Representations of Algebras and Related Topics (EMS Series of Congress Reports)

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Author: Andrzej Skowroński (Editor) | Kunio Yamagata (Editor)

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EMS Series of Congress Reports

EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field of pure or applied mathematics. The individual volumes include an introduction into their subject and review of the contributions in this context. Articles are required to undergo a refereeing process and are accepted only if they contain a survey or significant results not published elsewhere in the literature. Previously published: Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowroński (ed.) K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.) Classification of Algebraic Varieties, Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.) Surveys in Stochastic Processes, Jochen Blath, Peter Imkeller and Sylvie Rœlly (eds.)

Representations of Algebras and Related Topics Andrzej Skowronski ´ Kunio Yamagata Editors

Editors: Andrzej Skowronski ´ Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń Poland

Kunio Yamagata Department of Mathematics Tokyo University of Agriculture and Technology Nakacho 2-24-16, Koganei Tokyo 184-8588 Japan

E-mail: [email protected]

E-mail: [email protected]

2010 Mathematics Subject Classification: 13Dxx, 13Fxx, 14Bxx, 14Hxx, 14Lxx, 14Mxx, 14Nxx, 15Axx, 16Dxx, 16Exx, 16Gxx, 16Sxx, 16Wxx, 17Bxx, 18Exx, 19Kxx, 20Cxx, 20Jxx

ISBN 978-3-03719-101-9 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2011 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Preface The Fourteenth International Conference on Representations of Algebras and Workshop (ICRA XIV) was held at the National Olympics Memorial Youth Center (NYC) in Tokyo, Japan, from 6 to 15 August, 2010. The ICRA XIV was attended by the remarkable large number of 230 researchers and graduate students from 23 countries of all parts of the world. The Scientific Advisory ICRA XIV Committee consisted of R. Bautista (Morelia, Mexico), R.-O. Buchweitz (Toronto, Canada), M. C. R. Butler (Liverpool, United Kingdom), W. Crawley-Boevey (Leeds, United Kingdom), V. Dlab (Ottawa, Canada), Y. A. Drozd (Kiev, Ukraine), K. Erdmann (Oxford, United Kingdom), B. HuisgenZimmermann (Santa Barbara, United States), B. Keller (Paris, France), H. Lenzing (Paderborn, Germany), M.-P. Malliavin (Paris, France), H. Merklen (Sao Paulo, Brazil), J.A. de la Peña (Mexico City, Mexico), M. I. Platzeck (Bahia Blanca, Argentina), I. Reiten (Trondheim, Norway), C. M. Ringel (Bielefeld, Germany), D. Simson (Toruń, Poland), A. Skowroński (Toruń, Poland), S. O. Smalø (Trondheim, Norway), K. Yamagata (Tokyo, Japan), Y. Zhang (Beijing, China). The Local Organizing ICRA XIV Committee was formed by K. Yamagata (Chairman), H. Asashiba, O. Iyama, S. Koshitani, I. Mori, K. Nishida, M. Sato. We would like to thank the members of the Committees as well the leaders of research groups for the advices, help and cooperation making the ICRA XIV very successful. We are also grateful to the National Olympics Memorial Youth Center in Tokyo for the possibility to organize the ICRA XIV in this wonderful place and to the Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B) 21340003, for a substantial financial support. According to a tradition in the area, the ICRA XIV was divided into two parts: the Workshop and the Conference. The ICRA XIV Workshop consisted of seven mini-courses of three hours each given by M. Kashiwara (Kyoto), B. Keller (Paris), B. Leclerc (Caen), H. Lenzing (Paderborn), M. Linckelmann (Aberdeen), C. M. Ringel (Bielefeld), G. Zwara (Toruń). The ICRA XIV Conference comprised 124 talks (24 plenary talks, 100 talks in parallel sessions), among them 14 one hour plenary lectures given by R.-O. Buchweitz (Toronto), J. Chuang (London), A. Henke (Garching), O. Iyama (Nagoya), S. B. Iyengar (Nebraska), D. Kussin (Paderborn), I. Mori (Shizuoka), J. A. de la Peña (Mexico City), I. Reiten (Trondheim), J. Schröer (Bonn), A. Skowroński (Toruń), C. Xi (Beijing),Y.Yoshino (Okayama), A. Zelevinsky (Boston). The ICRA AWARD 2010 (for outstanding work by young mathematician working in the area of representation theory of algebras) was given to Claire Amiot for her original and influential work on 2-Calabi–Yau categories and in particular her construction of generalized cluster categories associated to quivers with potential and to algebras of global dimension two. This book contains eleven expository survey articles and two research articles on recent developments and trends in the area of representation theory of algebras and

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Preface

related topics, reflecting the topics of some lectures presented during the ICRA XIV Workshop and Conference held in Tokyo. We now briefly describe the contents of the articles. The article by Amiot surveys development and motivations on cluster categories leading from the cluster categories of finite dimensional hereditary algebras defined by Buan, Marsh, Reineke, Reiten and Todorov as well as the stable categories of modules over preprojective algebras of Dynkin type studied by Geiss, Leclerc and Schröer to her generalized cluster categories associated to the Jacobi-finite quivers with potential and to finite dimensional algebras of global dimension at most two. In the article, the general construction of these new triangulated 2-Calabi–Yau categories and links with other related categories are presented. Moreover, several interesting applications of the generalized cluster categories in the representation theory of finite dimensional algebras are outlined. The purpose of the article by Benson, Iyengar and Krause is to explain the recent work of the three authors on the classification of localizing and colocalizing subcategories of triangulated categories. In the article, the main attention is devoted to the stable module categories of group algebras of finite groups over algebraically closed fields of positive characteristic. A prominent role of thick subcategories of the considered triangulated categories as well as the support varieties of finitely generated modules over the group algebras of finite groups for the discussed classification problems is illuminated. Moreover, interesting applications of the authors classification of localizing subcategories are exhibited. The aim to the article by Keller is to give an introduction to quantum dilogarithm identities as well as explain connections to Fomin–Zelevinsky theory of cluster algebras. In the first part of the article Keller explains Reineke’s identities between products of quantum dilogarithm series associated with Dynkin quiver, extending the dilogarithm identities established for two quantum variables by Schützenberger, Faddeev–Volkov, and Faddeev–Kashaev. The second part of the article is devoted to similar quantum dilogarithm identifies for quivers with potential, following ideas due to Bridgeland, Fock–Goncharov, Kontsevich–Soilbelman and Nagao. Moreover, a prominent role of stability functions, Hall algebras and Jacobian algebras for proving the considered dilogarithm identities is exhibited. The article by Leclerc explains connections between quantum loop algebras of simple complex Lie algebras, Nakajima quiver varieties and cluster algebras of Fomin– Zelevinsky. In particular, an introduction to finite dimensional representations of quantum loop algebras and Nakajima’s geometric description of the irreducible q-characters in terms of graded quiver varieties is provided. The final part of the article is devoted to a recent attempt to understand the tensor structure of the category of finite dimensional representations of the quantum loop algebra of an arbitrary simple complex Lie algebra by means of cluster algebras. Here a general conjecture by Hernandez and Leclerc as well as well as its solution in a special case are discussed. The article by Lenzing serves as a guide to the theory of coherent sheaves over weighted projective lines and applications, since their introduction in 1987 by Geigle

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and the author to recent developments. In particular, a recent application of the theory to the analysis of the singularity category of triangle singularities, Kleinian and Fuchsian singularities is covered in some details. A prominent role of the stable categories over vector bundles over weighted projective lines in this analysis is exhibited. The author outlines the results of his recent joint work with Kussin, Meltzer and de la Peña as well as connections with the work by Buchweitz, Orlov, and the recent work by Kajiura, Saito and Takahashi. Further, an application of presented methods to the study of invariant subspace problem for nilpotent operators initiated by Ringel and Schmidmeier is also presented. The main general question discussed in the article by Linckelmann is a description of finite dimensional algebras which occur as indecomposable direct factors (blocks) of group algebras of finite groups over an algebraically closed field of positive characteristic p. The second main question discussed in the article is to what extent the blocks of finite group algebras are determined by their defect groups and fusion systems. The author provides an introduction to the basic theory of blocks of finite group algebras as well as surveys old and recent results concerning the two questions raised above, invoking the block cohomology and Hochschild cohomology of blocks. In particular, prominent conjectures in block theory (finiteness conjectures, counting conjectures, structural conjectures) and their partial confirmations are presented and discussed. The article by Malicki and Skowroński surveys old and new results on the structure and homological properties of Artin algebras whose Auslander–Reiten quiver admits a separating family of connected components. In the article, many important results on the structure of module categories of distinguished classes of Artin algebras (tilted algebras, quasitilted algebras, double tilted algebras, generalized double tilted algebras, generalized multicoil algebras) with separating families of Auslander–Reiten components, established during the last 30 years, as well as illustrating examples are presented in details. In the final part of the article, a complete description of the module categories of arbitrary Artin algebras having separating families of Auslander–Reiten components (equivalently, module categories with hearts) is presented. In particular, the generically tame Artin algebras with separating families of Auslander–Reiten components as well as their module categories are described completely. The article by Mori discusses classification problems in the noncommutative algebraic geometry and their strong connections with classification problems of homologically nice finite dimensional algebras over a field. The author starts with two major achievements in the noncommutative algebraic geometry: the classification of quantum projective planes and the classification of noncommutative projective curves. Then an interesting new nice class of finite dimensional algebras of finite global dimension, called quasi-Fano algebras, is introduced and investigated. Further, generalizations of Artin–Schelter regular algebras and the structure of Artin–Schelter Gorenstein algebras are exhibited. Moreover, interesting interactions between the classification of Artin– Schelter regular algebras and that of quasi-Fano algebras as well as the classification of Artin–Schelter Koszul algebras and that of Frobenius Koszul algebras (via Koszul duality) are explained. Applications of algebraic geometry techniques to classification problems of Fano algebras and graded Frobenius algebras are also presented.

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In the research article by Nakanishi two kinds of periodicities of mutations of general cluster algebras and their connections with systems of algebraic relations called T -systems and Y -systems as well as dilogarithm identities are discussed. The first one is the periodicity of exchange matrices (or quivers) under a sequence of mutations. The second one is the (stronger) periodicity of seeds under a sequence of mutations. For any sequence of mutations under which exchange matrices are periodic, Nakanishi defines the associated T -systems and Y -systems, which for regular sequences of mutations coincide with the known classical T -systems and Y -systems. Furthermore, for any sequence of mutations under which seeds are periodic, the associated dilogarithm identity is formulated. The author proves these dilogarithm identities when the exchange matrices are skew symmetric. The article by de la Peña and Skowroński surveys old and new results on the properties of the Tits quadratic forms of tame finite dimensional algebras over an algebraically closed field. One of the main aims of the article is to outline the crucial ingredients of the proof of the recent result by Brüstle, de la Peña and Skowroński asserting that the tameness of a strongly simply connected algebra is equivalent to the weak nonnegativity of the associated Tits quadratic form. In the article several important applications of this result are exhibited. Furthermore, criteria for a strongly simply connected algebra to be representation-finite, of finite growth, of polynomial growth are presented. The authors present also several results on the values of the Tits forms, as well as the related Euler forms, on the dimension vectors of finite dimensional indecomposable modules over tame strongly simply connected algebras. Finally, the realization of positive roots of the Tits forms of tame algebras as the dimension vectors of indecomposable modules is discussed. The main aim of the research article by Ringel is to give a complete classification of the minimal representation-infinite special biserial algebras over an algebraically closed field and to describe the structure of module categories of these algebras. In particular, Ringel proves that the minimal representation-infinite special biserial algebras without nodes are cycle algebras, the barbell algebras with nonserial bars and the wind wheel algebras. Furthermore, it is shown in the article that a minimal representation-infinite algebra is special biserial if and only if its universal Galois covering is interval-finite, with free Galois group, and any its finite convex subcategory is representation-finite. An interesting new phenomena discovered by Ringel is that some minimal representationinfinite special biserial algebras (namely the barbell algebras) are not of polynomial growth. In the article by Simson a current overview on the representation theory of coalgebras over a field is presented. Simson discusses the concepts of tame comodule type, of discrete comodule type, of polynomial growth, and of wild comodule type for a wide class of coalgebras intensively investigated during the last decade. In particular, the author shows that the tame-wild dichotomy holds for a wide class of coalgebras of infinite dimension over an algebraically closed field, including the semiperfect coalgebras and the incidence coalgebras of intervally finite partially ordered sets. Furthermore, basic tools and techniques applied in the study of coalgebras and their comodule categories

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are presented. Characterizations of large classes of coalgebras of tame comodule type are also given. The article by Zwara surveys old and recent results on singularities of Zariski closures of orbits in module varieties of finite dimensional algebras (and representations of quivers) over an algebraically closed field under the actions of general linear groups via conjugations. One of the main aims of the article is to discuss the types of singularities which may occur in the orbit closures of module varieties and links with classical types of singularities occurring in algebraic geometry (Schubert varieties of flag varieties). Zwara presents in the article several tools (degenerations, transversal slices, desingularizations, hom-controlled exact functors) allowing to study the singularities of orbit closures of module varieties. The author’s results on singularities for degenerations of modules of codimension at most two are presented in details. Furthermore, the equations of orbit closures and generic singularities are discussed. The article contains many examples and open problems, motivating further study of orbit closures of modules and their singularities. It is our hope that the wide scope of the collection of articles in the book will give a panoramic view of some recent trends in the representation theory of algebras and its exciting interaction with cluster algebras and categories, representation theory of finite groups, commutative and noncommutative geometry, commutative algebra, homological algebra, quantum algebras, algebraic combinatorics, theoretical physics, topology, and representation theory of coalgebras. This interaction was responsible for much of enormous progress we have seen during the last three decades in representation theory of algebras. The articles are self-contained and addressed to researchers and graduate students in algebra as well as a broader mathematical community. The large number of open problems posed in the articles gives also a perspective of further research. We express our gratitude to all authors contributing in this book and the referees for their assistance. Particular thanks are due to Jerzy Białkowski for his computer help in proper edition of the articles. We also thank the European Mathematical Society Publishing House for publication of this collection of articles and Manfred Karbe and Irene Zimmermann for very kind cooperation.

