Fontaine, Il n’y a pas de vari´ et´e ab´elienne sur Z, Invent. math. 81 (1985), 515–538. [7] D. Harbater, Galois Groups with Prescribed Ramiﬁcation, Contemporary Mathematics 174 (1994), 35–60. [8] Y. Kitaoka and H. Suzuki, Finite arithmetic subgroups of GLn , IV, Nagoya Math. J. 142 (1996), 183–188. [9] D.A. Malinin., On integral representations stable under Galois action., Preprint MSLU N 5, 27p. (1997). [10] D. A. Malinin, Integral representations of ﬁnite groups with Galois action, Dokl. Russ. Acad. Nauk 349 (1996), 303–305. (Russian) [11] D.A. Malinin, Integral representations of p-groups of given nilpotency class over local ﬁelds, Algebra i analiz 10 (1998), N 1, 58–67 (Russian); English translation in St. Petersburg Math. J. v. 10, N 1, 45–52. [12] D.A. Malinin, On integral representations of ﬁnite p-groups over local ﬁelds, Dokl. Akad. Nauk USSR 309 (1989), 1060–1063 (Russian); English transl. in Sov. Math. Dokl. v.40 (1990), N 3, 619–622. [13] D. A. Malinin, On integral representations of ﬁnite nilpotent groups, Vestnik Beloruss. State Univ. Ser. 1 (1993), N 1, 27–29. (Russian) [14] D. A. Malinin, On realization ﬁelds of integral matrix groups, Vesti Beloruss. Pedag. Univ. 2 (1994), 101–104. (Belarusian) [15] D. A. Malinin, Isometries of positive deﬁnite quadratic lattices, ISLC Math. Coll. Works Lie Lobachevsky Colloquium. Tartu (1992), 21–22. [16] D.A. Malinin, Arithmetic properties of ﬁnite groups with coeﬃcients in Dedekind rings, Dissertation, Saint-Petersburg State University, St. Petersburg, 1993, 164 pages. [17] D. A. Malinin, Galois stability for integral representations of ﬁnite groups, Algebra i Analiz 12 (2000), 106–145 (Russian); English translation in St. Petersburg Math. J. v. 12, N 3.

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[18] G. N. Markshaitis, On p-extensions with one critical number, Izvestija Akad. Nauk USSR 27 (1963), 463–466. (Russian) [19] M. Mazur, Finite Arithmetic Subgroups of GLn , Journal of Number Theory 75 (1999), 109–119. [20] M. Mazur, Finite Arithmetic Subgroups of GLN . The Normalizer of a Group in the Unit Group of its Group Ring and the Isomorphism Problem., Dissertation, Department of Mathematics, Chicago, Illinois, 1999, 112 pages. ¨ ¨ [21] H. Minkowski, Uber den arithmetischen Begriﬀ der Aquivalenz und u ¨ber die endlichen Gruppen linearer ganzzahliger Substitutionen, J. reine angew. Math. 100 (1887), 449–458. [22] L. Moret-Bailly, Extensions de corps globaux a ramiﬁcation et groupe de Galois donnes, C.R. Acad. Sci. Paris, Serie 1 311 (1990), 273–276. [23] A. Odlyzko, Discriminant bounds, unpublished Tables from November 29 (1976), see http://www.research.att.com/∼ amo/unpublished/discr.bound.table. [24] I. Schur, Elementarer Beweis eines Satzes von L. Stickelberger, Math. Z. 29 (1929), 464–465. [25] J.-P. Serre, Corps locaux, Hermann, Paris, 1962. [26] D.A. Suprunenko, R.I. Tyshkevich, Commutative Matrices, Academic Press, New York and London, 1968. [27] J. Tate, The Non-Existence of Certain Galois Extensions of Q Unramiﬁed Outside 2, Contemporary Mathematics 174 (1994), 153–156. [28] J. Tate, p-Divisible Groups (1967), in: Conf. Local Fields (Dreibergen), Springer Verlag, Berlin and New York, 158–183. [29] A. Weiss, Rigidity of p-adic p-torsion, Annals of Math. 127 (1988), 317–322. [30] A. Weiss, Torsion in integral group rings, J. f¨ ur die Reine und angew. Math. 145 (1991), 175–187. 1 Fakulta ¨t fu ¨r Mathematik und Informatik, Universita ¨t Mannheim, Seminar-geba ¨ude A5, D-68131 Mannheim, Germany E-mail address: [email protected] 2 Belarusian State Pedag. University, Sovetskaya str. 18, 220050 Minsk, Belarus E-mail address: [email protected]

DERIVED CATEGORIES FOR NODAL RINGS AND PROJECTIVE CONFIGURATIONS IGOR BURBAN AND YURIY DROZD

Contents Introduction 1. Backstr¨om rings 2. Nodal rings 3. Examples 3.1. Simple node 3.2. Dihedral algebra 3.3. Gelfand problem 4. Projective conﬁgurations 5. Conﬁgurations of type A and A˜ 6. Application: Cohen–Macaulay modules over surface singularities References

23 24 25 29 29 32 33 36 37 43 45

Introduction This paper is devoted to recent results on explicit calculations in derived categories of modules and coherent sheaves. The idea of this approach is actually not new and was eﬀectively used in several questions of module theory (cf. e.g. [10, 12, 13, 7]). Nevertheless it was somewhat unexpected and successful that the same technique could be applied to derived categories, at least in the case of rings and curves with “simple singularities.” We present here two cases: nodal rings and conﬁgurations of projective lines of types A and ˜ when these calculations can be carried out up to a result, which can be presented in A, more or less distinct form, though it involves rather intricate combinatorics of a special sort of matrix problems, namely “bunches of semi-chains” [4] (or, equivalently, “clans” [8]). In Sections 1 and 4 we give a general construction of “categories of triples,” which are a connecting link between derived categories and matrix problems, while in Sections 2 ˜ Section 3 and 5 this construction is applied to nodal rings and conﬁgurations of types A. contains examples of calculations for concrete rings and Section 5 also presents those for nodal cubic. We tried to choose typical examples, which allow to better understand the general procedure of passing from combinatorial data to complexes. Section 6 contains an application to Cohen–Macaulay modules over surface singularities, which was in fact the origin of investigations of vector bundles over projective curves in [13]. More detailed exposition of these results can be found in [5, 6, 14].

2000 Mathematics Subject Classification. 16E05, 16D90. It is a survey of a research supported by the CRDF Award UM 2-2094 and by the DFG Schwerpunkt “Globale Methoden in der komplexen Geometrie”.

24

IGOR BURBAN AND YURIY DROZD

¨ m rings 1. Backstro We consider a class of rings, which generalizes in a certain way local rings of ordinary multiple points of algebraic curves. Following the terminology used in the representations theory of orders, we call them Backstr¨ om rings. Since in the ﬁrst three sections we are investigating a local situation, all rings there are supposed to be semi-perfect [3] and noetherian. We denote by A-mod the category of ﬁnitely generated A-modules and by D(A) the derived category D− (A-mod) of right bounded complexes over A-mod. As usually, it can be identiﬁed with the homotopy category K − (A-pro) of (right bounded) complexes of (ﬁnitely generated) projective A-modules. Moreover, since A is semi-perfect, each complex from K − (A-pro) is homotopic to a minimal one, i.e. to such a complex C• = (Cn , dn ) that Im dn ⊆ rad Cn−1 for all n. If C• and C• are two minimal complexes, they are isomorphic in D(A) if and only if they are isomorphic as complexes; moreover, any morphism C• → C• in D(A) can be presented by a morphism of complexes, and f is an isomorphism if and only if the latter one is. Definition 1.1. A ring A is called a Backstr¨ om ring if there is a hereditary ring H ⊇ A (also semi-perfect and noetherian) and a (two-sided) H-ideal I ⊂ A such that both R = H/I and S = A/I are semi-simple. For Backstr¨om rings there is a convenient approach to the study of derived categories. Recall that for a hereditary ring H every object C• from D(H) is isomorphic to the direct sum of its homologies. Especially, any indecomposable object from D(H) is isomorphic to α a shift N [n] for some H-module N , or, the same, to a “short” complex 0 → P −→ P → 0, where P and P are projective modules and α is a monomorphism with Im α ⊆ rad P (maybe P = 0). Thus it is natural to study the category D(A) using this information about D(H) and the functor T : D(A) → D(H) mapping C• to H ⊗A C• .1 Consider a new category T = T (A) (the category of triples) deﬁned as follows: • Objects of T are triples (A• , B• , ι), where – A• ∈ D(H); – B• ∈ D(S); – ι is a morphism B• → R ⊗H A• from D(S) such that the induced morphism ιR : R ⊗S B• → R ⊗H A• is an isomorphism in D(R). • A morphism from a triple (A• , B• , ι) to a triple (A• , B• , ι ) is a pair (Φ, φ), where – Φ : A• → A• is a morphism from D(H); – φ : B• → B• is a morphism from D(S); – the diagram ι

B• −−−−→ R ⊗H A• 1⊗Φ φ ι

(1.1)

B• −−−−→ R ⊗H A• commutes in D(S). One can deﬁne a functor F : D(A) → T (A) setting F(C• ) = (H ⊗A C• , S ⊗A C• , ι), where ι : S ⊗A C• → R ⊗H (H ⊗A C• ) R ⊗A C• is induced by the embedding S → R. The values of F on morphisms are deﬁned in an obvious way. 1 Of course, we mean here the left derived functor of ⊗, but when we consider complexes of projective modules, it restricts indeed to the usual tensor product.

DERIVED CATEGORIES FOR NODAL RINGS

25

Theorem 1.2. The functor F is a full representation equivalence, i.e. it is • dense, i.e. every object from T is isomorphic to an object of the form F(C• ); • full, i.e. each morphism F(C• ) → F(C• ) is of the form F(γ) for some γ : C• → C• ; • conservative, i.e. F(γ) is an isomorphism if and only if so is γ; As a consequence, F maps non-isomorphic objects to non-isomorphic and indecomposable to indecomposable. Note that in general F is not faithful : it is possible that F(γ) = 0 though γ = 0 (cf. Example 3.1.3 below). Sketch of the proof. Consider any triple T = (A• , B• , ι). We may suppose that A• is a minimal complex from K − (A-pro), while B• is a complex with zero diﬀerential (since S is semi-simple) and the morphism ι is a usual morphism of complexes. Note that R ⊗H A• is also a complex with zero diﬀerential. We have an exact sequence of complexes 0 −→ IA• −→ A• −→ R ⊗H A• −→ 0. Together with the morphism ι : B• → R ⊗H A• it gives rise to a commutative diagram in the category of complexes Com− (A-mod) 0 −−−−→ IA• −−−−→ A• −−−−→ R ⊗H A• −−−−→ 0 ι α 0 −−−−→ IA• −−−−→ C• −−−−→

B•

−−−−→ 0,

where C• is the preimage in A• of Im ι. The lower row is also an exact sequence of complexes and α is an embedding. Moreover, since ιR is an isomorphism, IA• = IC• . It implies that C• consists of projective A-modules and H ⊗A C• A• , wherefrom T FC• . Let now (Φ, φ) : FC• → FC• . We suppose again that both C• and C• are minimal, while Φ : H⊗A C• → H⊗A C• and φ : S⊗A C• → S⊗A C• are morphisms of complexes. Then the diagram (1.1) is commutative in the category of complexes, so Φ(C• ) ⊆ C• and Φ induces a morphism γ : C• → C• . It is evident from the construction that F(γ) = (Φ, φ). Moreover, if (Φ, φ) is an isomorphism, so are Φ and φ (since our complexes are minimal). Therefore Φ(C• ) = C• , i.e. Im γ = C• . But ker γ = ker Φ ∩ C• = 0, thus γ is an isomorphism too. 2. Nodal rings We apply these considerations to the class of rings ﬁrst considered in [10], where the second author has shown that they are unique pure noetherian rings such that the classiﬁcation of their modules of ﬁnite length is tame (all others being wild). Definition 2.1. A ring A (semi-perfect and noetherian) is called a nodal ring if it is pure noetherian, i.e. has no minimal ideals, and there is a hereditary ring H ⊇ A, which is semi-perfect and pure notherian such that 1) rad A = rad H; we denote this common radical by R. 2) lengthA (H ⊗A U ) ≤ 2 for every simple left A-module U and lengthA (V ⊗A H) ≤ 2 for every simple right A-module V . Note that condition 2 must be imposed both on left and on right modules.

26

IGOR BURBAN AND YURIY DROZD

It is known that such a hereditary ring H is Morita equivalent to a direct product of rings H(D, n), where D is a discrete valuation ring (maybe non-commutative) and H(D, n) is the subring of Mat(n, D) consisting of all matrices (aij ) with non-invertible entries aij for i < j. Especially, H and A are semi-prime (i.e. without nilpotent ideals) Example 2.2. 1. The ﬁrst example of a nodal ring is the completion of the local ring of a simple node (or a simple double point) of an algebraic curve over a ﬁeld k. It is isomorphic to A = k[[x, y]]/(xy) and can be embedded into H = k[[x1 ]] × k[[x2 ]] as the subring of pairs (f, g) such that f (0) = g(0): x maps to (x1 , 0) and y to (0, x2 ). Evidently this embedding satisﬁes conditions of Deﬁnition 2.1. 2. The dihedral algebra A = k x, y /(x2 , y 2 ) is another example of a nodal ring. In this case H = H(k[[t]], 2) and the embedding A → H is given by the rule x →

0t , 00

00 y → . 10

3. The “Gelfand problem” is that of classiﬁcation of diagrams with relations x+

2

y+

1 x−

3 y−

x+ x− = y+ y− .

If we consider the case when x+ x− is nilpotent (the main part of the problem), such diagrams are just modules over the ring A, which is the subring of Mat(3, k[[t]]) consisting of all matrices (aij ) with a12 (0) = a13 (0) = a23 (0) = a32 (0) = 0. The arrows of the diagram correspond to the following matrices: x+ → te12 ,

x− → e21 ,

y+ → te13 ,

y− → e31 ,

where eij are matrix units. It is also a nodal ring with H being the subring of Mat(3, k[[t]]) consisting of all matrices (aij ) with a12 (0) = a13 (0) = 0 (it is Morita equivalent to H(k[[t]], 2)). 4. The classiﬁcation of quadratic functors, which play an important role in algebraic topology, reduces to the study of modules over the ring A, which is the subring of Z22 × Mat(2, Z2 ) consisting of all triples c 2c a, b, 1 2 c3 c4

with a ≡ c1 (mod 2) and b ≡ c4 (mod 2),

where Z2 is the ring of p-adic integers [11]. It is again a nodal ring: one can take for H the ring of all triples as above, but without congruence conditions; then H = Z22 × H(Z2 , 2). Certainly, a nodal ring is always Backstr¨ om, so Theorem 1.2 can be applied. Moreover, in nodal case the resulting problem belongs to a well-known type. For the sake of simplicity, we consider now the situation, when A is a D-algebra ﬁnitely generated as D-module, where D is a discrete valuation ring with algebraically closed residue ﬁeld k. We denote by U1 , U2 , . . . , Us indecomposable non-isomorphic projective (left) modules over A

DERIVED CATEGORIES FOR NODAL RINGS

27

and by V1 , V2 , . . . , Vr those over H. Condition 2 from Deﬁnition 2.1 implies that there are three possibilities: 1) H ⊗A Ui Vj for some j and Vj does not occur as a direct summand in H ⊗A Uk for k = i; 2) H ⊗A Ui Vj ⊕ Vj (j = j ) and neither Vj nor Vj occur in H ⊗A Uk for k = i; 3) there are exactly two indices i = i such that H ⊗A Ui H ⊗A Ui Vj and Vj does not occur in H ⊗A Uk for k ∈ / { i, i }. We denote by Hj the indecomposable projective H-module such that Hj /RHj Vj . Since H is a semi-perfect hereditary order, any indecomposable complex from D(H) is φ

isomorphic either to 0 → Hk −→ Hj → 0 or to 0 → Hj → 0 (it follows, for instance, from [9]). Moreover, the former complex is completely deﬁned by either j or k and the length l = lengthH Coker φ. We shall denote it both by C(j, −l, n) and by C(k, l, n + 1), while the latter complex will be denoted by C(j, ∞, n), where n denotes the place of Hj ˜ the set (Z \ { 0 }) ∪ { ∞ } and consider the (so the place of Hk is n + 1). We denote by Z ˜ which coincides with the usual ordering separately on positive integers ordering ≤ on Z, and on negative integers, but l < ∞ < −l for any l ∈ N. Note that for each j the submodules of Hj form a chain with respect to inclusion. It immediately implies the following result. Lemma 2.3. There is a homomorphism C(j, l, n) → C(j, l , n), which is an isomorphism ˜ Otherwise the n-th component of any on the n-th components, if and only if l ≤ l in Z. homomorphism C(j, l, n) → C(j, l , n) is zero modulo R. ˜ , so the ˜ to the set Ej,n = C(j, l, n) | l ∈ Z We transfer the ordering from Z latter becomes a chain with respect to this ordering. We also denote by Fj,n the set { (i, j, n) | Vj is a direct summand of H ⊗A Ui }. It has at most two elements. We always consider Fj,n with trivial ordering. Then a triple (A• , B• , ι) from the category T (A) is given by homomorphisms φijn jln : di,j,n Ui → rj,l,n Vj , where (i, j, n) ∈ Fjn , the left Ui comes from Bn and the right Vj comes from direct summands rj,l,n C(j, l, n) of A• . Note that if both C(j, −l, n) and C(k, l, n + 1) correspond to the same complex (then we write ijn C(j, −l, n) ∼ C(k, l, n + 1)), we have rj,−l,n = rk,l,n+1 . We present φijn jln by its matrix Mjln . Then Lemma 2.3 implies the following ijn ijn and Njln describe isomorphic triples Proposition 2.4. Two sets of matrices Mjln if and only if one of them can be transformed to the other by a sequence of the following “elementary transformations”: ijn ijn 1) For any given values of i, n, simultaneously Mjln

→ Mjln S for all j, l such that (ijn) ∈ Fj,n , where S is an invertible matrix of appropriate size. ijn ijn

→ S Mjln for all (i, j, n) ∈ Fjn 2) For any given values of j, l, n, simultaneously Mjln

i,k,n−sgn l i,k,n−sgn l and Mk,−l,n−sgn l → S Mk,−l,n−sgn l for all (i, k, n − sgn l) ∈ Fk,n−sgn l , where S is an invertible matrix of appropriate size and C(j, l, n) ∼ C(k, −l, n − sgn l). If l = ∞, it ijn ijn just means Mj∞n

→ SMj∞n . ijn ijn ijn

→ Mjln + RMjl 3) For any given values of j, l < l, n, simultaneously Mjln n for all (i, j, n) ∈ Fj,n , where R is an arbitrary matrix of appropriate size. Note that, unlike

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IGOR BURBAN AND YURIY DROZD i,k,n−sgn l the preceding transformation, this one does not touch the matrices Mk,−l,n−sgn l such that C(j, l, n) ∼ C(k, −l, n − sgn l).

This sequence must contain ﬁnitely many transformations for every ﬁxed values of j and n. Therefore we obtain representations of the bunch of semi-chains Ejn , Fjn in the sense of [4], so we can deduce from this paper a description of indecomposables in D(A). We arrange it in terms of strings and bands, often used in representation theory. Definition 2.5. 1. We deﬁne the alphabet X as the set j,n (Ej,n ∪ { (j, n) }). We deﬁne symmetric relations ∼ and − on X by the following exhaustive rules: (a) C(j, l, n) − (j, n) for all l ∈ Z; (b) C(j, −l, n) ∼ C(k, l, n + 1) deﬁned as above; (c) (j, n) ∼ (k, n) (k = j) if Vj ⊕ Vk H ⊗A Ui for some i; (d) (j, n) ∼ (j, n) if Vj H ⊗A Ui H ⊗A Ui for some i = i. 2. We deﬁne an X-word as a sequence w = x1 r1 x2 r2 x3 . . . rm−1 xm , where xk ∈ X, rk ∈ { −, ∼ } such that (a) xk rk xk+1 in X for 1 ≤ k < m; (b) rk = rk+1 for 1 ≤ k < m − 1. We call x1 and xm the ends of the word w. 3. We call an X-word w full if (a) r1 = rm−1 = − (b) x1 ∼ y for each y = x1 ; (c) xm ∼ z for each z = xm . Condition (a) reﬂects the fact that ιR must be an isomorphism, while conditions (b,c) come from generalities on bunches of semi-chains [4]. 4. A word w is called symmetric, if w = w∗ , where w∗ = xm rm−1 xm−1 . . . r1 x1 (the inverse word ), and quasisymmetric, if there is a shorter word v such that w = v ∼ v ∗ ∼ · · · ∼ v ∗ ∼ v. 5. We call the end x1 (xm ) of a word w special if x1 ∼ x1 and r1 = − (respectively, xm ∼ xm and rm−1 = −). We call a word w (a) usual if it has no special ends; (b) special if it has exactly one special end; (c) bispecial if it has two special ends. Note that a special word is never symmetric, a quasisymmetric word is always bispecial, and a bispecial word is always full. 6. We deﬁne a cycle as a word w such that r1 = rm−1 =∼ and xm − x1 . Such a cycle is called non-periodic if it cannot be presented in the form v − v − · · · − v for a shorter cycle v. For a cycle w we set rm = −, xqm+k = xk and rqm+k = rk for any q, k ∈ Z. 7. A (k-th) shift of a cycle w, where k is an even integer, is the cycle w[k] = xk+1 rk+1 xk+2 . . . rk−1 xk . A cycle w is called symmetric if w[k] = w∗ for some k. 8. We also consider inﬁnite words of the sorts w = x1 r1 x2 r2 . . . (with one end) and w = . . . x0 r0 x1 r1 x2 r2 . . . (with no ends) with restrictions (a) every pair (j, n) occurs in this sequence only ﬁnitely many times; (b) there is an n0 such that no pair (j, n) with n < n0 occurs. We extend to such inﬁnite words all above notions in the obvious manner. Definition 2.6 (String and band data). 1. String data are deﬁned as follows: (a) a usual string datum is a full usual non-symmetric X-word w; (b) a special string datum is a pair (w, δ), where w is a full special word and δ ∈ { 0, 1 };

DERIVED CATEGORIES FOR NODAL RINGS

29

(c) a bispecial string datum is a quadruple (w, m, δ1 , δ2 ), where w is a bispecial word that is neither symmetric nor quasisymmetric, m ∈ N and δ1 , δ2 ∈ { 0, 1 }. 2. A band datum is a triple (w, m, λ), where w is a non-periodic cycle, m ∈ N and λ ∈ k∗ ; if w is symmetric, we also suppose that λ = 1. The results of [4, 8] imply Theorem 2.7. Every string or band datum d deﬁnes an indecomposable object C• (d) from D(A), so that 1) Every indecomposable object from D(A) is isomorphic to C• (d) for some d. 2) The only isomorphisms between these complexes are the following: (a) C(w) C(w∗ ); (b) C(w, m, δ1 , δ2 ) C(w∗ , m, δ2 , δ1 ); (c) C(w, m, λ) C(w[k] , m, λ) C(w∗ [k] , m, 1/λ) if k ≡ 0 (mod 4); (d) C(w∗ , m, λ) C(w[k] , m, 1/λ) C(w∗ [k] , m, λ) if k ≡ 2 (mod 4). 3) Every object from D(A) uniquely decomposes into a direct sum of indecomposable objects. The construction of complexes C• (d) is rather complicated, especially in the case, when there are pairs (j, n) with (j, n) ∼ (j, n) (e.g. special ends are involved). So we only show several examples arising from simple node, dihedral algebra and Gelfand problem.

3. Examples 3.1. Simple node. In this case there is only one indecomposable projective A-module (A itself) and two indecomposable projective H-modules H1 , H2 corresponding to the ﬁrst and the second direct factors of the ring H. We have H ⊗A A H H1 ⊕ H2 . So the ∼-relation is given by: 1) (1, n) ∼ (2, n); 2) C(j, l, n) ∼ C(j, −l, n − sgn l) for any l ∈ Z \ { 0 }. Therefore there are no special ends at all. Moreover, any end of a full string must be of the form C(j, ∞, n). Note that the homomorphism in the complex corresponding to C(j, −l, n) and C(j, l, n + 1) (l ∈ N) is just multiplication by xlj . Consider several examples of strings and bands.

Example 3.1.

1. Let w be the cycle

C(2, 1, 1) ∼ C(2, −1, 0) − (2, 0) ∼ (1, 0) − C(1, −2, 0) ∼ C(1, 2, 1)− − (1, 1) ∼ (2, 1) − C(2, 4, 1) ∼ C(2, −4, 0) − (2, 0) ∼ (1, 0)− − C(1, −1, 0) ∼ C(1, 1, 1) − (1, 1) ∼ (2, 1) − C(2, −3, 1) ∼ C(2, 3, 2)− − (2, 2) ∼ (1, 2) − C(1, 2, 2) ∼ C(1, −2, 1) − (1, 1) ∼ (2, 1)

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IGOR BURBAN AND YURIY DROZD

Then the band complex C• (w, 1, λ) is obtained from the complex of H-modules x2

H2

H2

x21

H1

H1

x42

H2

H2

λ x1

H1 x32

H2

x21

H1

H1

H2

H1

by gluing along the dashed lines (they present the ∼ relations (1, n) ∼ (2, n)). All glueings are trivial, except the last one marked with ‘λ’; the latter must be twisted by λ. It gives the A-complex y

A λx2

y4

A

A

A

x2

A

(3.1)

x

y3

A Here each column presents direct summands of a non-zero component Cn (in our case n = 2, 1, 0) and the arrows show the non-zero components of the diﬀerential. According to the embedding A → H, we have to replace x1 by x and x2 by y. Gathering all data, we can rewrite this complex as

λx2 y 0 2 4 x y 0 0 x y3 A −−−−−→ A ⊕ A ⊕ A −−−−−−→ A ⊕ A ,

though the form (3.1) seems more expressive, so we use it further. If m > 1, one only has to replace A by mA, each element a ∈ A by aE, where E is the identity matrix,

DERIVED CATEGORIES FOR NODAL RINGS

31

and λa by aJm (λ), where Jm (λ) is the Jordan m × m cell with eigenvalue λ. So we obtain the complex 2 x Jm (λ) 0 3

yE

0 y4 E 0 xE y E mA −−−−−−−−−→ mA ⊕ mA ⊕ mA −−−−−−−−−→ mA ⊕ mA . 2 x E

2. Let w be the word

C(1, ∞, 1) − (1, 1) ∼ (2, 1) − C(2, 2, 1) ∼ C(2, −2, 0) − (2, 0) ∼ ∼ (1, 0) − C(1, −3, 0) ∼ C(1, 3, 1) − (1, 1) ∼ (2, 1) − C(2, −1, 1) ∼ ∼ C(2, 1, 2) − (2, 2) ∼ (1, 2) − C(1, 1, 2) ∼ C(1, −1, 1) − (1, 1) ∼ ∼ (2, 1) − C(2, 2, 1) ∼ C(2, −2, 0) − (2, 0) ∼ (1, 0) − C(1, ∞, 0)

Then the string complex C• (w) is

A

A

y

y2

A

x3

A

x

A

y2

A

Note that for string complexes (which are always usual in this case) there are no multiplicities m and all glueings are trivial. a 3. Set a = x + y. Then the factor A/aA is represented by the complex A −→ A, which is the band complex C• (w, 1, 1), where w = C(1, 1, 1) ∼ C(1, −1, 0) − (1, 0) ∼ (2, 0)− − C(2, −1, 0) ∼ C(2, 1, 1) − (2, 1) ∼ (1, 1).

Consider the morphism of this complex to A[1] given on the 1-component by multix plication A −→ A. It is non-zero in D(A), but the corresponding morphism of triples a is (Φ, 0), where Φ arises from the morphism of the complex H −→ H to H[1] given by multiplication with x1 . But Φ is homotopic to 0: x1 = e1 a, where e1 = (1, 0) ∈ H, thus (Φ, 0) = 0 in the category of triples.

32

IGOR BURBAN AND YURIY DROZD

4. The string complex C• (l, 0), where w is the word C(1, ∞, 0) − (1, 0) ∼ (2, 0) − C(2, −1, 0) ∼ C(2, 1, 1) − (2, 1) ∼ ∼ (1, 1) − C(1, −2, 1) ∼ C(1, 1, 2) − (1, 2) ∼ (2, 2) − C(2, −1, 2) ∼ ∼ C(2, 1, 3) − (2, 3) ∼ (1, 3) − C(1, −2, 3) ∼ C(1, 2, 4) − · · · , is x2

x2

y

y

. . . A −→ A −→ A −→ A −→ A −→ 0. Its homologies are not left bounded, so it does not belong to Db (A-mod). 3.2. Dihedral algebra. This case is very similar to the preceding one. Again there is only one indecomposable projective A-module (A itself) and two indecomposable projective Hmodules H1 , H2 corresponding to the ﬁrst and the second columns of matrices from the ring H, and we have H ⊗A A H H1 ⊕ H2 . The main diﬀerence is that now the unique maximal submodule of Hj is isomorphic to Hk , where k = j. So the ∼-relation is given by: 1) (1, n) ∼ (2, n); 2) C(j, l, n) ∼ C(j, −l, n−sgn l) if l ∈ Z\{ 0 } is even, and C(j, l, n) ∼ C(j , −l, n−sgn l), where j = j, if l ∈ Z \ { 0 } is odd. Again there are no special ends. The embeddings Hk → Hj are given by right multiplications with the following elements from H: H1 → H1 − by tr e11 r

H1 → H2 − by t e12 r

H2 → H1 − by t e21 r

H2 → H2 − by t e22

(colength 2r), (colength 2r − 1), (colength 2r + 1), (colength 2r).

When gluing H-complexes into A-complexes we have to replace them respectively tr e11 − by (xy)r , tr e22 − by (yx)r , tr e12 − by (xy)r−1 x, tr e21 − by (yx)r y. The glueings are quite analogous to those for simple node, so we only present the results, without further comments. Example 3.2.

1. Consider the band datum (w, 1, λ), where

w = C(1, −2, 0) ∼ C(1, 2, 1) − (1, 1) ∼ (2, 1) − C(2, −5, 1) ∼ ∼ C(1, 5, 2) − (1, 2) ∼ (2, 2) − C(2, 4, 2) ∼ C(2, −4, 1) − (2, 1) ∼ ∼ (1, 1) − C(1, 3, 1) ∼ C(2, −3, 0) − (2, 0) ∼ (1, 0).

DERIVED CATEGORIES FOR NODAL RINGS

33

The corresponding complex C• (w, m, λ) is mA 2

(yx)2 E

mA

xyxJm (λ)

(xy) xE

mA

xyE

mA

2. Let w be the word C(2, ∞, 0) − (2, 0) ∼ (1, 0) − C(1, −1, 0) ∼ C(2, 1, 1) − (2, 1) ∼ (1, 1) − C(1, 3, 1) ∼ ∼ C(2, −3, 0) − (2, 0) ∼ (1, 0) − C(1, −3, 0) ∼ C(2, 3, 1) − (2, 1) ∼ (1, 1) − C(1, ∞, 1). Then the string complex C• (w) is A

e21

A

2

t e12

A

te21

A

3. The factor A/R is described by the inﬁnite string complex C• (w) ...

e21

A

te12

A

e21

A.

te12

...

te12

A

e21

A

The corresponding word w is

· · · − C(2, 1, 2) ∼ C(1, −1, 1) − (1, 1) ∼ (2, 1)− − C(2, 1, 1) ∼ C(1, −1, 0) − (1, 0) ∼ (2, 0) − C(2, −1, 0) ∼ ∼ C(1, 1, 1) − (1, 1) ∼ (2, 1) − C(2, −1, 1) ∼ C(1, 1, 2) − · · ·

3.3. Gelfand problem. In this case there are 2 indecomposable projective H-modules H1 (the ﬁrst column) and H2 (both the second and the third columns). There are 3 indecomposable A-projectives Ai (i = 1, 2, 3); Ai correspond to the i-th column of A. We have H ⊗A A1 H1 and H ⊗A A2 H ⊗A A3 H2 . So the relation ∼ is given by: 1) (2, n) ∼ (2, n); 2) C(j, l, n) ∼ C(j, −l, n − sgn l) if l is even; 3) C(j, l, n) ∼ C(j , −l, n − sgn l) (j = j) if l is odd. So a special end is always (2, n).

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IGOR BURBAN AND YURIY DROZD

Example 3.3.

1. Consider the special word w:

(2, 0) − C(2, −2, 0) ∼ C(2, 2, 1) − (2, 1) ∼ (2, 1) − C(2, −4, 1) ∼ ∼ C(2, 4, 2) − (2, 2) ∼ (2, 2) − C(2, 2, 2) ∼ C(2, −2, 1)− − (2, 1) ∼ (2, 1) − C(2, −1, 1) ∼ C(1, 1, 2) − (1, 2) The complex C• (w, 0) is obtained by gluing from the complex of H-modules H2

H2

4

H2

H2

2

H2

H1

1

H2

2

H2

Here the numbers inside arrows show the colengths of the corresponding images. We mark dashed lines deﬁning glueings with arrows going from the bigger complex (with respect to the ordering in Ej,n ) to the smaller one. When we construct the corresponding complex of A-modules, we replace each H2 by A2 and A3 starting with A2 (since δ = 0; if δ = 1 we start from A3 ). Each next choice is arbitrary with the only requirement that every dashed line must touch both A2 and A3 . (Diﬀerent choices lead to isomorphic complexes: one can see it from the pictures below.) All horizontal mappings must be duplicated by slanting ones, carried along the dashed arrow from the starting point or opposite the dashed arrow with the opposite sign from the ending point (the latter procedure will be marked by ‘−’ near the duplicated arrow). So we get the A-complex −

A2

4

A3

2

A2

2

4

A3

2

A2

2 1

2

−

A2

2

A1

1

A3

All mappings are uniquely deﬁned by the colengths in the H-complex, so we just mark them with ‘l.’

DERIVED CATEGORIES FOR NODAL RINGS

35

2. Let w be the bispecial word (2, 2) − C(2, 2, 2) ∼ C(2, −2, 1) − (2, 1) ∼ (2, 1) − C(2, 2, 1) ∼ ∼ C(2, −2, 0) − (2, 0) ∼ (2, 0) − C(2, −4, 0) ∼ C(2, 4, 1)− − (2, 1) ∼ (2, 1) − C(2, 6, 1) ∼ C(2, −6, 0) − (2, 0) The complex C• (w, m, 1, 0) is the following one: aA3 ⊕ bA2

M1

mA3

−M1

2

−

mA2

2

2

mA3

2

mA3

−

4

−

mA2

4

mA2

aA2 ⊕ bA3

M2

where a = [(m + 1)/2], b = [m/2], so a + b = m. (The change of δ1 , δ2 transpose A2 and A3 at the ends.) All arrows are just αl E, where αl is deﬁned by the colength l, except of the “end” matrices Mi . To calculate the latter, write αl E for one of them (say, M1 ) and αl J for anothher one (say, M2 ), where J is the Jordan m × m cell with eigenvalue 1, then put the odd rows or columns into the ﬁrst part of Mi and the even ones to its second part. In our example we get

1 0 M1 = α2 0 0 0

0 0 1 0 0

0 0 0 0 1

0 1 0 0 0

0 0 0 1 0

,

M2 = α6

1 0 0 0 0

1 0 0 1 0

0 1 0 1 0

0 1 0 0 1

0 0 1 . 0 1

(We use columns for M1 and rows for M2 since the left end is the source and the right end is the sink of the corresponding mapping.) 3. The band complex C• (w, 1, λ), where w is the cycle (2, 1) ∼ (2, 1) − C(2, −2, 1) ∼ C(2, 2, 2) − (2, 2) ∼ (2, 2)− − C(2, 4, 2) ∼ C(2, −4, 1) − (2, 1) ∼ (2, 1) − C(2, 6, 1) ∼ ∼ C(2, −6, 0) − (2, 0) ∼ (2, 0) − C(2, −4, 0) ∼ C(2, 4, 1) is

36

IGOR BURBAN AND YURIY DROZD 2

mA2

mA2

2 4

mA3

4λ

mA2

6

24 −

2

−

mA3

−

6

mA3 −

4λ

4λ

mA2 −

4λ

mA3

Superscript ‘λ’ denotes that the corresponding mapping must be twisted by Jm (λ). 4. The projective resolution of the simple A-module U1 is −

A2

1

A1

1

1

A1

1

A3

It coincides with the usual string complex C• (w), where w is (1, 0) − C(1, −1, 0) ∼ C(2, 1, 1) − (2, 1) ∼ (2, 1) − C(2, −1, 1) ∼ C(1, 1, 2) − (1, 2). The projective resolution of U2 (U3 ) is A1 → A2 (respectively A1 → A3 ), which is the special string complex C• (w, 0) (respectively C• (w, 1)), where w = (2, 0) − C(2, −1, 0) ∼ C(1, 1, 1) − (1, 1). Note that gl.dim A = 2.

4. Projective configurations We can “globalize” the results of the preceding sections. The simplest way is to consider the so called projective conﬁgurations, which are a sort of global analogues of Backstr¨om rings. Definition 4.1. Let X be a projective curve over k, which we suppose reduced, but possibly ˜ → X its normalization; then X ˜ is a disjoint union of reducible. We denote by π : X ˜ are rational smooth curves. We call X a projective conﬁguration if all components of X curves (i.e. of genus 0) and all singular points p of X are ordinary. Thelatter means that m m if π −1 (p) = { y1 , y2 , . . . , ym }, the image of OX,p in i=1 OX,y ˜ i contains i=1 mi , where mi is the maximal ideal of OX,y ˜ i.

DERIVED CATEGORIES FOR NODAL RINGS

37

We denote by S = { p1 , p2 , . . . , ps } the set of singular points of X and by S˜ = ˜ We also put O = OX , O ˜ = O ˜ and denote by J the { y1 , y2 , . . . , yr } its preimage in X. X ˜ in O, i.e. the maximal sheaf of π∗ O-ideals ˜ conductor of O contained in O. Set S = O/J ˜ ˜ −1 J . Both these sheaves have 0-dimensional support S, so we O/π and R = π∗ O/J may (and shall) identify them with the algebras of their global sections. In the case of s projective conﬁgurations both these algebras are semi-simple, namely S = i=1 k(pi ) and r R = i=1 k(yi ). Let D(X) = D− (Coh X) be the right bounded derived category of coherent sheaves over X. As X is a projective variety, it can be identiﬁed with the category of fractions K − (VB X)[Q−1 ], where K − (VB X) is the category of right bounded complexes of vector bundles (or, the same, locally free coherent sheaves) over X modulo homotopy and Q is the set of quasi-isomorphisms in K − (VB X). So we always present objects from D(X) ˜ as complexes of vector bundles. We denote by T : D(X) → D(X) ˜ the and from D(X) ∗ left derived functor Lπ . Again if C• is a complex of vector bundles, T C• coincides with π ∗ C• . Just as in Section 1, we deﬁne the category of triples T = T (X) as follows: • Objects of T are triples (A• , B• , ι), where ˜ – A• ∈ D(X); – B• ∈ D(S); – ι is a morphism B• → R ⊗O˜ A• from D(S) such that the induced morphism ιR : R ⊗S B• → R ⊗O˜ A• is an isomorphism in D(R). • A morphism from a triple (A• , B• , ι) to a triple (A• , B• , ι ) is a pair (Φ, φ), where ˜ – Φ : A• → A• is a morphism from D(X); – φ : B• → B• is a morphism from D(S); – the diagram ι

B• −−−−→ R ⊗O˜ A• 1⊗Φ φ ι

(4.1)

B• −−−−→ R ⊗O˜ A• commutes in D(S). We deﬁne a functor F : D(X) → T (X) setting F(C• ) = (π ∗ C• , S ⊗O C• , ι), where ι : S ⊗O C• → R ⊗O˜ (π ∗ C• ) R ⊗O C• is induced by the embedding S → R. Just as in Section 1 the following theorem holds (with almost the same proof, see [6]). Theorem 4.2. The functor F is a representation equivalence, i.e. it is dense and conservative. Remark. We do not now whether it is full, though it seems to be true.

5. Configurations of type A and A˜ As it was shown in [13], even classiﬁcation of vector bundles is wild for almost all projective curves. Among singular curves the only exceptions are projective conﬁgurations of ˜ These curves only have ordinary double points (so no three components type A and A.

38

IGOR BURBAN AND YURIY DROZD

have a common point). Moreover, in A case irreducible components X1 , X2 , . . . , Xs and singular points p1 , p2 , . . . , ps−1 can be so arranged that pi ∈ Xi ∩ Xi+1 , while in A˜ case the components X1 , X2 , . . . , Xs and the singular points p1 , p2 , . . . , ps can be so arranged that pi ∈ Xi ∩ Xi+1 for i < s and ps ∈ Xs ∩ X1 . Note that in A case s > 1, while in A˜ case s = 1 is possible: then there is one component with one ordinary double point (a nodal plane cubic). These projective conﬁgurations are global analogues of nodal rings, and the calculations according Theorem 4.2 are quite similar to those of Section 2. We present here the A˜ case and add remarks explaining which changes should be done for A case. s If s > 1, the normalization of X is just a disjoint union i=1 Xi ; for uniformity, we ˜ if s = 1. We also denote Xqs+i = Xi . Note that Xi P1 for all i. write X1 = X ˜ we suppose that p ∈ Xi correEvery singular point pi has two preimages pi , pi in X; i 1 sponds to the point ∞ ∈ P and pi ∈ Xi+1 corresponds to the point 0 ∈ P1 . Recall that any indecomposable vector bundle over P1 is isomorphic to OP1 (d) for some d ∈ Z. ˜ is isomorphic either to 0 → Oi (d) → 0 So every indecomposable complex from D(X) or to 0 → Oi (−lx) → Oi → 0, where Oi = OXi , d ∈ Z, l ∈ N and x ∈ Xi . The latter complex corresponds to the indecomposable sky-scraper sheaf of length l and support { x }. We denote this complex by C(x, −l, n) and by C(x, l, n + 1). The complex 0 → Oi (d) → is denoted by C(pi , dω, n) and by C(pi−1 , dω, n). As before, n is the unique place, where the complex has non-zero homologies. We deﬁne the symmetric relation ∼ for these symbols setting C(x, −l, n) ∼ C(x, l, n + 1) and C(pi , dω, n) ∼ C(pi−1 , dω, n). Let Zω = (Z ⊕ { 0 }) ∪ Zω, where Zω = { dω | d ∈ Z }. We introduce an ordering on Zω , which is natural on N, on −N and on Zω, but l < dω < −l for each l ∈ N, d ∈ Z. Then an analogue of Lemma 2.3 can be easily veriﬁed. Lemma 5.1. There is a morphism of complexes C(x, z, n) → C(x, z , n) such that its nth component induces a non-zero mapping on Cn (x) if and only if z ≤ z in Zω . We introduce the ordered sets Ex,n = { C(x, z, n) | z ∈ Zω } with the ordering inherited from Zω , We also put Fx,n = { (x, n) } and (pi , n) ∼ (pi−1 , n) for all i, n. Lemma 5.1 shows that the category of triples T (X) can be again described in terms of the bunch of chains { Ex,n , Fx,n }. Thus we can describe indecomposable objects in terms of strings and bands just as for nodal rings. We leave the corresponding deﬁnitions to the reader; they are quite analogous to those from Section 2. If we consider a conﬁguration of type A, we have to exclude the points ps , ps and the corresponding symbols C(ps , z, n), C(ps , z, n), (ps , n), (ps , n). Thus in this case C(ps−1 , dω, n) and C(p1 , dω, n) are not in ∼ relation with any symbol. It makes possible ﬁnite or oneside inﬁnite full strings, while in A˜ case only two-side inﬁnite strings are full. Note that an inﬁnite word must contain a ﬁnite set of symbols (x, n) with any ﬁxed n; moreover there must be n0 such that n ≥ n0 for all entries (x, n) that occur in this word. If x ∈ / S and z ∈ / Zω, the complex C(x, z, n) vanishes after tensoring by R, so gives no essential input into the category of triples. It gives rise to the n-th shift of a sky-scraper sheaf with support at the regular point x. Therefore in the following examples we only consider complexes C(x, z, n) with x ∈ S. Moreover, we conﬁne most examples to the case s = 1 (so X is a nodal cubic). If s > 1, one must distribute vector bundles in the pictures ˜ below among the components of X.

DERIVED CATEGORIES FOR NODAL RINGS

39

Example 5.2. 1. First of all, even a classiﬁcation of vector bundles is non-trivial in A˜ case. They correspond to bands concentrated at 0 place, i.e. such that the underlying cycle w is of the form

(ps , 0) ∼ (ps , 0) − C(ps , d1 ω, 0) ∼ C(p1 , d1 ω, 0)− − (p1 , 0) ∼ (p1 , 0) − C(p1 , d2 ω, 0) ∼ C(p2 , d2 ω, 0)− − (p2 , 0) ∼ (p2 , 0) − C(p2 , d3 ω, 0) ∼ · · · ∼ C(ps , drs ω, 0) (obviously, its length must be a multiple of s, and we can start from any place pk , pk ). Then C• (w, m, λ) is actually a vector bundle, which can be schematically described as ˜ the following gluing of vector bundles over X.

•

•

•

d1

• λ

d2

•

d3

•

.. . •

drs

•

Here horizontal lines symbolize line bundles over Xi of the superscripted degrees, their left (right) ends are basic elements of these bundles at the point ∞ (respectively 0), and the dashed lines show which of them must be glued. One must take m copies of each vector bundle from this picture and make all glueings trivial, except one going from the uppermost right point to the lowermost left one (marked by ‘λ’), where the gluing must be performed using the Jordan m × m cell with eigenvalue λ. In other words, if e1 , e2 , . . . , em and f1 , f2 , . . . , fm are bases of the corresponding spaces, one has to identify f1 with λe1 and fk with λek + ek−1 if k > 1. We denote this vector bundle rover X by V(d, m, λ), where d = (d1 , d2 , . . . , drs ); it is of rank mr and of degree m i=1 di . If r = s = 1, this picture becomes

•

d

λ •

If they are V((d1 , d2 , . . . , ds ), 1, λ) (of degree rs = m = 1, we obtain all line bundles: s ∗ d ). Thus the Picard group is Z × k . i i=1

40

IGOR BURBAN AND YURIY DROZD

In A case there are no bands concentrated at 0 place, but there are ﬁnite strings of this sort:

C(p1 , d1 ω, 0) − (p1 , 0) ∼ (p1 , 0) − C(p1 , d2 ω, 0) ∼ ∼ C(p2 , d2 , 0) − (p2 , 0) ∼ (p2 , 0) − C(p2 , d3 , 0) ∼ · · · ∼ C(ps−1 , ds−1 ω, 0) − (ps−1 , 0) ∼ (ps−1 , 0) − C(ps−1 , ds ω, 0)

So vector bundles over such conﬁgurations are in one-to-one correspondence with integral vectors (d1 , d2 , . . . , ds ); in particular, all of them are line bundles and the Picard group is Zs . In the picture above one has to set r = 1 and to omit the last gluing (marked with ‘λ’). 2. From now on s = 1, so we write p instead of p1 . Let w be the cycle

(p , 1) ∼ (p , 1) − C(p , −2, 1) ∼ C(p , 2, 2) − (p , 2) ∼ (p , 2)− − C(p , 3ω, 2) ∼ C(p , 3ω, 2) − (p , 2) ∼ (p , 2) − C(p , 3, 2) ∼ ∼ C(p , −3, 1) − (p , 1) ∼ (p , 1) − C(p , 1, 1) ∼ C(p , −1, 0)− − (p , 0) ∼ (p , 0) − C(p , −2, 0) ∼ C(p , 2, 1).

Then the band complex C• (w, m, λ) can be pictured as follows:

◦

•

2

•

◦

λ •

◦

3

•

•

3

◦

•

•

◦

1

•

◦

◦

•

2

◦

•

˜ Bullets and circles correspond to Again horizontal lines describe vector bundles over X. the points ∞ and 0; circles show those points, where the corresponding complex gives no input into R⊗O˜ A• . Horizontal arrows show morphisms in A• ; the numbers l inside give the lengths of factors. Dashed and dotted lines describe glueings. Dashed lines (between bullets) correspond to mandatory glueings arising from relations (p , n) ∼

DERIVED CATEGORIES FOR NODAL RINGS

41

(p , n) in the word w, while dotted lines (between circles) can be drawn arbitrarily; the only conditions are that each circle must be an end of a dotted line and the dotted lines between circles sitting at the same level must be parallel (in our picture they are between the 1st and 3rd levels and between the 4th and 5th levels). The degrees of line bundles in complexes C(x, z, n) with z ∈ N ∪ (−N) (they are described by the levels containing 2 lines) can be chosen as d − l and d with arbitrary d (we set d = 0), otherwise (in the second row) they are superscripted over the line. Thus the resulting complex is V((−2, 3, −3), m, 1) −→ V((0, 0, −1, −2), m, λ) −→ V((0, 0), m, 1) (we do not precise mappings, but they can be easily restored). 3. If s = 1, the sky-scraper sheaf k(p) is described by the complex ···

◦

•

···

•

◦

···

◦

•

···

•

◦

◦

•

◦

•

1

•

◦

•

◦

1

•

◦

1

◦

•

◦

•

1

◦

•

•

◦

•

◦

1

1

which is the string complex corresponding to the word . . . C(p , −1, 2) − (p , 2) ∼ (p , 2) − C(p , 1, 2) ∼ C(p , −1, 1)− − (p , 1) ∼ (p , 1) − C(p , 1, 1) ∼ C(p , −1, 0) − (p , 0) ∼ ∼ (p , 0) − C(p , −1, 0) ∼ C(p , 1, 1) − (p , 1) ∼ (p , 1)− − C(p , −1, 1) ∼ C(p , 1, 2) − (p , 2) ∼ (p , 2) − C(p , −1, 2) . . . 4. The band complex C(w, m, λ) , where w is the cycle (p , 0) ∼ (p , 0) − C(p , −3ω, 0) ∼ C(p , −3ω, 0)− − (p , 0) ∼ (p , 0) − C(p , 0ω, 0) ∼ C(p , 0ω, 0) − (p , 0) ∼ ∼ (p , 0) − C(p , −1, 0) ∼ C(p , 1, 1) − (p , 1) ∼ (p , 1)− − C(p , 2, 1) ∼ C(p , −2, 0) − (p , 0) ∼ (p , 0) − C(p , −4, 0) ∼ ∼ C(p , 4, 1) − (p , 1) ∼ (p , 1) − C(p , 5, 1) ∼ C(p , −5, 0)− − (p , 0) ∼ (p , 0) − C(p , 0ω, 0) ∼ C(p , 0ω, 0) describes the complex

42

IGOR BURBAN AND YURIY DROZD

•

-3

•

0

•

λ

•

◦

•

1

◦

•

•

◦

2

•

◦

◦

•

4

◦

•

•

◦

5

•

◦

•

0

•

or V((0, 0), m, 1) ⊕ V((0, 0), m, 1) −→ V((−3, 0, 1, 2, 4, 5, 0), m, λ). Its homologies are zero except the place 0, so it correspond to a coherent sheaf. One can see that this sheaf is a “mixed” one (neither torsion free nor sky-scraper). Note that this time we could trace dotted lines another way, joining the ﬁrst free end with the last one and the second with the third. •

-3

•

0

•

λ

•

◦

•

1

◦

•

•

◦

2

•

◦

◦

•

4

◦

•

•

◦

5

•

◦

•

0

•

It gives an isomorphic object in D(X) V((0, 0, 0, 0), m, 1) −→ V((−3, 0, 1, 5, 0), m, λ) ⊕ V((2, 4), m, 1).

DERIVED CATEGORIES FOR NODAL RINGS

43

Remark. In [6] we used another encoding of strings and bands for projective conﬁgurations, which is equivalent, but uses more speciﬁcs of the situation. In this paper we prefer to use a uniform encoding, which is the same both for nodal rings and for projective conﬁgurations.

6. Application: Cohen–Macaulay modules over surface singularities The results on vector bundles over projective conﬁgurations can be applied to study Cohen–Macaulay modules over normal surface singularities. Recall some related notions. Let A be a noetherian local complete domain of Krull dimension 2, which is normal (i.e. integrally closed in its ﬁeld of fractions), X = Spec A and o be the unique closed points of X (corresponding to the maximal ideal m of A). We call A or X a normal surface singularity. A resolution of this singularity is a morphism of schemes π : Y → X such that • Y is smooth; • π is projective and birational; ˘ = X \ { o }. • the restriction of π onto Y˘ = Y \ π −1 (o) is an isomorphism Y˘ → X We denote by E = π −1 (o)red and call it the exceptional curve of the resolution. It is indeed a projective curve. Let E1 , E2 , . . . , Es be its sirreducible components. We call eﬀective cycles non-zero divisors on Y of the form Z = i=1 ki Ei with ki ≥ 0 and consider such a cycle as a projective curve (non-reduced if some ki > 1), namely the subscheme of Y deﬁned by the sheaf of ideals OY (−Z). Obviously Zred = ki >0 Ei . In [17] C. Kahn established a one-to-one correspondence between Cohen–Macaulay modules over A and some vector bundles over a special eﬀective cycle Z, called a reduction cycle. We shall not present here his result in full generality, but only in the case, when the singularity is minimally elliptic, which means, by deﬁnition, that A is Gorenstein and dimk H1 (Y, OY ) = 1 [19]. We also suppose that the resolution π : Y → X is minimal, i.e. cannot be factored through any other non-isomorphic resolution. Then Kahn’s result can be stated as follows Theorem 6.1 ([17]). Let A be a minimally elliptic surface singularity and Z be the fundamental cycle of its minimal resolution, i.e. the smallest eﬀective cycle such that (Z.Ei ) ≤ 0 for all i. There is one-to-one correspondence between Cohen–Macaulay modules over A and vector bundles F over Z such that F G ⊕ nOZ , where 1) G is generically spanned, i.e. global sections from Γ(E, G) generate G everywhere, except maybe ﬁnitely many closed points; 2) H1 (E, G) = 0; 3) n ≥ dimk H0 (E, G(Z)). Especially, indecomposable Cohen–Macaulay A-modules correspond to vector bundles F G ⊕ nOZ , where either G = 0, n = 1 or G is indecomposable, satisﬁes the above conditions (a,b) and n = dimk H0 (E, G(Z)). (The vector bundle OZ corresponds to the regular A-module, i.e. A itself.) Kahn himself deduced from this theorem and the results of Atiyah [1] a description of Cohen–Macaulay modules over simple elliptic singularities, i.e. such that E is an elliptic curve (smooth curve of genus 1). Using the results of Section 5, one can obtain an analogous ˜ description for cusp singularities, i.e. such that E is a projective conﬁguration of type A. Brieﬂy, one gets the following theorem (for more details see [14]).

44

IGOR BURBAN AND YURIY DROZD

Theorem 6.2. There is a one-to-one correspondence between indecomposable Cohen– Macaulay modules over a cusp singularity A, except the regular module A, and vector bundles V(d, m, λ), where d = (d1 , d2 , . . . , drs ) satisﬁes the following conditions2 : • d > 0, i.e. di ≥ 0 for all i and d = (0, 0, . . . , 0); • no shift of d, i.e. a sequence (dk+1 , . . . , drs , d1 , . . . , dk ), contains a subsequence (0, 1, 1, . . . , 1, 0), in particular (0, 0); • no shift of d is of the form (0, 1, 1, . . . , 1). Moreover, from Theorem 6.1 and the results of [13] one gets the following Theorem 6.3 ([14]). If a minimally elliptic singularity A is neither simple elliptic nor cusp, it is Cohen–Macaulay wild, i.e. the classiﬁcation of Cohen–Macaulay A-modules includes the classiﬁcation of representations of all ﬁnitely generated k-algebras. As a consequence of Theorem 6.2 and the Kn¨orrer periodicity theorem [18, 20], one also obtains a description of Cohen–Macaulay modules over hypersurface singularities of type Tpqr , i.e. factor-rings k[[x1 , x2 , . . . , xn ]]/(xp1 + xq2 + xr3 + λx1 x2 x3 + Q)

(n ≥ 3, 1/p + 1/q + 1/r ≤ 1),

where Q is a non-degenerate quadratic form of x4 , . . . , xn , and over curve singularities of type Tpq , i.e. factor-rings k[[x, y]]/(xp + y q + λx2 y 2 )

(1/p + 1/q ≤ 1/2).

The latter ﬁlls up a ﬂaw in the result of [12], where one has only proved that the curve singularities of type Tpq are Cohen–Macaulay tame, but got no explicit description of modules. Recall that a normal surface singularity A is Cohen–Macaulay ﬁnite, i.e. has only a ﬁnite number of non-isomorphic indecomposable Cohen–Macaulay modules, if and only if it is a quotient singularity, i.e. A k[[x, y]]G , where G is a ﬁnite group of automorphisms [2, 15]. Just in the same way one can show that all singularities of the form A = BG , where B is either simple elliptic or cusp, are Cohen–Macaulay tame, and obtain a description of Cohen–Macaulay modules in this case. We call such singularities elliptic-quotient. There is an evidence that all other singularities are Cohen–Macaulay wild, so Table 1 completely describes Cohen–Macaulay types of isolated singularities (we mark by ‘?’ the places, where the result is still a conjecture).

2 There was a mistake in the preprint [14], where we claimed that d > 0 is enough for V(d, m, λ) to satisfy Kahn’s conditions. It has been improved in the final version. We are thankful to Igor Burban who has noticed this mistake.

DERIVED CATEGORIES FOR NODAL RINGS

45

Table 1. Cohen–Macaulay types of singularities

CM type

curves

surfaces

hypersurfaces

ﬁnite

dominate A-D-E

quotient

simple (A-D-E)

tame

dominate Tpq

elliptic-quotient (only ?)

Tpqr (only ?)

wild

all other

all other ?

all other ?

References [1] M. Atiyah. Vector bundles over an elliptic curve. Proc. London Math. Soc. 7 (1957), 414–452. [2] M. Auslander. Rational singularities and almost split sequences. Trans. Amer. Math. Soc. 293 (1986), 511–531. [3] H. Bass. Finitistic dimension and a homological generalization of semi-primary rings. Trans. Amer. Math. Soc. 95 (1960), 466–488. [4] V. M. Bondarenko. Representations of bundles of semi-chained sets and their applications. Algebra i Analiz 3, No. 5 (1991), 38–61 (English translation: St. Petersburg Math. J. 3 (1992), 973–996). [5] I. I. Burban and Y. A. Drozd. Derived categories of nodal rings. J. Algebra 272 (2004), 46–94. [6] I. I. Burban and Y. A. Drozd. Coherent sheaves on rational curves with simple double points and transversal intersections. Duke Math. J. 121 (2004), 189–229. [7] I. I. Burban, Y. A. Drozd and G.-M. Greuel. Vector bundles on singular projective curves. Applications of Algebraic Geometry to Coding Theory, Physics and Computation. Kluwer Academic Publishers, 2001, 1–15. [8] W. Crawley-Boevey. Functorial filtrations, II. Clans and the Gelfand problem. J. London Math. Soc. 1 (1989), 9–30. [9] Y. A. Drozd. Modules over hereditary orders. Mat. Zametki 29 (1981), 813–816. [10] Y. A. Drozd. Finite modules over pure Noetherian algebras. Trudy Mat. Inst. Steklov Acad. Nauk USSR 183 (1990), 56–68. (English translation: Proc. Steklov Inst. of Math. 183 (1991), 97–108.) [11] Y. A. Drozd. Finitely generated quadratic modules. Manuscripta matem. 104 (2001), 239–256. [12] Y. A. Drozd and G.-M. Greuel. Cohen–Macaulay module type. Compositio Math. 89 (1993), 315–338. [13] Y. A. Drozd and G.-M. Greuel. Tame and wild projective curves and classification of vector bundles. J. Algebra 246 (2001), 1–54. [14] Y. A. Drozd, G.-M. Greuel and I. V. Kashuba. On Cohen–Macaulay modules on surface singularities. Preprint MPI 00–76. Max–Plank–Institut f´ ur Mathematik, Bonn, 2000 (to appear in Moscow Math. J.). ´ [15] H. Esnault. Reflexive modules on quotient surface singularities. J. Reine Angew. Math. 362 (1985), 63–71. [16] R. Hartshorn. Algebraic Geometry. Springer–Verlag, New York, 1977.

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[17] C. Kahn. Reflexive modules on minimally elliptic singularities. Math. Ann. 285 (1989), 141–160. [18] H. Kn¨ orrer. Cohen–Macaulay modules on hypersurface singularities. I. Invent. Math. 88 (1987), 153–164. [19] H. Laufer. On minimally elliptic singularities. Am. J. Math. 99 (1975), 1257–1295. [20] Y. Yoshino. Cohen–Macaulay Modules over Cohen–Macaulay Rings. Cambridge University Press, 1990. Kyiv Taras Shevchenko University, University of Kaiserslautern and Institute of Mathematics of the National Academy of Sciences of Ukraine E-mail address: [email protected] E-mail address: [email protected]

CROWNS IN PROFINITE GROUPS AND APPLICATIONS ELOISA DETOMI AND ANDREA LUCCHINI

In [6] Gasch¨ utz introduced the notion of crown associated with a complemented chief factor H/K of a ﬁnite soluble group G; the crown is a certain normal factor of G, which collects all complemented chief factors of G which are G-isomorphic to H/K. He employed this notion in the construction of a characteristic conjugacy class of subgroups, the prefrattini subgroups. Later this notion has been generalized to all ﬁnite groups (see for example [10] and [8]): it has been deﬁned the crown associated with a non-Frattini chief factor of an arbitrary ﬁnite group. In [4] the notion of crown have been applied to study some properties of the probabilistic zeta function of a ﬁnite group. Let we recall how this function is deﬁned. For a ﬁnite group G and a non-negative integer t let ProbG (t) be the probability that t random elements generate G. In [7] Hall proved that

ProbG (t) =

H≤G

µ(H) |G : H|t

where µ is the M¨ obius function of the subgroup lattice of G. Hence ProbG (t) can be exhibited as a ﬁnite Dirichlet series n∈N an n−t with an ∈ Z and an = 0 unless n divides |G|. So, in view of Hall’s formula, we can speak of ProbG (s) for an arbitrary complex number s. The function ProbG (s) is the multiplicative inverse of a zeta function for G, as described by Mann [11] and Boston [1]. What is shown in [4] is that the properties of the crowns of a ﬁnite group G can be used to study the factors of ProbG (s) in the ring of ﬁnite Dirichlet series with integer coeﬃcients. In the present paper we revise the notion of crown in the contest of proﬁnite groups. We prove that it is possible to extend the deﬁnitions and the results known in the ﬁnite case, to arbitrary proﬁnite groups. Moreover, when G is a ﬁnitely generated proﬁnite group, it is possible to associate to G an inﬁnite formal Dirichlet series, generalizing the deﬁnition given in the ﬁnite case: we apply the crowns to study some properties of this series.

1. G-equivalence and crowns Recall that a proﬁnite group is a compact Hausdorﬀ topological group whose open subgroups form a base for the neighborhoods of the identity; these groups are exactly those obtained as inverse limits of ﬁnite groups. In this paper we are mainly interested in proﬁnite groups, so, unless stated otherwise, “groups” means proﬁnite groups, “subgroups” means closed subgroups and the homomorphisms are assumed to be continuous. Recall that a (closed) subgroup is open if and only if it has ﬁnite index in G. In [8] an equivalence relation among irreducible G-groups is described in the particular case when G is a ﬁnite group; we generalize this notion to the case when G is a proﬁnite group.

48

ELOISA DETOMI AND ANDREA LUCCHINI

Definition 1. Let G be a proﬁnite group and let A and B be two ﬁnite irreducible G-groups. We say that they are G-equivalent and put A ∼G B, if there are two continuous isomorphisms φ : A → B and Φ : AG → BG such that the following diagram commutes: 1 −−−−→ A −−−−→ φ

AG −−−−→ Φ

G −−−−→ 1

1 −−−−→ B −−−−→ BG −−−−→ G −−−−→ 1 It is immediate that this is an equivalence relation. Notice that if φ : A → B is a G-isomorphism then (ag)Φ = aφ g, a ∈ A, g ∈ G, deﬁnes an isomorphism Φ : AG → BG which makes the above diagram commutative. That is, two G-isomorphic G-groups are G-equivalent. Conversely, if A and B are abelian and G-equivalent then A and B are also G-isomorphic. Indeed for any g ∈ G there exists bg ∈ B with g Φ = bg g, so for any a ∈ A Φ we have (ag )φ = (ag )Φ = (aΦ )g = (aφ )bg g = (aφ )g . But for nonabelian G-groups the G-equivalence is strictly weaker than G-isomorphism; for example the two minimal normal subgroups of G = Alt(5)2 are G-equivalent without being G-isomorphic. Now assume that A and B are ﬁnite irreducible G-groups and consider C = CG (A) ∩ CG (B). Since the actions of G on A and B are assumed to be continuous, CG (A) and CG (B) are open subgroups of G, so in particular C is an open normal subgroup of G and G/C is a ﬁnite group. The following lemma reduces the study of our equivalence relation to the case when G is a ﬁnite group. Lemma 2. A and B are G-equivalent if and only if they are G/C-equivalent. Proof. The statement is obvious when A and B are abelian, since in that case G-equivalent is the same as G-isomorphic. So we may assume that A and B are nonabelian. First assume A ∼G B. For any c ∈ C there exists bc ∈ B with cΦ = bc c. If a ∈ A, we Φ have aφ = (ac )φ = (ac )Φ = (aΦ )c = (aφ )bc c = (aφ )bc , hence bc ∈ Z(Aφ ) = Z(B) = 1. This proves that cφ = c for any c ∈ C. But then it is well deﬁned an isomorphism Ψ : AG/C → BG/C by the position (agC)Ψ = (ag)Φ C which makes commutative the following diagram: 1 −−−−→ A −−−−→ AG/C −−−−→ G/C −−−−→ 1 φ Ψ 1 −−−−→ B −−−−→ BG/C −−−−→ G/C −−−−→ 1. Hence A ∼G/C B. Now assume that A ∼G/C B and let Ψ : AG/C → BG/C be an isomorphism which makes commutative the diagram: 1 −−−−→ A −−−−→ AG/C −−−−→ G/C −−−−→ 1 φ Ψ 1 −−−−→ B −−−−→ BG/C −−−−→ G/C −−−−→ 1.

CROWNS IN PROFINITE GROUPS AND APPLICATIONS

49

For any g ∈ G, there exists bg ∈ B such that (gC)Ψ = bg gC. Deﬁne Φ : AG → BG by setting aΦ = aφ if a ∈ A, g Φ = bg g if g ∈ G; it is easy to check that Φ is well deﬁned and the following diagram is commutative: 1 −−−−→ A −−−−→ φ

AG −−−−→ Φ

G −−−−→ 1

1 −−−−→ B −−−−→ BG −−−−→ G −−−−→ 1. Hence A ∼G B. We will say that a section H/K is a chief factor of G if H and K are closed normal subgroups of G with K < H and for any closed normal subgroup X of G with K ≤ X ≤ H either X = K or X = H. Notice that if H/K is a chief factor of G, then there exists an open normal subgroup of N of G with H/K ∼ =G HN/KN ; indeed H (as well as K) is the intersection of all the open normal subgroups that contain it and so, as H = K, we get HN = KN for at least one open normal subgroup N of G. This implies that a chief factor H/K is ﬁnite and that the action of G on H/K is continuous and irreducible. Our ﬁrst aim is to study the G-equivalence relation between the chief factors of G. Recall that a ﬁnite group L is said to be primitive if it has a maximal subgroup with trivial normal core. The socle soc(L) of a primitive group L can be either an abelian minimal normal subgroup (I), or a nonabelian minimal normal subgroup (II), or the product of two nonabelian minimal normal subgroups (III); we say respectively that G is primitive of type I, II, III and in the ﬁrst two cases we say that L is monolithic. As in the case of ﬁnite groups (see [5], [8]) the G-equivalence relation on chief factors of G is strictly related to the primitive epimorphic images of G. We have: Lemma 3. Let G be a proﬁnite group. Two chief factors are G-equivalent as G-groups if and only if they are G-isomorphic either between them or to the two minimal normal subgroups of a ﬁnite primitive epimorphic image of type III of G. Proof. This is true if G is ﬁnite (see [8]) and Lemma 2 allows us to reduce the proof to the ﬁnite case. A chief factor H/K is called Frattini factor if H/K ≤ Frat(G/K). Notice that if H/K is a Frattini factor, then HN/KN is Frattini for every normal closed subgroup N of G. In particular, by considering a ﬁnite image of G, we get that a Frattini chief factor is abelian. Now we are ready to give two crucial deﬁnitions. Let A be a ﬁnite irreducible G-group. We set IG (A) = {g ∈ G | g induces an inner automorphism in A}. Notice that IG (A) contains CG (A), so it is an open normal subgroup of G. Next let XG (A) be the set of open normal subgroups N of G with the properties that N ≤ IG (A), IG (A)/N ∼G A and IG (A)/N is non-Frattini. We deﬁne RG (A) =

N ∈XG (A)

N

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if the set XG (A) is nonempty, otherwise we set RG (A) = IG (A). The quotient group IG (A)/RG (A) is called the A-crown of G or the crown of G associated with A. Note that two G-equivalent G-groups, A and B, deﬁne the same crown; indeed IG (A) = IG (B) and so RG (A) = RG (B). Moreover, since RG (A) and IG (A) are closed normal subgroups of G, the quotient groups G/RG (A) and IG (A)/RG (A) are proﬁnite groups; if XG (A) = ∅, then the family of subgroups N/RG (A) where N is an intersection of ﬁnitely many subgroups in XG (A) is a fundamental system of open neighborhoods of the identity in both G/RG (A) and IG (A)/RG (A). We want to study the structure of G/RG (A). First note that RG (A) = IG (A) if and only if A is equivalent to a non-Frattini chief factor of G; so we restrict our attention to this case. Let ρ : G → Aut(A) be deﬁned by g → g ρ , where g ρ : a → ag for all a ∈ A. The monolithic primitive group associated with A is deﬁned as Gρ A ∼ = (G/CG (A))A LG (A) = Gρ ∼ = G/CG (A)

if A is abelian, otherwise.

Observe that LG (A) is a ﬁnite primitive group of type I or II, and soc(LG (A)) ∼ = A. Note that two G-equivalent G-groups may have diﬀerent centralizers in G, but their associated monolithic primitive groups are isomorphic. To simplify our notation we identify A with soc(LG (A)) and we set I = IG (A), R = RG (A), L = LG (A), and X = XG (A). Moreover let Y be the set of normal subgroups of G obtained as intersection of ﬁnitely many subgroups in X ; we remark that, if X = ∅, then G/R is the inverse limit of the family of ﬁnite groups G/N for N ∈ Y (as well as I/R is the inverse limit of the family I/N for N ∈ Y). We want to describe the structure of G/N when N ∈ Y. To do that we recall a deﬁnition: Definition 4. (see [3]) Let now L be a monolithic primitive group and let A be its unique minimal normal subgroup. For each positive integer k, let Lk be the k-fold product of L. The crown-based power of L of size k is the subgroup Lk of Lk deﬁned by Lk = {(l1 , . . . , lk ) ∈ Lk | l1 ≡ · · · ≡ lk mod A}. Clearly, soc(Lk ) = Ak , Lk / soc(Lk ) ∼ = L/A and the quotient group of Lk over any minimal normal subgroup is isomorphic to Lk−1 , for k > 1. Moreover any normal subgroup of Lk either contains or is contained in soc(Lk ). The utility of the previous deﬁnition in our study of the group G/R is explained by the next lemma (see Proposition 9 in [4]): Lemma 5. If Y ∈ Y then G/Y ∼ = Lk where k is the smallest cardinality of a subset {N1 , . . . , Nk } of X with Y = N1 ∩ · · · ∩ Nk . Moreover I/Y = soc(G/Y ) and any chief factor H/K of G with Y ≤ K < H ≤ I is non-Frattini and G-equivalent to A. Corollary 6. If N is a closed normal subgroup of G and R ≤ N then either I ≤ N or N ≤ I. Moreover if N is open and R ≤ N < I then N ∈ Y. Proof. As N is closed and {Y /R}Y ∈Y is a fundamental system of open neighborhoods of the identity in G/R, we have N = Y ∈Y N Y . Now N Y /Y is a normal subgroup of the ﬁnite group G/Y which is isomorphic to Lk for an integer k by the previous lemma. It follows that I/Y = soc(G/Y ) and also either N Y ≤ I or N Y > I. In the ﬁrst case we conclude N ≤ I. Otherwise, N Y > I for every Y ∈ Y and thus N = Y ∈Y N Y ≥ I.

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For any set Ω, the cartesian product LΩ , endowed with the product topology, is a proﬁnite group. Let now consider the subgroup LΩ = {(lω )ω∈Ω ∈ LΩ | lω1 ≡ lω2 mod A for any ω1 , ω2 ∈ Ω}. LΩ is a closed subgroup of LΩ , so it can be viewed as a proﬁnite group; indeed, LΩ is the inverse limit of the family of ﬁnite groups LI , where I is a ﬁnite subset of Ω. Let now D be the set of subsets ∆ of Hom(G, L) satisfying: (1) for any φ ∈ ∆, ker φ ∈ X ; (2) for any ﬁnite subset I = {φ1 , . . . , φk } of ∆, the position g φI = (g φ1 , . . . , g φk ), deﬁnes an homomorphism φI : G → Lk ; in particular g φ1 ≡ g φ2 mod A for any g ∈ G and any φ1 , φ2 ∈ ∆; (3) for any ﬁnite subset I of ∆, the homomorphism φI is surjective. This deﬁnition implies that if ∆ ∈ D, then the functions φI , where I is a ﬁnite subset of ∆, are compatible surjections from G to the inverse system {LI }; thus the corresponding induced mapping of proﬁnite groups Φ : G → L∆ is onto. Moreover, ker φI ∈ Y and so ker Φ = φ∈∆ ker φ is an intersection of elements of X . We may order the elements of D by inclusion. By Zorn’s lemma, D has at least one maximal element. Lemma 7. If ∆ is a maximal element of D then φ∈∆ ker φ = R. Proof. For any φ ∈ ∆ let Nφ = ker φ ∈ X . Suppose that S = φ∈∆ Nφ = R. Then there exists N ∈ X with S ≤ N. Moreover there is an epimorphism α : G → L with ¯ ker α = N. Fix φ¯ ∈ ∆; the map G/(Nφ¯ ∩ N ) → L2 deﬁned by g(Nφ¯ ∩ N ) → (g φ , g α ) is injective; by Lemma 5, G/(Nφ¯ ∩ N ) ∼ = L2 , hence there exists β ∈ Aut(L) such that ¯ β −1 g α β ≡ g φ mod soc(L) for any g ∈ G. Let γ : G → L be deﬁned by g γ = β −1 g α β. ¯ = ∆ ∪ {γ}. We claim that ∆ ¯ ∈ D. The only thing that remains to prove is that Now let ∆ for any ﬁnite subset I = {φ1 , . . . , φk } of ∆, the homomorphism φ¯I : G → Lk+1 deﬁned by g → (g φ1 , . . . , g φk , g γ ) is surjective. By Lemma 5 and the fact that φI is surjective, either φ¯I is surjective or G/(Nφ1 ∩ · · · ∩ Nφk ) ∼ = G/(Nφ1 ∩ · · · ∩ Nφk ∩ N ) ∼ = Lk . But in the latter case S ≤ Nφ1 ∩ · · · ∩ Nφk ≤ N, a contradiction. Let w0 (G) denote the local weight of the proﬁnite group G, i.e. the smallest cardinality of a fundamental system of open neighborhoods of 1 in G. Theorem 8. G/R is homeomorphic to LΩ , for a suitable choice of the set Ω. If X is inﬁnite, then |Ω| = |X |. Proof. By Lemma 7, G/R is homeomorphic to LΩ , where Ω is a maximal element of D. Since a base of neighborhoods of 1 in G/R is given by the subgroups N/R, for N ∈ Y, if X is inﬁnite, then |X | = |Y| = w0 (G/R). On the other hand, w0 (G/R) = w0 (LΩ ) is the cardinality of the set of the ﬁnite subsets of Ω, which is precisely the cardinality of Ω whenever Ω is inﬁnite. In [4] it is proved that if G is a ﬁnite group, then the cardinality of the set Ω which appears in the previous theorem coincides with the number of non-Frattini factors G-equivalent to A in any chief series of G. We want to prove that a similar result holds for arbitrary proﬁnite groups.