Toruń and Tokyo, July 2011

Andrzej Skowroński and Kunio Yamagata Editors

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v On generalized cluster categories Claire Amiot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Module categories for finite group algebras David J. Benson, Srikanth B. Iyengar and Henning Krause . . . . . . . . . . . . . . . . . . . . . . 55 On cluster theory and quantum dilogarithm identities Bernhard Keller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Quantum loop algebras, quiver varieties, and cluster algebras Bernard Leclerc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Weighted projective lines and applications Helmut Lenzing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Cohomology of block algebras of finite groups Markus Linckelmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Algebras with separating Auslander–Reiten components Piotr Malicki and Andrzej Skowroński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Classification problems in noncommutative algebraic geometry and representation theory Izuru Mori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Periodicities in cluster algebras and dilogarithm identities Tomoki Nakanishi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 The Tits forms of tame algebras and their roots José Antonio de la Peña and Andrzej Skowroński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 The minimal representation-infinite algebras which are special biserial Claus Michael Ringel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

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Coalgebras of tame comodule type, comodule categories, and a tame-wild dichotomy problem Daniel Simson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 Singularities of orbit closures in module varieties Grzegorz Zwara . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

On generalized cluster categories Claire Amiot

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2-Calabi–Yau categories and cluster-tilting theory . . . . . . 3 From 3-Calabi–Yau DG-algebras to 2-Calabi–Yau categories 4 Stable categories as generalized cluster categories . . . . . . 5 On the Z-grading on the 3-preprojective algebra . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 2 8 17 32 38 48

Introduction In 2003, Marsh, Reineke and Zelevinsky attempted in [79] to understand Fomin– Zelevinsky cluster algebras (defined in [40]) in terms of representations of quivers. This article was immediately followed by the fundamental paper [24] of Buan, Marsh, Reineke, Reiten and Todorov which stated the definition of cluster categories. In this paper, the authors associate to each finite dimensional hereditary algebra a triangulated category endowed with a special set of objects called cluster-tilting. The combinatorics of these objects is closely related to the combinatorics of acyclic cluster algebras, and especially with the mutation of quivers. The same kind of phenomena appear naturally in the stable categories of modules over preprojective algebras of Dynkin type, and have been studied by Geiss, Leclerc and Schröer in [45], [47]. Actually most of the results on cluster-tilting theory of [24], [26], [27], [45], [47] have been proved in the more general setting of Homfinite, triangulated, 2-Calabi–Yau categories with cluster-tilting objects (see [70], [63], [22]). Since then, other categories with the above properties have been constructed and investigated. One finds stable subcategories of modules over a preprojective algebra of type Q associated with any element in the Coxeter group of Q (see [22], [23], [50], [51]); stable Cohen–Macaulay modules over isolated singularities (see [14], [60], [29]); and generalized cluster categories associated with finite dimensional algebras of global dimension at most two, or with Jacobi-finite quivers with potential (introduced in [2], [3]). The context of this survey is this last family of triangulated 2-Calabi– Yau categories: the generalized cluster categories. The aim here is first to give some motivation for enlarging the family of cluster categories (Sections 1 and 2). Then we will explain the general construction of these new cluster categories (Section 3), and link these new categories with the categories cited above (Section 4). Finally, we give

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C. Amiot

some applications in representation theory of the cluster categories associated with algebras of global dimension at most 2 (Section 5). Notation. k is an algebraically closed field. All categories considered in this paper are k-linear additive Krull–Schmidt categories. By Hom-finite categories, we mean, categories such that Hom.X; Y / is finite dimensional for any objects X and Y . We denote by D D Homk .; k/ the k-dual. The tensor products are over the field k when not specified. For an object T in a category C , we denote by add.T / the additive closure of T , that is, the smallest full subcategory of C containing T and stable under taking direct summands. All modules considered here are right modules. For a k-algebra A, we denote by Mod A the category of right modules and by mod A the category of finitely presented right A-modules. A quiver Q D .Q0 ; Q1 ; s; t / is given by a set of vertices Q0 , a set of arrows Q1 , a source map s W Q1 ! Q0 and a target map t W Q1 ! Q0 . We denote by ei ; i 2 Q0 the set of primitive idempotents of the path algebra kQ. By a Dynkin quiver, we mean a quiver whose underlying graph is of simply laced Dynkin type.

1 Motivation 1.1 Mutation of quivers. In the fundamental paper [40], Fomin and Zelevinsky introduced the notion of mutation of quivers as follows. Definition 1.1 (Fomin–Zelevinsky [40]). Let Q be a finite quiver without loops and oriented cycles of length 2 (2-cycles for short). Let i be a vertex of Q. The mutation of the quiver Q at the vertex i is a quiver denoted by i .Q/ and constructed from Q using the following rule: (M1) for any couple of arrows j ! i ! k, add an arrow j ! k; (M2) reverse the arrows incident with i ; (M3) remove a maximal collection of 2-cycles. This definition is one of the key steps in the definition of cluster algebras. Even if the initial motivation of Fomin and Zelevinsky to define cluster algebras was to give a combinatorial and algebraic setup to understand canonical basis and total positivity in algebraic groups (see [77], [78], [50]), these algebras have led to many applications in very different domains of mathematics: • representations of groups of surfaces (higher Teichmüller theory [37], [38], [39]); • discrete dynamical systems (Y-systems, integrable systems [43], [69]); • non commutative algebraic geometry (Donaldson–Thomas invariants [74], Calabi–Yau algebras [53]); • Poisson geometry [52]; • quiver and finite dimensional algebras representations.

3

On generalized cluster categories

This paper is focused on this last connection and on the new insights that cluster algebras have brought in representation theory. However, in order to make this survey not too long, we will not discuss cluster algebras and their precise links with representation theory, but we will concentrate on the link between certain categories and quiver mutation. We refer to [67], [25], [82] for nice overviews of categorifications of cluster algebras. Example 1.2.

1

@2= ===

2

3

/

2 ^= === /3 1

1

/

@2 1o

3

3

/

1

@2

/3:

One easily checks that the mutation at a given vertex is an involution. If i is a source (i.e. there are no arrows with target i ) or a sink (i.e. there are no arrows with source i ), the mutation i consists only in reversing arrows incident with i (step (M2)). Hence it coincides with the reflection introduced by Bernstein, Gelfand, and Ponomarov [17]. Therefore mutation of quivers can be seen as a generalization of reflections (which are just defined if i is a source or a sink). The BGP reflections have really nice applications in representation theory, and are closely related to tilting theory of hereditary algebras. For instance, combining the functorial interpretation of reflections functors by Brenner and Butler [20], the interpretation of tilting modules for derived categories by Happel (Theorem 1.6 of [56]) and its description of the derived categories of hereditary algebras (Corollary 4.8 of [56]), we obtain that the reflections characterize combinatorially derived equivalence between hereditary algebras. Theorem 1.3 (Happel). Let Q and Q0 be two acyclic quivers. Then the algebras kQ and kQ0 are derived equivalent if and only if one can pass from Q to Q0 using a finite sequence of reflections. One hopes therefore that the notion of quiver mutation has also nice applications and meaning in representation theory. In the rest of the section, we give two fundamental examples of categories where the combinatorics of the quiver mutation of 1 ! 2 ! 3 appear naturally. These categories come from representation theory, they are very close to categories of finite dimensional modules over finite dimensional algebras.

1.2 Cluster category of type A3 . Let Q be the following quiver 1 ! 2 ! 3. We consider the category mod kQ of finite dimensional (right) modules over the path algebra kQ. We refer to the books [12], [15], [44], [84] for a wealth of information on the representation theory of quivers and finite dimensional algebras. Since A3 is a Dynkin quiver, this category has finitely many indecomposable modules. The

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Auslander–Reiten quiver is the following: 3 2 1 2 1

1

3 2

2

3

3

Here the simples associated with the vertices are symbolized by 1 , 2 , 3 . The module 2 1 is an indecomposable module M such that there exists a filtration M1 M2 M3 D M with M1 ' 1 , M2 =M1 ' 2 and M3 =M2 ' 3 . In this situation, it determines the module M up to isomorphism. There are three projective indecomposable objects corresponding to the vertices of Q and which are P1 D e1 kQ D 1 , P2 D e2 kQ D 21 3

and P3 D e3 kQ D 2 . The dotted lines describe the AR-translation that we denote 1 by ; it induces a bijection between indecomposable non projective kA3 -modules and indecomposable non injective kA3 -modules. Dotted lines also correspond to minimal relations in the AR-quiver of mod kQ. The bounded derived category D b .A/, where A is a finite dimensional algebra of finite global dimension, is a k-category whose objects are bounded complexes of finite dimensional (right) A-modules. Its morphisms are obtained from morphisms of complexes by inverting formally quasi-isomorphisms. We refer to [57] (see also [66]) for more precise description. For an object M 2 D b .A/ we denote by M Œ1 the shift complex defined by M Œ1n ´ M nC1 and dM Œ1 D dM . This category is a triangulated category, with suspension functor M ! M Œ1 corresponding to the shift. The functor mod A ! D b .A/ which sends a A-module M on the complex ! 0 ! M ! 0 concentrated in degree 0 is fully faithful. Moreover for any M and N in mod.A/ and i 2 Z we have i HomD b .A/ .M; N Œi / ' ExtA .M; N /:

L N The category D b .A/ has a Serre functor S D A DA (which sends the projective A-module ei A on the injective A-module Ii D ei DA) and an AR-translation D L N A DAŒ1 which extends the AR-translation of mod A and that we still denote by .

In the case A D kQ, the indecomposable objects are isomorphic to stalk complexes, that is, are of the form X Œi , where i 2 Z and X is an indecomposable kQ-module.

5


The AR-quiver of D b .kQ/ is the following infinite quiver (cf. [56]): 2 [−1]

S2 ( 3 )

3 [−1] 2

3 2 [−1] 1

3 [−1]

3 1

S2 ( 21 [1])

S2 ( 1 [1])

3 2 1

S2 ( 2 [1])

2 1

1

1 [1]

3 2

2

2 [1]

S−1 2 (1)

2 [1] 1

3 2 [1] 1

3

The functor S2 ´ SŒ2 D Œ1 acts bijectively on the indecomposable objects of D b .kQ/. It is an auto-equivalence of D b .kQ/. The cluster category CQ is defined to be the orbit category of D b .kQ/ by the functor S2 : the indecomposable objects are the indecomposable objects of D b .kQ/. The space of morphisms between two objects in D b .kQ/ is given by M HomC .X; Y / D HomD .X; S2i Y /: i2Z

Hence the objects X and S2i X become isomorphic in CQ for any i 2 Z. We denote by .X / the S2 -orbit of an indecomposable X . One can see that there are finitely many indecomposable objects in CQ . They are of the form .X /, where X is an indecomposable kQ-module, or X ' P Œ1 where P is an indecomposable projective kQ-module. Hence the AR-quiver of the category CQ is the following:

3

3

( 2 [1])

( 2 )

1

( 21 ) ( 1 )

( 1 [1])

1

( 32 ) ( 2 )

( 1 ) ( 21 [1])

( 3 )

3

( 2 [1]) 1

where the two vertical lines are identified. Therefore one can view the category CQ as 3 the module category mod kQ with extra objects (. 1 Œ1/, . 21 Œ1/, . 2 Œ1/) and extra 1 morphisms: if M and N are kQ-modules, then HomC .M; N / ' HomkQ .M; N / ˚ Ext1kQ .M; N /:

Note that in the category CQ , since the functor S2 D SŒ2 D Œ1 is isomorphic to the identity, then the functors and Œ1 are isomorphic. We are interested in the

6

C. Amiot

objects of CQ which are rigid (i.e. satisfying HomC .X; X Œ1/ D 0), basic (i.e. with pairwise non-isomorphic direct summands), and maximal for these properties. An easy computation yields 14 maximal basic rigid objects which are as follows: 3

i .. 1 / ˚ . 21 / ˚ . 2 //; i D 0; : : : ; 5I .. 2 / ˚ i

. 21 /

˚

i .. 2 / ˚ . 32 / ˚ i .. 1 / ˚ . 3 / ˚

1 3 . 2 //; 1 3 . 2 //; 1 3 . 2 //; 1

i D 0; : : : ; 2I i D 0; : : : ; 2I i D 0; 1:

All these maximal rigid objects have the same number of indecomposable summands. The Gabriel quivers of their endomorphism algebras are the following: 3

/ o

/

for i .. 1 / ˚ . 21 / ˚ . 2 //; i D 0; : : : ; 5I /

for i .. 2 / ˚ . 21 / ˚ . 2 //; i D 0; : : : ; 2I

1 3 1 3

/ o

for i .. 2 / ˚ . 32 / ˚ . 2 //; i D 0; : : : ; 2I 1

? ?? ??? o

3

for i .. 1 / ˚ . 3 / ˚ . 2 //; i D 0; 1: 1

These objects satisfy a remarkable property: given a maximal basic rigid object, one can replace an indecomposable summand by another one in a unique way to find another maximal basic rigid object. This process can be understood as a mutation, and under this process the quiver of the endomorphism algebra changes according to the mutation rule. For instance, if we write the maximal rigid object together with its quiver we obtain

1

B

2 1

88 88

o 3 2 1

.2 1/ .3/

\8 / 888 8 / 1 3

.1/

o 3 2 1

.3 2/

/ 3 2

B3 o

o 3 2 1

3 .2/ 1 . 1 Œ1/

/ 3 2

B3 /

1 Œ1:

1.3 Stable module category of a preprojective algebra of type A3 . Let Q be the quiver 1 ! 2 ! 3. Then the preprojective algebra …2 .kQ/ (see [85]) is presented by the quiver a

1h

a

(

b

2h

b

(

3

with relations a a D 0; aa b b D 0; bb D 0:

7


It is a finite dimensional algebra. The projective indecomposable …2 .kQ/-modules 1 2 3 are (up to isomorphism) P1 D I3 D 2 , P2 D I2 D 3 1 and P3 D I1 D 2 . They are 3 2 1 also injective, since the algebra …2 .kQ/ is self-injective. The category mod …2 .kQ/ has finitely many indecomposable objects (up to isomorphism) and its AR-quiver is the following: 3 2 1 2 1

3

3

2

1

2

3

2

3 2

1

3

2

1

2 3

1

2

1

1 2

3

2 2

1

3

1 2 3

where the vertical dotted lines are identified. There are 14 different basic maximal rigid objects (that is, objects M satisfying Ext1…2 .kQ/ .M; M / D 0). They all have 6 direct summands, among which the three projective-injectives. Therefore, if we work in the stable category mod …2 .kQ/ (see [57]), where all morphisms factorizing through a projective-injective vanish, we obtain 14 maximal rigid objects with 3 direct summands (indeed the projective-injective objects become isomorphic to zero in mod …2 .kQ/). Moreover, the category mod …2 .kQ/ satisfies also the nice property that, given a maximal rigid object, one can replace any of its indecomposable summands by another one in a unique way to obtain another maximal rigid object. The category mod …2 .kQ/ has the same AR-quiver as the cluster category CQ , moreover if we look at the combinatorics of the quivers of the maximal rigid objects in mod …2 .kQ/, we obtain:

1

3 {= {{ { {

2

1C

CC CC !

o 2 1

.3 2 1/ / .1 2/

1

1 2

];; ;; /

.1/

o 2 1

.2/

/

A 2 o

1 2

o 2 1

.2 1/ .3 2/

/ 2

A

1 2

/

3 2

:

In conclusion, the categories CQ and mod …2 .kQ/ seem to be equivalent, and the combinatorics of their maximal rigid objects is closely related to the Fomin–Zelevinsky mutation of quivers. Our aim in this survey is to show how general these phenomena are, and to construct a large class of categories in which similar phenomena occur.

8

C. Amiot

2 2-Calabi–Yau categories and cluster-tilting theory The previous examples are not isolated. These two categories have the same properties, namely they are triangulated 2-Calabi–Yau categories, and have cluster-tilting objects (which in these cases are the same as maximal rigid). We will see in this section that most of the phenomena that appear in the two cases described in Section 1 still hold in the general setup of 2-Calabi–Yau categories with cluster-tilting objects. 2.1 Iyama–Yoshino mutation. A triangulated category which is Hom-finite is called d -Calabi–Yau (d -CY for short) if there is a bifunctorial isomorphism HomC .X; Y / ' D HomC .Y; X Œd /

where D D Homk .; k/ is the usual duality over k. Definition 2.1. Let C be a Hom-finite triangulated category. An object T 2 C is called cluster-tilting (or 2-cluster-tilting) if T is basic and if we have add.T / D fX 2 C j HomC .X; T Œ1/ D 0g D fX 2 C j HomC .T; X Œ1/ D 0g:

Note that a cluster-tilting object is maximal rigid (the converse is not always true cf. [29]), and that the second equality in the definition always holds when C is 2-Calabi– Yau. If there exists a cluster-tilting object in a 2-CY category C , then it is possible to construct others by a recursive process resumed in the following: Theorem 2.2 (Iyama–Yoshino, [63]). Let C be a Hom-finite 2-CY triangulated category with a cluster-tilting object T . Let Ti be an indecomposable direct summand of T ' Ti ˚ T0 . Then there exists a unique indecomposable Ti non isomorphic to Ti such that T0 ˚ Ti is cluster-tilting. Moreover Ti and Ti are linked by the existence of triangles Ti

u

/B

v

/ T i

w

/ Ti Œ1

and

Ti

u0

/ B0

v0

/ Ti

w0

/ T Œ1 i

where u and u0 are minimal left add.T0 /-approximations and v and v 0 are minimal right add.T0 /-approximations. These exchange triangles permit to mutate cluster-tilting objects and have been first described by Buan, Marsh, Reineke, Reiten and Todorov (see Proposition 6.9 in [24]) for cluster categories. The corresponding exchange short exact sequences in module categories over a preprojective algebra of Dynkin type appeared also in the work of Geiss, Leclerc and Schröer (see Lemma 5.1 in [45]). The general statement is due to Iyama and Yoshino [63], Theorem 5.3. This is why we decided to refer to this mutation as the IY-mutation of cluster-tilting objects in this article. This recursive process of mutation of cluster-tilting objects is closely related to the notion of mutation of quivers defined by Fomin and Zelevinsky [40] in the following sense.


9

Theorem 2.3 (Buan–Iyama–Reiten–Scott [22]). Let C be a Hom-finite 2-CY triangulated category with cluster-tilting object T . Let Ti be an indecomposable direct summand of T , and denote by T 0 the cluster-tilting object Ti .T /. Denote by QT (resp. QT 0 ) the Gabriel quiver of the endomorphism algebra End C .T / (resp. QT 0 ). Assume that there are no loops and no 2-cycles at the vertex i of QT (resp. QT 0 ) corresponding to the indecomposable Ti (resp. Ti ). Then we have QT 0 D i .QT /; where i is the Fomin–Zelevinsky quiver mutation. We illustrate this result by the following diagram T

cluster-tilting

QT o

o

IY-mutation

FZ-mutation

/

T0 cluster-tilting / QT 0 :

The corresponding results have been first shown in the setting of cluster categories in [26] and in the setting of preprojective algebras of Dynkin type in [45]. 2.2 2-Calabi–Yau-tilted algebras. The endomorphism algebras End C .T / where T is a cluster-tilting object are of importance, they are called 2-CY tilted algebras (or cluster-tilted algebras if C is a cluster category). The following result says that a 2CY category C with cluster-tilting objects is very close to a module category over a 2-CY-tilted algebra. Such a 2-CY category can be seen as a “recollement” of module categories over 2-CY tilted algebras. Proposition 2.4 (Keller–Reiten [70]). Let C be a 2-CY triangulated category with a cluster-tilting object T . Then the functor FT D HomC .T; / W C ! mod End C .T / induces an equivalence C = add.T Œ1/ ' mod End C .T /. If the objects T and T 0 are linked by an IY-mutation, then the categories mod End C .T / and mod End C .T 0 / are nearly Morita equivalent, that is, there exists a simple End C .T /-module S and a simple End C .T 0 /-module S 0 , and an equivalence of categories mod End C .T /= add.S / ' mod End C .T 0 /= add.S 0 /. These results have been first studied in [26] for cluster categories. The 2-CY-tilted algebras are Gorenstein (i.e. End C .T / has finite injective dimension as a module over itself) and are either of infinite global dimension, or hereditary (see [70]). Other nice properties of the functor HomC .T; / W C ! mod End C .T / are studied in [73].

10

C. Amiot

Example 2.5. Let Q be the quiver 1 ! 2 ! 3. The category CQ = add.T Œ1/ where 3 T D . 1 / ˚ . 21 / ˚ . 2 / has the same AR-quiver as mod kQ. (Note that kQ is an 1

3

hereditary algebra.) If we take T D . 1 / ˚ . 3 / ˚ . 2 /, then the associated 2-CY1

^>>b > with relations ab D bc D ca D 0. titled algebra is given by the quiver c /3 1 It is an algebra of infinite global dimension. The module category of this algebra has finitely many indecomposables and its AR-quiver is of the following form: a

2

2 1

1

1 3

2

1 3

3

3 2

It is the same quiver as the AR-quiver of CQ = add.T Œ1/.

3

3

( 2 [1])

( 2 )

1

( 21 ) ( 1 )

( 1 [1])

1

( 32 )

( 2 ) = ( 3 [1])

( 1 ) ( 21 [1])

( 3 )

3

( 2 [1]) 1

2.3 Two fundamental families of examples. The two examples studied in Section 1 are part of two fundamental families of 2-CY categories with cluster-tilting objects. We give here the general construction of these families. The acyclic cluster category. The first one is given by the cluster category CQ associated with an acyclic quiver Q. This category was first defined in [24], following the


11

decorated representations of acyclic quivers introduced in [79]. The acyclic cluster category is defined in [24] as the orbit category of the bounded derived category D b .kQ/ of finite dimensional modules over kQ by the autoequivalence S2 D SŒ2 ' Œ1, L N where S is the Serre functor kQ DkQ, and is the AR-translation of D b .kQ/. The objects of this category are the same as those of D b .kQ/ and the spaces of morphisms are given by M HomCQ .X; Y / ´ HomD b .kQ/ .X; S2i Y /: i2Z

The canonical functor W D .kQ/ ! CQ satisfies B S2 ' . Moreover the k-category CQ satisfies the property that, for any k-category T and functor F W D b .kQ/ ! T with F B S2 ' F , the functor F factors through . b

D b .kQ/ F II II I I$ S2 : CQ uuu u uu F b D .kQ/

$ /T :