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First we recall that any proﬁnite group G has a chain of closed normal subgroups Gµ = 1 ≤ · · · ≤ Gλ ≤ · · · ≤ G0 = G indexed by the ordinals λ ≤ µ such that • Gλ /Gλ+1 is a chief factor of G, for each λ < µ; • if λ is a limit ordinal then Gλ = ν<λ Gν . Such a chain will be called chief series of G. Note that |µ| is an invariant of G, since, if G is inﬁnite, then |µ| = w0 (G). Lemma 9. Let H/K be a chief factor of G. If R ≤ K < H ≤ I then H/K is non-Frattini and G-equivalent to A. Proof. Since Y induces a fundamental systemof open neighborhoods of the identity in I/R, we get that K = N ∈Y KN and H = N ∈Y HN . Thus there exists N ∈ Y such that KN = HN and so H/K ∼ =G HN/KN . By Lemma 5 HN/KN is non-Frattini and G-equivalent to A, and thus the same holds for H/K. Lemma 10. Let H/K be a chief factor of G. Then H/K is non-Frattini and G-equivalent to A if and only if RH/RK = 1 and RH ≤ I. Proof. If RK = RH ≤ I, then, by Lemma 9, RH/RK is non-Frattini and G-equivalent to A; hence also H/K satisﬁes these properties. Now assume that H/K is a non-Frattini chief factor G-equivalent to A. As H/K ≤ Frat(G/K), there exists a closed maximal subgroup, say M, which contains K but not H. Let N be the normal core of M in G; since H and K are normal in G, from K ≤ M and H ≤ M we deduce K ≤ N and H ≤ N. In particular HN/N is a minimal normal subgroup of the primitive group G/N and it is G-isomorphic to H/K, hence G-equivalent to A. Note that, by the deﬁnition of I = IG (A), I/N is the socle of the primitive group G/N . Then either I/N = HN/N and N ∈ X or G/N is primitive of type III and, by Lemma 3, N ∈ Y. In particular R ≤ N = N K < N H ≤ I and so RK = RH ≤ I. Theorem 11. Let {Gλ }λ≤µ be a chief series of G and let Θ be the set of factors Gλ /Gλ+1 which are non-Frattini and G-equivalent to A. The cardinality δG (A) of Θ does not depend on the choice of the chief series. Moreover if G/R ∼ = LΩ then |Ω| = δG (A). Proof. We obtain a chain of closed normal subgroups {Hλ }λ≤µ with H0 = G and Hµ = R by deﬁning Hλ = RGλ . For any λ ≤ µ, either Hλ = Hλ+1 or Hλ /Hλ+1 is a chief factor of G/R. Moreover the set of non trivial factors Hλ /Hλ+1 coincides with the set of factors of a chief series of G/R. By Corollary 6 either Hλ ≤ I or Hλ ≥ I. Let ν be the smallest ordinal with Hν ≤ I. Now Lemma 10 implies that if λ < ν, Gλ /Gλ+1 cannot be non-Frattini and G-equivalent to A; moreover, if λ ≥ ν, then Hλ /Hλ+1 = 1 if and only if Gλ /Gλ+1 is non-Frattini and G-equivalent to A. So, {Hλ /Hλ+1 | Hλ /Hλ+1 = 1} is a chief series of G/R which passes through I/R and has the property that the elements of Θ are in bijective correspondence with the non trivial factors Hλ /Hλ+1 contained in I/R; in particular |Θ| does not depend on the choice of the series. Finally, since G/R ∼ = AΩ , we conclude that |Θ| = |Ω|. = LΩ implies I/R ∼

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Theorem 12. If G is ﬁnitely generated then δG (A) is ﬁnite for every ﬁnite irreducible G-group A. Proof. If X ∈ XG (A), then by deﬁnition I/X ∼ = A and so |G : X| = |G : I| · |A| = n for an integer n. As G is ﬁnitely generated, the number of subgroups of index n is ﬁnite; thus |XG (A)| is ﬁnite and consequently R has ﬁnite index in G. Therefore G/R ∼ = LΩ for a ﬁnite set Ω and the result follows from the previous theorem. 2. Factors of the Dirichlet series PG (s) Let G be a ﬁnitely generated proﬁnite group, that is a proﬁnite group topologically generated by a ﬁnite number of elements. Recall that for every integer n, the number of open subgroups of index n in G is ﬁnite. So we can deﬁne a formal Dirichlet series PG (s) as follows: αn with αn := µ(H) PG (s) := s n n∈N

|G:H|=n

where µ(H) denotes the M¨ obius function of the lattice of open subgroups of G, deﬁned by µ (K) = 0 unless H = G, in which case the sum is 1. H≤K G When G is ﬁnite, PG (s) is actually the series ProbG (s) deﬁned in the introduction; in particular, when t is an integer, PG (t) is the probability that t random elements generate G. Given a normal subgroup N of G we deﬁne a formal Dirichlet series PG,N (s) as follows: PG,N (s) :=

bn ns

with

n∈N

bn :=

µ(H).

|G:H|=n HN =G

Notice that PG (s) = PG,G (s). Moreover, N admits a proper supplement in G if and only if it is not contained in the Frattini subgroup of G; it follows easily that PG,N (s) = 1 if and only if N ≤ Frat(G). Let A(s) = n an /ns and B(s) = n bn /ns be two formal Dirichlet series. We denotes by A(s) ∗ B(s) the convolution product of A(s) and B(s), i.e. the Dirichlet series n cn /n with cn = d|n ad bn/d . Theorem 13. If G is a ﬁnitely generated proﬁnite group and N is a closed normal subgroup of G then PG (s) = PG/N (s) ∗ PG,N (s). Proof. The result is already known when G is a ﬁnite group (see for example [2] section 2.2), so we need an argument to reduce to the ﬁnite case. Let n ∈ N; we have to prove that the coeﬃcients of 1/ns in PG (s) and PG/N (s) ∗ PG,N (s) are equal, that is: |G:H|=n

µ(H) =

d|n

N ≤H1 ≤G |G:H1 |=d

µ(H1 )

H2 N =G |G:H2 |=n/d

µ(H2 )

(2.1)

Let Xn be the intersection of the open subgroups of G with index at most n; as G is ﬁnitely generated, Xn has ﬁnite index in G. Thus PG/Xn (s) = PG/N Xn (s) ∗ PG/Xn ,N Xn /Xn (s).

(2.2)

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ELOISA DETOMI AND ANDREA LUCCHINI

Now (2.1) follows from (2.2) since the terms in (2.1) are equal to the coeﬃcients of 1/ns in the two series in (2.2); indeed if |G : H| ≤ n then Xn ≤ H and µ(H) = µ(H/Xn ). If G is a ﬁnite group, by taking a chief series Σ : 1 = Nl < . . . N1 < N0 = G and iterating the above formula, we obtain an expression of PG (s) as a product indexed by the non-Frattini chief factors in the series: PG (s) =

PG/Ni+1 ,Ni /Ni+1 (s).

(2.3)

Ni /Ni+1 ≤Frat(G/Ni+1 )

In [4] it is proved that if G is a ﬁnite group, the factors in (2.3) are independent of the choice of the series Σ. Moreover it is also described how the factors in (2.3) look like. Let L be a ﬁnite monolithic primitive group, and let A be its socle. We deﬁne PL,1 (s) = PL,A (s), (1 + q + · · · + q i−2 )γ PL,i (s) = PL,A (s) − , |A|s

(2.4)

where γ = |CAut A (L/A)| and q = | EndL A| if A is abelian, q = 1 otherwise. In [4] it is proved that if G is a ﬁnite group, then the factors of PG (s) corresponding to the nonFrattini factors in a chief series are all of the kind PL,i (s), for suitable choices of L and i. In particular: Theorem 14. ([4], Theorem 18) Let G be a ﬁnite group. Then

PG (s) =

A

PLA ,i (s)

(2.5)

1≤i≤δG (A)

where A runs over the set of irreducible G-groups G-equivalent to a non-Frattini chief factor of G, and LA is the monolithic primitive group associated with A. Moreover, the factorization of PG (s) corresponding to the non-Frattini factors in a chief series Σ of G is precisely (2.5), independently of the choice of Σ. We want to generalize this result to the case when G is a ﬁnitely generated proﬁnite group. First of all, we get the following result: Theorem 15. Let G be a ﬁnitely generated proﬁnite group and let A = H/K be a nonFrattini chief factor of G. If R = RG (A), we have: PG/K,H/K (s) = PG/RK,RH/RK (s) = P˜L ,k (s) A

where LA is the monolithic primitive group associated with A and k = δG/RK (A).

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Proof. Let Sn = {X ≤ = G, |G : X| = n and µ(X) = 0}; notice G | K ≤ sX ≤ G, XH α /n with α = that PG/K,H/K (s) = n n n∈N X∈Sn µ(X). But µ(X) = 0 only if X is an intersection of closed maximal subgroups of G. Moreover in the proof of Lemma 10 we have seen that if M is a closed maximal subgroup of G with K ≤ M and H ≤ M then R ≤ M ; hence RK ≤ X for any X ∈ Sn . This implies immediately that PG/K,H/K (s) = PG/RK,RH/RK (s). Now, by Theorems 12 and 11, we get that k = δG/RK (A) is ﬁnite and G/RK ∼ = Lk . Moreover RH/RK is a minimal normal subgroup of G/RK and is equivalent to A. Therefore, by [4] Theorem 17, we conclude that PG/RK,RH/RK (s) = P˜L,k (s). Let now G be an inﬁnite ﬁnitely generated proﬁnite group. In that case ω0 (G) = ℵ0 and G has a chief series of length ℵ0 Σ : G = G0 ≥ · · · ≥ Gi ≥ · · · ≥ Gℵ0 = 1;

(2.6)

to each chief factor Gi /Gi+1 it is associated the ﬁnite Dirichlet series Pi (s) = PG/Gi+1 ,Gi /Gi+1 (s).

(2.7)

Now, by Theorem 13, for any i ∈ N we get PG/Gi+1 (s) = P0 (s) ∗ P1 (s) ∗ · · · ∗ Pi (s). So, one is tempted to say that PG (s) is the product of the inﬁnite factors {Pi (s)}i∈N . Unfortunately, since the formal series PG (s) is not necessarily convergent, PG (s) is not a function and a sentence like “PG (s) = limi→∞ P0 (s) . . . Pi (s)” has no meaning if we think to a convergence of complex functions. However we can prove that the formal Dirichlet series PG (s) is uniquely determined as an “inﬁnite convolution” of the factors {Pi (s)}i∈N and that the set of these factors is independent of the choice of the chief series Σ. To do that we give some deﬁnitions. αωn /ns , with Let P = {Pω (s)}ω∈Ω be a family of ﬁnite Dirichlet series, let say Pω (s) = the property that αω1 = 1, for every ω. We say that the family P is suitable for convolution if for every m > 1 the set Ωm = {ω ∈ Ω | αωn = 0 for some 1 < n ≤ m} is ﬁnite. If P has this properties, then, for any m > 1, ∗ (s) = Pm

Pω (s)

ω∈Ωm ∗ (s) = is a well-deﬁned and ﬁnite Dirichlet series, say Pm (inﬁnite) convolution product of P to be

P ∗ (s) =

n∈N

γn /ns

s n cn,m /n .

Then we deﬁne the

where γ1 = 1 and γn = cn,n if n > 1.

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Note that if P is suitable for convolution and ∆ ⊆ Ω, then the family Q = {Pω (s)}ω∈∆ is again suitable for convolution and the following holds: Lemma 16. Let Q∗ (s) = n δn /ns and let m > 1. If Ωm ⊆ ∆, then γn = δn for any n ≤ m. ∗ For example, whenever n ≤ m, the ﬁrst n terms of Pn∗ (s) and Pm (s) are equal. ˜ Now let ΩG be the set of pairs (A, i) where A runs over a set of representatives for the ˜G equivalence classes of ﬁnite irreducible G-groups and 1 ≤ i ≤ δG (A). If ω = (A, i) ∈ Ω ˜ deﬁne Pω (s) = PLA ,i (s) as in (2.4); ﬁnally let

PG = {Pω (s)}ω∈Ω˜ G .

(2.8)

Given a chief series Σ of G, let ΩΣ be the set of non-Frattini chief factors in this series and let PΣ = {PG/K,H/K (s)}H/K∈ΩΣ . Theorem 17. Let G be a ﬁnitely generated proﬁnite group and let Σ be a chief series of G. ˜ G such that if H/K ∈ ΩΣ then PG/K,H/K (s) = Pφ(H/K) (s). There is a bijection φ : ΩΣ → Ω ∗ The two families PΣ and PG are suitable for convolution and PG (s) = PΣ∗ (s) = PG (s). Proof. Let A be a ﬁnite irreducible G-group with δG (A) = 0. By Theorem 11 there are exactly δ = δG (A) non-Frattini factors H1,A /K1,A , . . . , Hδ,A /Kδ,A in the chief series Σ with Hi,A /Ki,A ∼G A, for 1 ≤ i ≤ δ. We may assume Kδ,A < Hδ,A < · · · < Ki,A < Hi,A < · · · < K1,A < H1,A . ˜ G . Moreover, by Theorem 15, The map Hi,A /Ki,A → (A, i) induces a bijection φ : ΩΣ → Ω ˜ PHi,A /Ki,A (s) = PLA ,i (s) = Pφ(Hi,A /Ki,A ) (s). This proves the ﬁrst part of the statement and ∗ (s) for every chief series Σ. that PΣ∗ (s) = PG ¯ such that To complete the proof it now suﬃces to prove that there exists a chief series Σ ∗ PΣ¯ is suitable for convolution and that PG (s) = PΣ¯ (s). For any integer n deﬁne Xn to be the intersection of the open subgroups H of G with |G : H| ≤ n. Since G is ﬁnitely generated, Xn is an open normal subgroup of G. Moreover n Xn = 1, hence we may produce a chief ¯ by reﬁning the series {Xn }n∈N . series Σ Now ﬁx an integer m. Let H/K ∈ ΩΣ¯ and PG/K,H/K (s) = n βn /ns . If βn = 0 for some 1 = n ≤ m, then there exists an open subgroup Y /K of G/K with G = HY and |G : Y | = n; this implies Xn ≤ K, otherwise H ≤ Xn and, as Xn ≤ Y , we get G = HY = Y, a contradiction. Thus Xm ≤ Xn ≤ K. As G/Xm is ﬁnite, there are only ﬁnitely many factors H/K ∈ ΩΣ¯ with Xm ≤ K. This proves that the family PΣ¯ is suitable for convolution. Moreover, if Qm is the subfamily of PΣ¯ indexed by the factors H/K ∈ ΩΣ¯ satisfying Xm ≤ K, then Lemma 16 gives that the coeﬃcients bm and cm in the two series Q∗m (s) =

are equal.

bn , ns n

PΣ∗¯ (s) =

cn ns n

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On theother hand Q∗m (s) = PG/Xm (s); thus, by deﬁnition of Xm , the coeﬃcient am of PG (s) = m am /ms is am =

µ(X) =

|G:X|=m

µ(X/Xm ) = bm

|G/Xm :X/Xm |=m

and we conclude that am = cm . Since this holds for every integer m, the theorem is proved.

3. The probabilistic zeta function Since a proﬁnite group G has a natural compact topology, it has also a Haar measure, which is determined uniquely by the algebraic structure of G. We normalize this measure so that G has measure 1, and is thus a probability space. This allows us to deﬁne, for any positive integer t, ProbG (t) as the measure of the subset {(g1 , . . . , gt ) ∈ Gt | g1 , . . . , gt topologically generate G}. If G is ﬁnite, the function φG (t) = Prob(G)|G|t is the Eulerian function of G, which gives the number of ordered t-uples (g1 , . . . , gt ) generating G. This function was introduced and studied by P. Hall [7], who proved in particular φG (t) =

µ(H)|H|t .

(3.1)

H≤G

This implies that if G is ﬁnite, then the Dirichlet series PG (s) deﬁned in the previous section is a complex function with the property that, for any integer t, PG (t) = ProbG (t) = φG (t)/|G|t ; the complex function ζG (s) = 1/PG (s) is called the probabilistic zeta function associated to the ﬁnite group G. In [11] Mann proposed the problem of looking for a complex function interpolating the values {ProbG (t) | t ∈ N}. Of course in order to discuss this question one must focus his attention on ﬁnitely generated proﬁnite groups with the property that ProbG (t) > 0 for some t ∈ N (otherwise the interpolating function we are looking for is just the zero function); the groups with this property are called positively ﬁnitely generated (PFG). It is worth mentioning that a ﬁnitely generated proﬁnite group is not necessarily PFG. For example Kantor and Lubotzky [9] proved that the free proﬁnite group of rank d is not PFG if d ≥ 2. The conjecture proposed by Mann in [11] is the following: to each PFG group G there corresponds naturally a “zeta function” ζG (s) which is an analytic function deﬁned in some right half plane of the complex numbers, such that ζG (t) = ProbG (t)−1 , for all ˆ denotes the proﬁnite completion of a cyclic suﬃciently large integers t. For example, if Z inﬁnite group, then

ProbZˆ (t) =

µ(n) n

nt

=

1 −1 = ζ(t)−1 t n n

where ζ is the Riemann zeta function. Hence in this case ζ(s) is the function we are looking for.

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ELOISA DETOMI AND ANDREA LUCCHINI

Before discussing Mann’s conjecture, we want to prove that it is possible to decide whether a ﬁnitely generated proﬁnite group G is PFG from the knowledge of the family PG of ﬁnite Dirichlet series deﬁned in (2.8). First recall: Proposition 18 (Mann [11] Theorem 1). If G is a ﬁnitely generated proﬁnite group and t is an integer, then ProbG (t) = inf N ProbG/N (t), where N varies over all open normal subgroups of G. In particular, if Σ : G = G0 < · · · < Gℵ0 = 1 is a chief series of G then ProbG (t) = inf ProbG/Gn (t) = lim PG/Gn (t), n∈N

n→∞

(3.2)

since {Gn }n∈N is a base of neighborhoods of 1 in G, ProbG/Gn (t) ≥ ProbG/Gn+1 (t) and PG/Gn (t) = ProbG/Gn (t). This suggests that given an integer t there may be a relation between ProbG (t) and the inﬁnite product of the numbers Pω (t), where Pω (s) ∈ PG as deﬁned in (2.8). Theorem 19. A ﬁnitely generated proﬁnite group G is PFG if and only if the inﬁnite prod uct ω∈Ω˜ G Pω (t) is absolutely convergent for some positive integer t; in that case ProbG (t) = ˜ G Pω (t) for suﬃciently large positive integers. ω∈Ω Proof. If G is ﬁnite, the result follows from Theorem 14. So, let G be inﬁnite and let Σ : G = G0 > . . . > Gℵ0 = 1 be a chief series of G. Let us denote the Dirichlet series PG/Gi+1 ,Gi /Gi+1 (s) by Pi (s); notice that Pi (s) = 1 whenever Gi /Gi+1 is a Frattini factor. Now G is PFG if and only if for some integer t we have 0 = ProbG (t) = lim PG/Gn (t) = lim P0 (t) . . . Pn−1 (t). n→∞

n→∞

(3.3)

Since 0 < Pi (t) < 1 for any i ∈ N, the condition lim n→∞ P0 (t) . . . Pn−1 (t) = 0 is equivalent to the absolute convergence of the inﬁnite product n∈N Pn (t). As the value of an absolutely convergent product does not change if the factors are reordered, and the Frattini 17 we deduce factors do not inﬂuence the product, from Theorem that the inﬁnite product n∈N Pn (t) is absolutely convergent if and only if ω∈Ω˜ G Pω (t) is absolutely convergent. The previous theorem says that if G is PFG then the inﬁnite product ω∈Ω˜ GPω (t) is absolutely convergent for any suﬃciently large integer t. Unfortunately from this result no information can be obtained about the behaviour of the product ω∈Ω˜ G Pω (s) when s is a complex number. Mann [11] proved that if G is prosoluble then ω∈Ω˜ G Pω (s) is absolutely convergent in some right half plane of the complex plane. We conjecture that this holds for an arbitrary PFG group G. This would give the possibility of deﬁning the probabilistic zeta function of G as the multiplicative inverse of the inﬁnite product ω∈Ω˜ G Pω (s). 4. Recognizing PFG groups Mann and Shalev proved that PFG groups can be characterized by the behaviour of the function mn (G) which is deﬁned as the number of closed maximal subgroups of G with index n.

CROWNS IN PROFINITE GROUPS AND APPLICATIONS

59

Theorem 20 (Mann and Shalev [12] Theorem 4). A ﬁnitely generated proﬁnite group G is PFG if and only if G has polynomial maximal subgroup growth, i.e. there exists a constant c such that for all n, the number mn (G) is at most nc . This criterion can be translated in term of “multiplicity” of the chief factors of G. We need ˜ ﬁrst some deﬁnition. For a ﬁnite primitive group L let λ(L) be the minimum of the index |L : X| where X runs over the set of core-free maximal subgroups of L. Let M be the set of closed maximal subgroups of G; as M ∈ M is open, its normal core MG has ﬁnite index in G and G/MG is a primitive group. We deﬁne K = {N G | N = MG for some M ∈ M}, ˜ κn (G) = |{N ∈ K | λ(G/N ) = n}|. It was announced by Pyber that, using the classiﬁcation of the ﬁnite simple groups, the following result can be proved: Theorem 21 (Pyber). There exists a constant b such that for every ﬁnite group G and every n ≥ 2, G has at most nb core-free maximal subgroups of index n. In fact, b = 2 will do. Using this result we can deduce easily: Lemma 22. κn (G) ≤ mn (G) ≤ n2 m≤n κm (G). Proof. The ﬁrst inequality is trivial. We prove the second one. Let M ∈ Mn ; since ˜ ) ≤ n, the normal subgroup M can be chosen in at most λ(G/M G G m≤n κm (G) diﬀerent 2 ways. Given N = MG , by Theorem 21, there are at most n core-free maximal subgroups of index n containing N. Combining Theorem 20 and Lemma 22 we obtain: Corollary 23. Let G be a ﬁnitely generated proﬁnite group. The following are equivalent: (1) G is PFG; (2) there exists a constant c1 such that mn (G) ≤ nc1 for all n ∈ N; (3) there exists a constant c2 such that κn (G) ≤ nc2 for all n ∈ N. Now we study κn (G) using the G-equivalent relation among ﬁnite irreducible G-groups described in the ﬁrst section. In the following G will be a ﬁnitely generated proﬁnite group. Let N be an element of K; the quotient group G/N is a ﬁnite primitive group and its minimal normal subgroups are all equivalent to the same irreducible G-group, say A; indeed either G/N is monolithic or G/N is primitive of type III and, by Lemma 3, its two minimal normal subgroups are G-equivalent. By deﬁnition IG (A)/N is the socle of G/N and so either N ∈ XG (A) or N is the intersection of two diﬀerent elements of XG (A); anyway, RG (A) ≤ N and G/N ∼ = Li where L = LG (A) is the monolithic primitive group associated with A and i = 1, 2. When G/N ∼ = L2 , A is nonabelian and any faithful primitive representation of G/N has degree |A|. Given an irreducible G-group A, we deﬁne KA as the subset of K containing those normal subgroups N with the property that the minimal normal subgroups of G/N are equivalent to A. It is clear that Lemma 24. The set K is the disjoint union of the subsets KA , where A runs over the set of ﬁnite irreducible G-groups, up to equivalence.

60

ELOISA DETOMI AND ANDREA LUCCHINI

˜ Now deﬁne λ(A) = λ(L), where L = LG (A). Notice that when A is abelian λ(A) = |A| ˜ and λ(G/N ) = |A| for every N ∈ XG (A), since in this case G/N ∼ = L (see Lemma 5); 1 2 ∪ KA therefore, for n = |A|, Kn ⊇ KA = XG (A). When A is nonabelian we get KA = KA 1 2 1 ∼ ∼ with KA = {N ∈ KA | G/N = L} and KA = {N ∈ KA | G/N = L2 }. Notice that KA = 1 2 ˜ ) = λ(A) for every N ∈ KA ; also, KA is the set of the intersections XG (A) and so λ(G/N 2 ˜ . Set γA = of two diﬀerent normal subgroups of XG (A) and λ(G/N ) = |A| for N ∈ KA |CAut A (L/A)| and qA = | EndL A| if A is abelian, qA = 1 otherwise. Since, by Theorems 11 and 12, IG (A)/RG (A) ∼ = AδG (A) and δG (A) is ﬁnite, it can be easily proved that δ (A)−1

Lemma 25. Let n = λ(A). If A is abelian, then |Kn ∩ KA | = 1 + qA + · · · + qAG 1 2 is non abelian, then |Kn ∩ KA | = δG (A) and |Kn ∩ KA | = δG (A)(δG (A) − 1)/2.

; if A

Now deﬁne: κab n =

δ (A)−1

1 + qA + · · · + qAG

;

A =1,|A|=n

κ1n =

δG (A);

A =A,λ(A)=n

κ2n =

A =A,|A|=n

δG (A) . 2

1 2 ab 1 2 By Lemma 25 κn (G) = κab n + κn + κn . So G is PFG if and only if κn , κn , κn are polynomially bounded. Now let αn (G) be the number of ﬁnite abelian irreducible G-groups A, with |A| = n and δG (A) > 0.

Lemma 26. κab n is polynomially bounded if and only if αn (G) is polynomially bounded. Proof. Obviously αn (G) ≤ κab n . We have to prove that if αn (G) is polynomially bounded then the same is true for κab n . Assume that G can be generated by r elements and let A be a ﬁnite abelian irreducible G-group with |A| = n and δG (A) = 0. By [4] Theorem 18, P˜L,δG (A) (r) > 0, where L = LG (A). On the other hand δ (A)−2

(1 + qA + · · · + qAG P˜L,δG (A) (r) = PL,A (r) − |A|r

)γA

.

In particular δ (A)−2

(1 + qA + · · · + qAG |A|r

)γA

< PL,A (r) ≤ 1,

hence δ (A)−2

1 + qA + · · · + qAG

≤

|A|r ≤ |A|r = nr γA

and δ (A)−1

1 + qA + · · · + qAG

δ (A)−2

≤ (1 + qA )(1 + qA + · · · + qAG

r+1 since qA ≤ n. It follows that κab . n ≤ 2αn (G)n

) ≤ (1 + qA )nr = 2nr+1

CROWNS IN PROFINITE GROUPS AND APPLICATIONS

61

Now consider the number ξn (G) of ﬁnite nonabelian irreducible G-groups A, with |A| = n and δG (A) ≥ 2. Lemma 27. κ2n (G) is polynomially bounded if and only if ξn (G) is polynomially bounded. Proof. Obviously ξn (G) ≤ κ2n (G). We have to prove that if ξn (G) is polynomially bounded then the same is true for κ2n (G). Assume that G can be generated by r elements and let A be a ﬁnite nonabelian irreducible G-group with |A| = n and δG (A) > 1. By [4] Theorem 18, P˜L,δG (A) (r) > 0, where L = LG (A). On the other hand (δG (A) − 1)γA P˜L,δG (A) (r) = PL,A (r) − . |A|r In particular (δG (A) − 1)γA < PL,A (r) ≤ 1, |A|r hence δG (A) − 1 ≤

|A|r ≤ |A|r = nr . γA

It follows that κ2n (G) ≤ ξn (G)(n + 1)2r . Lemma 28. If κ1n (G) is polynomially bounded then ξn (G) and κ2n (G) are polynomially bounded. Proof. If A is nonabelian and |A| = n, then λ(A) ≤ n; so ξn (G) ≤

m≤n

κ1m (G) ≤ nκ1n (G).

So we conclude: Theorem 29. G is PFG if and only if αn (G) and κ1n (G) are polynomially bounded. We conclude with two question: Question 1. Does there exist a ﬁnitely generated proﬁnite group G such that αn (G) is not polynomially bounded? Question 2. Let βn (G) be the number of nonabelian irreducible G-groups A, with λ(A) = n and δG (A) = 0. Does there exist a ﬁnitely generated proﬁnite group G such that βn (G) is not polynomially bounded? We conjecture that both these questions have a negative answer. This would imply: Conjecture.A ﬁnitely generated proﬁnite group G is PFG if and only if ρG (n) = max{δG (A) | A nonabelian, λ(A) = n} is polynomially bounded.

(4.1)

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ELOISA DETOMI AND ANDREA LUCCHINI

References [1] N. Boston, ‘A probabilistic generalization of the Riemann zeta functions’, Analytic Number Theory 1 (1996), 155–162. [2] K. S. Brown, ‘The coset poset and probabilistic zeta function of a finite group’, J. Algebra 225 (2000), 989–1012. [3] F. Dalla Volta and A. Lucchini, ‘Finite groups that need more generators than any proper quotient’, J. Austral. Math. Soc., Ser. A 64 (1998), 82–91. [4] E. Detomi and A. Lucchini, ‘Crowns and factorization of the probabilistic zeta function of a finite group’, J. Algebra to appear. [5] P. F¨ orster, ‘Chief factors, crowns, and the generalized Jordan-Holder Theorem’, Comm. Algebra 16 (1988), 1627–1638. [6] W. Gasch¨ utz, ‘Praefrattinigruppen’, Arch. Math. 13 (1962), 418–426. [7] P. Hall, ‘The Eulerian functions of a group’, Quart. J. Math. 7 (1936), 134–151. [8] P. Jim´enez-Seral and J. Lafuente, ‘On complemented nonabelian chief factors of a finite group’, Israel J. Math. 106 (1998), 177–188. [9] W. M. Kantor and A. Lubotzky, ‘The probability of generating a finite classical group’, Geom. Ded. 36 (1990), 67–87. [10] J. Lafuente, ‘Crowns and centralizers of chief factors of finite groups’, Comm. Algebra 13 (1985), 657–668. [11] A. Mann, ‘Positively finitely generated groups’, Forum Math. 8 No. 4 (1996), 429–459. [12] A. Mann and A. Shalev, ‘Simple groups, maximal subgroups and probabilistic aspects of profinite groups’, Israel J. Math. 96 (1996), 449–46 8.

A. Lucchini Dipartimento di Matematica Universit`a di Brescia Via Valotti, 9 25133 Brescia, Italy [email protected] E. Detomi Dipartimento di Matematica Universit`a di Brescia Via Valotti 9 25133 Brescia, Italy [email protected] Current address: Dipartimento di Matematica Universit`a di Padova via Belzoni, 7 35131 Padova, Italy [email protected]

THE GALOIS STRUCTURE OF AMBIGUOUS IDEALS IN CYCLIC EXTENSIONS OF DEGREE 8 G. GRIFFITH ELDER

Abstract. In cyclic, degree 8 extensions of algebraic number ﬁelds N/K, ambiguous ideals in N are canonical Z[C8 ]-modules. Their Z[C8 ]-structure is determined here. It is described in terms of indecomposable modules and determined by ramiﬁcation invariants. Although inﬁnitely many indecomposable Z[C8 ]-modules are available (classiﬁcation by Yakovlev), only 23 appear.

1. Introduction We are concerned with the interrelationship between two basic objects in algebraic number theory: the ring of integers and the Galois group. In particular, we seek to understand the eﬀect of the Galois group upon the ring of integers. At the same time, we are also interested in the Galois action upon other fractional ideals. So that the action may be similar, we restrict ourselves to ambiguous ideals – those that are mapped to themselves by the Galois group. The setting for our investigation is the family of C8 -extensions. This choice is guided by by a result of E. Noether as well as results in Integral Representation Theory. Noether’s Normal Integral Basis Theorem. A ﬁnite Galois extension of number ﬁelds N/K is said to be at most tamely ramiﬁed (TAME) if the factorization of each prime ideal PK (of OK ) in ON results in exponents (degrees of ramiﬁcation) that are relatively prime to the ideal PK . A normal integral basis (NIB) is said to exist if there is an element α ∈ ON (in the ring of integers of N ) whose conjugates, {σα : σ ∈ Gal(N/K)}, provide a basis for ON over OK (the integers in K). Noether proved NIB ⇒ TAME; moreover, for local number ﬁelds NIB ⇔ TAME, tying the Galois module structure of the ring of integers to the arithmetic of the extension [?]. This is a nice eﬀect – NIB means that the integers are isomorphic to the group ring, OK [Gal(N/K)]. It is similar to the eﬀect of the Galois group on the ﬁeld itself (i.e. Normal Basis Theorem). The impact of her result is two-fold: (1) We are encouraged to localize. (2) We are directed away from tamely ramiﬁed extensions – toward wildly ramiﬁed extensions and p-groups (See [?]). Integral Representation Theory (Restricted to p-groups G). Classiﬁcation of Modules. The number of indecomposable modules over a group ring Z[G] is, in general, inﬁnite. Only Z[Cp ] and Z[Cp2 ] are of ﬁnite type. Still, among those of inﬁnite type, there are two whose classiﬁcations are somehow manageable. These are the ones of so– called tame type [?]: Z[C2 ×C2 ] (classiﬁcation by L. A. Nazarova [?]) and Z[C8 ] (classiﬁcation by A. V. Yakovlev [?]). Unique Decomposition. The Krull–Schmidt–Azumaya Theorem does not, in general, hold: although a module over a group ring will decompose into indecomposable modules, this

Date: October 6, 2002. 2000 Mathematics Subject Classiﬁcation. Primary 11S23; Secondary 20C10. Key words and phrases. Galois Module Structure, Wild Ramiﬁcation, Integral Representation.

64

G. GRIFFITH ELDER

decomposition may not be unique. Fortunately, it does hold for a few group rings, including Z[C2 × C2 ] and Z[C8 ] [?]. Topic. Let G = Gal(N/K). We are led to ask the following natural question: What is the Z[G]-module structure of ambiguous ideals when • the number theory is ‘bad’ (wild ramiﬁcation), while • the representation theory is ‘good’ (tame type, K–S–A)? In other words: What is the Z[G]-module structure of ambiguous ideals in wildly ramiﬁed C2 × C2 and C8 number ﬁeld extensions? Previous work solved this for C2 × C2 extensions [?], [?]. So our focus here is on C8 -extensions. (Note: This question has already been addressed for those group rings with ‘very good’ representation theory, those of ﬁnite type. See [?] and [?].) As with C2 × C2 -extensions, the Z[G]-module structure of ambiguous ideals in C8 extensions is completely determined by the structure at its 2-adic completion – our global question reduces to a collection of local ones. We leave it to the reader to ﬁll in the details. (One may follow [?, §2] using [?].) 1.1. Local Question, Answer. Let K0 be a ﬁnite extension of the 2-adic numbers Q2 and let Kn be a wildly ramiﬁed, cyclic, degree 2n extension of K0 with G = Gal(Kn /K0 ). The maximal ideal Pn in Kn is unique (therefore ambiguous). So every fractional ideal Pin is ambiguous. We ask: What is the Z2 [G]-module structure of Pin for n = 1, 2, 3? (Z2 denotes the 2-adic integers.) The answered is given by the following theorem and the description of the modules Ms (i, b1 , . . . , bs ). Let T denote the maximal unramiﬁed extension of Q2 in K0 . Following [?, Ch IV], let G = G−1 ⊇ G0 ⊇ G1 ⊇ · · · denote the ramiﬁcation ﬁltration. Use subscripts to denote ﬁeld of reference, so Ok denotes the ring of integers of k. Theorem 1.1. Let Kn /K0 be a cyclic extension of degree 2n and let k ⊆ K0 be an unramiﬁed extension of Q2 . Suppose that |G1 | ≤ 8 (i.e. s = 1, 2 or 3) and let b1 , . . . , bs be the break numbers in the ramiﬁcation ﬁltration of G1 . If Ms (i, b1 , . . . , bs ) is the Z2 [G1 ]-module deﬁned below, then Pin ∼ = Ok [G] ⊗Z2 [G1 ] Ms (i, b1 , . . . , bs )[T :k] as left Ok [G]-modules. 1.1.1. Ms (i, b1 , · · · , bs ). Indecomposable modules are listed in Appendix A and e0 denotes the absolute ramiﬁcation index of K0 . Following [?] and [?], M1 (i, b1 ) = (R0 ⊕ R1 )(i+b1 )/2−i/2 ⊕ Z2 [G1 ]e0 −((i+b1 )/2−i/2) if b2 + 2b1 > 4e0 , HbA ⊕ G cA ⊕ LdA a M2 (i, b1 , b2 ) = I ⊕ bB cB dB H ⊕M ⊕L if b2 + 2b1 < 4e0 .

(1.1) (1.2)

where a = (i+b2 )/4− (i+2b1 )/4, bA = e0 + i/4− (i+b2 )/4, bB = (i+b2 +2b1 )/4−

(i + b2 )/4, cA = (i + b2 + 2b1 )/4 − e0 − i/4, cB = e0 + i/4 − (i + b2 + 2b1 )/4, dA = e0 + (i + 2b1 )/4 − (i + b2 + 2b1 )/4, dB = (i + 2b1 )/4 − i/4. The description of M3 (i, b1 , b2 , b3 ) is given by Tables 1 and 2. Note the eight columns in each table. There are eight cases. Each module that appears in M3 (i, b1 , b2 , b3 ), except for R3 , is listed in the appropriate column of Table 1. The multiplicity of the module is appears in the corresponding spot in Table 2. The multiplicity of R3 follows the tables.

H2 H1,2 M1 L L3 I I2

d + e0 − d¯ a ¯ − d − 2e0 a + e0 − a ¯ c¯ − a c + e0 − c¯ ¯b − c − e0 b − ¯b + e0

Table 2. A ¯ d−b

H2 M M1 L L3 I I2

Table 1. A B H H D H H1 L H1 G H1 H1,2 G4 L L3 L2 I2

a ¯ − d¯ − e0 d + 2e0 − a ¯ a − d − e0 c¯ − a c + e0 − c¯ ¯b − c − e0 b − ¯b + e0

B ¯ d−b

H1 H1,2 G4 L L3 I I2

C H H1 L

F I1 H1 L H1 G H1 G G4 G3 L3 L2 I2

d¯ − w a − d¯ d + e0 − a c¯ − d − e0 z¯ + b1 − c¯ ¯b − c − e0 b − ¯b + e0

C a ¯ − b − e0 ¯ w + e0 − a

H1 G G4 G3 L3 I I2

E I1 H1 L H1 G G4 G3 G2 G1 L1

H I1

D a ¯ − b − e0 y¯ + m − a ¯ d − y¯ + e0 d¯ − d − m a − d¯ w ¯−m−a c¯ − d − e0 z¯ + b1 − c¯ c + e0 − ¯b b−c

H1 G G4 G3 G2 L2 L1

G I1

a−w d¯ − a c¯ − d¯ d + e0 − c¯ y−d ¯b − c − e0 a ¯ − ¯b

E b + e0 − a ¯ w−b

F b + e0 − a ¯ y¯ + m − e0 − b c¯ − y¯ a + e0 − c¯ − m d¯ − a y¯ − d¯ d + e0 − c¯ y + e0 − d c + e0 − ¯b a ¯ − c − e0

a−b d¯ − a c¯ − d¯ ¯b − c¯ d + e0 − ¯b d + e0 − ¯b c + e0 − a ¯

G b−c

a−b d¯ − a c¯ − d¯ ¯b − c¯ a ¯ − ¯b d + e0 − a ¯ c−d

H b−c

GALOIS STRUCTURE 65

66

Cases. A. B. C. D. E. F. G. H.

G. GRIFFITH ELDER

4e0 − 4b1 /3 < b2 (including Stable Ramiﬁcation, b1 ≥ e0 ). 4e0 − 2b1 < b2 < 4e0 − 4b1 /3 4e0 − 4b1 < b2 < 4e0 − 2b1 and b2 > (4e0 + 4b1 )/3 4e0 − 4b1 < b2 < 4e0 − 2b1 and b2 < (4e0 + 4b1 )/3 b2 < 4e0 − 4b1 and b3 > 8e0 + 4b1 − 2b2 b2 < 4e0 − 4b1 and 8e0 + 4b1 − 2b2 < b3 < 8e0 + 4b1 − 2b2 8e0 − 4b1 − 2b2 < b3 < 8e0 − 2b2 b3 < 8e0 − 4b1 − 2b2

A graphic representation of these cases appears in §3.2. Constants used in Table 2. a := (i − 2b2 )/8, a ¯ := (i + b3 − 2b2 )/8, b := (i − 2b2 − 4b1 )/8, ¯b := (i+b3 −2b2 −4b1 )/8, c := (i−4b2 )/8, c¯ := (i+b3 −4b2 )/8, d := (i−4b2 −4b1 )/8, ¯ := (i + b3 − 2b2 − 2b1 )/8, d¯ := (i + b3 − 4b2 − 4b1 )/8, w := (i − 2b2 − 2b1 )/8, w y := (i − 4b2 − 2b1 )/8, y¯ := (i + b3 − 4b2 − 2b1 )/8, z¯ := (i + b3 − 4b2 − 6b1 )/8, m := (b2 − b1 )/2. ¯ − (a + b + c + d) − 3e0 )f . The multiplicity of R3 . In Cases A and B it is ((¯ a + ¯b + c¯ + d) ¯ − (a + b + d) − 2e0 − (¯ In Case C, it is ((¯ a + ¯b + c¯ + d) z + b1 ))f . While in Case D it is ¯ − (a + b) − e0 + m − (w ¯ + z¯ + b1 ))f . In Case E, it is (¯b + d¯ − a − y − e0 )f . ((¯ a + ¯b + c¯ + d) In Case F it is ((¯b + d¯ + c¯) − (a + y + y¯) − e0 )f . Finally, in Cases G and H, the number of R3 that appear is (d¯ − a)f . 1.2. Discussion. Cyclic p-Extensions. The Galois module structure of the ring of integers in fully and wildly ramiﬁed, cyclic, local extensions of degree pn was studied in [?] and more recently in [?]. Both of these papers required a lower bound on the ﬁrst ramiﬁcation number b1 . In particular, [?] restricted b1 to about half of its possible values, under so-called strong ramiﬁcation. In this paper, by focusing on p = 2 we are able to remove this restriction. Our work sheds light (1) on strong ramiﬁcation and (2) on the structures that are possible outside of it. (1) Strong ramiﬁcation for p = 2 means b1 > e0 , a small part of Case A. The structure under strong ramiﬁcation given by [?, Thm 5.3], when restricted to p = 2, remains valid throughout Case A. What then should Case A be, for odd p? (2) Suppose that ‘nice’ refers to the structure under strong ramiﬁcation, indeed under Case A. Does the structure remain relatively ‘nice’ beyond Case A? This depends upon a precise deﬁnition. Let an indecomposable module be nice if it is made up of distinct irreducible modules. Note only nice modules appear in Case A. But then, as we leave Case A, the structure turns nasty immediately. At least one of H1,2 , H1 L and H1 G appears in every Case B through F . Induced Structure. The subﬁeld of Kn ﬁxed by the ﬁrst ramiﬁcation group G1 is tame over the base ﬁeld K0 . Miyata generalized Noether’s Theorem proving that each ideal is relatively projective over G1 [?]. In other words, the ideals are direct summands of modules that have been induced from G1 to G [?, §10]. We ﬁnd, in our situation, that ideals are relatively free over G1 . See [?, Thm 2] for a more general, related result. Extension of Ground Ring. When studying the structure of ideals in an extension Kn /K0 over a group ring, one must choose a ring of coeﬃcients. Does one study ‘ﬁne’ structure – over O0 [G] where the coeﬃcients are integers in K0 . Does one study ‘coarse’ structure – over Z2 [G]. We study a canonical intermediate structure – over OT [G] where the coeﬃcients belong to the Witt ring of the residue class ﬁeld. We determine this structure

GALOIS STRUCTURE

67

by listing generators and relations. Interestingly, the coeﬃcients in these relations always belong Z2 [G]. Therefore OT [G]-structure results, by extension of the ground ring, from Z2 [G]-structure [?, §30B]. Realizable Modules. Let SG denote the set of realizable indecomposable Z2 [G]-modules: Those indecomposable Z2 [G]-modules that appear in the decomposition of some ambiguous ideal in an extension N/K with Gal(N/K) ∼ = G. Chinburg asked whether SG could be inﬁnite. In [?], since SC2 ×C2 is inﬁnite, the answer was found to be yes. We determine here that although the set of indecomposable Z2 [C8 ]-modules is inﬁnite, SC8 is ﬁnite. The sequence |SC2n |, n = 0, 1, 2, . . . begins 1, 3, 7, 23 . . . 1.3. Organization of Paper. Preliminary results are presented in §2, main results in §3. There are two appendixes. Appendix A lists all necessary indecomposable modules. Appendix B lists bases for our ideals. Preliminary Results: In §2.1 we handle the special case when a ramiﬁcation break number is even. In §2.2, we present a strategy for handling odd ramiﬁcation numbers. To motivate our work in §3, we implement this strategy for |G1 | = 2 and 4, in §2.2.1 and §2.2.3 respectively. We conclude, in §2.3, with a reduction to totally ramiﬁed extensions. Main Results: We begin in §3.1 with a brief outline and discussion. Then, we catalog ramiﬁcation numbers and prove some technical lemmas in §3.2. All this sets the stage for our work in §3.3 determining the Galois structure of ideals in fully, though unstably, ramiﬁed C8 -extensions. This is our primary focus. Our work in §3.4 on stably ramiﬁed extensions is essentially contained in [?]. 2. Preliminary Results We continue to use the notation of §1.1. Let K0 be a ﬁnite extension of Q2 and Kn /K0 be a cyclic extension of degree 2n . Let σ generate G = Gal(Kn /K0 ) and use subscripts to i distinguish among subﬁelds. So Ki denotes the ﬁxed ﬁeld of σ 2 , Oi denotes the ring of integers of Ki and Pi denotes the maximal ideal of Oi . Let vi be the additive valuation in Ki , πi its prime element, so that vi (πi ) = 1. Let Tri,j denote the trace from Ki down to Kj . Recall the ramiﬁcation ﬁltration of G. Note G−1 = G0 if and only if Kn /K0 is fully ramiﬁed. Also since G is a 2-group and [G1 : G0 ] is odd, G0 = G1 . Furthermore since G is cyclic and Gi /Gi−1 is elementary abelian for i > 1, there are s = log2 |G1 | breaks in the ﬁltration of G1 [?, p 67]. Let b1 < b2 < · · · < bs denote these break numbers. (The break numbers of G may include −1 as well.) It is a standard exercise to show that b1 , . . . , bs are all either odd or even [?, Ex 3, p 71]. When they are even, we are in an extreme case, called maximal ramiﬁcation. The general case, when they are odd, will be our primary concern. 2.1. Even Ramiﬁcation Numbers. If b1 , . . . , bs are even, we use idempotent elements of the group algebra, Q2 [G], and Ullom’s generalization of Noether’s result [?, Thm 1], to determine the structure of each ideal. In doing so, we rely upon two observations: (1) Idempotent elements in Q2 [G] that map an ideal into itself, decompose the ideal. (2) Modules over a principal ideal domain are free. We illustrate this process in one case, leaving other cases to the reader. Suppose |G| = 8 and |G1 | = 4. So K3 /K0 is only partially ramiﬁed and s = 2. From [?, IV §2 Ex 3], b1 = 2e0 and b2 = 4e0 . Using [?, V §3], one ﬁnds that (1/2)(σ 4 + 1)Pi3 ⊆ Pi3 . As a result, the i/2 idempotent (σ 4 + 1)/2 decomposes the ideal Pi3 ∼ ⊕ M2 with (σ 4 + 1)M2 = 0. = P2

68

G. GRIFFITH ELDER

i/2 i/2 i/2 Meanwhile (1/2)(σ 2 + 1)P2 ⊆ P2 . So P2 decomposes as well. This yields Pi3 ∼ = i/4 ⊕ M1 ⊕ M2 with (σ 4 + 1)M2 = 0 and (σ 2 + 1)M1 = 0. Each Mi may be viewed as a P1 i i module over OTK [σ]/(σ 2 +1), a principal ideal domain. So Mi is free over OTK [σ]/(σ 2 +1). i/4 Ullom’s result provides a normal integral basis for P1 . Counting OT -ranks, we ﬁnd that

OT [σ] Pi3 ∼ = 2 (σ − 1)

e0

⊕

OT [σ] (σ 2 + 1)

e0

e

⊕

OT [σ] 0 . (σ 4 + 1)

2.2. Odd Ramiﬁcation Numbers. Henceforth the ramiﬁcation numbers will be odd. In this context we will use the following technical result (with Ki /Ki−1 ). Lemma 2.1. Let k be a ﬁnite extension of Q2 and K/k be a ramiﬁed quadratic extension. Let ek be the absolute ramiﬁcation index of k. Assume that σ generates the Galois group and that the ramiﬁcation number, b < 2ek , is odd. Then (1) vK ((σ ± 1)α) = vK (α) + b for vK (α) odd; (2) if τ ∈ k, there is a ρ ∈ K such that (σ + 1)ρ = τ and vK (ρ) = vK (τ ) − b; (3) if vK (α) is even and (σ + 1)α = 0, there is a θ ∈ K such that α = (σ − 1)θ and vK (θ) = vK (α) − b. Proof. These may be shown using [?, V §3], as in [?, Lem 3.12–14]. Our strategy is based upon the following observations: (1) Under wild ramiﬁcation, Galois action ‘shifts/increases’ valuation (Lemma 2.1(1)). So an element may be used to ‘construct’ other elements with distinct valuation. (2) Elements with distinct valuation may be used to construct bases. If the valuation map vn : Kn → Z is one–to–one on a subset A ⊆ Kn , while vn (A) is onto {i, i+1, . . . , i+vn (2)−1}; then A is a basis for Pin over the integers in the maximal unramiﬁed subﬁeld of Kn . If Kn /K0 is fully ramiﬁed, this subﬁeld is T . The strategy is illustrated below. It is: Use Galois Action to Create Bases. 2.2.1. First Ramiﬁcation Group of Order Two. Suppose that |G1 | = 2. To use Observation (1), we pick α ∈ Kn an element with vn (α) = b1 (e.g. α = πnb1 ). Let αm := α · π0m . Since n−1 vn (π0 ) = 2, vn (αm ) = b1 + 2m. Use Lemma 2.1 with Kn /Kn−1 . So vn ((σ 2 + 1)αm ) = n−1 2b1 + 2m. Since b1 is odd, the valuations of αm and (σ 2 + 1)αm have opposite parity. The valuations for all m lie in one–to–one correspondence with Z. Select those with valuation in {i, . . . , i + vn (2) − 1}. Replace π0e0 by 2 whenever possible. The result is n−1

+ 1)αm : (i − b1 )/2 ≤ m ≤ e0 + i/2 − b1 − 1 B := {αm , (σ 2 n−1 ∪ (σ 2 + 1)αm , 2αm : i/2 − b1 ≤ m ≤ (i − b1 )/2 − 1 .

(2.1)

Since Kn−1 /K0 is unramiﬁed, there is a root of unity ζ with Kn−1 = K0 (ζ). The maximal unramiﬁed extension Q2 in Kn is T (ζ). By Observation (2), B is a basis for Pin over OT (ζ) . n−1 + 1)αm = OT (ζ) · αm + OT (ζ) · σαm yields the group Note that OT (ζ) · αm + OT (ζ) · (σ 2 n−1 n−1 + 1)αm + OT (ζ) · 2αm = OT (ζ) · (σ 2 + 1)αm + OT (ζ) · ring, OT (ζ) [G1 ], while OT (ζ) · (σ 2 n−1 2 − 1)αm yields the maximal order of OT (ζ) [G1 ]. Restricting coeﬃcients and counting (σ leads to the Ok [G1 ]-module structure of Pin , and to M1 (i, b1 ) as in (1.1).

GALOIS STRUCTURE

69

Next, we extend B to a basis upon which the action of the whole group can be followed. Since Kn−1 /K0 is unramiﬁed, there is a normal ﬁeld basis for Pjn−1 /Pj+1 n−1 over O0 /P0 (for each j). Of course, [O0 /P0 : OT /PT ] = 1. So Pjn−1 /Pj+1 has a normal ﬁeld basis n−1 b1 over OT /PT . For j = b1 , this means that there is an element µ ∈ Pn−1 and basis n−1 µ, σµ, . . . , σ 2 −1 µ. Using Lemma 2.1(2), there is an α ∈ Kn with vn (α) = b1 such that n−1 n−1 (σ 2 + 1)α = µ. Then α, σα, . . . , σ 2 −1 α is a normal ﬁeld basis for Pbn1 /Pbn1 +1 over n−1 1 1 +1 OT /PT . Since {σ j (σ 2 + 1)α : j = 0, . . . , 2n−1 − 1} is a basis for Pbn−1 /Pbn−1 , it is 2b1 2b1 +1 also a basis for Pn /Pn over OT /PT . This together with the fact that {σ j α : j = n−1 0, . . . , 2n−1 − 1} is a basis for Pbn1 /Pbn1 +1 over OT /PT leads to ∪2j=0 −1 σ j B being a basis for Pin over OT , and Pin ∼ = OT [G] ⊗Z2 [G1 ] M1 (i, b1 ) as OT [G]-modules. 2.2.2. An Application of Nakayama’s Lemma. In the previous section we were able to follow the Galois action from one basis element to another explicitly. This level of detail becomes overwhelming as we generalize to |G1 | = 4, 8. Fortunately, Nakayama’s Lemma allows us to push some of these details into the background. Lemma 2.2. Let A be a Ok [C2n ]-module (torsion-free over Ok ) where C2n = σ and k denotes an unramiﬁed extension of Q2 . Let H denote the subgroup of order 2, H A the submodule ﬁxed by H, and TrH A the image under the trace. Then TrH A/ H is free over Ok/2Ok . Suppose that B ⊆ A such that TrH B is a (σ − 1)TrH A + 2A basis for TrH A/ (σ − 1)TrH A + 2AH then B can be extended to a Ok [C2n ]/ TrH -basis of A/AH . Proof. Since A/AH is a module over the principal ideal domain Ok [C2n ]/ TrH , it is a free. So C := A/AH ∼ = (Ok [C2n ]/ TrH ) for some exponent a. Now Ok [C2n ]/ TrH is a local ring with maximal ideal σ − 1 dividing 2. Therefore by Nakayama’s Lemma any collection of elements in A that serves as a Ok /2Ok -basis for C/(σ − 1)C will serve as an Ok [C2n ]/ TrH -basis for C. This leaves us to show that B can be extended to a H Ok /2Ok -basis for the − 1)C = A/(A + (σ − 1)A). But since TrH B is vector space C/(σ H a basis for TrH A/ (σ − 1)TrH A + 2A , the elements of B are linearly independent in A/(AH + (σ − 1)A) and therefore span a subspace. 2.2.3. First Ramiﬁcation Group of Order Four. Let |G1 | = 4. This case is important because it illustrates the utility of Lemma 2.2. (Recall that §2.2.1 and §2.2.3 are included in this paper to motivate considerations in §3.) Step 1: Collect |G1 | elements whose valuations are a complete set of residues modulo |G1 |. We begin with the elements used to determine the structure of ideals in Kn−1 (from §2.2.1), n−2 namely αm and (˜ σ + 1)αm ∈ Kn−1 (replacing n by n − 1, expressing σ 2 as σ ˜ ). Note that the ﬁrst ramiﬁcation number of Kn /Kn−2 is the (only) ramiﬁcation number of Kn−1 /Kn−2 (use [?, pg 64 Cor] or switch to upper ramiﬁcation numbers [?, IV §3]). So vn (αm ) = 2vn−1 (αm ) = 2b1 + 4m and vn ((˜ σ + 1)αm ) = 4b1 + 4m. We have two elements of even valuation. To get elements with odd valuation, we apply Lemma 2.1(2). For each X ∈ Kn−1 , Lemma 2.1(2) gives us a preimage X ∈ Kn (under the trace Trn,n−1 ), a preimage that satisﬁes vn (X) = 2vn−1 (X) − b2 . So Trn,n−1 X = (˜ σ 2 + 1)X = X. The integers vn (αm ), vn ((˜ σ + 1)αm ), vn (αm ) = 2b1 − b2 + 4m, vn ((˜ σ + 1)αm ) = 4b1 − b2 + 4m are a complete set of residues modulo 4. Step 2: Collect elements with valuation in {i, i+1, . . . , i+vn (2)−1}. To organize this process we use Wyman’s catalog of ramiﬁcation numbers [?]. If b1 ≥ e0 , the second ramiﬁcation

70

G. GRIFFITH ELDER

number is uniquely determined, b2 = b1 + 2e0 . If b1 < e0 , then either b2 = 3b1 , b2 = 4e0 − b1 , or b2 = b1 + 4t for some t with b1 < 2t < 2e0 − b1 [?, Thm 32]. In any case, we have the bound, 2b1 < b2 .

(2.2)

σ + 1)αm+ke0 , αm+ke0 , Now for a given m, list the inﬁnitely many elements, αm+ke0 , (˜ (˜ σ + 1)αm+ke0 , in terms of increasing valuation. Replace αm+ke0 by 2k αm and drop the subscripts m. So for b2 > 4e0 − 2b1 , beginning at α, we have: 1

2

3

2

· · · −→ α −→ 1/2(˜ σ + 1)α −→ (˜ σ + 1)α −→ α −→ 2α −→ · · · x

Each increase in valuation, denoted by −→, is justiﬁed as follows: For x = 1, the justiﬁcation depends upon the case either b2 > 4e0 − 2b1 or b2 < 4e0 − 2b1 . For x = 2, it is b2 < 4e0 . For x = 3, it is (2.2). If b2 < 4e0 − 2b1 , the list is as follows: 4

3

4

1

σ + 1)α −→ α −→ (˜ σ + 1)α −→ 2α −→ · · · · · · −→ α −→ (˜ Note x = 4 is justiﬁed by b1 > 0. Now collect those elements with valuation in {i, . . . , i + vn (2) − 1}. This will provide us with an OT (ζ) -basis for Pin . Begin with the smallest m such that i ≤ vn (αm ). Note then σ + 1)αm ) < i + vn (2). Associated with this particular m are four elements in that vn (2(˜ {i, . . . , i + vn (2) − 1}. They are listed in the ﬁrst row of the table below. Consider this interval to be a ‘window’. As we increase m, new elements appear (e.g. 2X) – appearance coincides with disappearance (namely of X). Four elements are in ‘view’ always. There are four ‘views’ (four sets). We list the ‘views’ as rows under the appropriate heading. D: The OT (ζ) -basis for Pin . A:

b2 < 4e0 − 2b1 (˜ σ + 1)α

B:

(1)

α

(2)

(˜ σ + 1)α

(3)

α

(˜ σ + 1)α

α

(˜ σ + 1)α

(4)

1/2(˜ σ + 1)α

α

(˜ σ + 1)α

α

2α

2(˜ σ + 1)α

(˜ σ + 1)α

α

b2 > 4e0 − 2b1 2α

(˜ σ + 1)α

2α

(˜ σ + 1)α α

1/2(˜ σ + 1)α α

α

2α

2(˜ σ + 1)α (˜ σ + 1)α

(˜ σ + 1)α

1/2(˜ σ + 1)α

α

(˜ σ + 1)α

2α α

Should we need to determine the subscripts (associated with a particular ‘view’), we can easily do so: For example the four elements listed in A(1) and B(1), appear for m with σ + 1)αm ) ≤ i + 4e0 − 1. In other words, (i − 2b1 )/4 ≤ m ≤ i ≤ vn (αm ) and vn (2(˜

(i + b2 )/4 − b1 − 1. i/2

Step 3: Identify a basis for the quotient module Pin /Pn−1 , and determine the precise image i/2 of each basis element under the trace Trn,n−1 (in terms of the basis for Pn−1 ). Observe i/2 that Pin /Pn−1 is, in a natural way, free over the principal ideal domain OT (ζ) [G]/ ˜ σ 2 + 1. We begin by identifying those elements listed in D, the OT (ζ) -basis from Step 2, that can σ 2 + 1-basis. Take D and partition it into two sets. Let D contain serve as a OT (ζ) [G]/ ˜ those elements X with a bar. Let D0 contain those elements X without a bar. So D is an

GALOIS STRUCTURE i/2

71 i/2

OT (ζ) -basis for Pin /Pn−1 , and D0 is an OT (ζ) -basis for Pn−1 . If we knew which elements i/2 from D provide us with OT (ζ) [G]/ ˜ σ 2 + 1-basis for Pin /Pn−1 we would be done, as it is easy to express the image (under the trace Trn,n−1 ) of each element in D in terms of elements of D0 (there is a one–to–one correspondence). Before we proceed further, note the following. We may assume without loss of generality that for X ∈ D, Trn,n−1 X = 0 if and only if X appears together with X (for the same subscript m) in D. Clearly if X and X appear together, then Trn,n−1 X = X = 0. However when 2X and X appear together, after a change of basis, we may assume that Trn,n−1 2X = 0. The reason for this is as follows: We can change an element of D by adding i/2 an element from D0 and still have a OT (ζ) -basis for Pin /Pn−1 . So whenever 2X and X appear together, replace 2X with 2X − X. Note Trn,n−1 (2X − X) = 0. If we perform this change throughout our basis, but relabel 2X − X as 2X, then we may continue to use the lists, A(1)–A(4) and B(1)–B(4), but assume that Trn,n−1 2X = 0 if 2X appears together with X. i/2 Our next step will be to provide an OT (ζ) [G]/ ˜ σ 2 +1-basis for Pin /Pn−1 . Consider those rows with an X such that Trn,n−1 X = 0 (namely A(2), A(3), A(4), B(2), B(4)). Let S ⊆ D denote the set of left–most X associated with those rows. So, for example, if b2 + 2b1 < 4e0 , then S is made up of the (˜ σ + 1)αm from A(2), and the αm from A(3) and A(4). Verify that i/2 Trn,n−1 S is a OT (ζ) /2OT (ζ) -basis for Trn,n−1 Pin /((˜ σ − 1)Trn,n−1 Pin + 2Pn−1 ) (observe i/2

that Trn,n−1 S generates Trn,n−1 Pin /2Pn−1 over OT (ζ) /2OT (ζ) [G]). Now use Lemma 2.2 i/2

i/2

to extend S to S , an OT (ζ) [G]/ ˜ σ 2 + 1-basis for Pin /Pn−1 . Since Pin /Pn−1 has rank e0 2 over OT (ζ) [G]/ ˜ σ + 1, we have |S | = e0 . σ 2 +1-basis, S , possesses two important properties. First, it contains S. This OT (ζ) [G]/ ˜ Second, without loss of generality we may assume that the elements in S − S are killed by the trace Trn,n−1 . These two properties are shared with another set: The set of all left–most X (an X for every value of m). Clearly the set of all left–most X contains S. Moreover, by an earlier assumption, the compliment of S in the set of all left–most X is mapped to zero under the trace. And so, because the sets have the same cardinality (namely e0 ), we can identify them. Without loss of generality, assume that S is the set of all left–most X. This allows us to use the lists, A(1)–A(4) and B(1)–B(4), in the ‘book-keeping’ necessary for determining the Galois module structure below. i/2 At this point, we know that Pin /Pn−1 is free over OT (ζ) [G]/ ˜ σ 2 + 1. Indeed, S (the i/2 set of all left–most X) provides us a OT (ζ) [G]/ ˜ σ 2 + 1-basis for Pin /Pn−1 . Of course, i/2 the OT (ζ) [G]-structure of Pn−1 is known from §2.2.1 (and can be read oﬀ of D0 ). So a ˜ 2 + 1 in terms of D0 will determine the Galois module description of the image of S under σ structure. See [?, §8]. The Result: For each m associated with A(1) or B(1) we decompose oﬀ an OT (ζ) [G1 ]-summand of OT (ζ) ⊗Z2 I, for A(2) or B(2) we get an OT (ζ) ⊗Z2 H, for A(3) we ﬁnd the group ring, OT (ζ) [G1 ] ∼ = OT (ζ) ⊗Z2 G. But, for B(3) we decompose oﬀ the maximal order of OT (ζ) [G1 ], OT (ζ) ⊗Z2 M. For A(4) and B(4) there is OT (ζ) ⊗Z2 L. All this and counting determines the OT (ζ) [G1 ]-module structure of Pin from which the Ok [G1 ]-module structure can be inferred. It also determines the module M2 (i, b1 , b2 ). To determine the OT [G]-module structure (from which the Ok [G]-module structure can be inferred), we need to take our OT (ζ) -bases for Pin and create OT -bases. 2.3. Partially Ramiﬁed Extensions. Let Ti denote the maximal unramiﬁed extension of Q2 contained in Ki . So T (ζ) of the previous section can be expressed at Tn , while T = T0 . Recall the steps in §2.2.1. We ﬁrst determined a OTn -basis B for Pin , one upon which the

72

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action of G1 could be explicitly followed. Then noting that we can identify G/G1 with the Galois group for Tn /T0 , we extended B to an OT0 -basis for Pin . This time the action of every element in the Galois group G could be followed. What were the important ingredients in this process? It was important that the elements of B lay in one–to–one correspondence, via valuation, with the integers i, . . . , i + vn (2) − 1. Using this fact and the fact that for had a normal ﬁeld basis over OT0 /PT0 , we were able to make an OT each t, Ptn /Pt+1 n basis for Pin , namely B = ∪σi ∈G/G1 σ i B. At that point we were done. The OT [G]-structure could simply be read oﬀ of this basis. This is not the case when |G1 | = 4. Nor is it the case when |G1 | = 8. At this point we still need to change our basis and use Nakayama’s Lemma, if only to determine OT [G1 ]-structure. We leave it to the reader to check that this process of basis change ‘commutes’ with the process of extending our OTn -basis to an OT0 basis. Simply follow the argument using elements of the form σ t αm , σ t (˜ σ + 1)αm , . . . with σ + 1)αm , . . .. As a consequence, the t = 0, . . . 2[G:G1 ]−1 instead of elements of the form αm , (˜ problem of determining the OT [G]-module structure reduces to the problem of determining the OTn [G1 ]-module structure. 3. Fully Ramified Cyclic Extensions of Degree Eight We consider fully ramiﬁed extensions K3 /K0 with odd ramiﬁcation numbers. 3.1. Outline. Our discussion here is focused on the unstably ramiﬁed case, b1 < e0 . (The stably ramiﬁed case will be addressed separately in §3.4.) Recall Step 1 of §2.2.3 (in reference to K2 /K0 ). But ﬁrst note that the ﬁrst two ramiﬁcation numbers of K3 /K0 are the (only) two ramiﬁcation numbers of K2 /K0 [?, pg 64 Cor]. We began with two elements, namely α, (σ +1)α in the subﬁeld K1 . (The Galois relationship between them was explicit.) Then we created α, (σ + 1)α ∈ K2 , preimages under the trace Tr2,1 . In this section, we will start with these four elements from K2 and use Lemma 2.1(2) to ﬁnd further preimages: of α, (σ + 1)α, α, (σ + 1)α under Tr3,2 . To avoid confusion (confusion resulting from additional bars denoting a preimage under Tr3,2 ), we relabel. Let α := α and let ρ := (σ + 1)α. So the four elements in K2 are labeled α, (σ 2 +1)α, ρ, (σ +1)(σ 2 +1)α (instead of α, α, (σ + 1)α, (σ +1)α respectively). The eight resulting elements (four from K2 along with their preimages) lie in one–to–one correspondence with the residues modulo 8. We would have accomplished all that was accomplished in Step 1 from §2.2.3 if we knew the Galois relationships among α, (σ 2 +1)α, ρ, (σ+1)(σ 2 +1)α explicitly. We need an explicit relationship between α and ρ. This is accomplished in §3.2.2 through a list of technical results – generalizations of Lemma 2.1. Note that ρ is an ‘approximation’ to (σ + 1)α – they have the same image under the trace Tr2,1 . Our results describe their diﬀerence, the ‘error’ in this ‘approximation’. As a prerequisite for the technical results of §3.2.2, and in preparation for the analog of Step 2 from §2.2.3 we use a result of Fontaine to provide a catalog of ramiﬁcation numbers in §3.2.1. We are then ready for Step 2: First we order the eight elements (that we inherit from Step 1) in terms of increasing valuation. This is accomplished in §3.3. There are eight orderings – eight cases. The result is eight diﬀerent bases, listed as A – H (as opposed to just two in D from §2.2.3). For the convenience of the reader, they are listed in Appendix B. We are now ready for the analog of Step 3 from §2.2.3. We are ready to determine those i/2 elements in each OT -basis that serve as an OT [G]/ Tr3,2 -basis, S, for Pi3 /P2 . We will i/2 then be able to describe the image, Tr3,2 S, in terms of our OT -basis for P2 (or more to i/2 the point, explicitly in terms of OT [G]-generators for P2 ). To do all this we will need, as in §2.2.3, to perform certain basis changes. The processes are similar, but there are a

GALOIS STRUCTURE

73

few very important diﬀerences. For the convenience of the reader, the results of this step are summarized in §3.3.1. The steps are then spelled out in §3.3.2 – §3.3.5. The structure of M3 (i, b1 , b2 , b3 ) (given in Tables 1 and 2) can then be read oﬀ of the bases in Appendix B. Note however, that we still need to determine the structure under b1 ≥ e0 (part of Case A). This situation is addressed in § 3.3.4. 3.2. Preliminary Results. We catalog the ramiﬁcation triples and generalize Lemma 2.1, describing the diﬀerence ρ − (σ + 1)α. 3.2.1. Ramiﬁcation Triples. There is stability and instability. Theorem 3.1 ([?, Prop 4.3]). Stability: b1 ≥ e0 ⇒ b2 = b1 + 2e0 ,

and

b1 + b2 ≥ 2e0 ⇒ b3 = b2 + 4e0 .

Instability: b1 < e0 ⇒ 3b1 ≤ b2 ≤ 4e0 − b1 ,

b1 + b2 < 2e0 ⇒ 3b2 + 2b1 ≤ b3 ≤ 8e0 − b2 − 2b1 .

In particular, when b1 < e0 , either b2 = 3b1 , b2 = 4e0 − b1 , or b2 = b1 + 4t for b1 < 2t < 2e0 − b1 , while if b1 + b2 < 2e0 , then either b2 = 3b2 + 2b1 , b2 = 8e0 − b2 − 2b1 , or b3 = 8s − b2 + 2b1 for b2 < 2s < 2e0 − b1 . Plot these ramiﬁcation triples (b1 , b2 , b3 ) in 3 , and project this plot to the ﬁrst two coordinates, (x, y, z) → (x, y, 0), thus creating Figure 1 (next page). This projection is partly a line: for b1 ≥ e0 , each point (b1 , b2 ) is restricted to b2 = b1 + 2e0 . It is partly a triangular region: for b1 < e0 , each point (b1 , b2 ) is bound between the lines b2 = 3b1 and b2 = 4e0 − b1 . The signiﬁcance of the regions A, B, C, . . . will be explained later. Note that for points, (b1 , b2 ), above the line b2 = −b1 + 2e0 , the plot of the (b1 , b2 , b3 ) in 3 will be a plane – b3 is uniquely determined. In Figure 2 we have plotted a slice, at a particular value of b1 , through our plot of ramiﬁcation triples in 3 . Part of this slice is a line – when b3 is uniquely determined. Thus the line from (2e0 − b1 , 6e0 − b1 ) to (4e0 − b1 , 8e0 − b1 ). Indeed, as drawn, Figure 2 implicitly assumes that the slice was taken at b1 for b1 < e0 /2. Otherwise there would be no triangular region. Observe that in Figure 1, the lines b2 = 2e0 − b1 and b2 = 3b1 intersect at b1 = e0 /2. If b1 ≥ e0 /2, the third ramiﬁcation number is uniquely determined by b2 . The triangular region bound by the lines b2 = 3b1 , b3 = 3b2 + 2b1 and b3 = 8e0 − b2 − 2b1 exists only for b1 < e0 /2. Because the ramiﬁcation numbers are odd, the triangular part of Figure 1 can be partitioned as follows: A. B. C. D. E. F.

4e0 − 4b1 /3 < b2 4e0 − 2b1 < b2 < 4e0 − 4b1 /3 4e0 − 4b1 < b2 < 4e0 − 2b1 and b2 > (4e0 + 4b1 )/3 4e0 − 4b1 < b2 < 4e0 − 2b1 and b2 < (4e0 + 4b1 )/3 2e0 − b1 ≤ b2 < 4e0 − 4b1 and b2 > (4e0 + 4b1 )/3 2e0 − b1 ≤ b2 < 4e0 − 4b1 and b2 < (4e0 + 4b1 )/3

74

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Assuming that b1 < e0 /2, there is a triangular region in Figure 2. This can be partitioned into the following cases: E. F. G. H.

8e0 + 4b1 − 2b2 < b3 8e0 − 2b2 < b3 < 8e0 + 4b1 − 2b2 8e0 − 4b1 − 2b2 < b3 < 8e0 − 2b2 b3 < 8e0 − 4b1 − 2b2

Note that if b1 > 8e0 /17, region G is empty; if b1 > 8e0 /21, region H is empty; if b1 > 8e0 /28, region E is empty. So as drawn, we have assumed that b1 < 2e0 /7. If however the slice were taken for a value 8e0 /17 < b1 < 8e0 /16, note that the triangular region would consist of only one case, namely F . The relationship between E, F and E, F will be explained in §3.3. 3.2.2. Technical Lemmas. The diﬀerence ρ − (σ + 1)α depends upon ramiﬁcation. Unstable Ramiﬁcation. Assume that b1 < e0 . These results may be thought of as consequences of indirect ‘routes’ from α to ρ. For example, we may begin with α ∈ K2 , create (σ 2 + 1)α, then (σ + 1)(σ 2 + 1)α and let ρ be the inverse image of (σ + 1)(σ 2 + 1)α under Tr2,1 . This results in an expression for the ρ − (σ + 1)α. Lemma 3.2. If b2 ≡ b1 mod 4 (equivalently 3b1 < b2 < 4e0 − b1 ), let t = (b2 − b1 )/4. There are elements αm ∈ K2 with v2 (αm ) = b2 + 4m, such that ρm = (σ + 1)αm + (σ 2 ± 1)αm−t has valuation v2 (ρm ) = b2 + 2b1 + 4m. The ‘+’ or ‘−’ depends on our needs. Proof. Let α ∈ K2 with valuation, v2 (α) ≡ b2 mod 4. Using Lemma 2.1, v2 ((σ + 1)α) = v2 (α) + b1 , v2 ((σ 2 + 1)α) = v2 (α) + b2 . Since (σ 2 + 1)α ∈ K1 and v1 ((σ 2 + 1)α) = (v2 (α) + b2 )/2 ≡ b2 mod 2, v1 ((σ + 1)(σ 2 + 1)α) = (v2 (α) + b2 )/2 + b1 . Using Lemma 2.1(2), there is a ρ ∈ K2 with v2 (ρ) = v2 (α) + 2b1 such that (σ 2 + 1)ρ = (σ + 1)(σ 2 + 1)α. Since (σ 2 +1) [ρ − (σ + 1)α] = 0. Using Lem 2.1(3), there is a θ ∈ K2 with v2 (θ) = (v2 (α)−b2 )+b1 and ρ = (σ +1)α+(σ 2 −1)θ. Since b1 < e0 , v2 (2θ) > v2 (ρ). We may replace ρ by ρ := ρ+2θ (they have the same valuation), and get ρ = (σ + 1)α + (σ 2 + 1)θ. Once αm is chosen, we let αm−t := θ. Lemma 3.3. If b2 ≡ −b1 mod 4 (equivalently b2 = 3b1 or b2 = 4e0 −b1 ), let s := (b2 +b1 )/4. There are elements αm ∈ K2 with v2 (αm ) = b2 + 4m, such that ρm = (σ + 1)αm + (σ + 1)(σ 2 + 1)αm−s has valuation, v2 (ρm ) = 2b2 − b1 + 4m. Note if b2 = 3b1 , v2 (ρm ) = b2 + 2b1 + 4m.

GALOIS STRUCTURE

75

Proof. There is a τ ∈ K0 with v0 (τ ) = (b2 − b1 )/2. Using Lemma 2.1(2), let ρ ∈ K2 with v2 (ρ) = b2 − 2b1 such that (σ 2 + 1)ρ = τ . Clearly (σ 2 + 1) · (σ − 1)ρ = 0, so there is a θ ∈ K2 with v2 (θ) = −b1 such that (σ − 1)ρ = (σ 2 − 1)θ. Since (σ − 1) · [ρ − (σ + 1)θ] = 0, τ := ρ − (σ + 1)θ is a unit in K0 . Let ρ = ρ/τ and θ = θ/τ , so 1 = ρ − (σ + 1)θ . Now let β = (σ + 1)(σ 2 + 1)θ . Clearly (σ 2 + 1)θ ∈ K1 and v1 ((σ 2 + 1)θ ) = (b2 − b1 )/2 is odd. Therefore v2 (β) = b2 + b1 . Replacing 1 with the expression, (σ + 1)(σ 2 + 1)(θ /β), yields ρ = (σ + 1)θ + (σ + 1)(σ 2 + 1)(θ /β).