L For any X and Y in D b .kQ/, the infinite sum i2Z HomD .X; S2i Y / has finitely many non zero summands, hence the category CQ is Hom-finite. Moreover the shift functor, and the Serre functor of D b .kQ/ yield a shift functor and a Serre functor for CQ . The acyclic cluster category CQ satisfies the 2-CY property by construction since the functor S2 D SŒ2 becomes isomorphic to the identity in the orbit category D b .kQ/=S2 . Furthermore, the category CQ has even more structure as shown in the following fundamental result. Theorem 2.6 (Keller [65]). Let Q be an acyclic quiver. The acyclic cluster category has a natural structure of triangulated category making the functor W D b .kQ/ ! CQ a triangle functor. The triangles in D b .kQ/ yield natural candidates for triangles in CQ , but checking that they satisfy the axioms of triangulated categories (see e.g. [57]) is a difficult task. A direct proof (by axioms checking) that CQ is triangulated is especially difficult for axiom .TR1/: Given a morphism in CQ , how to define its cone if it is not liftable into a morphism in D b .kQ/? Keller obtains Theorem 2.6 using different techniques. He first embeds CQ in a triangulated category (called triangulated hull), and then shows that this embedding is dense. Link with tilting theory. If T is a tilting kQ-module, then .T / is a cluster-tilting object as shown in [24]. Moreover the mutation of the cluster-tilting objects is closely related to mutation of tilting kQ-modules in the following sense (see [24]): If

12

C. Amiot

T ' Ti ˚ T0 is a tilting kQ-module with Ti indecomposable and if there exists an indecomposable module Ti such that T0 ˚ Ti is a tilting module, then .Ti ˚ T0 / is the IY-mutation of .T / at .Ti /. The exchange triangles are the images of exchange sequences in the module category. Furthermore, the 2-CY-tilted algebra End C ..T // (called cluster-tilted in this case) is isomorphic to the trivial extension of the algebra 2 B D EndkQ .T / by the B-B-bimodule ExtB .D.B/; B/ as shown in [11]. An advantage of cluster-tilting theory over tilting theory is that in cluster-tilting theory it is always possible to mutate. In a hereditary module category, an almost complete tilting module does not always have 2 complements (cf. [58], [83]). Using strong results of tilting theory due to Happel and Unger [59], one has the following property. Theorem 2.7 ([24], Proposition 3.5). Let Q be an acyclic quiver. Any cluster-tilting objects T and T 0 in CQ are IY-mutation-equivalent, that is, one can pass from T to T 0 using a finite sequence of IY-mutations. This fact is unfortunately not known to be true or false in general 2-CY categories (see also Remark 3.15). The category CQ provides a first and very important example which permits to categorify the acyclic cluster algebras. We refer to [24], [30], [26], [28], [33], [34] for results on the categorification of acyclic cluster algebras. Detailed overviews of this subject can be found in [25], [82] and in the first sections of [67]. Remark 2.8. The construction of CQ can be generalized to any hereditary category. Therefore, it is possible to consider CH D D b .H /=S2 , where H D cohX, and X is a weighted projective line. This orbit category is still triangulated by [65] (the generalization from acyclic quivers to hereditary categories was first suggested by Asashiba). These cluster categories are studied in details by Barot, Kussin and Lenzing in [16]. Recognition theorem for acyclic cluster categories. The following result due to Keller and Reiten ensures that the acyclic cluster categories are the only categories satisfying the properties needed to categorify acyclic cluster algebras. Theorem 2.9 (Keller–Reiten [71]). Let C be an algebraic triangulated 2-CY category with a cluster-tilting object T . If the quiver of the endomorphism algebra End C .T / is acyclic, then there exists a triangle equivalence C ' CQ . Note that this result implies in particular that the cluster-tilting objects of the category C are all mutation equivalent. The equivalence of the two categories described in Sections 1.2 and 1.3 is a consequence of Theorem 2.9. Geometric description of the acyclic cluster category. Another description of the cluster category has been given independently by Caldero, Chapoton and Schiffler in [31], [32] in the case where the quiver Q is an orientation of the graph An . The description is geometric: in this situation, indecomposable objects in CQ correspond

13


to homotopy classes of arcs joining two non consecutive vertices of an .n C 3/-gon, and cluster-tilting objects correspond to ideal triangulations of the .n C 3/-gon. One can associate a quiver to any ideal triangulation : the vertices are in bijection with the inner arcs of , and two vertices are linked by an arrow if and only if the two corresponding arcs are part of the same triangle. Mutation of cluster-tilting objects corresponds to the flip of an arc in an ideal triangulation. A similar description of CQ zn. exists in the case where Q is some (acyclic) orientation of the graphs Dn , Azn and D In these cases, the surface with marked points is respectively an n-gon with a puncture, an annulus without puncture, and an n-gon with two punctures (see [86], [88], [87]). Q T D Q o B O

FZ-mutation

O

flip

ideal triang.

o

/ Q 0 D Q T 0 O \

O

/

ideal triang.

1W1

1W1

T

cluster-tilting

o

/

IY-mutation

T0

cluster-tilting

Example 2.10. Here is the correspondence between cluster-tilting objects of the example in Section 1.2 and ideal triangulations of the hexagon. 2 1 3 2 1

1

1

2

3

2

3

3 2 1

1

1

3 2

3

1

3 2 1

3 2

1 2

3

3

3

3 2

1 [1]

3 1 2

Preprojective algebras of Dynkin type. A very different approach is given by module categories over preprojective algebras. Let be a finite graph, and Q be an acyclic x is obtained from Q by adding to each arrow orientation of . The double quiver Q a W i ! j of Q an arrow a W j ! i . The preprojective algebra …2 .kQ/ P is then defined to be the quotient of the path algebra kQ by the ideal of relations a2Q1 aa a a. This algebra is finite dimensional if and only if Q is of Dynkin type, and its module category is closely related to the theory of the Lie algebra of type Q. When Q is Dynkin, the algebra …2 .kQ/ is selfinjective and the stable category mod …2 .kQ/ is triangulated (see [57]) and 2-Calabi–Yau. Moreover it does not depend on the choice

14

C. Amiot

of the orientation of Q, so we denote it by …2 ./. In their work [45], [46] about cluster algebras arising in Lie theory, Geiss, Leclerc and Schröer have constructed special cluster-tilting objects in the module category mod …2 ./, which helped them to categorify certain cluster algebras. Further developments of this link between cluster algebras and preprojective algebras can be found in [47], [48], [50], [51] (see also [49] for an overview). Theorem 2.11 (Geiss–Leclerc–Schröer [46]). Let be a simply laced Dynkin diagram. Then for any orientation Q of , there exists a cluster-tilting object TQ in the category mod …2 ./ so that the Gabriel quiver of the cluster-tilting object TQ is obtained from the AR quiver of mod kQ by adding arrows corresponding to the AR-translation. Example 2.12. In the example of the first section with D A3 , with the three orientations 1 ! 2 ! 3, 1 2 ! 3 and 1 ! 2 3 of , we obtain respectively the following three cluster-tilting objects 3 2 1 2 1 1

3 1 2

3 2 1

3 2 2 2

2

1 1 2 3

3 1 2

2 2

3 2 1

3

1

1 1 2 3

1

2

1

3

2 2

3

1 2 3

Correspondence. The link between cluster categories and stable module categories of preprojective algebras of Dynkin type is given in the following. Theorem 2.13. Let be a simply laced Dynkin diagram and Q be an acyclic quiver. There is a triangle equivalence mod …2 ./ ' CQ if and only if we are in one of the cases D A2 and Q is of type A1 , D A3 and Q is of type A3 , D A4 and Q is of type D6 . Remark 2.14. In the case D A5 , the category mod …2 .A5 / is equivalent to the cluster category CH where H is of tubular type E8.1;1/ . Even though the categories mod …2 ./ and CQ are constructed in a very distinct way, this theorem shows that these categories are not so different in a certain sense. This observation allows us to ask the following. Question 2.15. Is it possible to generalize the constructions above so that the 2-Calabi– Yau categories mod …2 ./ and CQ are part of the same family of categories? There are two different approaches to this problem. One consists in generalizing the construction mod …2 ./ and the other consists in generalizing the construction of the cluster category. A construction of the first type is given in [47] and in [22] (see Section 4.2) and a construction of the second type is given in [2] and [3] (see Section 3).

15


2.4 Quivers with potential. Theorem 2.3 links the quivers of the 2-CY-tilted algebras appearing in the same IY-mutation class of cluster-tilting object. This leads to the natural question: Is there a combinatorial way to deduce the relations of the 2-CY-tilted algebra after mutation? This question (among others) brought Derksen, Weyman and Zelevinsky to introduce in [36] the notions of quiver with potential (QP for short), Jacobian algebra and mutation of QP.

3

ŒkQ; kQ where Definition 2.16. A potential W on a quiver Q is an element in kQ= is the completion of the path algebra kQ for the J -adic topology (J being the ideal kQ generated by of kQ generated by the arrows) where ŒkQ; kQ is the subspace of kQ the commutators of the algebra kQ.

3

In other words, a potential is a (possibly infinite) linear combination of cycles of Q, up to cyclic equivalence (a1 a2 : : : an a2 a3 : : : an a1 ).

3

ŒkQ; kQ ! kQ Definition 2.17. Let Q be a quiver. The partial derivative @ W kQ= is defined to be the unique continuous linear map which sends the class of a path p P to the sum pDuav vu taken over all decompositions of the path p. Let .Q; W / be a quiver with potential. The Jacobian algebra of .Q; W / is defined to be Jac.Q; W / ´ kQ=h@ a W; a 2 Q1 i, where h@a W; a 2 Q1 i is the ideal of kQ generated by @a W for all a 2 Q1 . A QP .Q; W / is Jacobi-finite if its Jacobian algebra is finite dimensional. In [36] the authors introduced the notion of reduction of a QP. The reduction of .Q; W / consists in finding a QP .Qr ; W r / whose key properties are that the Jacobian algebras Jac.Q; W / and Jac.Qr ; W r / are isomorphic and W r has no 2-cycles as summands. In the case of a Jacobi-finite QP, the quiver Qr is the Gabriel quiver of the Jacobian algebra and is then uniquely determined. The potential W r is not uniquely determined. The notion of reduction is defined up to an equivalence relation of QPs called right equivalence (see [36]). Derksen, Weyman and Zelevinsky also refined the notion of the FZ-mutation of a quiver into the notion of mutation of a QP (that we will call DWZ-mutation) at a vertex of Q without loop. Steps (M1) and (M2) are the same as in FZ-mutation. The new potential is defined to be a sum ŒW C W , constructed from W using the new arrows. Step (M30 ) consists in reducing the QP we obtained. For generic QPs, step (M30 ) of the DWZ-mutation coincides with step (M3) of the FZ-mutation when restricted to the quiver. Let us illustrate this process in an example. Example 2.18. Let us define .Q; W / as follows: 2 @ ===b == c /3 Q D 1 ^= == = e = d 4 a

with W D edc:

16

C. Amiot

The Jacobian ideal is defined by the relations ed D ce D dc D 0. One easily checks that .Q; W / is Jacobi-finite. Let us mutate .Q; W / at the vertex 3. After steps (M1) and (M2) we obtain the QP .Q0 ; W 0 / defined as follows:

a

@ 2 ^>> >> >>b >> Œdb >>

Q0 D 1 ^>o >> c >>>> >>>>Œdc > >> e >>> > >> 4

@3

with W 0 D eŒdc C d Œdcc C d Œdbb :

d

After Step (M30 ) we obtain the QP: @ 2 ^= ===ˇ == = .Q0 /r D 1 o 3 @ ı 4 a

with .W 0 /r D ıˇ:

The map kQ0 ! k.Q0 /r sending a, b , c , d e, Œdc, Œdb on respectively ˛, ˇ, , ı, ı, 0, , sends W 0 on .W 0 /r and induces an isomorphism between Jac.Q0 ; W 0 / and Jac..Q0 /r ; .W 0 /r /. The mutation of .Q; W / at 3 is defined to be ..Q0 /r ; .W 0 /r / (up to right equivalence). If there do not occur 2-cycles at any iterate mutation of .Q; W /, then the potential is non-degenerate [36] and the DWZ-mutation (when restricted to the quiver) coincides with the FZ-mutation of quivers. This happens in particular when the potential is rigid, that is, when all cycles of Q are cyclically equivalent to an element in the Jacobian ideal h@a W; a 2 Q1 i. This notion of rigidity is stable under mutation [36]. Moreover the following theorem gives an answer to the previous question in the case where the 2-CY-tilted algebra is Jacobian. Theorem 2.19 (Buan–Iyama–Reiten–Smith [23]). Let T be a cluster-tilting object in an Hom-finite 2-CY category C and let Ti be an indecomposable direct summand of T . Assume that there is an algebra isomorphism End C .T / ' Jac.Q; W /, for some QP .Q; W / and assume that there are no loops nor 2-cycles at vertex i (corresponding to Ti ) in the quiver of End C .T / and a technical assumption called glueing condition. Then there is an isomorphism of algebras Jac.i .Q; W // ' End C .Ti .T //:

17


T _

cluster-tilting

o

IY-mutation

T0 cluster-tilting _

End C .T 0 / ' Jac.Q0 ; W 0 / O

End C .T / ' Jac.Q; W / O _ .Q; W / o

/

DWZ-mutation

_ / .Q0 ; W 0 /

This result applies in particular for acyclic cluster categories. If Q is an acyclic quiver, the endomorphism algebra of the canonical cluster-tilting object .kQ/ 2 CQ is isomorphic to the Jacobian algebra Jac.Q; 0/ and 0 is a rigid potential for the quiver Q. Moreover the categories CQ satisfy the glueing condition [23]. Therefore, combining Theorems 2.19 and 2.7, all cluster-tilted algebras are Jacobian algebras associated with a rigid QP and can be deduced one from each other by mutation. The endomorphism algebras of the canonical cluster-tilting object of the category mod …2 ./, where is a Dynkin graph, are also Jacobian algebras with rigid QP and satisfy the glueing condition ([23], Theorem 6.5). The two natural questions are now the following: Question 2.20. 1. Are all 2-CY-tilted algebras Jacobian algebras? 2. Are all Jacobian algebras 2-CY-tilted algebras? The first question is still open, but the answer is expected to be negative by recent results of Davison [35] and Van den Bergh [92], which pointed out the existence of algebras which are bimodule 3-CY but which do not come from a potential. The second one has a positive answer in the Jacobi-finite case (see Section 3.3). Derksen Weyman and Zelevinsky also describe the mutation of (decorated) representation of Jacobian algebras in [36]. Their aim was to generalize the construction of mutations of decorated representations [79]. Given a Jac.Q; W /-module M , they associate a Jac.i .Q; W //-module i .M /. This mutation preserves indecomposability.

3 From 3-Calabi–Yau DG-algebras to 2-Calabi–Yau categories 3.1 Graded algebras and DG algebras. In this section, we recall some definitions concerning differential graded algebras, differential graded modules, and derived categories. We refer to [64] for definitions and properties on differential graded algebras and associated triangulated categories. Graded algebras. We denote by Gr k the categoryLof graded k-vector spaces. ReL call that given two Z-graded vector spaces M D p2Z ML N p and N D p2Z p , form f D .f / where fp 2 a morphism f 2 HomGr k .M; N / is of the p p2Z L Homk .Mp ; Np / for any p 2 Z. Let A D A be a Z-graded algebra. A Zi2Z i L graded A-module M is a Z-graded k-vector space M D i2Z Mi with a morphism

18

C. Amiot

L of Z-graded algebras A ! p2Z HomGr k .M; M.p//, where M.p/ is the Z-graded k-module such that M.p/i D MpCi for any i 2 Z. In other words, for any n; p 2 Z there is a k-linear map Mn Ap ! MnCp sending .m; a/ to m:a such that m:1 D m for all m 2 M and such that the following diagram commutes: Mn Ap Aq

/ Mn ApCq

MnCp Aq

/ MnCpCq ;

where the maps are induced by the multiplication in A for the first row and by the action of A on M respectively for the others. Then the degree shift M.1/ of a graded module M is still a graded A-module. A morphism of graded A-modules is a morphism f W M ! L N homogeneous of degree 0, that is, f D n2Z fn where fn 2 Homk .Mn ; Nn / and which commutes with the action of A, that is, for any a 2 Ap there is a commutative diagram fn

Mn :a

MnCp

fnCp

/ Nn

:a

/ NnCp :

The category of graded A-modules Gr A is an abelian category. Therefore we can define the derived category of graded A-modules D.Gr A/ as usual (see [66]). For a complex of graded A-modules, we denote by M Œ1 the complex such that M Œ1n D M nC1 and dM Œ1 D dM . Definition 3.1. Let d 2. We say that A is bimodule d -Calabi–Yau of Gorenstein parameter 1 if there exists an isomorphism RHomAe .A; Ae /Œd .1/ ' A

in D.Gr Ae /

where Ae is the graded algebra Aop ˝ A. DG algebras Definition 3.2. AL differential graded algebra (= DG algebra for short) A is a Z-graded n k-algebra A D n2Z A with a differential dA , that is, a k-endomorphism of A homogeneous of degree 1 satisfying the Leibniz rule: for any a 2 Ap and any b 2 A we have dA .ab/ D dA .a/b C .1/p adA .b/. L n A DG A-module M is a graded A-module M D n2Z M , endowed with a differential dM , that is, dM 2 HomGr A .M; M.1// such that dM B dM D 0. Moreover the differential dM of the complex M is compatible with the differential of A in the following sense: dM .m:a/ D m:dA .a/ C .1/n dM .m/:a

for all m 2 M n , a 2 Ap I


M n ApQQ QQQ Q( .1;dA /

M n ApC1

.dM ;1/

19

/ M nC1 Ap

M nCp QQdM QQQ ( / M nCpC1 .

Note that this equality for p D 0 implies that M has a structure of complex of Z 0 .A/modules. A morphism of DG A-modules is a morphism of graded A-modules which is a morphism of complexes. Note that if we endow the graded A-module M.1/ with the differential dM.1/ D dM , it is a DG A-module and there is a canonical isomorphism of DG A-modules M.1/ ' M Œ1. The derived category D.A/ is defined as follows. The objects are DG A-modules, and morphisms are equivalence classes of diagrams s 1 f W M D DDD " f

N0

{ }{{s

N

where f is a morphism of DG A-modules, and s is a morphism of DG A-modules such that for any n 2 Z, the morphism H n .s/ W H n .N / ! H n .N 0 / is an isomorphism of H 0 .A/-module (s is a quasi-isomorphism). This is a triangulated category. We denote by per A the thick subcategory (= the smallest triangulated category stable under direct summands) generated by A. We denote by D b .A/ the subcategory of D.A/ whose objects are the DG A-modules with finite dimensional total cohomology. Remark 3.3. If A is a k-algebra, we can view it as a DG algebra concentrated in degree 0, and with differential 0. Then we recover the usual derived categories D.A/, per A and D b .A/. Definition 3.4. A DG algebra is bimodule d -Calabi–Yau, if there exists an isomorphism RHomAe .A; Ae / ' AŒd in D.Ae / where Ae is the DG algebra Aop ˝ A. The next proposition says that the bimodule d -Calabi–Yau property for A implies the d -CY property for the bounded derived category D b .A/. Proposition 3.5 ([53, 67]). If A is a DG algebra which is bimodule d -Calabi–Yau, then there exists a functorial isomorphism HomDA .X; Y / ' D HomDA .Y; X Œd /;

for any X 2 DA and Y 2 D b .A/:

Remark 3.6. If A is a graded algebra, we can view it as a DG algebra with differential 0. Then if L X is a complex of graded L A-modules, for any n 2 Z, the differential dX W X n D i2Z Xin ! X nC1 D i2Z XinC1 is homogeneous of degree 0, it goes L ni from Xin to XinC1 . If we set M n ´ , then the complex .M; dX / has i2Z Xi

20

C. Amiot

naturally a structure of DG A-module. This yields a canonical functor D.Gr A/ ! DA, which induces a fully faithful functor per Gr A=.1/ ! per A. The category per.A/ is generated, as a triangulated category, by the image of the orbit category per.Gr A/=.1/ through this functor. More precisely, per A is the triangulated hull of the orbit category per.Gr A/=.1/. If the graded algebra A is bimodule d -Calabi–Yau of Gorenstein parameter 1, then the DG algebra A endowed with the zero differential is bimodule d -Calabi–Yau. 3.2 General construction. The next result is the main step in the construction of new 2-CY categories with cluster-tilting object which generalize the acyclic cluster categories. Theorem 3.7 ([3], Theorem 2.1). Let … be a DG-algebra with the following properties: (a) (b) (c) (d)

… is homologically smooth (i.e. … 2 per.…e /), H p .…/ D 0 for all p 1, H 0 .…/ is finite dimensional as k-vector space, … is bimodule 3-CY.

Then the triangulated category C .…/ D per …=D b .…/ is Hom-finite, 2-CY and the object … is a cluster-tilting object with End C .…/ ' H 0 .…/. Let us make a few comments on the hypotheses and on the proof of this theorem. Hypothesis (a) implies that D b .…/ per … ([67]), so that the category C.…/ D per …=D b .…/ exists. Hypothesis (b) implies the existence of a natural t -structure (coming from the usual truncation) on per …, which is an essential ingredient for the proof. Moreover, hypotheses (c) and (d) imply that H p .…/ is finite dimensional over k for all p 2 Z. Thus for any X 2 per …, there is a triangle n X

/X

/ >n X

/ n X Œ1,

where >n X is in D b .…/. Moreover we have HomC .X; Y / ' lim Homper … .n X; n Y /; n!1

which implies the Hom-finiteness of the category C. Now we define a full subcategory F .…/ of per … by F .…/ D .per …/0 \ ..per …/2 /? ; where .per …/p (resp. .per …/p ) is the full subcategory of per … consisting of objects having their homology concentrated in degrees p (resp. p). An important step of the proof consists in showing that the composition / per … / per …=D b … D C .…/ F .…/ is an equivalence. Notice that the subcategory F .…/ is not stable under the shift functor. This equivalence implies in particular the following.


21

Proposition 3.8. Let … be as in Theorem 3.7. Then the following diagram is commutative: / C .…/ per … F .…/ QQQ r r QQQ r QQQ rrr rHom QQQ r r H0 C .…;/DF… ( xr 0 mod H .…/. 3.3 Ginzburg DG algebras. The general theorem above applies to Ginzburg DG algebras associated with a Jacobi-finite QPs. Definition 3.9 (Ginzburg [53]). Let .Q; W / be a QP. Let QG be the graded quiver with the same set of vertices as Q and whose arrows are: • the arrows of Q (of degree 0); • an arrow a W j ! i of degree 1 for each arrow a W i ! j of Q; • a loop ti W i ! i of degree 2 for each vertex i 2 Q0 . y The completed Ginzburg DG algebra .Q; W / is the DG algebra whose underlying graded algebra is the completion (for the J -adic topology) of the graded path algebra y kQG . The differential of .Q; W / is the unique continuous linear endomorphism homogeneous of degree 1 which satisfies the Leibniz rule (i.e. d.uv/ D .du/v C .1/p udv for all homogeneous u of degree p and all v), and takes the following values on the arrows of QG : d.a/ D 0 and d.a / D @a W P d.ti / D ei Œa; a ei

for all a 2 Q1 ; for all i 2 Q0 .

a2Q1

y Theorem 3.10 (Keller [68]). The completed Ginzburg DG algebra .Q; W / is homologically smooth and bimodule 3-Calabi–Yau. y It is immediate to see that .Q; W / is non zero only in negative degrees, and that y H . .Q; W // ' Jac.Q; W /. Therefore by Theorem 3.7 we get the following. 0

Corollary 3.11. Let .Q; W / be a Jacobi-finite QP. Then the category y y W // W /=D b . .Q; C.Q;W / ´ per .Q; is Hom-finite, 2-Calabi–Yau, and has a canonical cluster-tilting object whose endomorphism algebra is isomorphic to Jac.Q; W /. This category C.Q;W / is called the cluster category associated with a QP. This corollary gives in particular an answer to Question 2.20 (2) in the Jacobi-finite case.

22

C. Amiot

Remark 3.12. If .Q; W / is not Jacobi-finite, a generalization of the category C.Q;W / , which is not Hom-finite, is constructed in [81]. Note that in a recent paper [91], Van den Bergh has shown that complete DG algebras in negative degrees which are bimodule 3-Calabi–Yau are quasi-isomorphic to deformations of Ginzburg DG algebras in the sense of [68]. Hence a complete DG algebra … as in Theorem 3.7 is a (deformation) of a Ginzburg DG algebra associated with a Jacobi-finite QP. The following two results give a link between Ginzburg DG algebras associated with QPs linked by mutation. Theorem 3.13. Let .Q; W / be a QP without loops and i 2 Q0 not on a 2-cycle in y y i .Q; W // the completed Ginzburg DG Q. Denote by ´ .Q; W / and 0 ´ . algebras. (a) ([72], [68]). There are triangle equivalences per O

? D b

/ per 0 O

/

? D b :

Hence we have a triangle equivalence C .Q; W / ' C .i .Q; W //. (b) ([81]). We have a diagram

per H0

mod Jac.Q; W / o

/ per 0 H0

DWZ-mutation for representations

/ mod Jac.i .Q; W //:

Combining (b) together with Proposition 3.8, we obtain that in the Jacobi-finite case, for any cluster-tilting object T 2 C.Q;W / which is IY-mutation equivalent to the canonical one, we have: IY-mutation

( C.Q;W / R 3 T0 RRR ll RRRFT FT 0 llll RRR ll RRR lll l R) l ul DWZ-mutation / mod End C .T 0 /: mod End C .T / o

T 2

v

(3.3.1)

for representations

It is not clear from the definition whether the cluster categories C.Q;W / satisfy the glueing condition of Theorem 2.19. However Theorem 3.13 (a) ensures that if .Q; W / is a non degenerate Jacobi-finite QP, then IY-mutation is compatible with DWZ-mutation for cluster-tilting objects mutation equivalent to the canonical one .Q; W /.