(3.1)

By choosing τ ∈ K0 with other valuations, the result follows. Unfortunately, if b2 = 4e0 − b1 then s = e0 (valuation can not distinguish between αm /2 and αm−s ). To avoid this confusion, we include the following. Lemma 3.4. Let b2 = 4e0 − b1 . There are αm ∈ K2 with v2 (αm ) = b2 + 4m, so 1 1 ρm := (σ + 1)αm − (σ + 1)(σ 2 + 1)αm + (σ + 1)(σ 2 + 1)αm+e0 −b1 2 2 has valuation, v2 (ρm ) = 2b2 − b1 + 4m. Proof. From (3.1) we have ρ = (σ + 1)θ + (σ + 1)(σ 2 + 1)(θ /β). Apply (σ 2 + 1)/β to both sides. So (σ 2 + 1)(ρ /β) = 1 + 2/β. Since v2 ((σ 2 + 1)ρ ) = 8e0 − 4b1 and v2 (β) = 4e0 , then v0 (1 + 2/β) = e0 − b1 . Replace θ /β with (1/2) · [−θ + θ (1 + 2/β)], and distribute (σ + 1)(σ 2 + 1). Remark 3.5. Note (σ − 1)ρm = (σ 2 − 1)αm and (σ 2 + 1)ρm = (σ + 1)(σ 2 + 1)αm+e0 −b1 , using Lemma 3.4. Apparently, ρm is ‘torn’ between αm and αm+e0 −b1 . We chose to emphasize ρm ’s tie to αm . If we relabel ρm−e0 +b1 as ρm (keep the αm the same), Lemma 3.4 reads 1 1 ρm := (σ + 1)αm−e0 +b1 − (σ + 1)(σ 2 + 1)αm−e0 +b1 + (σ + 1)(σ 2 + 1)αm 2 2 has valuation, v2 (ρm ) = b2 + 2b1 + 4m – thus tying ρm to (1/2)(σ + 1)(σ 2 + 1)αm . This valuation of ρm is as in Lemmas 3.2 and 3.3 (for b2 = 3b1 ). Stably Ramiﬁed Extensions. Assume that b1 ≥ e0 . The results may be seen as direct routes from α to ρ. We create ρ immediately from (σ +1)α ∈ K2 . For discussion and generalization, see [?]. Lemma 3.6. Let b1 > e0 . For every odd integer, a, there are elements α, ρ ∈ K2 with v2 (α) = a, v2 (ρ) = a + (b2 − b1 ). such that (σ + 1)α − ρ = µ ∈ K1 , with v2 (µ) = v2 (α) + b1 . Furthermore µ ∈ K0 for v2 (µ) = v2 (α) + b1 ≡ 0 mod 4. Proof. Since v2 ((σ + 1)α) = v2 (α) + b1 is even, we may express (σ + 1)α as a sum µ + ρ with µ ∈ K1 , ρ ∈ K2 , v2 (µ) = v2 (α) + b1 and odd v2 (ρ). Apply (σ − 1). So (σ 2 − 1)α = (σ − 1)µ + (σ − 1)ρ. Since b2 = b1 + 2e0 < 3b1 , v2 ((σ 2 − 1)α) = v2 (α) + b2 < v2 (α) + 3b1 ≤ v2 ((σ − 1)µ). So v2 ((σ 2 − 1)α) = v2 ((σ − 1)ρ) and v2 (ρ) = v2 (α) + (b2 − b1 ). If v2 (µ) ≡ 0 mod 4, we may

76

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choose α so that µ ∈ K0 . Pick a µ∗ ∈ K0 with v2 (µ∗ ) = v2 (µ). Relabel α as α0 . Choose αi ∈ K i = µi + ρi . Clearly 2∞with v2 (αi ) = v2 (α0 ) + 2i. As before, generate ∞ µi and ρi with α ∞ µ∗ = i=0 ai µi for some units ai ∈ K0 . Let α∗ = i=0 ai αi and ρ∗ = i=0 ai ρi . Lemma 3.7. Let b1 = e0 be odd. For every odd integer, a, there are elements α, ρ ∈ K2 with v2 (α) = a, v2 (ρ) = a + (b2 − b1 ) such that (σ − 1)α − ρ = µ1 ∈ K1 if a ≡ e0 mod 4, (σ + 1)α − ρ = µ0 ∈ K0 if a ≡ 3e0 mod 4. with v2 (µi ) = v2 (α) + b1 . Proof. Let τ ∈ K0 be a unit. From Lemma 2.1(2), there is a ρ ∈ K2 with v2 (ρ) = −b2 and (σ 2 + 1)ρ = τ . So (σ 2 + 1) · (σ − 1)ρ = 0. Use Lemma 2.1(3) to ﬁnd θ ∈ K2 with v2 (θ) = b1 − 2b2 and (σ 2 − 1)θ = (σ − 1)ρ. For a ≡ e0 mod 4, we may assume that α = ρπ0m for some m. Let µ1 = (σ 2 + 1)θπ0m ∈ K1 and ρ = −2θπ0m ∈ K2 . The statement follows. For a ≡ 3e0 mod 4, (σ 2 − 1)θ = (σ − 1)ρ can be interpreted to mean that ρ − (σ + 1)θ ∈ K0 . Multiplying by an appropriate power of π0 , we let α = θπ0m , µ0 = −(ρ − (σ + 1)θ)π0m ∈ K0 and ρ = ρπ0m ∈ K2 . 3.3. The Galois module structure under unstable ramiﬁcation. Assume b1 < e0 . First we determine the OT -bases in Appendix B. From Lemmas 3.2, 3.3, 3.4 we have αm , ρm , (σ 2 +1)αm , (σ +1)(σ 2 +1)αm ∈ K2 , with valuations (measured in v2 ) for every residue class modulo 4. Recall v2 (αm ) = b2 +4m, v2 ((σ 2 +1)αm ) = 2b2 +4m, v2 ((σ+1)(σ 2 +1)αm ) = 2b2 + 2b1 +4m and v2 (ρm ) = 8e0 −3b1 +4m if b2 = 4e0 −b1 , otherwise v2 (ρm ) = b2 +2b1 +4m. Using Lemma 2.1(2) we determine elements αm , ρm , (σ 2 + 1)αm , (σ + 1)(σ 2 + 1)αm ∈ K3 , with (σ 4 + 1)X = X and v3 (X) = 2v2 (X) − b3 . These eight elements have valuations (measured in v3 ) in one–to–one correspondence with the residue classes modulo 8. By varying m, it is possible to choose those with valuation i ≤ v3 (x) < 8e0 + i. To organize this process, we list these elements in terms of increasing valuation. There are eight orderings – eight cases. In each case X (or X), an increase in valuation is denoted by an arrow, −→, and justiﬁed by an inequality assigned a number. Numbers above an arrow apply to X. Numbers below the arrow apply to X. As we see below, the ordering of the elements in E is the same as in E (also in F as in F ). This explains the use of similar notation. A.

1

2

1

1

1

ρ −→ 2ρ −→ (σ 2 + 1)α −→ 2(σ 2 + 1)α −→ 2α −→ 6

1

4

4α −→ (σ + 1)(σ 2 + 1)α −→ 2(σ + 1)(σ 2 + 1)α −→ 2ρ In Case A, the valuation of ρm depends upon whether or not b2 = 4e0 − b1 . If b2 = 4e0 − b1 , 0 < b1 justiﬁes 2 while b1 < 2e0 justiﬁes 4. All other increases, including 2 and 4 for b2 = 4e0 − b1 , are justiﬁed by the inequalities listed below. In Cases B through H, there is only one valuation of ρm . 1

2

1

4

5

B. ρ −→ 2ρ −→ (σ 2 + 1)α −→ 2(σ 2 + 1)α −→ 2α −→ 6

5

4

(σ + 1)(σ 2 + 1)α −→ 4α −→ 2(σ + 1)(σ 2 + 1)α −→ 2ρ

GALOIS STRUCTURE

C.

1

2

1

77

7

5

7

5

ρ −→ 2ρ −→ (σ 2 + 1)α −→ 2(σ 2 + 1)α −→ (σ + 1)(σ 2 + 1)α −→ 5

7

7

2α −→ 2(σ + 1)(σ 2 + 1)α −→ 4α −→ 2ρ D.

2

9

9

ρ −→ (σ 2 + 1)α −→ 2ρ −→ 2(σ 2 + 1)α −→ (σ + 1)(σ 2 + 1)α −→ 5

7

7

2α −→ 2(σ + 1)(σ 2 + 1)α −→ 4α −→ 2ρ E = E.

7

8

2

8

7

11

8

13

8

11

ρ −→ 2α −→ 2ρ −→ (σ 2 + 1)α −→ (σ + 1)(σ 2 + 1)α −→ 8

7

8

8

14

8

2(σ 2 + 1)α −→ 2(σ + 1)(σ 2 + 1)α −→ 2α −→ 2ρ 7

10

2

12

13

7

10

F = F . ρ −→ 2α −→ (σ 2 + 1)α −→ 2ρ −→ (σ + 1)(σ 2 + 1)α −→ 11

8

12

11

7

8

14

8

2(σ 2 + 1)α −→ 2(σ + 1)(σ 2 + 1)α −→ 2α −→ 2ρ 8

G.

9

12

15

12

9

8

8

9

8

ρ −→ (σ 2 + 1)α −→ 2α −→ (σ + 1)(σ 2 + 1)α −→ 2ρ −→ 2(σ 2 + 1)α −→ 14

8

2(σ + 1)(σ 2 + 1)α −→ 2α −→ 2ρ 9

15

8

H. ρ −→ (σ 2 + 1)α −→ (σ + 1)(σ 2 + 1)α −→ 2α −→ 2ρ −→ 2(σ 2 + 1)α −→ 14

8

2(σ + 1)(σ 2 + 1)α −→ 2α −→ 2ρ Numbered Inequalities: (1) b1 < 2e0 , b2 < 4e0 , b3 < 8e0 . (2) 3b2 > 4e0 + 4b1 . (2 ) 3b2 < 4e0 +4b1 . (3) 4e0 −4b1 < 3b2 (true for A–F since b2 ≥ 2e0 −b1 ). (4) 2b2 < b3 . (5) 4e0 −2b1 < b2 . (5 ) 4e0 − 2b1 > b2 . (6) 4e0 − 4b1 /3 < b2 . (6 ) 4e0 − 4b1 /3 > b2 . (7) 4e0 − 4b1 < b2 . (7 ) 4e0 − 4b1 > b2 . (8) b1 > 0. (9) b2 > 2b1 . (10) b2 > 4e0 /3 (true for A–F , since b2 ≥ 3e0 /2). (11) Since b2 > 2b1 and b3 ≤ 8e0 − 2b1 − b2 , b3 < 8e0 − 4b1 . (12) 8e0 − 2b2 < b3 . (12 ) 8e0 − 2b2 > b3 . (13) 8e0 + 4b1 − 2b2 < b3 . (13 ) 8e0 + 4b1 − 2b2 > b3 . (14) Since b2 > 2b1 and b3 ≥ 3b2 + 2b1 , b3 > 2b1 + 4b2 . (15) 8e0 − 4b1 − 2b2 < b3 . (15 ) 8e0 − 4b1 − 2b2 > b3 . We leave it to the reader to verify Appendix B. 3.3.1. Summary: Results of Basis Changes and Nakayama’s Lemma. Basis Changes. Except in four rows, C(2), D(2), E(2), F (2),

(3.2)

we ﬁnd we may change the OT -bases in Appendix B so that the Galois action upon each basis is as if ρ and ρ had been everywhere replaced by (σ + 1)α and (σ + 1)α. In the four exceptional cases there are nontrivial Galois relationships among the basis elements. This is explained in §3.3.5. Nakayama’s Lemma. We ﬁnd, without loss of generality, that the set S of ‘left–most’ elements X (as in S of §2.2.3) from each basis in Appendix B will serve as a OT [G]/ Tr3,2 i/2 basis for Pi3 /P2 , except that S contains both (σ + 1)(σ 2 + 1)α and 2α in B(3), C(3), D(3). At this point, the reader can skip the veriﬁcation of these assertions, ignore Cases C through F , replace ρ with (σ + 1)α, and lift the Galois module structure oﬀ of the bases listed in Appendix B. See [?, §8] The result of the readers eﬀort will be the statement of our main result in every case except those associated with (3.2).

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3.3.2. Trivial Diﬀerence. The elements αm , ρm (or ρm , 2αm ) from each basis in Appendix B i/2 i/4 i/4 provide a OT -basis for P2 /P1 . We can change ρm by an element in P1 and still i/4 have a OT -basis. So when ρm −(σ +1)αm ∈ P1 , the diﬀerence between ρm and (σ +1)αm is trivial. i/4 is equivSince v2 ((σ + 1)α) = v2 (ρ − (σ + 1)α), checking ρm − (σ + 1)αm ∈ P1 alent to checking v3 ((σ + 1)α) ≥ i. In Case A, because b2 + b1 ≤ 4e0 we ﬁnd that v3 ((1/2) · (σ + 1)(σ 2 + 1)αm ) ≤ v3 ((σ + 1)αm ). Therefore, in A(3) through A(8), we may replace ρm by (σ + 1)αm . We refrain from doing so in A(8) as it may hamper our ability to determine the eﬀect of Tr3,2 on ρ. We will return to this issue in §3.3.4. In Case B, because b2 > 4e0 − 2b1 we ﬁnd v3 (2α) < v3 ((σ + 1)α). We may replace ρ in B(3) through B(8). For similar reasons, we refrain in B(8). In Cases C and D, b3 > 2b2 +2b1 (since b3 = b2 +4e0 and b2 < 4e0 − 2b1 ). As a consequence, v3 ((σ + 1)(σ 2 + 1)α) < v3 ((σ + 1)α). We may replace ρ in C(3) through C(8), and in D(3) through D(6). In Cases E through H, we clearly have v3 (α) < v3 ((σ + 1)α). We may replace ρ in E(1) or E(3) – E(8), F (1) or F (3) – F (8), G(1) or G(3) – G(8), H(1) or H(3) – H(8). We replace ρ everywhere that we may, except that we refrain for A(8), B(8), C(8), D(7), D(8), E(8), F (7), F (8), G(6), G(7), H(6).

(3.3)

Now we consider the diﬀerence between ρ and (σ + 1)α and replace ρ with (σ + 1)α (2ρ with (σ + 1)2α) in E(1), F (1), G(1), G(8), H(1), H(7), H(8).

(3.4)

Since (σ 4 + 1) · [ρ − (σ + 1)α] = 0, we may use Lem 2.1(2) and ﬁnd an element ω ∈ K3 with v3 (ω) = 2b2 + b1 − 2b3 so that (σ 4 − 1)ω = ρ − (σ + 1)α. As long as b3 < 8e0 − 3b1 , which holds in Cases E through H, we have v3 (ρ) = v3 (ρ + 2ω). On the basis of valuation, we may replace ρ with ρ + 2ω and still have a basis (i.e. Observation (2)). Now since (ρ + 2ω) − (σ + 1)α = (σ 4 + 1)ω ∈ K2 , we may replace (ρ + 2ω) with (σ + 1)α and still have a basis. All we need is v3 ((σ + 1)α) ≥ i. But this clearly holds since v3 (α) ≥ i. i/2

3.3.3. Nakayama’s Lemma and an OT [G]/ Tr3,2 -basis for Pi3 /P2 . The collection of i/2 X in our bases provide an OT -basis for Pi3 /P2 . As in §2.2.3, whenever X and (1/2)· X appear in the same row, we may replace X with X − (1/2) · X and still have a OT -basis. Since Tr3,2 (X − (1/2) · X) = 0, we relabel and assume, without loss of generality, that for these X’s, Tr3,2 X = 0. Let T=0 denote this set (trace zero). Let T=0 denote the set of X’s with X in the same row. For each such X ∈ T=0 , Tr3,2 X ≡ 0 mod 2. This is the set i/2 of trace not zero. Note that Tr3,2 T=0 is an OT /2OT -basis for Tr3,2 Pi3 /2P2 . Following §2.2.3, we select from T=0 a set S (notation as in §2.2.3) such that Tr3,2 S is a OT /2OT -basis i/2 for Tr3,2 Pi3 /((σ − 1)Tr3,2 Pi3 + 2P2 ). It turns out that just as in §2.2.3, S is the set of left-most X for which X appears in the same row, except that S contains both X’s in T=0 from B(3), C(3), D(3). Note that σ acts trivially (modulo 2) upon (σ + 1)(σ 2 + 1)α and 2α in B(3), C(3) and D(3). These elements are linearly independent over OT /2OT [G]. Since both contribute to i/2 the OT /2OT -basis for Tr3,2 Pi3 /2P2 , both (σ + 1)(σ 2 + 1)α and 2α are in S. When a row contributes exactly one X to T=0 , the phrase ‘left–most’ is unnecessary. Indeed σ acts trivially (modulo 2) on the lone X = Tr3,2 X, and since X is needed for the OT /2OT -basis for i/2 Tr3,2 Pi3 /2P2 , X must appear in S. Note this is the only situation to consider in Case A.

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In the other cases, we need to show that each X, corresponding to the left–most X of T=0 , generates over OT /2OT [G] all other elements in the same row (in Tr3,2 T=0 ). This is easy to see for rows E(1), E(5), F (1), F (5), G(1), G(5), G(8) and H(1), H(5), H(7), H(8). More work is required for rows D(7), F (7), G(6), G(7), H(6). Note that ρ − (σ + 1)αm = (σ 2 + 1)αm−t or (σ + 1)(σ 2 + 1)αm−s depending upon b2 > 3b1 or b2 = 3b1 , respectively. If i/2 ρ−(σ+1)αm = (σ+1)(σ 2 +1)αm−s , then (σ−1)ρ = (σ 2 +1)α−2α ≡ (σ 2 +1)α mod 2P2 . 2 2 So ρ generates (σ + 1)α. If ρ − (σ + 1)αm = (σ + 1)αm−t the analysis is a little more i/2 involved. Note (σ − 1)ρ − (σ 2 + 1)α ≡ (σ − 1)(σ 2 + 1)αm−t mod 2P2 . For m associated with D(7), F (7), G(6), G(7), H(6), check that m − t lies in D(3), F (4), G(4), H(4) or later. In any case (σ + 1)(σ 2 + 1)αm−t ∈ Pi3 . So ρm and another X, namely (σ + 1)(σ 2 + 1)αm−t , combine together to generate (σ 2 + 1)αm . i/2 Apply Lemma 2.2 and extend S to an OT [G]/ Tr3,2 -basis for Pi3 /P2 . Except in Cases B, C, D (where a row contributes more than one element), we may assume that this basis is the set of left–most elements X, one from each row. 3.3.4. Essentially Trivial Diﬀerence. In §3.3.2 we did not replace ρ by (σ + 1)α in rows A(1), A(2), B(1), B(2), C(1), C(2), D(1), D(2), E(2), F (2), G(2), H(2). It was not clear i/4 that the diﬀerence ρ − (σ + 1)α lay in P1 . Neither did we replace ρ by (σ + 1)α in the rows listed in (3.3). In this section we remedy this situation. We show, except in four cases, C(2), D(2), E(2), F (2), we may change our basis so that the Galois action is as if ρ had been replaced by (σ + 1)α (ρ by (σ + 1)α). We begin with Case A, explaining why the diﬀerence between ρ and (σ+1)α is essentially trivial and then determine the Galois module structure (to illustrate the process). Consider A(1), A(2) and A(8). Recall there are three expressions for ρm corresponding to 3b1 < b2 < 4e0 − b1 , b2 = 3b1 , and b2 = 4e0 − b1 . Suppose 3b1 < b2 < 4e0 − b1 , and ρm = (σ + 1)αm + (σ 2 − 1)αm−t . Consider ρm in A(8). Since b1 + b2 < 4e0 , v3 (ρm ) ≤ v3 (2αm−t ). So for m in A(8), m − t is in A(4) or later. In any case, (σ − 1)αm−t = (σ + 1)αm−t − 2αm−t ∈ Pi3 and (1/2)(σ−1)(σ 2 +1)αm−t = (1/2)(σ+1)(σ 2 +1)αm−t −(σ 2 +1)αm−t ∈ Pi3 (i.e. these elements are available). We replace αm with x = αm + (σ − 1)αm−t − (1/2)(σ − 1)(σ 2 + 1)αm−t . Note (σ + 1)x = ρ and (σ 2 + 1)x = (σ 2 + 1)αm . The Galois action on x and ρm is the same as the Galois action on αm and (σ + 1)αm . It is as if ρm had been replaced by (σ + 1)αm and ρm by (σ + 1)αm . Now consider A(1) and A(2), ρm = (σ + 1)αm + (1/2)(σ 2 − 1)αm−t+e0 . Since v2 (ρm ) < v2 ((1/2)(σ + 1)(σ 2 + 1)αm−t+e0 ), for m in A(1) or A(2), m − t + e0 lies in A(3) or later. In any case, (1/2)(σ − 1)(σ 2 − 1)αm−t+e0 is available. So in A(1) and A(2), we replace 2αm by 2αm − (1/2)(σ − 1)(σ 2 − 1)αm−t+e0 . The eﬀect of this replacement on the Galois action is, again, the same as if we replaced ρm by (σ + 1)αm . Now suppose b2 = 3b1 and ρm = (σ + 1)αm + (σ + 1)(σ 2 + 1)αm−s . Note s = b1 . Starting with the smallest m such that i ≤ v3 (ρm ) we replace αm by αm + (1/2)(σ + 1)αm+e0 −b1 so long as m + e0 − b1 is associated with A(8). If i ≤ v3 (ρm−b1 ), we replace αm by αm + (σ 2 + 1)αm−b1 . In any case, we can systematically replace αm by x = αm + (1/2)(σ 2 + 1)αm+e0 −b1 or αm + (σ 2 + 1)αm−b1 (1/2)(σ 2 + 1)αm by (1/2)(σ 2 + 1)x and (1/2)(σ +1)(σ 2 +1)αm by (1/2)(σ +1)(σ 2 +1)x. The Galois action after this change of basis is as if ρm = (σ + 1)αm and ρm = (σ + 1)αm . Consider A(1) and A(2). Note (σ − 1)ρm = (σ −1)·(σ +1)αm . Moreover, for m associated with these two cases, (σ +1)(σ 2 +1)αm+e0 −b1 and (σ 2 + 1)αm+e0 −b1 are available elsewhere in our basis. So we replace (σ 2 + 1)αm by (σ 2 + 1)(αm + αm+e0 −b1 ) and (σ + 1)(σ 2 + 1)αm by (σ + 1)(σ 2 + 1)(αm + αm+e0 −b1 ). Note for m associated with A(2), m + e0 − b1 is associated with A(3) or later. We achieve the desired eﬀect by replacing (σ + 1)(σ 2 + 1)αm with (σ + 1)(σ 2 + 1)αm + (σ + 1)(σ 2 + 1)αm+e0 −b1 .

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This leaves b2 = 4e0 −b1 . Because this case is more complicated (recall Remark 3.5: ρm is ‘torn’ between αm and αm+e0 −b1 ), we ﬁrst determine the Galois module structure for b2 < 4e0 −b1 . Each m in A(1) results in an OT ⊗Z2 (R3 ⊕H); m in A(2) in an OT ⊗Z2 H2 ; m in A(3) in an OT ⊗Z2 (R3 ⊕ M); m in A(4) in an OT ⊗Z2 M1 ; m in A(5) in an OT ⊗Z2 (R3 ⊕ L); m in A(6) in an OT ⊗Z2 L3 ; m in A(7) in an OT ⊗Z2 (R3 ⊕ I); m in A(8) in an OT ⊗Z2 I2 . Counting the number of m associated with each A(j) yields the ﬁrst column of Table 2. Now consider b2 = 4e0 − b1 . Because v2 (ρm ) = 2b2 − b1 + 4m, the number of m associated with A(1) and A(7) are diﬀerent. The number for A(7) is e0 − b1 too low, while A(1) is e0 − b1 too high. We seem to be missing e0 − b1 of OT ⊗Z2 I and have e0 − b1 too many of OT ⊗Z2 H. Let us look at this more carefully. Note ρm in A(8) maps (via Tr3,2 ) to ρm

(1/2)(σ + 1)(σ 2 + 1)αm+e0 −b1 = (σ + 1)(αm − (1/2)(σ 2 + 1)αm ) + (σ + 1)(σ 2 + 1)αm−b1

So ρm maps into the OT -module spanned by αm − (1/2)(σ 2 + 1)αm and (σ + 1)(αm − (1/2)(σ 2 +1)αm ) along with either (1/2)(σ 2 +1)αm+e0 −b1 and (1/2)(σ +1)(σ 2 +1)αm+e0 −b1 or (σ 2 + 1)αm−b1 and (σ + 1)(σ 2 + 1)αm−b1 . In any case, the elements (1/2)(σ 2 + 1)αm and (1/2)(σ + 1)(σ 2 + 1)αm for (i + b3 − 4b2 + 2b1 )/8 ≤ m ≤ (i + b3 − 4b2 + 2b1 )/8 + e0 − b1 − 1 are not associated with a ρm in A(8). The ρm in A(1) map to (σ 2 − 1)αm (under (σ − 1)) and so (σ +1)(σ 2 +1)αm+e0 −b1 (under (σ 2 +1)) yielding a H, unless m+e0 −b1 is associated with A(2). In fact, there are e0 − b1 ρm that map into A(2) under (σ 2 + 1). For each m in A(2) we have (σ 4 + 1)(σ + 1)(σ 2 + 1)αm = (σ 2 + 1)ρm−e0 +b1 = (σ + 1)(σ 2 + 1)αm , yielding a copy of H2 . But for the last e0 − b1 elements ρm in A(2), namely those m such that m + e0 − b1 is in A(3) we may replace ρm by ρm − (1/2)(σ + 1)(σ 2 + 1)αm+e0 −b1 . For each of these m we have the OT [G]-submodule spanned by ρm − (1/2)(σ + 1)(σ 2 + 1)αm+e0 −b1 and (σ 2 − 1)αm . These e0 − b1 together with the elements left out of a module in A(8) yield a e0 − b1 copies of I, precisely making up the counts. Cases B – H: In the remaining cases, we only have two situations: b2 = 3b1 and 3b1 < b2 < 4e0 − b1 . Consider 3b1 < b2 < 4e0 − b1 ﬁrst, and ρm = (σ + 1)αm + (σ 2 ± 1)αm−t where we may choose between ± as we like. We are concerned with the image of the trace, Tr3,2 , in particular Tr3,2 ρm = (σ + 1)αm + (σ 2 + 1)αm−t , for ρm appearing in B(8), C(8), D(7), D(8), E(8), F (7), F (8), G(6), G(7), and H(6). Note if (σ 2 + 1)αm−t ∈ Pi3 , we may replace ρm with ρm − (σ 2 + 1)αm−t . So if (σ 2 + 1)αm−t appears in B(6), C(6), D(6), E(5), F (5), G(5), H(5) or later we may replace ρm with (σ + 1)αm and ρm with ρm − (σ 2 + 1)αm−t . The later replacement exhibits the same Galois action as a replacement of ρm by (σ + 1)αm . Without loss of generality we will call it a replacement of ρm by (σ + 1)αm . Since b2 ≤ 4e0 − b1 , v3 (2αm ) ≥ v3 (ρm ). What happens when (σ 2 + 1)αm−t appears in B(3) – B(5), C(3) – C(5), D(3) – D(5), E(4), F (6), G(4), H(6)? In this case (σ − 1)αm−t = (σ + 1)αm−t − 2αm−t ∈ Pi3 . In B(8), C(8), D(7), D(8), E(8), F (8), G(6), G(7), H(6), we replace αm with αm + (σ − 1)αm−t , and (σ 2 + 1)αm with (σ 2 + 1)αm + (σ − 1)(σ 2 + 1)αm−t . Note ρm = (σ + 1) · [αm + (σ − 1)αm−t ]. The Galois action upon these basis elements: Tr3,2 ρm = ρm = (σ + 1) · [αm + (σ − 1)αm−t ], (σ 2 + 1) · [αm + (σ − 1)αm−t ] = (σ 2 + 1) αm + (σ − 1)(σ 2 + 1)αm−t , and (σ + 1) · [(σ 2 + 1)αm + (σ − 1)(σ 2 + 1)αm−t ] = (σ 2 + 1) ρm = (σ + 1)(σ 2 + 1)αm , is similar to the Galois action upon: (σ + 1)αm , αm , (σ + 1)αm , (σ 2 + 1)αm , (σ + 1)(σ 2 + 1)αm . We may assume (σ + 1)αm and (σ + 1)αm appear instead of ρm and ρm . Now consider the appearance of ρ in B(1), B(2), C(1), D(1), G(2), H(2). Suppose ρm = (σ+1)αm +(1/2)·(σ 2 −1)αm+e0 −t . One may check v3 (ρm ) ≤ v3 ((1/2)(σ+1)(σ 2 +1)αm+e0 −t

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and v3 (2ρm+e0 −t ) ≤ v3 (4αm ). So (1/2)(σ+1)(σ 2 +1)αm+e0 −t appears in B(4) – B(7), C(6) – C(8) or D(6) – D(8). Note in these sets of elements, ρm+e0 −t has already been replaced by (σ + 1)αm+e0 −t . Importantly, (1/2)(σ − 1)(σ 2 + 1)αm+e0 −t along with (σ − 1)αm+e0 −t are available to us. We replace 2αm with 2αm −(1/2)(σ −1)(σ 2 +1)αm+e0 −t +(σ −1)αm+e0 −t = 2αm − (1/2)(σ − 1)(σ 2 − 1)αm+e0 −t in B(1), B(2), C(1) and D(1). The eﬀect of this change of basis is the same as if we replaced ρm by (σ + 1)αm . Now consider G(2) and H(2). Again ρm = (σ+1)αm +(1/2)(σ 2 −1)αm+e0 −t . In G and H, b3 ≤ 8e0 − 2b2 . As a result, v3 (ρm ) ≤ v3 ((σ − 1)αm+e0 −t ). Note we refer to (σ − 1)αm+e0 −t and not (σ − 1)αm+e0 −t . The valuation of the ﬁrst is b1 more than the valuation of the second. As one may check v3 (ρm ) ≤ v3 ((1/2)(σ +1)(σ 2 +1)αm+e0 −t ), so (1/2)(σ +1)(σ 2 +1) αm+e0 −t appears in G(7), G(8) or H(8). If (σ + 1)(σ 2 + 1)α appeared in G(1) or H(1), (σ 2 + 1)αm−t would be available and so we would replace ρm with ρm − (σ 2 + 1)αm−t . If (1/2)(σ + 1)(σ 2 + 1)αm+e0 −t appears in G(7), then we may assume (σ − 1)αm+e0 −t appears there instead of ρm+e0 −t , because v3 ((σ 2 + 1)αm+e0 −2t ) = v3 ((σ − 1)αm+e0 −2t ) ≥ i, and we would have replaced ρm+e0 −t previously in our discussion with ρm+e0 −t −(σ 2 + 1)αm+e0 −2t . We may now replace 2αm with 2αm − (σ − 1)αm+e0 −t . We replace (σ 2 + 1)αm with (σ 2 + 1) αm + (1/2)(σ − 1)(σ 2 )αm+e0 −t . We may assume without loss of generality that (σ + 1)αm appears in G(2) and H(2) instead of ρm . Now we work with Cases B through H under the assumption b2 = 3b1 . So ρm = (σ + 1) · [αm + (σ 2 + 1)αm−b1 ]. First note if (σ + 1)(σ 2 + 1)αm−b1 appears in B(2), C(3), D(3), E(4), F (4), G(4), H(4), or later we may replace ρm in B(8), C(8), D(7), D(8), E(7), F (7), G(5), G(6), H(5) with ρm − (σ + 1)(σ 2 + 1)αm . Suppose (σ 2 + 1)αm−b1 appears elsewhere. In B, these elements can appear in B(1), B(2), or as (1/2)·(σ 2 +1)αm+e0 −b1 elsewhere in B(8). In cases C through H, since b1 < 4e0 /5, v3 (ρm ) ≤ v3 (ρm−b1 ). So (σ 2 +1)αm−b1 appears in C(1), C(2), D(1), D(2), E(2), E(3), F (2), F (3), G(2), G(3), H(2), H(3). In these cases, we may either replace αm with αm +(1/2)·(σ 2 +1)αm+e0 −b1 or αm +(σ 2 +1)αm+−b1 . If for example, we replace αm with αm + (σ 2 + 1)αm−b1 , (σ 2 + 1)αm with (σ 2 + 1)αm + 2(σ 2 + 1)αm−b1 , and (σ + 1)(σ 2 + 1)αm with (σ + 1)(σ 2 + 1)αm + 2(σ + 1)(σ 2 + 1)αm−b1 , then the Galois action on this new basis is the same as if (σ + 1)αm and (σ + 1)αm appear instead of ρm and ρm . We now concern ourselves with B(1), B(2), C(1), D(1), G(2) and H(2). Check v3 ((σ 2 + 1) αm+e0 −b1 ) ≥ v3 (ρm ). We replace (σ 2 +1)αm with (σ 2 +1)αm +(σ 2 +1)αm+e0 −b1 , and (σ + 1)(σ 2 + 1)αm with (σ + 1)(σ 2 + 1)αm + (σ + 1)(σ 2 + 1)αm+e0 −b1 . In B(2), v3 ((σ + 1)(σ 2 + 1)αm+e0 −b1 ) ≥ v3 ((σ + 1)(σ 2 + 1)αm ), we replace (σ + 1)(σ 2 + 1)αm with (σ + 1)(σ 2 + 1)αm + (σ + 1)(σ 2 + 1)αm+e0 −b1 . All this has the same eﬀect upon the Galois action as a replacement of ρm by (σ + 1)αm . 3.3.5. Non-Trivial Diﬀerence. We consider ρ in C(2), D(2), E(2), F (2). First consider the case b2 = 3b1 where ρm = (σ + 1)αm + (1/2)(σ + 1)(σ 2 + 1)αm+e0 −b1 . Note C and E do not intersect the line b2 = 3b1 . We focus on D(2), F (2). In D with b2 = 3b1 , we have b1 < 4e0 /5. So v3 (2αm ) ≤ v3 (αm+e0 −b1 ). Since v3 (2(σ + 1)(σ 2 + 1)αm ) ≤ v3 ((σ + 1)(σ 2 + 1)αm+e0 −b1 ), for m associated with D(2), (σ + 1)(σ 2 + 1)αm+e0 −b1 appears in D(4), or (1/2)(σ + 1)(σ 2 + 1)αm+e0 −b1 appears in D(5) or later. If (1/2)(σ + 1)(σ 2 + 1) αm+e0 −b1 is available, we may replace ρm with (σ + 1)αm . The Galois action when m is in D(2) and m + e0 − b1 is in D(4) is our primary concern. But ﬁrst consider F (or F ) with b2 = 3b1 . Note then b3 ≤ 8e0 + 2b2 − 8b1 . So v3 (ρm ) ≤ v3 ((σ + 1)(σ 2 + 1)αm+e0 −b1 ). Since b3 ≤ 8e0 +2b2 −8b1 , v3 (αm ) ≤ v3 (2(σ 2 + 1)αm+e0 −b1 ). So for m associated with F (2), (σ+1) (σ 2 + 1)αm+e0 −b1 appears in F (4), or in F (5) or later. If m + e0 − b1 is associated with F (5) or later, we have (σ 2 + 1)αm+e0 −b1 available. We replace 2αm with 2αm +(σ 2 + 1)αm+e0 −b1 .

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We replace (σ 2 + 1)αm and (σ + 1)(σ 2 + 1)αm with (σ 2 + 1)αm + (σ 2 + 1)αm+e0 −b1 and (σ + 1)(σ 2 + 1)αm + (σ + 1)(σ 2 + 1)αm+e0 −b1 . The eﬀect of these changes upon the Galois action is the same as the replacement of ρm by (σ + 1)αm . This leaves the situation when m belongs to D(2), F (2) while m + e0 − b1 belongs to D(4), F (4). In both of these cases, we replace (σ + 1)(σ 2 + 1)αm+e0 −b1 with (σ + 1)(σ 2 + 1)αm+e0 −b1 + (σ + 1)2αm − ρm . This new basis element has trace, Tr3,2 , zero. For each such pair (m, m + e0 − t) we get a copy of H1 G ⊕ R3 . Let us now turn to the case where 3b1 < b2 < 4e0 −b1 and ρm = (σ +1)αm +(1/2)(σ 2 +1) αm+e0 −t . Consider cases C and E. Because v3 (2α) ≤ v3 ((σ 2 + 1)α), if m appears in C(2), then m + e0 − t appears in C(6) or later. Since v3 ((σ 2 + 1)αm+e0 −t > v3 (2(σ + 1)(σ 2 + 1)α), not every m + e0 − t is in C(6) when m is in C(2). Since v3 (ρ) ≤ v3 ((1/2)(σ 2 + 1)α), if m appears in E(2), then m+e0 −t appears in E(6) or later. Since v3 ((σ 2 +1)αm+e0 −t > v3 (2α), some m + e0 − t spill over into C(7). Consequently, whenever a pair (m, m + e0 − t) has m in C(2), E(2) while m + e0 − t is in C(6), E(6) we get a copy of H1 L ⊕ R3 . Consider cases D and F (including F ). Consider D ﬁrst. Since v3 (2αm ) < v3 ((σ 2 + 1)αm+e0 −t ), for m in D(2), m + e0 − t lands in D(6) or later. Note since v3 (2αm ) > v3 (ρm+e0 −t ), some m + e0 − t land in D(6). Since v3 ((σ 2 + 1)αm+e0 −t ) > v3 (2(σ 2 + 1)αm ), the collection of m + e0 − t overlap into D(8). When m + e0 − t is in D(8), the element (1/2)(σ 2 + 1)αm+e0 −t is available and we replace ρm by ρm − (1/2)(σ 2 + 1)αm+e0 −t = (σ + 1)αm . For each pair (m, m + e0 − t) such that m is associated with D(2) and m + e0 − t is associated with D(6), we get a copy of H1 L ⊕ R3 . What we are principally concerned with is what happens when for m in D(2), m + e0 − t is in D(7). In this case, because ρm+e0 −t = (σ + 1)αm+e0 −t + (σ 2 + 1)αm+e0 −2t , there is some new interaction to consider. Suppose m is in D(2), while m + e0 − t is in D(7). Since v3 (ρm+e0 −t ) ≤ v3 (αm+e0 −2t ) and v3 (2(σ + 1)(σ 2 + 1)αm ) ≤ v3 ((σ + 1)(σ 2 + 1)αm+e0 −2t ), for m in D(2) and m + e0 − t in D(7), we ﬁnd m + e0 − 2t is associated with D(4), or D(5) or later. Consider m in D(2), m + e0 − t in D(7), and m + e0 − 2t in D(4). Perform change of basis: Replace 2αm with 2αm + 2αm+e0 −t − 2αm+e0 −t , ρm with ρm − αm+e0 −t , (σ 2 + 1)αm with (σ 2 + 1) αm +(σ 2 +1)αm+e0 −2t +1/2(σ−1)(σ 2 +1)αm+e0 −t , and (σ+1)(σ 2 +1)αm with (σ+1)(σ 2 +1) αm + (σ + 1)(σ 2 + 1)αm+e0 −2t . The eﬀect of these base changes upon the Galois action is the same as if we were to replace ρm with (σ + 1)αm − (1/2)(σ + 1)(σ 2 + 1)αm+e0 −2t . Notice the similarity between this expression and the expression for ρm used when b2 = 3b1 . Consequently, this scenario results in copies of H1 G ⊕ R3 . (Note if b2 = 3b1 , then 2t = b1 .) In the alternative situation, when m is in D(2), m+e0 −t in D(7), and m+e0 −2t is in D(5) or later, we perform the same basis changes. Except, since the element (1/2)(σ + 1)(σ 2 + 1) αm+e0 −2t is available, we replace ρm with ρm − αm+e0 −t + (1/2)(σ + 1)(σ 2 + 1)αm+e0 −2t . The eﬀect of this alternative basis change upon the Galois action is the same as a simple replacement of ρm with (σ + 1)αm . We now turn our attention to Cases F and F . Since 0 < 2b1 , v3 (ρm ) < v3 ((1/2)(σ +1)(σ 2 +1)αm+e0 −t . So for m associated with F (2), m+e0 −t is associated with F (6) or later. We leave it to the reader to check that m + e0 − t lands in F (6) or F (7). If m + e0 − t is associated with F (7), then m + e0 − 2t lands in F (4) or F (5). In any case, all this is analogous to D. 3.4. The Galois module structure under stable ramiﬁcation. For p = 2, stable ramiﬁcation b1 ≥ e0 is nearly strong ramiﬁcation b1 > (1/2) · pe0 /(p − 1), (the conditions diﬀer only when e0 is odd – K0 tame over Q2 ). In [?], the structure of the ring of integers was determined under strong ramiﬁcation for any prime p. We revisit that argument extending it to ambiguous ideals and the case b1 = e0 .

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e0 i/2 i/4 ∼ i/2 i/4 Following §2.1, P2 /P1 = OT [σ]/ σ 2 + 1 . So e0 elements generate P2 /P1 over OT [G]. Use Lemmas 3.6, 3.7 to select elements, α, with odd valuation a such that

i/2 ≤ a ≤ i/2 + 2e0 − 1. Each of these e0 elements gives rise (via the action of (σ ± 1)) to another element, ρ in K2 , with odd valuation, a + (b2 − b1 ) = a + 2e0 . These α along i/4 with their Galois translates, ρ ≡ (σ ± 1)α mod P1 , have valuations in one–to–one correspondence (via v2 ) with the odd integers in i/2, . . . , 4e0 + i/2 − 1, and as a result i/4 i/2 serve as a OT -basis for P2 /P1 . The α provide a OT [G]/ Tr2,1 -basis. i/2 i/4 i/4 to be compatible with our OT -basis for P1 We need this basis for P2 /P1 i/2 (as determined as in §2.2.1), as well as our OT [G]/ Tr3,2 -basis for Pi3 /P2 . First we i/4 i/4 consists of pairs: either ((σ + consider compatibility with P1 . The OT -basis for P1 1)η, η) or ((σ+1)η, 2η) ∈ K0 ×K1 where v1 (η) is odd. Because of Lemma 2.1 each coordinate uniquely determines the other. Now consider pairs where the valuation v3 of both elements is bound between i and 8e0 + i − 1. For example, pairs of the form ((σ + 1)η, η) appear for i/4 ≤ v1 (η) ≤ 2e0 + i/4 − b1 − 1, while pairs of the form ((σ + 1)η, 2η) appear for

i/4 − b1 ≤ v1 (η) ≤ i/4 − 1. The coordinates of all pairs provides us with an OT basis i/4 for P1 . Each α with v2 ((σ 2 + 1)α) ≤ 4e0 + i/2 − 1 determines (via (σ 2 + 1)α ∈ K1 ) i/4 a pair of elements in the OT -basis for P1 . If v1 ((σ 2 + 1)α) is odd, then α determines a pair of the form ((σ + 1)η, 2η). If even, it determines a pair of the form ((σ + 1)η, η). In general for α with v2 ((σ 2 + 1)α) ≥ 4e0 + i/2, v2 (1/2(σ 2 + 1)α) ≥ i/2. So 1/2(σ 2 + 1)α is available and we may replace α in by α − 1/2(σ 2 + 1)α and still have a basis. Note 2 2 (σ +1) α − 1/2(σ + 1)α = 0. So we can assume, without loss of generality, (σ 2 +1)α = 0. This posses no complication, unless (σ ± 1)α = µ + ρ with ρ in the image of Tr3,2 Pi3 . In other words, v2 (ρ) ≥ (b3 + i + 1)/2. (Note for α with v2 ((σ 2 + 1)α) ≤ 4e0 + i/2 − 1 and (σ ± 1)α = µ + ρ, we have v2 (ρ) < (b3 + i + 1)/2.) For these α (actually α − 1/2(σ 2 + 1)α), µ (actually µ − (σ ± 1)1/2(σ 2 + 1)α) will determine a pair ((σ + 1)η, 2η) or ((σ + 1)η, η) i/4 in our OT -basis for P1 . We need simply to show µ and µ − (σ ± 1)1/2(σ 2 + 1)α have the same properties. We leave it to the reader to do this (use Lemma 3.6 and 3.7 to show that the valuations are the same, that µ − (σ ± 1)1/2(σ 2 + 1)α ∈ K0 if and only if µ ∈ K0 ). The only issue that remains is whether there can be any conﬂict between a pair of basis i/4 determined directly, via (σ 2 + 1)α, and a pair determined indirectly via elements for P1 µ = (σ ± 1)α − ρ. Note any element in the image of the trace, Tr2,1 , has valuation that is larger than the valuation of every µ ∈ K1 that arises from the expression for a Galois translate ρ = (σ ± 1)α − µ. i/2 We select our OT [G]-basis for Pi3 /P2 now. There is one element X in our OT -basis i/2 for P2 for each valuation v2 in (i + b3 + 1)/2, . . . , 4e0 + i/2 − 1.

(3.5)

The reader may check for v2 (X) even, X = (σ 2 + 1)α for some α in our OT [G]-basis for i/4 i/2 P2 /P1 . For v2 (X) odd, since i/2+(b2 −b1 ) < (i+b3 +1)/2, X = ρ = (σ ±1)α−µ also for some α. Use Lem 2.1 to create elements X ∈ Pi3 such that Tr3,2 X = X and v3 (X) = v3 (X) − b3 . Note the elements (σ 2 + 1)α and µ (from each case) have expressions i/4 in terms of our OT -basis for P1 . These expressions depend solely upon the valuations 2 of (σ + 1)α and µ. i/2 Before we move on to our result, we should say something about our basis for Pi3 /P2 . i/2 Since OT [σ]/ σ 4 + 1 is a principal ideal domain, Pi3 /P2 is free over OT [σ]/ σ 4 + 1 of

84

G. GRIFFITH ELDER

rank e0 . Given elements of K2 with valuation v2 listed in (3.5) we may use Lem 2.1(2) to ﬁnd elements, ρ ∈ Pi3 , whose images under the trace, Tr3,2 , lie one–to–one correspondence (via valuation) with (3.5). Refer to this set of elements in Pi3 as S. One can check b1 + (i + i/2 b3 + 1)/2 > 4e0 + i/2. Therefore (σ − 1)Tr3,2 Pi3 ⊆ 2P2 . Since Tr3,2 S is an OT -basis i/2 i/2 for Tr3,2 Pi3 ⊆ 2P2 and σ acts trivially upon Tr3,2 Pi3 ⊆ 2P2 we may use Lemma 2.2 i/2 and extend S to an OT [G]/ σ 4 + 1-basis for Pi3 /P2 . At this point we may put the preceding discussion together with our work in §2.2.3 (that i/2 determines the structure of P2 ) and determine the Galois module structure of Pi3 . We i/2 need to express the image of S under the trace, Tr3,2 , in terms of our OT [G]-basis for P2 . This is the same as a determination of the expression (in terms of Galois generators of i/2q P2 ) for each valuation in (3.5). First note under stable ramiﬁcation, b2 > 4e0 −2b1 so the i/2 structure of P2 is determined by the basis listed as Case B in §2.2.3. However it is more convenient for us to use the basis listed as Case A in Appendix B. To translate between the two bases, note in the elements α, (σ + 1)α, α, (σ + 1)α from §2.2.3 are referred to as α, ρ, (σ 2 +1)α, (σ+1)(σ 2 +1)α in §3.1 and then in Appendix B. So row B(1) in §2.2.3 corresponds with a pair of rows A(7) and A(8) in Appendix B. Moreover B(2) corresponds to rows A(1) and A(2), B(3) corresponds to A(3) and A(4), and B(4) corresponds to A(5) and A(6). There are four types of expression with valuation listed in (3.5). If the valuation a satisﬁes a − (b2 − 2b1 ) ≡ 0 mod 4 then a is the valuation of a Galois translate ρ where the diﬀerence i/4 between (σ ± 1)α and ρ is an element (σ + 1)µ ∈ K0 where µ is in the basis for P1 . Note each such a corresponds with the appearance of I2 in the OT [G] decomposition of Pi3 . Counting such a one ﬁnds the same count as in A(8). Note therefore A(7) counts the number of I that are not mapped to under the trace, Tr3,2 , from Pi3 . Each valuation a satisfying a ≡ 0 mod 4 is the valuation of (σ 2 + 1)α = (σ + 1)µ for some i/2 i/4 i/8 α in the basis for P2 and µ in the basis for P1 P0 . So each such a, corresponds with the appearance of an H2 . A count of such a equals the count in A(2). Note A(1) counts the number of H not interacted with. Each valuation a satisfying a − (b2 − 2b1 ) ≡ 2 mod 4 is the valuation of a Galois translate ρ where the diﬀerence between (σ ± 1)α and ρ is an element i/4 i/8 2µ ∈ P1 where (σ + 1)µ is in the basis for P0 . Each such a, therefore corresponds with the appearance of an M1 . The count of such a is the same as the count for A(4). The number of M that appear in Pi3 is the same as the count for A(3). Finally each valuation a satisfying a ≡ 2 mod 4 is the valuation of (σ 2 + 1)α = 2µ for some α in the basis for i/2 i/8 P2 . Also (σ + 1)µ is in the basis for P0 , so each such a, therefore corresponds with the appearance of an L3 . The count of such a is the same as the count for A(6). Again, A(5) counts the number of L in Pi3 . Note the structure of Pi3 under stable ramiﬁcation is consistent with the structure of Pi3 under unstable ramiﬁcation so long as b2 > 4e0 − 4b1 /3. Appendix A. The Modules In this section we introduce twenty–three indecomposable Z2 [C8 ]-modules. It is left to the interested reader to translate our notation into Yakovlev’s [?]. Irreducibles: Four of the Z2 [C8 ]-modules are irreducible: R0 , R1 , R2 , and R3 where Rn := Z2 [ζ2n ], ζ2n denotes a primitive 2n root of unity, and σ the generator of C8 acts via multiplication by ζ2n . The other nineteen modules are ‘compounds’. They are organized according to ﬁxed part – those ﬁxed by σ 2 are listed ﬁrst, followed by those ﬁxed by σ 4 , etc.

GALOIS STRUCTURE

85

Z2 [C2 ]-modules: Besides the two irreducibles R0 , R1 , the group ring Z2 [σ]/ σ 2 is the only other indecomposable module that is ﬁxed by σ 2 . Notation for ‘compounds’: The group ring, Z2 [σ]/ σ 2 , is made up of two irreducibles. To make the relationships between irreducibles and their ‘compounds’ explicit, we will use diagrams like R1 → 1 ∈ R0 (instead of Z2 [σ]/ σ 2 ). These diagrams are to be interpreted as follows: The number of Z2 [σ]-generators is the number of irreducible modules that appear in the diagram. For example, R1 → 1 ∈ R0 means two generators. Let us call them c and d. (Think: c generates R1 while d generates R0 .) Relations determine the module. If there is no ‘arrow’ leaving an irreducible Ri , then the trace Φ2i (σ) maps the generator to zero. So Φ20 (σ)d = 0. Note Φ2i (x) denotes the cyclotomic polynomial and x8 − 1 = Φ20 (x) · Φ21 (x) · Φ22 (x) · Φ23 (x). If there is an ‘arrow’ leaving an irreducible Ri (pointing to an element), then the trace Φ2i (σ) maps the generator to that element. In this case Φ21 (σ)c = 1 · d. Z2 [C4 ]-modules: There are three indecomposable modules ﬁxed by σ 4 (yet not ﬁxed by σ 2 ). Notation for two other decomposable modules is included as it will be needed to describe certain modules later (those not ﬁxed by σ 4 ). For three (of these ﬁve), the submodule ﬁxed by σ 2 is the group ring Z2 [σ]/ σ 2 (note how their diagams include R1 → 1 ∈ R0 ): (G) : R2 → 1 ∈ R1 → 1 ∈ R0 ,

R2 (H) : , R1 → 1 ∈ R0

(I) : R2 ⊕ (R1 → 1 ∈ R0 ).

Denote the three generators by b, c, d. (Think: generating R2 , R1 , R0 , respectively.) Recall (σ − 1)d = 0 while (σ + 1)c = d. In G, we have Φ22 (σ)b = 1 · c. So G is the group ring Z2 [σ]/ σ 4 . In H, we have Φ22 (σ)b = 1 · d. While in I, Φ22 (σ)b = 0. For two (of these ﬁve), the submodule ﬁxed by σ 2 is the maximal order of Z2 [σ]/ σ 2 (note how R1 ⊕ R0 appears): (L) : R2 →

1 ∈ R1 ⊕ , 1 ∈ R0

(M) : R2 ⊕ R1 ⊕ R0 .

Denote the three generators by b, c, d where (σ − 1)d = 0 and (σ + 1)c = 0. In L, we have Φ22 (σ)b = 1 · c + 1 · d. In M, we have Φ22 (σ)b = 0. So M is the maximal order of Z2 [σ]/ σ 4 . Z2 [C8 ]-modules: The remaining ﬁfteen indecomposable modules can now be listed. They are collected according to submodule ﬁxed by σ 4 . Fixed part G. R3

R2 → 1 ∈ R1 → 1 ∈ R0 R3 (G2 ) : R3 → λ ∈ R2 → 1 ∈ R1 → 1 ∈ R0 (G4 ) : R2 → 1 ∈ R1 → 1 ∈ R0 (G1 ) : R3 → 1 ∈ R2 → 1 ∈ R1 → 1 ∈ R0 (G3 ) :

Call the generators a, b, c, d, where the Z2 [σ]-relations among b, c, d are as in G. In G1 , we have Φ23 (σ)a = 1 · b. So G1 is the group ring Z2 [σ]. In G2 , we have Φ23 (σ)a = λ · b where λ = σ − 1. In G3 , Φ23 (σ)a = 1 · c. In G4 , Φ23 (σ)a = 1 · d.

86

G. GRIFFITH ELDER

Fixed part H. (H1 ) : R3 →

R3 λ ∈ R2 ⊕ (H2 ) : R2 → 1 ∈ R0 1 ∈ R1 → 1 ∈ R0 R1

Call the generators a, b, c, d, where the Z2 [σ]-relationships among b, c, d are as in H. In H1 , Φ23 (σ)a = λ · 1 · b + 1 · c. In H2 , Φ23 (σ)a = d. Fixed part I. (I1 ) : R3 →

1 ∈ R2 1 ∈ R2 ⊕ (I2 ) : R3 → ⊕ 1 ∈ R1 → 1 ∈ R0 R1 → 1 ∈ R0

Each module is generated by a, b, c, d, where the Z2 [σ]-relationships among b, c, d are as in I. In I1 , Φ23 (σ)a = 1 · b + 1 · c. In I2 , Φ23 (σ)a = 1 · b + 1 · d. Fixed part L or M. R3 1 ∈ R1 1 ∈ R1 (L3 ) : (L1 ) : R3 → 1 ∈ R2 → ⊕ 1 ∈ R0 R2 → ⊕ 1 ∈ R0 λ ∈ R2 ⊕ 1 ∈ R1 (L2 ) : R3 → λ ∈ R2 → ⊕ (M1 ) : R3 → 1 ∈ R1 1 ∈ R0 ⊕ 1 ∈ R0

The generators are a, b, c, d, where the Z2 [σ]-relationships among b, c, d are as in L or M respectively. In L1 , Φ23 (σ)a = b. In L2 , Φ23 (σ)a = λ · b. In L3 , Φ23 (σ)a = 1 · c + 1 · d. In M1 , Φ23 (σ)a = 1 · b + 1 · c + 1 · d. Hybrids of H1 . The next three modules result from the linking of an H1 with either another R3 , or with a G, or with a L.

(H1,2 ) :

R3 →

R3

1 ∈ R1 → 1 ∈ R0 ⊕ λ ∈ R2

This module is generated by a1 , a2 , b, c, d with the Z2 [σ]-relationships among b, c, d as in H, while Φ23 (σ)a1 = λ · b + 1 · c and Φ23 (σ)a2 = d. If Φ23 (σ)a1 = 0, H2 would decompose oﬀ. If Φ23 (σ)a2 = 0, H1 would decompose oﬀ. It is a mixture of H1 and H2 , hence the name. (H1 G) :

R3 →

1 ∈ R1 → 1 ∈ R0 ⊕ λ ∈ R2 → ⊕

R2 →

1 ∈ R1 → 1 ∈ R0

GALOIS STRUCTURE

87

This module is generated by a1 , b1 , c1 , d1 and b2 , c2 , d2 . The Z2 [σ]-relationships among b2 , c2 , d2 are as in G. The Z2 [σ]-relationships among a1 , c1 , d1 are as in H1 with (σ 2 + 1)b1 = 1 · d1 + 1 · d 2 . R3 → (H1 L) :

1 ∈ R1 → 1 ∈ R0 ⊕ λ ∈ R2 → ⊕ 1 ∈ R0 R2 → ⊕ 1 ∈ R1

This module is generated by a1 , b1 , c1 , d1 and b2 , c2 , d2 . The Z2 [σ]-relationships among b2 , c2 , d2 are as in L. The Z2 [σ]-relationships among a1 , c1 , d1 are as in H1 with (σ 2 + 1)b1 = 1 · d1 + (1 · c2 + 1 · d2 ).

Appendix B. The Bases by Case, A through H From §3.4, we inherit sequences of elements ordered in terms of increasing valuation (for Case A, we have . . . ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α, 4α, (σ + 1)(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2ρ, . . .). Following §2.2.3, we are interested in those elements ‘in view’ (i.e. with valuation in i, i + 1, . . . , i + v3 (2) − 1). As we vary m the ‘view’ changes. Indeed, for each case, there are eight views (eight sets). They are listed below. Recall from §2.2.3 it is easy to determine the subscripts m associated with a particular ‘view’. For example, the elements in A(2) appear for i ≤ v3 (σ + 1)(σ 2 + 1)α and v3 (σ + 1)(σ 2 + 1)α ≤ 8e0 + i − 1. In other words, (i + b3 − 4b1 − 4b2 )/8 ≤ m ≤ (i + 8e0 − 4b1 − 4b2 )/8 − 1.

Case A (1)

2

ρ, 2ρ, (σ + 1)α,

2(σ 2

+ 1)α, 2α, 4α, (σ + 1)(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α

(σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α, 4α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α, 4α (3) 2 1 (4) 2α, (σ + 1)(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α 2 1 (5) α, 2α, (σ + 1)(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α 2 1 (6) (σ 2 + 1)α, α, 2α, (σ + 1)(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α 2 1 1 2 (σ + 1)α, (σ 2 + 1)α, α, 2α, (σ + 1)(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ (7) 2 2 1 1 (8) ρ, (σ 2 + 1)α, (σ 2 + 1)α, α, 2α, (σ + 1)(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, ρ 2 2 (2)

88

G. GRIFFITH ELDER Case B 2

+ 1)α, 2α, (σ + 1)(σ 2 + 1)α, 4α, 2(σ + 1)(σ 2 + 1)α

(1)

ρ, 2ρ, (σ + 1)α,

(2)

(σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, 4α

(3)

2α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, 2α 2 1 α, (σ + 1)(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α 2 1 (σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ, (σ 2 + 1)α 2 1 1 2 (σ + 1)α, (σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, ρ, 2ρ 2 2 1 2 1 2 ρ, (σ + 1)α, (σ + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, ρ 2 2

(4) (5) (6) (7) (8)

2(σ 2

Case C (1)

ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2(σ + 1)(σ 2 + 1)α, 4α

(2)

2α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2(σ + 1)(σ 2 + 1)α

(3)

(σ + 1)(σ 2 + 1)α, 2α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α

(4)

α, (σ + 1)(σ 2 + 1)α, 2α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ, 2ρ, (σ 2 + 1)α, 2(σ 2 + 1)α 2 1 (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ, 2ρ, (σ 2 + 1)α 2 1 2 1 (σ + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ, 2ρ 2 2 1 2 1 ρ, (σ + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ 2 2

(5) (6) (7) (8)

Case D (1)

ρ, (σ + 1)α, 2ρ, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2(σ + 1)(σ 2 + 1)α, 4α

(2)

2α, ρ, (σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2(σ + 1)(σ 2 + 1)α

(3)

(σ + 1)(σ 2 + 1)α, 2α, ρ, (σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α

(4)

α, (σ + 1)(σ 2 + 1)α, 2α, ρ, (σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ, (σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α 2 1 (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ, (σ 2 + 1)α, 2ρ 2 1 ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ, (σ 2 + 1)α 2 1 2 1 (σ + 1)α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, (σ + 1)(σ 2 + 1)α, 2α, ρ 2 2

(5) (6) (7) (8)

2

Case E 2

(1)

2α, 2ρ, (σ + 1)α, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α, 2ρ

(2)

ρ, 2α, 2ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α

(3)

α, ρ, 2α, 2ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α

GALOIS STRUCTURE (4)

(σ + 1)(σ 2 + 1)α, α, ρ, 2α, 2ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α

(5)

(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α, 2ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α, 2ρ, (σ 2 + 1)α 2 1 1 2 (σ + 1)α, (σ + 1)(σ 2 + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α, 2ρ 2 2 1 2 1 ρ, (σ + 1)α, (σ + 1)(σ 2 + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α 2 2 Case F

(6) (7) (8)

(1)

2α, (σ 2 + 1)α, 2ρ, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α, 2ρ

(2)

ρ, 2α, (σ 2 + 1)α, 2ρ, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α

(3)

α, ρ, 2α, (σ 2 + 1)α, 2ρ, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α

(4)

(σ + 1)(σ 2 + 1)α, α, ρ, 2α, (σ 2 + 1)α, 2ρ, (σ + 1)(σ 2 + 1)α, 2(σ 2 + 1)α

(5)

(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α, (σ 2 + 1)α, 2ρ, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α, (σ 2 + 1)α, 2ρ 2 1 ρ, (σ + 1)(σ 2 + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α, (σ 2 + 1)α 2 1 2 1 (σ + 1)α, ρ, (σ + 1)(σ 2 + 1)α, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, 2α 2 2 Case G

(6) (7) (8)

(1)

(σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α, 2ρ

(2)

ρ, (σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α

(3)

α, ρ, (σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α

(4)

(σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, 2ρ, 2(σ 2 + 1)α

(5)

(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α, 2ρ

(6)

ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, 2α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, 2α 2 1 α, (σ + 1)(σ 2 + 1)α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α 2 Case H

(7) (8)

(1)

(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2ρ, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α, 2ρ

(2)

ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2ρ, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α, 2α

(3)

α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2ρ, 2(σ 2 + 1)α, 2(σ + 1)(σ 2 + 1)α

(4)

(σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2ρ, 2(σ 2 + 1)α

(5)

(σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α, 2ρ

(6)

ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, 2α

(7)

α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α 1 (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α, (σ + 1)(σ 2 + 1)α, α, ρ, (σ 2 + 1)α 2

(8)

89

Department of Mathematics, University of Nebraska at Omaha, Omaha, Nebraska 68132-0243 E-mail address: [email protected]

AN INTRODUCTION TO NONCOMMUTATIVE DEFORMATIONS OF MODULES EIVIND ERIKSEN

Abstract. Let k be an algebraically closed (commutative) field, let A be an associative k-algebra, and let M = {M1 , . . . , Mp } be a finite family of left A-modules. We study the simultaneous formal deformations of this family, described by the noncommutative deformation functor DefM : ap → Sets introduced in Laudal [8]. In particular, we prove that this deformation functor has a pro-representing hull, and describe how to calculate this hull using the cohomology groups Extn A (Mi , Mj ) and their matric Massey products.

Introduction In this paper, I shall give an elementary introduction to the noncommutative deformation theory for modules, due to Laudal. This theory, which generalizes the classical deformation theory for modules, was introduced by Laudal in [8]. Earlier versions of this material appeared in the preprints Laudal [3], [4], [5], [6], [7]. This noncommutative deformation theory has several applications: In the paper Laudal [8], Laudal used it to construct algebras with a prescribed set of simple modules, and also to study the moduli space of iterated extensions of modules. In the preprint Laudal [7], he also showed that this theory is a useful tool in the study of algebras, and in establishing a noncommutative algebraic geometry. These applications are an important part of the motivation for the noncommutative deformation theory. But we shall not go into the details of these applications in this elementary introduction. Instead, we refer to the papers and preprints of Laudal mentioned above for applications and further developments of the theory. Throughout this paper, we shall ﬁx the following notations: Let k be an algebraically closed (commutative) ﬁeld, let A be an associative k-algebra, and let M = {M1 , . . . , Mp } be a ﬁnite family of left A-modules. Notice that this notation diﬀers from Laudal’s: While Laudal considers families of right modules in all his paper, I consider families of left modules. Of course, the diﬀerence is only in the appearance — the resulting theories are obviously equivalent. We shall present a noncommutative deformation functor DefM : ap → Sets, which describes the simultaneous formal deformations of the family M of left A-modules. Furthermore, we shall prove that this deformation functor has a pro-representable hull (H, ξ) when the family M satisfy a certain ﬁniteness condition. We shall also describe a method for ﬁnding the pro-representable hull explicitly. In section 1, we describe the category ap . It is a full sub-category of the category Ap of p-pointed k-algebras. The objects of Ap are the k-algebras R equipped with k-algebra homomorphisms k p → R → k p , such that the composition k p → k p is the identify. For any

This research has been supported by a Marie Curie Fellowship of the European Community programme “Improving Human Research Potential and the Socio-economic Knowledge Base” under contract number HPMF-CT-2000-01099.

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such object, R = (Rij ) is a k-algebra of p × p matrices. The radical of this object is the ideal I(R) = ker(R → k p ) ⊆ R. The category ap is the full sub-category of Ap consisting of objects such that R is Artinian and complete in the I(R)-adic topology. In section 2, we describe the noncommutative deformation functor associated to the family M of left A-modules, DefM : ap → Sets It is constructed in the following way: Let R be an object of ap , and consider the vector space MR = (Mi ⊗k Rij ), equipped with the natural right R-module structure induced by the multiplication in R. A deformation of M to R consists of the following data: • A left A-module structure on MR making MR a left A ⊗k Rop -module, • Isomorphisms ηi : MR ⊗R ki → Mi of left A-modules for 1 ≤ i ≤ p. The set of equivalence classes of such deformations is denoted DefM (R), and this deﬁnes the covariant functor DefM . Notice that the fact that MR ∼ = (Mi ⊗k Rij ) as right R-modules replaces the ﬂatness condition in classical deformation theory. If p = 1 and R is commutative, the above condition is of course equivalent to the ﬂatness condition, so the noncommutative deformation functor generalizes the classical one. In section 3, we look at noncommutative deformations from the point of view of resolutions. Let R be any object of ap . An M-free module over R is a left A ⊗k Rop -module F of the form F = (Li ⊗ Rij ), where L1 , . . . , Lp are free left A-modules. M-free complexes and M-free resolutions are deﬁned similarly. Let us ﬁx a free resolution of Mi the form 0 ← Mi ← L0,i ← · · · ← Lm,i ← · · · for 1 ≤ i ≤ p. We prove that there is a bijective correspondence between deformations of M to R and complexes of M-free modules over R of the form (L0,i ⊗k Rij ) ← · · · ← (Lm,i ⊗k Rij ) ← · · · In fact, each such complex of M-free modules is an M-free resolution of the corresponding deformation MR of M to R. In section 4, we recall some general facts about pointed functors and their representability. In section 5, we consider the special case of the noncommutative deformation functor DefM . From this point in the text, we assume that the family M satisfy the ﬁniteness condition (FC)

dimk ExtnA (Mi , Mj ) is ﬁnite for 1 ≤ i, j ≤ p, n = 1, 2.

When this condition holds, we deﬁne T1 , T2 to be the formal matrix rings (in the sense of section 1) given by the families of k-vector spaces Vij = ExtnA (Mj , Mi )∗ for n = 1, 2.

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Assuming condition (FC), we show the following theorem of Laudal, which generalizes the corresponding theorem for the classical deformation functor: ˆ T2 k p Theorem 0.1. There exists an obstruction morphism o : T2 → T1 , such that H = T1 ⊗ is a pro-representable hull for the noncommutative deformation functor DefM : ap → Sets. In the rest of the paper, we show how to construct the hull H explicitly, which can be accomplished by using matric Massey products. In section 6, we introduce the immediately deﬁned matric Massey products. In section 7, we deﬁne the matric Massey products in general, and show that the hull H of the noncommutative deformation functor DefM is determined by the vector spaces ExtnA (Mi , Mj ) for n = 1, 2 and 1 ≤ i, j ≤ p and their matric Massey products. We also describe a general method for calculating the hull H in concrete terms. In appendix A, we describe the Yoneda and Hochschild representations of the cohomology groups ExtnA (Mi , Mj ). In this paper, we have chosen to express the matric Massey products using the Yoneda representation and M-free resolutions. It is also possible to express the matric Massey products using the Hochschild representation, see for instance Laudal [8]. 1. Categories of pointed algebras Let p be a ﬁxed natural number, and consider the ring k p . This ring has a natural kalgebra structure given by the map α → (α, . . . , α) for α ∈ k. Let pri : k p → k p be the i’th projection, and consider the ideal ki = pri (k p ) ⊆ k p as a k p -module for 1 ≤ i ≤ p. Clearly, k p is an Artinian k-algebra and {k1 , . . . , kp } is the full set of isomorphism classes of simple k p -modules, each of them of dimension 1 over k. This simple example will serve as a model for the p-pointed algebras that we shall consider in this section. A p-pointed k-algebra is a triple (R, f, g), where R is an associative ring and f : k p → R, g : R → k p are ring homomorphisms such that g ◦ f = id. A morphism u : (R, f, g) → (R , f , g ) of p-pointed k-algebras is a ring homomorphism u : R → R such that the natural diagrams commute (that is, such that u ◦ f = f and g ◦ u = g). We shall denote the category of p-pointed k-algebras by Ap . Notice that if (R, f, g) is an object of Ap , then f is injective and g is surjective, and we shall identify k p with its image in R. We often write R for the object (R, f, g) to simplify notation. Let (R, f, g) be an object in Ap . We deﬁne the radical of R to be I(R) = ker(g), which is an ideal in R. Furthermore, we denote by J(R) the Jacobson radical of R J(R) = {x ∈ R : xM = 0 for all simple left R-modules M }, which is also an ideal in R. We shall write I, J for the radicals I(R), J(R) when there is no danger of confusion. Notice that the Jacobson radical J depends only on the ring R, while the radical I depends on the structural morphism g as well. For all objects R in Ap , we have an inclusion J(R) ⊆ I(R): We have J(k p ) = 0 since p k is semi-simple, and g(J(R)) ⊆ J(k p ) = 0 since g : R → k p is a surjection. In general, we know that R and R/J(R) have the same simple left modules. So if we consider ki as a left R-module via the morphism g : R → k p for 1 ≤ i ≤ p, we see that {k1 , . . . , kp } is contained in the set of isomorphism classes of simple left R-modules, and the equality J(R) = I(R) holds if and only if {k1 , . . . , kp } is the full set of isomorphism classes of simple left R-modules. Equivalently, the equality I(R) = J(R) holds if and only if there are exactly p isomorphism classes of simple left R-modules.

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It is therefore clear that the equality I(R) = J(R) does not hold in general: It is easy to ﬁnd examples where R has ‘too many’ simple modules. For instance, consider R = k[x]/(x − x2 ) with the natural k-algebra structure f : k → R and let g : R → k be given by x → 0. Then R is an object of A1 , but J(R) = I(R) because R has two non-isomorphic simple left R-modules (given by x → 0 and x → 1). Let ei be the idempotent (0, 0, . . . , 1, . . . , 0) ∈ k p for 1 ≤ i ≤ p. Notice that ei ej = 0 if i = j, and that e1 + · · · + ep = 1. For any object R in Ap , we identify {e1 , . . . , ep } with idempotents in R via the inclusion k p → R. Denote by Rij the k-linear sub-space ei Rej ⊆ R. We immediately see, using the properties of the idempotents, that the following relations hold for 1 ≤ i, j, l, m ≤ p: (1) Rij Rlm ⊆ δjl Rim , (2) Rij ∩ Rlm = 0 if (i, j) = (l, m), (3) Rij = R. In particular, we have that R = ⊕Rij , so every element r ∈ R may be written in matrix form r = (rij ) with rij ∈ Rij for 1 ≤ i, j ≤ p. Furthermore, elements of R multiply as matrices when we write them in this form. It is therefore reasonable to call an object R in Ap a matrix ring, and to write it R = (Rij ). Notice that Rii is an associative ring (with identity ei ), and that Rij is a (unitary) Rii − Rjj bimodule for 1 ≤ i, j ≤ p. For any ideal K ⊆ R, we see that ei Kej = K ∩ Rij , and we shall denote this k-linear subspace Kij for 1 ≤ i, j ≤ p. Since K = ⊕ Kij , we write K = (Kij ). Let R be an object of Ap , so R = (Rij ) is a matrix ring in the above sense. The following standard result gives useful information on when R is an Artinian or Noetherian ring: Proposition 1.1. Let R = (Rij ) be an object in Ap . Then R is Noetherian (Artinian) if and only if the following conditions hold: i) Rii is Noetherian (Artinian) for 1 ≤ i ≤ p, ii) Rij is a Noetherian (Artinian) left Rii -module and a Noetherian (Artinian) right Rjj module for 1 ≤ i = j ≤ p. We recall that a ﬁnitely generated, associative k-algebra is not necessarily Noetherian. That is, Hilbert’s basis theorem does not hold for associative rings. For a counter-example, let R = k{x1 , . . . , xn } be the free associative k-algebra on n generators. It is well-known that R is Noetherian only if n = 1. However, we know from the Hopkins-Levitzki theorem that an associative Artinian ring is Noetherian. A k-algebra R of ﬁnite dimension as vector space over k is Artinian. This is clear, since every one-sided ideal is a vector space over k of ﬁnite dimension. We have a converse statement under the following conditions: Lemma 1.2. Let R be an object of Ap . If R is Artinian and I(R) is nilpotent, then R has finite dimension as a vector space over k. Proof. We write I = I(R). Since R is Artinian and therefore Noetherian, I m is ﬁnitely generated as a left R-module for all m. Consequently, I m /I m+1 is a ﬁnitely generated R/I-module for all m, and hence has ﬁnite k-dimension. But I n = 0 for some n, so I m has ﬁnite kdimension for all m ≥ 0. In particular, R has ﬁnite dimension as a vector space over k. We deﬁne the category ap to be the full sub-category of Ap consisting of objects R in Ap such that R is Artinian and I(R) = J(R). The condition I(R) = J(R) might equivalently be replaced by the condition that I(R) is a nilpotent ideal, since the Jacobson radical is the

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largest nilpotent ideal in an Artinian ring. So by lemma 1.2, all objects R in ap have ﬁnite k-dimension. Since R is Artinian, the condition that I(R) is nilpotent is also equivalent to ∩ I(R)n = 0. Finally, there is a geometric interpretation of the condition I(R) = J(R): By the comment earlier in this section, I(R) = J(R) if and only if {k1 , . . . , kp } is the full set of isomorphism classes of simple left R-modules (or equivalently, that the number of such isomorphism classes is exactly p). Lemma 1.3. Let R be an associative ring. Then there exists morphisms f : k p → R and g : R → k p making (R, f, g) an object of ap if and only if R is an Artinian k-algebra with exactly p isomorphism classes of simple left R-modules, each of them of dimension 1 over k. Proof. One implication follows from the comments above. For the other, assume that R is Artinian with the prescribed isomorphism classes of simple left R-modules. This deﬁnes a morphism g : R → k p . Clearly, I = ker(g) = J(R) by the comments above. So it is enough to lift the idempotents {e1 , . . . , ep } of k p to idempotents {r1 , . . . , rp } in R such that r1 + · · · + rp = 1 and ri rj = 0 when i = j. But R is Artinian and therefore I = J(R) is nilpotent, so this is clearly possible. Let R be an object in Ap with radical I = I(R). Then the I-adic ﬁltration deﬁnes a topology on R compatible with the ring operations, and we shall always consider R a topological ring in this way. We say that the topology on R is Hausdorﬀ (or separated) if and only if ∩I n = 0. ˆ of R and a canonical morphism For all objects R in Ap , there is an I-adic completion R ˆ is deﬁned by the projective limit ˆ in Ap . The I-adic completion R R→R ˆ = lim R/I n , R ←

ˆ is the natural one induced by this projective limit. Notice that and the morphism R → R the kernel of this morphism is ∩I n . We say that R is complete (or separated complete) if ˆ is an isomorphism in Ap . In particular, this implies that the the natural morphism R → R morphism is injective, so R is Hausdorﬀ (or separated). This gives a new characterization of the category ap : Lemma 1.4. The category ap is the full sub-category of Ap consisting of objects such that R is Artinian and I-adic complete. We deﬁne the pro-category ˆ ap of ap to be the full sub-category of Ap consisting of objects such that R is complete and R/I(R)n belongs to ap for all n ≥ 1. It is clear that we have an inclusion of (full) sub-catgories ap ⊆ ˆ ap . Let R be an object in ˆ ap with radical I = I(R). To ﬁx notation, we write grn (R) = I n /I n+1 for n ≥ 0 (with I 0 = R). We also write gr R = ⊕ grn (R), this is the graded ring associated to the I-adic ﬁltration of R. The tangent space of R is deﬁned to be the k-linear space dual to gr1 (R), tR = Homk (I/I 2 , k) = (I/I 2 )∗ , which is clearly of ﬁnite dimension over k. In particular, we have (tR )∗ ∼ = I/I 2 . Let u : R → S be a morphism in ˆ ap . As usual, we consider R and S with the I-adic ﬁltrations, where I is I(R) and I(S) respectively. Since u preserves these ﬁltrations, it induces a morphism of graded rings gr(u) : gr R → gr S. This morphism is homogeneous of degree 0, so u also induces morphisms of k-vector spaces grn (u) : grn (R) → grn (S) for all

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n ≥ 0. In particular, we have a morphism of k-vector spaces gr1 (u) : gr1 (R) → gr1 (S), and a dual morphism tu : tS → tR . Proposition 1.5. Let u : R → S be a morphism in ˆ ap . Then u is a surjection if and only if gr1 (u) is a surjection. Furthermore, u is injective if gr(u) is injective. Proof. If u is surjective, then clearly gr1 (u) is also surjective. To prove the other implication, let us consider the map gr(u) : gr(R) → gr(S). Since gr S is generated by the elements in gr1 S as an algebra, it follows that if gr1 (u) is surjective, then gr(u) is also surjective. From Bourbaki [1], chapter III, §2, no. 8, corollary 1 and 2, we have that u is surjective (injective) if gr(u) is surjective (injective), and the result follows. Let n be any natural number. We deﬁne the category ap (n) to be the full sub-category of ap consisting of objects R in ap such that I(R)n = 0. Notice that ap (n) ⊆ ap (n + 1) for all n ≥ 1. Furthermore, each object R in ap belongs to a sub-category ap (n) for some integer n. Let u : R → S be a morphism in ap , and denote by K = ker(u) the kernel of u. We say that u is a small morphism if we have I(R) · K = K · I(R) = 0. We prove the following important fact about small surjections: Lemma 1.6. Let u : R → S be a surjection in ap . Then u can be factored into a finite number of small surjections. Proof. Let I = I(R), then I n K = 0 for some n ≥ 0. Consider the surjection uq : R/I q K → R/I q−1 K for 1 ≤ q ≤ n. Clearly I(R/I q K) ker(uq ) = 0 for all q. Moreover, u1 ◦ · · · ◦ un = u when u1 : R/IK → R/K is considered as a morphism onto S ∼ = R/K. It is therefore enough to prove the lemma for a surjection u : R → S with IK = 0. In this situation, KI n = 0 for some n ≥ 0. Now consider the surjection vq : R/KI q → R/KI q−1 for 1 ≤ q ≤ n. Clearly, vq is a small surjection for all q. Moreover, u = v1 ◦ · · · ◦ vn when v1 : R/KI → R/K is considered as a morphism onto S ∼ = R/K. It follows that u can be factorized in a ﬁnite number of small surjections in ap . We conclude this section with an important family of examples: Let Vij be a ﬁnite dimensional k-vector space for 1 ≤ i, j ≤ p, with dimk Vij = dij . Let furthermore {rij (l) : 1 ≤ l ≤ dij } be a basis of Vij for 1 ≤ i, j ≤ p (or simply {rij } if dij = 1). We deﬁne the free matrix ring R = R({Vij }) deﬁned by the vector spaces Vij in the following way: We say that a monomial in R of type (i, j) and degree n is an expression of the form ri0 i1 (l1 )ri1 i2 (l2 ) . . . rin−1 in (ln ) with i0 = i, in = j. To these, we add the monomials ei for 1 ≤ i ≤ p, which we consider to be of type (i, i) and degree 0. We deﬁne R to be the k-linear space generated by all monomials in R, with the obvious multiplication: If M is a monomial of type (i, j), and M is a monomial of type (l, m), then M M = 0 if j = l, and M M is the monomial obtained by juxtapositioning M and M (possibly after having erased unnecessary ei ’s) if j = l. We see that (R, f, g) is an object of the category Ap , where f, g are the obvious maps k p → R → k p . In fact, Rij is the k-linear subspace generated by monomials in R of type (i, j), and the ideal I = I(R) is the k-linear subspace generated by all monomials of positive degree. ˆ = R({V ˆ We denote by R ij }) the completion of R = R({Vij }), and call this the formal ˆ ij is an inﬁnite matrix ring deﬁned by the vector spaces Vij . Explicitly, every element in R k-linear sum of monomials in R of type (i, j). Let I = I(R), then we have that Rn = R/I n ∼ =

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ˆ R) ˆ n belongs to ap for n ≥ 1: Clearly, Rn has ﬁnite dimension as k-vector space, so Rn R/I( ˆ clearly is complete, it is Artinian, and I(Rn ) = I/I n , so the radical is nilpotent. Since R ˆ belongs to ˆ follows that R ap . ˆ is Noetherian in Notice that neither the free matrix ring R nor the formal matrix ring R general. For a counter-example, it is enough to consider the case when p = 1 and d11 = 2, or the case when p = 2 and d11 = d12 = d21 = 1, d22 = 0. In the ﬁrst case, R ∼ = k{x, y}, which we know is not Noetherian. In the second case, we have that R11 = k{r11 , r12 r21 } ∼ = k{x, y}, which again is not Noetherian. So by proposition 1.1, R is not Noetherian in this case either. ˆ is not Noetherian in any of the two cases. A similar argument shows that R 2. Noncommutative deformations of modules We recall that k is an algebraically closed (commutative) ﬁeld, A is an associative k-algebra, and M = {M1 , . . . , Mp } is a ﬁnite family of left A-modules. In this section, we shall deﬁne the noncommutative deformation functor DefM : ap → Sets describing the simultaneous formal deformations of the family M. Let R be an object of ap . A lifting of the family M of left A-modules to R is a left A ⊗k Rop -module MR , together with isomorphisms ηi : MR ⊗R ki → Mi of left A-modules for 1 ≤ i ≤ p, such that MR ∼ = (Mi ⊗k Rij ) as right R-modules. We remark that a left A⊗k Rop -module is the same as an A-R bimodule such that the left and right k-vector space structures coincide. Furthermore, the notation (Mi ⊗k Rij ) refers to the k-vector space (Mi ⊗k Rij ) = ⊕ (Mi ⊗k Rij ) i,j

with the natural right R-module structure coming from the multiplication in R. The condition that MR ∼ = (Mi ⊗k Rij ) as right R-modules generalizes the ﬂatness condition in commutative deformation theory. Let MR , MR be two liftings of M to R. We say that these two liftings are equivalent if there exists an isomorphism τ : MR → MR of left A ⊗k Rop -modules such that the natural diagrams commute (that is, such that ηi ◦ (τ ⊗R ki ) = ηi for 1 ≤ i ≤ p). We let DefM (R) denote the set of equivalence classes of liftings of M to R, and we refer to these equivalence classes as deformations of M to R. We shall often denote a deformation represented by (MR , ηi ) by MR to simplify notation. Let u : R → S be a morphism in ap , and let MR be a lifting of M to R, representing an element in DefM (R). We deﬁne MS = MR ⊗R S, which has a natural structure as a left A ⊗k S op -module. Since u is a morphism in ap , we have natural isomorphisms of left A-modules (MR ⊗R S) ⊗S ki ∼ = MR ⊗R ki , inducing isomorphisms of left A-modules ρi : MS ⊗S ki → Mi via ηi for 1 ≤ i ≤ p. A straight-forward calculation shows that MS together with the isomorphisms ρi for 1 ≤ i ≤ p constitutes a lifting of M to S, and furthermore that the equivalence class of this lifting is independent upon the representative of the equivalence class of MR . Hence, we obtain a map DefM (u) : DefM (R) → DefM (S), and we see that DefM : ap → Sets is a covariant functor.

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Let R = (Rij ) be an object in ap . We shall describe how one, in principle, could attempt to calculate DefM (R) explicitly: We may assume that every element of DefM (R) is represented by a lifting MR , such that MR = (Mi ⊗k Rij ) considered as a right R-module. In order to describe this lifting completely, it is enough to describe the left action of A on MR . Furthermore, it is enough to describe this action on elements of the form mi ⊗ ei with mi ∈ Mi , since we have a(mi ⊗ rij ) = (a(mi ⊗ ei ))rij for all a ∈ A, mi ∈ Mi , rij ∈ Rij. For a ﬁxed a ∈ A, mi ∈ Mi , assume that a(mi ⊗ ei ) = (mj ⊗ rjl ) with mj ∈ Mj , rjl ∈ Rjl . Then multiplication by ei on the right gives the equality a(mi ⊗ ei ) = (mj ⊗ rji ), j

and the isomorphism ηi gives a further restriction on the left action of A, expressed by the formula (1)

a(mi ⊗ ei ) = (ami ) ⊗ ei +

mj ⊗ rji ,

j where a ∈ A, mi ∈ Mi , mj ∈ Mj , rji ∈ I(R)ji . Consequently, the set DefM (R) consists of all possible choices of left A-actions on elements of the form mi ⊗ ei , fulﬁlling condition (1) and the associativity condition, up to equivalence. Let R be any object in ap . Then the formula a(mi ⊗ ei ) = (ami ) ⊗ ei for a ∈ A, mi ∈ Mi deﬁnes a left A-module structure on (Mi ⊗ Rij ) compatible with the right R-module structure. Hence, there exists a trivial lifting MR to R for all objects R in ap , and DefM (R) is non-empty. Notice that in the case R = k p , we have I = I(R) = 0, so this trivial lifting is the only one possible. Consequently, we have DefM (k p ) = {∗}, where ∗ denotes the equivalence class of the trivial lifting. Let u : R → S be a morphism in ap , and let MS ∈ DefM (S) be a given deformation. We say that a deformation MR ∈ DefM (R) is a lifting of MS or is lying over MS if DefM (u)(MR ) = MS . Given any object R in ap and a deformation MR ∈ DefM (R), we see that MR is a lifting of the trivial deformation ∗ in DefM (k p ) in the above sense via the structural morphism g : R → k p . Hence, our notation is consistent. For another example, consider the test algebras R(α, β) for 1 ≤ α, β ≤ p, constructed in the following way: Let R be the free matrix algebra deﬁned by the k-vector spaces Vij with dimensions dα,β = 1 and dij = 0 when (i, j) = (α, β). We deﬁne R(α, β) = R/I(R)2 , which is an object in ap (2) by construction. We know that any lifting of M to R(α, β) is deﬁned by a left A-action

a(mβ ⊗ eβ ) = (amβ ) ⊗ eβ + ψ(a)(mβ ) ⊗ εα,β for all a ∈ A, mβ ∈ Mβ , where ψ : A × Mβ → Mα is a k-bilinear map and εα,β is the class of rα,β . Clearly, we must have a(mi ⊗ ei ) = (ami ) ⊗ ei for all a ∈ A, mi ∈ Mi when i = β. Moreover, ψ deﬁnes an associative A-module structure if and only if ψ ∈ Derk (A, Homk (Mβ , Mα )). In this case, we shall denote the corresponding lifting by M(ψ) ∈ DefM (R(α, β)). Given two derivations ψ, ψ , we see that M(ψ) and M(ψ ) are equivalent liftings if and only if there is a φ ∈ Homk (Mβ , Mα ) such that (ψ − ψ )(a)(mβ ) = aφ(mβ ) − φ(amβ )

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for all a ∈ A, mβ ∈ Mβ . Lemma 2.1. There is a bijective correspondence DefM (R(α, β)) ∼ = Ext1A (Mβ , Mα ) for 1 ≤ α, β ≤ p. Proof. From the deﬁnition of Hochschild cohomology (see appendix A), we see that ψ → M(ψ) induces a bijective correspondence between HH1 (A, Homk (Mβ , Mα )) and DefM (R(α, β)). Moreover, HH1 (A, Homk (Mβ , Mα )) ∼ = Ext1A (Mβ , Mα ) by proposition A.3.

3. M-free resolutions and noncommutative deformations We recall that k is an algebraically closed (commutative) ﬁeld, A is an associative kalgebra, and M = {M1 , . . . , Mp } is a ﬁnite family of left A-modules. In this section, we shall deﬁne M-free resolutions and relate them to noncommutative deformations of modules. In particular, we shall show that M-free resolutions are useful computational tools in order to study the deformation functor DefM . Let R be any object of ap . An M-free module over R is a left A ⊗k Rop -module F of the form F = (Li ⊗k Rij ), where L1 , . . . , Lp are free left A-modules, and the left A-module structure on F is the trivial one. In other words, F is the trivial lifting of a family {L1 , . . . , Lp } of free left A-modules to R. Although an M-free module over R is not free considered as a left A ⊗k Rop -module, it behaves as a free module when interpreted as a module of matrices in the correct way: Lemma 3.1. Let u : R → S be a surjection in ap , and consider a left A ⊗k Rop -module MR = (Mi ⊗k Rij ) and a left A ⊗k S op -module MS = (Mi ⊗k Sij ) such that the natural map v : MR → MS induced by u is left A-linear. If F S is any M-free module over S given by the free left A-modules L1 , . . . , Lp and fS : F S → MS is any left A ⊗k S op -linear map, then there exists a left A ⊗k Rop -linear map fR : F R → MR making the diagram MR

fR

v

MS

FR (id ⊗u)

fS

FS

commutative, where F R is the M-free module over R given by the free left A-modules L1 , . . . , Lp . Proof. Clearly, the map fS is determined by its values on Li ⊗ ei , and therefore by the corresponding left A-linear maps Lj → ⊕(Mi ⊗k Sij ). Since each left A-module Lj is projective, we can lift these maps to left A-linear maps Lj → ⊕(Mi ⊗k Rij ), and these maps determine fR . Let R be any object of ap , and let MR = (Mi ⊗k Rij ) ∈ DefM (R) be a lifting of M to R. An M-free resolution of MR is an exact sequence of left A ⊗k Rop -linear maps R 0 ← MR ← F0R ← F1R ← · · · ← Fm ← ···

NONCOMMUTATIVE DEFORMATIONS OF MODULES

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R R where Fm is an M-free module over R for m ≥ 0. So we have Fm = (Lm,i ⊗k Rij ) where Lm,i are free left A-modules for 1 ≤ i ≤ p, m ≥ 0. We shall denote the diﬀerentials by R R dR m : Fm+1 → Fm for m ≥ 0. We ﬁx a k-linear basis {rij (l) : 1 ≤ l ≤ dimk Rij } of Rij for 1 ≤ i, j ≤ p such that ei is contained in the basis of Rii for 1 ≤ i ≤ p. Consider the diﬀerential dR m in the M-free uniquely in the form resolution of MR above. Clearly, we can write dR m

dR m =

(2)

α(rij (l))m ⊗ rij (l)

i,j,l

for all m ≥ 0, where α(rij (l))m : Lm+1,j → Lm,i is a homomorphism of left A-modules for 1 ≤ i, j ≤ p, 1 ≤ l ≤ dimk Rij . In particular, the M-free resolution of MR deﬁnes a family of 1-cochains α(rij (l)) ∈ Hom1 (L∗j , L∗i ), indexed by a k-linear basis for R. From now on, we ﬁx a free resolution (L∗i , d∗i ) of Mi considered as left A-module for 1 ≤ i ≤ p. These free resolutions correspond to an M-free resolution (F∗ , d∗ ) of the trivial p deformation (Mi ⊗k (k p )ij ) ∈ Def M (k ). In fact, the M-free resolution (F∗ , d∗ ) is given by p Fm = (Lm,i ⊗k (k )ij ) and dm = dm,i ⊗ ei for m ≥ 0. We have therefore ﬁxed an M-free resolution (F∗ , d∗ ) of the trivial lifting of M to k p . R Let R be any object of ap . We say that a complex (F∗R , dR ∗ ) of M-free modules Fm = (Lm,i ⊗k Rij ) over R is a lifting of the complex (F∗ , d∗ ) if the following diagram commutes F0R

dR 0

v0

F0

F1R

dR 1

v1 d0

F1

F2R

...

v2 d1

F2

...

R where vm : Fm → Fm are the natural maps induced by R → k p .

Lemma 3.2. Let R be any object of ap , and let (F∗R , dR ∗ ) be a lifting of the complex (F∗ , d∗ ). Then we have: (1) H m (F∗R , dR ∗ ) = 0 for all m ≥ 0, (2) H 0 (F∗R , dR ∗ ) is a lifting of the family M to R. Proof. Clearly, the lemma holds for R = k p . We shall consider a small surjection u : R → S in ap and liftings of complexes (F∗U , dU ∗ ) of (F∗ , d∗ ) to U for U = R, S such that the following diagram commutes: F0R

dR 0

v0

F0S

F1R

dR 1

v1

dS 0

F1S

F2R

...

v2

dS 1

F2S

...

In this situation, we shall prove that if the conclusion of the lemma holds for S, it holds for R as well. This is clearly enough to prove the lemma. Let K = ker(u), then we clearly have ker(vm ) = (Fm,i ⊗k Kij ) with the trivial left K , then (F∗K , dK A-action for all m ≥ 0. We denote this kernel by Fm ∗ ) is a complex of

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R left A ⊗k Rop -modules, where dK ∗ is the restriction of d∗ . Moreover, it is clear that vm is 0 U U surjective for m ≥ 0. Deﬁne MU = H (F∗ , d∗ ) for U = R, S, let v : MR → MS be the induced map, and denote the kernel by MK = ker(v). Then clearly v is surjective, and we have the following commutative diagram of complexes:

0

0

0

MK

ρK

R

MR

ρ

MS

0

ρS

F1K

0 dK 1

i1

F0R

dR 0

v0

v

0

dK 0

i0

i

0

F0K

0

0

dS 0

...

i2

F1R

dR 1

v1

F0S

F2K

F2R

...

v2

F1S

0

dS 1

F2S

...

0

Clearly all columns are exact, so the diagram gives a short exact sequence of complexes. By assumption, the bottom row is exact and MS = (Mi ⊗k Sij ) is a lifting of M to S. Let us ﬁrst show that H m (F∗K , dK ∗ ) = 0 for m ≥ 1: This follows since the complex is a lifting of (F∗ , d∗ ) and because I(R)K = 0 (since u : R → S is small). The long exact sequence of cohomologies of the complexes above now implies that H m (F∗R , dR ∗ ) = 0 for all m ≥ 1 and that we have a short exact sequence 0 → H 0 (F∗K , dK ∗ ) → MR → MS = (Mi ⊗k Sij ) → 0, of left A-modules, so in particular MK ∼ = H 0 (F∗K , dK ∗ ). But since I(R)K = 0, it follows 0 K K ∼ 0 that H (F∗ , d∗ ) = (H (L∗,i , d∗,i ) ⊗k Kij ) = (Mi ⊗k Kij ) with the trivial left A-module structure. It follows that MR ∼ = (Mi ⊗k Rij ) considered as a k-vector space, and therefore MR is a lifting of M to R. Lemma 3.3. Let R be any object of ap , and let MR be a lifting of M to R. Then there exists an M-free resolution of MR which lifts the complex (F∗ , d∗ ) to R. Proof. Clearly, the lemma holds for R = k p . We shall consider a small surjection u : R → S in ap , deformations MU ∈ DefM (U ) for U = R, S such that MR lifts MS to R, and an M-free resolution (F∗S , dS∗ ) of MS which lifts the complex (F∗ , d∗ ) to S. In this situation, we shall prove that there exists an M-free resolution (F∗R , dR ∗ ) of MR compatible with the M-free resolution of MS . This is clearly enough to prove the lemma. R K = (Lm,i ⊗k Rij ) for all m ≥ 0. Moreover, we write Fm = (Lm,i ⊗k Kij ) for all Let Fm m ≥ 0, where K = ker(u). To complete the proof, we have to ﬁnd the diﬀerentials dR m for m ≥ 0 and the augmentation map ρR : By lemma 3.1, we can ﬁnd a homomorphism ρR : F0R → MR lifting ρS . Denote by ρK : F0K → MK its restriction, where MK = ker(MR → MS ). Since u is small, ρK is surjective, and this implies that the induced map ker(ρR ) → R R S ker(ρS ) is surjective. By lemma 3.1, we can ﬁnd a homomorphism dR 0 : F1 → F0 lifting d0 R R K R K K such that ρ d0 = 0. Let d0 be the restriction of d0 , then clearly ker(ρ ) = Im(d0 ) since

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101

u is small. An easy induction argument shows that we can construct a complex (F∗R , dR ∗) lifting the complex (F∗S , dS∗ ) in such a way that the restriction (F∗K , dK ∗ ) is a resolution of MK . By the proof of lemma 3.2, it follows that H m (F∗R , dR ∗ ) = 0 for m ≥ 1 and that there is an exact sequence 0 → MK → H 0 (F∗R , dR ∗ ) → MS → 0. R R This implies that MR = H 0 (F∗R , dR ∗ ), and (F∗ , d∗ ) is the required M-free resolution of MR compatible with the given M-free resolution of MS .

Proposition 3.4. Let u : R → S be a surjection in ap , and consider a deformation MS ∈ DefM (S) and any M-free resolution (F∗S , dS∗ ) of MS which lifts the complex (F∗ , d∗ ) to S. There is a bijective correspondence between the set of liftings {MR ∈ DefM (R) : DefM (u)(MR ) = MS } S S and the set of M-free complexes (F∗R , dR ∗ ) which lift the resolution (F∗ , d∗ ) to R, up to equivalence.

Proof. For a small surjection, this follows from lemma 3.2 and lemma 3.3. But any surjection in ap is a composition of small surjections. Let R be any object in ap . In section 2, we described how to, in principle, calculate DefM (R) by considering the possible left A-module structures on the right R-module (Mi ⊗k Rij ). The M-free resolutions give us another way of viewing deformations in DefM (R): By proposition 3.4, we can view DefM (R) as the set of liftings of the complex (F∗ , d∗ ) to R, up to equivalence. Using equation 2, each lifting of complexes corresponds to a family of 1-cochains α(rij (l)) ∈ Hom1 (L∗j , L∗i ), parametrized by a k-basis for R. We leave it as an exercise for the reader to use this approach to calculate DefM (R) in the case R = Rα,β — this will give a new proof of lemma 2.1 via the Yoneda representation of Ext1A (Mβ , Mα ).

4. Pro-representing hulls of pointed functors We say that a covariant functor F : ap → Sets is pointed if F(k p ) = {∗}. In this section, we shall consider pointed functors deﬁned on the category ap , and study their representability. Of course, the motivation for this is the fact that DefM is such a pointed functor. Let R be any object of ˆ ap , and consider the functor hR : ap → Sets given by hR (S) = Mor(R, S) for all objects S in ap . The notation Mor(R, S) denotes the set of morphisms from R to S in the pro-category ˆ ap . Then hR is clearly a pointed functor deﬁned on ap . We say that a pointed functor F : ap → Sets is representable is F is isomorphic to hR for some object R in ap , and pro-representable if F is isomorphic to hR for some object R in ˆ ap . However, it is well-known that deformation functors seldom are representable or even pro-representable. So a weaker notion is required, and we shall deﬁne the notion of a pro-representing hull of a pointed functor on ap . We start by introducing some notation: ˆ :ˆ ap → Sets deﬁned Any pointed functor F : ap → Sets has an extension to a functor F on the pro-category ˆ ap . This extension is deﬁned by the formula ˆ F(R) = lim F(R/I n ) ←

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for any object R in ˆ ap with I = I(R). Clearly, any pointed functor F : ap → Sets also has a restriction to the sub-category ap (n) ⊆ ap for all n ≥ 1. We shall denote this restriction by Fn : ap (n) → Sets. Lemma 4.1. Let R be an object in ˆ ap , and let F : ap → Sets be a pointed functor. Then ˆ there is a natural isomorphism of sets α : F(R) → Mor(hR , F). ˆ Proof. Let ξ ∈ F(R), then ξ = (ξn ) with ξn ∈ F(R/I n ) for all n ≥ 1. For any object S in ap , we construct a map of sets α(ξ)S : Mor(R, S) → F(S): Let u : R → S be a morphism in ˆ ap , then u(I(R)) ⊆ I(S), and I(S) is nilpotent since S is in ar , so there exists n ≥ 1 such that u factorizes through un : R/I(R)n → S. We deﬁne α(ξ)S (u) = F(un )(ξn ), and a straight-forward calculation shows that this expression is independent upon the choice of n, and gives rise to a natural transformation of functors. Conversely, let φ : hR → F be a natural transformation of functors on ap . Then we deﬁne ξn ∈ F(R/I(R)n ) to be ξn = φR/I(R)n (R → R/I(R)n ), where R → R/I(R)n is the natural morphism. Again, a ˆ straight-forward calculation shows that ξ = (ξn ) deﬁnes an element in F(R), and that this map of sets deﬁnes an inverse to α. There is also a version of lemma 4.1 for the category ap (n): For an object R in ap (n), and a pointed functor F : ap (n) → Sets, there is a natural isomorphism of sets αn : F(R) → Mor(hR , F). The construction of this isomorphism is similar to the construction in lemma 4.1. We recall that a morphism φ : F → G of pointed functors F, G : ap → Sets is smooth if the following condition holds: For all surjective morphisms u : R → S in ap , the natural map of sets (3)

F(R) → F(S) × G(R), G(S)

given by x → (F(u)(x), φR (x)) for all x ∈ F(R), is a surjection. Clearly, it is enough to check this for small surjections in ap . Also notice that any morphism φ : F → G of functors ˆ → G ˆ of functors on ˆ naturally extends to a morphism φˆ : F ap , and if φ is a smooth ˆ ˆ ˆ morphism, then φR : F(R) → G(R) is surjective for all objects R in ˆ ap . Similarly, we say that a morphism φ : F → G of functors F, G : ap (n) → Sets on ap (n) is smooth if the map of sets (3) is surjective for all surjective morphisms u : R → S in ap (n). Clearly, a morphism φ : F → G of functors on ap is smooth if and only if the restriction φn : Fn → Gn is smooth for all n ≥ 1. Let F be a pointed functor on ap . A pro-couple for F is a pair (R, ξ), where R is an ˆ A morphism u : (R, ξ) → (R , ξ ) of pro-couples is a morphism object in ˆ ap and ξ ∈ F(R). ˆ ap such that F(u)(ξ) = ξ . If (R, ξ) is a pro-couple for F such that R is also u : R → R in ˆ an object of ap , then it is called a couple for F. We say that a pro-couple (R, ξ) pro-represents F if α(ξ) : hR → F is an isomorphism of functors on ap . If (R, ξ) pro-represents F and (R, ξ) is also a couple for F, then we say that (R, ξ) represents F. It is clear that if the couple (R, ξ) represents F, then (R, ξ) is unique up to a unique isomorphism of couples. Similarly, let F be a pointed functor on ap (n). A couple for F is a pair (R, ξ), where R is an object of ap (n) and ξ ∈ F(R). We say that the couple (R, ξ) represents F if and only if αn (ξ) is an isomorphism of functors deﬁned on ap (n). It is clear that if this is the case, the couple (R, ξ) is unique up to a unique isomorphism of couples.

NONCOMMUTATIVE DEFORMATIONS OF MODULES

103

Let F be a functor on ap , and let (R, ξ) be a pro-couple for F. For all n ≥ 1, let (Rn , ξn ) be given by Rn = R/I(R)n and ξn = F(un )(ξ), where un : R → Rn is the natural surjection. Then (Rn , ξn ) is a couple for the restriction Fn : ap (n) → Sets of F for all n ≥ 1. Notice that αn (ξn ) is the restriction of the morphism α(ξ) to ap (n) for all n ≥ 1. Consequently, (R, ξ) pro-represents F if and only if (Rn , ξn ) represents Fn for all n ≥ 1. In particular, it follows that if (R, ξ) pro-represents F, then (R, ξ) is unique up to a unique isomorphism of pro-couples. Let F : ap → Sets be a pointed functor on ap . A pro-representing hull of F is a pro-couple (R, ξ) of F such that the following conditions hold: (1) α(ξ) : hR → F is a smooth morphism of functors on ap (2) α2 (ξ2 ) : hR2 → F2 is an isomorphism of functors on ap (2) To simplify notation, we sometimes call the pro-representing hull (R, ξ) a hull of F. Proposition 4.2. Let F : ap → Sets be a pointed functor on ap , and assume that (R, ξ), (R , ξ ) are pro-representing hulls of F. Then there exists an isomorphism of procouples u : (R, ξ) → (R , ξ ). Proof. Let φ = α(ξ), φ = α(ξ ). Since φ, φ are smooth morphisms, we have that φR and φR are surjective. So we can ﬁnd morphisms u : (R, ξ) → (R , ξ ) and v : (R , ξ ) → (R, ξ) of pro-couples of F. The restriction to ap (2) gives us morphisms u2 : (R2 , ξ2 ) → (R2 , ξ2 ) and v2 : (R2 , ξ2 ) → (R2 , ξ2 ). But both (R2 , ξ2 ) and (R2 , ξ2 ) represent F2 , so u2 and v2 are inverses. In particular, gr1 (u2 ) and gr1 (v2 ) are inverses, and (v ◦ u)2 = v2 ◦ u2 = id. From the proof of proposition 1.5, we see that gr(v ◦ u) is surjective. This means that grn (v ◦ u) is a surjective endomorphims of a ﬁnite dimensional k-vector space for all n ≥ 1, so gr(v ◦ u) is an isomorphism. By proposition 1.5, v ◦ u is an isomorphism as well, and the same holds for u ◦ v by a symmetric argument. It follows that u and v are isomorphisms. So if there exists a pro-representing hull of a pointed functor F, we know that it is unique, and we shall denote it by (H, ξ). Notice that (H, ξ) is only unique up to non-canonical isomorphism. By abuse of language, we shall sometimes omit ξ from the notation, and say that H is the hull of F. 5. Hulls of noncommutative deformation functors We recall that k is an algebraically closed (commutative) ﬁeld, A is an associative k-algebra, and M = {M1 , . . . , Mp } is a ﬁnite family of left A-modules. In this section, we prove that if the family M satisfy the ﬁniteness condition (FC), then there exists a hull H = H(M) of the noncommutative deformation functor DefM . The proof follows Laudal [8], and the essential point is the following obstruction calculus: Proposition 5.1. Let u : R → S be a small surjective morphism in ap with kernel K = ker(u), and let MS ∈ DefM (S) be a deformation. Then there exists a canonical obstruction o(u, MS ) ∈ (Ext2A (Mj , Mi ) ⊗k Kij ), such that o(u, MS ) = 0 if and only if there exists a deformation MR ∈ DefM (R) lifting MS . If this is the case, the set of deformations in DefM (R) lifting MS is a torsor under the k-vector space (Ext1A (Mj , Mi ) ⊗k Kij ).

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Proof. We recall from section 2 that up to equivalence, we may assume that MS has the following form: MS = (Mi ⊗k Sij ) with right S-module structure given by the multiplication in S, and with left A-module structure given by k-linear homomorphisms ai : Mi → ⊕(Mj ⊗k Sji ) for all a ∈ A. Via the natural projections, the map ai gives rise to k-linear maps aji : Mi → Mj ⊗k Sji for a ∈ A, 1 ≤ i, j ≤ p. Since u is surjective, we may choose k-linear maps L(a)ji : Mi → Mj ⊗k Rji such that (id ⊗ u) ◦ L(a)ji = aji for a ∈ A, 1 ≤ i, j ≤ p. Let L(a) = (L(a)ij ) ∈ (Homk (Mj , Mi ⊗k Rij )), this deﬁnes a k-linear left action of A on MR = (Mi ⊗k Rij ), lifting the left A-module structure on MS . We let Q = (Homk (Mj , Mi ⊗k Rij )), and remark that this is an associative k-algebra in a natural way: We compose the k-linear morphisms in Q by using the multiplication in R. For a, b ∈ A, consider the expression L(ab)−L(a)L(b) ∈ Q . By the associativity of the left A-module structure on MS , we see that L(ab)−L(a)L(b) ∈ Q, where Q = (Homk (Mj , Mi ⊗k Kij )) ⊆ Q . Furthermore, we notice that Q ⊆ Q is an ideal, and Q has a natural structure as an A-A bimodule via L, since K 2 = 0. We deﬁne ψ ∈ Homk (A ⊗k A, Q) to be given by ψ(a, b) = L(ab) − L(a)L(b) for all a, b ∈ A. A straight-forward calculation shows that ψ is a 2-cocycle in HC∗ (A, Q), so ψ gives rise to an element o(u, MS ) ∈ HH2 (A, Q) — see appendix A for the deﬁnition of the Hochschild complex and its cohomology. Since K 2 = 0, it follows that if L is another k-linear lifting of the left A-module structure on MS , then the A-A bimodule structures of Q given by L and L coincide. Therefore, HH∗ (A, Q) is independent upon the choice of L, and a straight-forward calculation shows that the same holds for the obstruction o(u, MS ). We remark that there exists a deformation MR ∈ DefM (R) lifting MS if and only if there exists some k-linear lifting L : A → Q of the left A-module structure of MS such that L (ab) = L (a)L (b) for all a, b ∈ A. Let τ = L − L, then τ : A → Q is a k-linear map, and a straight-forward calculation shows that L (ab) = L (a)L (b) if and only if the relation L(ab) − L(a)L(b) = L(a)τ (b) − τ (ab) + τ (a)L(b) + τ (a)τ (b) holds. Since K 2 = 0, the last term vanishes. The fact that the above relation holds for all a, b ∈ A is therefore equivalent to the fact that o(u, MS ) = 0 in HH2 (A, Q). So we have established that there exists a canonical obstruction o(u, MS ) ∈ HH2 (A, Q) such that o(u, MS ) = 0 if and only if there is a lifting of MS to R. Assume that L : A → Q is such that L(ab) = L(a)L(b) for all a, b ∈ A, that is, such that it deﬁnes a deformation MR lying over MS . For any other k-linear lifting L : A → Q of the left A-module structure on MS , we may consider the diﬀerence τ = L − L : A → Q. A straight-forward calculation shows that τ is a 1-cocycle in HC∗ (A, Q) if and only if L (ab) = L (a)L (b) for all a, b ∈ A, that is, if and only if L deﬁnes a left A-module structure on MR . Furthermore, we have that L and L give rise to equivalent deformations if and only if τ is a 1-coboundary: It is clear that any equivalence between the left A-module structures of MR = (Mi ⊗k Rij ) given by L and L has the form id + ψ, where ψ ∈ Q. Furthermore, the map id + ψ : MR → MR (with the left A-module structure from L and L respectively) is a left A-module homomorphism if and only if L(r)(id + ψ) = (id + ψ)L (r) holds for all a ∈ A, and this last condition is equivalent with the fact that τ = d(ψ), so that τ is a 1-coboundary. If τ is a 1-boundary in HC∗ (A, Q), it is also clear that id + ψ deﬁnes an equivalence between the two deformations given by L and L . Therefore, the set of deformations MR lying over MS is a torsor under the k-vector space HH1 (A, Q).

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To end the proof, we have to show that there are isomorphisms of k-vector spaces HHn (A, Q) ∼ = (ExtnA (Mj , Mi ) ⊗k Kij ) for n = 1, 2: Since L(a) is a lifting to MR of the left multiplication of a on MS (satisfying equation 1), L(a) satisﬁes equation 1 as well. That is, we have L(a)ji (mi ) − δij (ami ) ⊗ ei ∈ Mj ⊗k Iji for all a ∈ A, mi ∈ Mi , 1 ≤ i, j ≤ p. Since K 2 = 0, this means that the A-A bimodule structure of Q deﬁned via L coincides with the following natural one: Since Mi , Mj ⊗k Kji are left A-modules, we have that Qij = Homk (Mj , Mi ⊗k Kij ) and Q = ⊕Qij has natural A-A bimodule structures. Clearly, we have HHn (A, Q) ∼ = ⊕ HHn (A, Qij ) = (HHn (A, Qij )). i,j

∼ Extn (Mj , Mi ⊗k Kij ) for By appendix A, proposition A.3, we have that HHn (A, Qij ) = A n n n ≥ 0. Moreover, ExtA (Mj , Mi ⊗k Kij ) ∼ = ExtA (Mj , Mi ) ⊗k Kij since Kij is a k-vector space of ﬁnite dimension. This completes the proof of the proposition. We remark that it is easy to ﬁnd an alternative proof of proposition 5.1 using resolutions and the Yoneda representation of ExtnA (Mi , Mj ). This is straight-forward, but makes essential use of proposition 3.4. Also notice that the obstruction calculus is functorial in the following sense: Let u : R → S and u : R → S be two small surjections in ap , and write K = ker(u) and K = ker(u ). Assume that v : R → R and w : S → S are morphisms such that u ◦ v = w ◦ u. Then v(K) ⊆ K , and the map v induces a k-linear map of obstruction spaces ). (Ext2A (Mj , Mi ) ⊗k Kij ) → (Ext2A (Mj , Mi ) ⊗k Kij

If MS is a deformation of M to S and MS = DefM (w)(MS ) is the corresponding deformation to S , then this map of obstruction spaces maps o(u, MS ) to o(u , MS ). This follows from the proof of proposition 5.1. Let us start the construction of the pro-representing hull (H, ξ) of DefM , using the obstruction calculus for DefM given above. From now on, we shall assume that the family M satisfy the ﬁniteness condition (FC)

dimk ExtnA (Mi , Mj )is ﬁnite for 1 ≤ i, j ≤ p, n = 1, 2.

We ﬁx the following notation: Let {xij (l) : 1 ≤ l ≤ dij } be a basis for Ext1A (Mj , Mi )∗ and let {yij (l) : 1 ≤ l ≤ rij } be a basis for Ext2A (Mj , Mi )∗ for 1 ≤ i, j ≤ p, with dij = dimk Ext1A (Mj , Mi ) and rij = dimk Ext2A (Mj , Mi ). Moreover, we consider the formal ˆ matrix rings in ˆ ap corresponding to these vector spaces, and denote them by T1 = R 1 2 2 ∗ ∗ ˆ ({ExtA (Mj , Mi ) }) and T = R({ExtA (Mj , Mi ) }). First, let us show that DefM restricted to ap (2) is representable: We deﬁne H2 to be the object H2 = T12 = T1 /I(T1 )2 in ap (2). For all objects R in ap (2), we get Mor(H2 , R) ∼ = (Homk (Ext1A (Mj , Mi )∗ , I(R)ij )) ∼ = (Ext1A (Mj , Mi ) ⊗k I(R)ij ), and 1 ∼ DefM (R) = (ExtA (Mj , Mi ) ⊗k I(R)ij ) by proposition 5.1 applied to the small surjection R → k p . The isomorphisms we obtain in this way are compatible, so they induce an isomorphism φ2 : hH2 → DefM of functors on ap (2). From the version of lemma 4.1 for the category ap (2), we see that there is a unique deformation ξ2 ∈ DefM (H2 ) such that α2 (ξ2 ) = φ2 . By deﬁnition, (H2 , ξ2 ) represents the deformation functor DefM restricted to ap (2).

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Let us also give an explicit description of the deformation ξ2 : We have H2 = T12 , so let us denote by ij (l) the image of xij (l) in H2 for 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij . In this notation, ξ2 is represented by the right H2 -module (Mi ⊗k (H2 )ij ), with left A-module structure deﬁned by a(mj ⊗ ej ) = amj ⊗ ej +

l ψij (a)(mj ) ⊗ ij (l)

i,l l for all a ∈ A, mj ∈ Mj , 1 ≤ j ≤ p, where ψij ∈ Derk (A, Homk (Mj , Mi )) is a representative 1 ∗ of xij (l) ∈ ExtA (Mj , Mi ) via Hochschild cohomology. There is also an alternative description of ξ2 using M-free resolutions and the Yoneda representation of Ext1A (Mi , Mj ): Let α(ij (l)) ∈ Hom1 (L∗j , L∗i ) be a 1-cocycle representing xij (l)∗ ∈ Ext1A (Mj , Mi ) for 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij . Then by construction, the formula 2 dH m =

dm,i ⊗ ei +

i

α(ij (l))m ⊗ ij (l)

i,j,l

deﬁnes a diﬀerential which lifts the complex (F∗ , d∗ ) to H2 . By proposition 3.4, the lifted complex is in fact an M-free resolution of some deformation of M to H2 , and this deformation is ξ2 ∈ DefM (H2 ). Theorem 5.2. Assume that dimk ExtnA (Mi , Mj ) is finite for 1 ≤ i, j ≤ p, n = 1, 2. Then ˆ T2 k p is a pro-representing ap such that H(M) = T1 ⊗ there exists a morphism o : T2 → T1 in ˆ hull for DefM . Proof. For simplicity, let us write I for the ideal I = I(T1 ), and for all n ≥ 1, let us write T1n for the quotient T1n = T1 /I n , and tn : T1n+1 → T1n for the natural morphism. From the paragraphs preceding this theorem, we know that (H2 , ξ2 ) represents DefM restricted to ap (2). Let o2 : T2 → T12 be the trivial morphism given by o2 (I(T2 )) = 0 and let a2 = I 2 , then H2 = T1 /a2 ∼ = T12 ⊗T2 k p . Using o2 and ξ2 as a starting point, we shall construct on and ξn for n ≥ 3 by an inductive process. So let n ≥ 2, and assume that the morphism on : T2 → T1n and the deformation ξn ∈ DefM (Hn ) is given, with Hn = T1n ⊗T2 k p . We shall also assume that tn−1 ◦ on = on−1 and that ξn is a lifting of ξn−1 . Let us now construct the morphism on+1 : T2 → T1n+1 : We let an be the ideal in 1 Tn generated by on (I(T2 )). Then an = an /I n for an ideal an ⊆ T1 with I n ⊆ an , and Hn ∼ = T1 /an . Let bn = Ian + an I, then we obtain the following commutative diagram: T2

T1n+1

T1 /bn

T1n

Hn = T1 /an ,

on

Observe that T1 /bn → T1 /an is a small surjection. So by proposition 5.1, there is an obstruction on+1 = o(T1 /bn → Hn , ξn ) for lifting ξn to T1 /bn , and we have on+1 ∈ (Ext2A (Mj , Mi ) ⊗k (an /bn )ij ) ∼ = (Homk (gr1 (T2 )ij , (an /bn )ij )).

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Consequently, we obtain a morphism on+1 : T2 → T1 /bn . Let an+1 be the ideal in T1 /bn generated by on+1 (I(T2 )). Then an+1 = an+1 /bn for an ideal an+1 ⊆ T1 with bn ⊆ an+1 ⊆ an . We deﬁne Hn+1 = T1 /an+1 and obtain the following commutative diagram: on+1

T2

T1n+1

T1 /bn

T1n

Hn = T1 /an

Hn+1 = T1 /an+1

on

By the choice of an+1 , the obstruction for lifting ξn to Hn+1 is zero. We can therefore ﬁnd a lifting ξn+1 ∈ DefM (Hn+1 ) of ξn to Hn+1 . The next step of the construction is to ﬁnd a morphism on+1 : T2 → T1n+1 which commutes with on+1 and on : We know that tn−1 ◦ on = on−1 , which means that an−1 = I n−1 + an . For simplicity, let us write O(K) = (Homk (gr1 (T2 )ij , Kij )) for any ideal K ⊆ T1 . Consider the following commutative diagram of k-vector spaces, in which the columns are exact: 0

0

O(bn /I n+1 )

O(an /I n+1 )

jn

O(bn−1 /I n )

kn

O(an−1 /I n )

rn+1

O(an /bn )

0

rn ln

O(an−1 /bn−1 )

0

We may consider consider on as an element in O(an−1 /I n ), since an ⊆ an−1 . On the other hand, on+1 ∈ O(an /bn ). Let on = rn (on ), then the natural map T1 /bn → T1 /bn−1 maps the obstruction on+1 to the obstruction on by the second remark following proposition 5.1. This implies that on+1 commutes with on , so ln (on+1 ) = on = rn (on ). But we have on (I(T2 )) ⊆ an , so we can ﬁnd an element on+1 ∈ O(an /I n+1 ) such that kn (on+1 ) = on . Since an−1 = an + I n−1 , jn is surjective. Elementary diagram chasing using the snake lemma implies that we can ﬁnd on+1 ∈ O(an /I n+1 ) such that rn+1 (on+1 ) = on+1 and kn (on+1 ) = on . It follows that the obstruction on+1 deﬁnes a morphism on+1 : T2 → T1n+1 compatible with on such that T1n+1 ⊗T2 k p ∼ = Hn+1 . By induction, it follows that we can ﬁnd a morphism on : T2 → T1n and a deformation ξn ∈ DefM (Hn ), with Hn = T1n ⊗T2 k p , for all n ≥ 1. From the construction, we see that tn−1 ◦ on = on−1 for all n ≥ 2, so we obtain a morphism o : T2 → T1 by the universal property of the projective limit. Moreover, the induced morphisms hn : Hn+1 → Hn are such that ξn+1 ∈ DefM (Hn+1 ) is a lifting of ξn ∈ DefM (Hn ) to Hn+1 . Notice that I(Hn )n = 0

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and that Hn /I(Hn )n−1 ∼ = Hn−1 for all n ≥ 2. It follows that H/I(H)n = Hn for all n ≥ 1, so H is an object of the pro-category ˆ ap . Let ξ = (ξn ), then clearly ξ ∈ DefM (H), so (H, ξ) is a pro-couple for DefM . It remains to show that (H, ξ) is a pro-representable hull for DefM . It is clearly enough to show that (Hn , ξn ) is a pro-representing hull for DefM restricted to ap (n) for all n ≥ 3. So let φn = αn (ξn ) be the morphism of functors on ap (n) corresponding to ξn . We shall prove that φn is a smooth morphism. So let u : R → S be a small surjection in ap (n), and assume that MR ∈ DefM (R) and v ∈ Mor(Hn , S) are given such that DefM (u)(MR ) = DefM (v)(ξn ) = MS . Let us consider the following commutative diagram: T1 /bn

T1

Hn+1

R u

Hn

v

S

Let v : T1 → R be any morphism making the diagram commutative. Then v (an ) ⊆ K, where K = ker(u), so v (bn ) = 0. But the induced map T1 /bn → R maps the obstruction on+1 to o(u, MS ), and we know that o(u, MS ) = 0. So we have v (an+1 ) = 0, and v induces a morphism v : Hn+1 → R making the diagram commutative. Since v (I(Hn+1 )n ) = 0, we may consider v a map from Hn+1 /I(Hn+1 )n ∼ = Hn . So we have constructed a map v ∈ Mor(Hn , R) such that u ◦ v = v. Let MR = DefM (v )(ξn ), then MR is a lifting of MS to R. By proposition 5.1, the diﬀerence between MR and MR is given by an element d ∈ (Ext1A (Mj , Mi ) ⊗k Kij ) = (Homk (gr1 (T1 )ij , Kij )). Let v : T1 → R be the morphism given by v (xij (l)) = v (xij (l))+d(xij (l)) for 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij . Since an+1 ⊆ I(T1 )2 , we have v (an+1 ) ⊆ v (an+1 ) + I(R)K + KI(R) + K 2 . But u is small, so v (an+1 ) = 0 and v induces a morphism v : Hn → R. Clearly, u ◦ v = u ◦ v = v, and DefM (v )(ξn ) = MR by construction. It follows that φn is smooth for all n ≥ 3. We remark that the conclusion of the theorem still holds if we relax the ﬁniteness condition (FC). If we only assume that dimk Ext1A (Mi , Mj )is ﬁnite for 1 ≤ i, j ≤ p, then the object T2 is in Ap , but not necessarily in ˆ ap . However, the rest of the proof is still valid as stated, so the ﬁniteness condition on Ext2A (Mi , Mj ) is clearly not essential. In general, it is possible to generalize theorem 5.2 to the case when ExtnA (Mi , Mj ) has countable dimension as a vector space over k for 1 ≤ i, j ≤ p, n = 1, 2, see Laudal [2]. However, we shall always assume (FC) in this paper.

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109

Assume that M satisfy (FC). If Ext2A (Mi , Mj ) = 0 for 1 ≤ i, j ≤ p, we say that the deformation functor DefM is unobstructed. For instance, DefM is unobstructed for any ﬁnite family M of left A-modules satisfying (FC) if A is left hereditary (that is, the left global homological dimension of A is at most 1). If DefM is unobstructed, H = T1 is the hull of DefM . In general, DefM can be obstructed, and there is no simple formula for the hull H of DefM if this is the case. However, there exists an algorithm for calculating the hull H using matric Massey products. In the next sections, we shall introduce the matric Massey products and explain how the hull can be calculated when M satisfy (FC). 6. Immediately defined matric Massey products We recall that k is an algebraically closed (commutative) ﬁeld, A is an associative k-algebra, and M = {M1 , . . . , Mp } is a ﬁnite family of left A-modules. From now on, we also assume that the family M satisfy the ﬁniteness condition (FC). In this section, we shall deﬁne the immediately deﬁned matric Massey products and their deﬁning system, and show how to calculate these products using matrices. Let us ﬁx a monomial X ∈ I(T1 ) of type (i, j) and degree n ≥ 2. Then we can write X uniquely in the form X = xi0 i1 (l1 )xi1 i2 (l2 ) . . . xin−1 in (ln ), where (i0 , in ) = (i, j). Let X be another monomial in T1 . We shall say that X divides X if there exist monomials X(l), X(r) ∈ T1 such that X = X(l)X X(r), and write X | X if this is the case. Consider the set of monomials {X ∈ I(T1 ) : X | X}, and denote by J(X) the ideal in 1 T generated by these monomials. We deﬁne R(X) = T1 /J(X) and S(X) = R(X)/(X) = T1 /(J(X), X). Then the natural map π(X) : R(X) → S(X) is a small surjection in ap , and it has a 1-dimensional kernel which is generated by the monomial X. We write I(X) = I(S(X)) and S(X)n = S(X)/I(X)n for all n ≥ 1. Let us consider the set B(X) = {(i, j, l) : 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij , xij (l) | X}, and denote by vij (l) the image of xij (l) in S(X)2 for all (i, j, l) ∈ B(X). Then the set {vij (l) : (i, j, l) ∈ B(X)} is a natural k-basis for I(X)/I(X)2 . Assume that a morphism φ(X) : H → S(X) is given, and denote the composition of φ(X) with the natural morphism S(X) → S(X)2 by φ(X)2 : H → S(X)2 . This morphism can be written uniquely in the form φ(X)2 =

αij (l) ⊗ vij (l),

(i,j,l)∈B(X)

where αij (l) ∈ Ext1A (Mj , Mi ) for all (i, j, l) ∈ B(X).

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Conversely, consider a family {αij (l) ∈ Ext1A (Mj , Mi ) : (i, j, l) ∈ B(X)} of extensions indexed by B(X), corresponding to a morphism φ(X)2 : H → S(X)2 given by φ(X)2 = αij (l) ⊗ vij (l). If there exists a lifting of φ(X)2 to a morphism φ(X) : H → S(X), we say that the matric Massey product α; X = αi0 ,i1 (l1 ), αi1 ,i2 (l2 ), . . . , αin−1 ,in (ln ) is deﬁned, and that φ(X) is a defining system for this matric Massey product. If this is the case, we denote the deformation induced by the deﬁning system φ(X) by MX ∈ DefM (S(X)), and by proposition 5.1, the obstruction for lifting MX to R(X) is an element o(π(X), MX ) ∈ (Ext2A (Mj , Mi ) ⊗k K(X)ij ) ∼ = Ext2A (Mj , Mi ), where K(X) = ker(π(X)) ∼ = kX. In general, this element depends upon the deformation MX , and therefore on the deﬁning system φ(X). We deﬁne the value of the matric Massey product to be α; X = αi0 ,i1 (l1 ), αi1 ,i2 (l2 ), . . . , αin−1 ,in (ln ) = o(π(X), MX ). Consequently, the value of the matric Massey product α; X will in general depend upon the chosen deﬁning system. Let us ﬁx the monomial X. Then the matric Massey product α → α; X is a not everywhere deﬁned k-linear map Ext1A (Mi1 , Mi0 ) ⊗k · · · ⊗k Ext1A (Min , Min−1 )

Ext2A (Min , Mi0 ).

In fact, this map is deﬁned for α if and only if the morphism φ(X)2 : H → S(X)2 corresponding to α can be lifted to a morphism φ(X) : H → S(X). Moreover, even when this map is deﬁned for α, it is not necessarily uniquely deﬁned: In general, its value α; X depends upon the chosen lifting φ(X), the deﬁning system. The matric Massey products α; X deﬁned above are called the immediately defined matric Massey products. We remark that if X is a monomial of degree n = 2, then the situation is much simpler: We have S(X) = S(X)2 , so the matric Massey product α; X is uniquely deﬁned for any family of extensions {αij (l) : (i, j, l) ∈ B(X)}. In fact, the matric Massey product is just the usual cup product in this case. Let us ﬁx a monomial X ∈ I(T1 ) of degree n ≥ 2. Then there exists a natural family of extensions indexed by B(X) given by αij (l) = xij (l)∗ , {xij (l)∗ ∈ Ext1A (Mj , Mi ) : (i, j, l) ∈ B(X)}. The matric Massey products of these extensions are the ones that we shall use for the construction of the hull H of DefM in the next section. We therefore introduce the notation x∗ ; X = xi0 i1 (l1 )∗ , xi1 i2 (l2 )∗ , . . . , xin−1 in (ln )∗ for their immediately deﬁned matric Massey products.

NONCOMMUTATIVE DEFORMATIONS OF MODULES

111

The matric Massey products are called matric because these products (and their deﬁning systems) can be described completely in terms of linear algebra and matrices. We shall end this section by giving such a description. Let {αij (l) ∈ Ext1A (Mj , Mi ) : (i, j, l) ∈ B(X)} be a family of extensions indexed by B(X), and consider the corresponding matric Massey product α; X = αi0 i1 (l1 ), αi1 i2 (l2 ), . . . , αin−1 in (ln ).

(4)

We assume that there exists a deﬁning system φ(X) : H → S(X) for this matric Massey product. Then φ(X) induces a deformation MX ∈ DefM (S(X)). We notice that the matric Massey product (4) only depends upon this deformation. By abuse of language, we shall therefore let the notion defining system refer to the deformation MX as well as the morphism φ(X) : H → S(X) which induces MX . We know that any deformation MX ∈ DefM (S(X)) can be described by a complex which lifts (F∗ , d∗ ) to S(X). Such a complex is given by diﬀerentials of the form dS(X) : (Lm+1,i ⊗k S(X)ij ) → (Lm,i ⊗k S(X)ij ). m We write v(X ) for the image of X in S(X) whenever X is a monomial in T1 , and deﬁne B(X) = {X ∈ I(T1 ) : X is a monomial such that X | X} ∪ {e1 , . . . , ep }. Then the set {v(X ) : X ∈ B(X)} is a natural k-basis for S(X), and B(X) contains {xij (l) : (i, j, l) ∈ B(X)} and {e1 , . . . , ep } as subsets. Let us write B(X)ij = B(X) ∩ S(X)ij for 1 ≤ i, j ≤ p. With this notation, the above diﬀerentials have the form dS(X) = m

1≤i≤p

dm,i ⊗ ei +

α(X )m ⊗ v(X ),

X ∈B(X)

where α(X ) ∈ Hom1A (L∗j , L∗i ) is a 1-cochain whenever X ∈ B(X )ij . S(X) Let dm be arbitrary maps between M-free modules over S(X) deﬁned by a family of 1-cochains {α(X ) : X ∈ B(X)} as above. These maps lifts the complex (F∗ , d∗ ) if α(ei ) = d∗i for 1 ≤ i ≤ p. Moreover, these maps are diﬀerentials if and only if the following condition holds: For all monomials Z ∈ B(X) and for all integers m ≥ 0, we have (5)

X ,X ∈B(X) X X =Z

α(X )m ◦ α(X )m+1 =

α(X )m+1 α(X )m = 0.

X ,X ∈B(X) X X =Z

In the ﬁrst sum, the symbol ◦ denotes composition of maps. We recall that each of the maps involved can be considered as right multiplication by a matrix. In the second summation, we identify the maps with such matrices, and re-write the composition of maps as multiplication of the corresponding matrices. Assume that these conditions hold. Then the family {α(X ) : X ∈ B(X)} of 1-cochains deﬁnes a lifting of complexes of (L∗ , d∗ ) to S(X) given by the diﬀerentials dS(X) as above,

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EIVIND ERIKSEN

and this lifting corresponds to a deformation MX ∈ DefM (S(X)). The deformation MX is a deﬁning system for the matric Massey product (4) if and only if α(X ) is a 1-cocycle which represents αij (l) ∈ Ext1A (Mj , Mi ) whenever X = xij (l) for some (i, j, l) ∈ B(X). In this case, we shall refer to the family of 1-cochains {α(X ) : X ∈ B(X)} as a defining system for the matric Massey product (4). Finally, assume that the family of 1-cochains {α(X ) : X ∈ B(X)} is a deﬁning system of the matric Massey product (4). Then the value of this matric Massey product is given by

α; Xm =

(6)

α(X )m+1 α(X )m

X ,X ∈B(X) X X =X

for all m ≥ 0, where the multiplication denotes matrix multiplication of the corresponding matrices. Proposition 6.1. Let {αij (l) ∈ Ext1A (Mj , Mi ) : (i, j, l) ∈ B(X)} be a family of extensions. A defining system for the matric Massey product α; X = αi0 i1 (l1 ), . . . , αin−1 in (ln ) corresponds to a family {α(X ) ∈ Hom1A (L∗j , L∗i ) : 1 ≤ i, j ≤ p, X ∈ B(X)ij } of 1-cochains satisfying the following conditions: • α(ei ) = d∗i for 1 ≤ i ≤ p, • α(X ) is a 1-cocycle representing αij (l) whenever X = xij (l) for some (i, j, l) ∈ B(X), • For all Z ∈ B(X) and for all m ≥ 0, we have

α(X )m+1 α(X )m = 0.

X ,X ∈B(X) X X =Z

Moreover, given such a family of 1-cochains, the matric Massey product α; X is represented by the 2-cocyle given by α; Xm =

α(X )m+1 α(X )m

X ,X ∈B(X) X X =X

for all m ≥ 0. Hence we have described the immediately deﬁned matric Massey products and their deﬁning systems in terms of linear algebra and matrices, as we set out to do. We remark that the description given in proposition 6.1 is extremely useful for doing concrete calculations with matric Massey products, and even for implementing such computations on computers. It also justiﬁes the name matric.

NONCOMMUTATIVE DEFORMATIONS OF MODULES

113

7. Calculating hulls using matric Massey products We recall that k is an algebraically closed (commutative) ﬁeld, A is an associative k-algebra, and M = {M1 , . . . , Mp } is a ﬁnite family of left A-modules. We also assume that the family M satisfy the ﬁniteness condition (FC). In this section, we show how to calculate the hull H of the deformation functor DefM using matric Massey products. By theorem 5.2, there exists an obstruction morphism o : T2 → T1 in ˆ ap such that ˆ T2 k p is a hull for the deformation functor DefM . We shall write I = I(T1 ) and H = T1 ⊗ 2 by fij (l) = o(yij (l)) for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . Then fij (l) is a formal power series in Iij 1 construction. Let us deﬁne a ⊆ T to be the ideal generated by {fij (l) : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij }. Then a ⊆ I 2 , and we have ˆ T2 k p ∼ H = T1 ⊗ = T1 /a. We shall use the matric Massey products from section 6 to calculate the coeﬃcients of the power series fij (l). Clearly, this is suﬃcient to determine the hull H. Let us ﬁx an integer N ≥ 2 such that a ⊆ I N . This is always possible, since a ⊆ I 2 . So fij (l) ∈ I N for all fij (l), and we can write fij (l) in the form fij (l) =

alij (X) · X +

|X|=N

alij (X) · X

|X|>N

for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij , with alij (X) ∈ k for all monomials X ∈ I N . As usual, we use the notation |X| to denote the degree of the monomial X. Let 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij and let n ≥ N . Then we agree to write fij (l)n for the truncated power series n alij (X) · X. fij (l)n = |X|=N

Moreover, let an+1 = I n+1 + (f n ) for all n ≥ N , where (f n ) ⊆ T1 is the ideal generated by {fij (l)n : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij }, and let an = I n for 2 ≤ n ≤ N . We write Hn = H/I(H)n as usual, then Hn = T1 /an for all n ≥ 2, in accordance with the notation in the proof of theorem 5.2. Recall that H2 = T12 and that ξ2 ∈ DefM (H2 ) denotes the universal deformation with the property that the couple (H2 , ξ2 ) represents DefM restricted to ap (2). We have assumed that a ⊆ I N , and this means that there exists a lifting of ξ2 to HN = T1 /aN = T1N . Let us proceed to ﬁnd such a lifting MN ∈ DefM (HN ) explicitly. We choose to describe the deformation MN in terms of M-free resolutions. Let us deﬁne B(N − 1) to be the set of all monomials in T1 of degree at most N − 1. Then {X : X ∈ B(N − 1)} is a monomial basis of HN , and any M-free resolution of MN can be described by a family {α(X) : X ∈ B(N − 1)} of 1-cochains satisfying the following conditions: • α(ei ) = d∗i for 1 ≤ i ≤ p, • α(xij (l)) is a 1-cocycle representing xij (l)∗ for 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij , • For all Z ∈ B(N − 1) and for all m ≥ 0, we have X ,X ∈B(N −1) X X =Z

α(X )m+1 α(X )m = 0.

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We know that a family of 1-cochains with the above properties exists, since we can ﬁnd a lifting MN of ξ2 to HN and this deformation must have some M-free resolution. So we choose one such family {α(X) : X ∈ B(N −1)} and ﬁx this choice. This means that we have ﬁxed a deformation MN ∈ DefM (HN ) with an M-free resolution given by the corresponding diﬀerentials. So (HN , MN ) is a pro-representing hull for DefM restricted to ap (N ). Lemma 7.1. Let π : R → S be any small surjection in ap , let φ : H → S be any morphism, and denote by Mφ ∈ DefM (S) the deformation induced by φ. Then we can lift φ to a morphism φ : T1 → R making the diagram φ

T1

R π

H

φ

S

commutative, and the obstruction o(π, Mφ ) for lifting Mφ to R is given by o(π, Mφ ) =

yij (l)∗ ⊗ φ(fij (l)).

i,j,l

Proof. By construction and functoriality, the obstruction o(π, Mφ ) is given as the restriction of the composition φ ◦ o to the k-linear subspace (Ext2A (Mj , Mi )∗ ) ⊆ T2 . Since {yij (l)} is a k-linear basis for this subspace, we get the desired expression for the obstruction. Let us deﬁne bN ⊆ T1 to be the ideal bN = IaN + aN I = I N +1 , and consider the natural map rN : RN → HN , where RN = T1 /bN = T1N +1 . By construction, rN is a small surjection in ap , and the natural surjection φN : T1 → RN makes the diagram o

T2

T1

φN

RN rN

H

φN

HN

commutative. Let B (N ) be the set of all monomials in T1 of degree N . Since ker(rN ) = I N /I N +1 , we see that {X : X ∈ B (N )} is a monomial basis for ker(rN ). Moreover, let B (N ) = B (N ) ∪ B(N − 1). Then clearly {X : X ∈ B (N )} is a monomial basis for RN . Since rN is a small surjection, there is an obstruction o(rN , MN ) for lifting MN to RN , and we see from lemma 7.1 that this obstruction can be expressed as o(rN , MN ) = yij (l)∗ ⊗ φN (fij (l)) i,j,l

=

i,j,l

=

yij (l)∗ ⊗ f ij (l)

yij (l)∗ ⊗ (alij (X) · X),

i,j,l X∈B (N )

where f ij (l) and X denote the images of fij (l) and X in RN .

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115

We say that the family D(N ) = {α(X) : X ∈ B(N − 1)} of 1-cochains is a defining system for the matric Massey products of order N , x∗ ; X for X ∈ B (N ). Let X ∈ B (N ) be any monomial of type (i, j). We deﬁne the matric Massey product x∗ ; X to be the coeﬃcient of X in the obstruction o(rN , MN ) above. Then we immediately see that this matric Massey product has value ∗

x ; X =

rij

alij (X) · yij (l)∗ .

l=1

In other words, the coeﬃcient of X in the power series fij (l) is given by the matric Massey product x∗ ; X above as alij (X) = yij (l)(x∗ ; X) for 1 ≤ l ≤ rij . We notice that the matric Massey products of order N deﬁned above are immediately defined. In other words, they can be expressed in terms of the matric Massey products of section 6. In fact, the deﬁning system D(N ) induces a deﬁning system {α(X ) : X | X, X = X} in the sense of section 6, and the value of the corresponding matric Massey product x∗ ; X is exactly the coeﬃcient of X in the obstruction o(rN , MN ). On the other hand, we can calculate the obstruction o(rN , MN ) using the deﬁning system D(N ), and therefore also the coeﬃcient of X in this obstruction for each X ∈ B (N ). A straight-forward calculation show that this coeﬃcient is given by the 2-cocycle y(X) deﬁned by α(X )m+1 α(X )m y(X)m = X ,X ∈B(N −1) X X =X

for all m ≥ 0. This means that the matric Massey product x∗ ; X is represented by y(X), so we can easily calculate all matric Massey products of order N using the deﬁning system D(N ). This determines the truncated power series fij (l)N , since we have fij (l)N =

alij (X) · X =

X∈B (N )

yij (l)(x∗ ; X) · X

X∈B (N )

for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . Let hN : HN +1 → HN be the natural map. Then ker(hN ) = I N /aN +1 , so we can ﬁnd a subset B(N ) ⊆ B (N ) of monomials in T1 of degree N such that {X : X ∈ B(N )} is a monomial basis for ker(hN ). Let B(N ) = B(N ) ∪ B(N − 1), then clearly {X : X ∈ B(N )} is a monomial basis for HN +1 . So for each monomial X ∈ T1 with |X| ≤ N , we have a unique relation in HN +1 of the form X=

X ∈B(N )

β(X, X ) X ,

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with β(X, X ) ∈ k for all X ∈ B(N ). Since we have o(hN , MN ) = 0, we deduce that

x∗ ; X β(X, X ) = 0

|X|=N

for all X ∈ B(N ). Notice that β(X, X ) = 0 if the monomials X and X do not have the same type. Therefore, it makes sense to consider the 1-cocycle

β(X, X ) y(X),

|X|=N

and by the relation above, this is a 1-coboundary. It follows that we can ﬁnd a 1-cochain α(X ) such that d α(X ) = − β(X, X ) y(X), |X|=N

and we ﬁx such a choice. Consider the family {α(X) : X ∈ B(N )}. This deﬁnes an M-free complex over HN +1 if and only if we have

β(X, Z)

|X|=N

α(X ) α(X ) = 0

X ,X ∈B(N ) X X =X

for all Z ∈ B(N ). By the deﬁnition of α(X ) when X ∈ B(N ), this condition holds, and we denote by MN +1 ∈ DefM (HN +1 ) the deformation with the complex deﬁned by {α(X) : X ∈ B(N )} as M-free resolution. It is clear from the construction that MN +1 is a lifting of MN , so (HN +1 , MN +1 ) is a pro-representing hull for DefM restricted to ap (N + 1). Let bN +1 ⊆ T1 be the ideal bN +1 = IaN +1 + aN +1 I = I N +2 + I(f N )+(f N )I, and consider the natural map rN +1 : RN +1 → HN +1 , where RN +1 = T1 /bN +1 . By construction, rN +1 is a small surjection in ap , and it is clear that the natural morphism φN +1 : T1 → RN +1 makes the diagram T2

o

T1

φN +1

RN +1 rN +1

H

φN +1

φN

HN +1 hN

HN commutative. We see that ker(rN +1 ) = aN +1 /bN +1 , which we can re-write in the following way: ker(rN +1 ) = (I N +1 + (f N ))/(I N +2 + I(f N ) + (f N )I) = (f N )/(I(f N ) + (f N )I) ⊕ I N +1 /(I N +2 + I(f N ) + (f N )I)

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117

Let us write c(N + 1) = I N +1 /(I N +2 + I(f N ) + (f N )I). Then c(N + 1) ⊆ ker(rN +1 ) is an ideal, and we can clearly ﬁnd a set B (N + 1) of monomials in T1 of degree N + 1 such that {X : X ∈ B (N + 1)} is a monomial basis for cN +1 . Let us choose B (N + 1) such that for every X ∈ B (N + 1), there is a monomial X ∈ B(N ) such that X | X, this is clearly possible. We let B (N + 1) = B (N + 1) ∪ B(N ), then {X : X ∈ B (N + 1)} ∪ {fij (l)N : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij } is a basis for RN +1 . So for each monomial X ∈ T1 with |X| ≤ N + 1, we have a unique relation in RN +1 of the form X=

β (X, X )X +

β (X, i, j, l)f ij (l)N ,

i,j,l

X ∈B (N +1)

with β (X, X ), β (X, i, j, l) ∈ k for all X ∈ B (N + 1), 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . Since rN +1 is a small surjection, there is an obstruction o(rN +1 , MN +1 ) for lifting MN +1 to RN +1 , and we see from lemma 7.1 that this obstruction can be expressed as o(rN +1 , MN +1 ) =

yij (l)∗ ⊗ φN +1 (fij (l))

i,j,l

=

yij (l)∗ ⊗ f ij (l)

i,j,l

=

yij (l)∗ ⊗ (f ij (l)N +

i,j,l

alij (X) · X),

X∈B (N +1)

where f ij (l), f ij (l)N and X denote the images of fij (l), fij (l)N and X in RN +1 . We say that the family D(N + 1) = {α(X) : X ∈ B(N )} is a defining system for the matric Massey products of order N + 1, x∗ ; X for X ∈ B (N + 1) Let X ∈ B (N + 1) be any monomial of type (i, j). We deﬁne the matric Massey product x∗ ; X to be the coeﬃcient of X in the obstruction o(rN +1 , MN +1 ) above. Then we immediately see that this matric Massey product has value ∗

x ; X =

rij

alij (X) · yij (l)∗ .

l=1

In other words, the coeﬃcient of X in the power series fij (l) is given by the matric Massey product x∗ ; X above as alij (X) = yij (l)(x∗ ; X) for 1 ≤ l ≤ rij . On the other hand, we can calculate the obstruction o(rN +1 , MN +1 ) using the deﬁning system D(N + 1), and therefore also the coeﬃcient of X in this obstruction for each

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EIVIND ERIKSEN

X ∈ B (N + 1). A straight-forward calculation show that this coeﬃcient is given by the 2-cocycle y(X) deﬁned by y(X)m =

β (Z, X)

|Z|≤N +1

α(X )m+1 α(X )m

X ,X ∈B(N ) X X =Z

for all m ≥ 0. This means that the matric Massey product x∗ ; X is represented by y(X), so we can easily calculate all matric Massey products of order N + 1 using the deﬁning system D(N + 1). By the construction in the proof of theorem 5.2, we have that HN +2 is the quotient of RN +1 by the ideal generated by the obstruction o(rN +1 , MN +1 ). On the other hand, we know that HN +2 = T1 /(I N +2 + (f N +1 ). This implies that for all monomials X ∈ B (N + 1) of degree N + 1, the coeﬃcient alij (X) = 0 for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . In other words, the truncated power series fij (l)N +1 is determined by the matric Massey products of order N + 1 above, since we have fij (l)N +1 = fij (l)N +

alij (X) · X

X∈B (N +1)

= fij (l)N +

yij (l)(x∗ ; X) · X

X∈B (N +1)

for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . Let hN +1 : HN +2 → HN +1 be the natural map, and consider its kernel. By deﬁnition, we have ker(hN +1 ) = aN +1 /aN +2 = ((f N ) + I N +1 )/((f N +1 ) + I N +2 ), so we can clearly ﬁnd a subset B(N + 1) ⊆ B (N + 1) of monomials of degree N + 1 such that {X : X ∈ B(N + 1)} ∪ {f ij (l)N : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij } is a basis for ker(hN +1 ). Let B(N + 1) = B(N + 1) ∪ B(N ), then clearly {X : X ∈ B(N + 1)} ∪ {f ij (l)N : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij } is a monomial basis for HN +2 . So for each monomial X ∈ T1 with |X| ≤ N + 1, we have a unique relation in HN +2 of the form X=

β(X, X ) X +

X ∈B(N +1)

β(X, i, j, l) fij (l)N ,

i,j,l

with β(X, X ), β(X, i, j, l) ∈ k for all X ∈ B(N + 1), 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . Since we have o(hN +1 , MN +1 ) = 0, we deduce that |X|≤N +1

x∗ ; X β(X, X ) = 0

NONCOMMUTATIVE DEFORMATIONS OF MODULES

119

for all X ∈ B(N + 1). Notice that β(X, X ) = 0 if the monomials X and X do not have the same type. Therefore, it makes sense to consider the 1-cocycle

β(X, X ) y(X),

|X|≤N +1

and by the relation above, this is a 1-coboundary. It follows that we can ﬁnd a 1-cochain α(X ) such that dα(X ) = − β(X, X ) y(X), |X|≤N +1

and we ﬁx such a choice. Consider the family {α(X) : X ∈ B(N + 1)}. This deﬁnes an M-free complex over HN +2 if and only if we have |X|≤N +1

β(X, Z)

α(X ) α(X ) = 0

X ,X ∈B(N +1) X X =X

for all Z ∈ B(N + 1). By the deﬁnition of α(X ) when X ∈ B(N + 1), this condition holds, and we denote by MN +2 ∈ DefM (HN +2 ) the deformation with the complex deﬁned by {α(X) : X ∈ B(N + 1)} as M-free resolution. It is clear from the construction that MN +2 is a lifting of MN +1 , so (HN +2 , MN +2 ) is a pro-representing hull for DefM restricted to ap (N + 2). It is clear that we can continue in this way. For every k ≥ 1, we can calculate the coeﬃcients in the truncated power series fij (l)N +k , and therefore ﬁnd HN +k+1 . At the same time, we ﬁnd the deﬁning systems {α(X) : X ∈ B(N + k)} necessary to calculate the matric Massey products of order N + k + 1, and these deﬁning systems completely determine the deformation MN +k+1 . We have described how to do this in the case k = 1, and the general case is similar. We conclude that the method that we have described above can be used to calculate the pro-representing hull (Hn , Mn ) for the deformation functor DefM restricted to ap (n) for any n ≥ N . We can therefore, in principle, ﬁnd the hull H = lim Hn ←

of DefM , and also the corresponding versal family deﬁned over H, ξ = M = lim Mn . ←

It follows that the pro-representing hull (H, ξ) of the deformation functor DefM can be calculated using matric Massey products. 8. An example Let k be an algebraically closed ﬁeld of characteristic 0, and let A = A2 (k) be the second Weyl algebra over k. We shall think of A as the ring of diﬀerential operators in the plane deﬁned over k with coordinates x and y. Thus, we can write A = k[x, y]∂x, ∂y, where

120

EIVIND ERIKSEN

∂x = ∂/∂x and ∂y = ∂/∂y. In other words, A is the k-algebra generated by x, y, ∂x, ∂y with relations [∂x, x] = [∂y, y] = 1. Let us consider the family of left A-modules M = {M1 , M2 , M3 , M4 }, where Mi = A/Ii for 1 ≤ i ≤ 4 and Ii ⊆ A are left ideals given by I1 = A(∂x, ∂y)

I2 = A(∂x, y)

I3 = A(x, ∂y)

I4 = A(x, y)

We immediately notice that the left A-modules in the family M have the following free resolutions:

∂x ∂y ∂y −∂x 2 0 ←M1 ← A ←−−−− A ←−−−−−−− A ← 0 ∂x y y −∂x 0 ←M2 ← A ←−−−− A2 ←−−−−−− A ← 0

x ∂y ∂y −x 2 0 ←M3 ← A ←−−−− A ←−−−−−− A ← 0

x y y −x 2 0 ←M4 ← A ←−−− A ←−−−−− A ← 0

We consider the elements of the free A-modules An as row vectors, and the maps in the free resolutions above as right multiplication of these row vectors by the given matrices. Notice that for 1 ≤ i ≤ 4, the free A-module Lm,i in the free resolution of Mi does not depend upon i. We shall therefore write Lm = Lm,i for all m ≥ 0, 1 ≤ i ≤ 4. It is known that M is a family of simple holonomic left A-modules, so this family satisfy the ﬁniteness condition (FC). Therefore, there exists a pro-representing hull (H, ξ) for the deformation functor DefM : a4 → Sets by theorem 5.2. We shall use the methods from section 7 to construct this hull explicitly. Let us start by calculating ExtnA (Mi , Mj ) for n = 1, 2, 1 ≤ i, j ≤ 4. We need both the dimensions and k-linear bases for these vector spaces, where each basis vector is represented by a cocycle in the corresponding Yoneda complex. The calculations are straight-forward, so we only state the results here: 1 if i = 1 or i = 4 and j = 2 or j = 3, or 1 dimk ExtA (Mi , Mj ) = if i = 2 or i = 3 and j = 1 or j = 4, 0 otherwise dimk Ext2A (Mi , Mj )

1 if (i, j) = (1, 4), (2, 3), (3, 2), (4, 1), = 0 otherwise

NONCOMMUTATIVE DEFORMATIONS OF MODULES

121

We denote the basis vectors of Ext1A (Mj , Mi ) by x∗ij since there is at most one for each pair of indices (i, j). From the dimensions listed above, we see that we have the following basis vectors: x∗12 , x∗13 , x∗21 , x∗24 , x∗31 , x∗34 , x∗42 , x∗43 We choose a Yoneda representative for each vector x∗ij in this list, and we denote this representative by α(xij ). From the free resolutions above, we see that we can write each of these representatives in the form α(X) = {α(X)0 , α(X)1 }, where α(X)0 : L1 → L0 is right multiplication by a matrix ( ab ) with entries a, b ∈ A, and α(X)1 : L2 → L1 is right multiplication by a matrix ( c d ) with entries c, d ∈ A for each monomial X = xij . We ﬁnd the following representatives: α(x12 ) = α(x21 ) = α(x34 ) = α(x43 ) = {( 01 ) , ( 1 0 )} α(x13 ) = α(x31 ) = α(x24 ) = α(x42 ) = {( 10 ) , ( 0 −1 )} ∗ Similarly, we denote the basis vectors of Ext2A (Mj , Mi ) by yij since there is at most one for each pair of indices (i, j). From the dimensions listed above, we see that we have the following basis vectors: ∗ ∗ ∗ ∗ y14 , y23 , y32 , y41 ∗ We choose a Yoneda representative for each vector yij in this list, and we denote this representative α(yij ). From the free resolutions above, we see that we can write each of these representatives in the form

α(Y ) = {α(Y )0 }, where α(Y )0 : L2 → L0 is given by right multiplication of an element a ∈ A for each monomial Y = yij . We ﬁnd the following representatives: α(y14 ) = α(y23 ) = α(x32 ) = α(x41 ) = {( 1 )} This completes the calculations of ExtnA (Mi , Mj ) for n = 1, 2 and 1 ≤ i, j ≤ 4. We know that these calculations determine the hull at the tangent level, (H2 , ξ2 ). The next step is to ﬁnd the the hull H and the versal family ξ, and we shall employ the notations and methods of section 7 to accomplish this. Let N = 2, we know that this choice is always possible. As usual, we let T1 be the formal matrix algebra generated by the monomials xij in the above list, and let I = I(T1 ) be its radical. Furthermore, denote 2 n for (i, j) = (1, 4), (2, 3), (3, 2), (4, 1), and by fij the corresponding by fij = o(yij ) ∈ Iij truncated power series for each n ≥ N . First, we have to ﬁnd a deﬁning system {α(X) : |X| < 2} for the matric Massey products x∗ ; X when X is any monomial of degree 2 in T1 . This is easily done: The 1-cocycle α(ei ) is the free resolution of Mi for 1 ≤ i ≤ 4, and the 1-cocycle α(X) was chosen above for each monomial X = xij of degree 1.

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Let us calculate the matric Massey products of order 2: Using the deﬁning system given above, we ﬁnd that the cocycles y(X) representing the matric Massey products x∗ ; X are given by −1 if X = x12 x24 , x21 x13 , x34 x42 , x43 x31 , y(X)0 = 1 if X = x13 x34 , x24 x43 , x31 x12 , x42 x21 , 0 otherwise for all monomials X of degree 2 in T1 . This means that the corresponding matric Massey products are given by x12 , x24 = −y14

x13 , x34 = y14

x21 , x13 = −y23

x24 , x43 = y23

x31 , x12 = y32

x34 , x42 = −y32

x42 , x21 = y41

x43 , x31 = −y41 ,

and all other matric Massey products of order 2 are zero. This translates to the following 2 : truncated power series fij 2 = x13 x34 − x12 x24 f14 2 f23 = x24 x43 − x21 x13 2 f32 = x31 x12 − x34 x42 2 f41 = x42 x21 − x43 x31 2 2 2 2 By the general theory, we therefore have H3 = T1 /(f14 , f23 , f32 , f41 ) + I 3 . We know that we can ﬁnd a lifting ξ3 of ξ2 to H3 , and that (H3 , ξ3 ) is a pro-representing hull of DefM restricted to a4 (3). In order to ﬁnd ξ3 , we let B(2) = {X : |X| = 2} \ {x13 x34 , x24 x43 , x31 x12 , x42 x21 }. We also let B(2) = B(2) ∪ B(1), where B(1) = {X : |X| ≤ 1}. Then {X : X ∈ B(2)} is a monomial basis for H3 . We observe that if we choose α(X) = 0 for all X ∈ B(2), the family {α(X) : X ∈ B(2)} deﬁnes an M-free complex over H3 . In other words, this family completely deﬁnes the deformation ξ3 ∈ DefM (H3 ) lifting ξ2 . Clearly, we could continue in this way. But after the last computations, it is tempting 2 for (i, j) = (1, 4), (2, 3), (3, 2), (4, 1). Let us check if this is the case: to think that fij = fij 1 2 2 2 2 , f23 , f32 , f41 ), and choose a monomial basis B of T containing B(2). We put T = T /(f14 Furthermore, we let α(X) be as before when X ∈ B(2) and let α(X) = 0 for all monomials X ∈ B of degree at least 3. This choice corresponds to maps dT0 , dT1 of M-free modules over T , and a computation shows that 2 2 2 2 + f23 + f32 + f41 )) = 0. dT0 ◦ dT1 = (1 ⊗ (f14

So the family {α(X) : X ∈ B} deﬁnes an M-free complex over T , and therefore a deformation ξ ∈ DefM (T ) lifting ξ3 . This proves that H = T , or in other words, that H = T1 /(x13 x34 − x12 x24 , x24 x43 − x21 x13 , x31 x12 − x34 x42 , x42 x21 − x43 x31 ) 2 for all i, j. Moreover, the family is a pro-representing hull of DefM . In particular, fij = fij {α(X) : X ∈ B} deﬁnes the versal family ξ ∈ DefM (H).