23


3.4 Application to surfaces with marked points. Let .†; M / be a pair consisting of a compact Riemann surface † with non-empty boundary and a set M of marked points of †, with at least one marked point on each boundary component. By an arc on †, we mean the homotopy class of a non crossing simple curve on † with endpoints in M (which may coincide), which does not cut out an unpunctured monogon, or an unpunctured digon. To each ideal triangulation of the surface .†; M /, Labardini associates in [75] a QP .Q ; W / which is rigid and Jacobi-finite. Moreover he shows that the flip of the triangulation coincides with the DWZ-mutation of the quiver with potential. Since any two triangulations are linked by a finite sequence of flips, the generalized cluster categories obtained from the QPs associated with the triangulations are all equivalent. More precisely, combining results in [75] with Corollary 3.11 and Theorem 3.13 (a) we get the following. Corollary 3.14. Let .†; M / be a surface with marked points with non empty boundary. Then there exists a Hom-finite triangulated 2-CY category C.†;M / with a cluster-tilting object T corresponding to each ideal triangulation such that we have the following commutative diagram: _

triangulation

o

T

cluster-tilting

o

0 triangulation /

flip

IY-mutation

/

_

T 0

cluster-tilting:

All the cluster-tilting objects T for an ideal triangulation are IY-mutation equivalent. In [76] Labardini moreover associates to each arc j on the surface a module over the Jacobian algebra Jac.Q ; W / for each triangulation , in a way compatible with DWZ-mutation for representations. More precisely, if X .j / (resp. X 0 .j //) is the Jac.Q ; W /-module (resp. Jac.Q 0 ; W 0 /-module), where 0 is a flip of , then one can pass from X .j / to X 0 .j / using the DWZ-mutation for representations on the corresponding vertex: _

triangulation

X .j / o

o

flip

DWZmutation for representations

/

0 triangulation _

/ X 0 .j /:

Denote by IY.†; M / C.†;M / the set of all cluster-tilting objects of C.†;M / which are in the IY-mutation class of a cluster-tilting object T , where is an ideal triangulation (it does not depend on the choice of by Corollary 3.14). If a triangulation contains a self-folded triangle, it is not possible to flip at the inside radius of the self-folded triangle. To avoid this problem, Fomin, Shapiro and Thurston introduced in

24

C. Amiot

[39] the notions of tagged arcs and tagged triangulations which generalize the notions of arcs and triangulation. Then it is possible to mutate any tagged arc of any tagged triangulations. Hence we obtain a bijection: IY.†; M / o

/ ftagged triangulations of .†; M /g:

1W1

Let ind IY.†; M / be the set of all indecomposable summands of IY.†; M /. Then since any (tagged) arc on .†; M / can be completed into a (tagged) triangulation, there is a bijection: ind IY.†; M / o

/ ftagged arcs on .†; M /g:

1W1

Now fix an ideal triangulation of .†; M / and denote by T the associated clustertilting of C.†;M / . Using results of [76] together with the diagram 3.3.1 we obtain the following commutative diagram: ind IY.†; M / n ind.T / o _

C.†;M /

farcs on .†; M / not in g t tt tt t t tt tt t t TTTT tt X TTTT tt t TTTT tt tt FT Œ1 TTTT ztt ) mod Jac.Q ; W /,

where FT Œ1 D HomC .T Œ1; / ' HomC .T ; Œ1/. Remark 3.15. In the case of an unpunctured surface .†; M /, Brüstle and Zhang study the category C.†;M / very precisely in [21] (see also [10]), and show an analogue of Theorem 2.7, that is, any cluster-tilting object of C.†;M / is in IY.†; M /. Example 3.16. Let † be a surface of genus 0 with boundary having two connected components, and let M be a set of marked points on † consisting of three marked points on the boundary, and one puncture. Let be the following ideal triangulation:

1

2

6 3

5

4

25


The quiver with potential .Q ; W / is the following: a

1 c

i Q =

2 b

6 f

h

W D cba Cf ed Cgdb.

g

3 d e

5

4

Let j be an arc of .†; M / which is not in . In the most simple cases, the composition series of the module X .j / correspond to the arcs of crossed transversally by j . For instance, if j is the following arc

6

1

2 3

5

4

6

the module X .j / has the following composition series X .j / D 2 5 . Let j be 4 an arc of and j 0 be the flip of this arc with respect to . Then the arc j 0 crosses the triangulation uniquely through the arc j , thus the corresponding Jac.Q ; W /module X .j 0 / is the simple module associated with the vertex of Q corresponding to the arc j of . Remark 3.17. Labardini also associates in [75] QPs to ideal triangulations of surfaces without boundary. For instance if .†; M / is a once punctured torus, then all triangulations give the same QP: 2 ^=^== ======b 0 ==== ; Q D a = c b // 1 3 a0

W D abc C a0 b 0 c 0 C ab 0 ca0 bc 0 :

c0

This QP is not rigid, but is non-degenerate and Jacobi-finite (cf. Section 8 in [76]). Using Corollary 3.11, it is possible to associate a generalized cluster category with clustertilting objects to this surface. However the non-degeneracy and Jacobi-finiteness are not known to be true or false for general surfaces without boundary.

26

C. Amiot

3.5 Derived preprojective algebras. Derived preprojective algebras give another application of Theorem 3.7. Definition 3.18 (Keller [68]). Let ƒ be a finite dimensional algebra of global dimension at most 2. Denote by ‚2 a cofibrant resolution of RHomƒe .ƒ; ƒe /Œ2 2 D.ƒe /. Then the derived 3-preprojective algebra is defined as the tensor DG algebra …3 .ƒ/ ´ Tƒ ‚2 D ƒ ˚ ‚2 ˚ .‚2 ˝ƒ ‚2 / ˚ and the algebra …3 .ƒ/ ´ H 0 .…3 .ƒ// is called the 3-preprojective algebra. Since the algebra ƒ is a finite dimensional algebra of finite global dimension, there is an isomorphism of ƒ-ƒ-bimodules RHomƒe .ƒ; ƒe / ' RHomƒ .Dƒ; ƒ/. Moreover the functor L N ƒ RHomƒ .Dƒ; ƒ/ W D b .ƒ/ ! D b .ƒ/ b Chapter 1 of [2]). Hence is a quasi inverse for the Serre functor Lof D .ƒ/ (see [64] or p we have an isomorphism …3 .ƒ/ ' p0 HomD b ƒ .ƒ; S2 ƒ/: Since ƒ is of global dimension at most 2, then HomD b ƒ .ƒ; S2p ƒ/ vanishes for p 1. Therefore we have isomorphisms of algebras (cf. [3], Proposition 4.7): M p HomD b ƒ .ƒ; S2 ƒ/ ' Tƒ Ext2ƒ .Dƒ; ƒ/: (3.5.1) …3 .ƒ/ ' p2Z

Definition 3.19 (Iyama). An algebra of global dimension at most 2 is called 2 -finite if …3 .ƒ/ is finite dimensional. This is equivalent to the fact that the endofunctor 2 ´ H 0 .S2 / of mod ƒ is nilpotent. Note that if ƒ is an hereditary algebra, then …3 .ƒ/ ' ƒ since Ext2ƒ .Dƒ; ƒ/ vanishes. Hence for any acyclic quiver Q, the algebra kQ is 2 -finite. Theorem 3.20 (Keller, [68]). Let ƒ be an algebra of global dimension at most 2. Then the derived preprojective algebra …3 .ƒ/ is homologically smooth and bimodule 3-CY. Applying then Theorem 3.7 we get the following construction. Corollary 3.21. Let ƒ be as in Theorem 3.20. If ƒ is moreover 2 -finite, then the category C2 .ƒ/ ´ per …3 .ƒ/=D b .…3 .ƒ// is Hom-finite and 2-Calabi–Yau. Moreover the object …3 .ƒ/ is cluster-tilting in C2 .ƒ/ with endomorphism algebra …3 .ƒ/. This category is called the cluster category associated with an algebra of global dimension at most 2. The next result gives the link between the category C2 .ƒ/ and the orbit category D b .ƒ/=S2 . Theorem 3.22. Let ƒ be a finite dimensional algebra of global dimension at most 2.


27

(1) For any X in D b .ƒ/, there is an isomorphism S2 .X /

L N ƒ

…3 .ƒ/ ' X

L N ƒ

…3 .ƒ/

in C2 .ƒ/ functorial in X . Thus there is a commutative diagram

D b .ƒ/

L N ƒ

…3 .ƒ/

D b .ƒ/=S2

/ per …3 .ƒ/ / C2 .ƒ/ .

Moreover, the factorization D b .ƒ/=S2 ! C2 .ƒ/ is fully faithful. (2) The smallest triangulated subcategory of C2 .ƒ/ containing the orbit category D b .ƒ/=S2 is C2 .ƒ/. Let J be the ideal J D ‚2 ˚ .‚2 ˝ƒ ‚2 / ˚ of …3 .ƒ/. Then using the L L N N isomorphism 3.5.1, we obtain an isomorphism S2 .X / ƒ J ' X ƒ …3 .ƒ/. The morphism in (1) of the above theorem is induced by the inclusion J ! …3 .ƒ/ in per.…3 .ƒ/e /. The cone of J ! …3 .ƒ/ is in D b .…3 .ƒ/e / since ƒ is finite dimensional. Therefore the morphism S2 .X /

L N ƒ

…3 .ƒ/ ' X

L N ƒ

J !X

L N ƒ

…3 .ƒ/ in per …3 .ƒ/

has its cone in D .…3 .ƒ//, so is an isomorphism in C2 .ƒ/: Point .2/ of this theorem says that C2 .ƒ/ can be understood as the triangulated hull of the orbit category D b .ƒ/=S2 . Keller introduced this notion with DG categories. A DG category is a category where morphisms have a structure of k-complexes. We refer to [64], [65] for precise definitions and constructions (see also [62], Appendix, and Section 6 in [7]). The philosophy is the following: any algebraic triangulated category is triangle equivalent to H 0 .X/, where X is a DG category. But for a given DG category X, the category H 0 .X/ is not always triangulated. However the category H 0 .X/ can be viewed (via the Yoneda functor) as a full subcategory of H 0 .DGModX/ which is a triangulated category. Here DGModX is the DG category of the DG X-modules. The triangulated hull of H 0 .X/ is defined to be the smallest triangulated subcategory of H 0 .DGModX/ containing H 0 .X/. In our situation, the category D b .ƒ/ has a canonical enhancement in a DG category X (that is, D b .ƒ/ is equivalent to some category H 0 .X/ for a canonical X ). The functor S2 can be canonically lifted to a DG functor S of X. One can define the orbit category X=S ; it is a DG category. The orbit category D b .ƒ/=S2 is equivalent to the category H 0 .X=S /. Its triangulated hull is defined to be the triangulated hull of H 0 .X=S / following the previous construction. Since the enhancement X and the lift S are canonical, one can speak of the triangulated hull. But in general, triangulated hulls depend on the choice of the enhancement. From Theorem 3.22 one deduces the following fact. b

28

C. Amiot

Corollary 3.23. Let ƒ D kQ for an acyclic quiver Q. Then kQ is 2 -finite and there is a triangle equivalence C2 .kQ/ ' CQ . This corollary associated with Theorem 3.22 shows that C2 .ƒ/ is the most natural generalization of the cluster category when replacing the algebra kQ of global dimension 1 by the algebra ƒ of global dimension 2. Remark 3.24. If T is a tilting kQ-module, where Q is an acyclic quiver, then the algebra B D EndkQ .T / has global dimension at most 2 and is 2 -finite. The triangle functor L N B T W D b .B/ ! D b .kQ/ is an equivalence (see [56]) sending B 2 D b .B/ on T 2 D b .kQ/. This equivalence induces a triangle equivalence between the cluster categories C2 .B/ and CQ sending .B/ on .T /. Moreover the tensor product Ext2 .DB; B/˝B Ext2 .DB; B/ vanishes, hence the algebra …3 .B/ is isomorphic to the trivial extension B ˚ Ext2 .DB; B/. The isomorphism …3 .B/ ' EndCQ .T / recovers then a result of [11]. Note that in general, a 2 -finite algebra B of global dimension 2 can satisfy Ext2 .DB; B/ ˝B Ext2 .DB; B/ ¤ 0, even if C2 .B/ is an acyclic cluster category, as shown in the following example. Let B be the Auslander algebra of mod kQ, where Q is the linear orientation of A3 3 2

5

1

4

6.