NONCOMMUTATIVE DEFORMATIONS OF MODULES

123

Appendix A. Yoneda and Hochschild representations Let k be an algebraically closed (commutative) ﬁeld, let A be an associative k-algebra, and let M, N be left A-modules. In this appendix, we recall several diﬀerent descriptions of the k-vector space ExtnA (M, N ) for n ≥ 0. In particular, we show how to realize this cohomology group using the Yoneda and Hochschild complexes. A.1. The Yoneda representation. Fix free resolutions (L∗ , d∗ ) of M and (L∗ , d∗ ) of N . We shall write di : Li+1 → Li and di : Li+1 → Li for the diﬀerentials, and denote the augmentation morphisms by ρ : L0 → M and ρ : L0 → N . For all integers n ≥ 0, the cohomology group ExtnA (M, N ) is deﬁned to be the n’th cohomology group of the complex HomA (L∗ , N ), ExtnA (M, N ) = H n (HomA (L∗ , N )). Notice that in general, this Abelian group does not have a left A-module structure, but only a left C(A)-module structure, where C(A) is the centre of A. In particular, if A is commutative, then ExtnA (M, N ) has the structure of an A-module, and if A is a k-algebra, then ExtnA (M, N ) has the structure of a k-vector space. We denote by Hom∗ (L∗ , L∗ ) the Yoneda complex given by the given free resolutions. This complex is deﬁned in the following way: For each integer n ≥ 0, let Homn (L∗ , L∗ ) be the left A-module Homn (L∗ , L∗ ) = i HomA (Li+n , Li ). Moreover, let the diﬀerential dn : Homn (L∗ , L∗ ) → Homn+1 (L∗ , L∗ ) for n ≥ 0 be the A-linear map given by the formula dn (φ)i = φi dn+i + (−1)n+1 di φi+1 for all i ≥ 0, where we write φ = (φi ) with φi ∈ HomA (Li+n , Li ) for all i ≥ 0. It is easy to check that this map is a well-deﬁned diﬀerential, so the Yoneda complex is a complex of Abelian groups. We shall write H n (Hom(L∗ , L∗ )) for the cohomology groups of the Yoneda complex. Since the diﬀerential d = dn is left C(A)-linear, these cohomology groups have a natural structure as left C(A)-modules. Lemma A.1. For all integers n ≥ 0, there is a canonical isomorphism of left C(A)-modules H n (Hom(L∗ , L∗ )) ∼ = ExtnA (M, N ). Proof. There is a natural map fn : Homn (L∗ , L∗ ) → HomA (Ln , N ), given by f (φ) = ρ φ0 , where φ = (φi ) ∈ Homn (L∗ , L∗ ). It is easy to see that these maps are compatible with the diﬀerentials, and a small calculation show that fn induces an isomorphism on cohomology H n (Hom(L∗ , L∗ )) → ExtnA (M, N ) for all integers n ≥ 0. A.2. Definition of Hochschild cohomology. Let Q be an A-A bimodule. We deﬁne the Hochschild complex of A with values in Q in the following way: Let HCn (A, Q) = Homk (⊗nk A, Q) for all n ≥ 0. So any ψ ∈ HCn (A, Q) corresponds to a k-multilinear map from n copies of A into Q, and we shall therefore write ψ(a1 , . . . , an ) in place of ψ(a1 ⊗ · · · ⊗

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an ) for ψ ∈ HCn (A, Q), a1 , . . . , an ∈ A. Moreover, let dn : HCn (A, Q) → HCn+1 (A, Q) for n ≥ 0 be the k-linear map given by the formula (7)

dn (ψ)(a0 , . . . , an ) = a0 ψ(a1 , . . . , an ) +

n

(−1)i ψ(a0 , . . . , ai−1 ai , . . . , an )

i=1

+(−1)n+1 ψ(a0 , . . . , an−1 )an for all ψ ∈ HCn (A, Q), a0 , . . . , an ∈ A. Lemma A.2. HC∗ (A, Q) is a complex of k-vector spaces. Proof. Let ψ ∈ HCn (A, Q). Then ψ = dn (ψ) is a sum of n + 1 summands, and we denote these by ψ0 , . . . , ψn , in the order they appear in formula 7. We let ψ = dn+1 ψ = dn+1 dn ψ. for Each dn+1 ψi for 0 ≤ i ≤ n is a sum of n + 2 summands, and we denote these by ψij 0 ≤ j ≤ n + 1 in the order they appear in formula 7. A straight-forward calculation shows + ψj,i+1 = 0 for all indices i, j with 0 ≤ j ≤ n + 2, j ≤ i ≤ n + 1. Since that we have ψi,j ψ = ψij , it follows that ψ = 0 in HCn+2 (A, Q). Consequently, HC∗ (A, Q) is a complex of k-vector spaces. We deﬁne the Hochschild cohomology of A with values in Q to be the cohomology of the Hochschild complex HC∗ (A, Q), so we have HHn (A, Q) = H n (HC∗ (A, Q)) = ker(dn )/ Im(dn−1 ) for all n ≥ 0. In particular, the cohomology groups HHn (A, Q) have a natural structure as k-vector spaces. Let ψ ∈ HC1 (A, Q), then ψ is a 1-cocycle if and only if ψ(ab) = aψ(b) + ψ(a)b for all a, b ∈ A. So we have ker(d1 ) = Derk (A, Q). We say that a derivation ψ ∈ Derk (A, Q) is trivial if there is an element q ∈ Q such that ψ is of the form ψ(a) = aq − qa for all a ∈ A. Clearly, the set of trivial derivations is the image Im(d0 ). So HH1 (A, Q) ∼ = Derk (A, Q)/T where T is the trivial derivations of A into Q. A.3. The Hochschild representation. We remark that Q = Homk (M, N ) is an A-A bimodule in a natural way: For any a ∈ A, let La : M → M denote left multiplication on M by a, and La : N → N left multiplication on N by a. The bimodule structure is given by aφ = La φ, φa = φLa for a ∈ A, φ ∈ Homk (M, N ). We shall consider the Hochschild cohomology of A with values in Q = Homk (M, N ). By deﬁnition, we have that HH0 (A, Q) = HomA (M, N ) when Q = Homk (M, N ). So we have a natural isomorphism of k-vector spaces Ext0A (M, N ) ∼ = HH0 (A, Q). Notice that since n k ⊆ C(A), ExtA (M, N ) has a natural k-vector space structure for all n ≥ 0. It is possible to extend the above isomorphism to the higher cohomology groups: Proposition A.3. For all integers n ≥ 0, there is an isomorphism of k-vector spaces σn : ExtnA (M, N ) → HHn (A, Homk (M, N )). Proof. From Weibel [9], lemma 9.1.9, there is an isomorphism of k-vector spaces between HHn (A, Homk (M, N )) and ExtnA/k (M, N ) for n ≥ 0. But since k is a commutative ﬁeld, there is a canonical isomorphism between ExtnA/k (M, N ) and ExtnA (M, N ), see theorem 8.7.10 in Weibel [9].

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We shall give an explicit identiﬁcation of k-vector spaces between Ext1A (M, N ) and HH1 (A, Homk (M, N )): Let (L∗ , d∗ ) be a free resolution of M , with augmentation morphism ρ : L0 → M , and let τ : M → L0 be a k-linear section of ρ. For any 1-cocycle φ ∈ HomA (L1 , N ), let ψ = ψ(φ) ∈ Derk (A, Homk (M, N )) be the following derivation: For any a ∈ A, m ∈ M , let x = x(a, m) ∈ L1 be such that d0 (x) = aτ (m) − τ (am). Notice that such an x exists, and is uniquely deﬁned modulo the image Im d1 . We deﬁne ψ by the equation ψ(a)(m) = φ(x) with x = x(a, m). Since φ is a cocycle, ψ is a well-deﬁned homomorphism in Homk (A, Homk (M, N )), and a straight-forward calculation shows that ψ is a derivation. Lemma A.4. Assume that Ext1A (M, N ) is a finite dimensional k-vector space. Then the assignment φ → ψ(φ) defined in the above paragraph induces an isomorphism σ1 : Ext1A (M, N ) → HH1 (A, Homk (M, N )). Proof. Assume that φ is a co-boundary, so φ = d0 (φ ), where φ ∈ HomA (L0 , N ). Then ψ = d0 (φ ), where ψ = φ τ ∈ Homk (M, N ), so φ is a trivial derivation. Consequently, the assignment induces a well-deﬁned map of k-linear spaces. This map is furthermore injective: Assume that ψ is a trivial derivation, so ψ = d0 (ψ ), where ψ ∈ Homk (M, N ). Then, we can construct an A-linear map φ ∈ HomA (L0 , N ) in the following way: Choose a basis for L0 , and for each basis vector y ∈ L0 , choose y ∈ L1 such that d0 (y ) = y − ψ ρ(y). Then we deﬁne φ (y) = ψ ρ(y) + φ(y ) for each basis vector y ∈ L0 . We obtain a morphism φ ∈ HomA (L0 , N ) by A-linear extension, and d0 (φ ) = φ, so φ is a co-boundary. To show that σ1 is an isomorphism as well, it is enough to notice that dimk Ext1A (M, N ) = dimk HH1 (A, Homk (M, N )) by proposition A.3, since Ext1A (M, N ) has ﬁnite k-dimension. The identiﬁcation σn : ExtnA (M, N ) → HHn (A, Homk (M, N )) for n ≥ 2 can be constructed in a similar way. References ´ ements de math´ [1] N. Bourbaki, Alg`ebre commutative, El´ ematique, Masson, 1985. [2] Olav Arnfinn Laudal, Formal moduli of algebraic structures, Lecture notes in mathematics, no. 754, Springer-Verlag, 1979. [3] ——, A generalized burnside theorem, Preprint Series no. 42, University of Olso, 1995. [4] ——, Noncommutative deformations of modules, Preprint Series no. 2, University of Oslo, 1995. [5] ——, Noncommutative algebraic geometry, Preprint Series no. 28, University of Olso, 1996. [6] ——, Noncommutative algebraic geometry II, Preprint Series no. 12, University of Olso, 1998. [7] ——, Noncommutative algebraic geometry, Preprint Series no. 115, Max Planck Institute of Mathematics, 2000. [8] ——, Noncommutative deformations of modules, Homology, Homotopy and Applications 4 (2002), no. 2, 357–396. [9] Charles A. Weibel, An introduction to homological algebra, Cambridge studies in advanced mathematics, no. 38, Cambridge University Press, 1994. Institute of Mathematics, University of Warwick, Coventry CV4 7AL, UK E-mail address: [email protected]

SYMMETRIC FUNCTIONS, NONCOMMUTATIVE SYMMETRIC FUNCTIONS AND QUASISYMMETRIC FUNCTIONS II by MICHIEL HAZEWINKEL CWI, POBox 94079, 1090GB Amsterdam, The Netherlands

Abstract. Like its precursor this paper is concerned with the Hopf algebra of noncommutative symmetric functions and its graded dual, the Hopf algebra of quasisymmetric functions. It complements and extends the previous paper but is also selfcontained. Here we concentrate on explicit descriptions (constructions) of a basis of the Lie algebra of primitives of NSymm and an explicit free polynomial basis of QSymm. As before everything is done over the integers. As applications the matter of the existence of suitable analogues of Frobenius and Verschiebung morphisms is discussed. MSCS: 16W30, 05E05, 05E10, 20C30, 14L05 Key words and key phrases: symmetric function, quasisymmetric function, noncommutative symmetric function, Hopf algebra, divided power sequence, endomorphism of Hopf algebras, automorphism of Hopf algebras, Frobenius operation, Verschiebung operation, Adams operator, power sum, Newton primitive, Solomon descent algebra, cofree coalgebra, free algebra, dual Hopf algebra, lambda-ring, Leibniz Hopf algebra, Lie Hopf algebra, Lie polynomial, formal group, primitive of a Hopf algebra shuﬄe algebra, overlapping shuﬄe algebra.

1. Introduction As said before, [24], the symmetric functions are an exceedingly fascinating object of study; they are best studied from the Hopf algebraic point of view (in my opinion), although they carry quite a good deal more important structures, indeed so much that whole books do not suﬃce, but see [26, 27, 31, 33, 34]. The ﬁrst of the two generalizations to be discussed is the Hopf algebra, NSymm, of noncommutative symmetric functions (over the integers). As an algebra, more precisely a ring, this is simply the free associative ring over the integers, Z, in countably many indeterminates N Symm = ZZ1 , Z2 , . . .

(1.1)

and the coalgebra structure is given by the comultiplication determined by µ : Zn →

Zi ⊗ Zj ,

where

Z0 = 1

(1.2)

i+j=n

and i and j are in N ∪ {0} = {0, 1, 2, · · · }. The augmentation is given by ε(Zn ) = 0,

n = 1, 2, 3, . . .

(1.3)

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127

(and, of course ε(Z0 ) = ε(1) = 1). The Hopf algebra NSymm is a noncommutative covering generalization of the Hopf algebra of symmetric functions, Symm = Z[z1 , z2 , . . .]

(1.4)

where the zn are seen as either the elementary symmetric functions en or the complete symetric functions hn . The interpretation of the zn as the hn seems to work out somewhat nicer, for instance in obtaining the standard inner product autoduality of Symm in terms of the natural duality between NSymm and QSymm, the Hopf algebra of quasisymmetric functions, see [24], section 6. QSymm will be described and discussed later in this paper. The projection is given by N Symm −→ Symm,

Zn → Zn

(1.5)

and is a morphism of Hopf algebras. The systematic investigation of NSymm as a noncommutative generalization of Symm was started in [14] and continued in a whole slew of subsequent papers, e.g. [7, 8, 9, 20, 21, 22, 23, 25, 28, 29, 30, 32, 46]. It is amazing how much of the theory of Symm has natural noncommutative analogues. This includes Newton primitives, Schur functions, representation theoretic interpretations, determinental formulas now involving the quasideterminants of Gel’fand - Retakh, [12, 13]), Capelli and Sylvester identities, and much more. And, not rarely, the noncommutative versions are more elegant than their commutative counterparts. Note, however, that in most of these papers the noncommutative symmetric functions are studied over a ﬁxed ﬁeld K of characteristic zero and not over the integers (or a ﬁeld of positive characteristic). This makes quite a diﬀerence, see section 3 below. The papers [19, 20, 21, 22, 23] focuss on the case over the integers, as does the present paper. It should be stressed that NSymm attracts a lot of attention not only as a natural generalization of Symm. It turns up spontaneously. For instance in terms of representations of the Hecke algebras at zero, [8, 24, 30, 46] and as the direct sum of the Solomon descent algebras of the symmetric groups, [1, 10, 14, 35, 43, 44] and [39], Ch. 9. Moreover there are e.g. applications to noncommutative continued fractions, Pad´e approximants, and a variety of interrelations with quantum groups and quantum enveloping algebras, [2, 14, 29, 37]. Further, the duals, the quasisymmetric functions, ﬁrst turned up (under that name) in the theory of plane partitions and counting permutations with given descent sets, [15, 16, 45]. Actually, QSymm, precisely as the graded dual of NSymm, goes back at least to 1972 in the theory of noncommutative formal groups, [5]. See [20] for an outline of the role played by QSymm in that context. An application of NSymm to chromatic polynomials is in [11]. Given a Hopf algebra H, with multiplication m and comultiplication µ, a primitive in H is an element P of H such that µ(P ) = 1 ⊗ P + P ⊗ 1

(1.6)

The primitives of a Hopf algebra form a Lie algebra under the commutator product [P1 , P2 ] = P1 P2 − P2 P1

(1.7)

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MICHIEL HAZEWINKEL

which is denoted Prim(H). For any Hopf algebra there is strong interest ina description of its Lie algebra of primitives. For instance because of the Milnor - Moore theorem, [36], that says that a graded connected cocommutative Hopf algebra over a ﬁeld of characteristic zero is isomorphic to to the universal enveloping algebra of its Lie algebra of primitives. Also, far from unrelated, let Q(H) = I(H)/I(H)2 be the module of indecomposables of a graded Hopf algebra H. Here I(H) is the augmentation ideal of H. Then there is an induced duality between Q(H) and Prim(H ∗ ), and there is the (classical) Leray theorem that says that for a connected commutative graded Hopf algebra H over a characteristic zero ﬁeld any section of I(H) −→ Q(H) induces an isomorphism of the free commutative algebra over Q(A) to H. This last theorem now has been considerably generalized to the setting of operads, see [38], and the references quoted there. The ﬁrst main topic that is treated in some detail (but without proofs) in this survey is an explicit and algorithmic description of a basis over the integers of Prim(NSymm). A divided power sequence in a Hopf algebra H is a sequence of elements d = (d(0) = 1, d(1), d(2), . . .)

(1.8)

such that for all n µH (d(n)) =

d(i) ⊗ d(j)

i, j ∈ {1, 2, 3, . . .}

(1.9)

i+j=n

Note that d(1) is a primitive. Is is sometimes useful to write a DPS (divided power sequence) as a power series in a counting variable t: d(t) = 1 + d(1)t + d(2)t2 + d(3)t3 + · · ·

(1.10)

That makes it easier to talk about the inverse of a DPS (inverse power series), the product of two DPS’s (multiplication of power series) and shifted DPS’s: d(t) → d(tn ), all operations that give new DPS’s from old ones. When written in the form (1.10) a DPS is often called a curve. It turns out that each primitive of Prim(NSymm) can be extended to a divided power sequence. This is important because it implies that as a coalgebra NSymm is the cocommutative cofree graded coalgebra over the module Prim(NSymm). Now let QSymm be the graded dual Hopf algebra (over the integers) of NSymm. For an explicit description of QSymm, the Hopf algebra of quasisymmetric functions, see below in section 2. A most important question concerning QSymm is whether it is free polynomial as a commutative algebra. This has been an important issue since 1972, since it is crucial for the development of certain parts of the theory of noncommutative formal groups, [5, 6, 17]. The matter was ﬁnally settled in 1999, [21], in the positive sense that it is indeed free. A second proof follows from the cofreeness of NSymm. However, both these proofs fail to produce explicit generators. This has now also been taken care of, [23], and is the second main topic that will be discussed in some detail below. One most interesting and important aspect of the structure of Symm is the presence of two families of Hopf algebra morphisms that are called Frobenius and Verschiebung morphisms. They satisfy a large number of beautiful relations. The third main topic of this survey is to what extent these can be lifted to NSymm, respectively, extended to QSymm.

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There are both positive and negative results. However, the matter has not yet been quite completely settled. This paper is an expanded write-up of two talks that I gave on the subject: in Krasnoyarsk in August 2002 at the occasion of the International Conference “Algebra and its applications” in honour of the 70-th anniversary of V P Shunkov and the 65-th anniversary of V M Busarkin, and at the Z. Borewicz memorial conference in Skt Petersburg in September 2002.

2. The Hopf algebra QSymm of quasisymmetric functions Above, in the introduction, the graded Hopf algebra NSymm of noncommutative symmetric functions was deﬁned. The grading is deﬁned by wt(Zn ) = n

(2.1)

and, more generally, if α = [a1 , a2 , . . . , am ] is a nonempty word over the positive integers N = {1, 2, . . .}, let Zα be the noncomutative monomial Zα = Za1 Za2 · · · Zam

(2.2)

wt(Zα ) = wt(α) = a1 + · · · + am

(2.3)

then

Let Z[ ] = 1, where [ ] is the empty word, then the Zα , α ∈ N∗ , the monoid of words over N form a basis of NSymm (as a graded Abelian group). The empty word, and also Z[ ] = 1, has weight zero. As a free Abelian graded group QSymm, the graded dual of NSymm can be taken to be the free Abelian group with as basis N∗ , the words over the set of natural numbers. The duality is then < Zα , β >= δαβ

(2.4)

The duality induced comultiplication is easy to describe. It is ‘cut’:

[a1 , a2 , . . . , am ] →

m

[a1 , . . . , ai ] ⊗ [ai+1 , . . . , am ]

(2.5)

i=0

where of course [a1 , . . . , ai ] = [ ] = 1 if i = 0 and [ai+1 , . . . am ] = [ ] = 1 if i = m. The duality induced multiplication is more diﬃcult to describe. It is the socalled ‘overlapping shuﬄe multiplication’ which can be described as follows. Let α = [a1 , a2 , . . . , am ] and β = [b1 , b2 , . . . , bn ] be two compositions or words. Take a ‘sofar empty’ word with n + m − r slots where r is an integer between 0 and min{m, n}, 0 ≤ r ≤ min{m, n}. Choose m of the available n + m − r slots and place in it the natural numbers from α in their original order; choose r of the now ﬁlled places; together with the remaining n+m−r−m = n−r places these form n slots; in these place the entries from β in their orginal order; ﬁnally, for those slots which have two entries, add them. The product of

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MICHIEL HAZEWINKEL

the two words α and β is the sum (with multiplicities) of all words that can be so obtained. So, for instance, [a, b] ×osh [c, d] = [a, b, c, d] + [a, c, b, d] + [a, c, d, b] + [c, a, b, d] + [c, a, d, b] + [c, d, a, b]+ + [a + c, b, d] + [a + c, d, b] + [c, a + d, b] + [a, b + c, d] + [a, c, b + d]+ (2.6) + [c, a, b + d] + [a + c, b + d] and [1] ×osh [1] ×osh [1] = 6[1, 1, 1] + 3[1, 2] + 3[2, 1] + [3]. There is a concrete realization of QSymm much like the standard realization of Symm as the ring of symmetric functions in inﬁnitely many indeterminates x1 , x2 , . . . . See [34], Chapter 1 for some detail on how to work with inﬁnitely many indeterminates in this context. Let X be a ﬁnite or inﬁnite set (of commuting variables) and consider the ring of polynomials, R[X], and the ring of power series, R[[X]], over a commutative ring R with unit element in the commuting variables from X. A polynomial or power series f (X) ∈ R[[X]] is called symmetric if for any two ﬁnite sequences of indeterminates x1 , x2 , . . . , xn and y1 , y2 , . . . , yn from X and any sequence of exponents i1 , i2 , . . . , in ∈ N, the coeﬃcients in f (X) of xi11 xi22 . . . xinn and y1i1 y2i2 . . . ynin are the same. The quasi-symmetric formal power series are a generalization introduced by Gessel, [15], in connection with the combinatorics of plane partitions. This time one takes a totally ordered set of indeterminates, e.g. V = {v1 , v2 , . . .}, with the ordering that of the natural numbers, and the condition is that the coeﬃcients of xi11 xi22 . . . xinn and y1i1 y2i2 . . . ynin are equal for all totally ordered sets of indeterminates x1 < x2 < · · · < xn and y1 < y2 < · · · < yn . Thus, for example, x1 x22 + x2 x23 + x1 x23

(2.7)

is a quasi-symmetric polynomial in three variables that is not symmetric. Products and sums of quasi-symmetric polynomials and power series are again quasisymmetric (obviously), and thus one has, for example, the ring of quasi-symmetric power series QSymm∧ in countably many commuting variables over the integers and its subring QSymm

(2.8)

of quasi-symmetric polynomials in ﬁnite of countably many indeterminates, which are the quasi-symmetric power series of bounded degree. The notation is justiﬁed. The quasisymmetric functions in {x1 , x2 , . . .} in this sense are a concrete realization of the quasisymmetric functions as introduced above as the graded dual of NSymm. In detail, given a word α = [a1 , a2 , . . . , am ] over N, also called a composition in this context, consider the quasi-monomial function Mα =

i1 <···

xai11 xai22 . . . xaimm

(2.9)

deﬁned by α. It is now an easy exercise to verify that as power series in the {x1 , x2 , . . .} the Mα satisfy Mα Mβ = Mα×osh β where ×osh is the overlapping shuﬄe product of words just deﬁned above (2.6).

(2.10)

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131

3. NSymm and QSymm over a field of characteristic zero Let K be a ﬁeld of characteristic zero, in particular the ﬁeld of rational numbers Q. The Hopf algebras of noncommutative symmetric functions and quasisymmetric functions over K are denoted N SymmK = N Symm ⊗Z K,

QSymmK = QSymm ⊗Z K

(3.1)

As remarked in the introduction, working over a ﬁeld of characteristic zero tends to simplify things considerably. Not that then everything becomes clear and easy and straightforward. Very far from it; witness the many papers quoted in the introduction. However, it is certainly true, that for the three groups of questions which form the main topic of this paper: primitives of noncommutative symmetric functions, freeness of the algebra of quasisymmetric functions, existence of Frobenius and Verschiebung morphisms, things either reduce to known things, or become fairly straightforward. At least up to fairly explicitly given isomorphisms. And in that connection it is well to reﬂect that knowing something well up to an isomorphism can be not all the same as really controlling things. To start with, consider another Hopf algebra over the integers, the Lie Hopf algebra U = ZU1 , U2 , . . .,

µ(Un ) = 1 ⊗ Un + Un ⊗ 1,

ε(Un ) = 0,

n = 1, 2, . . .

(3.2)

That is, as an algebra U is the free algebra over the integers in the noncommuting variables Un and the coalgebra structure is given by requiring that the two right hand formulas of (3.2) are algebra homomorphisms. The primitives of U are called Lie polynomials and they form the free Lie algebra over Z in the alphabet {U1 , U2 , · · · }. Also the universal enveloping algebra of Prim(U) is U. Now, over the rationals (and hence over any ﬁeld of characteristic zero or ring containing Q) N Symm1 and U are isomorphic. One particularly beautiful isomorphism ϕ N SymmQ −→UQ is given by setting 1 + Z1 t + Z2 t2 + Z3 t3 + · · · = exp(U1 t + U2 t2 + U3 t3 + · · · ) =

∞

(U1 t + U2 t2 + U3 t3 + · · · )i

(3.3)

i=0

where t is a (counting) variable commuting with everything in sight. Equating equal powers of t on the left and right hand sides of (3.3) gives Zn =

i1 + · · · ik = n ij ∈ N

Ui1 Ui2 · · · Uik k!

(3.4)

For a proof that the algebra isomorphism deﬁned by setting ϕ(Zn ) equal to the right hand side of (3.4) deﬁnes an isomorphism of Hopf algebras see [18]. Thus, up to the isomorphism ϕ, the matter of describing Prim(NSymm), comes down to describing Prim(U), or, equivalently writing down some explicit bases for the free Lie algebra 1 The

Hopf algebra NSymm is sometimes called the Leibniz Hopf algebra.

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generated by a countable set of indeterminates {U1 , U2 , . . .}. The matter of constructing bases for free Lie algebras has had a lot af attention. There are many more or less diﬀerent bases such as Hall bases, Shirshov bases, Lyndon bases, see [3, 39, 41, 42, 47]. Here, partly because most of the concepts will be needed below anyway, is a description of the Lyndon basis (also called Chen - Fox - Lyndon basis). Consider the free monoid N* of words on the alphabet N (or any other totally ordered alphabet for that matter). The lexicographic order (also called dictionary order, or alphabetical order) on N* is deﬁned as follows. If α = [a1 , a2 , . . . , am ] and β = [b1 , b2 , . . . , bn ] are two words of length m and n respectively, lg(α) = m, lg(β) = n,

α >lex

∃k ≤ min{m, n} such that a1 = b1 , . . . , ak−1 = bk−1 and ak > bk β⇔ or lg(α) = m > lg(β) = n and a1 = b1 , . . . , an = bn

(3.5)

The empty word is smaller than any other word. This deﬁnes a total order. Of course, if one accepts the dictum that anything is larger than nothing, the second clause of (3.5) is superﬂuous. The proper tails (suﬃxes) of the word α = [a1 , a2 , . . . , am ] are the words [ai , ai+1 , . . . am ], i = 2, 3, . . . , m. Words of length 1 or 0 have no proper tails. The preﬁx corresponding to a tail α = [ai , ai+1 , . . . am ] is α = [a1 , . . . , ai−1 ] so that α = α ∗ α where * denotes concatenation of words. A word is Lyndon iﬀ it is lexicographically smaller than each of its proper tails. For instance [4], [1, 3, 2], [1, 2, 1, 3] are Lyndon and [1, 2, 1] and [2, 1, 3] are not Lyndon. For each Lyndon word α of length > 1 consider the lexicographically smallest proper tail α of α. Let α be the corresponding preﬁx to α . Then α and α are both Lyndon and α = α ∗ α is called the canonical factorization of α. A basis of the free Lie algebra on {U1 , U2 , . . .}, i.e. a basis of Prim(U) ⊂ U, is now obtained as follows. For each word α = [a1 , a2 , . . . , am ] let Uα = Ua1 Ua2 . . . Uam be the corresponding monomial. Now, by recursion in length, deﬁne for a word of length 1 Q[i] = Ui

(3.6)

and for α Lyndon and of length lg(α) ≥ 2 let α = α ∗ α be its canonical factorization and set Qα = [Qα , Qα ]

(3.7)

then the {Qα : α Lyndon} form a basis of Prim(U) ⊂ U. For a proof see e.g. [39], p. 105ﬀ. The next topic to be taken up is the matter of the freeness of QSymm over the rationals. The graded dual of U is the socalled shuﬄe algebra. As a free module over Z it has the words over N as a basis and the product is the shuﬄe product which is like the overlapping shuﬄe product except that the overlap terms, i.e. those which involve additions of entries are left out. Thus for example [a, b] ×sh [c, d] = [a, b, c, d] + [a, c, b, d] + [a, c, d, b] + [c, a, b, d] + [c, a, d, b] + [c, d, a, b] (compare (2.6) above). It is well known that the shuﬄe algebra is free polynomial with as generators (for example) the Lyndon words. See, for example, [39]. p. 111 for a proof. Thus via the

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isomorphism ϕ, or rather its graded dual, it follows that QSymmQ is a free commutative algebra. But the description of the generators is rather involved and they do not look very nice. Actually the situation is rather better and a modiﬁcation of the proof of the freeness of the shuﬄe algebra (using a diﬀerent ordering on words) gives that in fact QSymmQ is commutative free polynomial on the Lyndon words. The ordering to be used is the wll-ordering. The acronym stands for weight ﬁrst, than length, than lexicographic. See [20] for details. The third main topic of this survey is the existence of Frobenius and Verschiebung type Hopf algebra endomorphisms of NSymm and QSymm which lift, respectively extend, those on Symm. Again, over the rationals, this is a relatively straightforward matter. Though there are some unanswered questions. Recall the situation for Symm, see [17, 24] for more details. On Symm there are two families of Hopf algebra endomorphims, called Frobenius and Verschiebung morphisms, denoted fn , vn , n ∈ N, which among others have the following beautiful properties: (i) f1 = v1 = id (ii) fn is homogeneous of degree n, i.e. fn (Symmk ) ⊂ Symmnk Here, for any graded Hopf algebra, H, Hn is the homogeneous part of of weight n of H. (iii) vn is homogenous of degree n−1 , i.e. vn (Symmk ) ⊂ Symn−1 k if n divides k, and vn (Symmk ) = 0 if n does not divide k. (iv) fn fm = fnm for all n, m ∈ N (v) vn vm = vnm for all n, m ∈ N (vi) fn vm = vm fn provided n and m are relatively prime, gcd(m, n) = 1 (vii) vn fn = n, where n is the n-fold convolution of the identiy. Now there is the natural projection N Symm −→ Symm, Zn → hn

(3.8)

and the natural (graded dual) inclusion Symm ⊂ QSymm

(3.9)

obtained by regarding a symmetric function as a special kind of quasisymmetric function. The question is whether there are lifts, respectively extensions, on NSymm, respectively QSymm, which also have the properties (i) - (vii). Retaining property (vii) can be ruled out immediately for trivial reasons. The simple fact is that n on either Qsymm or NSymm simply is not a Hopf algebra endomorphism. So it is natural to concentrate on the other six properties. And then the answer over the rationals is yes. But, as will be stated below, the answer over the integers is no. But there are interesting substitutes. Let pn = xn1 + xn2 + xn3 . . .

(3.10)

denote the power sums in Symm. They are related to the complete symmetric functions by the recursion relation nhn = pn + pn−1 h1 + pn−2 h2 + · · · p1 hn−1

(3.11)

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MICHIEL HAZEWINKEL

The Frobenius and Verschiebung morphisms on Symm are characterized by

fn pk = pnk ,

npk/n if n divides k, vn p k = 0 if n does not divide k.

(3.12)

On the polynomial generators hn this characterization of vn works out as hk/n if n divides k, vn h k = 0 otherwise.

(3.13)

Deﬁne the (noncommutative) Newton primitives in NSymm by Pn (Z) =

(−1)k+1 rk Zr1 Zr2 . . . Zrk ,

ri ∈ N = {1, 2, . . .}

(3.14)

r1 +···rk =n

or, equivalently, by the recursion relation nZn = Pn (Z) + Z1 Pn−1 (Z) + Z2 Pn−2 (Z) + · · · + Zn−1 P1 (Z)

(3.15)

Note that under the projection Zn → hn by (3.15) and (3.11) Pn (Z) goes to pn . It is easily proved by induction, using (3.15), or directly from (3.14), that the Pn (Z) are primitives of NSymm, and it is also easy to see from (3.15) that over the rationals NSymm is the free associative algebra generated by the Pn (Z). Thus over the rationals the Lie algebra of primitives of NSymm is simply the free Lie algebra generated by the Pn (Z), giving a second description of Prim(N SymmQ ). There are obvious candidate lifts of the vn on Symm to Hopf algebra endomorphisms on NSymm., viz Zk/n if k is divisible by n vn (Zk ) = (3.16) 0 otherwise By (3.14) or (3.15) this implies nPk/n if n divides k vn (Pk ) = 0 otherwise

(3.17)

Now on N SymmQ deﬁne the Frobenius morphisms as the algebra morphisms given by fn (Pk (Z)) = Pnk (Z)

(3.18)

It is now easily checked that the vn and fn as deﬁned by (3.16) and (3.18) are Hopf algebra endomorphisms of N SymmQ , that they satisfy (the analogues on N SymmQ of) properties (i)-(vi) and that they descend to the usual Frobenius and Verschiebung morphisms on Symm.

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A priori, the fn as deﬁned by (3.18) are only deﬁned over the rationals and indeed nontrivial denominators show up almost immediately. For instance f2 (Z1 ) = 2Z2 − Z12 f2 (Z2 ) = 2Z4 − 32 Z1 Z3 − 12 Z3 Z1 + Z22 + 12 Z1 Z2 Z1 + 12 Z12 Z2

(3.19)

On Symm a certain amount of coeﬃcient magic sees to it that all coeﬃcients become integral. But of course over Symm there are much better deﬁnitions of the Frobenius morphisms that immediately show that they are deﬁned over the integers, see [24] or [17], §17. As we shall see later, over the integers there are even no algebra endomorphisms fn of NSymm that lift the fn on Symm such that together with the vn as deﬁned by (3.16) they satisfy (i)-(vi). Note there is nothing unique about this solution (3.18) of the Frobenius-Verschiebung lifting problem over the rationals. For instance one could work instead with the seond set of Newton primitives deﬁned by Pn (Z) =

(−1)k+1 r1 Zr1 Zr2 . . . Zrk ,

ri ∈ N = {1, 2, . . .}

(3.20)

r1 +···rk =n

and satisfying the recursion relation nZn = Pn (Z) + Pn−1 (Z)Z1 + Pn−2 (Z)Z2 + · · · + P1 (Z)Zn−1

(3.20)

4. The primitives of NSymm Above, some primitives of NSymm were already written down and they generate a free graded Lie algebra contained in Prim(NSymm). Denote this Lie algebra by FrLie(P ) and its homogeneous part of weight n by FrLie(P )n . The Lie algebra Prim(NSymm) is also graded of course. Let Prim(N Symm)n ⊂ N Symmn be the homogeneous part of degree n of Prim(NSymm). Both Prim(N Symm)n and FrLie(P )n are free Abelian groups of rank βn , the number of weight n Lyndon words2 . The index of FrLie(P )n ⊂ Prim(N Symm)n as a function of n measures how large FrLie(P ) is in Prim(NSymm). As it turns out FrLie(P ) is only a tiny part. Indeed, the value of the index alluded to is

Index of FrLie(P )n in Prim(N Symm)n =

α∈LYN,

wt(α)=n

k(α) g(α)

(4.1)

where for a word α = [a1 , a2 , . . . , am ] over the natural numbers g(α) is the gcd (greatest common divisor) of its entries a1 , a2 , . . . , am and k(α) is the product of its entries. Thus the values of (4.1) for the ﬁrst six n are 1, 1, 2, 6, 576, 69120. Thus taking iterated commutators of the known Newton primitives is not nearly good enough. One can see this coming very quickly. Indeed [P1 , P2 ] = 2(Z1 Z2 − Z2 Z1 ). It also follows that Prim(NSymm) is not a free Lie algebra over the integers. Rather it tries to be something like a free divided power Lie algebra (though I do not know what such a thing might be). 2 The

numbers βn are given by the identity (1 − t)−1 (1 − 2t) = Witt, [48].

∞

n=1 (1

− tn )βn which goes back to

136

MICHIEL HAZEWINKEL

Instead of taking commutators of primitives it turns out to be a good idea to work with whole DPS’s (divided power sequences, see (1.8)). Accordingly, the next thing to be described are techniques for producing new divided power sequences from known ones. There are two more techniques for this (besides the ones mentioned in the introduction, which do not suﬃce) coming from two socalled isobaric decomposition theorems. For the ﬁrst isobaric decomposition theorem consider the Hopf algebra

2N Symm = ZX1 , Y1 , X2 , Y2 , · · · , µ(Xn ) =

Xi ⊗ Xj ,

µ(Yn ) =

i+j=n

Yi ⊗ Yj (4.2)

i+j=n

and the two natural curves X(s) = 1 + X1 s + X2 s2 + · · · ,

Y (t) = 1 + Y1 t + Y2 t2 + · · ·

(4.3)

and consider the commutator product X(s)−1 Y (t)−1 X(s)Y (t)

(4.4)

On the set of pairs of nonnegative integers consider the ordering (u, v) <wl (u , v )

⇔

u + v < u + v or (u + v = u + v and u < u )

(4.5)

(Here the index wl on <wl is supposed to be a mnemonic for weight ﬁrst, then lexicographic.) 4.6 Theorem. (ﬁrst bi-isobaric decomposition theorem, Shay [40], Ditters). There are ‘higher commutators’ (or perhaps better ‘corrected commutators’) Lu,v (X, Y ) ∈ ZX, Y , (u, v) ∈ N × N

(4.7)

such that X(s)−1 Y (t)−1 X(s)Y (t) =

→

(1 + La,b (X, Y )sa tb + L2a,2b (X, Y )s2a t2b + · · · )

(4.8)

gcd(a,b)=1

where the product is an ordered product for the ordering <wl just introduced, (4.5). Moreover (i) Lu,v (X, Y ) = [Xu , Yv ] + (terms of length ≥ 3) (ii) Lu,v (X, Y ) is homogeneous of weight u in X and of weight v in Y . (iii) For gcd(a, b) = 1, 1 + La,b (X, Y )sa tb + L2a,2b (X, Y )s2a t2b + · · · is a 2-curve.

(4.9) (4.10)

SYMM, NSYMM AND QSYMM FUNCTIONS

137

Here a 2-curve is a two dimensional version of a curve. A power series in two variables with constant term 1 d(i, j)si tj (4.11) d(s, t) = i,j

is a 2-curve iﬀ

µ(d(m, n)) =

d(m1 , n1 ) ⊗ d(m2 , n2 )

(4.12)

m1 +m2 =m n1 +n2 =n

This is not at all diﬃcult to prove. The only thing needed is to observe that pure powers of s or t do not occur on the LHS of (4.8) and that each pair of nonnegative integers (u, v) occurs just once in one of the factors on the right of (4.8). That gives the decomposition. The fact that the factors are two curves then follows easily with induction from the observation that the LHS of (4.8) is a 2-curve. Also (4.8) implies an explicit recursion formula for the Lu,v (X, Y ). 4.13. Theorem. (Second bi-isobaric decomposition theorem, Hazewinkel [22]). There are unique homogeneous noncommutative polynomials Nu,v (Z) ∈ N Symm such that

Z(s)−1 Z(t)−1 Z(s + t) =

→

(1 + Na,b (Z)sa tb + N2a,2b (Z)s2a t2b + · · · ).

(4.14)

a,b∈N gcd(a,b)=1

Moreover

u+v Zu+v + (terms of length ≥ 2) u (ii) Nu,v (Z) is homogeneous of weight u + v (iii) For each a, b ∈ N2 , gcd(a, b) = 1, 1 + Na,b (Z)sa tb + N2a,2b (Z)s2a t2b + · · · is a 2-curve. (iv) For each n ≥ 2, N1,n−1 (Z) = Pn (Z) (i) Nu,v (Z) =

(4.15) (4.16) (4.17) (4.18)

Again, the decomposition is not at all diﬃcult to prove and the ﬁnal observation results directly from the recursion formula implied by (4.14) compared to the recursion formula (3.15) for the Pn (Z). There is now a suﬃciency of tools to describe a basis of Prim(NSymm) and more. The essential fact is that if d1 , d2 are two divided power sequences in some Hopf algebra H, than so is Da,b (d1 , d2 ) = (1, La,b (d1 , d2 ), L2a,2b (d1 , d2 ), . . .)

(4.19)

where, as the notation suggests, Lu,v (d1 , d2 ) is obtained from the Lu,v (X, Y ) of theorem 4.6 by substituting d1 (k) for Xk and d2 (l) for Yl . This follows immediately from the fact that for gcd(a, b) = 1 1 + La,b (X, Y )sa tb + L2a,2b (X, Y )s2a t2b + · · ·

138

MICHIEL HAZEWINKEL

is a 2-curve. Similarly, if d is a curve in any Hopf algebra H Na,b (d) = 1 + Na,b (d)t + N2a,2b (d)t2 + · · ·

(4.20)

is another curve.3 Let LYN denote the set of Lyndon words over the natural numbers N = {1, 2, 3, . . .}. Then to each α = [a1 , . . . , am ] ∈ LYN there are associated three things (i) A number g(α) = gcd{a1 , . . . , am } (ii) A divided power sequence dα (iii) A primitive Pα The items (ii) and (iii) are deﬁned recursively as follows. If lg(α) = m is 1, d[n] = (1, Z1 , Z2 , . . .) and P[n] = Pn (d[n] ) = Pn (Z). If lg(α) ≥ 2, let α = α ∗ α be the canonical factorization of α (see just above (3.6)). Then dα = (1.Lg(α )/g(α),g(α )/g(α) (dα , dα ), L2g(α )/g(α),2g(α )/g(α) (dα , dα ), . . .) = Dg(α )/g(α),g(α )/g(α) (dα , dα )

(4.21)

and Pα = Pg(α) (dα )

(4.22)

Note that the divided power sequences associated to [a1 , . . . , am ] and [ra1 , . . . , ram ], r ∈ N are the same. 4.23. Theorem. The Pα , α ∈ LYN form a basis over the integers of Prim(NSymm). Each of the Pα is the ﬁrst term of a DPS. The second property of theorem 4.23 guarantees that NSymm is the cofree cocommutative graded coalgebra over the graded module Prim(NSymm), see the appendix of [22] for a proof of that. It follows that the graded dual QSymm is a free commutative algebra, and implicitly speciﬁes a set of generators for QSymm. However, this does not give a convenient description of such a set of generators. The original proof of theorem 4.23, [22], is rather long and intricate. Fortunately there is now a much shorter proof, which will be discussed in the next section. The basis Pα , α ∈ LYN of Prim(NSymm) has a number of nice properties, particularly with respect to the Verschiebung endomorphisms vn of NSymm, see (3.16). Consider the following ordering on words: α <wll β if and only if (wt(α) < wt(β) or (wt(α) = wt(β) and lg(α) < lgβ)) or (wt(α) = wt(β) and lg(α) = lgβ) and α

3 The operations on curves defined by (4.19) and (4.20) are functorial. Thus assigning to a Hopf algebra H its group of curves gives a group valued functor with many operations on it, viz 2-ary operations La,b for every pair a, b with gcd(a, b) = 1, and also unary operations Na,b for any such pair. In addition there are the operations which come from the Verschiebung Hopf algebra endomorphisms on NSymm (see section 6) which commute with the La,b and Na,b . And then there are also Frobenius type operations (see also section 6). It is almost totally uninvestigated whether some subset of all this structure can be used for classification purposes, say, of noncommutative formal groups.

SYMM, NSYMM AND QSYMM FUNCTIONS

139

4.24. Theorem. For each Lyndon word α = [a1 , . . . , am ] (i) Pα = g(α)Zα + (wll larger terms)

(ii)

(4.25)

nP[n−1 a1 ,··· ,n−1 am ] if n divides g(α) vn (Pα ) = 0 otherwise.

(4.26)

The ﬁrst property of theorem 4.24 results directly from the construction. The second Nra,rb (Z)sra trb property comes from the fact that the curves Lra,rb (X, Y )sra trb and of the two isobaric decomposition theorems 4.6 and 4.13 are particularly nice curves in that Ln−1 u,n−1 v (X, Y ) if n divides u, v, vn (Lu,v (X, Y )) = 0 otherwise. Nn−1 u,n−1 v (Z) if n divides u, v vn (Nu,v (Z)) = 0 otherwise.

(4.27)

(4.28)

where vn is deﬁned on 2NSymm just like (3.16) for both the X’s and the Y ’s. 5. Free polynomial generators for QSymm over the integers As in section 2 above, consider Symm and QSymm as symmetric functions in an inﬁnity of ideterminates Symm = Z[e1 , e2 , . . .] ⊂ QSymm ⊂ Z[x1 , x2 , . . .]

(5.1)

where the ei are the elementary symmetric functions in the xj . There is a well known λ-ring structure on Z[x1 , x2 , . . .] given by xj λi (xj ) = 0

if i = 1 , if i ≥ 2

j = 1, 2, . . .

(5.2)

For information on λ-rings see [27]. The asscociated Adams operators, determined by the formula ∞ d (−1)n fn (a)tn (5.3) t log λt (a) = dt n=1 where λt (a) = 1 +

∞

λn (a)tn

(5.4)

n=1

are the ring endomorphisms fn : xj → xnj

(5.5)

140

MICHIEL HAZEWINKEL

There are well-known determinantal relations between the λn and the fn as follows 0 ··· 0 .. f2 (a) f1 (a) 2 . . . . . . .. .. n!λn (a) = det .. .. . . 0 fn−1 (a) fn−2 (a) · · · f1 (a) n − 1 fn (a) fn−1 (a) · · · f2 (a) f1 (a)

(5.6)

··· 0 .. .. . λ1 (a) 1 . λ2 (a) . . . . fn (a) = det .. .. .. .. 0 (n − 1)λn−1 (a) λn−2 (a) · · · λ1 (a) 1 λn−1 (a) · · · λ2 (a) λ1 (a) nλn (a)

(5.7)

f1 (a)

λ1 (a)

1

1

0

(which come from the Newton relations between the elementary symmetric functions and the power sum symmetric functions). It follows that the subrings Symm and QSymm are stable under the λn and fn because λn (QSymm ⊗z Q) ⊂ QSymm ⊗z Q by (1.6) and (QSymm ⊗z Q) ∩ Z[x1 , x2 , . . .] = QSymm, and similarly for Symm. It follows that QSymm and Symm have induced λ-ring structures. On Symm this is of course the standard one. It follows immediately from (5.5) that fn ([a1 , a2 , . . . , am ]) = [na1 , na2 , . . . , nam ]

(5.8)

when α = [a1 , a2 , . . . , am ] is seen as a monomial quasisymmetric function as in the realization of QSymm described in section 2. It is more customary to denote the Adams operations (power operations) associated to a λ-ring structure by Ψ‘s. However, in the present case they coincide with the Frobenius endomorphisms on Symm ([17], section E.2, p. 144ﬀ), and their natural extensions to QSymm; so it seems natural to use f ’s instead in this case. For any λ − ring R there is an associated mapping Symm × R −→ R, (ϕ, a) → ϕ(λ1 (a), λ2 (a), . . . , λn (a), . . .)

(5.9)

I.e. write ϕ ∈ Symm as a polynomial in the elementary symmetric functions e1 , e2 , . . . and then substitute λi (a) for ei , i = 1, 2, . . .. For a ﬁxed a ∈ R this is obviously a homomorphism of rings Symm −→ R. We shall often simply write ϕ(a) for ϕ(λ1 (a), λ2 (a), . . . , λn (a), . . .). Another way to see (5.9) is to observe that for ﬁxed a ∈ R(ϕ, a) → ϕ(λ1 (a), λ2 (a), . . .) = ϕ(a) is the unique homomorphism of λ-rings that takes e1 into a. (Symm is the free λ-ring on one generator, see also [27].) Note that en (α) = λn (α),

pn (α) = fn (α) = [na1 , na2 , . . . nam ]

(5.10)

The ﬁrst formula of (5.10) is by deﬁnition and the second follows from (5.7) because the relations between the en and pn are precisely the same as between the λn (a) and the fn (a).

SYMM, NSYMM AND QSYMM FUNCTIONS

141

Now let P ∈ N Symm be a primitive. Then, by duality P, αβ = µ(P ), α ⊗ β = 1 ⊗ P + P ⊗ 1, α ⊗ β = 0

(5.11)

for any words of length ≥ 1. Using the Newton relations pn (α) = pn−1 (α)e1 (α) − pn−2 (α)e2 (α) + · · · (−1)n−2 p1 (α)en−1 (α) + (−1)n−1 nen (α) (5.12) it follows that for any primitive in NSymm P, pn (α) = ±nP, en (α)

(5.13)

Now for any α ∈ LYN , α = [a1 , a2 , . . . am ] let αred = [g(α)−1 a1 , g(α)−1 a2 , . . . , g(α)−1 am ] and deﬁne Eα = eg(α) (αred )

(5.14)

5.15. Theorem. The Eα , α ∈ LYN form a free polynomial basis of QSymm over the integers For a rather simple direct proof of this (based on Chen - Fox - Lyndon factorization, [4]), see [23]. Sometimes it is useful to relabel the Eα , α ∈ LYN a bit. Let eLYN be the set of elementary Lyndon words, i.e those Lyndon words α for which g(α) = 1. Then the en (α), α ∈ eLYN, n = 1, 2, 3, . . .

(5.16)

are a sometimes convenient relabeling of the free polynomial basis Eα , α ∈ LYN . Note that for a ﬁxed α ∈ eLYN , the en (α) generate a subalgebra of QSymm that is isomorphic to Symm. Now, for all weights n, consider the matrices of integers (Pα , Eβ )α,β∈LYNn

(5.17)

where LYNn is the set of Lyndon words of weight n and the columns and rows of (5.17) are ordered by increasing wll-order. Claim: this matrix is diagonal with entries ±1 on the main diagonal. The triangularity follows from theorem 4.24 (ii). As to the diagonal part: g(α)Pα , Eα = g(α)Pα , λg(α) (αred ) by deﬁnition = ±Pα , pg(α) (αred ) by (5.13) = ±Pα , α by (5.10) = ±g(α) by (4.25) Using that there are precisely βn = #LY Nn Pα ‘s and Eα ‘s with α of weight n, the invertibility (over the integers) of the matrix (5.17) immediately implies both that the Pα are a basis of Prim(NSymm) and that the Eα are a free polynomial basis for QSymm over the integers

142

MICHIEL HAZEWINKEL

6. Frobenius and Verschiebung on NSymm and QSymm To ﬁx notations let fnSymm and vnSymm be the classical Frobenius and Verschiebung Hopf algebra endomorphisms over the integers of Symm, characterized by fnSymm (pk ) = pnk ,

npk/n if n divides k vnSymm (pk ) = 0 otherwise

(6.1)

Now NSymm comes with a canonical projection N Symm −→ Symm, Zn → hn and there is the inclusion Symm ⊂ QSymm (see section 2). The question arises whether ther are lifts fnNSymm , vnNSymm on NSymm and extensions fnQSymm , vnQSymm on QSymm that satisfy respectively (i) (ii) (iii) (iv) (v) (vi)

?Symm ?Symm = fnm fn?Symm fm fn is homogeneous of degree n, i.e. fn (?Symmk ) ⊂? Symmnk f1 = v1 = id fm vn = vn fm if (n, m) = 1 ?Symm ?Symm = vnm vn?Symm vm vn is homogeneous of degree n−1 , ?Symmn−1 k if n divides k i.e. vn (? Symmk ) ⊂ 0 otherwise.

(6.2)

Here ‘?’ can be ‘N’, or ‘Q’. Now there exist a natural lifts of the vnSymm to NSymm given by the Hopf algebra endomorphisms Zk/n if n divides k NSymm vn (Zk ) = (6.3) 0 otherwise and there exist natural extension of the Frobenius morphisms on Symm to QSymm ⊃ Symm given by the Hopf algebra endomorphisms fnQSymm ([a1 , . . . , am ]) = [na1 , . . . , nam ]

(6.4)

which, moreover, have the Frobenius-like property fpQSymm (α) = αp

mod p

(6.5)

for each prime number p. These two families are so natural and beautiful that nothing better can be expected and in the following these are ﬁxed as the Verschiebung morphisms on NSymm and Frobenius morphisms on QSymm. They are also dual to each other. The question to be examined in this section is whether there are supplementary families of morphisms fn on NSymm, respectively vn on QSymm, such that (6.2) holds. The ﬁrst result is negative

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6.6. Theorem. There are no (Verschiebung-like) coalgebra endomorphisms vn of QSymm that extend the vn on Symm, such that parts (iii)-(vi) of (6.2) hold. Dually there are no (Frobenius-like) algebra homomorphisms of NSymm that lift the fn on Symm such that parts (i)-(iv) of (6.2) hold. It is the coalgebra morphism property which makes it diﬃcult for (6.2) (iii)-(vi) to hold. It is not particularly diﬃcult to ﬁnd algebra endomorphisms of QSymm that do the job. For instance deﬁne the vn on QSymm as the algebra endomorphisms given on the generators (5.16) by ek/n (α) if n divides k vn (ek (α)) = (6.6) 0 otherwise It then follows from (5.12) that npk/n (α) if n divides k vn (pk (α)) = 0 otherwise

(6.7)

and all the properties (6.2) follow. The last topic I would like to discuss is whether there are Hopf algebra endomorphisms of QSymm (and dually, NSymm) such that some weaker versions of (6.2) hold. To this end we ﬁrst discuss a ﬁltration by Hopf subalgebras of QSymm. Deﬁne

Gi (QSymm) =

Zα ⊂ QSymm

(6.8)

α, lg(α)≤i

the free subgroup spanned by all α of length ≤ i, and let Fi (QSymm) = Z[en (α) : lg(α) ≤ i]

(6.9)

the subalgebra spanned by those generators en (α), α ∈ eLYN , lg(α) ≤ i of length less or equal to i. Note that this does not mean that the elements of Fi (QSymm) are bounded in length. For instance F1 (QSymm) = Symm ⊂ QSymm contains the elements [1, 1, . . . , 1] = en = en ([1])

(6.10)

n

for any n. 6.11. Theorem. Gi (QSymm) ⊂ Fi (QSymm) This is a consequence of the proof of the free generation theorem 5.15. Moreover, Fi (QSymm) ⊗z Q = Z[pn (α) : lg(α) ≤ i] Fi (QSymm) = Z[pn (α) : lg(α) ≤ i] ∩ QSymm

(6.11)

It follows from (6.11) and (6.12) that the Fi (QSymm) are not only subalgebras but sub pn (α ) ⊗ pn (α )).4 Hopf algebras (because µ(pn (α)) = α ∗α =α

4I

believe the corresponding Hopf ideals in NSymm to be the iterated commutator ideals.

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Now consider a coalgebra endomorphism of QSymm. Because of the commutative cofreeness of QSymm as a coalgebra over the module tZ[t] for the projection QSymm −→ tZ[t],

[ ] → 0, [n] → tn ,

α → 0 if lg(α) ≥ 2

∞ or, equivalently, because of the freeness of NSymm over its submodule i=1 ZZi , a homogeneous coalgebra morphism of degree n−1 of QSymm is necessarily given by an expression of the form vϕ (α) =

ϕ(α1 ) · · · ϕ(αr )[n−1 wt(α1 ), . . . , n−1 wt(αr )]

(6.12)

α1 ∗···∗αr =α

for some morphism of Abelian groups ϕ : QSymm −→ Z. The endomorphism vϕ is a Hopf algebra endomorphism iﬀ ϕ is a morphism of algebras. 6.13. Proposition. vϕ (Fi (QSymm)) ⊂ Fi (QSymm) One particularly interesting family of ϕ’s is the family of ring morphisms given by τn (en ([1]) = τn (en ) = (−1)n−1 τn (ek (α)) = 0 for k = n or lg(α) ≥ 2(α ∈ eLYN )

(6.14)

Let vn be the Verschiebung type Hopf algebra endomorphism deﬁned by τn according to formula (6.12). Then 6.15. Theorem. nm [n−1 a1 , . . . n−1 am ] mod(length m − 1) if n|ai ∀i (i) vn ([a1 , . . . , am ]) ≡ 0 mod(length m − 1) otherwise (ii) vn extends vnSymm on Symm = F1 (QSymm) ⊂ QSymm (iii) vp vq = vq vp on F2 (QSymm) (iv) vn fn (α) = nlg(α) α mod(Flg(α)−1 (QSymm)) And of course there is a corresponding dual theorem concerning Frobenius type endomorphisms of NSymm. This seems about the best one can do. One unsatisfactory aspect of theorem 6.15 is that there are also other families vn that work. To conclude I would like to conjecture a stronger version of theorem 6.6, viz that there is no family fn of algebra endomorphisms of NSymm over the integers that satisﬁes (6.2) (i)-(iii) and such that these fn descend to the fnSymm on Symm. References [1] F Bergeron, A Garsia, C Reutenauer, Homomorphisms between Solomon descent algebras, J. of Algebra 150 (1992), 503–519. [2] F Bergeron, D Krob, Acyclic complexes related to noncommutative symmetric functions, J. of algebraic combinatorics 6 (1997), 103–117. [3] N Bourbaki, Groupes et alg`ebres de Lie. Chapitres 2 et 3 (Alg`ebres de Lie libres; Groupes de Lie), Hermann, 1972. [4] K T Chen, R H Fox, R C Lyndon, Free diﬀerential calculus IV, Ann. Math. 68 (1958), 81–95. [5] E J Ditters, Curves and formal (co) groups, Inv. Math. 17 (1972), 1–20.

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[6] E J Ditters, Groupes formels, Lecture notes, Universit´e de Paris XI, Orsay, 1974. Chapitre II, §5, p. 29 [7] G´erard Duchamp, Alexander Klyachko, Daniel Krob, Jean-Yves Thibon, Noncommutative symmetric functions III: deformations of Cauchy and convolution algebras, Preprint, Universit´e la Marne-la-Vall´ee, 1996. [8] G Duchamp, D Krob, B Leclerc, J-Y Thibon, Fonctions quasi-symm´ etriques, fonctions symm´ etriques noncommutatives, et alg`ebres de Hecke ` a q = 0, CR Acad. Sci. Paris 322(1996), 107–112. [9] G Duchamp, D Krob, E A Vassilieva, Zassenhaus Lie idempotents, q-bracketing, and a new exponential/logarithmic correspondence, J algebraic combinatorics 12:3(2000), 251–278. [10] A M Garsia, C Reutenauer, A decomposition of Solomon’s descent algebra, Adv. in Mathematics 77(1989), 189–262. [11] David Gebhard, Noncommutative symmetric functions and the chromatic polynomial, Preprint, Dept. Math., Michigan State University, 1994. [12] I M Gel’fand, V S Retakh, Determinants of matrices over noncommutative rings, Funct. Anal. and Appl. 25(1991), 91–102. [13] I M Gel’fand, V S Retakh, A theory of noncommutative determinants and characteristic functions of graphs, Funct. Anal. and Appl. 26 (1992), 1–20. [14] Israel M Gel’fand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir S Retakh, JeanYves-Thibon, Noncommutative symmetric functions, Adv. Math. 112 (1995), 218–348. [15] Ira M Gessel, Multipartite P-partitions and inner product of skew Schur functions. In: 1984, 289–301. [16] Ira M Gessel, Christophe Reutenauer, Counting permutations with given cycle-structure and descent set, J Combinatorial Theory, Series A :64 (1993), 189–215. [17] Michiel Hazewinkel, Formal groups and applications, Acad. Press, 1978. [18] Michiel Hazewinkel, The Leibniz-Hopf algebra and Lyndon words, preprint, CWI, Amsterdam, 1996. [19] Michiel Hazewinkel, Generalized overlapping shuﬄe algebras. In: S M Aseev and S A Vakhrameev (ed.), Proceedings Pontryagin memorial conference, Moscow 1998, VINITI, 2000, 193–222. [20] Michiel Hazewinkel, Quasisymmetric functions. In: D Krob, A A Mikhalev and A V Mikhalev (ed.), Formal series and algebraic combinatorics. Proc of the 12-th international conference, Moscow, June 2000, Springer, 2000, 30–44. [21] Michiel Hazewinkel, The algebra of quasi-symmetric functions is free over the integers, Advances in Mathematics 164 (2001), 283–300. [22] Michiel Hazewinkel, The primitives of the Hopf algebra of noncommutative symmetric functions over the integers, Preprint, CWI, 2001. [23] Michiel Hazewinkel, Explicit polynomial generators for the ring of quasisymmetric functions over the integers, submitted C.R. Acad. Sci. Paris (2002), [24] Michiel Hazewinkel, Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions, Acta Appl. Math. 75 (2003), 55–83. [25] Florent Hivert, Hecke algebras, diﬀerence operators and quasi symmetic functions, Adv. Math. 155 (2000), 181–238. [26] P Hoﬀman, Exponential maps and λ-rings, J. pure and appl. Algebra 27 (1983), 131–162. [27] D Knutson, λ-Rings and the representation theory of the symmetric group, Springer, 1973. [28] D Krob, B Leclerc, J-Y Thibon, Noncommutative symmetric functions II: transformations of alphabeths, Int. J. Algebra and Computation 7:2 (1997), 181–264. [29] Daniel Krob, Jean-Yves Thibon, Noncommutative symmetric functions IV: quantum linear groups and Hecke algebras at q = 0 , J. of algebraic combinatorics 6 (1997), 339–376. [30] D Krob, J-Y Thibon, Noncommutative symmetric functions V: a degenerate version of Uq (glN ), Int. J. Algebra and Computation 9:3&4 (1997), 405–430. [31] A Lascoux, Polynˆ omes symm´ etriques, foncteurs de Schur et Grassmanniens, Th`ese, Univ. de Paris VII, 1977.

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[32] Bernard Leclerc, Thomas Scharf, Jean-Yves Thidbon, Noncommutative cyclic characters of the symmetric groups, J Combinatorial theory, Series A 75:1 (1996), 55–69. [33] A Liulevicius, Arrows, symmetries and representation rings, J. pure and appl. Algebra 19 (1980), 259–273. [34] I G Macdonald, Symmetric functions and Hall polynomials, Clarendon Press, 2-nd edition Edition, 1995. [35] C Malvenuto, Chr Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. of Algebra 177 (1994), 967–982. [36] John W Milnor, John C. Moore, On the structure of Hopf algebras, Ann. Math. 81:2 (1965), 211–264. [37] Alexander Molev, Noncommutative symmetric functions and Laplace operators for classical Lie algebras, Lett. Math. Phys. 35:2 (1995), 135–143. [38] Fr´ed´eric Patras, A Leray theorem for the generalization to operads of Hopf algebras of divided powers, J of Algebra 218 (1999), 528–542. [39] Chr Reutenauer, Free Lie algebras, Oxford University Press, 1993. [40] P. Brian Shay, An obstruction theory for smooth formal group structure, preprint, Hunter College, New York City Univ., undated, probably 1974. [41] A I Shirshov, Subalgebras of free Lie algebras, Mat. Sbornik 33 (1953), 441–452. [42] A I Shirshov, On bases for a free Lie algebra, Algebra i Logika 1 (1962), 14–19. [43] L Solomon, On the Poincar´e-Birkhoﬀ-Witt theorem, J. Combinatorial Theory 4 (1968), 363–375. [44] L Solomon, A Mackey formula in the group ring of a Coxeter group, J. of Algebra 41 (1976), 255–268. [45] R P Stanley, On the number of reduced decompositions of elements of Coxeter groups, Eur. J. Combinatorics 5 (1984), 359–372. [46] J-Y Thibon, B-C-V Ung, Quantum quasisymmetric functions and Hecke algebras, J. Phys. A: Math. Gen. 29 (1996), 7337–7348. [47] G´erard Xavier Viennot, Alg´ebres de Lie libres et mono¨ıdes libres, Springer, 1978. [48] E Witt, Die Unterringe der freien Lieschen Ringe, Math. Zeitschrift 64 (1956), 195–216.

QUOTIENT GROTHENDIECK REPRESENTATIONS J.NDIRAHISHA AND F.VAN OYSTAEYEN Dept. of Mathematics and Computer Science, University of Antwerp (UIA) B-2610 Wilrijk, Belgium @-mail: [email protected]

Abstract. We generalize the concept of a Grothendieck representation to quotient Grothendieck representations; this allows the construction of a noncommutative projective geometry for geometrically graded rings that need not be positively graded. Mathematics Subject Classifications (2000): 16 B 50; 14 A 15; 14 A 22. Key words: (Quotient) Grothendieck representations; topological nerve; spectral representation and scheme

Introduction A Grothendieck representation of an arbitrary category B has been introduced in [9], with an aim to explain the appearance of canonical noncommutative topologies in noncommutative geometry. This point of view also provided a natural framework for a theory of schemes for noncommutative algebras, including a general categorical version of Serre’s global sections theorem. Now the noncommutative theory of P roj may also be ﬁt into the context of a Grothendieck representation but if one wants to relate the projective to the corresponding “aﬃne” theory, two Grothendieck representations of the same category have to be compared. In this note we introduce the quotient of a Grothendieck representation with respect to a so-called topological nerve with respect to B. We introduce some restricting condition, i.e. a spectral representation, allowing to retranslate certain properties holding in the representing Grothendieck categories into properties related to a notion of spectrum in B. The main result in Section 1, states that the quotient Grothendieck representation of a spectral representation is again a spectral representation. In Section 2, we study an application to noncommutative algebra (or geometry). As an example of the general construction in Section 1, we provide the construction and basic noncommutative scheme theory for the scheme P roj of an arbitrary ZZ-graded ring (noncommutative and not necessarily positively graded). The main result following from these considerations is that P roj may indeed be viewed as a gluing of aﬃne noncommutative schemes.

1. Quotient Grothendieck Representations Let B be a general category allowing products and coproducts. To R in B we associate a Grothendieck category Rep(R), to f : S → R in B we associate an exact functor F : Rep(R) → Rep(S) commuting with products and coproducts (we sometimes

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write F = f 0 ) such that: (IR )0 = IRep(R) for every R ∈ B, (f ◦ g)0 = g 0 ◦ f 0 for g: T → S, f : S → R in B. A Grothendieck categorical representation (G. C.-representation) is a contravariant functor Rep : B → R where R is the class consisting of the objects Rep(R), R ∈ B with HomR (Rep(R), Rep(S)) = HomB (S, R)0 . The functor Rep associates to f : S → R in B the exact functor f 0 : Rep(R) → Rep(S) given before. We say that Rep: B → R measures B if for every R ∈ B there exists an object R R in Rep(R) such that to any f : S → R in B there is an associated morphism f ∗ : S S → Rep(f )(R R) in Rep(S), we put Rep(f )(R R) = S R, such that 1∗S is the identity of S S in Rep(S) and for g: T → S, f : S → R in B, (f ◦ g)∗ = Rep(g)(f ∗ ) ◦ g ∗ : T T →T S →T R, where TR = Rep(g)Rep(f )(R R). For detail on torsion theory (i.e localization theory, quotient categories, Serre localizing subcategories, . . .) in a Grothendieck category, we refer to [2], [12]. For an arbitrary Grothendieck category G, the set of all torsion theories of G will be denoted by T ors(G). It is known that T ors(G) is a complete distributive lattice. We use notation as in [9], e.g we write κ, τ, γ, . . ., for torsion theories of G and Tκ , Tτ , Tγ , . . . resp., for the classes of torsion objects in G. A partial order, ≤, is deﬁned on T ors(G) by putting σ ≤ τ if and only if Tσ ⊂ Tτ . An M ∈ G is in Tσ∧τ if and only if M is both in Tσ and Tτ ; on the other hand Tσ∨τ is the torsion class generated by Tσ and Tτ . Recall that torsion classes in G are exactly those that are closed under taking: subobjects, images, ﬁnite products, extensions. For τ ∈ T ors(G), Tτ : G → G is the torsion functor corresponding to τ , i.e for M ∈ G, Tτ (M ) is the largest object of Tτ contained in M . Any subfunctor T of the identity functor in G is of type Tτ (-) if and only if it is left exact and idempotent in the sense that T (M/T (M )) = 0. As in [9] we abbreviate T orsRep to T op. To τ ∈ T ors(G) we associate the Serre quotient category (G, τ ) together with the canonical functors: iτ : (G, τ ) → G aτ : G → (G, τ ), the reflector for τ , with aτ iτ the identity of (G, τ ) and iτ aτ = Qτ the localization functor G → G associated to τ . To any exact functor F : Rep(R) → Rep(S) that commutes with direct sums (coproducts) there corresponds a functor F 0 : T op(S) → T op(R), deﬁning F 0 γ by letting its torsion class consist of those objects X in Rep(R) such that F (X) ∈ Tγ . In case F derives from a morphism f : S → R in B, we write F 0 = f˜. For any γ ∈ Top(A), A ∈ B, we put gen(γ) = {τ, τ ≥ γ}; for U ⊂ T op(A):

gen(∧U ) ⊃ ∪ gen(τ ) τ ∈U

gen(∨U ) = ∩ gen(τ ) τ ∈U

Hence, the subsets of the type gen(τ ) for τ in Top(A) deﬁne a topology on T op(A). For an exact functor F : Rep(R) → Rep(S) as above, F 0 (γ) ≤ F 0 (τ ) if γ ≤ τ and F 0 (∧U ) = ∧F 0 (U ) for any U ⊂ T op(S). Moreover, (F 0 )−1 (gen(τ )) = gen(ξτ ) for τ ∈ T op(R), where ξτ is such that Tξτ is the torsion class generated by F (Tτ ). Note that F 0 is not necessarily a lattice morphism (F 0 does not necessarily respect ∨) but it is a continuous map in the gen-topologies. In case γ ∈ T op(A) is perfect (Qγ is exact and commutes with arbitrary direct sums), then a0γ deﬁnes a homeomorphism (in the gen topologies): T ors(Rep(A), γ) gen(γ) ( see also Lemma 1.4 in [9]).

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The foregoing expresses that open sets of type gen(τ ) with perfect τ , behave as aﬃne open sets in scheme theory, i.e. viewed as topological spaces with the induced topologies they are homeomorphic to the topological space of a localized object. In the sequel, we only consider G.C.-representations that are faithful in the sense that for any S −→ R in B, Rep(f )(X) = 0 if and only if X = 0 (we write 0 for the zero object f

in the category considered). 1.1. Definition. A faithful G.C.-representation that measures B is said to be spectral if for every A ∈ B, γ ∈ T op(A) and τ ∈ gen(γ) we are given the following: A(γ) and fγ : A → A(γ) in B such that the morphism in Rep(A), fγ∗ :A A → Rep(fγ )(A(γ) A(γ)), is exactly the localization morphism A A → Qγ (A); a morphism fτγ : A(γ) → A(τ ) in B fitting into a commutative triangle:

Moreover, if ξ0 (A) stands for the trivial element of T op(A) (i.e. the one having only the zeroobject for its torsion class) then A(ξ0 (A)) = A and the identity of A is the corresponding morphism in B. 1.2. Proposition. Suppose Rep is spectral and γ ∈ T op(A) for A ∈ B, τ ∈ gen(γ). Write τ ), τ˜ for f˜γ (τ ) ∈ T op(A(γ)) and fτ˜ for the corresponding morphism in B, fτ˜ : A(γ) → A(γ)(˜ existing because of the spectral property of Rep (applied to A(γ) ∈ B). With notation as in Definition 1.1. we have: i) Rep(fτγ )(A(τ ) A(τ )) = Qτ˜ (A(γ) A(γ)). ii) Rep(fγ )(Qτ˜ (A(γ) A(γ))) = Qτ (A A). Proof. Since Rep is spectral there exists an A(γ)(˜ τ ) in B together with a morphism τ ) with corresponding morphism fτ˜∗ in Rep(A(γ)), fτ˜∗ : A(γ) A(γ) → A(γ) −→ A(γ)(˜ fτ˜

Rep(fτ˜ )(A(γ)(˜τ ) A(γ)(˜ τ )), which is nothing but the localization homomorphism A(γ) A(γ) → Qτ˜ (A(γ) A(γ)), in Rep(A(γ)). On the other hand we may look at Rep(fγ )(fτ˜∗ ): Qγ (A A) → Rep(fγ )(Qτ˜ (A(γ) A(γ))), where the latter is in fact equal to: Rep(fγ )Rep(fτ˜ )(A(γ)(˜τ ) A(γ)(˜ τ )) = Rep(fτ˜ fγ )(A(γ)(˜τ ) A(γ)(˜ τ )). Looking at A −→ A(γ) −→ A(γ)(˜ τ ), we obtain: fγ

fτ˜

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Now A(τ ) A(τ ) in Rep(A(τ )) is such that the localization morphism → Qτ (A) is exactly given by fτ∗ , fτ∗ : A A → Rep(fτ )(A(τ ) A(τ )). Look at the object Mτ (γ) in Rep(A(γ)) deﬁned by Mτ (γ) = Rep(fτγ )(A(τ ) A(τ )) and fτ∗γ : A(γ) A(γ) → Mτ (γ) the morphism in Rep(A(γ)) given by the fact that Rep is measuring. We have τ Mτ (γ)) τ˜(Mτ (γ)) ⊂ Mτ (γ) in Rep(A(γ)). By the deﬁnition of τ˜ we have that Rep(fγ )(˜ τ Mτ (γ)) ⊂ is τ -torsion in Rep(A), moreover the exactness of Rep(fγ ) yields that Rep(fγ )(˜ Rep(fγ )(Mτ (γ)) = Rep(fτ )(A(τ ) A(τ )), the latter being τ -torsionfree because it is equal to Qτ (A A). Consequently Rep(fγ )(˜ τ Mτ (γ)) = 0 and τ˜Mτ (γ) = 0 then follows from faithfulness τ (A(γ) A(γ)). We have of Rep. Hence fτ∗γ :A(γ) A(γ) → Mτ (γ) factorizes over Bτ (γ)=A(γ) A(γ)/˜ a sequence in Rep(A):A A −→ Qγ (A A) −−−−−−−− → Qτ (A A). So it is clear that in Rep(A), ∗ ∗ AA

fγ

Rep(fγ )(fτγ )

Rep(fγ )(Mτ (γ)/Bτ (γ)) is τ -torsion because the cokernel of Rep(fγ )(fτ∗γ ) is τ -torsion, therefore Mτ (γ)/Bτ (γ) is τ˜-torsion in Rep(A(γ)). It follows that Mτ (γ) ⊂ Qτ˜ (A(γ) A(γ)) in Rep(A(γ)). By deﬁnition Qτ˜ (A(γ) A(γ)) is τ˜-torsion over Bτ (γ) in Rep(A(γ)), hence Rep(fγ )(Qτ˜ (A(γ) A(γ))) is contained in Qτ (A A) as it is τ -torsion over A A/τ (A A). Since Mτ (γ) ⊂ Qτ˜ (A(γ) A(γ)) the functor Rep(fγ ) takes the value Qτ (A) for both objects and again from exactness and faithfulness of Rep(fγ ) it follows that: Mτ (γ) = Qτ˜ (A(γ) A(γ)). This proves i) and() yields that the morphisms Rep(fγ )(fτ˜∗ )fγ∗ and Rep(fγ )(fτ∗γ )fγ∗ are the same. Now ii) is obtained from i) by applying Rep(fγ ) to both members, so it follows again from the exact faithfulness of Rep(fγ ) that i) holds to. 1.3. Corollary. With conventions and notation as before, consider δ ≤ τ and γ ≤ τ with τ˜1 ∈ T op(A(δ)) and τ˜2 ∈ T op(A(γ)). We obtain the following diagram of morphisms in B:

with objects: τ1 ) in Rep(A(δ)(˜ τ1 )) M1 =A(δ)(˜τ1 ) A(δ)(˜ τ2 ) in Rep(A(γ)(˜ τ2 )) M2 =A(δ)(˜τ2 ) A(γ)(˜ M =A(τ ) A(τ ) in Rep(A(τ )) such that: i) Rep(fτ˜1 )(M1 ) = Qτ˜ (A(δ) A(δ)) = Rep(fτδ )(M ) ii) Rep(fτ˜2 )(M2 ) = Qτ˜ (A(γ) A(γ)) = Rep(fτγ )(M ) iii) Rep(fδ )Rep(fτ˜1 )(M1 ) = Qτ (A A) = Rep(fγ )Rep(fτ˜2 )(M2 ) = Rep(fτ )(A(τ ) A(τ )). This may be compared to the classical fact for localization in R-mod for a ring R, Qτ (R) = Qτ˜ (Qγ (R)) for τ ≥ γ, where Qτ˜ is a localization in Qγ (R)-mod. In our abstract setting Rep(A(γ)) replaces Qγ (R)-mod and we have traced the basic properties of a

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G.C.-representation (i.e. spectral) necessary to generalize the foregoing classical fact to the more general situation. τ2 ) as in the corollary are a priori not related so not necesNote that A(δ)(˜ τ1 ) and A(γ)(˜ sarily isomorphic in B. Up to introducing restrictive conditions on B and Rep, e.g. in terms of properties making A A a generator for Rep(A) etc. . ., one could study this further, however we do not aim to push category theoretical generality to the limit, contenting ourselves with the observation that the example(s) considered later will have an even better behaviour. Let us write T op(A)0 for the opposite lattice of T op(A). It would be natural to consider the functor A P: T op(A)0 → Rep(A), A P(τ ) = Qτ (A A), as a structural presheaf (in fact: sheaf) of A A with values in Rep(A). The spectral property of Rep allows to “realize” this structure sheaf inside B, in some sense, by considering P: T op(A)0 → B, deﬁned by P(τ ) = A(τ ) with structure morphism fτ : A → A(τ ). We have P(ξ0 (A)) = A with IA : A → A as the corresponding morphism. Moreover for γ ≤ τ we take fτγ : A(γ) → A(τ ) as the restriction morphism from γ to τ (note that in T op(A)0 the partial order is reversed when viewing γ and τ in T op(A) as open sets). For P to be a presheaf we need an extra property of Rep, as follows. A spectral Rep is said to be schematic if in addition to the properties in Deﬁnition 1.1. we have for every triple γ ≤ τ ≤ δ in T op(A) for any A ∈ B, a commutative diagram:

1.4. Corollary. If Rep is schematic then P: T op(A)0 → B is a presheaf of B-objects over the lattice T op(A), for every A ∈ B. Proof. The composition property of “sections” follows from fδγ = fδτ fτδ

(in the proof of the foregoing proposition one may start from a triangle

then use the same arguments for the triangles

and

and

to retract information

further to the Rep(A)-level). If f : S → R is a morphism in B then we have deﬁned F 0 = f˜ : T op(S) → T op(R),

and we observed that f˜ is a continuous map with respect to the gen-topologies on T op(S)and T op(R), (see remarks before Deﬁnition 1.1).

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A topological nerve η associates to each A ∈ B a ηA ∈ Top(A) such that for any f : A → B in B we have ηB ≤ f˜(ηA ) in T op(B). The fact that the latter domination relation is assumed for every f : A → B makes the topological nerve rather narrowly deﬁned; in practical situations the choice of ηA in each T op(A) follows from a radical-type of construction. A perfect example will be given by the torsion theory associated to the positive part A+ of a positively graded K-algebra or ring, where B is then the category of positively graded K-algebras, resp. rings. We adopt the following notation: iA : (Rep(A), ηA ) → Rep(A), aA : Rep(A) → (Rep(A), ηA ), QA : QηA = iA aA . To a G.C.-representation Rep and a topological nerve η we associate (Rep, η) such that for A ∈ B we have (Rep, η)(A) = (Rep(A), ηA ) and for any morphism f : A → B in B we consider the functor aA F iB : (Rep(B), ηB ) → (Rep(A), ηA ) where F = f 0 . For g: B → C in B we obtain that the composition (aA F iB )(aB GiC ): (Rep(C), ηC ) → Rep(A, ηA ) corresponds to g ◦ f , so we have to relate this to aA F GiC . This will follow from the following proposition. 1.5. Proposition. With notation as before, for any f : A → B in B we have aA F iB aB (M ) = aA F (M ) for every M ∈ Rep(B). Proof. Clearly QB (M )/M is ηB -torsion in Rep(B), hence it is η˜A -torsion because ηB ≤ η˜A and η˜A is deﬁned by f as usual, η˜A = f˜(ηA ). By deﬁnition of η˜A (and using exactness of F ) we obtain that F iB aB (M )/F (M ) is ηA -torsion in Rep(A). The exact sequence 0 → ηB (M ) → M → iB aB (M ) in Rep(B) yields an exact sequence in Rep(A): 0 → F ηB (M ) → F M → F iB aB (M ) → T → 0,

()

where T is ηA -torsion in Rep(A) in view of the foregoing. Clearly ηB (M ) is η˜A -torsion, hence F ηB (M ) is ηA -torsion and it follows that QA (F (M )) = QA (F iB aB (M )), or also aA F (M ) = aA F iB aB (M ). We introduce a notion slightly more general than a G.C.-representation, say a generalized G.C.-representation by weakening the deﬁnition of a G.C.-representation such that for f : S → R in B, F : Rep(R) → Rep(S) is only assumed to commute with ﬁnite products. 1.6. Corollary. With notation as before: aA F iB aB GiC = aA F GiC corresponds to A −→ f

B −→ C. Hence, associating f 2 = aA F iB to f : A → B, defines a generalized G.C.g

representation (Rep, η) as before (observe in particular that aA IM iA is IM by definition of iA and aA , for all M ∈ (Rep(A), ηA ). Proof. The ﬁrst statement is clear from the Proposition 1.5, so we only have to verify that aA F iB is exact and commutes with ﬁnite products (and coproducts i.e direct sums). This is obvious, except perhaps the exactness needs a word of explanation, since iB is not an exact functor in general. Exactness in the quotient categories may be interpreted as exactness up to torsion so the presence of aA in aA F iB (and the exactness of F ) makes aA F iB : (Rep(B), ηB ) → (Rep(A), ηA )

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exact. Explicitely, if 0 → M → M → M → 0 is exact in (Rep(B), ηB ) then: 0 → iB M → iB M → iB M → TB → 0 is exact in Rep(B) where TB is ηB -torsion in Rep(B). Applying the exact functor F yields an exact sequence in Rep(A): 0 → F iB (M ) → F iB (M ) → F iB (M ) → F (TB ) → 0. Since TB is ηB -torsion it is η˜A -torsion and thus F (TB ) is ηA -torsion in Rep(A). Applying the exact functor aA : Rep(A) → (Rep(A), ηA ) we obtain an exact sequence in (Rep(A), ηA ): 0 → aA F iB (M ) → aA F iB (M ) → aA F iB (M ) → 0.

The representation (Rep, η) associated to a topological nerve η is called the quotient generalized G.C.-representation of Rep. 1.7. Observation. If η 1 and η 2 are two topological nerves then η 1 ∧η 2 deﬁned by associating 1 2 ∧ ηA to A ∈ B is again a topological nerve but this does not hold for ∨, see the remarks ηA preceding Deﬁnition 1.1. (i.e. F 0 (∧U ) = ∧F 0 (U ) for any U ⊂ T op(S)). 1.8. Theorem. Let Rep and η be as before, then we have: i) If Rep is measuring B then so is (Rep, η). ii) If Rep is weakly spectral, i.e same definition as spectral but without assuming the faithfulness, then so is (Rep, η).

Proof. i) Since Rep is measuring, there are A A ∈ Rep(A) for A ∈ B and for f : S → R in B, a corresponding f ∗ : S S →S R = Rep(f )(R R) such that IS∗ = IS S and (f ◦g)∗ = Rep(g)(f ∗ )◦g ∗ for g: T → S in B. Now take aA (A A) ∈ (Rep(A), ηA ) for A ∈ B. To f : A → B in B we associate f 2 = aA f 0 iB as in the Corollary 1.6. and then f : aA (A A) → f 2 (aB (B B)). In view of Proposition 1.5. we now have f 2 (aB (B B)) = aA f 0 (B B) = aA (A B) and for f we just take aA (f ∗ ), the localized map of f ∗ . Obviously (IA ) = IaA (A) and the composition rule follows as in Corollary 1.6. ii) For every γ ∈ T op(A), A ∈ B, we have A(γ) ∈ B and fγ : A → A(γ) in B such that fγ∗ : 0 A A → fγ (A(γ) A(γ)) is exactly the localization morphism A A → Qγ (A A) on Rep(A). Now if we consider some γ ∈ T ors(Rep(A), ηA ) then this γ corresponds to a γ ∈ T orsRep(A) = T op(A) such that γ ≥ ηA . (Since aA is exact and commutes with ﬁnite products, we may

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take γ to be given by its torsion class consisting of M ∈ Rep(A) such that aA (M ) is γ-torsion in (Rep(A), ηA ). We obtain aA (fγ∗ ) = fγ as follows: fγ : aA (A A) aA fγ0 iA(γ) aA(γ) (A(γ) A(γ)) || aA fγ0 (A(γ) A(γ)) || aA (Qγ (A A))

(Proposition 1.5) (choice of A(γ))

Now since γ ≥ ηA we have QA Qγ (A A) = Qγ (A A) = Qγ QA (A A) (this is also a special case of compatibility of localization functors cf. [13]) = Qγ QA (A A). So we arrive at aA (Qγ (A A)) = aγ QA (A A)) = aγ (A A), and fγ is the localization map corresponding to γ. 1.9. Remarks. 1. The whole theory of G.C.-representations including the results of [9] extends to the case of generalized G.C.-representations. Since the case where the functors F are “restrictions of scalars” functors over rings is very interesting we have made the deﬁnition of G.C.-representation as we did. 2. Given f : A → B then aA f 0 iB (M ) = 0 if and only if iB (M ) is η˜A -torsion. Since ηB ≤ η˜A this means that M is η A -torsion, η A induced on (Rep(B), ηB ) by η˜A . Thus even if Rep is faithful, (Rep, η) need not be. However since f 2 (M ) = 0 exactly if M is η˜A -torsion, we see for τ ≥ γ in T ors(Rep(A), ηA ), corresponding to τ ≥ γ ≥ ηA in T op(A), that f˜γ (τ ) ≥ f˜γ (γ) ≥ f˜γ (ηA ) ≥ ηA(γ) in notation of Proposition 1.2; in particular we see that a γ˜ = f˜γ (γ)-torsion object is certainly a τ˜ = f˜γ (τ )torsion object. This allows to obtain an equivalent of Proposition 1.2 for (Rep, η). 1.10. Proposition. Let Rep be a spectral G.C.-representation then (Rep, η) is a weakly spectral generalized G.C.-representation such that: i) Qη˜A (fτ2γ (A(τ ) A(τ ))) = Qτ˜ (A(γ) A(γ)) ii) fγ2 (aτ˜ (A(γ) A(γ)) = Qτ (A A) = Qτ˜ (QA (A A)) Proof. Follow the lines of proof of the Proposition 1.2 when deﬁning Mτ (γ) as fτ2γ (A(τ ) A(τ )), and replacing fγ0 = Rep(fγ ) by fγ2 , fτ∗ by fτ , etc. . ., we obtain that f 2 (Mτ (γ)) is ηA torsionfree by deﬁnition, hence Mτ (γ) is η˜A -torsionfree. Thus fγ2 (˜ τ Mτ (γ)) = 0 leads to τ˜Mτ (γ) being η˜A -torsion, contradicting the foregoing unless τ˜Mτ (γ) = 0. As in the proof of Proposition 1.2, one then arrives at fγ2 (Qτ˜ (A(γ) A(γ))/Mτ (γ)) = 0, hence Qτ˜ (A(γ) A(γ)) is η˜A -torsion over Mτ (γ) and this yields i). We know already (the weakly spectral property) that fγ2 fτ2γ (A(τ ) A(τ )) = Qτ˜ (QA (A A)) hence fγ2 Mτ (γ) is τ˜-closed so fγ2 (Qτ˜ (A(γ) A(γ))) = fγ2 Mτ (γ) follows, or ii) follows too. 1.11. Example. Let B be the category of positively ZZ-graded rings with graded morphisms of degree zero for the morphisms. For A ∈ B we let Rep(A) be the category of graded (left) A-modules and graded A-linear morphisms. Full detail on the theory of graded rings and modules may be found in [7], [8]. We shall write Gr(A) = A − gr for the category as deﬁned above. To a graded ring morphism f : A → B in B we associate the restriction of scalars functor F = f 0 : B-gr → A-gr. It is easily seen that we have obtained a G.C.-representation

QUOTIENT GROTHENDIECK REPRESENTATIONS

155

Gr with Gr(A) = A-gr for A ∈ B. In T ors(A-gr) we may consider the graded torsion theory κA (+) by taking for its torsion class the graded A-modules such that every element may be annihilated by some power An+ , n ∈ IN , of the ideal A+ = A1 ⊕ . . . ⊕ An ⊕ . . .. For any f : A → B in B we have that f (A+ ) ⊂ B+ it follows easily that κB (+) ≤ f˜(κA (+)); consequently we have deﬁned a topological nerve κ(+) for Gr. The quotient generalized G.C.-representation (Gr, κ(+)) is denoted by P roj. It is not hard to see that Gr is measuring, just take A A = A viewed as a left graded A-module, and also spectral (for a graded (rigid) τ in T ors(Gr(A)) we have a graded ring of quotients Qgτ (A) and a graded morphism of rings jτg : A → Qgτ (A), then we look at Qgτ (A) Qgτ (A) i.e Qgτ (A) as a left module over itself). Detail on rigid graded localization may be found in [7]; a ﬁrst attempt at a projective scheme theory already appeared in [15], using results of [17]. The following section is devoted to a generalization of the foregoing example, to a more general useful P roj.

Let R =

⊕ Rn

n∈Z Z

2. Proj of a ZZ-graded ring be a ZZ-graded ring. Then δ = R−n Rn is an ideal of R0 and n=0

I = δ ⊕ ( ⊕ Rn ) is an ideal of R that is obviously graded and we have: I0 = δ. An ideal I n=0

of R0 is said to be invariant if for all n ∈ ZZ, Rn IR−n ⊂ I. Obviously, δ is an invariant ideal. In fact, one easily veriﬁes that an ideal I of R0 is invariant if and only if I ⊕ ( ⊕ Rn ) n=0

is an ideal of R. Observe that every ideal I of R0 that contains δ is necessarily invariant! Indeed, for all n ∈ ZZ we have that: Rn IR−n ⊂ Rn R−n ⊂ δ ⊂ I. On R0 -mod we may deﬁne a torsion theory deﬁned by the Gabriel ﬁlter L(δ), L(δ) = {L left ideal of R0 , L contains an ideal I of R0 such that δ ⊂ rad(I)}. Such torsion theories were called (symmetric) radical torsion theories in [16]. An arbitrary -mod, say σ is given by its ﬁlter L(σ) and is said to be invariant if torsion theory σ on R0 Rn IR−n ∈ L(σ). I ∈ L(σ) entails that n=0

In this section we assume that R0 is a (left) Noetherian ring. 2.1. Observation. L(δ) is invariant. Proof. Pick I ∈ L(δ) and consider I =

n=0

Rn IR−n . Look at rad(I) and let m ∈ IN be

such that: rad(I)m ⊂ I. m ) ⊂ Rn rad(I)m R−n it follows that P ⊂I for some prime ideal ¿From (Rn rad(I)R−n Rn rad(I)R−n and consequently: P ⊃ Rn δR−n . In particular P of R0 , yields P ⊃ n=0

n=0

P ⊃ Rn R−n Rn R−n , hence P ⊃ Rn R−n , and this holds for all n = 0 in ZZ. Therefore we arrive at P ⊃ δ. Hence rad(I ) ⊃ δ, or I ∈ L(δ). An ideal I of R0 is said to be strongly invariant whenever Rn I = IRn for all n ∈ ZZ. For example, if R is a centralizing extension of R0 then every ideal I of R0 is strongly invariant. Clearly a strongly invariant ideal is invariant in general. The properties of interest in this section will all follow in case R is a centralizing extension of R0 ; in fact, an acceptable generalization of projective scheme theory for a positively

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graded algebra A with A0 = k, a ﬁeld in the centre of A would already be obtained by looking at ZZ-graded rings with A0 ⊂ Z(A). Nevertheless it is somewhat remarkable that a much less restrictive condition will suﬃce. We say that the ZZ-graded ring R is 0-normal if for all a ∈ ZZ we have: R−a (Ra R−a ) ⊂ (Ra R−a )R−a . 2.2. Proposition. Let R be a 0-normal graded ring with R0 being (left) Noetherian. If δ = R0 then there exists a d ∈ ZZ such that Rd R−d = R0 = R−d Rd . Proof. Fix an a = 0 in ZZ and look at (Ra R−a )n for n ∈ IN . Assuming that n ≥ 1, we obtain: (Ra R−a )n = (Ra R−a )(Ra R−a )(Ra R−a ) . . . ⊂ Ra (Ra R−a )R−a (Ra R−a ) . . . . 2 The latter equals Ra2 R−a (Ra R−a ) . . . and repeated application of the passing Ra R−a over some R−a , to the left, we arrive at n (Ra R−a )n ⊂ Ran R−a ⊂ Rna R−na .

Consequently Ra R−a ⊂ rad(Rna R−na ), for any n ∈ IN . Now observe that: (R−a Ra )(R−a Ra ) = R−a (Ra R−a )Ra ⊂ Ra R−a R−a Ra and the latter is contained in Ra R−a since this is an ideal of R0 . Consequently R−a Ra ⊂ rad(Ra R−a ). Now if δ = R0 the 1 ∈ δ and thus there exist ﬁnitely many a1 , a2 , . . . , am ∈ ZZ such that 1 ∈ Ra1 R−a1 + Ra2 R−a2 . . . + Ram R−am

()

Since we have observed that R−a Ra ⊂ rad(Ra R−a ) for all a ∈ ZZ, it follows that we may interchange ai and −ai in the expression () and obtain 1 ∈ rad(Ra1 R−a1 + · · · + Ram R−am ), where we now have ai ∈ IN for i = 1, . . . , m (up to renaming). Since Ra R−a ⊂ rad(Rna R−na ) for any n ∈ IN we obtain that: rad(Ra1 R−a1 + · · · + Ram R−am ) ⊂ rad(Rd R−d ) where d ∈ IN is the lowest common multiple of a1 , . . . , am ∈ IN . It follows then that R0 = rad(Rd R−d ) or R0 = Rd R−d . Now look at R−d Rd and calculate: R−d Rd = R−d (Rd R−d )Rd = R−d (Rd R−d Rd ) ⊂ R−d (R−d Rd Rd )

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the latter following from 0-mormality applied to −d. ¿From R−d Rd ⊂ R−d R−d Rd Rd , we obtain: Rd (R−d Rd )R−d ⊂ Rd (R−d R−d Rd Rd )R−d , but applying Rd R−d = R0 then leads to: R0 ⊂ R−d Rd , hence R0 = R−d Rd , as well.

For d ∈ IN we let R(d) be the ZZ-graded ring deﬁned by R(d)n = Rnd (the dth Veronese subring of R). A ZZ-graded ring R is said to be d-strongly graded if R(d) is strongly graded (i.e R(d)1 R(d)−1 = R(d)−1 R(d)1 = R(d)0 ). 2.3. Corollary. Let R be a 0-normal (left) Noetherian ZZ-graded ring. If δ = R0 then R is d-strongly graded for some d ∈ IN . 2.4. Proposition. Let R be a left Noetherian ZZ-graded 0-normal ring and consider a perfect rigid torsion theory τ on R-gr, say with graded filter Lg (τ ). If δ = 0 and Rδ ∈ Lg (τ ) then the graded localization Qgτ (R) is a d-strongly graded ring for some d ∈ IN . Proof. Since R is left Noetherian, R0 is left Noetherian too. Hence rad(δ) is ﬁnitely generated as a left ideal and so we obtain that: rad(δ) = rad(Ra1 R−a1 + · · · + Ram R−am ), with a1 , . . . , am ∈ IN , using arguments formally similar to those used in the proof of Proposition 2.2. Then we also obtain that rad(δ) = rad(Rd R−d ) for some d ∈ IN , in fact we may take for d the lowest common multiple of a1 , . . . , am ∈ IN . Since τ is perfect and Rδ ∈ Lg (τ ), we obtain that Sδ = S, where we put S = Qgτ (R). Looking at the parts of degree zero, we obtain S0 δ = S0 , thus S0 rad(Rd R−d ) = S0 . Then for any m ∈ IN we also have that S0 rad(Rd R−d )

m

= S0

and because for some m0 ∈ IN , (rad(Rd R−d ))

m0

⊂ Rd R−d ,

it also follows that S0 Rd R−d = S0 . ¿From the obvious inclusions Sd ⊃ Rd , S−d ⊃ R−d , it then follows that Sd S−d = S0 .

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Applying the foregoing argument to R−d Rd , noting that rad(Rd R−d ) = rad(R−d Rd ), we arrive at Sd S−d = S−d Sd = S0 .

The equivalent of Proposition 2.4 in case δ = 0 and assuming I ∈ Lg (τ ) does hold when S is 0-normal graded; however at this moment we do not know whether the 0-normality of S follows from the 0-normality of R. 2.5. Lemma. In case R is a domain, δ = 0 if and only if R is positively or negatively graded, i.e resp. R = R≥0 or R = R≤0 . n Proof. Rn = 0 for some n > 0, then for any −m < 0 we have R−m Rnm = 0 because m n R−nm Rnm = 0 as δ = 0. Now Rn = 0 and thus R−m and R−m are zero.

For homogeneous Ore sets T in R, putting T (d) = T ∩ R(d), one easily veriﬁes that T (d) is an Ore set in R(d); indeed, for t ∈ T (d) and r ∈ R(d) we ﬁnd t ∈ T, r ∈ R, such that t r = r t hence (t )d r = (t )d−1 r t, with (t )d ∈ T (d) and (t )d−1 r ∈ R(d). However, not every Ore set of R(d) is of the form T (d). Such problems may be circumvented by developing a “weighted” space theory generalizing the commutative case but we do not go into that here. We content ourselves to pointing out an interesting case, allowing the noncommutative scheme theory and an interpretation in terms of quotient Grothendieck representations as in Section 1. The ZZ-graded ring R is said to be geometrically graded if R is a Noetherian, R0 is central in R (hence certainly 0-normalizing) and R is generated over R0 by R1 ∪ R−1 as a ring. 2.6. Proposition. Consider a geometrically graded ring R and a perfect rigid torsion theory τ on R-gr given by its graded filter Lg (τ ). 1. If δ = 0 and Rδ ∈ Lg (τ ) then S = Qgτ (R) is strongly graded. 2. If δ = 0 and I ∈ Lg (τ ) then S0 = S−1 S1 = S−n Sn for n ≥ 0. In case τ corresponds to an Ore set T of R that is homogeneous and not contained in R0 , then S is strongly graded. Proof. An arbitrary r ∈ R can be written as a sum of monomials of type r0 x1 x2 . . . xn where r0 ∈ R0 and each xi is either in R1 or in R−1 . In case xi ∈ R1 and xi+1 ∈ R−1 , or conversely, then xi xi+1 ∈ R0 and therefore it is central in R. Consequently, such a monomial e e or in R−1 R1d for suitable e and d in IN . In fact, if r ∈ Rn with n ≥ 0, then is in R1d R−1 we see in the same way that r ∈ R1n by putting factors in degree zero at the beginning −m of the monomials in the expression of r as above; for m ≤ 0 we ﬁnd that Rm = R−1 . n n Summarizing, for n ≥ 0 we have Rn = R1 , R−n = R−1 . 1. When δ = 0 then we establish for some d ∈ IN that S0 Rd R−d = S0 R−d Rd = S0 , just as in the proof of Proposition 2.4. However: d Rd R−d = R1d R−1 = (R1 R−1 )d

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159

d

follows from foregoing remarks. Then it follows from S0 (R1 R−1 ) = S0 that S0 (R1 R−1 ) = S0 . Similarly we arrive at S0 (R−1 R1 ) = S0 from S0 R−d Rd = S0 . Clearly we then obtain that S1 S−1 = S0 = S−1 S1 . 2. In this situation δ = 0 if and only if R1 R−1 = R−1 R1 = 0 or Rn Rm = 0 for n > 0, m < 0. Consequently δ = 0 if and only if R is either positively or negatively graded depending whether R1 = 0, or resp. R−1 = 0. Let us treat the positively graded case (the negatively graded case may be treated in a similar way, note however that the statement in the proposition should then be given as S1 S−1 = S0 ). So R = ⊕ Rn , I = ⊕ Rn . The assumption I ∈ Lg (τ ) and τ being a perfect n≥0

n>0

torsion theory, leads to S = SI, hence by looking at the parts of degree zero: S0 =

n>0

S−n In =

S−n Rn .

n>0

Look at a typical element s−n rn with s−n ∈ S−n , rn ∈ Rn . For some I ∈ Lg (τ ) we have: Ip s−n rn ⊂ Rp−n Rn ⊂ Rp ,

()

because Ip s−n ⊂ Rp−n for all p. Thus, for any p > 0; S−p Ip s−n rn ⊂ S−p Rp = S−p R1p−1 R1 ⊂ S−1 R1 ⊂ S−1 S1 . Now s−n

S−p Ip = (SI)0 = S0 because I ∈ Lg (τ ), this leads to S0 s−n rn ⊂ S−1 S1 where ∈ S−n , rn ∈ Rn as well as n were arbitrary. ¿From S0 = S−n Rn it thus follows p>0

n>0

that S0 = S−1 S1 . Note that in () we may indeed use that I0 = 0 because if I ∈ Lg (τ ) then I ∩ I ∈ Lg (τ ) has (I ∩ I)0 = 0 and we may replace I by the smaller I ∩ I in (). Note also that from the foregoing information it does not follow that S1 S−1 = S0 ! However if τ is associated to a (left)Ore Set, T say, then the strongly graded condition does follow. Indeed, for y ∈ S0 look at ytm for some tm ∈ T ∩ Rm , m > 0. Since tm −1 is invertible in S and t−1 m ∈ S−m we may consider (ytm )tm = y ∈ Sm S−m . ¿From m S−1 S1 = S0 we may derive Sm = S1 , indeed Sm = Sm S0 = Sm S−1 S1 yields Sm = Sm−1 S1 and by repetition of the argument (Sm−1 = Sm−2 S1 etc. . .) we obtain Sm = S1m . Now y ∈ S1m S−m = S1 (S1m−1 S−m ) ⊂ S1 S−1 . Thus S0 = S1 S−1 . To a ZZ-graded ring R we associate a torsion theory on R-gr, denoted by κR , deﬁned by taking for its graded ﬁlter Lg (κR ) the graded ﬁlter generated by Rδ and I. Note that the ideal I = δ ⊕ ( ⊕ Rn ) is automatically in Lg (κR ) because it contains Rδ, in case n=0

δ = 0. We can now deﬁne schematically graded rings by looking at the class of graded ring R such that there is a ﬁnite number of homogeneous Ore sets T1 , . . . , Tm such that κR = κT1 ∧ . . . ∧ κTm (if so desired one may weaken the deﬁnition to κTi that are only perfect rigid torsion theories).

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We entend to use the κR in deﬁning a topological nerve as mentioned after the proof of Corollary 1.4 and used in Proposition 1.5 and Theorem 1.8. For that we need the following easy lemma. 2.7. Lemma. Let R and S be ZZ-graded rings with either δR = 0 and δS = 0 or else δR = δS = 0. If f : R → S is a morphism of graded rings then κS ≤ f˜(κR ). Proof. Since f (Rn ) ⊂ Sn for every n ∈ ZZ, it is clear that f (δR ) ⊂ δS and also that f (IR ) ⊂ IS . Now L ∈ L(f˜(κR )) means that S/L is a κR -torsion as an R-module, i.e L contains some I ∈ L(κR ). The statement now follows easily. Consider the category B, consisting of ZZ-graded rings R with δR = 0 and graded ring morphisms. The association of R-gr to R deﬁnes a Grotendieck representation. The foregoing lemma entails that κR deﬁnes a nerve and therefore also a quotient representation with respect to the nerve κR . Applying the methods of [15] we obtain a satisfactory theory for P rojR which is deﬁned by the noncommutative topology on (R-gr, κR ) (see Section 1) and the corresponding sheaf theory. 2.8. Conclusion. If R is geometrically graded and schematic then P rojR deﬁned on (Rgr, κR ) satisﬁes all the properties valid in the positively graded case, in particular the schematic property in combination with Proposition 2.6 yields the existence of an aﬃne covering (in the sense of [15], [17]), moreover the proof of Serre’s global section theorem given for P rojR of a positively graded ring carries over this situation too. All this follows from a trivial modiﬁcation of the proofs given in the positively graded case, taking into account the results included in this section, so we omit this repetition here.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13] [14]

P. Gabriel, Des cat´egories ab´eliennes, Bull. Soc. Math. France 90, (1962), 323–448. J. Golan, Localization of Noncommutative Rings, M. Dekker, New York, 1975. O. Goldman, Rings and Modules of Quotients, J. of Algebra 13, (1969), 10–47. L. Le Bruyn, M. Van den Bergh and F. Van Oystaeyen, Graded Orders, Birkhauser Verlag, Basel 1988. H. Li and F. Van Oystaeyen, Zariskian Filtrations, K-Monogr. Math. 2, Kluwer Acad. Publ. Dordrect 1996. S. Maclane, Categories for the Working Mathematician, Springer-Verlag, New York, 1974. C. Nˇ astˇ asescu, F. Van Oystaeyen, Graded Ring Theory, Library of Math. 28, North-Holland, Amsterdam 1982. C. Nˇ astˇ asescu, F. Van Oystaeyen, Dimensions of Ring Theory, D. Reidel Publ. Co., 1987. J. Ndirahisha, F. Van Oystaeyen, Grothendieck Representations of Categories and Canonical Noncommutative Topologies, J.of K- Theory, to appear. R. Sallam, F. Van Oystaeyen, A microstructure Sheaf and Quantum Sections over a Projective Scheme, J. Algebra 158(1), 1993, 201–225. J.P. Serre, Faisceaux Alg´ebriques Coh´ erents, Ann. Math. 61, 1955, 197–278. B. Stenstr¨ om, An Introduction to Methods of Ring Theory, Die Grundlehren der Mathematischen Wissenschaften, Vol. 217, Springer, Berlin, 1975. F. Van Oystaeyen, Compatibility of Kernel Functors and Localization Functors, Bull. Soc. Math. Belg., XVIII, 1976, 131–137 F. Van Oystaeyen, Prime Spectra in Noncommutative Algebra, Lect. Notes in Math. 444, Springer-Verlag, Berlin 1978.

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[15] F. Van Oystaeyen, Algebraic Geometry for Associative Algebras, Pure and Applied Mathematics, Vol. 232, M. Dekker, New York, 2000. [16] F. Van Oystaeyen, A. Verschoren, Non-commutative Algebraic Geometry, LNM 887, Springer-Verlag, Berlin 1981. [17] F. Van Oystaeyen, L. Willaert, Grothendieck Topology, Coherent Sheaves and Serre’s Theorem for Schematic Algebras, J. Pure Applied Algebra 104, 1995, 109–122.

ON THE STRONG RIGIDITY OF SOLVABLE LIE ALGEBRAS TOUKAIDDINE. PETIT1 Departement Wiskunde en Informatica, Universiteit Antwerpen, B-2020 (Belgium)

Abstract. We call a ﬁnite-dimensional complex Lie algebra g strongly rigid if its universal enveloping algebra Ug is rigid as an associative algebra, i.e. every formal associative deformation is equivalent to the trivial deformation. The aim of this paper is to study the strong rigidity properties of solvable Lie algebras. First, we show that a strongly rigid Lie algebra has to be rigid as Lie algebra, this restricts the research to rigid Lie algebras. In addition the second scalar cohomology group has to vanish. Therefore the nilpotent Lie algebras of dimension greater or equal than two are not strongly rigid and the torus’s dimension of strongly rigid solvable Lie algebra has to be one. Moreover, the Kontsevitch’s theory of deformation quantization helps to see that every polynomial deformation of the linear Poisson structure on g∗ which induces a nonzero cohomology class of g leads to a nontrivial deformation of Ug. Since the rigidity is intimately related to cohomology, the cohomology groups are characterized. At last, we classify the n-dimensional strongly rigid solvable Lie algebras where n ≤ 6 and give some remarks on linearizability of their corresponding Poisson structure.

1. Introduction The deformation of of rings and algebras was introduced by M. Gerstenhaber in 1964 ([12]). He gave a tool to deform algebraic structure based on formal power series. The interest on deformation has grown with the development of quantum groups related to quantum mechanics ([2]). Examples of quantum groups may be obtained as Hopf algebra deformation of enveloping algebra of Lie algebra. A formal deformation of an associative (resp. Lie) algebra (A, µ) is an associative (resp. Lie) algebra A[[t]] with a multiplication µt deﬁned by µt (p, q) = µ(p, q) + tµ1 (p, q) + t2 µ2 (p, q) + · · ·

(1.1)

where p, q ∈ A The algebra is said rigid if every formal deformation is isomorphic to a trivial deformation. The rigidity theorem of Gerstenhaber [12] (resp. of Nijenhuis-Richardson [20]) insure that if the 2nd Hochschild cohomological group H2H (A, A) (resp. Chevalley-Eilenberg H2CE (g, g)) of an associative algebra A (resp. a Lie algebra g) vanishes then the algebra (rep. Lie algebra) is rigid. Therefore the semisimple associative (resp. Lie) algebras are rigid because their second cohomology groups are trivial ([14]). The rigidity of n-dimensional complex rigid Lie algebras was studied by R.Carles, Y.Diakit´e, M.Goze and J.M. Ancochea-Bermudez. Carles and Diakit´e established the classiﬁcation for n ≤ 7 ([6],[4]), and Ancochea with Goze did the classiﬁcation for solvable 1 Author supported by the Scientific Programme NOG of the European Science Foundation, e-mail:[email protected]

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Lie algebras for n = 8 and some classes ([10], [1]). The classiﬁcation of associative rigid algebras are known up to n ≤ 6 (see [19]). In this paper we are interested in the deformation and rigidity of enveloping algebras associated to solvable Lie algebras. We have introduced in ([3]) the notion of strong rigidity of a Lie algebra. A Lie algebra is said strongly rigid if its enveloping algebra is rigid as an associative algebra. The paper is organized as follows. In Section 2 we summarize the deﬁnitons and recall some important and useful results, namely the Cartan-Eilenberg theorem and HochschildSerre factorization theorem. In Section 3 we introduce the strong rigidity of a Lie algebra and give some properties. We show that a strongly rigid Lie algebra has to be rigid as a Lie algebra. In addition, the scalar second cohomology group has to vanish. Therefore, it permits to construct some classes of non strongly rigid Lie algebras. As an example of a strongly rigid Lie algebra, we consider the 2-dimensional non abelian Lie algebra. We show by a direct calculation that the second Hochschild cohomology group of its enveloping algebra with values in the algebra is trivial. Thus this Lie algebra is strongly rigid. Section 4 is devoted to the deformation of the enveloping algebra via the Poisson structures. We recall the result of [3] that every nontrivial polynomial deformation of the linear Poisson structure associated the the Lie algebra induces a nontrivial deformation of the enveloping algebra. In Section 5 we characterize the cohomology groups HnCE (g, Ug)). At last, we apply the previous results to classify the strongly rigid Lie algebras in small dimensions and deduce some remarks on the linearization of Poisson structures. 2. Preliminaries 1. Let g be a ﬁnite dimensional decomposable solvable Lie algebra, i.e g = t ⊕ n where n is the nilradical and t is an exterior torus of derivations in Malcev’s sense; that is t is an abelian subalgebra of g such that adX is semisimple for all X ∈ t. This class of solvable Lie algebra contains the rigid Lie algebras ([4]). 2. Let K be a commutative ring and g be a Lie algebra over K. Recall that a (left) g-representation of g is a K-module M and a K-homomorphism g ⊗ M → M x ⊗ a → xa

(2.1)

such that x(ya) − y(xa) = [x, y]a. To each Lie algebra g, we associate an associative K-algebra Ug such that every (left) g-representation may be viewed as (left) Ugrepresentation and vice-versa. The algebra Ug is constructed as follows Let T g be a tensor algebra of K-module g , T g = T 0 ⊕ T 1 ⊕ · · · ⊕ T n ⊕ · · · where T n = g ⊗ g ⊗ · · · ⊗ g (n times). In particular T 0 = K1 and T 1 = g. The multiplication in T g is the tensor product. Every K-linear map g ⊗ M → M has a unique extension to a map T g ⊗ M → M. The g-module is a g-representation if and only if the elements of T g of the form x ⊗ y − y ⊗ x − [x, y] where x, y ∈ g annihilate M. Consequently, we are led to introduce the two-sided ideal I generated by the elements x ⊗ y − y ⊗ x − [x, y] where x, y ∈ g. We deﬁne the enveloping algebra Ug of g as T g/I. Thus, g-representations and the Ug-modules may be identiﬁed. Recall that every bimodule M is a g-module by (x, m) → xm − mx, denoted by Ma . Assume that g is a free Lie algebra. Let {xi } be a ﬁxed basis of g and yi be the image of xi by the K-homomorphism i : g → Ug. We set yI = yi1 · · · yip with I a ﬁnite

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sequence of indices i1 , . . . , ip and yI = 1 if I = ∅. The Poincar´e-Birkhoﬀ-Witt theorem insures that the enveloping algebra Ug is generated by the elements yI corresponding to the increasing sequences I. We denote by SV the symmetric algebra over a K-module V . If Q ∈ K, then there exists a canonical bijection between Sg and Ug which is a g-module isomorphism between Sg and Uga ( [9, pp.78–79] ) 3. Unless otherwise stated, K denotes an algebraically closed ﬁeld of characteristic 0. Let K[[t]] be the power series ring with coeﬃcients in K. For a K-vector space E we denote by E[[t]] the K[[t]]-module of the power series with coeﬃcients in E. Let (A, µ0 ) be an associative (resp Lie) K-algebra, then (A[[t]], µ0 ) is an associative (resp. Lie) K[[t]]-algebra. (a) A formal deformation of an associative (resp. Lie algebra) A is an associative (resp. Lie) K[[t]]-algebra (A[[t]], µt ) such that µt = µ0 + tµ1 + t2 µ2 + · · · + tn µn + · · · , where µn ∈ HomK (A ⊗K A, A). (resp. µn ∈ HomK (A ∧K A, A)). (b) Two deformations (A[[t]], µt ) and (A[[t]], µt ) are said equivalent if there exists a formal isomorphism ϕt = ϕ0 + ϕ1 t + · · · + ϕn tn + · · · , with ϕ0 = IdA (Identity map on A) and ϕn ∈ End(A) such that µt (a, b) = ϕ−1 t (µt (ϕt (a), ϕt (b)) ∀a, b ∈ A.

(c) A deformation of A is said trivial if it is equivalent to (A[[t]], µ0 ). (d) An associative (resp. Lie) algebra A is said rigid if every deformation of A is trivial. 4. The deformation theory is related to Hochschild cohomology in the case of associative algebra and Chevalley-Eilenberg cohomology in the case of Lie algebra. We denote by HnH (A, M) the n-th Hochschild cohomology group of an associative algebra A with values in the bimodule M and by HnCE (g, M) the n-th Chevalley-Eilenberg cohomology group of a Lie algebra A with values in a g-module M. The second Hochschild cohomology group of an associative algebra (resp. Chevalley-Eilenberg cohomology group of a Lie algebra ) with values in the algebra may be interpreted as the group of inﬁnitesimal deformations. It follows that if this group is trivial then the algebra is rigid. The third cohomology group corresponds to the obstructions to extend a deformation of order n to a deformation of order n + 1 ([12],[13] and [20]). 5. The following classical theorem due to H.Cartan et S.Eilenberg, ([7, pp.277]) gives a link between the Hochschild cohomology of an enveloping algebra with values in an Ug-bimodule M (in particular M = Ug) and the Chevalley-Eilenberg cohomology of the Lie algebra with values in the same module. Theorem 2.1. Let g be a ﬁnite dimensional Lie algebra over K. Then HnH (Ug, M) HnCE (g, Ma ) ∀n ∈ N

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In particular, if Q ⊂ K HnH (Ug, Ug) HnCE (g, Uga ) HnCE (g, Sg) ∀n ∈ N 6. The Hochschild-Serre theorem [17] gives the following factorization of the ChevalleyEilenberg cohomology groups in the case of a decomposable solvable Lie algebra. Theorem 2.2. Let g = n⊕t be a ﬁnite dimensional solvable Lie algebra over K, where n is the largest nilpotent ideal of g and t the supplementary subalgebra of n, reductive in g, such that the t-module induced on Ua g is semisimple, then for all positive integers p, we have Hp CE (g, U a (g))

i+j=p

t

HiCE (t, K) ⊗ HjCE (n, Ua g) .

where HjCE (n, Ua g)t denotes the subspace of the t-invariant elements.

3. Strongly rigid Lie algebras and properties We recall here the notion of strong rigid Lie algebra introduced in [3]. Definition 3.1. A Lie algebra g is said strongly rigid if its enveloping algebra Ug is rigid as an associative algebra. The semisimple Lie algebras give examples of strongly rigid Lie algebras. In fact, the Whitehead lemmas induce that the ﬁrst and second cohomology groups of a Lie algebra g with values in every ﬁnite dimensional K-module vanish. Therefore these Lie algebras are rigid as Lie algebra. Using the ﬁltration of Sg and the Cartan-Eilenberg theorem we obtain 2 HH (Ug,Ug) = 0. Therefore, the enveloping algebra of a semisimple Lie algebra is rigid. 3.1. The rigidity of the Lie algebra. Theorem 3.1. If g is a ﬁnite dimensional strongly rigid Lie algebra over K, then g is rigid as a Lie algebra. Proof. We suppose that the enveloping algebra Ug of g is rigid, but not theLie algebra g. ∞ Then there exists a nontrivial formal deformation (g[[t]], µt ) of g with µt = n=0 µn tn and the cohomology class of µ1 is nontrivial in H2CE (g, g). Since g is ﬁnite dimensional, then the K[[t]]-module g[[t]] is isomorphic to the free module g ⊗K K[[t]]. Let yI := yi1 · · · yik be the generators of the PBW basis of Ug, let yI := yi1 • · · · • yik be the generators of PBW basis of U g[[t]] over K[[t]] and that • is the multiplication in U g[[t]] . The map Φ : Ug ⊗K K[[t]] → U g[[t]] deﬁned by Φ(yI ) := yI is a K[[t]]-module isomorphism. Let on the module Ug⊗K K[[t]] πt : Ug⊗K K[[t]] × Ug⊗K K[[t]] → Ug⊗KK[[t]] the multiplication induced by • and Φ, i.e. πt (a, b) := Φ−1 Φ(a) • Φ(b) . The restriction of πt to elements of Ug × Ug deﬁned a K-bilinear map Ug × Ug → Ug ⊗K K[[t]] ⊂ Ug[[t]] which we denote also ∞ by πt , i.e. πt (u, v) = n=0 tn πn (u, v) for all u, v ∈ Ug where πn ∈ HomK (Ug ⊗ Ug, Ug). The K-bilinear map πt deﬁned naturally a K[[t]]-bilinear associative multiplication over the

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K[[t]]-module Ug [[t]] (which contains Ug ⊗K K[[t]] as a dense submodule with the t-adique topology) : ∞ ∞ ∞ s s πt t us , t vs := tr πs (us , vs ) s

r=0

s =0

s,s ,s ≥0

s+s +s =r

In particular, the map π0 deﬁned an associative multiplication over the vector space Ug, and (Ug [[t]], πt ) is a formal associative deformation of (Ug, π0 ). For a ﬁnite increasing sequence I, J we have π0 (yI , yJ ) = Φ−1 (yI • yJ )|t=0 . By ordering the product yI • yJ we obtain that π0 is the multiplication of Ug and (Ug [[t]], πt ) is a formal deformation of Ug. It follows that π1 is a Hochschild 2-cocycle of Ug, and the restriction of π1 to X, Y ∈ g satiﬁes µ1 (X, Y ) = π1 (X, Y ) − π1 (Y, X)

∀X, Y ∈ g.

(3.1)

because the Lie algebra (g[[t]], µt ) is a Lie subalgebra of U(g[[t]]) which may considered as an associative subalgebra ∞of (Ug [[t]], πt ). The rigidity of Ug implies that there exists a formal isomorphism ϕt = r=0 ϕr tr , where ϕ0 = IdU g and ϕn ∈ HomK (Ug, Ug) such that ϕt (πt (u, v)) = πt (ϕt (u), ϕt (v)),

∀u, v ∈ Ug,

which is equivalent to ∞ r=0

tr

ϕa (πb (u, v)) =

∞ r=0

a,b≥0

a+b=r

tr

πa (ϕb (u), ϕc (v))

∀u, v ∈ Ug.

(3.2)

a,b,c≥0

a+b+c=n

If r = 1, the relation becomes π1 (u, v) = (δH ϕ1 )(u, v)

∀u, v ∈ Ug

(3.3)

where δH is a Hochschild cobord operator (see [16]) with respect the multiplication π0 of the enveloping algebra. Then the formulae (3.1) and (3.3) imply µ1 (X, Y ) = (δH ϕ1 )(X, Y ) − (δH ϕ1 )(Y, X) = Xϕ1 (Y ) − ϕ(XY ) + ϕ(X)Y − Y ϕ1 (X) + ϕ(Y X) − ϕ(Y )X = (δCE ϕ1 )(X, Y ) ∀X, Y ∈ g

(3.4)

where δCE is the Chevalley-Eilenberg cobord operator. (see [7]). Therefore the class of µ1 in H2CE (g, g) is trivial. contradiction. This result show that the class of strongly rigid Lie algebras is contained in the class of rigid Lie algebras.

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3.2. Second scalar cohomology group. In this Section we give a necessary condition on the scalar Chevalley-Eilenberg cohomology group for the strong rigidity of a Lie algebra. Let ω ∈ Z2CE (g, K) be a scalar 2-cocycle of the Lie algebra g. Let gω = g ⊕ Kc be a central extension of g with ω such that the new bracket [ , ] is deﬁned as usually by [X + ac, Y + bc] := [X, Y ] + ω(X, Y )c ∀X, Y ∈ g; a, b ∈ K.

(3.5)

Theorem 3.2. Let g be a ﬁnite dimensional Lie algebra over K such that the second scalar cohomology group H2CE (g, K) is diﬀerent from 0, then g is not strongly rigid. Proof. Let ω ∈ Z2CE (g, K) be a 2-cocycle with a nonzero class and let gtω [[t]] be the onedimensional central extension of the Lie algebra g[[t]] = g ⊗K K[[t]] over K = K[[t]] (see (3.5)). The multiplication of the enveloping algebra U(gtω [[t]]) of gtω [[t]] is denoted by •. Let consider the two-sided ideal I := (1 − c ) • U(gtω [[t]]) = U(gtω [[t]]) • (1 − c ) (where c denote the image of c in U gtω [[t]] ) and the quotient algebra Utω g := U(gtω [[t]])/I. Let e1 , . . . , en be the K-basis of g. Then c, e1 , . . . , en is a K[[t]]-basis of gtω [[t]]. Let y1 , . . . , yn be the images of the basis vectors in Ug and c , y1 , . . . , yn be the images of the basis vectors in U gtω [[t]] . Let yI := yi1 . . . yik in Ug over K be the generators of the PBW basis. The elements c •i0 • yI (where i0 ∈ N and c •i0 : = 1) form a basis of U gtω [[t]] over K[[t]]. (the Lie algebra is a free module over a commutative ring, see [7], p.271). In the quotient algebra Utω g, the element c •i0 is identiﬁed to 1. We denote the multiplication in Utω g by · and by the canonical projection, the images of y1 , . . . , yn by y1 , . . . , yn the elements yI give yI := yi1 · . . . · yin . It follows that the elements yI form a basis of the quotient algebra Utω g. As in the proof of the previous theorem 3.1, the map Φ : Ug ⊗K K[[t]] → U gtω [[t]] given by yI → yI deﬁnes an isomorphism of free K[[t]]-modules. In a similar way we show that the multiplicationinduced on Ug ⊗K K[[t]] by the multiplication · of Utω g and Φ deﬁne a ∞ sequence of πt = r=0 πr tr , where πr ∈ HomK (Ug ⊗ Ug, Ug) with the following properties: 1. πt deﬁnes a formal associative deformation of (Ug, π0 ), 2. π0 is the usual multiplication of the enveloping algebra Ug of g. Therefore, π1 is a Hochschild 2-cocycle of Ug, and for all X, Y ∈ g ⊂ Ug we have the relation: ω(X, Y )1 = π1 (X, Y ) − π1 (Y, X) because the Lie algebra gtω [[t]] is injected in the quotient algebra Utω g, then in Ug ⊗K K[[t]] ⊂ Ug [[t]]. Suppose that Ug is rigid, then the deformation πt is trivial. Therefore there exists a Hochschild 1-cocycle ϕ1 ∈ C1H (Ug, Ug) such that π1 = δH (ϕ1 ). It follows ∀X, Y ∈ g: ω(X, Y )1 = π1 (X, Y ) − π1 (Y, X) = δH (ϕ1 )(X, Y ) − δH (ϕ1 )(Y, X) = δCE (ϕ1 )(X, Y ). Then ω is a Chevalley-Eilenberg cobord and its class is trivial in H2CE (g, K), contradiction. 3.3. Examples of non strongly rigid Lie algebras. The previous theorems permit to show that some classes of solvable Lie algebras are not strongly rigid. Corollary 3.1. The following Lie algebras are not strongly rigid : 1. Every n-dimensional nilpotent Lie algebra g with n greater or equal than 2. 2. Every Lie algebra g = t ⊕ n where the dimension of the torus t is greater or equar than 2. Proof. The ﬁrst assertion is a consequence of a classical result of Dixmier concerning the nilpotent Lie algebras ([8]): H2CE (g, K) = 0 if dim(g ≥ 2). For the second, we have H2CE (t, K) = {0} for an abelian Lie subalgebra.

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3.4. Example of a strongly rigid Lie algebra. In this section we prove that the 2-dimensional non abelian solvable Lie algebra is strongly rigid. We denote by r2 the solvable Lie algebra generated by X, Y such that [X, Y ] = Y . Lemma 3.1. 1. ∀n, m ∈ N : Y X n = (X − 1)n Y , [X, Y m ] = mY m and ∀n ∈ N, ∀m ∈ ∗ N (m − 1)X n Y m = [X, X n Y m ] − X n Y m . 2. There exists a polynomial Pn+1 (X) in X of degree n + 1 such that : n+1 k Pn+2−k (X), if n ≥ 1. (a) P1 (X) = X and Pn+1 (X) = X n+1 + k=2 (−1)k Cn+2 n (b) (n + 1)X Y = [Pn+1 (X), Y ]. Proof. The ﬁrst assertion may easily be proved by induction. Let us prove the property (2) by induction on n. It is true for n = 0, because [P1 (X), Y ] = [X, Y ] = Y . Assume that it is true until n. We have (a): [X n+2 , Y ] = X n+2 Y − Y X n+2 = X n+2 Y − (X − 1)n+2 Y following (1) =X

n+2

Y −

n+2

k (−1)k Cn+2 X n+2−k Y

k=0

= (n + 2)X n+1 Y −

n+2

k (−1)k Cn+2 X n+2−k Y

k=2

Applying the induction hypothesis on n + 2 − k with k ≥ 2, we obtain X n+2−k Y = [Pn+3−k (X), Y ], (the degree of Pn+3−k (X) = n + 3 − k ≤ n + 1). Then (b) becomes : n+2 k Pn+3−k (X), Y ] = [Pn+2 (X), Y ] (n + 2)X n+1 Y = [X n+2 + k=2 (−1)k Cn+2 In the following we show by a direct calculation, for the Lie algebra r2 , that the second Hochschild cohomology group of its enveloping algebra with values in the algebra is trivial. Thus this Lie algebra is strongly rigid. Proposition 3.1. Let r2 be the 2-dimensional non abelian Lie algebra. We have H2H (Ur2 , Ur2 ) H2CE (r2 , Ur2 ) = 0 Thus, the Lie algebra r2 is strongly rigid. Proof. By Cartan-Eilenberg theorem we have H2H (Ur2 , Ur2 ) H2CE (r2 , Ur2 ). We will show that ∀Φ ∈ Z2CE (r2 , Ur2 ) ∃f ∈ C1CH (r2 , Ur2 ) s.t. δCE (f ) = Φ

(∗)

Let {X n Y m : n, m ∈ N} be the Poincar´e-Birkhoﬀ-Witt basis of Ur2 . Let Φ be an element of Z2CE (r2 , Ur2 ). It is deﬁned by Φ(X, Y ) =: u =: n,m∈N un,m X n Y m where un,m ∈ K

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are nonzero for a ﬁnite number of n, m. Let f be an element of C1CH (r 2 , Ur2 ). It is deﬁned by two elements f (X) =: v =: n,m∈N vn,m X n Y m and f (Y ) = w = n,m∈N wn,m X n Y m where vn,m , wn,m ∈ K are nonzero for a ﬁnite number of n, m. Then ∀u =

un,m X n Y m ∈ Ur2 ∃v, w ∈ Ur2 tels que u = [X, w] − w + [v, Y ] (∗∗)

n,m∈N

We study two cases Case 1: m = 1. un,m We set wn,m = m−1 , then vn,m = 0 if m = 1 and v, w satisfy (∗∗) by lemma (3.1,(2)). Case 2: m = 1. 1 We set vn,m = n+1 un,1 Pn+1 (X) then wn,m = 0 if m = 1 and v, w satisfy (∗∗) by lemma (3.1,(1)). We conclude that the relation (∗∗) is satisﬁed. Therefore H2CE (r2 , Ur2 ) = 0, and the Lie algebra r2 is strongly rigid.

4. Deformation of enveloping algebras by quantification In this section, we recall a result of [3] which said that a nontrivial polynomial deformation of the linear Poisson structure associated to the Lie algebra induces a nontrivial deformation of the enveloping algebra. We recall ﬁrst the Poisson structure. We refer to Vaisman’s book ([21]) for a complete description. 1. A Poisson algebra is a commutative associative algebra A over K with a bilinear map {, } : A × A → A satisfying for f, g, h ∈ A (1) {f, g} = −{g, f } (2) {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0 (Jacobi identity ) (3) {h, f g} = {h, f }g + f {h, g} ( Leibniz relation) We denote by (A, ·, {, }) such an algebra. A manifold M is called a Poisson manifold if the algebra of functions, C∞ (M ), has a Poisson structure. A Poisson structure is determined by a skew-symmetric bilinear form on T ∗ M . In other word there exists a tensor ﬁeld P∈ Γ(M, Λ2 T M ) (with T M the ﬁbre bundle of M ) such that {f, g} = P (df, dg) = i,j P ij ∂i f ∂j g, where ∂i denotes the partial derivative with respect to the local coordinate xi . The tensor ﬁeld P is called the Poisson bivector of (M, { , }). A Poisson structure on M is given by a bivector P ∈ Γ(M, Λ2 T M ) satisfying h

P ih ∂h P jk + P jh ∂h P ki + P kh ∂h P ij = 0

i = Γ(M, Λi T M ) be the space of all skew symmetric tensor ﬁelds of rank 2. Let Tpoly 0 i i on a manifold M , Tpoly = C ∞ (M ) and Tpoly = (⊕n≥0 Tpoly , ∧) the algebra of multivectors on M .

A bivecteur P ∈ T2poly deﬁned a Poisson structure if and only if the SchoutenNijenhuis bracket [P, P ]s = 0. The operator δP := [P, −]s determines the so-called Poisson cohomology.

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3. Let g be a ﬁnite dimensional Lie algebra over K and g∗ its algebraic dual. The symmetric algebra Sg is identiﬁed to the algebra of polynomial functions on g∗ . The Lie algebra structure of g induces a linear Poisson structure on g∗ . {f, g}(x) = x([df (x), dg(x)]) with f, g ∈ Sg and x ∈ g∗ . Let (ei )i=1...n be a basis of g, (ei )i=1...n the dual basis and n x = i=1 xi ei ∈ g∗ , {f, g} = P0 (df, dg) with P0 the bivector deﬁned by P0 =

1 ij k P0 ∂i ∧ ∂j o` u P0ij (x) = Cij xk 2 i,j

(4.1)

k

k where Cij are the structure constants of g. Therefore, the Poisson algebra structure on S(g). 4. In the theory of deformation by quantiﬁcation ([2]), one associates to the Poisson structure a formal deformation of the associative commutative algebra C ∞ (M ), called star product, see e.g. [21] for the deﬁnition. The existence of a star product for every Poisson structure was established by Kontsewitsh see [18]. Using this result we have proved in [3] the following theorem

Theorem 4.1. Let g be a ﬁnite dimensional Lie algebra over K. Let P0 be the bivector deﬁning the linear Poisson structure on g∗ . Assume that it exists a sequence (Pn )n∈N of polynomial bivectors (Pn ∈ Sg ⊗ ∧2 g∗ ) such that i+j=n [Pi , Pj ]s = 0 for n ∈ N and P1 is not cohomologeous to 0. Then Pt = n≥0 tn Pn is a nontrivial deformation of the Poisson structure P0 and it induces a nontrivial deformation of the enveloping algebra Ug. Therefore, the Lie algebra g is not strongly rigid.

5. Some cohomological properties Let g = t ⊕ n be a ﬁnite-dimensional decomposable solvable Lie algebra, where n is the largest nilpotent ideal and t is an exterior torus of derivations of g such that the center of t is t (this condition holds for rigid Lie algebras ([4])). The group HH (Ug, Ug) is isomorphic to HCE (g, Uga ) using the Cartan-Eilenberg theorem 2.1 where the enveloping algebra Ug is considered as Ug-bimodule and Ua g is considered as an adjoint g-module with X.u := [X, u] := Xu − uX, where X ∈ g, u ∈ Ug [9]. Let Un be the two-sided ideal of Ug generated by n and Z(Ug) be the center of the enveloping algebra. We denote by Ugt (resp. Unt , Utt ), the t-invariant elements of Ug (resp. Un , Ut). The group HCE (g, Uga ) may be deduced from the t-invariant cohomology group HCE (n, Un )t under some assumptions on the torus t (over g). The g-adjoint module Uga is an inductive limit of adjoint sub-g-modules (Uk g)k≥0 where (Uk g)k≥0 is the canonical ﬁltration Ug ([9]). In order to simplify the notation, we denote next the adjoint module Uga by Ug. By the Hochschild-Serre factorization theorem 2.2 we obtain : Proposition 5.1. Let g = t ⊕ n be a decomposable solvable Lie algebra. Then H1CE (g, Ug) t∗ ⊗ Z(Ug) ⊕ H1CE (n, Ug)t H2CE (g, Ug) (∧2 t∗ ) ⊗ Z(Ug) ⊕ t∗ ⊗ H1CE (n, Ug)t ⊕ H2CE (n, Ug)t

(5.1)

ON THE STRONG RIGIDITY OF SOLVABLE LIE ALGEBRAS

171

The diﬀerent t-modules deduced canonically from the t action on Ug are locally semisimple. The exact sequence of t-modules : 0 → Un → Ug → Ug/n → 0

(5.2)

implies a cohomological exact sequence, which corresponds if we restrict to t-invariant groups the exact sequence : 0 → H0CE (n, Un )t → H0CE (n, Ug)t → H0CE (n, Ug/n)t → H1CE (n, Un )t p+1 t t → · · · → HpCE (n, Ug/n)t → Hp+1 CE (n, Un ) → HCE (n, Ug) → t → Hp+1 CE (n, Ug/n) → · · ·

(5.3)

Assume that HpCE (n, Ug/n)t = 0 for p ≥ 1, and (Un )t = 0. This conditions holds if it exists an element X of t such that the eigenvalues of adX|n are positive in an ordered subﬁeld of K. Recently, M. Goze and E. Remm showed that H2CE (g, C) = 0 if and only if λ = 0 is an eigenvalue ([11]). The sequence (5.3) implies the exact sequence 0 → Ugt → Utt → H1CE (n, Un )t → H1CE (n, Ug)t → 0

(5.4)

and the isomorphisms : p+1 t t Hp+1 CE (n, Un ) HCE (n, Ug) for all p ≥ 1

(5.5)

Then, the Chevalley-Eilenberg cohomology groups of g with values in Ug become: H1CE (g, Ug) t∗ ⊗ Z(Ug) ⊕ H1CE (n, Un )t /(Ut/Z(Ug)).

(5.6)

H2CE (g, Ug) ∧2 t∗ ⊗ Z(Ug) ⊕ t∗ ⊗ H1CE (n, Un )t /(Ut /Z(Ug)) ⊕ H2CE (n, Un )t

(5.7)

HnCE (g, Ug) ∧n t∗ ⊗ Z(Ug) + ∧n−1 t∗ ⊗ H1CE (n, Un )t /(Ut )/Z(Ug) ∧i t∗ ⊗ HjCE (n, Un )t + i+j=n; j≥2

∀n ≥ 2

(5.8)

Now, we characterize the center. Suppose that there exists X0 ∈ t such that the eigenvalues of adX|n are positive in an ordered subﬁeld of K and let Y0 , . . . , Yr be a basis of n and X0 , . . . , Xs be a basis of t. Suppose that the action of X0 on an element ...js j0 u = aji00...i X0 . . . Xsjs Y0i0 . . . Yrir in Ug vanishes. If X0 Yk = λk Yk then 0 = X0 u = r ...js j0 (λ1 i1 + · · · + λr ir )aji00...i X0 . . . Xsjs Y0i0 . . . Yrir . Since λk > 0 then the center of Ug is K. r One can see that Theorem 5.1. Let g = t ⊕ n be a decomposable solvable Lie algebra. We suppose that there exists an element X of t such that the eigenvalues of adX|n are positive in an ordered subﬁeld of K.

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Then, we have the following properties: Z(Ug) = K

(5.9)

H1CE (g, Ug) t∗ ⊕ H2CE (g, Ug) ∧2 t∗

(Der(n, Un ) /Ut+ ) ⊕ (Der(n, Un )t /Ut+ ) t

(5.10) ⊗t

∗

⊕

H2CE (n, Un )t

(5.11)

Where Der(n, Un )t denote the t-invariant exterior derivations and Ut+ = Ut /Z(Ug) Since the center is nontrivial, we ﬁnd again, the following necessary condition : Corollary 5.1. If the solvable Lie algebra g = t⊕n is strongly rigid with trivial H2H (Ug, Ug) then dim(t) ≤ 1. Since Ug and Sg are isomorphic as g-module. We can replace in the previous cohomological characterization the algebra Ug by Sg and the two-sided ideal Un by Sn . 6. The classification of strongly rigid solvable Lie algebras in low dimensions Let K be the complex ﬁeld. The classiﬁcation of n-dimensional rigid Lie algebras is known until n ≤ 8 [10]. For 2-dimensional Lie algebras, there is one isomorphism class, namely the Lie algebra r2 which is strongly rigid (see proposition 3.1) In dimension 3, there is no solvable rigid Lie algebras. In dimension 4, there is only one rigid Lie algebras, r2 + r2 . Since the torus is 2-dimensional, then according to corollary (3.1) this algebra is not strongly rigid. In dimension 5, there is only one rigid class with 2-dimensional torus. There is no strongly rigid Lie algebra. In dimension 6, there is 3 isomorphism classes of 6-dimensional rigid solvable Lie algebras. Only one has a one-dimensional torus. Let us consider this Lie algebra, it is denoted in [10] by t1 ⊕ n5,6 . Setting the basis {X0 , X1 , X2 , X3 , X4 , X5 } the Lie algebra is deﬁned by [X0 , Xi ] = iXi

i = 1, . . . , 5

(6.1)

[X1 , Xi ] = Xi+1 [X2 , X3 ] = X5

i = 2, 3, 4

(6.2) (6.3)

The other bracket are equal to 0 or deduced by skew-symmetry from the previous one. In the following we give a nontrivial deformation of the linear Poisson structure associated to the Lie algebra t1 ⊕ n5,6 . Proposition 6.1. Let P0 be the Poisson structure associated to the Lie algebra t1 ⊕ n5,6 and P1 ∈ Sg ⊗ ∧2 g∗ deﬁned by (α, β, γ ∈ C3 \ {(0, 0, 0)}): P1 = βX22

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∧ + γ(−X2 X3 ∧ + X2 X5 ∧ ) + αX1 X5 ∧ ∂X1 ∂X3 ∂X1 ∂X4 ∂X3 ∂X4 ∂X2 ∂X4

Then [P1 , P1 ]s = 0 = [P0 , P1 ]s and the cohomology class of P1 is not 0. Thus, g is not strongly rigid.

ON THE STRONG RIGIDITY OF SOLVABLE LIE ALGEBRAS

173

Proof. Straightforward computation. Theorem 4.1 implies that the Lie algebra is not strongly rigid. Theorem 6.1. There is only one n-dimensional solvable strongly rigid Lie algebra for n ≤ 6, namely the 2-dimensional Lie algebra r2 . Given a Poisson structure, if there exists a formal isomorphism such that this Poissons structure is isomorphic to its linear part then one says that this Poisson structure is linearizable. This problem was formulated ﬁrst by A.Weinstein (based on considerations by Sophus Lie) ([22]). Using the theorem 4.1, we may deduce : Proposition 6.2. Every Poisson structure which is a deformation of linear Poisson structure of n-dimensional strong rigid solvable Lie algebra is linearizable. It follows that every Poisson structure which is a deformation of linear Poisson structure of n-dimensional solvable Lie algebra, with 3 ≤ n ≤ 6, is linearizable. The Poisson structure P0 + P1 (deﬁned in proposition 6.1) is not linearizable.

References ´dez, M. Goze, algebras de Lie rigides dont le nilradical est filiforme. [1] J. M. Ancochea Bermu Notes aux C.R.A.Sc.Paris, 312 (1991), 21–24. [2] F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz, D. Sternheimer Deformation theory and quantization I/ II, Ann. Phys. 111 (1978), 61–110, 111–151. [3] M. Bordemann, A. Makhlouf, T. Petit, D´eformation par quantification et rigidit´ e des alg`ebres enveloppantes, Journal of Algebra (to appear). [4] R. Carles, Sur la structure des alg`ebres de Lie rigides, Ann. Inst. Fourier 34 (1984), 65–82. [5] R. Carles, Weight systems for complex Lie algebras. Preprint Universit´e de Poitiers, 96 (1996). ´: Sur les vari´et´es d’alg`ebres de Lie de dimension 7. J. of Algebra. 91, [6] R. Carles, Y. Diakite 53–63 (1984). [7] H. Cartan, S. Eilenberg Homological algebra. Princeton University Press (1946). [8] J. Dixmier Cohomologie des alg` ebres de Lie nilpotentes, Acta Sci. Math. 16, Nos.3–4 (1955), 246–250. [9] J. Dixmier, alg`ebres enveloppantes, Gauthier-Villars, Paris, (1974). enveloping algebras, GSM AMS, (1996). ´dez, On the classification of Rigid Lie algebras. J. Algebra, [10] M. Goze, J. M. Ancochea Bermu 245 (2001), 68–91. [11] M. Goze, E. Remm, valued deformation of Lie algebra. Preprint (2002). [12] M. Gerstenhaber, On the deformation of rings and algebras II, Ann. of Math., 79 (1964), pp.59–103. [13] M. Gerstenhaber, The cohomology structure of an associative ring. Ann.of Math. 78, 2, 267–288 (1963). [14] M. Gerstenhaber, S. D. Shack: Relative Hochschild cohomology, rigid algebras, and the Bockstein, J. of Pure and Appl. Alg. 43, 53–74 (1986). [15] P. J. Hilton, U. Stammbach, A Course in Homological Algebra, Springer, New York/Berlin, 1996. [16] G. Hochschild On the cohomology groups of an associative algebra. Ann. Math. 46 (1945), 58–87. [17] G. Hochschild, J-P. Serre, Cohomology of Lie algebras, Ann. Math. 57 (1953), 72–144. [18] M. Kontsevitch Deformation quantization of Poisson manifolds, arXiv:q-alg/9709040, 1997.

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[19] A. Makhlouf, M. Goze, Classification of rigid associative algebras in low dimensions, in: Lois d’algebras et vari´ et´es alg´ebriques Hermann, Collection travaux en cours 50 (1996). [20] A. Nijenhuis, R. W. Richardson, Cohomology and deformations in graded Lie Algebras, Bull. Amer. Math. Soc. 72, 1, (1966). [21] I. Vaisman, Lectures on the geometry of Poisson manifolds, Birkh¨ auser (1994). [22] A. Weinstein, The local structure of Poisson manifold, J. of diﬀ geometry. 18, 3, (1983).

THE ROLE OF A THEOREM OF BERGMAN IN INVESTIGATING IDENTITIES IN MATRIX ALGEBRAS WITH SYMPLECTIC INVOLUTION∗ TSETSKA GRIGOROVA RASHKOVA University of Rousse ”A.Kanchev” E-mail: [email protected]

The talk is a survey on a series of results considering as applications of a theorem of Bergman [1], which is connected with investigating a class of identities for matrix algebras. These applications are based both on the essential use of an analogue of the stated theorem concerning matrix algebras with symplectic involution and its elegant proof via graph theory as a method for proving other results as well related to the same theorem. We recall that in the matrix algebra over a ﬁeld Kmof characteristics zero M2n (K, ∗) the symplectic involution ∗ is deﬁned by

AB CD

∗ =

Dt −B t −C t At

,

where A, B, C, D are n × n matrices and t is the usual transpose. For an algebra R with involution ∗ we have (R, ∗) = R+ ⊕ R− , where R+ = {r ∈ R | ∗ r = r} and R+ = {r ∈ R | r∗ = −r}. Let KX be the free associative algebra. We call f (x1 , . . . , xn ) ∈ KX a ∗-polynomial identity for (R, ∗) in symmetric variables if f (r1+ , . . . , rn+ ) = 0 for all r1+ , . . . , rn+ ∈ R+ . Analogously f (x1 , . . . , xs ) ∈ KX is a ∗-polynomial identity for (R, ∗) in skew-symmetric variables if f (r1− , . . . , rs− ) = 0 for all r1− , . . . , rs− ∈ R− . Some of the investigations concerning such identities are based on the classical P.I. theory as every identity in symmetric (or skew-symmetric) variables for M2n (K, ∗) is an ordinary identity for Mn (K). Constructive results however in the symplectic case need stronger tools. They take into account the following considerations. The algebra R+ is a Jordan algebra with respect to the multiplication r1+ ◦ r2+ = r1+ r2+ + + + r2 r1 ; r1+ , r2+ ∈ R+ and the identities in symmetric variables are weak polynomial identities for the pair (R, R+ ). Similarly, the algebra R− is a Lie algebra with respect to the new multiplication [r1− , r2− ] = − − r1 r2 − r2− r1− ; r1− , r2− ∈ R− and the identities in skew-symmetric variables for (R, ∗) are weak polynomial identities for the pair (R, R− ). For polynomials in symmetric variables the Cayley-Hamilton theorem gives an identity 2 in two variables for M2n (K, ∗) of degree n +3n . For n = 3 this identity appears to be of 2 minimal degree [5]. A partial linearization of it gives rise to a Bergman type identity, namely

∗ Partially

supported by Grant MM1106/2001 of the Bulgarian Foundation for Scientific Research.

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TSETSKA GRIGOROVA RASHKOVA

a homogeneous (of degree k) and multilinear in y1 , . . . , yn polynomial f (x, y1 , . . . , yn ) from the free associative algebra Kx, y1 , . . . , yn which can be written as (1)

f (x, y1 , . . . , yn ) =

v(gi )(x, yi1 , . . . , yin ),

i=(i1 ,...,in )∈Sym(n)

where gi ∈ K[t1 , . . . , tn+1 ] are homogeneous (of degree k − n) polynomials in commuting variables gi (t1 , . . . , tn+1 ) =

p

n+1 αp tp11 . . . tn+1

and (2)

v(gi ) = v(gi )(x, yi1 , . . . , yin ) =

αp xp1 yi1 . . . xpn yin xpn+1 .

For polynomials of type (1) A. Giambruno and A. Valenti [2] gave a lower bound of their degree as identities in skew-symmetric variables for M2n (K, ∗). For any n they constructed a special multilinear polynomial of degree 4n − 1 and found a polynomial of minimal degree for M4 (K, ∗). It leads to the existence of an identity of minimal degree 7 of type (1) for the considered algebra. In [3] on its base a full description of the Bergman type identities in skew-symmetric variables for M4 (K, ∗) was made. In the survey we deﬁne polynomials of type (1) of minimal degree for M6 (K, ∗). Two diﬀerent classes of Bergman type identities in skew-symmetric variables are given for n = 3. One of the class is related to the existence of central polynomials in skew-symmetric variables described in [4]. A polynomial c(x1 , . . . , xm ) ∈ KX is central in skew-symmetric variables for the algebra − − − (R, ∗) if it is non-zero in R− and [c(r1− , . . . , rm ), rm+1 ] = 0 for any r1− , . . . , rm+1 ∈ R− . The existence of Bergman type identities in the general case is discussed in the talk as well. In the sequel we use the following notation: g2n,0 =

(t2p − t2q )(t1 − tn+1 ).

1≤p

Before stating the main results we formulate the theorem of Bergman and its analogue for the symplectic case. Proposition 1. [1, Section 6, (27)] (i) The polynomial v(gi ) from (2) is an identity for Mn (K) if and only if

(tp − tq )

1≤p

divides gi (t1 , . . . , tn+1 ) for all i = (i1 , . . . , in ). (ii) The polynomial f (x, y1 , . . . , yn ) from (1) is an identity for Mn (K) if and only if every summand v(gi ) is also an identity for Mn (K).

ROLE OF A THEOREM OF BERGMAN IN INVESTIGATING IDENTITIES

177

Proposition 2. [7, Theorem 3] Considered in M2n (K, ∗), the polynomial f from (1) satisfies f (a, r1 , . . . , rn ) = 0 for any skew-symmetric matrix a and all matrices r1 , . . . , rn if and only if (t1 + tn+1 )g2n,0 divides the polynomials gi (t1 , . . . , tn+1 ) for all i = (i1 , . . . , in ). Some of the main results included in the talk are the following: Proposition 3. The linearization in y of the standard polynomial S3 ([x3 , y], [x2 , y], [x, y]) is an identity in symmetric variables for M6 (K, ∗) of minimal degree. This proposition follows straightforward from [5, Theorem 2.2, Corollary 2.1]. Proposition 4. [3, Theorem 3] A polynomial f is a Bergman type identity in skewsymmetric variables for M4 (K, ∗) if and only if it has the form f = α(v(g1 )(x, y1 , y2 ) + v(g2 )(x, y2 , y1 )) + βv(g3 )(x, y1 , y2 ) + γv(g4 )(x, y2 , y1 ), where 1. g1 = g4,0 i (ai t1 + bi t2 + ci t3 ), g2 = g4,0 i (di t1 − bi t2 + (ci + di − ai )t3 ) and t1 + t3 is not a factor of the polynomials g1 and g2 ; 2. The polynomial (t1 + t3 )g4,0 divides g3 and g4 and 3. The identity v(g1 )(x, y1 , y2 )+v(g2 )(x, y2 , y1 ) = 0 follows from the identity f0 (x, y1 , y2 ) = σ∈Sym(2) v(g4,0 )(x, yσ(1) , yσ(2) ) = 0. Proposition 5. [6, Theorem 3] All Bergman type identities in skew-symmetric variables of degree 14 for M6 (K, ∗) are consequences of the identity P (x, y1 , y2 , y3 ) = v(g6,0 )(x, yσ(1) , yσ(2) , yσ(3) ) + v(g6,0 )(x, yσ(3) , yσ(2) , yσ(1) ) = 0, σ ∈ Sym(3). Theorem 1. A Bergman type polynomial of degree 15 is a ∗-identity in skew-symmetric variables for M6 (K, ∗) if and only if it has the form v(gi )(x, yi1 , yi2 , yi3 ) + β v(gkk )(x, yk1 , yk2 , yk3 ), f =α i

k

where 1. gi = (ai t1 + bi t2 + ci t3 + di t4 )g6,0 , gi+3 = −(di t1 + ci t2 + bi t3 + ai t4 )g6,0 , i = 1, . . . , 3 and t1 + t4 is not a factor of these polynomials; 2. The polynomial (t1 + t4 )g6,0 divides gkk and 3. The identity v(gi )(x, yi1 , yi2 , yi3 ) = 0 follows from the identity P (x, y1 , y2 , y3 ) = 0. For proving the theorem we need the following lemma: Lemma 1. The identities in skew-symmetric variables v(gi )(x, yi1 , yi2 , yi3 ) + v(gi+3 )(x, yi3 , yi2 , yi1 ) = 0

178

TSETSKA GRIGOROVA RASHKOVA

for gi = (ai t1 + bi t2 + ci t3 + di t4 )g6,0 gi+3 = −(di t1 + ci t2 + bi t3 + ai t4 )g6,0 , i = 1, 2, 3 follow from the identity P (x, y1 , y2 , y3 ) = 0. Proof. For simplicity from now on we write f1 = v(g1 )(x, y1 , y2 , y3 ), f2 = v(g2 )(x, y1 , y3 , y2 ), f3 = v(g3 )(x, y2 , y1 , y3 ), f4 = v(g4 )(x, y3 , y2 , y1 ), f5 = v(g5 )(x, y2 , y3 , y1 ) and f6 = v(g6 )(x, y3 , y1 , y2 ), where the polynomials gi will be further speciﬁed when needed. We start with the identity (3)

F = f1 + f4 = 0,

following from Proposition 5 for σ being the identity permutation. The identity (4)

α(F (y1 = [y1 , x]) + β(F (y2 = [y2 , x]) + γ(F (y3 = [y3 , x]) = 0

could be written as αf1 (x, y1 x, y2 , y3 ) − αf1 (x, xy1 , y2 , y3 ) + αf4 (x, y3 , y2 , y1 x) − αf4 (x, y3 , y2 , xy1 ) +βf1 (x, y1 , y2 x, y3 ) − βf1 (x, y1 , xy2 , y3 ) + βf4 (x, y3 , y2 x, y1 ) − βf4 (x, y3 , xy2 , y1 ) +γf1 (x, y1 , y2 , y3 x) − γf1 (x, y1 , y2 , xy3 ) + γf4 (x, y3 x, y2 , y1 ) − γf4 (x, xy3 , y2 , y1 ) = v(g1 (x, y1 , y2 , y3 )) + v(g4 (x, y3 , y2 , y1 )). Thus g1 = α(t2 − t1 ) + β(t3 − t2 ) + γ(t4 − t3 ) = −αt1 + (α − β)t2 + (β − γ)t3 + γt4 , g4 = α(t4 − t3 ) + β(t3 − t2 ) + γ(t2 − t1 ) = −γt1 + (γ − β)t2 + (β − α)t3 + αt4 . Summing (4) for α = 1, β = γ = −1 and α = β = −1, γ = 1 and taking into account the zero characteristics of the ﬁeld we get that v(g10 (x, y1 , y2 , y3 )) + v(g40 (x, y3 , y2 , y1 )) = 0 for g10 = (a1 t1 + b1 t2 + c1 t3 + d1 t4 )g6,0 and g40 = (a1 t1 − c1 t2 − b1 t3 + d1 t4 )g6,0 . Writing g40 = −(d1 t1 + c1 t2 + b1 t3 + a1 t4 )g6,0 + ((a1 + d1 )t1 + (d1 + a1 )t4 )g6,0 we get the validity of Lemma 1 for i = 1. Making in (3) the substitution y2 ↔ y3 we continue with the identity f2 + f5 = 0. Analogous to the above operations prove the statement for i = 2. The substitution y1 ↔ y2 in (3) ﬁnishes the proof of Lemma 1.