2 2 Then an easy computation gives ExtB .e4 DB; e1 B/ ¤ 0 and ExtB .e6 DB; e4 B/ ¤ 0, hence we have 2 2 2 2 .DB; B/ ˝B ExtB .DB; B/e6 D ExtB .DB; e1 B/ ˝B ExtB .e6 DB; B/ ¤ 0: e1 ExtB

However, the category C2 .B/ is equivalent to the acyclic cluster category CD6 , since the algebra B is an iterated tilted algebra of type D6 and of global dimension 2. From Keller’s results on triangulated hulls [65], one deduces the following property for the category C2 .ƒ/ (see Appendix of [62] for details). Theorem 3.25. Let ƒ be a finite dimensional algebra of global dimension at most 2. Let E be a Frobenius category. Let M be an object in D b .ƒop ˝ E/. Assume that L N there is a morphism f W RHomƒe .ƒ; ƒe / ƒ M Œ2 ! M in D b .ƒop ˝ E/ such that the cone of f , viewed as an object in D b .E/ is perfect. Then there is triangle functor F W C2 .ƒ/ ! E making the following diagram commutative: D b .ƒ/

L N ƒ

C2 .ƒ/

F

M

/ D b .E/

/ D b .E/= per.E/ ' E:


29

Remark 3.26. A very powerful application of cluster categories associated with algebras of global dimension at most 2 is made in [69], and allows to prove the periodicity conjecture for pairs of Dynkin diagrams (see also last sections of [67] for an overview of this result). 3.6 Link between the two constructions. In the last two subsections we have generalized the notion of cluster category, first using the Ginzburg DG algebra, and then derived 3-preprojective algebra. So a natural question is the following: What is the link between the categories C.Q;W / , where .Q; W / is a Jacobi-finite QP, and the categories C2 .ƒ/, where ƒ is a 2 -finite algebra of global dimension at most 2? Let ƒ D kQ=I (where I is an admissible ideal of kQ) be a finite dimensional algebra of global dimension at most 2. Let R be a minimal set of relations, i.e. the lift to I of a basis of I =.IJ C JI / (where J L is the ideal of kQ generated by the arrows) x compatible with the decomposition I D i;j ej Iei . Let Q be the quiver obtained from Q by adding an arrowPar W t .r/ ! s.r/ for each relation r W s.r/ ! t .r/. Let Sƒ be the potential W Sƒ D W r2R rar . Example 3.27. Let ƒ be the algebra presented by the quiver 2 ^ j . Hence the algebra EndGr S G .T / is presented by 1 x

x

0

y

z

z

2

y

zy

x

4

x

3

with the commutativity relations. Therefore the stable endomorphism algebra A D EndGr S G .T / ' EndGr S G .T1 ˚ ˚ T4 / is presented by the quiver x

1

2

x z y 4

zy x

3

with the commutativity relations. By the previous theorem the category CM.S G / is triangle equivalent to the generalized cluster category C2 .A/ .

38

C. Amiot

The proof of this theorem uses again Theorem 3.25. One fundamental step in the proof consists in using the fact that the skew-group algebra with the grading defined as above is bimodule d -Calabi–Yau of Gorenstein parameter 1 by [19], and then in showing the following: Theorem 4.10 ([4]). Let B be a Z-graded algebra bimodule d -Calabi–Yau of Gorenstein parameter 1 such that dimk Bi is finite for all i , and such that B 2 per B e . Then the DG algebra …d .A/ has its homology concentrated in degree 0. Moreover, there is an isomorphism of Z-graded algebras …d .A/ ' B: We end this section by asking the following intriguing questions: Question 4.11. 1. What can we say about the stable category CM.S G / when G is not cyclic? 2. Is there an analogue of the categories Sub …w in higher CY-dimensions?

5 On the Z-grading on the 3-preprojective algebra Throughout the section ƒ is a finite dimensional algebra of global dimension at most two. As we already saw in Section 3, the 3-preprojective algebra …3 .ƒ/ is isomorphic to the tensor algebra Tƒ Ext2ƒ .Dƒ; ƒ/ (see 3.5.1) and thus is naturally positively Zgraded as a tensor algebra. The algebra ƒ can easily be recovered from the graded algebra …3 .ƒ/ as its degree zero subalgebra. We already used this remark to prove equivalences between certain stable 2-CY categories and cluster categories C2 .ƒ/ in Sections 4.2 and 4.3. In this section, we study this grading, and outline applications in representation theory. 5.1 Grading of …3 .ƒ/. Let us describe more explicitly this grading. The algebra xƒ ; W Sƒ /, where Q x ƒ and W S are …3 .ƒ/ is isomorphic to the Jacobian algebra Jac.Q obtained from the quiver Qƒ and the minimal relations of ƒ (cf. isomorphism 3.6.1). x ƒ defined as follows: This natural grading comes from a grading on Q • the arrows of Qƒ are of degree 0 (indeed, they correspond to elements of ƒ which is the degree 0 subalgebra of …3 .ƒ/); • the new arrows, corresponding to minimal relations, are of degree 1 (indeed, they correspond to elements of the bimodule Ext2ƒ .D.ƒ/; ƒ/). S homogeneous of degree 1. Hence the x ƒ makes the potential W This grading on Q S for a 2 Q1 are homogeneous. relations @a W Example 5.1. Let ƒ be as in Example 3.27. The 3-preprojective algebra …3 .ƒ/ is

39


x ƒ ; W; d / with isomorphic, as a graded algebra, to the graded Jacobian algebra Jac.Q 2 ^===b == ; xƒ D Q c // 1 3 a

d.a/ D d.b/ D d.c/ D 0;

d.rab / D 1 and W D rab ab:

rab

Since we have an isomorphism of graded algebras M p HomD b .ƒ/ .ƒ; S2 ƒ/ …3 .ƒ/ ' p2Z

(cf. 3.5.1), the subcategory Uƒ ´ 1 .ƒ/ D fS2p ƒ; p 2 Zg D b .ƒ/ provides a Z-covering of the graded algebra …3 .ƒ/. Hence the study of the category Uƒ D 1 .ƒ/ is one of the fundamental tools for the study of the grading of …3 .ƒ/. Theorem 5.2 ([61], Theorem 1.22; [2], Proposition 5.4.2 ). If ƒ is 2 -finite, then the subcategory Uƒ is a cluster-tilting subcategory of D b .ƒ/. The next result shows that, moreover, the category D b .ƒ/ is determined by the category Uƒ . Theorem 5.3 (Recognition Theorem, [7], Theorem 3.5). Let T be an algebraic triangulated category, with Serre functor S and with a cluster-tilting subcategory V . Assume that there exist a 2 -finite algebra ƒ and an equivalence f W V ' Uƒ D p addfS2 ƒ; p 2 Zg such that f B S2 ' S B f Œ2. Then T is triangle equivalent to b D .ƒ/. This result is the key step for the proof of all results presented in the next two sections. 5.2 Mutation of graded QPs. In this section, we assume that ƒ is a 2 -finite algebra of global dimension at most 2. The observation that …3 .ƒ/ is a graded Jacobian algebra leads us to study the notion of graded QP .Q; W; d / with W homogeneous of degree 1. We can define the notion of reduction similarly to [36] (see Section 6 in [7]). Definition 5.4 ([7], see also [89]). Let .Q; W; d / be a graded QP such that Q does not have any loops. Let i be a vertex of Q. Then the left mutation L i .Q; W; d / (resp. right mutation R .Q; W; d /) of .Q; W; d / at vertex i is defined to be the reduction of i the graded QP .Q0 ; W 0 ; d 0 / constructed as follows: (M1gr) for each pair of arrows j d 0 .Œba/ D d.a/ C d.b/,

a

/i

b

/ k , add an arrow j

Œba

/ k and put

40

C. Amiot

a

a / (M2gr) replace each arrow j i by an arrow j o 0 1 d.a/ (resp. d .a / D d.a/),

b

b / replace each arrow i k by an arrow i o d.b/ (resp. d 0 .b / D 1 d.b/).

i and put d 0 .a / D k and put d 0 .b / D

All other arrows remain with the same degree and the potential W 0 D ŒW C W (as defined in [36]) and is again homogeneous of degree 1. In the derived category D b .ƒ/, if T is an object such that .T / 2 C2 .ƒ/ is clustertilting, then one can lift the exchange triangles of Theorem 2.2 to triangles in D b .ƒ/. Then we obtain for any indecomposable direct summand Ti of T D T0 ˚ Ti the following triangles in D b .ƒ/ Ti

/B

/ TL i

/ Ti Œ1

/ B0

TiR

and

/ Ti

/ T R Œ1 : i

The objects TiL and TiR satisfy that .TiL / D .TiR / D .Ti / is the unique complement non isomorphic to .Ti / of the almost complete cluster-tilting object .T0 /. We call the object T0 ˚ TiL (resp. TiR ) the left mutation (resp. right mutation) of T at Ti . Then we can formulate the graded analogue of Theorem 2.19, which is a first motivation for the introduction of mutation of graded QP. Theorem 5.5 ([7]). Let ƒ and T 2 D b .ƒ/ as above. Assume that there exist a graded QP .Q; W; d / with potential homogeneous of degree 1 such that we have an isomorphism of graded algebras L / p Jac.Q; W; d /: p2Z HomD .T; S2 T / Z

Let Ti be an indecomposable summand of T ' Ti ˚ T0 and assume that there are neither loops nor 2-cycles incident to i (corresponding to Ti ) in the quiver of End C .T /. Then there is an isomorphism of Z-graded algebras L / p L L Jac.L p2Z HomD .T0 ˚ Ti ; S2 .T0 ˚ Ti // i .Q; W; d //: Z

This result can be illustrated by the following picture: T 2 D b .ƒ/ o

.T / cluster-tilting

_

L i2Z

left mutation right mutation

HomD .T; S2i T / ' Jac.Q; W; d /

O

_

Z

.Q; W; d / o

/ T 0 2 D b .ƒ/ .T 0 / cluster-tilting _

L i2Z

left graded mutation right graded mutation

HomD .T 0 ; S2i T 0 / ' Jac.Q0 ; W 0 ; d 0 /

O

Z

_ / .Q0 ; W 0 ; d 0 /:

41


Our main application of the graded mutation is given by the following result, which gives a combinatorial criterion to see when two algebras of global dimension at most two are derived equivalent. Theorem 5.6 ([7]). Let ƒ and ƒ0 be two algebras of global dimension 2, which are x W S ; d / associated with 2 -finite. Assume that one can pass from the graded QP .Q; 0 S0 0 x …3 .ƒ/ to the graded QP .Q ; W ; d / (up to graded right equivalence) associated with …3 .ƒ0 / using a finite sequence of left and right mutations. Then the algebras ƒ and ƒ0 are derived equivalent. This result can be seen as a generalization (in one direction) of Happel’s theorem (Theorem 1.3). So we could ask what is the meaning of the hypothesis of 2 -finiteness for the algebras ƒ and ƒ0 , and whether this hypothesis is necessary or not. Indeed, cluster categories and cluster-tilting theory are hidden in the statement. However, the proof uses strongly Theorems 5.2, 5.3 and 5.5, where the hypothesis of 2 -finiteness is fundamental. Example 5.7. Let us illustrate this result with an example. The following graded quivers with potential are linked by sequences of left and right graded mutations:

2 1

0

2

L 2

0

0

3

1

0

0

2

R L 2 ◦ 2

1

0

3

1

1

0 0

0

3.

Here, the potential is W D 0 for the first quiver, and is W D c, where c is one of the two 3-cycles, for the next two quivers. Hence, using Theorem 5.6, one deduces that the following algebras of global dimension at most 2 (presented by a quiver with relations) are in the same derived equivalence class. In this picture, a dotted line between two vertices means that a path between these two vertices is zero:

2 1

2 3

1

2 3

1

3.