ROLE OF A THEOREM OF BERGMAN IN INVESTIGATING IDENTITIES

179

Remark 1. The proof of Lemma 1 does not use the speciﬁc form of g6,0 . Thus we come to its generalization: Lemma 2. Let f = v(g1 )(x, yi1 , yi2 , yi3 ) + v(g2 )(x, yi3 , yi2 , yi1 ) = 0 be a ∗-identity in skewsymmetric variables for M6 (K, ∗). Then the identity v(g3 )(x, yi1 , yi2 , yi3 ) + v(g4 )(x, yi3 , yi2 , yi1 ) = 0 for g3 = (at1 + bt2 + ct3 + dt4 )g1 g4 = − (dt1 + ct2 + bt3 + at4 )g2 is a consequence of the identity f = 0. Proof of Theorem 1: ⇒ We write a Bergman type identity of degree 14 in the form f = fi = i v(gi )(x, yi1 , yi2 , yi3 ) according to the notations at the beginning of Lemma 1. Then commutative to a consequence of f of degree 15 will 4 polynomials corresponding 4 the 4 4 be ( i=1 ai ti )g1 , ( i=1 bi ti )g2 , . . . , ( i=1 ei ti )g5 and ( i=1 li ti )g6 , respectively. Applying Lemma 1 we get that the part Q of f , corresponding to the commutative polynomials 4 4 4 ( ai ti )g1 − ( a5−i ti )g4 + ( bi ti )g2 − i=1

i=1

i=1

4 4 4 ( b5−i ti )g5 + ( ci ti )g3 − ( c5−i ti )g6 , i=1

i=1

i=1

is a ∗-identity in skew-symmetric variables. We consider the remaining part of f corresponding to {(a4 + d1 )t1 + (a3 + d2 )t2 + (a2 + d3 )t3 + (a1 + d4 )t4 }g4 + {(b4 + e1 )t1 + (b3 + e2 )t2 + (b2 + e3 )t3 + (b1 + e4 )t4 }g5 + {(c4 + l1 )t1 + (c3 + l2 )t2 + (c2 + l3 )t3 + (c1 + l4 )t4 }g6 . This part is related to the identity F = A + B + C = 0, where A = (a4 + d1 )xf4 + (a3 + d2 )f4 (y3 = y3 x) +(a2 + d3 )f4 (y2 = y2 x) + (a1 + d4 )f4 x, B = (b4 + e1 )xf5 + (b3 + e2 )f5 (y2 = y2 x) +(b2 + e3 )f5 (y3 = y3 x) + (b1 + e4 )f5 x, C = (c4 + l1 )xf6 + (c3 + l2 )f6 (y3 = y3 x) +(c2 + l3 )f6 (y1 = y1 x) + (c1 + l4 )f6 x. We calculate F (e22 −e55 +2(e33 −e66 )−3(e11 −e44 ), e21 −e45 −e36 , e15 +e24 +e14 , e53 +e62 ) and F (e22 − e55 + 2(e33 − e66 ) − 3(e11 − e44 ), e21 − e45 + e31 − e46 , e15 + e24 − 2e25 , e53 + e62 ). The resulting system 3(b4 + e1 ) + (b3 + e2 ) − 2(b2 + e3 ) + 2(b1 + e4 ) = 0 − 3(b4 + e1 ) − (b3 + e2 ) + 2(b2 + e3 ) − 3(b1 + e4 ) = 0 leads to b1 + e4 = 0.

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Then we make three more calculations, namely F (e11 − e44 − 2(e22 − e55 ) + 5(e33 − e66 ), e12 − e54 + e32 − e56 , e23 − e65 , e16 + e34 ), F (e11 − e44 + 2(e22 − e55 ) − 5(e33 − e66 ) + 3(e21 − e45 ) + e32 − e56 , e23 − e65 , e16 + e34 + e63 , e12 − e54 ) and F (e11 − e44 − 2(e22 − e55 ) + 3(e33 − e66 ), e36 , e43 + e61 , e12 − e54 ). The resulting system for the unknowns a4 + d1 , a3 + d2 , a2 + d3 and a1 + d4 is the following: 2 (a4 + d1 ) − (a3 + d2 ) − 3(a2 + d3 ) + 2(a1 + d4 ) = 0, 2 (a4 + d1 ) − (a3 + d2 ) + 3(a2 + d3 ) − 3(a1 + d4 ) = 0, (a4 + d1 ) + (a3 + d2 ) + 5(a2 + d3 ) + 5(a1 + d4 ) = 0, (a4 + d1 ) − 5(a3 + d2 ) + 2(a2 + d3 ) − (a1 + d4 ) = 0. It has only the trivial solution. Now we calculate B(b1 + e4 = 0) + C for x = 3(e22 − e55 ) − 5(e33 − e66 ) − e11 + e44 , y1 = e12 − e54 − e23 + e65 − e16 − e34 + e42 + e51 + e43 + e61 , y2 = e12 − e54 + e23 − e65 − e16 − e34 + e42 + e51 − e43 − e61 , y3 = e12 − e54 − e23 + e65 − 4(e16 + e34 ) − 2(e42 + e51 ) + e43 + e61 . Thus we come to a homogeneous system for the unknowns (b4 + e1 ), (b3 + e2 ), (b2 + e3 ), (c4 + l1 ), (c3 + l2 ), (c2 + l3 ) and (c1 + l4 ), whose matrix of the coeﬃcients is

−1 5 −3 −4 20 −12 −4 2 −60 −44 83 −465 199 −83 −12 20 −4 3 −5 1 3 5 −1 −3 1 −2 −6 1 . −1 −3 5 2 6 −10 2 6 2 −10 3 1 −5 3 5 −93 −89 5 −93 −89 −5 −10 6 2 −5 3 1 −5

The rank of this matrix is 7 meaning that the considered system has only a trivial solution, i.e. d1 + a4 = d2 + a3 = d3 + a2 = d4 + a1 = 0, e1 + b4 = e2 + b3 = e3 + b2 = e4 + b1 = 0, l1 + c4 = l2 + c3 = l3 + c2 = l4 + c1 = 0. These relations and Proposition 5 show that the part Q is an identity in skew-symmetric variables and the identity f has the form stated in the theorem. All calculations were made using the computer algebra system Mathematica. ⇐ Let f be a Bergman type polynomial of degree 15 of the stated form. We consider its part A = f1 (x, y1 , y2 , y3 ) + f4 (x, y3 , y2 , y1 ) in which g1 = (a1 t1 + a2 t2 + a3 t3 + a4 t4 )g6,0 ,

g4 = −(a4 t1 + a3 t2 + a2 t3 + a1 t4 )g6,0 .

Lemma 1 shows that the polynomial A is an ∗-identity in skew-symmetric variables. Applying the same lemma to the other two parts f2 + f5 and f3 + f6 of the polynomial f we get that f is a ∗-identity, as a consequence of the identity P (x, y1 , y2 , y3 ) = 0. This ends the proof of Theorem 1. An easy corollary of it using Lemma 2 is the following

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Proposition 6. Every Bergman type polynomial of degree k of the form f =α

fi (x, yi1 , yi2 , yi3 ) + β

v(gss )(x, ys1 , ys2 , ys3 ),

s

i

where k−14 k−14 1. gi = g6,0 j=1 (aji t1 + bji t2 + cji t3 + dji t4 ), gi+3 = g6,0 j=1 (−(aji t1 + cji t2 + bji t3 + dji t4 )), i=1, . . . ,3 and t1 + t4 is not a factor of these polynomials; 2. The polynomial (t1 + t4 )g6,0 divides gss and 3. The identity fi (x, yi1 , yi2 , yi3 ) = 0 follows from the identity P (x, y1 , y2 , y3 ) = 0, is a ∗-identity in skew-symmetric variables for M6 (K, ∗). Now we could deﬁne another class of Bergman type identities. Theorem 2. The polynomials f = σ∈Sym(3) (−1)σ v(g)(x, yσ(1) , yσ(2) , yσ(3) ) for g = (t22 − t23 )(t1 t2 + t2 t3 + t3 t4 )k g6,0 are Bergman type identities in skew-symmetric variables for M6 (K, ∗) of degree 16+2k. Proof. According to [4, Theorem 12] the polynomials f = σ∈Sym(3) (−1)σ v(g1 )(x, yσ(1) , yσ(2) , yσ(3) ) for

g1 =

(t2p − t2q )(t22 − t23 )(t1 t2 + t2 t3 + t3 t4 )k

1≤p

are central polynomials in skew-symmetric variables for M6 (K, ∗) of degree 15+2k. Now Theorem 2 is straightforward due to the deﬁnition of central polynomials. Now we are able to formulate a result for M2n (K, ∗) generalizing the “only if” part of Proposition 4 for n = 2 and Proposition 6 for n = 3. Theorem 3. For n ≡ 2,3 (mod 4) every Bergman type polynomial of degree k of the form f =α

i

v(gi )(x, yi1 , . . . , yin ) + β

v(gj )(x, yj1 , . . . , yjn ),

j

where k−n2 −2n+1 n k−n2 −2n+1 n (l) (l) 1. gi = g2n,0 l=1 = g2n,0 l=1 (− m=1 ai,n+1−m tm ), m=1 ai,m tm , gi+ n! 2 i = 1, . . . , n! 2 and t1 + tn+1 is not a factor of these polynomials; 2. The polynomial (t1 + tn+1 )g2n,0 divides gj and 3. The identity v(gi )(x, yi1 , . . . , , yin ) = 0 follows from the identity v(g2n,0 )(x, yi1 , yi2 , . . . , yin ) + v(g2n,0 )(x, yin , yin−1 = 0, . . . , yi1 ), (i1 , i2 , . . . , in ) ∈ Sym(n), is a ∗identity in skew-symmetric variables for M2n (K, ∗).

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Proof. The proof follows the ideas of presenting a polynomial and its consequences of the next degree as exposed at the beginning of the proof of Theorem 1 using the next Lemma (instead of Lemma 2) and Remark 2 as well. Lemma 3. For n ≡ 2,3 (mod 4) the polynomial C(x, y1 , y2 , . . . , yn ) = v(g2n,0 )(x, yi1 , yi2 , . . . , yin ) + v(g2n,0 )(x, yin , yin−1 , . . . , yi1 ) for any (i1 , i2 , . . . , in ) ∈ Sym(n) is a ∗-identity in skew-symmetric variables for M2n (K, ∗). Proof. Considering the factors tk ± tl of g(t1 , . . . , tn ) (except t1 + tn+1 ) we follow the proof of n[7, Proposition 2]. It shows that it is suﬃcient to consider only the case when x = i=1 ρi (eii − en+i,n+i ) and to ﬁnd an analogue of the stated proof concerning the factor t1 + tn+1 which is not the case here. For this analogue we evaluate n ¯ C = C( ρi (eii − en+i,n+i ), e12 − en+2,n+1 , . . . , en−1,n − e2n,2n−1 , en,n+1 + e1,2n ). i=1

The result is C¯ = Ae1,n+1 = (g2n,0 (ρ1 , . . . , ρn , −ρ1 ) + (−1)n g2n,0 (ρ1 , −ρn , −ρn−1 , . . . , −ρ1 ))e1,n+1 . The properties of the polynomial g2n,0 (t1 , . . . , tn+1 ) allow us to write A = g2n,0 (ρ1 , . . . , ρn , −ρ1 ) + (−1)n−1 g2n,0 (ρ1 , ρn , ρn−1 , . . . , ρ2 , −ρ1 ). Changing the places of two neighbouring variables causes (−1) as a factor of g2n,0 . Thus A = g2n,0 (ρ1 , . . . , ρn , −ρ1 ) + (−1)n−1+n−2+···+1 g2n,0 (ρ1 , ρ2 , . . . , ρn , −ρ1 ) = g2n,0 (ρ1 , . . . , ρn , −ρ1 ) + (−1)n−1+ = g2n,0 (ρ1 , . . . , ρn , −ρ1 ) + (−1) For n ≡ 2,3 (mod 4) A ≡ 0.

(n−1)(n−2) 2

n(n−1) 2

g2n,0 (ρ1 , ρ2 , . . . , ρn , −ρ1 )

g2n,0 (ρ1 , ρ2 , . . . , ρn , −ρ1 ).

1 2 ... n 1 2 ... n Remark 2. Let denote ρi = and ρi−1 = . Then for the i1 i2 . . . in in in−1 . . . i1 symmetric group Sym(n) the following holds: Sym(n) = ∪i {ρi , ρi−1 }. Proof. For any ρi ∈ Sym(n) the map ϕ(ρi ) = ρi−1 is an automorphism of second order and {ϕ, e} divides the elements of Sym(n) into orbits. Obviously each orbit contains 2 elements. No orbits share common elements and any element of Sym(n) is in an orbit. Remark 3. It could be easily seen that modulo the identity )(x, yσ(1) , yσ(2) ) the polynomial g2 in Proposition 4 could be written as g2 = 4,0 σ v(g −g4,0 i (ci ti + bi ti + ai ti ) meaning that Theorem 2 generalizes the partial cases for n investigating in Propositions 4 and 6.

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References [1] G.M. Bergman, Wild automorphisms of free P.I. algebras and some new identities (1981), preprint. [2] A. Giambruno, A. Valenti, On minimal ∗-identities of matrices, Linear Multilin. Algebra 39 (1995), 309–323. [3] T.G. Rashkova, Bergman type identities in matrix algebras with involution, Proceedings of the Union of Scientists - Rousse, ser. 5 Mathematics, Informatics and Physics 1 (2001), 26–31. [4] T. Rashkova, Matrix algebras with involution and central polynomials, J. of Algebra 248 (2002), 132–145. [5] T.G. Rashkova, ∗-identities of minimal degree in matrix algebras of low order, Periodica Mathematica Hungarica 34 (3) (1998), 229–233. [6] T.G. Rashkova, One conjecture for the identities in matrix algebras with involution, Ann. of Sofia Univ. 94 (2002), to appear. [7] Ts. Rashkova, V. Drensky, Identities of representations of Lie algebras and ∗-polynomial identities, Rendiconti del Circolo Matematico di Palermo 48 (1999), 153–162.

THE TRIANGULAR STRUCTURE OF LADDER FUNCTORS WOLFGANG RUMP Institut f¨ ur Algebra und Zahlentheorie, Universit¨ at Stuttgart Pfaﬀenwaldring 57, D-70550 Stuttgart, Germany e-mail: [email protected]

Ladder functors were introduced in [14] and [16] as a tool for the structural analysis of categories A with Auslander-Reiten sequences. The original idea of a ladder goes back to Igusa and Todorov [6] who considered chains of maps λ

λ

0 1 · · · −→ a0 −→ a1 −→ a2 −→ · · ·

between two-termed complexes ai such that the mapping cone of each λi is an almost split sequence. Being rare objects, such ladders were diﬃcult to handle, but Igusa and Todorov successfully applied them to obtain their characterization of the Auslander-Reiten quivers of representation-ﬁnite artinian algebras. The corresponding problem in dimension one, i. e. for orders over a complete discrete valuation domain, was solved 15 years later by Iyama [8], who showed that (modiﬁed) ladders of arbitrary length are obtained when the starting morphism a0 ∈ A is special, i. e. if its isomorphism class is invariant modulo Rad2 A. For this improvement, it has to be payed in return that at each step an of the ladder, trivial direct summands 0 → A have to be discarded before passing to an+1 . In [16] we showed that the homotopy category M(A) of two-termed complexes is rather useful for the study of ladders. We constructed a functor L: M(A) → M(A) together with an augmentation λ: L → 1 such that any morphism a ∈ A, regarded as an object in M(A), gives rise to a ladder λ

λ

La a La −→ a · · · −→ L2 a −→

which generalizes the above mentioned constructions. There is no restriction on a, and there are no trivial summands to be discarded. Moreover, the components of λ, regarded as commutative squares in A, are exact, i. e. they are simultaneous pullbacks and pushouts. The dual construction leads to an endofunctor L− with a natural transformation λ− : 1 → L− , which produces ladders in the reverse direction. Both functors L, L− determine each other, forming an adjoint pair L L− . The use of these ladder functors [14, 16] led to a better understanding and improvement of previous work on categories with almost split sequences. In particular, Iyama’s criterion [8] for ﬁnite Auslander-Reiten quivers in dimension one was replaced by a characterization in terms of additive functions [16]. As the homotopy category M(A) can be viewed as a certain part of a triangulated category, there remains some fragment of triangulated structure in M(A). Our ﬁrst aim in this article will be to show how the ladder functors are related to triangles (Theorem 1). We prove that each object a in M(A) determines a “triangle” (0)

σ

λ

π

a a a La −→ a −→ Sa, T Sa −→

2000 Mathematics Subject Classiﬁcation. Primary: 16G70, 16G30, 16D90, 18E30. Secondary: 16G60. Key words and phrases. Ladder functor, triangulated category, Auslander-Reiten sequence.

THE TRIANGULAR STRUCTURE OF LADDER FUNCTORS

185

where πa belongs to the localization M(A)PrL of M(A) by the full subcategory Pr L = {a ∈ Ob M(A) | λa invertible}, and σa belongs to the corresponding localization M(A)InL with respect to L− . The objects Sa and T Sa are semisimple in M(A)PrL and M(A)InL , respectively. (We call an object S left semisimple if every monomorphism S → S splits. Left and right semisimple objects are just called semisimple.) The functor T provides an equivalence between semisimple objects of M(A)PrL and M(A)InL , respectively. Since T is not an endofunctor, it cannot be iterated. However, with that grain of salt, the sequences (0) behave just like ordinary triangles. For example, if ϕ: b → a is a morphism in M(A) with πa ϕ = 0 in M(A)PrL , then ϕ factors uniquely through λa (as a morphism of M(A)!). In addition to the triangle property, the morphism πa ∈ M(A)PrL is universal in the sense that any morphism ψ: a → s in M(A)PrL with s semisimple factors uniquely through πa . Therefore, the ladder functor L is uniquely determined by the full subcategory Pr L. Let us remark that for a category A of maximal Cohen-Macaulay modules over an isolated singularity Λ in the sense Auslander [2], if Λ is of dimension > 2, then projective objects P in A are no longer characterized by the property τ P = 0. Fortunately, the construction of our ladder functor L makes no use of right almost split sequences 0 → ϑP → P . On the other hand, this means that the ladder functors do not tell the whole story about A. However, the missing structure is suggested by the triangles (0), as they deﬁne a connection between semisimple objects in M(A)PrL and M(A)InL . While left and right semisimple objects coincide in these localisations, they form two diﬀerent full subcategories Sl M(A) and Sr M(A) in M(A). We will show (Proposition 8) that an object a: A → P in M(A) is left semisimple if and only if a is a right almost split morphism in A with τ P = 0. Consequently, there are full embeddings of A into the categories of one-sided semisimple objects in M(A): Sl M(A) ← A → Sr M(A). As a byproduct, it follows that A can be recovered from M(A). In case ind A is ﬁnite, we show (Theorem 4) that the essential condition for A to be representable as a category A-mod over an artinian ring A, or as a category Λ-lat of lattices over an order Λ, consists in a speciﬁc correspondence between Sl M(A) and Sr M(A).

1. Preliminaries Let A be an additive category. For a full subcategory C, the ideal generated by the identity morphisms 1C with C ∈ Ob C will be denoted by [C]. By Mor(A) we denote the category with morphisms in A as objects and commutative squares as morphisms. There is a natural full embedding A → Mor(A) which maps A ∈ Ob A to 1A ∈ Ob Mor(A). If we regard the objects of Mor(A) as two-termed complexes 0 → A1 → A0 → 0, then the ideal [A] of Mor(A) consists of the morphisms which are homotopic to zero. We deﬁne mod(A) (resp. com(A)) as the factor category of Mor(A) modulo the ideal of morphisms

(1)

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WOLFGANG RUMP

such that f0 factors through b (resp. f1 factors through a). Then mod(A) is equivalent to the category of coherent functors Aop → Ab into the category of abelian groups, and com(Aop ) = mod(A)op .

(2)

If A = R-proj, the category of ﬁnitely generated projective left modules over a ring R, then mod(A) is equivalent to the category R-mod of ﬁnitely presented left R-modules. Note that mod(A) need not be abelian. Now let A be a Krull-Schmidt category, that is, an additive category such that every object is a ﬁnite direct sum of objects with local endomorphism rings. A morphism f : A → B in A is said to belong to the radical Rad A if 1 − gf is invertible for all g: B → A. (Of course, this concept is left-right symmetric.) Every morphism f ∈ A admits a decomposition f = e ⊕ r into an isomorphism e and a morphism r ∈ Rad A. Therefore, the full subcategory M(A) of objects a: A1 → A0 in Mor(A)/[A] with a ∈ Rad A is equivalent to Mor(A)/[A]. If A+ (resp. A− ) denotes the full subcategory of objects A+ : 0 → A (resp. A− : A → 0) in M(A), then (3)

mod(A) ≈ M(A)/[A− ];

com(A) ≈ M(A)/[A+ ].

This follows since a morphism in M(A) belongs to [A− ] if and only if it vanishes in mod(A). In order to work with mod(A), an intrinsic characterization will be useful. Recall that an object P of an additive category A is said to be projective if for every cokernel A B in A, the natural map HomA (P, A) → HomA (P, B) is surjective. Arrows (resp. ) stand for cokernels (kernels). We say that A has enough projectives if for each object A there is an epimorphism P → A with P projective. Injective objects and the property of having enough injectives are deﬁned dually. The full subcategory of projective (injective) objects in A will be denoted by Proj(A) (resp. Inj(A)). For a full subcategory C of A, we deﬁne add C as the full subcategory of objects C ∈ Ob A with 1C ∈ [C]. If idempotents split in A, this coincides with the usual deﬁnition (see [3]). There is a natural embedding A → mod(A)

(4) which maps A ∈ Ob A to 0 → A.

Proposition 1. Let A be an additive category. Then (5)

addA = Proj(mod(A)).

An additive category M is equivalent to mod(A) for some additive category A if and only if the following are satisfied. (a) (b) (c) (d)

Every morphism in mod(A) has a cokernel. Every epimorphism in M is a cokernel. M has enough projectives. c a If A → B C is a sequence of morphisms in M with c = cok a, and a morphism p: P → B with P projective satisfies cp = 0, then p factors through a. a

c

The proof makes use of the following lemma. A sequence of morphisms P1 → P0 M with P0 , P1 projective and c = cok a is said to be a projective presentation of M .

THE TRIANGULAR STRUCTURE OF LADDER FUNCTORS

187

Lemma 1. Let M be an additive category such that every object has a projective presentation. Then condition (c) of Proposition 1 is satisfied whenever it holds with A, B projective. a

c

Proof. For a given sequence A → B C as in (c), consider a projective presentation b

d

Q1 → Q0 B and a cokernel q: Q A with Q projective. Then there exists a morphism f : Q → Q0 with aq = df , and we obtain a commutative diagram

where cd is a cokernel of (f b). Now the assertion follows immediately. Proof of Proposition 1. For a morphism (1) in mod(A), it is easily checked that a cokernel is given by

(6)

This yields (a) and implies that the objects of A are projective in mod(A). Moreover, we infer that any object a: A1 → A0 in mod(A) has a projective presentation

(7)

If a ∈ Proj(mod(A)), this gives a split epimorphism A0 a. Hence (5) holds. Therefore, (7) satisﬁes the property stated in (d). By Lemma 1, this implies that (d) holds in mod(A). Furthermore, (7) yields (c) for mod(A). To verify (b), let (1) be an epimorphism in mod(A). By (6) this means that (b f0 ) has a section hs : B0 → B1 ⊕ A0 , i. e. bh + f0 s = 1. In fact, this implies that (1) is a cokernel:

(The composition is zero by virtue of the map (hf0 1 − hb): A0 ⊕ B1 → B1 .)

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WOLFGANG RUMP

Conversely, let M be an additive category which satisﬁes (a), (b), (c), and (d). Then a every object M of M has a projective presentation P1 → P0 M , and it is readily seen ∼ that M → a yields an equivalence M −→ mod(Proj(M)). Recall that a monic and epic morphism is said to be regular. Corollary. Let A be an additive category. Then every regular morphism in mod(A) is invertible. Proof. Let r: M → N be regular in mod(A). Since r is epic, we have r = cok f for some f ∈ mod(A). Since r is monic, this gives f = 0. Hence r is invertible. 2. Semisimple objects In this section we study existence properties for semisimple objects in mod(A), as far as needed in connection with ladder functors. Let A be an additive category. We call an object S left (right) semisimple if every monomorphism A → S (epimorphism S → A) in A splits. If both conditions hold, we call S semisimple. The full subcategory of semisimple objects will be denoted by S(A). We call A (co-)semilocal if S(A) is a (co-)reﬂective subcategory of A, i. e. if the inclusion S(A) → A has a left (right) adjoint. If this adjoint functor maps non-zero objects to non-zero objects, we call A strongly (co-)semilocal. Example. For a ring R, let R-Mod be the category of left R-modules. Then S(R-Mod) consists of the direct sums of simple modules ([1], Theorem 9.6), and there is no diﬀerence between left and right semisimple objects. Furthermore, R-Mod is semilocal if and only if R is a semilocal ring. However, R-Mod is always co-semilocal. By [1], Lemma 28.3, R-Mod is strongly semilocal if and only if R is left perfect. By [17], VIII, Proposition 2.5, R-Mod is strongly co-semilocal if and only if R is left semi-artinian. Recall that an additive functor S: A → A together with a natural transformation π: 1 → S is said to be a pointed functor [10]. For a reﬂective full subcategory C of A with reﬂector S: A → C and inclusion I: C → A, the unit η: 1 → IS deﬁnes a pointed functor IS. This will be called the reflection of C. In contrast to the above example, the components of η need not be epic, in general. Dually, an additive endofunctor S with a natural transformation S → 1 will be called an augmented functor, and if S comes from a coreﬂective full subcategory C, we call S the coreflection of C. Proposition 2. Let A be an additive category. Then every right semisimple object in mod(A) is semisimple. Proof. Let S be right semisimple in mod(A), and let f : A → S be a monomorphism. Then f has a cokernel c: S C. Since S is right semisimple, c has a section s: C → S, that is, cs = 1. Now it is easy to verify that (f s): A ⊕ C → S is regular. By the Corollary of Proposition 1, it follows that (f s) is invertible, whence f is split monic. Remark. It can be shown that for a strict τ -category (see §4), left and right semisimplicity is equivalent in mod(A). Let A be a Krull-Schmidt category. A morphism u: A → B in A is said to be right almost split if u ∈ Rad A, and every f : C → B in Rad A factors through u. When such a morphism u exists for each B ∈ Ob A, we simply say that A has right almost split morphisms. The left-hand notions are deﬁned in a dual fashion. The following proposition has a well-known version for (semi-)simple functors.

THE TRIANGULAR STRUCTURE OF LADDER FUNCTORS a

189

p

Proposition 3. Let A be a Krull-Schmidt category, and let P1 → P0 S be a projective presentation in mod(A) with a ∈ Rad A. Then S is semisimple if and only if a is right almost split in A. Proof. Suppose that S is semisimple. Let r: P → P0 be in Rad A. Then the cokernel c: S C of pr has a section s: C → S. Now scp: P0 → S lifts along p, say, scp = pe. Hence cp(1 − e) = cp − cscp = 0, and thus p(1 − e) = prg for some g: P0 → P . Therefore, 1 − e − rg factors through a. So we get 1 − e ∈ Rad A, whence e is invertible. Since (1 − sc)pe = scp − scpe = scp(1 − e) = 0, we get 1 − sc = 0. Thus pr = scpr = 0, which proves that r factors through a. Conversely, let a be right almost split, and let e: S → B be an epimorphism in mod(A). q

b

Consider a projective presentation Q1 → Q0 B with b ∈ Rad A. Since e is a cokernel, we get a morphism f : Q0 → P0 with ep · f = q. Since a is right almost split, we obtain a commutative diagram

This yields a morphism s: B → S with sq = pf . Hence (1 − es)q = q − epf = 0, and thus es = 1. Lemma 2. Let A be a Krull-Schmidt category, and let r: P → Q be a morphism in Rad A. If S is a semisimple object in mod(A), then every morphism s: Q → S in mod(A) satisfies sr = 0. a

p

Proof. Choose a projective presentation P1 → P0 S with a ∈ Rad A. By Proposition 3, a is right almost split in A. Therefore, we get a commutative diagram

which yields sr = 0. Lemma 3. If an object M of an additive category admits a projective presentation, then every cokernel q: Q0 M with Q0 projective can be completed to a projective presentation. a

p

Proof. Let P1 → P0 M be a projective presentation. There are morphisms f, g between P0 and Q0 such that p = qf and q = pg. Then q is a cokernel of (1−f g f a): Q0 ⊕ P1 → Q0 . Proposition 4. Let A be a Krull-Schmidt category. Then mod(A) is strongly semilocal if and only if A has right almost split morphisms. If mod(A) is semilocal with reflection η: 1 → S, then the components of η are epic.

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Proof. Let mod(A) be semilocal with reﬂection η: 1 → S. For an object M of mod(A), the cokernel c: SM C of ηM has a section s: C → SM . Hence (1 + sc)ηM = ηM , and thus 1 + sc = 1 by the uniqueness property of η. So we get c = 0, which shows that ηM is ηA a epic. By Lemma 3, every A ∈ Ob A gives rise to a projective presentation A → A SA in mod(A). Assume that mod(A) is strongly semilocal. Then A has no non-zero direct summand B with ηB = 0. Hence a ∈ Rad A. By Proposition 3, this yields a right almost split morphism a. Conversely, suppose that A has right almost split morphisms. For a given p a object M in mod(A), consider a projective presentation P1 → P0 M with a ∈ Rad A, and a right almost split morphism u: P → P0 in A. Then a factors through u, and so we get a commutative diagram

with c = cok u. By Proposition 3, SM is semisimple. Let f : M → S be any morphism in mod(A) with S semisimple. Then f pu = 0 by Lemma 2. Hence f p = gc for some g: SM → S. Thus (f − gηM )p = 0, which gives f = gηM . This proves that η: 1 → S is a reﬂection. If SM = 0, then u is epic, and thus a cokernel in mod(A). Hence u is split epic. So we get P0 = 0, and therefore, M = 0. This proves that mod(A) is strongly semilocal. 3. Quotient categories For our purpose we need a generalization of Grothendieck’s quotient category ([5], 1.11). Let A be an additive category. We shall say that A has a quotient category if the regular morphisms in A admit a calculus of left and right fractions [4]. The corresponding quotient category will be denoted by Q(A). Up to equivalence, the faithful embedding Q: A → Q(A) is characterized by the property that Q makes regular morphisms invertible, and that every morphism in Q(A) can be written in the form r−1 a = bs−1 with a, b, r, s ∈ A and r, s regular. More generally, let S be a full subcategory of A. We deﬁne Σ(S) as the class of morphisms in A which are regular in A/[S]. We call an additive functor (8)

Q:A→B

with S := Ker Q := {A ∈ Ob A | Q(A) = 0} a quotient functor if the following are satisﬁed. (a) For any r ∈ Σ(S), Q(r) is invertible in B. (b) Q(f ) = 0 if and only if f ∈ [S]. (c) Every morphism in B is of the form Q(r)−1 Q(f ) = Q(g)Q(s)−1 with f, g ∈ A and r, s ∈ Σ(S). It is easy to see that for a quotient functor (8), up to equivalence, the category B merely ∼ Q(A/[S]) such that EQ coincides depends on S, namely, there is an equivalence E: B −→ with the natural composition A → A/[S] → Q(A/[S]). Hence, for a full subcategory S of A, a quotient functor (8) with Ker Q = add S exists if and only if A/[S] has a quotient category. Therefore, if Q(A/[S]) exists, we write AS := B. In particular, A0 ≈ Q(A) when A has a quotient category.

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Remark. Obviously, a quotient functor Q: A → AS solves the universal problem of making the morphisms of Σ(S) invertible. We call a full subcategory S of A thick if Σ(S) admits a calculus of left and right fractions. When A is abelian, it can be shown that this concept of thickness coincides with the usual one (then S is also called a Serre subcategory), and that AS is equivalent to Grothendieck’s quotient category A/S. Proposition 5. Let A be an additive category and P a full subcategory. If Gen P denotes the full subcategory of objects M in mod(A) admitting a cokernel P M with P ∈ P, then (9)

mod(A/[P]) = mod(A)Gen P .

Proof. We have to show that the natural functor Q : mod(A)/[Gen P] −→ mod(A/[P]) is a faithful quotient functor. Thus let f : M → N be a morphism in mod(A) which is zero in mod(A/[P]). Consider a commutative diagram

p

q

in mod(A) with cokernels a, b and A, B ∈ A. Then we have morphisms A → P → B with P ∈ P such that b(g − qp) = 0. The pushout

in mod(A) yields an object C ∈ Gen P and bq · p = f · a implies that f factors through c. This proves that Q is faithful. q p Now let (1) be a morphism f ∈ mod(A/[P]). Then there are morphisms A1 → P → B0 a with P ∈ P such that f0 a − bf1 = qp. Consider the morphisms r: p → a and s: b → (b q) in mod(A)/[Gen P] given by

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respectively. Then Q(r) and Q(s) are invertible, whence r, s are regular. Moreover, there are morphisms g, h ∈ mod(A) with f · Q(r) = Q(g) and Q(s) · f = Q(h). Hence f = Q(g)Q(r)−1 = Q(s)−1 Q(h). It remains to show that Q makes regular morphisms invertible. More generally, let e ∈ mod(A)/[Gen P] be monic. Assume that f = Q(g)Q(r)−1 ∈ mod(A/[P]) satisﬁes Q(e) · f = 0. Then Q(e)Q(g) = 0, which gives eg = 0, and thus g = 0. Hence Q(e) is monic. Dually, we infer that Q preserves epimorphisms. Now the Corollary of Proposition 1 completes the proof. 4. Ladder functors and triangles Ladder functors were introduced [14, 16] in order to an analyse the Auslander-Reiten structure of a strict τ -category A. We will show now that ladder functors are closely related to a kind of triangular structure of the homotopy category M(A). Recall that a sequence of morphisms v

u

τ A ϑA → A

(10)

in a Krull-Schmidt category A is said to be right almost split if u is right almost split, v is left almost split, and v = ker u. Up to isomorphism, such a sequence is uniquely determined by the object A. In a dual fashion, left almost split sequences A → ϑ− A τ − A

(11)

are deﬁned. If left and right almost split sequences (10) and (11) exist for each object A of A, then A is said to be a strict τ -category [7]. Let A be a strict τ -category. Every object a: A1 → A0 of M(A) admits a left standard form [16], that is, a representation as a matrix

(12)

a=

b ft s p

:B⊕U →C ⊕P

with τ P = 0 and t ∈ Rad A, such that there exists a left and right almost split sequence (13) Then a morphism λa : La → a in M(A) is given by the commutative diagram

(14)

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By [16], Proposition 4, this yields an additive functor L: M(A) → M(A) together with a natural transformation λ: L → 1. The augmented functor L will be called the left ladder functor. In a dual way, we get the right ladder functor λ− : 1 → L− . In [16] we showed that the components λa and λ− a are regular. Notice that the vertical morphisms in (14) are in a sense dual to each other. By this property, it follows ([16], Proposition 5) that L is left adjoint to L− , i. e. there is a natural isomorphism ∼ HomM(A) (a, L− b). Φ : HomM(A) (La, b) −→ Let us write ϕ− := Φ(ϕ) and ψ + := Φ−1 (ψ). Then the correspondence ϕ → ϕ− is given by the commutative diagram

(15)

Let us denote the full subcategory of objects A in A with τ A = 0 (resp. τ − A = 0) by Projτ (A) (resp. Injτ (A)). In [16] we deﬁned the full subcategories Fix L and Fix L− of objects a in M(A) for which λa (resp. λ− a ) is invertible. An object x ∈ Ob M(A) will be called L-projective (L-injective) if (λa )∗ : HomM(A) (x, La) → HomM(A) (x, a) (resp. λ∗a : HomM(A) (a, x) → HomM(A) (La, x)) is surjective for all a ∈ Ob M(A). The full subcategory of L-projective (L-injective) objects will be denoted by Pr L (resp. In L). The relationship between these categories is given by Proposition 6. Let A be a strict τ -category. Then Pr L = Fix L and In L = Fix L− . Proof. The inclusion Fix L ⊂ Pr L is trivial. Conversely, assume that a ∈ Pr L. Then λa has a section ϕ: a → La. Since λa is regular, this implies that a ∈ Fix L. By (15), we get Fix L− ⊂ In L and a commutative diagram

(16)

for any b ∈ Ob M(A). Suppose that b ∈ In L. Then there is a morphism ϕ: L− b → b with εb = ϕλL− b . Since λL− b is componentwise epic, this gives λ− b ϕ = 1, and thus b ∈ Fix L− . Next we will prove that the localizations M(A)S with S = Pr L or S = In L exist for a strict τ -category A. For a morphism ϕ ∈ M(A) we deﬁne the local (co-)kernel kerS ϕ (resp. cokS ϕ) as the (co-)kernel of ϕ in M(A)S . As a certain counterpart, we call ϕ ∈ M(A) a

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global kernel of ψ ∈ M(A)S if ψϕ = 0 holds in M(A)S , and for any ϕ ∈ M(A) with ψϕ = 0 in M(A)S there exists a unique ω ∈ M(A) with ϕω = ϕ .

(17)

By deﬁnition, the global kernel, as well as its dual, the global cokernel, is unique up to isomorphism. We write ϕ = kerS ψ (resp. cokS ψ) for the global (co-)kernel of ψ. Proposition 7. Let A be a strict τ -category with a full subcategory P, and let SP (mod(A)) denote the full subcategory of objects S in mod(A) with HomA (P, S) = 0. Then there is a natural equivalence (18)

∼ S(mod(A/[P])). SP (mod(A)) −→ u

Proof. According to Proposition 3, let ϑA → A S be a projective presentation of an object S in SP (mod(A)). Since HomA (P, S) = 0, it follows that A has no nonzero direct summand in P. As u is right almost split in A/[P], it determines an object S in S(mod(A/[P])), and so S → S deﬁnes an additive functor F : SP (mod(A)) → S(mod(A/[P])). If F (ϕ) = 0 for a morphism ϕ ∈ SP (mod(A)), then ϕ ∈ [Gen P] by u

Proposition 5. Hence ϕ = 0, and thus F is faithful. Let ϑA → A S be a projective presentation of another object S in SP (mod(A)). A morphism F (S) → F (S ) is given by a commutative diagram

(19)

in A/[P]. Then f can be modiﬁed modulo [P] such that (19) becomes commutative in A. Thus F is full. By Proposition 3, any semisimple object in mod(A/[P]) is given by a right almost split morphism a: A1 → A0 in A/[P], where we may assume that a ∈ Rad A, and that A0 has no non-zero direct summand in P. Since the right almost split morphism u: ϑA0 → A0 in A belongs to Rad(A/[P]), there is a morphism f : ϑA0 → A1 with u−af ∈ [P]. So if we replace a by some a : A1 ⊕ P → A0 with P ∈ P, we may assume that u factors through a . Since a ∈ Rad A, it follows that a is right almost split in A. This proves that F is dense, and thus an equivalence. Theorem 1. Let A be a strict τ -category. Then M(A)PrL ≈ mod(A/[Projτ (A)]) and M(A)InL ≈ com(A/[Injτ (A)]). For any a ∈ Ob M(A), there is a sequence of morphisms (20)

σ

λ

π

a a a T Sa −→ La −→ a −→ Sa

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with πa ∈ M(A)PrL and σa ∈ M(A)InL such (a) (b) (c) (d)

πa = cokPr L λa and σa = kerIn L λa . λa = kerPr L πa = cokIn L σa . M(A)Pr L is semilocal with reflection π: 1 → S. T : S(M(A)PrL ) → S(M(A)InL ) is an equivalence.

Remarks. 1. The theorem shows that (20) shares some properties with the distinguished triangles of a triangulated category. The main diﬀerence lies in the fact that (20) belongs to three diﬀerent categories. 2. The natural transformation λ is close to an isomorphism since the end terms of (20) are semisimple in M(A)PrL resp. M(A)InL . 3. The ladder functors are determined by the full subcategory Pr L, or equivalently, by In L. Proof. Consider a cokernel (1) in mod(A) with a ∈ P := Projτ (A) ⊂ A ⊂ mod(A), i. e. A1 = 0 and A0 ∈ P. We may assume that b ∈ Rad A. Then (6) shows that f0 is split epic. Hence B0 ∈ P. Conversely, every object a: A → P in mod(A) with P ∈ P belongs to Gen P. By (3) and Proposition 5, this implies that M(A)PrL ≈ mod(A/[P]). The second equivalence follows by duality. Let us write out (20) explicitly.

(21)

Since (1 0)

f0 01

= (b f )

00 10

, we have πa λa = 0 in mod(A), hence also in mod(A/[P]).

Moreover, πa is epic in mod(A/[P]) by (6). Now let

be a morphism ϕ: a → x in mod(A/[P]) with ϕλa = 0 in mod(A/[P]). Then x(v w) − (y z) sb fpt ∈ [P], and there is a morphism (g h): B ⊕ P → X1 with (y z) f0 10 − x(g h) ∈ [P]. Hence we get a morphism

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in mod(A/[P]). As (y z) − y(1 0) − x(0 h) ∈ [P], this shows that ϕ factors through πa , whence πa = cokPrL λa . Note that the preceding proof remains valid if P is replaced by any full subcategory of A. Therefore, the dual argument yields (a). Next let

be a morphism ϕ: x → a in M(A) with πa ϕ = 0 in mod(A/[P]). Then there is a morphism h: X0 → B ⊕ B = ϑC with y − (b f )h ∈ [P]. Thus y ∈ Rad A. By [16], Proposition 3, this implies that ϕ factors uniquely through λa . Hence λa = kerPrL πa . Again, the dual argument gives (b). Since Sa is right almost split in A/[P], we get Sa ∈ S(mod(A/[P])) by Proposition 3. To prove (c), let ρ−1 ϕ: a → s ← s be a morphism in M(A)PrL with ϕ, ρ ∈ M(A) such that ρ is regular modulo [Pr L] and s semisimple. Then s is semisimple as well. Regarding s: S1 → S0 as an object of mod(A/[P]), we may assume that S0 ∼ = τ − τ S0 −1 and S1 = ϑS0 by Proposition 7. Hence ϕλa = 0 in mod(A/[P]), and thus ρ ϕ factors (uniquely) through πa . This proves (c). By Proposition 7, the objects of S(mod(A/[P])) are of the form u: ϑX X. This gives v u a left and right almost split sequence τ X ϑX X in A. If u : ϑX X is another such object, then Proposition 7 implies that there is a one-to-one correspondence between morphisms u → u in mod(A/[P]) and morphisms of complexes

(22)

modulo homotopy. Therefore, by symmetry, u → v deﬁnes an equivalence T : S(M(A)PrL ) → with(21) is achieved by a slight modiﬁcation of (21), replacS(M(A)InL ). The compatibility f b 10 ing f by −b and 0 s by 0s −1 . 0 λa

πa

Corollary 1. For a triangle (20) there is a short exact sequence La a Sa in M(A)PrL σa

λa

and a short exact sequence T Sa La a in M(A)InL . Proof. It suﬃces to prove that λa is monic in M(A)PrL and epic in M(A)InL . Thus let ϕρ−1 : x ← x → La be a morphism in M(A)PrL with λa (ϕρ−1 ) = 0 in M(A)PrL . Then λa ϕ ∈ [Pr L], say, λa ϕ = αβ with α: p → a, β: x → p, and p ∈ Pr L. Hence α factors through λa . Since λa is monic, we infer that ϕ ∈ [Pr L]. Thus λa is monic in M(A)PrL . The dual argument shows that λa is epic in M(A)InL .

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Corollary 2. Let A be a strict τ -category. For an object a of M(A), the following equivalences hold. (a) a is semisimple in M(A)PrL ⇔ πa is invertible. (b) a ∈ Pr L ⇔ λa is invertible. (c) a ∈ In L ⇔ σa is invertible. Proof. (a) follows by Theorem 1(c), and (b) is trivial, whereas (c) follows by Corollary 1 . By Theorem 1, every morphism ϕ: a → b in M(A) gives rise to a diagram

(23)

with commutative squares in M(A)InL , M(A), and M(A)PrL , respectively. We will show that (23) is in fact a triangle morphism, that is, ψ = T Sϕ. Theorem 2. Let A be a strict τ -category. Every morphism ϕ: a → b in M(A) induces a triangle morphism (23). f Proof. We modify (21) as indicated in the proof of Theorem 1, thus replacing fb by −b . Up to isomorphism, any semisimple object in M(A)PrL comes from an object u: ϑA A in M(A). Then (21) looks as follows.

(24)

If u : ϑA A is another such object, the theorem obviously holds for morphisms ϕ: u → u . Thus it remains to show that it also holds in the special case ϕ = πa . Here we get the following commutative diagram (23).

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00 : 10 10 f 0 0t 0 1

The homotopy

(in M(A)): T Sϕ = 1 in this case.

B ⊕ P → B ⊕ B yields the commutativity of the middle square f 00 b t − −b and (1 0) f0 10 = (b f ) 01 00 . Thus ψ = (1 0) = 10 sf p

As an immediate consequence of Theorem 1 and Theorem 2 we have Corollary 3. If the two triangles in (23) together with any pair of adjacent vertical morphisms α, β are given, such that α and β make up a commutative diagram in M(A)InL or M(A) or M(A)PrL , respectively, then (23) can be completed with ψ = T Sϕ in a unique way. 5. Semisimplicity in M(A) The preceding results make no use of one-sided almost split sequences in A. Moreover, for concrete strict τ -categories A (e. g., module categories over an artinian algebra, categories of socle-projective modules, or of lattices over an order), there is, in addition, a speciﬁc relationship between Projτ (A) and Injτ (A) (cf. Theorem 4.2 of [8], Theorem 4.4 of [9], Theorem 1 of [14], Theorem 4 of [16]). We will show that this connection is intimately related to the one-sided semisimple objects of the homotopy category M(A). Proposition 8. Let A be a strict τ -category. An object a: A1 → A0 of M(A) is left (right) semisimple if and only if the morphism a ∈ A is right (left) almost split with τ A0 = 0 (resp. τ − A1 = 0). Proof. Let a: A → P be a left semisimple object in M(A). Then the monomorphism λa : La → a splits. Hence a ∈ Fix L, i. e. τ P = 0. As a factors through the right almost split map u: ϑP → P , say, a = ue, we get a morphism

(25)

in Mor(A), and it is easy to verify that (25) is monic in Mor(A)/[A]. Since a is left semisimple in M(A), we infer that (25) is split monic. Hence e is split epic, and thus u factors through a. This implies that a is right almost split in A. Conversely, let a: A → P be an object in M(A) with τ P = 0 and a ∈ A right almost split. Consider a monomorphism

(26)

in M(A). Every split monomorphism d: D X0 in A with gd ∈ Rad A induces a morphism δ: D+ → x in M(A) such that the composition of (26) with δ is zero. Hence δ = 0, which implies that g is split monic. So there is a morphism p: P → X0 with pg = 1.

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The composition pu of p with u: ϑP → P induces a morphism ϕ: (ϑP )+ → x which is annihilated by the monomorphism (26). Hence ϕ = 0, i. e. pu = xq for some q: ϑP → X1 . As above, a = ue for some e: A → ϑP . Since a is right almost split, there is a morphism s: ϑP → A with u = as. Hence u(es − 1) = 0, and thus es = 1 by virtue of τ P = 0. Now x(1−qef ) = x − puef = x − paf = x − pgx = 0. So we have a commutative diagram

(27)

in Mor(A). Moreover, the composition (27) is homotopic to zero. In fact, uef (1−qef ) = af (1 − qef ) = 0 yields ef (1 − qef ) = 0. Therefore, the morphism 1−se: A → A satisﬁes (1 − se) · f (1 − qef ) = f · (1−qef ) and a · (1−se) = ue(1−se) = 0 = g · 0. As (26) is monic, the left-hand morphism in (27) belongs to [A], i. e. there is a morphism h: A → X1 with h · f (1−qef ) = 1 − qef and x · h = 0. This implies that

is a retraction of (26). Namely, x · (qe + h(1−f qe)) = xqe = pue = p · a and (qe + h(1−f qe)) · f = qef + hf (1−qef ) = qef + (1−qef ) = 1. Hence a is left semisimple. Let A be a strict τ -category. By the preceding result, every left semisimple object in u M(A) is of the form u ⊕ V − with a right almost split sequence 0 → ϑP → P . Let us denote the full subcategory of M(A), consisting of these objects u (resp. of the dual objects I → ϑ− I with τ − I = 0) by Ap (resp. Ai ). The objects of A− (resp. Ap ) will be called left semisimple of type 0 (resp. of type 1). Dually, the objects of A+ (resp. Ai ) are called right semisimple of type 0 (resp. 1). Proposition 8 also shows that the full subcategory Sl M(A) (resp. Sr M(A)) of left (right) semisimple objects in M(A) is a Krull-Schmidt category, containing A ≈ A− ≈ A+ as a full subcategory. Therefore, the endomorphism ring of an indecomposable one-sided semisimple object in M(A) need not be a skew-ﬁeld. We deﬁne a source object (sink object) [13] of an additive category as an object A = 0 such that every non-zero morphism X → A (resp. A → X) is split epic (split monic). For a strict τ -category A, this means that A is indecomposable with ϑA = 0 (resp. ϑ− A = 0). The one-sided semisimple objects of type 0 or 1 admit the following intrinsic characterization in M(A). Proposition 9. Let A be a strict τ -category without simultaneous source and sink objects. A left semisimple object s in M(A) is of type 0 if and only if HomM(A) (s, Ap ) = 0. An indecomposable left semisimple object is of type 1 if and only if it is a source object in Sl M(A).

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Proof. The ﬁrst assertion follows since HomM(A) (A− , Ap ) = 0. Let

be a morphism ϕ: u → u in Ap with u indecomposable. If g factors through u, then ϕ = 0 since u is monic. Otherwise, there is a section i: P → P of g. As iu ∈ Rad A, we ﬁnd a morphism j: ϑP → ϑP with iu = u j, hence a morphism ψ: u → u . Moreover, u(f j −1) = gu j − u = giu − u = 0 gives f j = 1, whence ϕψ = 1. Conversely, let s be a source object in Sl M(A). If s = A− for some A ∈ Ob A, then A is a source object in A. Hence A is not a sink object. So there is an irreducible morphism A → P in A with P indecomposable. Since A is a source object, this implies that τ P = 0. Furthermore, A is a direct summand of ϑP . Therefore, we get a non-zero morphism

which cannot be split epic since HomM(A) (A− , u) = 0. This contradiction shows that s ∈ Ap . Definition. Let A be a strict τ -category. We say that a left semisimple object s is connected to a right semisimple object t in M(A) if there exists a morphism γ: s → t such that every s → t with s left semisimple factors uniquely through γ, and every s → t with t right semisimple factors uniquely through γ. The deﬁnition implies that up to isomorphism, the objects s and t determine each other. If s and t are indecomposable, then either s or t has to be of type 1. This follows since HomM(A) (A− , A+ ) = 0. Hence there are three types of connections between indecomposables.

(28)

Recall that a simultaneous pullback and pushout is said to be an exact square. We will say that M(A) is ladder-finite if for each object a of M(A), there exists an n ∈ N with Ln a ∈ Fix L and (L− )n ∈ Fix L− (cf. [16], §4). The following theorem shows how connections in M(A) are related to invertible ladders. Theorem 3. Let A be a strict τ -category. For a connection between indecomposable one-sided semisimple objects in M(A), the corresponding commutative square (28) is

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exact. If M(A) is ladder-finite, then a morphism s → t in M(A) with indecomposable s ∈ Sl (mod(A)) and t ∈ Sr (mod(A)) is a connection if and only if it is of the form (29)

λ

λ

λ

1 2 n s = s0 −→ s1 −→ · · · −→ sn = t

with λi = λsi = λ− si−1 for i ∈ {1, . . . , n}. Proof. Let γ: s → t be a connection (28). By deﬁnition, every morphism ϕ: A− → t factors uniquely through γ. For the third square in (28), this is tantamount to the pullback property. For the ﬁrst two squares in (28), the fact that ϑP → P is monic implies that ϕ = 0. Hence we have a pullback by [16], Proposition 7. By duality, the commutative square (28) is exact. Now let M(A) be ladder-ﬁnite. If γ corresponds to the middle square in (28), the assertion follows by [16], Theorem 1. For the third square in (28), assume that Ln t = S − with n minimal. If Li t: Bi ⊕ Ui → Ci ⊕ Pi is the left standard form of Li t, then Pn = 0 implies that Pi = 0 for all i ∈ {1, . . . , n}. By [7], 3.6.1, it follows that Ui− is a direct summand of Li t for these i. Since t is indecomposable, this gives Un = S and Ui = 0 for i < n. Hence γ is of the form (29). Conversely, let γ: s → t be a morphism in M(A) with indecomposable s ∈ Sl (mod(A)), t ∈ Sr (mod(A)), such that γ is of the form (29). By [16], Proposition 2, γ is regular, and every A− → t factors through γ. Since Ap ⊂ Pr L, Proposition 8 implies that every morphism s → t with s ∈ Sl (mod(A)) factors uniquely through γ. Hence, by duality, γ is a connection. There are two important special types of connections in M(A). Definition. Let A be a strict τ -category. We say that M(A) has a connection of type 1 if there is a one-to-one connection between the indecomposable left and right semisimple objects of type 1 in M(A). If every indecomposable left (right) semisimple object of type 1 is connected to an indecomposable right (left) semisimple object of type 0, we say that M(A) has a connection of type 0. Let A be a strict τ -category. By ind A we denote any ﬁxed representative system of the isomorphism classes of indecomposable objects in A. We conclude this article with a result which shows that existence of a connection of type 0 or 1 is the essential condition for A to be equivalent to a category of representations of an artinian ring or an order in case ind A is ﬁnite. Definition. (cf. [15, 16]) Let R be a ring. An R-module E in R-mod is said to be an R-lattice if E has no simple submodules. The full subcategory of R-lattices in R-mod will be denoted by R-lat. If R is a subring of a ring R , then R is said to be a left overorder of R if SocR R = 0, and the left R-module R /{a ∈ R | R a ⊂ R} is of ﬁnite length. Similarly, we deﬁne a right overorder of R. A left and right overorder s will be called an overorder. We call R a one-dimensional order if R has an overorder i=1 Mni (Ωi ) with (non-commutative) discrete valuation domains Ωi . Theorem 4. Let A be a strict τ -category with |ind A| < ∞. (a) There exists an artinian ring R with R-mod ≈ A if and only if the radical of A is nilpotent, and M(A) has a connection of type 0. (b) There exists a one-dimensional order R with R-lat ≈ A if and only if A is ladder-finite ∞ with i=1 Radi A = 0, and M(A) has a connection of type 1.

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Proof. (a) follows by [14], Corollary of Proposition 6, using [7], Theorem 6.4. Part (b) is a consequence of Theorem 3, together with [16], Theorem 4 and its corollary. Remark. It can be shown (see [16]) that a lattice-ﬁnite one-dimensional order is a noetherian order in a semisimple ring. References [1] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules, Springer New York Heidelberg - Berlin 1974. [2] M. Auslander: Isolated singularities and existence of almost split sequences, Springer Lecture Notes in Mathematics, Vol. 1178 (1986), 194–242. [3] M. Auslander, I. Reiten, S. O. Smalø: Representation Theory of Artin Algebras, Cambridge University Press 1995. [4] P. Gabriel, M. Zisman: Calculus of Fractions and Homotopy Theory, Berlin - Heidelberg New York 1967. [5] A. Grothendieck: Sur quelques points d’alg`ebre homologique, Tˆ ohoku math. J. 9 (1957), 119–221. [6] K. Igusa, G. Todorov: Radical Layers of Representable Functors, J. Algebra 89 (1984), 105–147. [7] O. Iyama: τ -categories I: Radical Layers Theorem, Algebras and Representation Theory, to appear. [8] O. Iyama: τ -categories III: Auslander Orders and Auslander-Reiten quivers, Algebras and Representation Theory, to appear. [9] O. Iyama: The relationship between homological properties and representation theoretic realization of artin algebras, Preprint. [10] G. M. Kelly: A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on, Bull. Austral. Math. Soc. 22 (1980), 1–83. [11] B. Keller: Derived Categories and Their Uses, in: Handbook of Algebra (ed. M. Hazewinkel), vol. 1, Elsevier Science B. V., 1996, 671–701. [12] S. Mac Lane: Categories for the Working Mathematician, Springer, New York - Heidelberg Berlin, 1971. [13] W. Rump: Derived orders and Auslander-Reiten quivers, An. S ¸ t. Univ. Ovidius Constant¸a 8 (2000), 125–142. [14] W. Rump: Ladder functors with an application to representation-finite artinian rings, An. S ¸ t. Univ. Ovidius Constant¸a 9 (2001), 107–124. [15] W. Rump: Lattice-finite rings and their Auslander orders, Proc. 34th Symposium on Ring Theory and Representation Theory (2001), 89–99. [16] W. Rump: The category of lattices over a lattice-finite ring, Algebras and Representation Theory, to appear. [17] B. Stenstr¨ om: Rings of Quotients, Springer-Verlag, New York - Heidelberg - Berlin, 1975.

NON-COMMUTATIVE ALGEBRAIC GEOMETRY AND COMMUTATIVE DESINGULARIZATIONS LIEVEN LE BRUYN University of Antwerp (Belgium)

1. Introduction Ever since the dawn of non-commutative algebraic geometry in the mid seventies, see for example the work of P. Cohn [11], J. Golan [17], C. Procesi [38], F. Van Oystaeyen and A. Verschoren [47],[48], it has been ringtheorists’ hope that this theory may one day be relevant to commutative geometry, in particular to the study of singularities and their resolutions. Over the last decade, non-commutative algebras have been used to construct canonical (partial) resolutions of quotient singularities. That is, take a ﬁnite group G acting on Cd freely away from the origin then the orbit-space Cd /G is an isolated singularity. Resolutions Y Cd /G have been constructed using the skew group algebra C[x1 , . . . , xd ]#G which is an order with center C[Cd /G] = C[xi , . . . , xd ]G or deformations of it. In dimension d = 2 (the case of Kleinian singulariies) this gives us minimal resolutions via the connection with the preprojective algebra, see for example [14]. In dimension d = 3, the skew group algebra appears via the superpotential and commuting matrices setting (in the physics literature) or via the McKay quiver, see for example [13]. If G is Abelian one obtains from this study crepant resolutions but for general G one obtains at best partial resolutions with conifold singularities remaining. In dimension d > 3 the situation is unclear at this moment. Usually, skew group algebras and their deformations are studied via homological methods as they are Regular orders, see for example [46]. Here, we will follow a diﬀerent approach. My motivation was to ﬁnd a non-commutative explanation for the omnipresence of conifold singularities in partial resolutions of three-dimensional quotient singularities. Of course you may argue that they have to appear because they are somehow the nicest singularities. But then, what is the corresponding list of ‘nice’ singularities in dimension four? or ﬁve, six. . . ?? If my conjectural explanation has any merit the nicest partial resolutions of C4 /G should contain only singularities which are either polynomials over the conifold or one of the following three types C[[a, b, c, d, e, f ]] (ae − bd, af − cd, bf − ce)

C[[a, b, c, d, e]] (abc − de)

C[[a, b, c, d, e, f, g, h]] I

where I is the ideal of all 2 × 2 minors of the matrix

a b c d e f g h

In dimension d = 5 the conjecture is that another list of ten new speciﬁc singularities will appear, in dimension d = 6 another 63 new ones appear and so on.

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How do we come to these outlandish conjectures and speciﬁc lists? The hope is that any quotient singularity X = Cd /G has associated to it a ‘nice’ order A with center R = C[X] such that there is a stability structure θ with the scheme of all θ-semistable representations of A being a smooth variety (all these terms will be explained in the main text). If this is the case, the associated moduli space will be a partial resolution -X = Cd /G moduliθα Aand has a sheaf of Smooth orders A over it, allowing us to control its singularities in a combinatorial way as depicted in the frontispiece. If A is a Smooth order over R = C[X] then its non-commutative variety max A of maximal twosided ideals is birational to X away from the ramiﬁcation locus. If P is a point of the ramiﬁcation locus ram A then there is a ﬁnite cluster of inﬁnitesimally close noncommutative points lying over it. The local structure of the non-commutative variety max A near this cluster can be summarized by a (marked) quiver setting (Q, α) which in turn allows us to compute the ´etale local structure of A and R in P . The central singularities which appear in this way have been classiﬁed in [6] up to smooth equivalence giving us the small lists of conjectured singularities. In these talks I have tried to include background information which may or may not be useful to you. I suggest to browse through the notes by reading the ‘notes’ ﬁrst. If the remark seems obvious to you, carry on. If it puzzles you this may be a good point to enter the main text. More information can be found in the never-ending bookproject [28]. Acknowledgement These notes are (hopefully) a streamlined version of three talks given at the workshop “Sch´emas de Hilbert, alg`ebre noncommutative et correspondance de McKay” held at CIRM in Luminy (France), October 27–31, 2003. I like to thank the organizers, Jacques Alev, Bernhard Keller and Thierry Levasseur for the invitation (and the possibility to have a nice vacation with part of my family) and the participants for their patience. 2. Non-commutative algebra The organizers of this conference on “Hilbert schemes, non-commutative algebra and the McKay correspondence” are perfectly aware of my ignorance on Hilbert schemes and McKay correspondence. I therefore have to assume that I was hired in to tell you something about non-commutative algebra and that is precisely what I intend to do in these three talks. 2.1. Why non-commutative algebra? Let me begin by trying to motivate why you might get interested in non-commutative algebra if you want to understand quotient singularities and their resolutions. So let us take a setting which will be popular this week: we have a ﬁnite group G acting on d-dimensional aﬃne space Cd and this action is free away from the origin. Then the orbit-space, the so called quotient singularity Cd /G, is an isolated singularity

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and we want to construct ‘minimal’ or ‘canonical’ resolutions of this singularity. The buzzword seems to be ‘crepant’ in these circles. In his Bourbaki talk [40] Miles Reid asserts that McKay correspondence follows from a much more general principle Miles Reid’s Principle: Let M be an algebraic manifold, G a group of automorphisms - X a resolution of singularities of X = M/G. Then the answer to any well of M , and Y posed question about the geometry of Y is the G-equivariant geometry of M . Applied to the case of quotient singularities, the content of his slogan is that the - Cd /G. G-equivariant geometry of Cd already knows about the crepant resolution Y Men having principles are an easy target for abuse. So let us change this principle slightly: assume we have an aﬃne variety M on which a reductive group (and for deﬁniteness take P GLn ) acts with algebraic quotient variety M//P GLn Cd /G

then, in favorable situations, we can argue that the P GLn -equivariant geometry of M knows about good resolutions Y . This brings us to our ﬁrst entry in our note: One of the key lessons to be learned from this talk is that P GLn -equivariant geometry of M is roughly equivalent to the study of a certain non-commutative algebra over Cd /G. In fact, an order in a central simple algebra of dimension n2 over the function ﬁeld of the quotient singularity. Hence, if we know of good orders over Cd /G, we might get our hands at ‘good’ resolutions Y by non-commutative methods. 2.2. What non-commutative algebras? For the duration of these talks, we will work in the following, quite general, setting : • X will be a normal aﬃne variety, possibly having singularities. • R will be the coordinate ring C[X] of X. • K will be the function ﬁeld C(X) of X. If you are only interested in quotient singularities, you should replace X by Cd /G, R by the invariant ring C[x1 , . . . , Xd ]G and K by the invariant ﬁeld C(x1 , . . . , Xd )G in all statements below. If you are an algebraist, you have my sympathy and our goal will be to construct lots of R-orders A in a central simple K-algebra Σ.

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If you do not know what a central simple algebra is, take any non-commutative K-algebra Σ with center Z(Σ) = K such that over the algebraic closure K of K we obtain full n × n matrices Σ ⊗K K Mn (K) There are plenty such central simple K-algebras: Example 2.1. For any non-zero functions f, g ∈ K ∗ , the cyclic algebra Σ = (f, g)n

deﬁned by (f, g)n =

Kx, y (xn − f, y n − g, yx − qxy)

with q is a primitive n-th root of unity, is a central simple K-algebra of dimension n2 . Often, (f, g)n will even be a division algebra, that is a non-commutative algebra such that every non-zero element has an inverse. For example, this is always the case when E = K[x] is a (commutative) ﬁeld extension of dimension n and if g has order n in the quotient K ∗ /NE/K (E ∗ ) where NE/K is the norm map of E/K. See for example [37, Chp. 15] for more details, but if your German is a´ point I strongly suggest you to read Ina Kersten’s book [21] instead. Now, ﬁx such a central simple K-algebra Σ. An R-order A in Σ is a subalgebras A ⊂ Σ with center Z(A) = R such that A is ﬁnitely generated as an R-module and contains a K-basis of Σ, that is A ⊗R K Σ The classic reference for orders is Irving Reiner’s book [41] but it is hopelessly outdated and focusses too much on the one-dimensional case. Here is a gap in the market for someone to ﬁll. . . Example 2.2. In the case of quotient singularities X = Cd /G a natural choice of Rorder might be the skew group ring: C[x1 , . . . , Xd ]#G which consists of all formal sums g∈G rg #g witn multiplication deﬁned by (r#g)(r #g ) = rφg (r )#gg where φg is the action of g on C[x1 , . . . , Xd ]. The center of the skew group algebra is easily veriﬁed to be the ring of G-invariants R = C[Cd /G] = C[x1 , . . . , xd ]G Further, one can show that C[x1 , . . . , Xd ]#G is an R-order in Mn (K) with n the order of G. If we ever get to the third lecture, we will give another description of the skew group algebra in terms of the McKay-quiver setting and the variety of commuting matrices. However, there are plenty of other R-orders in Mn (K) which may or may not be relevant in the study of the quotient singularity Cd /G.

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Example 2.3. If f, g ∈ R −0, then the free R-submodule of rank n2 of the cyclic K-algebra Σ = (f, g)n of example 2.1 A=

n−1

Rxi y j

i,j=0

is an R-order. But there is really no need to go for this ‘canonical’ example. Someone more twisted may take I and J any two non-zero ideals of R, and consider

AIJ =

n−1

I i J j xi y j

i,j=0

which is an R-order too in Σ and which is far from being a projective R-module unless I and J are invertible R-ideals. For example, in Mn (K) we can take the ‘obvious’ R-order Mn (R) but one might also take the subring R I J R which is an R-order if I and J are non-zero ideals of R. If you are a geometer, our goal is to construct lots of aﬃne P GLn -varieties M such that the algebraic quotient M//P GLn is isomorphic to X and, moreover, such that there is a Zariski open subset U ⊂ X

for which the quotient map is a principal P GLn -ﬁbration, that is, all ﬁbers π −1 (u) P GLn for u ∈ U . The connection between such varieties M and orders A in central simple algebras may not be clear at ﬁrst sight. To give you at least an idea that there is a link, think of M as the aﬃne variety of n-dimensional representations repn A and of U as the Zariski open subset of all simple n-dimensional representations. Naturally, one can only expect the R-order A (or the corresponding P GLn -variety M ) to be useful in the study of resolutions of X if A is smooth in some appropriate noncommutative sense. Now, there are many characterizations of commutative regular domains R: • R is regular, that is, has ﬁnite global dimension • R is smooth, that is, X is a smooth variety and generalizing either of them to the non-commutative world leads to quite diﬀerent concepts.

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We will call an R-order A is a central simple K-algebra Σ: • Regular if A has ﬁnite global dimension together with some extra features such as Auslander regularity or Cohen-Macaulay property, see for example [33]. • Smooth if the corresponding P GLn -aﬃne variety M is a smooth variety as we will clarify later in this talk. For applications of Regular orders to desingularizations we refer to the talks by Michel Van den Bergh at this conference or to his paper [46] on this topic. I will concentrate on the properties of Smooth orders instead. Still, it is worth pointing out the strengths and weaknesses of both deﬁnitions right now note: Regular orders are excellent if you want to control homological properties, for example if you want to study the derived categories of their modules. At this moment there is no local characterization of Regular orders if dimX ≥ 2. Smooth orders are excellent if you want to have smooth moduli spaces of semi-stable representations. As we will see later, in each dimension there are only a ﬁnite number of local types of Smooth orders and these are classiﬁed. The downside of this is that Smooth orders are less versatile as Regular orders. In applications to canonical desingularizations, one often needs the good properties of both so there is a case for investigating SmoothRegular orders better than has been done in the past. In general though, both theories are quite diﬀerent. Example 2.4. The skew group algebra C[x1 , . . . , xd ]#G is always a Regular order but we will see in the next lecture, it is virtually never a Smooth order. Example 2.5. Let X be the variety of matrix-invariants, that is X = Mn (C) ⊕ Mn (C)//P GLn where P GLn acts on pairs of n × n matrices by simultaneous conjugation. The trace ring of two generic n × n matrices A is the subalgebra of Mn (C[Mn (C) ⊕ Mn (C)]) generated over C[X] by the two generic matrices x11 · · · x1n .. X = ... . xn1 · · · xnn

y11 · · · y1n .. and Y = ... . yn1 · · · ynn

Then, A is an R-order in a division algebra of dimension n2 over K, called the generic division algebra. Moreover, A is a Smooth order but is Regular only when n = 2, see [30]. 2.3. Constructing orders by descent. note: French mathematicians have developed in the sixties an elegant theory, called descent theory, which allows one to construct elaborate examples out of trivial ones by bringing in topology. This theory allows to classify objects which are only locally (but not necessarily globally) trivial. For applications to orders there are two topologies to consider: the well-known Zariski topology and the perhaps lesser-known ´etale topology. Let us try to give a formal deﬁnition of Zariski and ´etale covers aimed at ringtheorists.

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A Zariski cover of X is a ﬁnite product of localizations at elements of R

Sz =

k

R fi

such that

(f1 , . . . , fk ) = R

i=1

and is therefore a faithfully ﬂat extension of R. Geometrically, the ringmorphism R −→ Sz deﬁnes a cover of X = spec R by k disjoint sheets spec Sz = i spec Rfi , each corresponding to a Zariski open subset of X, the complement of V(fi ) and the condition is that these closed subsets V(fi ) do not have a point in common. That is, we have the picture of ﬁgure 1.1: k Zariski covers form a Grothendieck topology, that is, two Zariski covers Sz1 = i=1 Rfi l and Sz2 = j=1 Rgj nave a common reﬁnement Sz = SZ1 ⊗R Sz2 =

l k

R fi gj

i=1 j=1

For a given Zariski cover Sz =

k

i=1

Rfi a corresponding ´etale cover is a product

Se =

k

Rfi [x(i)1 , . . . , x(i)ki ] (g(i)1 , . . . , g(i)ki ) i=1

with

∂g(i)1 ∂x(i)1

···

∂g(i)ki ∂x(i)1

···

.. .

∂g(i)1 ∂x(i)ki

.. .

∂g(i)ki ∂x(i)ki

a unit in the i-th component of Se . In fact, for applications to orders it is usually enough to consider special etale extensions

Se =

k

RFi [x] (xki − ai ) i=1

where

ai is a unit in Rfi

Figure 1: A Zariski cover of X = spec R

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LIEVEN LE BRUYN

Geometrically, an ´etale cover determines for every Zariski sheet spec Rfi a locally isomorphic (for the analytic topology) multi-covering and the number of sheets may vary with i (depending on the degrees of the polynomials g(i)j ∈ Rfi [x(i)1 , . . . , x(i)ki ]. That is, the mental picture corresponding to an ´etale cover is given in ﬁgure 1.2 below. Again, ´etale covers form a Zariski topology as the common reﬁnement Se1 ⊗R Se2 of two ´etale covers is again ´etale because its components are of the form Rfi gj [x(i)1 , . . . , x(i)ki , y(j)1 , . . . , y(j)lj ] (g(i)1 , . . . , g(i)ki , h(j)1 , . . . , h(j)lj ) and the Jacobian-matrix condition for each of these components is again satisﬁed. Because of the local isomorphism property many ringtheoretical local properties (such as smoothness, normality etc.) are preserved under ´etale covers. Now, ﬁx an R-order B in some central simple K-algebra Σ, then a Zariski twisted form A of B is an R-algebra such that A ⊗R Sz B ⊗R Sz for some Zariski cover Sz of R. If P ∈ X is a point with corresponding maximal ideal m, then P ∈ specRfi for some of the components of Sz and as Afi Bfi we have for the local rings at P Am Bm

Figure 2: An ´etale cover of X = spec R

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that is, the Zariski local information of any Zariski-twisted form of B is that of B itself. Likewise, an ´etale twisted form A of B is an R-algebra such that A ⊗R Se B ⊗R Se for some ´etale cover Se of R. This time the Zariski local information of A and B may be diﬀerent at a point P ∈ X but we do have that the m-adic completions of A and B ˆm Aˆm B ˆ m -algebras. are isomorphic as R note: The Zariski local structure of A determines the localization Am , the ´etale local structure determines the completion Aˆm . Descent theory allows to classify Zariski- or ´etale twisted forms of an R-order B by means of the corresponding cohomology groups of the automorphism schemes. For more details on this please read the book [23] by M. Knus and M. Ojanguren if you are a ringth-eorist and that of S. Milne [35] if you are more of a geometer. If one applies descent to the most trivial of all R-orders, the full matrix algebra Mn (R), one arrives at 2.4. Azumaya algebras. A Zariski twisted form of Mn (R) is an R-algebra A such that

A ⊗R Sz Mn (Sz ) =

k

Mn (Rfi )

i=1

Conversely, you can construct such twisted forms by gluing together the matrix rings Mn (Rfi ). The easiest way to do this is to glue Mn (Rfi ) with Mn (Rfj ) over Rfi fi via the natural embedding Rfi → Rfi fj ← Rfj Not surprisingly, we obtain in this way Mn (R) back. But there are more clever ways to perform the gluing by bringing in the noncommutativity of matrix-rings. We can glue −1 gij ·gij

Mn (Rfi ) → Mn (Rfi fj ) −−−−−→ Mn (Rfi fj ) ← Mn (Rfj )

over their intersection via conjugation with an invertible matrix gij ∈ GLn (Rfi fi ). If the elements gij for 1 ≤ i, j ≤ k satisfy the cocycle condition (meaning that the diﬀerent possible gluings are compatible over their common localization Rfi fj fl ), we obtain a sheaf of non-commutative algebras A over X = spec R such that its global sections are not necessarily Mn (R).

212

LIEVEN LE BRUYN

Proposition 2.6. Any Zariski twisted form of Mn (R) is isomorphic to EndR (P ) where P is a projective R-module of rank n. Two such twisted forms are isomorphic as R-algebras EndR (P ) EndR (Q)

iﬀ

P Q⊗I

for some invertible R-ideal I. Proof. [sketch] We have an exact sequence of groupschemes 1 −→ Gm −→ GLn −→ PGLn −→ 1 (here, Gm is the sheaf of units) and taking Zariski cohomology groups over X we have a sequence 1 1 1 (X, Gm ) −→ HZar (X, GLn ) −→ HZar (X, PGLn ) 1 −→ HZar

where the ﬁrst term is isomorphic to the Picard group P ic(R) and the second term classiﬁes projective R-modules of rank n upto isomorphism. The ﬁnal term classiﬁes the Zariski twisted forms of Mn (R) as the automorphism group of Mn (R) is P GLn . Example 2.7. Let I and J be two invertible ideals of R, then R I −1 J EndR (I ⊕ J) ⊂ M2 (K) IJ −1 R

and if IJ −l = (r) then I ⊕ J (Rr ⊕ R) ⊗ J and indeed we have an isomorphism

1 0 0 r−1

R I −1 J −1 R IJ

1 0 0 r

=

R R R R

Things get a lot more interesting in the ´etale topology. Deﬁnition 2.8. An n-Azumaya algebra over R is an ´etale twisted form A of Mn (R). If A is also a Zariski twisted form we call A a trivial Azumaya algebra. From the deﬁnition and faithfully ﬂat descent, the following facts follow: Lemma 2.9. If A is an n-Azumaya algebra over R, then: 1. The center Z(A) = R and A is a projective R-module of rank n2 . 2. All simple A-representations have dimension n and for every maximal ideal m of R we have A/mA Mn (C)

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Proof. For (2) take M ∩R = m where M is the kernel of a simple representation A Mk (C), ˆ m ) it follows that then as Aˆm Mn (R A/mA Mn (C) and hence that k = n and M = Am. It is clear from the deﬁnition that when A is an n-Azumaya algebra and A is an mAzumaya algebra over R, A ⊗R A is an mn-Azumaya and also that A ⊗R Aop EndR (A) where Aop is the opposite algebra (that is, equipped with the reverse multiplication rule). These facts allow us to deﬁne the Brauer group BrR to be the set of equivalence classes [A] of Azumaya algebras over R where [A] = [A ]

iﬀ

A ⊗R A EndR (P )

for some projective R-module P and where multiplication is induced from the rule [A] · [A ] = [A ⊗R A ] One can extend the deﬁnition of the Brauer group from aﬃne varieties to arbitrary schemes and A. Grothendieck has shown that the Brauer group of a projective smooth variety is a birational invariant, see [19]. Moreover, he conjectured a cohomological description of the Brauer group BrR which was subsequently proved by O. Gabber in [16]. Theorem 2.10. The Brauer group is an ´etale cohomology group 2 BrR Het (XGm )torsion

where Gm is the unit sheaf and where the subscript denotes that we take only torsion 2 elements. If R is regular, then Het (X, Gm ) is torsion so we can forget the subscript. This result should be viewed as the ringtheory analogon of the crossed product theorem for central simple algebras over ﬁelds, see for example [37]. Observe that in Gabber’s result there is no sign of singularities in the description of the Brauer group. In fact, with respect to the desingularization problem, Azumaya algebras are only as good as their centers. Proposition 2.11. If A is an n-Azumaya algebra over R, then 1. A is Regular iﬀ R is commutative regular. 2. A is Smooth iﬀ R is commutative regular. Proof. (1) follows from faithfully ﬂat descent and (2) from lemma 2.9 which asserts that the P GLn -aﬃne variety corresponding to A is a principal P GLn -ﬁbration in the ´etale topology, which shows that both n-Azumaya algebras and principal P GLn -ﬁbrations are classiﬁed by 1 (X, PGLn ). the ´etale cohomology group Het note: In the correspondence between R-orders and P GLn -varieties, Azumaya algebras correspond to principal P GLn -ﬁbrations over X. With respect to desingularizations, Azumaya algebras are therefore only as good as their centers.

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LIEVEN LE BRUYN

2.5. Reﬂexive Azumaya algebras. So let us bring in ramiﬁcation in order to construct orders which may be more useful in our desingularization project. Example 2.12. Consider the R-order in M2 (K)

R R A= I R

where I is some ideal of R. If P ∈ X is a point with corresponding maximal ideal m we have that: For I not contained in m we have Am M2 (Rm ) whence A is an Azumaya algebra in P . For I ⊂ m we have Am

Rm Rm M2 (Rm ) = Im Rm

whence A is not Azumaya in P . Deﬁnition 2.13. The ramiﬁcation locus of an R-order A is the Zariski closed subset of X consisting of those points P such that for the corresponding maximal ideal m A/mA Mn (C) That is, ram A is the locus of X where A is not an Azumaya algebra. Its complement azu A is called the Azumaya locus of A which is always a Zariski open subset of X. Deﬁnition 2.14. An R-order A is said to be a reﬂexive n-Azumaya algebra iﬀ 1. ram A has codimension at least two in X, and 2. A is a reﬂexive R-module that is, A HomR (HomR (A, R), R) = A∗∗ . The origin of the terminology is that when A is a reﬂexive n-Azumaya algebra we have that Ap is n-Azumaya for every height one prime ideal p of R and that A = ∩p Ap where the intersection is taken over all height one primes. For example, in example 2.12 if I is a divisorial ideal of R, then A is not reﬂexive Azumaya as Ap is not Azumaya for p a height one prime containing I and if I has at least height two, then A is often not a reﬂexive Azumaya algebra because A is not reﬂexive as an R-module. For example take C[x, y] C[x, y] A= (x, y) C[x, y] then the reﬂexive closure of A is A∗∗ = M2 (C[x, y]). Sometimes though, we get reﬂexivity of A for free, for example when A is a CohenMacaulay R-module. An other important fact to remember is that for A a reﬂexive Azumaya, A is Azumaya if and only if A is projective as an R-module. If you want to know more about reﬂexive Azumaya algebras you may want to read [36] or my Ph.D. thesis [24]. Example 2.15. Let A = C[x1 , . . . , xd ]#G then A is a reﬂexive Azumaya algebra whenever G acts freely away from the origin and d ≥ 2. Moreover, A is never an Azumaya algebra as its ramiﬁcation locus is the isolated singularity.

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In analogy with the Brauer group one can deﬁne the reﬂexive Brauer group β(R) whose elements are the equivalence classes [A] for A a reﬂexive Azumaya algebra over R with equivalence relation [A] = [A ]

iﬀ

A ⊗R A EndR (M )

where M is a reﬂexive R-module and with multiplication induced by the rule [A] · [A ] = [(A ⊗R A )∗∗ ] In [26] it was shown that the reﬂexive Brauer group does have a cohomological description Proposition 2.16. The reﬂexive Brauer group is an ´etale cohomology group 2 β(R) Het (Xsm , Gm )

where Xsm is the smooth locus of X. This time we see that the singularities of X do appear in the description so perhaps reﬂexive Azumaya algebras are a class of orders more suitable for our project. This is even more evident if we impose non-commutative smoothness conditions on A. Proposition 2.17. Let A be a reﬂexive Azumaya algebra over R, then: 1. if A is Regular, then ram A = Xsing , and 2. if A is Smooth, then Xsing is contained in ram A. Proof. (1) was proved in [27] the essential point being that if A is Regular then A is a Cohen-Macaulay R-module whence it must be projective over a smooth point of X but then it is not just an reﬂexive Azumaya but actually an Azumaya algebra in that point. The second statement can be further reﬁned as we will see in the next lecture. Many classes of well-studied algebras are reﬂexive Azumaya algebras, • Trace rings Tm,n of m generic n × n matrices (unless (m, n) = (2, 2)), see [25]. • Quantum enveloping algebras Uq (g) of semi-simple Lie algebras at roots of unity, see for example [8]. • Quantum function algebras Oq (G) for semi-simple Lie groups at roots of unity, see for example [9]. • Symplectic reﬂection algebras At,c , see [10]. note: Many interesting classes of Regular orders are reﬂexive Azumaya algebras. As a consequence their ramiﬁcation locus coincides with the singularity locus of the center. 2.6. Cayley-Hamilton algebras. It is about time to clarify the connection with P GLn equivariant geometry. We will introduce a class of non-commutative algebras, the so called Cayley-Hamilton algebras which are the level n generalization of the category of commutative algebras and which contain all R-orders. A trace map tr is a C-linear function A −→ A satisfying for all a, b ∈ A tr(tr(a)b) = tr(a)tr(b)

tr(ab) = tr(ba)

and tr(a)b = btr(a)

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so in particular, the image tr(A) is contained in the center of A. If M ∈ Mn (R) where R is a commutative C-algebra, then its characteristic polynomial χM = det(tn − M ) = tn + a1 tn−1 + a2 tn−2 + · · · + an has coeﬃcients ai which are polynomials with rational coeﬃcients in traces of powers of M ai = fi (tr(M ), tr(M 2 ), . . . , tr(M n−1 ) Hence, if we have an algebra A with a trace map tr we can deﬁne a formal characteristic polynomial of degree n for every a ∈ A by taking χa = tn + f1 (tr(a), . . . , tr(an−1 )tn−1 + · · · + fn (tr(a), . . . , tr(an−1 ) which allows us to deﬁne the category [email protected] of Cayley-Hamilton algebras of degree n. Deﬁnition 2.18. An object A in [email protected] is a Cayley-Hamilton algebra of degree n, that is, a C-algebra with trace map tr : A −→ A satisfying tr(1) = n and

∀a ∈ A : χa (a) = 0

Morphisms A −→ B in [email protected] are trace preserving C-algebra morphisms, that is,

is a commutative diagram. Example 2.19. Azumaya algebras, reﬂexive Azumaya algebras and more generally every R-order A in a central simple K-algebra of dimension n2 is a Cayley-Hamilton algebra of degree n. For, consider the inclusions

Here, tr : Mn (K) −→ K is the usual trace map. By Galois descent this induces a trace map, the so called reduced trace, tr : Σ −→ K. Finally, because R is integrally closed in K and A is a ﬁnitely generated R-module it follows that tr(a) ∈ R for every element a ∈ A.

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217

If A is a ﬁnitely generated object in [email protected], we can deﬁne an aﬃne P GLn -scheme, φ

trepn A, classifying all trace preserving n-dimensional representations A−→Mn (C) of A. The action of P GLn on trepn A is induced by conjugation in the target space, that is g.φ is the trace preserving algebra map g.−.g −1

φ

A−→Mn (C) −→ Mn (C) Orbits under this action correspond precisely to isomorphism classes of representations. The scheme trepn A is a closed subscheme of repn A the more familiar P GLn -aﬃne scheme of all n-dimensional representations of A. In general, both schemes may be diﬀerent. Example 2.20. Let A be the quantum plane at −1, that is

A=

Cx, y (xy + yx)

then A is an order with center R = C[x2 , y 2 ] in the quaternion algebra (x, y)2 = K1 ⊕ Ku ⊕ Kυ ⊕ Kuυ over K = C(x, y) where u2 = x.v 2 = y and uv = −vu. Observe that tr(x) = tr(y) = 0 as the embedding A → (x, y)2 → M2 (C[u, y]) is given by x →

u 0 0 −u

and

y →

0 1 y 0

Therefore, a trace preserving algebra map A −→ M2 (C) is fully determined by the images of x and y which are trace zero 2 × 2 matrices

a b φ(x) = c −a

d e φ(y) = f −d

and

satisfying

bf + ce = 0

That is, trep2 A is the hypersurface V(bf +ce) ⊂ A6 which has a unique isolated singularity at the origin. However, rep2 A contains more points, for example φ(x) =

a 0 0 b

and φ(y) =

0 0 0 0

is a point in rep2 A–trep2 A whenever b = −a. A functorial description of trepn A is given by the following universal property proved by C. Procesi [39] Theorem 2.21. Let A be a C-algebra with trace map trA , then there is a trace preserving algebra morphism jA : A −→ Mn (C[trepn A])

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LIEVEN LE BRUYN

satisfying the following universal property. If C is a commutative C-algebra and there is a ψ trace preserving algebra map A−→Mn (C) (with the usual trace on Mn (C)), then there is a φ

unique algebra morphism C[trepn A]−→C such that the diagram

is commutative. Moreover, A is an object in [email protected] if and only if jA is a monomorphism. The P GLn -action on trepn A induces an action of P GLn by automorphisms on C[trepn A]. On the other hand, P GLn acts by conjugation on Mn (C) so we have a combined action on Mn (C[trepn A]) = Mn (C) ⊗ C[trepn A] and it follows from the universal property that the image of jA is contained in the ring of P GLn -invariants jA

A−→Mn (C[trepn A])P GLn which is an inclusion if A is a Cayley-Hamilton algebra. In fact, C. Procesi proved in [39] the following important result which allows to reconstruct orders and their centers from P GLn -equivariant geometry. Theorem 2.22. The functor trepn : [email protected] −→ PGL(n)-affine has a left inverse A : PGL(n)-affine −→ [email protected] deﬁned by Ay = Mn (C[Y ])P GLn . In particular, we have for any A in [email protected] A = Mn (C[trepn A])P GLn

and

tr(A) = C[trepn A]P GLn

That is the central subalgebra tr(A) is the coordinate ring of the algebraic quotient variety trepn A//P GLn = tissn A classifying isomorphism classes of trace preserving semi-simple n-dimensional representations of A. However, these functors do not give an equivalence between [email protected] and P GLn -equivariant aﬃne geometry. There are plenty more P GLn -varieties than Cayley-Hamilton algebras.

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219

Example 2.23. Conjugacy classes of nilpotent matrices in Mn (C) correspond bijective to partitions λ = (λ1 ≥ λ2 ≥ · · · ) of n (the λi determine the sizes of the Jordan blocks). It follows from the Gerstenhaber-Hesselink theorem that the closures of such orbits Oλ = ∪µ≤λ Oµ where ≤ is the dominance order relation. Each Oλ is an aﬃne P GLn -variety and the corresponding algebra is AOλ = C[x]/(xλ1 ) whence many orbit closures (all of which are aﬃne P GLn -varieties) correspond to the same algebra. note: The category [email protected] of Cayley-Hamilton algebras is to noncommutative [email protected] what commalg, the category of all commutative algebras is to commutative algebraic geometry. In fact, [email protected] commalg by taking as trace maps the identity on every commutative algebra. Further we have a natural commutative diagram of functors

where the bottom map is the equivalence between aﬃne algebras and aﬃne schemes and the top map is the correspondence between Cayley-Hamilton algebras and aﬃne P GLn schemes, which is not an equivalence of categories. 2.7. Smooth orders. To ﬁnish this talk let us motivate and deﬁne the notion of a Smooth order properly. Among the many characterizations of commutative regular algebras is the following due to A. Grothendieck. Theorem 2.24. A commutative C-algebra A is regular if and only if it satisﬁes the following lifting property: if (B,I) is a test-object such that B is a commutative algebra and I is a nilpotent ideal of B, then for any algebra map φ, there exists a lifted algebra morphism φ˜

making the diagram commutative.

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LIEVEN LE BRUYN

As the category commalg of all commutative C-algebras is just [email protected] it makes sense to deﬁne Smooth Cayley-Hamilton algebras by the same lifting property. This was done ﬁrst by W. Schelter [42] in the category of all algebras satisfying all polynomial identities of n × n matrices and later by C. Procesi [39] in [email protected] Deﬁnition 2.25. A Smooth Cayley-Hamilton algebra A is an object in [email protected] satisfying the following lifting property. If (B, I) is a test-object in [email protected], that is, B is an object in [email protected], I is a nilpotent ideal in B such that B/I is an object in [email protected] and such that the π natural map B −→ −→B/I is trace preserving, then every trace preserving algebra map φ has a lift φ˜

making the diagram commutative. If A is in addition an order, we say that A is a Smooth order. Next talk we will give a large class of Smooth orders but again it should be stressed that there is no connection between this notion of non-commutative smoothness and the more homological notion of Regular orders (except in dimension one when all notions coincide). Still, in the context of P GLn -equivariant aﬃne geometry this notion of non-commutative smoothness is quite natural as illustrated by the following result due to C. Procesi [39]. Theorem 2.26. An object A in [email protected] is Smooth if and only if the corresponding aﬃne P GLn -scheme trepn A is smooth (and hence, in particular, reduced). Proof. (One implication) Assume A is Smooth, then to prove that trepn A is smooth we have to prove that C[trepn A] satisﬁes Grothendieck’s lifting property. So let (B, I) be a test-object in commalg and take an algebra morphism φ : C[trepn A] −→ B/I. Consider the following diagram

the morphism (1) follows from Smoothness of A applied to the morphism Mn (φ) ◦ jA . From the universal property of the map jA it follows that there is a morphism (2) which is of the form Mn (ψ) for some algebra morphism ψ : C[trepn A] −→ B. This ψ is the required lift.

NON-COMMUTATIVE ALGEBRAIC GEOMETRY

221

Example 2.27. Trace rings Tm,n are the free algebras generated by m elements in [email protected] and as such trivially satisfy the lifting property so are Smooth orders. Alternatively, because 2

trepn Tm,n Mn (C) ⊕ · · · ⊕ Mn (C) = Cmn is a smooth P GLn -variety, Tm,n is Smooth by the previous result.

Example 2.28. Any commutative algebra C can be viewed as an element of [email protected] via the diagonal embedding C → Mn (C). However, if C is a regular commutative algebra it is not true that C is Smooth in [email protected] For example, take C = C[x1 , . . . , xd ] and consider the 4-dimensional non-commutative local algebra B=

Cx, y = C ⊕ Cx ⊕ Cy ⊕ Cxy (x2 , y 2 , xy + yx)

with the obvious trace map so that B ∈ [email protected] B has a nilpotent ideal I = B(xy − yx) such that the quotient B/I is a 3-dimensional commutative algebra. Consider the algebra map φ

C[x1 , . . . , xd ]−→

B I

deﬁned by

x1 → x

x2 → y

and xi → 0

for i ≥ 3

This map has no lift as for any potential lifted morphism φ˜ we have ˜ ˜ [φ(x), φ(y)] = 0 whence C[x1 , . . . , xd ] is not Smooth in [email protected] Example 2.29. Consider again the quantum plane at −1 A=

Cx, y (xy + yx)

then we have seen that trep2 A = V(bf + ce) ⊂ A6 has a unique isolated singularity at the origin. Hence, A is not a Smooth order. note: Under the correspondence between [email protected] and PGL(n)-aff, Smooth CayleyHamilton algebras correspond to smooth P GLn -varieties. 3. Non-commutative geometry Last time we introduced [email protected] as a level n generalization of commalg, the variety of all commutative algebras. Today we will associate to any A ∈ [email protected] a non-commutative variety max A and argue that this gives a non-commutative manifold if A is a Smooth order. In particular we will show that for ﬁxed n and central dimension d there are a ﬁnite number of ´etale types of such orders. This fact is the non-commutative analogon of the fact that every manifold is locally diﬀeomorphic to aﬃne space or, in ringtheory terms, that the m-adic completion of a regular algebra C of dimension d has just one ´etale type: Cˆm C[[x1 , . . . , xd ]].

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LIEVEN LE BRUYN

3.1. Why non-commutative geometry? note: There is one new feature that non-commutative geometry has to oﬀer compared to commutative geometry: distinct points can lie inﬁnitesimally close to each other. As desingularization is the process of separating bad tangents, this fact should be useful somehow in our project. Recall that if X is an aﬃne commutative variety with coordinate ring R, then to each point P ∈ X corresponds a maximal ideal mP R and a one-dimensional simple representation Sp =

R mp

A basic tool in the study of Hilbert schemes is that ﬁnite closed subschemes of X can be decomposed according to their support. In algebraic terms this means that there are no extensions between diﬀerent points, that if P = Q then Ext1R (SP , SQ ) = 0

whereas

Ext1R (SP , SP ) = TP X

In more plastic lingo: all inﬁnitesimal information of X near P is contained in the selfextensions of SP and distinct points do not contribute. This is no longer the case for non-commutative algebras. Example 3.1. Take the path algebra A of the quiver ←− , that is

C C A 0 C

Then A has two maximal ideals and two corresponding one-dimensional simple representations C C C 0 C = S1 = 0 0 C 0 C

and

0 C C C C S2 = = C 0 C 0 0

Then, there is a non-split exact sequence with middle term the second column of A C C 0 −→ M = −→ S2 = −→ 0 0 −→ S1 = 0 C C Whence Ext1A (S2 , S1 ) = 0 whereas Ext1A (S1 , S2 ) = 0. It is no accident that these two facts are encoded into the quiver. Deﬁnition 3.2. For A an algebra in [email protected], deﬁne its maximal ideal spectrum max A to be the set of all maximal twosided ideals M of A equipped with the non-commutative Zariski topology, that is, a typical open set of max A is of the form X(I) = {M ∈ max A|I ⊂ M }

NON-COMMUTATIVE ALGEBRAIC GEOMETRY

223

Recall that for every M ∈ max A the quotient A Mk (C) M

for some k ≤ n

that is, M determines a unique k-dimensional simple representation SM of A. As every maximal ideal M of A intersects the center R in a maximal ideal mP = M ∩ R we get, in the case of an R-order A a continuous map c

max A−→X

deﬁned by M → P

where M ∩ R = mP

Ringtheorists have studied the ﬁbers c−1 (P ) of this map in the seventies and eighties in connection with localization theory. The oldest description is the Bergman-Small theorem, see for example [2] Theorem 3.3. (Bergman-Small) If c−1 (P ) = {M1 , . . . , Mk } then there are natural numbers ei ∈ N+ such that

n=

k

ei di

where di = dimC SMi

i=1

In particular, c−1 (P ) is ﬁnite for all P. Here is a modern proof of this result based on the results of the previous lecture. Because X is the algebraic quotient trepn A//GLn , points of X correspond to closed GLn -orbits in repn A. By a result of M. Artin [1] closed orbits are precisely the isomorphism classes of semi-simple n-dimensional representations, and therefore we denote the quotient variety X = trepn A//GLn = tissn A So, a point P determines a semi-simple n-dimensional A-representation MP = S1⊕e1 ⊕ . . . ⊕ Sk⊕ek with the Si the distinct simple components, say of dimension di =dimC Si and occurring in Mp with multiplicity ei ≥ 1. This gives n = Σei di and clearly the annihilator of Si is a maximal ideal Mi of A lying over mP . Another interpretation of c−1 (P ) follows from the work of A. V. Jategaonkar and B. M¨ ullen Deﬁne a link diagram on the points of max A by the rule M M

⇐⇒

Ext1A (SM , SM ) = 0

In fancier language, M M if and only if M and M lie inﬁnitesimally close together in max A. In fact, the deﬁnition of the link diagram in [20, Chp. 5] or [18, Chp. 11] is slightly diﬀerent but amounts to the same thing.

224

LIEVEN LE BRUYN

Theorem 3.4. (Jategaonkar-M¨ uller) The connected components of the link diagram on max A are all ﬁnite and are in one-to-one correspondence with P ∈ X. That is, if {M1 , . . . , Mk } = c−1 (P ) ⊂ max A then this set is a connected component of the link diagram. note: In max A there is a Zariski open set of Azumaya points, that is those M ∈ max A such that A/M Mn (C). It follows that each of these maximal ideals is a singleton connected component of the link diagram. So on this open set there is a one-to-one correspondence between points of X and maximal ideals of A so we can say that max A and X are birational. However, over the ramiﬁcation locus there may be several maximal ideals of A lying over the same central maximal ideal and these points should be thought of as lying inﬁnitesimally close to each other.

One might hope that the cluster of inﬁnitesimally points of max A lying over a central singularity P ∈ X can be used to separate tangent information in P rather than having to resort to the blowing-up process to achieve this. 3.2. What non-commutative geometry? As an R-order A in a central simple K-algebra Σ of dimension n2 is a ﬁnite R-module, we can associate to A the sheaf OA of noncommutative OX -algebras using central localization. That is, the section over a basic aﬃne open piece X(f ) ⊂ X are Γ(X(f ), OA ) = Af = A ⊗R Rf which is readily checked to be a sheaf with global sections Γ(X, OA ) = A. As we will investigate Smooth orders via their (central) ´etale structure, that is information about Aˆmp , we will only need the structure sheaf OA over X. In the ’70-ties F. Van Oystaeyen [47] and A. Verschoren [48] introduced genuine noncommutative structure sheaves associated to an R-order A. It is not my intention to promote nc on max A deserve nostalgia here but perhaps these non-commutative structure sheaves OA renewed investigation. nc is deﬁned by taking as the sections over the typical open set X(I) Deﬁnition 3.5. OA (for I a twosided ideal of A) in max A nc ) = {δ ∈ Σ | ∃l ∈ N : I l δ ⊂ A} Γ(X(I), OA

By [47] this deﬁnes a sheaf of non-commutative algebras over max A with global sections nc ) = A. The stalk of this sheaf at a point M ∈ max A is the symmetric localΓ(max A, OA ization nc OA,M = QA−M (A) = {δ ∈ Σ | Iδ ⊂ A for some ideal I ⊂ P }

NON-COMMUTATIVE ALGEBRAIC GEOMETRY

225

Often, these stalks have no pleasant properties but in some examples, these noncommutative stalks are nicer than those of the central structure sheaf. Example 3.6. Let X = A1 , that is, R = C[x] and consider the order

R R A= m R

where m = (x) R. A is an hereditary order so is both a Regular order and a Smooth order. The ramiﬁcation locus of A is P0 = V(x) so over any P0 = P ∈ A1 there is a unique maximal ideal of A lying over mp and the corresponding quotient is M2 (C). However, over m there are two maximal ideals of A

m R M1 = m R

and

R R M2 = m m

Both M1 and M2 determine a one-dimensional simple representation of A, so the BergmanSmall number are e1 = e2 = 1 and d1 = d2 = 1. That is, we have the following picture

There is one non-singleton connected component in the link diagram of A namely

with the vertices corresponding to {M1 , M2 }. The stalk of OA at the central point P0 is clearly Rm Rm OA,P0 = mm R m nc On the other hand the stalks of the non-commutative structure sheaf OA in M1 resp. M2 can be computed to be

nc = OA,M 1

Rm Rm Rm Rm

and

nc = OA,M 2

Rm x−1 Rm xRm Rm

and hence both stalks are Azumaya algebras. Observe that we recover the central stalk OA,P0 as the intersection of these two rings in M2 (K). Hence, somewhat surprisingly, the non-commutative structure sheaf of the hereditary non-Azumaya algebra A is a sheaf of Azumaya algebras over max A.

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LIEVEN LE BRUYN

3.3. Marked quiver and Morita settings. Consider the continuous map for the Zariski topology c

max A−→ X and let for a central point P ∈ X the ﬁber be {M1 , . . . , Mk } where the Mi are maximal ideals of A with corresponding simple di -dimensional representation Si . In the previous section we have introduced the Bergman-Small data, that is

α = (e1 , . . . , ek )

and

β = (d1 , . . . , dk ) ∈ Nk+

satisfying

α.β =

k

ei di = n

i=1

(recall that ei is the multiplicity of Si in the semi-simple n-dimensional representation corresponding to P . Moreover, we have the Jategaonkar-M¨ uller data which is a directed connected graph on the vertices {υ1 , . . . , υk } (corresponding to the Mi ) with an arrow υi υ j

iﬀ

Ext1A (Si , Sj ) = 0

We now want to associate combinatorial objects to this local data. To start, introduce a quiver setting (Q, α) where Q is a quiver (that is, a directed graph) on the vertices {υ1 , . . . , υk } with the number of arrows from υi to υj equal to the dimension of Ext1A (Si , Sj ), # (υi −→ υj ) = dimC Ext1A (Si , Sj ) and where α = (e1 , . . . , ek ) is the dimension vector of the multiplicities ei . Recall that the representation space repα Q of a quiver-setting is ⊕a Mei ×ej (C) where the sum is taken over all arrows a : υj −→ υi of Q. On this space there is a natural action by the group GL(α) = GLe1 × · · · × GLek by base-change in the vertex-spaces Vi = Cei (actually this is an action of P GL(α) which is the quotient of GL(α) by the central subgroup C∗ (1e1 , . . . , 1ek )). The ringtheoretic relevance of the quiver-setting (Q, α) is that repα Q Ext1A (MP , MP )

as GL(α)-modules

where Mp is the semi-simple n-dimensional A-module corresponding to P MP = S1⊕e1 ⊕ · · · ⊕ Sk⊕ek and because GL(α) is the automorphism group of Mp there is an induced action on Ext1A (Mp , MP ). Because Mp is n-dimensional, an element ψ ∈ Ext1A (Mp , Mp ) deﬁnes an algebra morphism ρ

A −→ Mn (C[])

NON-COMMUTATIVE ALGEBRAIC GEOMETRY

227

where C[] = C[x]/(x2 ) is the ring of dual numbers. As we are working in the category [email protected] we need the stronger assumption that ρ is trace preserving. For this reason we have to consider the GL(α)-subspace tExt1A (MP , MP ) ⊂ Ext1A (MP , MP ) of trace preserving extensions. As traces only use blocks on the diagonal (corresponding to loops in Q) and as any subspace Mei (C) of repα Q decomposes as a GL(α)-module in simple representations Mei (C) = Me0i (C) ⊕ C where Me0i (C) is the subspace of trace zero matrices, we see that repα Q∗ tExt1A (MP , MP )

as GL(α)-modules

where Q∗ is a marked quiver that has the same number of arrows between distinct vertices as Q has, but may have fewer loops and some of these loops may acquire a marking meaning that their corresponding component in repα Q∗ is Me0i (C) instead of Mei (C). note: Let the local structure of the non-commutative variety max A near the ﬁber c−1 (P ) of a point P ∈ X be determined by the Bergman-Small data α = (e1 , . . . , ek )

and

β = (d1 , . . . , dk )

and by the Jategoankar-M¨ uller data which is encoded in the marked quiver Q∗ on k-vertices. Then, we associate to P the combinatorial data type(P ) = (Q∗ , α, β) We call (Q∗ , α) the marked quiver setting associated to A in P ∈ X. The dimension vector β = (d1 , . . . , dk ) will be called the Morita setting associated to A in P . Example 3.7. If A is an Azumaya algebra over R. then for every maximal ideal m corresponding to a point P ∈ X we have that A/mA = Mn (C) so there is a unique maximal ideal M = mA lying over m whence the Jategaonkar-M¨ uller data are α = (1) and β = (n). If Sp = R/m is the simple representation of R we have Ext1A (MP , MP ) Ext1R (Sp , SP ) = TP X and as all the extensions come from the center, the corresponding algebra representations A −→ Mn (C[]) are automatically trace preserving. That is, the marked quiver-setting associated to A in P is

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LIEVEN LE BRUYN

where the number of loops is equal to the dimension of the tangent space TP X in P at X and the Morita-setting associated to A in P is (n). Example 3.8. Consider the order of example 3.6 which is generated as a C-algebra by the elements 1 0 0 1 0 0 0 0 a= b= c= d= 0 0 0 0 x 0 0 1 and the 2-dimensional semi-simple representation MP0 determined by m is given by the algebra morphism A −→ M2 (C) sending a and d to themselves and b and c to the zero matrix. A calculation shows that u

Ext1A (MP0 , MP ) = repα Q

for

(Q, α) = 1

1 υ

and as the correspondence with algebra maps to M2 (C[]) is given by

1 0 0 υ 0 0 0 0 a → b → c → d → 0 0 0 0 u 0 0 1 each of these maps is trace preserving so the marked quiver setting is (Q, α) and the Moritasetting is (1,1). 3.4.

Local classiﬁcation.

note: Because the combinatorial data type(P ) = [Q∗ , α, β) encodes the inﬁnitesimal information of the cluster of maximal ideals of A lying over the central point P ∈ X, (repα Q∗ , β) should be viewed as analogous to the usual tangent space TP X. If P ∈ X is a singular point, then the tangent space is too large so we have to impose additional relations to describe the variety X in a neighborhood of P , but if P is a smooth point we can recover the local structure of X from TP X. Here we might expect a similar phenomenon: in general the data (repα Q∗ , β) will be too big to describe AˆmP unless A is a Smooth order in P in which case we can recover AˆmP . We begin by deﬁning some algebras which can be described combinatorially from (Q∗ , α, β). For every arrow a : υi −→ υj deﬁne a generic rectangular matrix of size ej × ei x11 (a) . . . . . . x1ei (a) .. Xa = ... . xej1 (a) . . . . . . xej ei (a)

(and if a is a marked loop take xei ei (a) = −x11 (a) − x22 (a) − . . . − xei −1ei −1 (a)) then the coordinate ring C[repα Q∗ ] is the polynomial ring in the entries of all Xa . For an oriented path p in the marked quiver Q∗ with starting vertex υi and terminating vertex υj p

a

a

al−1

a

l 1 2 υi υj = υi −→ υi1 −→ . . . −→ υil −→ υj

we can form the square ej × ei matrix Xp = Xal Xal−1 · · · Xa2 Xa1

NON-COMMUTATIVE ALGEBRAIC GEOMETRY

229

which has all its entries polynomials in C[repα Q∗ ]. In particular, if the path is an oriented cycle c in Q∗ starting and ending in υi then Xc is a square ei × ei matrix and we can take its trace tr(Xc ) ∈ C[repα Q∗ ] which is a polynomial invariant under the action of GL(α) on repα Q∗ . In fact, it was proved in [31] that these traces along oriented cycles generate the invariant ring α ∗ GL(α) RQ ⊂ C[repα Q∗ ] ∗ = C[repα Q ]

Next we bring in the Morita-setting β = (d1 , . . . , dk ) and deﬁne a block-matrix ring Md1 ×d1 (P11 ) . . . Md1 ×dk (P1k ) .. .. ∗ = ⊂ Mn (C[repα Q ]) . .

Aα,β Q∗

Mdk ×d1 (Pk1 ) . . . Mdk ×dk (Pkk ) α ∗ where Pij is the RQ ∗ -submodule of Mej ×ei (C[repα Q ]) generated by all Xp where p is an ∗ oriented path in Q starting in υi and ending in υk . Observe that for triples (Q∗ , α, β1 ) and (Q∗ , α, β2 ) we have that 1 Aα,β Q∗

is Morita-equivalent to

2 Aα,β Q∗

whence the name Morita-setting for β. Before we can state the next result we need the Euler-form of the underlying quiver Q of Q∗ (that is, forgetting the markings of some loops) which is the bilinear form χQ on Zk a determined by the matrix having as its (i,j)-entry δij − #{a : υi −→υj }. The statements below can be deduced from those of [31] Theorem 3.9. For a triple (Q∗ , α, β) with α.β = n we have α 1. Aα,β Q∗ is an RQ -order in [email protected] if and only if α is the dimension vector of a simple representation of Q∗ , that is, for all vertex-dimensions δi we have

χQ (α, δi ) ≤ 0

and

χQ (δi , α) ≤ 0

unless Q∗ is an oriented cycle of type A˜k−1 then α must be (1, . . . , 1). α 2. If this condition is satisﬁed, the dimension of the center RQ ∗ is equal to α ∗ dim RQ ∗ = 1 − χQ (α, α) − #{marked loops in Q }

These combinatorial algebras determine the ´etale local structure of Smooth orders as was proved in [29]. The principal technical ingredient in the proof is the Luna slice theorem, see for example [45] or [34]. Theorem 3.10. Let A be a Smooth order over R in [email protected] and let P ∈ X with corresponding maximal ideal m. If the marked quiver setting and the Morita-setting associated to A in P is

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given by the triple (Q∗ , α, β), then there is a Zariski open subset X(fi ) containing P and an α ´etale extension S of both Rfi and the algebra RQ ∗ such that we have the following diagram

In particular, we have ˆm R ˆα ∗ R Q

and

Aˆm Aˆα,β Q∗

where the completions at the right hand sides are with respect to the maximal (graded) ideal α of RQ ∗ corresponding to the zero representation. Example 3.11. From example 3.7 we recall that the triple (Q∗ , α, β) associated to an Azumaya algebra in a point P ∈ X is given by

and β = (n) where the number of arrows is equal to dimC TP X. In case P is a smooth point of X this number is equal to d = dim X. Observe that GL(α) = C∗ acts trivially on repα Q∗ = Cd in this case. Therefore we have that α RQ and Aαβ ∗ C[x1 , . . . , xd ] Q∗ = Mn (C[x1 , . . . , xd ])

Because A is a Smooth order in such points we get that AˆmB Mn (C[[x1 , . . . , xd ]]) consistent with our ´etale local knowledge of Azumaya algebras. note: Because α.β = n, the number of vertices of Q∗ is bounded by n and as d = 1 − χQ (α, α) − #{marked loops} the number of arrows and (marked) loops is also bounded. This means that for a particular dimension d of the central variety X there are only a ﬁnite number of ´etale local types of Smooth orders in [email protected] This fact might be seen as a non-commutative version of the fact that there is just one ´etale type of a smooth variety in dimension d namely C[[x1 , . . . , xd ]]. At this moment a similar result for Regular orders seems to be far out of reach.

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3.5. A two-person game. Starting with a marked quiver setting (Q∗ , α) we will play a two-person game. Left will be allowed to make one of the reduction steps to be deﬁned below if the condition on Leaving arrows is satisﬁed, Red on the other hand if the condition on aRRiving arrows is satisﬁed. Although we will not use combinatorial game theory in any way, it is a very pleasant topic and the interested reader is referred to [12] or [3]. The reduction steps below were discovered by R. Bocklandt in his Ph.D. thesis [4] (see also [5]) in which he classiﬁes quiver settings having a regular ring of invariants. These steps were slightly extended in [6] in order to classify central singularities of Smooth orders. All reductions are made locally around a vertex in the marked quiver. There are three types of allowed moves Vertex removal Assume we have a marked quiver setting (Q∗ , α) and a vertex υ such that the local structure of (Q∗ , α) near υ is indicated by the picture on the left below, that is, inside the vertices we have written the components of the dimension vector and the subscripts of an arrow indicate how many such arrows there are in Q∗ between the indicated vertices. Deﬁne the new marked quiver setting (Q∗R , αR ) obtained by the operation RVυ which removes the vertex υ and composes all arrows through υ, the dimensions of the other vertices are unchanged:

where cij = ai bj (observe that some of the incoming and outgoing vertices may be the same so that one obtains loops in the corresponding vertex). Left (resp. Right) is allowed to make this reduction step provided the following condition is met (Lef t)

χQ (α, υ ) ≥ 0

⇔

αυ ≥

l

aj ij

j=1

(Right)

χQ (υ , α) ≥ 0

⇔

αυ ≥

l

bj u j

j=1

(observe that if we started oﬀ from a marked quiver setting (Q∗ , α) coming from an order, then these inequalities must actually be equalities). loop removal If υ is a vertex with vertex-dimension αυ = 1 and having k ≥ 1 loops. Let (Q∗R , αR ) be the marked quiver setting obtained by the loop removal operation Rlυ

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LIEVEN LE BRUYN

removing one loop in υ and keeping the same dimension vector. Both Left and Right are allowed to make this reduction step. Loop removal If the local situation in υ is such that there is exactly one (marked) loop in υ, the dimension vector in υ is k ≥ 2 and there is exactly one arrow Leaving υ and this to a vertex υ indicated below with dimension vector 1, then Left is allowed to make the reduction RL

Similarly, if there is one (marked) loop in υ and αυ = k ≥ 2 and there is only one arrow aRRiving at υ coming from a vertex of dimension vector 1, then Right is allowed to make υ the reduction RL

In accordance with combinatorial game theory we call a marked quiver setting (Q∗ , α) a zero setting if neither Left nor Right has a legal reduction step. The relevance of this game on marked quiver settings is that if (Q∗1 , α1 ) (Q∗2 , α2 ) is a sequence of legal moves (both Left and Right are allowed to pass), then α1 α2 RQ ∗ RQ∗ [y1 , . . . , yz ] 1

2

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where z is the sum of all loops removed in Rlυ reductions plus the sum of αυ for each υ involving a genuine loop and the sum of αυ − 1 for each reduction step reduction step RL υ RL involving a marked loop. That is, marked quiver settings which below to the same game tree have smooth equivalent invariant rings. In general games, a position can reduce to several zero-positions depending on the chosen moves. For this reason the next result, proved in [6] is somewhat surprising Theorem 3.12. Let (Q∗ , α) be a marked quiver setting, then there is a unique zero-setting (Q∗0 , α0 ) far which there exists a reduction procedure (Q∗ , α) (Q∗0 , α0 ) We will denote this unique zero-setting by Z(Q∗ , α). note: Therefore it is suﬃcient to classify the zero-positions if we want to characterize all central singularities of a Smooth order in a given central dimension d. 3.6. Central singularities. Let A be a Smooth R-order in [email protected] and P a point in the central variety X with corresponding maximal ideal m R. We now want to classify the ˆm. types of singularities of X in P , that is to classify R To start, can we decide when P is a smooth point of X? In the case that A is an Azumaya algebra in P , we know already that A can only be a Smooth if R is regular in P . Moreover we have seen for A a Regular reﬂexive Azumaya algebra that the non-Azumaya points in X are precisely the singularities of X. For Smooth orders the situation is more delicate but as mentioned before we have a complete solution in terms of the two-person game by a slight adaptation of Bocklandt’s main result [5]. Theorem 3.13. If A is a Smooth R-order and (Q∗ , α, β) is the combinatorial data associated to A in P ∈ X. Then, P is a smooth point of X if and only if the unique associated zero-setting

The Azumaya points are such that Z(Q∗ , α) = 1 hence the singular locus of X is contained in the ramiﬁcation locus ram A but may be strictly smaller. To classify the central singularities of Smooth orders we may reduce to zero-settings (Q∗ , α) = Z(Q∗ , α). For such a setting we have for all vertices υi the inequalities χQ (α, δi ) < 0

and

χQ (δi , α) < 0

and the dimension of the central variety can be computed from the Euler-form χQ . This gives us an estimate of d = dimX which is very eﬃcient to classify the singularities in low dimensions.

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LIEVEN LE BRUYN

Theorem 3.14. Let (Q∗ , α) = Z(Q∗ , α) be a zero-setting on k ≥ 2 vertices. Then,

In this sum the contribution of a vertex υ with αυ = a is determined by the number of (marked) loops in υ. By the reduction steps (marked) loops only occur at vertices where αυ > 1. Let us illustrate this result by classifying the central singularities in low dimensions Example 3.15. (dimension 2) When dim X = 2 no zero-position on at least two vertices satisﬁes the inequality of theorem 3.14, so the only zero-position possible to be obtained from a marked quiver-setting (Q∗ , α) in dimension two is Z(Q∗ , α) = 1 and therefore the central two-dimensional variety X of a Smooth order is smooth. Example 3.16. (dimension 3) If (Q∗ , α) is a zero-setting for dimension ≤ 3 then Q∗ can have at most two vertices. If there is just one vertex it must have dimension 1 (reducing again to 1 whence smooth) or must be

which is again a smooth setting. If there are two vertices both must have dimension 1 and both must have at least two incoming and two outgoing arrows (for otherwise we could perform an additional vertex-removal reduction). As there are no loops possible in these vertices for zero-settings, it follows from the formula d = 1 − χQ (α, α) that the only possibility is

α ∗ The ring of polynomial invariants RQ so ∗ is generated by traces along oriented cycles in Q in this case it is generated by the invariants

x = ac,

y = ad,

u = bc

and

υ = bd

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and there is one relation between these generators, so α RQ ∗

C(x, y, u, υ) (xy − uυ)

Therefore, the only ´etale type of central singularity in dimension three is the conifold singularity. Example 3.17. (dimension 4) If (Q∗ , α) is a zero-setting for dimension 4 then Q∗ can have at most three vertices. If there is just one, its dimension must be 1 (smooth setting) or 2 in which case the only new type is

which is again a smooth setting. If there are two vertices, both must have dimension 1 and have at least two incoming and outgoing arrows as in the previous example. The only new type that occurs is

for which one calculates as before the ring of invariants to be α RQ ∗ =

C[a, b, c, d, e, f ] (ae − bd, af − cd, bf − ce)

If there are three vertices all must have dimension 1 and each vertex must have at least two incoming and two outgoing vertices. There are just two such possibilities in dimension 4

The corresponding rings of polynomial invariants are α RQ ∗ =

C[x1 , x2 , x3 , x4 , x5 ] (x4 x5 − x1 x2 x3 )

resp.

α RQ ∗ =

C[x1 , x2 , x3 , x4 , y1 , y2 , y3 , y4 ] R2

where R2 is the ideal generated by all 2 × 2 minors of the matrix

x1 x2 x3 x4 y1 y2 y3 y4

In [6] it was proved that there are exactly ten types of Smooth order central singularities in dimension d = 5 and 53 in dimension d = 6. The strategy to prove such a result is as follows.

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LIEVEN LE BRUYN

First one makes a full list of all zero-settings (Q∗ , α) = Z(Q∗ , α) such that d = 1 − χQ (α, α) − # marked loops, using theorem 3.14. Next, one has to weed out zero-settings having isomorphic rings of polynomial invariα ants (or rather, having the same m-adic completion where m RQ ∗ is the unique graded maximal ideal generated by all generators). There are two invariants to separate two rings of invariants. One is the sequence of numbers dimC

mn mn+1

which can sometimes be computed easily (for example if all dimension vector components are equal to 1). The other invariant is what we call the ﬁngerprint of the singularity. In most cases, there will be other types of singularities (necessarily also of Smooth order type) in the variety α corresponding to RQ ∗ and the methods of [29] allow us to determine their associated marked quiver settings as well as the dimensions of these strata. In most cases these two methods allow to separate the diﬀerent types of singularities. In the few remaining cases it is then easy to write down an explicit isomorphism. We refer to (the published version of) [6] for the full classiﬁcation of these singularities in dimension 5 and 6. ˆ m of a note: In low dimensions there is a full classiﬁcation of all central singularities R ˆ Smooth order in [email protected] However, at this moment no such classiﬁcation exists for Am . That is, under the game rules it is not clear what structural results of the orders Aα Q∗ are preserved. 3.7. Isolated singularities. In the classiﬁcation of central singularities of Smooth orders, isolated singularities stand out as the ﬁngerprinting method to separate them clearly fails. Fortunately, we do have by [7] a complete classiﬁcation of these (in all dimensions). Theorem 3.18. Let A be a Smooth order over R and let (Q∗ , α, β) be the combinatorial data associated to a A in a point P ∈ X. Then, P is an isolated singularity if and only if Z(Q∗ , α) = T (k1 , . . . , kl ) where

with d = dim X = i ki − l + 1. Moreover, two such singularities, corresponding to T (k1 , . . . , kl ) and T (k1 , . . . , kl ), are isomorphic if and only if l = l for some permutation σ ∈ Sl .

and

ki = kσ(i)

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237

The results we outlined in this talk are good as well as bad news. note: On the positive side we have very precise information on the types of singularities which can occur in the central variety of a Smooth order (certainly in low dimensions) in sharp contrast to the case of Regular orders. However, because of the scarcity of such types most interesting quotient singularities Cd /G will not have a Smooth order over their coordinate ring R = C[Cd /G]. So, after all this hard work we seem to have come to a dead end with respect to the desingularization problem as there are no Smooth orders with center C[Cd /G]. Fortunately, we have one remaining trick available: to bring in a stability structure.

4. Non-commutative desingularizations In the ﬁrst talk I claimed that in order to ﬁnd good desingularizations of quotient singularities Cd /G we had to ﬁnd Smooth orders in [email protected] with center R = C[Cd /G]. Last time we have seen that Smooth orders can be described and classiﬁed locally in a combinatorial way but also that there can be no Smooth order with center C[Cd /G]. What we will see today is that there are orders A over R which may not be Smooth but are Smooth on a suﬃciently large Zariski open subset of repα A. Here ‘suﬃciently large’ means determined by a stability structure. Whenever this is the case we can apply the results of last time to construct nice (partial) desingularizations of Cd /G and if you are in for non-commutative geometry, even a genuine non-commutative desingularization. 4.1. Quotient singularities. Last time we associated to a combinatorial triple (Q∗ , α, β) α a Smooth order Aα,β Q∗ with center the ring of polynomial quiver-invariants RQ∗ . As we were able to classify the quiver-invariants it followed that there is no triple such that the center d of Aα,β Q∗ is the coordinate ring R = C[C /G] of the quotient singularity. However, there are nice orders of the form A=

Aα,β Q∗ I

for some ideal I of relations which do have center R are have been used in studying quotient singularities. Example 4.1. (Kleinian singularities) For a Kleinian singularity, that is, a quotient singularity C2 /G with G ⊂ SL2 (C) there is an extended Dynkin diagram D associated. x Let Q be the double quiver of D, that is to each arrow 2−→2 in D we adjoin an arrow ∗ x 2←−2 in Q in the opposite direction and let α be the unique minimal dimension vector such that χD (α, α) = 0. Further, consider the moment element m=

[x, x∗ ]

x∈D

in the order Aα Q then A=

Aα Q (m)

238

LIEVEN LE BRUYN

is an order with center R = C[C2 /G] which is isomorphic to the skew-group algebra C[x, y]#G. Moreover, A is Morita equivalent to the preprojective algebra which is the quotient of the path algebra of Q by the ideal generated by the moment element

= CQ/( [x, x∗ ])

0

For more details we refer to the lecture notes by W. Crawley-Boevey [14]. Example 4.2. Consider a quotient singularity X = Cd /G with G ⊂ SLd (C) and Q be the McKay quiver of G acting on V = Cd . That is, the vertices {υ1 , . . . , υk } of Q are in one-to-one correspondence with the irreducible representations {R1 , . . . , Rk } of G such that R1 = Ctriv is the trivial representation. Decompose the tensorproduct in irreducibles V ⊗C Rj = R1⊗j1 ⊗ . . . ⊗ Rk⊗jk then the number of arrows in Q from υi to υj #(υi −→ υj ) = ji is the multiplicity of Ri in V ⊗ Rj . Let α = (e1 , . . . , ek ) be the dimension vector where ei = dimC Ri . The relevance of this quiver-setting is that repα Q = HomG (R, R ⊗ V ) where R is the regular representation, see for example [13]. Consider Y ⊂ repα Q the aﬃne subvariety of all α-dimensional representations of Q for which the corresponding G-equivariant map B ∈ HomG (R, V ⊗ R) satisﬁes B ∧ B = 0 ∈ HomG (R, ∧2 V ⊗ R) Y is called the variety of commuting matrices and its deﬁning relations can be expressed as linear equations between paths in Q evaluated in repα Q, say (l1 , . . . , lz ). Then, A=

Aα Q (l1 , . . . , lz )

is an order with center R = C[Cd /G]. In fact, A is just the skew group algebra A = C[x1 , . . . , xd ]#G Let us give one explicit example illustrating both approaches to the Kleinian singularity C2 /Z3 .

NON-COMMUTATIVE ALGEBRAIC GEOMETRY

239

Example 4.3. Consider the natural action of Z3 on C2 via its embedding in SL2 (C) sending the generator to the matrix ρ 0 0 ρ−1 where ρ is a primitive 3-rd root of unity. Z3 has three one-dimensional simples R1 = Ctriv , R2 = Cρ and R2 = Cρ2 . As V = C2 = R2 ⊗ R3 it follows that the McKay quiver setting (Q, α) is

Consider the matrices 0 0 x3 X = x1 0 0 0 x2 0

and

0 y1 0 Y = 0 0 y2 y3 0 0

then the variety of commuting matrices is determined by the matrix-entries of [X, Y] that is I = (x3 y3 − y1 x1 , x1 y1 − y2 x2 , x2 y2 − y3 x3 ) so the skew-group algebra is the quotient of the Smooth order Aα Q (which incidentally is one of our zero-settings for dimension 4) C[x, y]#Z3

Aα Q (x3 y3 − y1 x1 , x1 y1 − y2 x2 , x2 y2 − y3 x3 )

Taking yi = x∗i this coincides with the description via preprojective algebras as the moment element is

m=

3

[xi , x∗i ] = (x3 y3 − y1 x1 )e1 + (x1 y1 − y2 x2 )e2 + (x2 y2 − y3 x3 )e3

i=1

where the ei are the vertex-idempotents. note: Many interesting examples of orders are of the following form: A=

Aα Q∗ I

240

LIEVEN LE BRUYN

satisfying the following conditions: • α = (e1 , . . . , ek ) is the dimension vector of a simple representation of A, and • the center R = Z(A) is an integrally closed domain. These requirements (which are often hard to verify!) imply that A is an order over R in [email protected] where n is the total dimension of the simple representation, that is |α| = Σi ei . Observe that such orders occur in the study of quotient singularities (see above) or as the ´etale local structure of (almost all) orders. From now on, this will be the setting we will work in. 4.2. Stability structures. For A = Aα Q∗ /I we deﬁne the aﬃne variety of α-dimensional representations repα A = {V ∈ repα Q∗ |r(V = 0 ∀r ∈ I} The action of GL(α) = i GLei by basechange on repα Q∗ induces an action (actually of P GL(α)) on repα A. Usually, repα A will have singularities but it may be smooth on the Zariski open subset of θ-semistable representations which we will now deﬁne. A character of GL(α) is determined by an integral k-tuple θ = (t1 , . . . , tk ) ∈ Zk χθ : GL(α) −→ C∗

(g1 , . . . , gk ) → det(g1 )t1 · · · det(gk)tk

Characters deﬁne stability structures on A-representations but as the acting group on repα A is really P GL(α) = GL(α)/C∗ (1e1 , . . . , 1ek ) we only consider characters θ satisfying θ.α = i ti ei = 0. If V ∈ repα A and V ⊂ V is an A-subrepresentation, that is V ⊂ V as representations of Q∗ and in addition I(V ) = 0, we denote the dimension vector of V by dimV . Deﬁnition 4.4. For θ satisfying θ.α = 0, a representation V ∈ repα A is said to be • θ-semistable if and only if for every proper A-subrepresentation 0 = V ⊂ V we have θ.dimV ≥ 0. • θ-stable if and only if for every proper A-subrepresentation 0 = V ⊂ V we have θ.dimV > 0. For any setting θ.α = 0 we have the following inclusions of Zariski open GL(α)-stable subsets of repα A A ⊂ repθ−stable A ⊂ repθ−semist A ⊂ repα A repsimple α α α but one should note that some of these open subsets may actually be empty! Recall that a point of the algebraic quotient variety issα A = repα //GL(α) represents the orbit of an α-dimensional semi-simple representation V and such representations can be separated by the values f (V ) where f is a polynomial invariant on repα A. This follows because the coordinate ring of the quotient variety C[issα A] = C[repα A]GL(α) and points correspond to maximal ideals of this ring. Recall from [31] that the invariant ring is generated by taking traces along oriented cycles in the marked quiver-setting (Q∗ , α).

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note: For θ-stable and θ-semistable representations there are similar results and morally one should view θ-stable representations as corresponding to simple representations whereas θ-semistables are arbitrary representations. For this reason we will only be able to classify direct sums of θ-stable representations by certain algebraic varieties which are called the moduli spaces of semistables representations. The notion corresponding to a polynomial invariant in this more general setting is that of a polynomial semi-invariant. A polynomial function f ∈ C[repα A] is said to be a θ-semiinvariant of weight l if for all g ∈ GL(α) we have g·f = χθ (g)l f where χθ is the character of GL(α) corresponding to θ. A representation V ∈ repα A is θ-semistable if and only if there is a θ-semi-invariant f of some weight l such that f (V ) = 0. Clearly, θ-semi-invariants of weight zero are just polynomial invariants and the multiplication of θ-semi-invariants of weight l resp. l has weight l + l . Hence, the ring of all θ-semiinvariants l C[repα A]GL(α),θ = ⊕∞ l=0 {f ∈ C[repα A]|∀g ∈ GL(α) : g·f = χθ (g) f }

is a graded algebra with part of degree zero C[issα A]. But then we have a projective morphism π

proj C[repα A]GL(α),θ −→ −→ issα A such that all ﬁbers of π are projective varieties. The main properties of π can be deduced from [22] Theorem 4.5. Points in proj C[repα A]GL(α),θ are in one-to-one correspondence with isomorphism classes of direct sums of θ-stable representations of total dimension α. If α is such that there are α-dimensional simple A-representations, then π is a birational map. Deﬁnition 4.6. We call proj C[repα A]GL(α),θ the moduli space of θ-semistable representations of A and denote it with moduliθα A. Example 4.7. In the case of Kleinian singularities, see example 4.1, if we take θ to be a generic character such that θ.α = 0, then the projective map moduliθα A X = C2 /G is a minimal resolution of singularities. Note that the map is birational as α is the dimension vector of a simple representation of A = 0 , see [14]. Example 4.8. For general quotient singularities, see example 4.2, assume that the ﬁrst vertex in the McKay quiver corresponds to the trivial representation. Take a character θ ∈ Zk such that t1 < 0 and all ti > 0 for i ≥ 2, for example take

θ = (−

k i=2

dimRi , 1, . . . , 1)

242

LIEVEN LE BRUYN

Then, the corresponding moduli space is isomorphic to moduliθα A G − Hilb Cd the G-equivariant Hilbert scheme which classiﬁes all #G-codimensional ideals I C[x1 , . . . , xd ] where C[x1 , . . . , xd ] CG I as G-modules, hence in particular I must be stable under the action of G. It is well known that the natural map G − Hilb Cd X = Cd /G is a minimal resolution if d = 2 and if d = 3 it is often a crepant resolution, for example whenever G is Abelian. In non-Abelian cases it may have remaining singularities though which often are of conifold type. See [13] for more details. note: My motivation for this series of talks was to look for a non-commutative explanation for the omnipresence of conifold singularities in partial resolutions of three dimensional quotient singularities as well as to have a conjectural list of possible remaining singularities for higher dimensional quotient singularities. Example 4.9. In the C2 /Z3 -example one can take θ = (−2, 1, 1). The following representations

are all nilpotent and are θ-stable. In fact if bc = 0 they are representants of the exceptional ﬁber of the desingularization moduliθα A issα A = C2 /Z3 4.3. Partial resolutions. It is about time we state the main result of these notes which was proved in [32]. Theorem 4.10. Let A = Aα Q∗ /(R) be an R-order in [email protected] Assume that there exists a A of all θ-semistable stability structure θ ∈ Zk such that the Zariski open subset repθ−semist α α-dimensional representations of A is a smooth variety. Then there exists a sheaf A of Smooth orders over moduliθα A such that the diagram below is commutative

NON-COMMUTATIVE ALGEBRAIC GEOMETRY

243

Here, spec A is a non-commutative variety obtained by gluing aﬃne non-commutative varieties spec Ai together and c is the map which intersects locally a maximal ideal with the center. As A is a sheaf of Smooth orders, φ can be viewed as a non-commutative desingularization of X. If you are only interested in commutative desingularizations, π is a partial resolution of X and we have full control over the remaining singularities in moduliθα A, that is, all remaining singularities are of the form classiﬁed in the previous lecture. Moreover, if θ is such that all θ-semistable A-representations are actually θ-stable, then A is a sheaf of Azumaya algebras over moduliθα A and in this case π is a commutative desingularization of X. If, in addition, also gcd(α) = 1, then A End P for some vectorbundle of rank n over moduliθα A. A is a smooth variety is note: It should be stressed that the condition that repθ−semist α very strong and is usually hard to verify in concrete situations. Example 4.11. In the case of Kleinian singularities, see example 4.1, there exists a suitable stability structure θ such that repθ−semist Π0 is smooth. For consider the moment map α µ

repα Q −→ lie GL(α) = Mα (C) = Me1 (C) ⊗ . . . ⊗ M∈k (C) deﬁned by sending V = (Va , Va∗ ) to

The diﬀerential dµ can be veriﬁed to be surjective in any representation V ∈ repα Q which has stabilizer subgroup C∗ (1e1 , . . . , 1ek ) (a so called Schur representation) see for example [15, lemma 6.5]. Further, any θ-stable representation is Schurian. Moreover, for a generic stability structure θ ∈ Zk we have that every θ-semistable α-dimensional representation is θ-stable as the gcd(α) = 1. Combining these facts it follows that µ−1 (0) = repα Π0 is smooth in all θ-stable representations. A is evident is when Example 4.12. Another case where smoothness of repθ−semist α is a Smooth order as then rep A itself is smooth. This observation can be used A = Aα ∗ α Q to resolve the remaining singularities in the partial resolution. If gcd(α) = 1 then for a suﬃciently general θ all θ-semistable representations are actually θ-stable whence the quotient map A moduliθα A repθ−semist α is a principal P GL(α)-ﬁbration and as the total space is smooth, so is moduliθα A. Therefore, the projective map π

moduliθα A issα A is a resolution of singularities in this case.

244

LIEVEN LE BRUYN

However, if l = gcd(α), then moduliθα A will usually contain singularities which are as bad as the quotient variety singularity of tuples of l×l matrices under simultaneous conjugation. Fortunately, the proof of the theorem will follow from the hard work we did in last lecture provided we can solve two problems. A minor problem is that we classiﬁed central singularities of Smooth orders in [email protected] but here we are working with α-dimensional representations and with the action of GL(α) rather than GLn . This problem we will address immediately. A is not an aﬃne variety in general so we A more serious problem is that repθ−semist α will have to cover it with aﬃne varieties Xi and consider associated orders Ai . But then we have to clarify why θ-semistable representations of A correspond to all representations of the Ai . This may not be clear at ﬁrst sight. 4.4. Going from. [email protected] to [email protected]α If Q∗ is a marked quiver on k vertices, then the subalgebra generated by the vertexidempotents Ck is a subalgebra of A = Aα Q∗ /(R) hence we have a morphism repn A −→ repn Ck =

GLn /GL(α)

|α|=n

where the last decomposition follows from the fact that Ck is semi-simple whence every n-dimensional representation is fully determined by the multiplicities of the simple 1-dimensional components. Further, we should consider trepn A the subvariety of trace preserving A-representations but a trace map on A ﬁxes the trace on Ck and hence determines the component GLn /GL(α). That is, we have that trepn A = GLn ×GL(α) repα A the variety is a principal ﬁber bundle. φ That is, if V is any n-dimensional trace preserving A-representation A −→ Mn (C) then the images φ(υi ) of the vertex-idempotents are a full set of orthogonal idempotents so they can be conjugated to a set of matrices

..

. 1 .. φ (υi ) = . 1

..

.

i−1 i with only 1’s from place j=1 ej + 1 to place j=1 ej . But using these idempotents we see that the representation φ : A −→ Mn (C) has block-matrices coming from a representation in repα A. As is the case for any principal ﬁber bundle, this gives a natural one-to-one correspondence between • GLn -orbits in trepn A, and • GL(α)-orbits in repα A.

NON-COMMUTATIVE ALGEBRAIC GEOMETRY

245

Moreover the corresponding quotient varieties tissn A = trepn A//GLn and issα A = repα A//GL(α) are isomorphic so we can apply all our (P )GLn -results to this setting. note: Alternatively, we can deﬁne [email protected]α to be the subcategory of [email protected] with objects the algebras A ∈ [email protected] which are Ck -algebras via the embedding given by the matrices φ (υi ) above and with morphism the Ck -algebra morphisms in [email protected] It is then clear that a Smooth order in [email protected]α (that is, having the lifting property with respect to nilpotent ideals in [email protected]α) is a Smooth order in [email protected] which is an object in [email protected]α. 4.5. The aﬃne opens XD . To solve the second problem, we claim that we can cover the moduli space XD moduliθα A = D

where XD is an aﬃne open subset such that under the canonical quotient map π

repθ−semist A moduliθα A α we have that π −1 (XD ) = repα AD for some C[XD ]-order AD in [email protected] A is a smooth variety, each of the repα AD are smooth aﬃne If in addition repθ−semist α GL(α)-varieties whence the orders AD are all Smooth and the result will follow from the results of last lecture. Because moduliθα A = projC[repα A]GL(α),θ we need control on the generators of all θ-semi-invariants. Such a generating set was found by Aidan Schoﬁeld and Michel Van den Bergh in [44]: determinantal semi-invariants. In order to deﬁne them we have to introduce some notation ﬁrst. Reorder the vertices in Q∗ such that the entries of θ are separated in three strings θ = (t1 , . . . , ti ), ti+1 , . . . , tj , tj+1 , . . . , tk >0

=0

<0

and let θ be such that θ.α = 0. Fix a weight l ∈ N+ and take arbitrary natural numbers {li+1 , . . . , lj }. Consider a rectangular matrix L with • lt1 + · · · + lti + li+1 + · · · + lj rows and • li+1 + · · · + lj − ltj+1 − · · · − ltk columns li+1

lt1 { L1,i+1 .. . L = lt { L i i,i+1 li+1 { Li+1,i+1 .. . lj { Lj,i+1

...

lj

L1,j

−ltj+1

L1,j+1

Li,j Li,j+1 Li+1,j Li+1,j+1 Lj,j

Lj,j+1

...

−lt

k L1,k

Li,k Li+1,k Lj,k

246

LIEVEN LE BRUYN

in which each entry of Lr,c is a linear combination of oriented paths in the marked quiver Q∗ with starting vertex υc and ending vertex υr . The relevance of this is that we can evaluate L at any representation V ∈ repα A and obtain a square matrix L(V) as θ.α = 0. More precisely, if Vi is the vertex-space of V at vertex υi (that is, Vi has dimension ei ), then evaluating L at V gives a linear map ⊕l

⊕lj

⊕−lt

⊕ Vj+1 j+1 ⊕ · · · ⊕ Vk⊕−ltk L(V ) ↓ ⊕l ⊕l ⊕ · · · ⊕ Vi⊕lti ⊗ Vi+1i+1 ⊕ · · · ⊕ Vj j

Vi+1i+1 ⊕ · · · ⊕ Vj V1⊕lt1

and L(V) is a square N × N matrix where li+1 + · · · + lj − ltj+1 − · · · − ltk = N = lt1 + · · · + lti + li+1 + · · · + lj So we can consider D(V ) = detL(V ) and verify that D is a GL(α)-semi-invariant polynomial on repα A of weight χlθ . The result of [44] asserts that these determinantal semiinvariants are algebra generators of the graded algebra C[repα A]GL(α),θ Observe that this result is to semi-invariants what the result of [31] is to invariants. In fact, one can deduce the latter from the ﬁrst. We have seen that a representation V ∈ repα A is θ-semistable if and only if some semi-invariant of weight χlθ for some l is non-zero on it. This proves Theorem 4.13. The Zariski open subset of θ-semistable α-dimensional A-representations can be covered by aﬃne GL(α)-stable open subsets repθ−semist A= α

{V |D(V ) = detL(V ) = 0}

D

and hence the moduli space can also be covered by aﬃne open subsets moduliθα A =

XD

D

where XD = {[V ] ∈ moduliθα A|D(V ) = detL(V ) = 0}. Example 4.14. In the C2 /Z3 example, the θ-semistable representations

NON-COMMUTATIVE ALGEBRAIC GEOMETRY

247

with θ = (−2, 1, 1) all lie in the aﬃne open subset XD where L is a matrix of the form L=

x1 0 ∗ y3

where ∗ is any path in Q starting in x1 and ending in x3 . 4.6. The C[XD ]- orders AD . Analogous to the rectangular matrix L we deﬁne a rectangular matrix N with • lt1 + · · · + lti + li+1 + · · · + lj columns and • li+1 + · · · + lj − ltj+1 − · · · − ltk rows lt

1 li+1 { Ni+1,1 .. . N= l { Nj,1 j −ltj+1 { Nj+1,1 .. . −ltk { Nk,1

...

lt

l

i i+1 Ni+1,i Ni+1,i+1

Nj,i Nj,i+1 Nj+1,i Nj+1,i+1 Nk,i

Nk,i+1

...

lj

Ni+1,j Nj,j Nj+1,j Nk,j

ﬁlled with new variables and deﬁne an extended marked quiver Q∗D where we adjoin for each entry in Nr,c an additional arrow from υc to υr and denote it with the corresponding variable from N . Let I1 (resp. I2 be the set of relations in CQ∗D determined from the matrix-equations

respectively

where (υi )nj is the square nj × nj matrix with υi on the diagonal and zeroes elsewhere.

248

LIEVEN LE BRUYN

Deﬁne a new non-commutative order AD =

Aα Q∗

D

I, I1 , I2

then AD is a C[XD ]-order in [email protected] Example 4.15. In the setting of example 4.14 with ∗ = y3 , the extended quiver-setting (QD , α) is

Hence, with

L=

x1 0 y3 y3

N=

n1 n3 n2 n4

the deﬁning equations of the order AD become I = (x3 y3 − y1 x1 , x1 y1 − y2 x2 , x2 y2 − y3 x3 ) I1 = (n1 x1 + n3 y3 − υ1 , n3 y3 , n2 x1 + n4 y3 , n4 y3 − υ1 ) I2 = (x1 n1 − υ2 , x1 n3 , y3 n1 + y2 n2 , y3 n3 + y3 n4 − υ3 ) note: This construction may seem a bit mysterious at ﬁrst but what we are really doing is to construct the universal localization as in for example [43] associated to the map between projective A-modules determined by L, but this time not in the category alg of all algebras but in [email protected]α. That is, take Pi = υi A be the projective right ideal associated to vertex υi , then L determines an A-module morphism ⊕l

L

⊕lj

P = Pi+1i+1 ⊕ · · · ⊕ Pk⊕−ltk −→ P1⊕lt1 ⊕ · · · ⊕ Pj

=Q

φ

The algebra map A −→ AD is universal in [email protected]α with respect to L⊗φ being invertible, that ψ

is, if A −→ S is a morphism in [email protected]α such that L ⊗ ψ is an isomorphism of right S-modules, u then there is a unique map in [email protected]α AD −→S such that ψ = u ◦ φ. The proof of the main result follows from the following result:

NON-COMMUTATIVE ALGEBRAIC GEOMETRY

249

Theorem 4.16. The following statements are equivalent 1. V ∈ repθ−semist A lies in π −1 (XD ), and α 2. There is a unique extension V˜ of V such that V˜ ∈ repα AD . Proof. 1 ⇒ 2: Because L(V ) is invertible we can take N (V ) to be its inverse and decompose it into blocks corresponding to the new arrows in Q∗D . This then deﬁnes the unique extension V˜ ∈ repα Q∗D of V . As V˜ satisﬁes R (because V does) and R1 and R2 (because N (V ) = L(V )− 1) we have that V˜ ∈ repα AD . 2 ⇒ 1: Restrict V˜ to the arrows of Q to get a V ∈ repα Q. As V˜ (and hence V ) satisﬁes R, V ∈ repα A. Moreover, V is such that L(V ) is invertible (this follows because V˜ satisﬁes R1 and R2 ). Hence, D(V ) = 0 and because D is a θ-semiinvariant it follows that V is an α-dimensional θ-semistable representation of A. An alternative method to see this is as follows. Assume that V is not θ-semistable and let V ⊂ V be a subrepresentation such that θ.dimV < 0. Consider the restriction of the linear map L(V ) to the subrepresentation V and look at the commuting diagram

As θ.dimV < 0 the top-map must have a kernel which is clearly absurd as we know that L(V ) is invertible. Example 4.17. In the setting of example 4.14 with ∗ = y3 we have that the uniquely determined extension of the A-representation

Observe that this extension is a simple AD -representation for every b, c ∈ C. 4.7. Non-commutative desingularizations. There is just one more thing to clarify: how are the diﬀerent AD ’s glued together to form a sheaf A of non-commutative algebras over moduliθα A and how can we construct the non-commutative variety spec A? The solution to both problems follows from the universal property of AD . Let AD1 (resp. AD2 ) be the algebra constructed from a rectangular matrix L1 (resp. L2 ), then we can construct the direct sum map L = L1 ⊕ L2 for which the corresponding

250

LIEVEN LE BRUYN

semi-invariant D = D1 D2 . As A −→ AD makes the projective module morphisms associated to L1 and L2 an isomorphism we have uniquely determined maps in [email protected]α

As repα AD = π −1 (XD ) (and similarly for Di ) we have that i∗j are embeddings as are the ij . This way we can glue the sections Γ(XD1 , A) = AD1 with Γ(XD2 , A) = AD2 over their intersection XD = XD1 ∩ XD2 via the inclusions ij . Hence we get a coherent sheaf of non-commutative algebras A over moduliθα A. Observe that many of the orders AD are isomorphic. In example 4.14 all matrices L with ﬁxed diagonal entries x1 and y3 but with varying ∗-entry have isomorphic orders AD (use the universal property). In a similar way we would like to glue max AD1 with max AD2 over max AD using the algebra maps ij to form a non-commutative variety spec A. However, the construction of max A and the non-commutative structure sheaf is not functorial in general. Example 4.18. Consider the inclusion map map in [email protected] A=

R R R R → = A I R R R

then all twosided maximal ideals of A are of the form M2 (m) where m is a maximal ideal of R. If I ⊂ m then the intersection m m R R m m ∩ = m m I R I m which is not a maximal ideal of A as m R R R m m = I R I m I m and so there is no natural map max A −→ max A, let alone a continuous one. note: Associating to a non-commutative algebra A its prime ideal spectrum spec A is only f

functorial for extensions A −→ B, that is, satisfying B = f (A)ZB (A)

with

ZB (A) = {b ∈ B|bf (a) = f (a)b∀a ∈ A}

f

In [38] it was proved that if A −→ B is an extension then the map spec B −→ spec R

P −→ f −1 (P )

is well-deﬁned and continuous for the Zariski topology.

NON-COMMUTATIVE ALGEBRAIC GEOMETRY

Fortunately, in the case of interest to us, that is for the maps ij presents no problem as they are even central extensions, that is

:

251

ADj −→ AD this

AD = ADj Z(AD ) which follows again from the universal property by localizing ADj at the central element D. Hence, we can deﬁne a genuine non-commutative variety spec A with central scheme moduliθα A, ﬁnishing the proof of the main result and these talks. References [1] Michael Artin, On Azumaya algebras and finite-dimensional representations of rings, J. Algebra 11 (1969), 523563. [2] George Bergman and Lance Small, P.I. degrees and prime ideals J. Algebra 33 (1975), 435–462 [3] Berlekamp, Conway and Guy, Winning Ways for your mathematical plays [4] Raf Bocklandt, Quotient varieties of representations of quivers, Ph.D. thesis UA-Antwerp (2002) [5] Raf Bocklandt, Smooth quiver quotient varieties, Linear Alg. Appl. (2002) (to appear) arXiv:math.RT/0204355 [6] Raf Bocklandt, Lieven Le Bruyn and Geert Van de Weyer, Smooth order singularities, Journal Algebra & Appl. (to appear) arXiv:math.RA/0207250 [7] Raf Bocklandt, Lieven Le Bruyn and Stijn Symens, Isolated singularities, smooth orders and Auslander regularity, Comm. Algebra arXiv:math.RA/0207251 [8] Ken Brown and Iain Gordon, The ramification of the centres: Lie algebras in positive characteristic and quantised enveloping algebras, Math.Zeit. 238 (2001), 733–779. [9] Ken Brown and Iain Gordon, The ramification of the centres: quantised function algebras at roots of unity, Proc.L.M.S. 84 (2002), 147–178. [10] Ken Brown and Iain Gordon, Poisson orders, symplectic reflection algebras and representation theory, J. reine angew. Math. 559 (2003), 193–216. [11] Paul Cohn, Skew field constructions London Mathematical Society Lecture Note Series 27 Cambridge University Press, Cambridge-New York-Melbourne, (1977) [12] John Horton Conway, On numbers and games, A.K. Peters Ltd. (2001) [13] Alastair Craw, The McKay correspondence and representations of the McKay quiver Warwick Ph.D. thesis (2001), available from http://www.maths.Warwick.ac.uk/ miles/McKay/ [14] Bill Crawley-Boevey, Representations of quivers, preprojective algebras and deformations of quotient singularities, Lectures from a DMV Seminar in May 1999 on “Quantizations of Kleinian singularities”, available from http://www.amsta.leeds.ac.uk/Pure/staff/ crawley b/ [15] Bill Crawley-Boevey, Geometry of the moment map for representations of quivers, available from http://www.amsta.leeds.ac.uk/Pure/staff/crawley b/ [16] Ofer Gabber, Some theorems on Azumaya algebras The Brauer group (Sem., Les Plans-surBex, 1980) 129–209, Lecture Notes in Math., 844, Springer, Berlin-New York, (1981) [17] Jonathan Golan, Structure sheaves over a noncommutative ring Lecture Notes in Pure and Applied Mathematics 56 Marcel Dekker, Inc., New York, (1980). [18] Ken Goodearl and Robert Warfield, An introduction to noncommutative Noetherian rings, LMS Student Texts 16 (1989) [19] Alexandre Grothendieck, Le groupe de Brauer. I. Algbres d’Azumaya et interprtations diverses (1968) Dix Exposs sur la Cohomologie des Schmas 46–66 North-Holland, Amsterdam; Masson, Paris [20] A. V. Jategaonkar, Localization in Noetherian rings, LMS Lect. Note Series 98 (1986) [21] Ina Kersten, Brauergruppen von K¨ orpern, Aspects of Math. D6 Vieweg (1990) [22] Alastair King, Moduli of representations of finite dimensional algebras, Quat. J. Math. Oxford 45 (1994) 515–530

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[23] Max-Albert Knus and Manuel Ojanguren, Thorie de la descente et algbres d’Azumaya Lecture Notes in Mathematics, Vol. 389 Springer-Verlag, Berlin-New York, (1974) [24] Lieven Le Bruyn, Class groups of maximal orders over Krull domains, Ph. D. thesis, Antwerp (1983) [25] Lieven Le Bruyn, The Artin-Schofield theorem and some applications Comm. Algebra 14 (1986), no. 8, 1439–1455. [26] Lieven Le Bruyn, A cohomological interpretation of the reflexive Brauer group, J. Alg. 105 (1987) 250–254 [27] Lieven Le Bruyn, Central singularities of quantum spaces, J. Alg. 177 (1995) 142–153 [28] Lieven Le Bruyn, noncommutative [email protected], forgotten book (2000), available upon request [29] Lieven Le Bruyn, Local structure of Schelter-Procesi smooth orders, Trans. AMS 352 (2000) 4815–4841 [30] Lieven Le Bruyn and Michel Van den Bergh, Regularity of trace rings of generic matrices, J. Alg 117 (1988) 19–29 [31] Lieven Le Bruyn and Claudio Procesi, Semi-simple representations of quivers, Trans. AMS 317 (1990) 585–598 [32] Lieven Le Bruyn and Stijn Symens, Partial desingularizations arising from non-commutative algebras, to appear [math.RA/0401?] [33] Thierry Levasseur, Some properties of noncommutative regular graded rings Glasgow Math. J. 34 (1992), no. 3, 277–300 [34] Domingo Luna, Slices etales, Bull. Soc. Math. France Suppl. Mem. 33 (1973), 81105. [35] J. S. Milne, Etale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton New Jersey (1980). [36] Morris Orzech, Brauer groups and class groups for a Krull domain in: Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), pp. 66–90, Lecture Notes in Math. 917 Springer, Berlin-New York, (1982). [37] Richard S. Pierce, Associative algebras, Graduate Texts in Mathematics, vol. 88, SpringerVerlag, Berlin Heidelberg New York (1982). [38] Claudio Procesi, Rings with polynomial identitis, Marcel Dekker (1973) [39] Claudio Procesi, A formal inverse to the Cayley-Hamilton theorem, J. Algebra 107 (1987) 143–176 [40] Miles Reid, La correspondance de McKay, S´eminaire Bourbaki (1999) Ast´erisque (2000) arXiv:math.AG/9911165 [41] Irving Reiner, Maximal orders, Corrected reprint of the 1975 original. With a foreword by M. J. Taylor. London Mathematical Society Monographs. New Series 28. The Clarendon Press, Oxford University Press. Oxford, (2003). [42] William Schelter, Smooth algebras, J. Algebra 103 (1986) 677–685 [43] Aidan Schofield, Representations of rings over skew fields, LMS Lect. Notes Series 92 (1985) [44] Aidan Schofield and Michel Van den Bergh, Semi-invariants of quivers for arbitrary dimension vectors (1999) arXiv:math.RA/9907174 [45] Peter Slodowy, Der scheibensatzfur algebraische transformationsgruppen, Algebraische Transformationsgruppen und Invariantentheorie, Algebraic Transformation Groups and Invariant Theory (Hanspeter Kraft, Peter Slodowy, and Tonny A. Springer, eds.), DMV Seminar, vol. 13, Birkhauser, Basel Boston Berlin (1989) 89114. [46] Michel Van den Bergh, Non-commutative crepant resolutions, Proceedings of the Abel Bicentennial Conference (to appear), arXiv:math.RA/0211064 [47] Fred Van Oystaeyen, Prime spectrum of non-commutative rings, Lecture Notes Math. 444 (1975) [48] Fred Van Oystaeyen and Alain Verschoren, Non-commutative algebraic geometry, Lecture Motes Math. 887 (1980)

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Author Index

Bartels, Hans-Jochen ...........................................1 Burban, Igor.......................................................23 Detomi, Eloisa ...................................................47 Drozd, Yuriy.......................................................23 Elder, G. Griffith................................................63 Eriksen, Eivind ..................................................90

Hazewinkel, Michiel........................................126 Le Bruyn, Lieven.............................................203 Lucchini, Andrea ...............................................47 Malinin, D.A........................................................1 Ndirahisha, Janvière ........................................147 Petit, Toukaiddine............................................162 Rashkova, Tsetska Grigoriva ...........................175 Rump, Wolfgang..............................................184 Van Oystaeyen, Freddy....................................147

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