5.3 Application to cluster equivalence. In this subsection, we describe another application of graded mutation to the notion of cluster equivalence. Definition 5.8. Two finite dimensional algebras ƒ and ƒ0 of global dimension at most 2 which are 2 -finite are called cluster equivalent if there exists a triangle equivalence C2 .ƒ/ ' C2 .ƒ0 / between their cluster categories. The notion of cluster equivalence seems to be a reasonable way to relate the homological algebras of two algebras of global dimension 2 (see Theorem 5.10 below). This notion is more general than derived equivalence: two derived equivalent algebras of global dimension at most two are cluster equivalent, the converse is not true, as shown in the following example.

42

C. Amiot

Example 5.9. We take ƒ as in Example 3.27. It is a 2 -finite algebra of global S D rab ab is rigid. The object .ƒ/ D .P1 ˚ dimension at most 2. The potential W xƒ ; W S /. P2 ˚P3 / is a cluster-tilting object with endomorphism algebra …3 .ƒ/ ' Jac.Q By Theorem 2.19, if we mutate .ƒ/ at .P2 /, we obtain a new cluster-tilting object whose endomorphism algebra is the Jacobian algebra Jac.Q; 0/, where Q is the acyclic quiver 2 @ 0. With a few exceptions in characteristic two, if the Sylow p-subgroups of G are non-cyclic then the group algebra kG has wild representation type. We therefore do not hope to classify the indecomposable finitely generated kG-modules, and so we are left with several options. We can restrict the kinds of modules that we are interested in; we can look for properties of modules that we can prove without the need for a

56

D. J. Benson, S. B. Iyengar and H. Krause

classification; or we look for coarser classification theorems. Examples abound of all three types of approach; we shall be examining the last option. The first indication that it might be possible to make some sort of categorical classification theorem came with Mike Hopkins’ 1987 classification [21] of the thick subcategories of the derived category Db .proj.R// of perfect complexes over a commutative Noetherian ring R. His interest was in the nilpotence theorem for the stable homotopy category, for which he regarded the derived category of perfect complexes as a toy model. The parametrisation of the thick subcategories was by specialisation closed sets of prime ideals in R. Neeman’s 1992 paper [25] took up Hopkins’ work, clarified it and went on to classify the localising subcategories of the unbounded derived category of R-modules D.Mod.R//. This is the appropriate big category whose compact objects are the perfect complexes Db .proj.R//. The parametrisation for the localising subcategories was by all subsets of the set of prime ideals of R, not just the specialisation closed ones. In 1997, the first author together with Carlson and Rickard [6] proved the analogue of Hopkins’ theorem for the stable category stmod.kG/ of finitely generated kGmodules, over an algebraically closed field k of characteristic p. The parametrisation for the thick subcategories was by specialisation closed subsets of homogeneous primes in the cohomology ring H .G; k/, ignoring the maximal ideal of all positive degree elements. The corresponding big category is the stable category StMod.kG/ of all kGmodules; the full subcategory of compact objects in this is the finitely generated stable module category stmod.kG/. It was expected by analogy with D.Mod.R// that the classification of localising subcategories of StMod.kG/ would be by all subsets of the set of non-maximal homogeneous primes in H .G; k/. But this turned out to be very hard to prove, and it was not until a couple of years ago that we managed to achieve this classification [9]. More recently, in 2009 Neeman [26] classified the colocalising subcategories of D.Mod.R//; the corresponding classification for StMod.kG/ is given in [11]. Other classifications follow similar models. The work of Hovey, Palmieri and Strickland [22] and the series of papers [8], [10], [9], [11] lay the general foundations. For background on the theory of support varieties we refer to Solberg’s survey [32].

1 The stable module category We write Mod.kG/ for the module category of kG. The objects are the left kG-modules and the morphisms are the module homomorphisms. This is an abelian category. We write mod.kG/ for the full subcategory of finitely generated kG-modules. We refer the reader to [2] for basic constructions and facts concerning kG-modules. The following statement is well known for finitely generated modules, but is true more generally. Lemma 1.1. Projective and injective kG-modules coincide.

Module categories for finite group algebras

57

Proof. It is an easy exercise using the group basis to show that kG is self-dual as a kG-module. This implies that every projective module is injective. For the converse, we note that every kG-module M embeds in a free kG-module, via the map M ! P kG ˝k M sending m to g2G g ˝g 1 m. Here, we make kG ˝k M into a kG-module via g.g 0 ˝ m/ D gg 0 ˝ m; thus a k-basis of M gives a free kG-basis for kG ˝k M . If M is injective then this embedding splits and M is a summand of a free module, hence projective. Projective kG-modules are well understood. Every projective is a direct sum of finitely generated projective indecomposables, and every projective indecomposable is the projective cover of a simple module. To work “modulo projectives”, we use the stable module category StMod.kG/. This has the same objects as the module category, but the morphisms are given by HomkG .M; N / D HomkG .M; N /= PHomkG .M; N / where PHomkG .M; N / is the linear subspace of maps that factor through some projective module. We write stmod.kG/ for the full subcategory of StMod.kG/ whose objects are the modules stably isomorphic to finitely generated modules. Notice that if a map between finitely generated modules factors through some projective module then it factors through a finitely generated projective module. The subcategory stmod.kG/ is distinguished categorically as the compact objects in StMod.kG/. DefinitionL 1.2. An object M in StMod.kG/ is said to be compact if given any small coproduct ˛ M˛ , the natural map M M HomkG .M; M˛ / ! HomkG .M; M˛ / ˛

˛

is an isomorphism. Although StMod.kG/ and stmod.kG/ are not abelian categories, they come with a natural structure of triangulated category. The “shift” is given by 1 , the functor assigning to each module M the cokernel of an embedding of M into an injective module. At the level of Mod.kG/ this is not functorial, but for StMod.kG/ it is a functorial self-equivalence whose inverse is the functor taking a module M to the kernel of a surjection from a projective module onto M . The distinguished triangles A ! B ! C ! 1 .A/ are by definition those isomorphic to diagrams coming from a short exact sequence 0!A!B!C !0

58


of modules as follows. Given such a short exact sequence, we embed B into an injective module I and obtain the following commutative diagram with exact rows: 0

/A

/B

/C

/0

0

/A

/I

/ 1 .A/

/ 0.

This gives us a map C ! 1 A which is well defined in StMod.kG/. The right hand square in this diagram gives us a short exact sequence 0 ! B ! C ˚ I ! 1 .A/ ! 0: Since C is isomorphic to C ˚ I in StMod.kG/, applying the construction again gives rise to the rotated triangle B ! C ! 1 .A/ ! 1 .B/: Most of the axioms for a triangulated category are easy to verify. The only one that needs any comment is the octahedral axiom, which is the expression in StMod.kG/ of the third isomorphism theorem in Mod.kG/. More details can be found in Buchweitz [18], and in Happel [20].

2 Thick subcategories of stmod.kG / Definition 2.1. For any triangulated category T, a thick subcategory S is a full triangulated subcategory that is closed under taking finite direct sums and summands. It is worth spelling out what this means for the stable module category. Being a full triangulated subcategory means that the subcategory is closed under and 1 , and it has the two in three property: given a triangle in the stable module category for which two of the objects are in S, so is the third. Pulling back to mod.kG/, thick subcategories of stmod.kG/ are in one to one correspondence with the subcategories c of mod.kG/ with the following properties: (i) (ii) (iii) (iv)

All projective modules are in c. c is closed under finite sums and summands. If M is in c then so are .M / and 1 .M /. If 0 ! A ! B ! C ! 0 is a short exact sequence of modules with two of A, B and C in c then so is the third.

The clue to finding thick subcategories of stmod.kG/ comes from the theory of varieties for modules, which we now briefly describe. A fuller treatment can be found in [3]. The cohomology ring H .G; k/ is defined to be Ext kG .k; k/ where k denotes


59

the trivial representation. This is a finitely generated graded commutative ring, and we write VG for its maximal ideal spectrum. If M is a finitely generated kG-module, we have a ring homomorphism M ˝k

H .G; k/ D Ext kG .k; k/ ! ExtkG .M; M / given by tensoring Yoneda extensions with M . As usual, we are tensoring over k with diagonal group action, which is exact. We write IM for the kernel of this homomorphism, and VG .M / for the subvariety of VG determined by this ideal. Since IM is a homogeneous ideal, VG .M / is a closed homogeneous subvariety of the homogeneous variety VG . The following theorem summarises some of the main properties of these varieties; not all of them are easy to prove. Theorem 2.2. Let M , M1 , M2 , M3 be finitely generated kG-modules. (i) VG .M / D f0g if and only if M is a projective module. (ii) More generally, the dimension of the variety VG .M / determines the polynomial rate of growth of a minimal resolution of M . (iii) VG .Homk .M; k// D VG .M /. (iv) VG .M1 ˚ M2 / D VG .M1 / [ VG .M2 /. (v) VG .M1 ˝k M2 / D VG .M1 / \ VG .M2 /. (vi) If 0 ! M1 ! M2 ! M3 ! 0 is a short exact sequence then for i D 1; 2; 3, VG .Mi / is contained in the union of the varieties of the other two modules. (vii) If 0 ¤ 2 H n .G; k/ is represented by a cocycle O W n .k/ ! k, we write L O Then we have VG .L / D VG hi, the variety determined by for the kernel of . the principal ideal hi. (viii) Given a closed homogeneous subvariety V VG , there exists a finitely generated module M with VG .M / D V . Namely, if V D VG h1 ; : : : ; n i then we may take M D L1 ˝k ˝k Ln and use properties (v) and (vii). Definition 2.3. We write VG for the collection of closed homogeneous irreducible subvarieties of VG (including f0g/, or equivalently the spectrum of homogeneous prime ideals p H .G; k/ (including the maximal ideal m of positive degree elements). We say that a subset V VG is specialisation closed if p 2 V , q 2 VG and q p imply q 2 V . If V is a specialisation closed subset of VG then we write cV for the full subcategory of stmod.kG/ consisting of modules M such that VG .M / is a finite union of elements of V (i.e., each irreducible component of VG .M / is in V ). Lemma 2.4. If V is a specialisation closed subset of VG then cV is a thick subcategory of stmod.kG/. Proof. This follows directly from Theorem 2.2.

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We now come to an issue which will reappear in later contexts. Namely, the thick subcategories of stmod.kG/ appearing in Lemma 2.4 are closed under tensor products with finitely generated kG-modules, by part (iii) of Theorem 2.2. Lemma 2.5. If c is a thick subcategory of stmod.kG/ then the following are equivalent: (i) c is closed under tensor product with finitely generated kG-modules. (ii) c is closed under tensor product with simple kG-modules. These conditions are automatically satisfied if G is a finite p-group. Proof. The equivalence of (i) and (ii) follows from the fact that every finitely generated module has a finite filtration in which the filtered quotients are simple modules. If G is a finite p-group then the trivial module is the only simple kG-module. Definition 2.6. We say that a thick subcategory of stmod.kG/ is tensor ideal, or tensor closed, if the equivalent conditions of the lemma are satisfied. We can now state the classification theorem. This was proved in Theorem 3.4 of Benson, Carlson and Rickard [6] in case k is an algebraically closed field, and in Theorem 11.4 of Benson, Iyengar and Krause [9] for general fields k. Theorem 2.7. There is a one to one correspondence between tensor ideal thick subcategories of stmod.kG/ and non-empty specialisation closed subsets of VG . Under this correspondence, V corresponds to cV . Note that the condition that V is non-empty is equivalent to the condition that V contains f0g. This subset plays no role for stmod.kG/ but will make an appearance later. If we remove the tensor ideal condition, then the classification becomes much harder. One example of a thick subcategory which is usually not tensor ideal is the full subcategory of modules in the principle block. The principal block B0 .kG/ is the block containing the trivial module. So one might ask whether the thick subcategories of stmod.B0 .kG// are all of the form cV \ stmod.B0 .kG//. The obstruction to this being true is given in terms of the nucleus, defined as follows. Definition 2.8. The nucleus YG of a finite group G is the union of the images of resG;H W VH ! VG , as H runs over the subgroups of G such that CG .H / is not pnilpotent. Recall that a finite group is said to be p-nilpotent if it has a normal subgroup whose order is prime to p and whose index is a power of p. Theorem 2.9 (Benson, Carlson and Robinson [7]; Benson [4]). The subvariety YG of VG is equal to the union of the VG .M / as M runs over the finitely generated modules M in B0 .kG/ satisfying H .G; M / D 0. In particular, every non-projective module in the principal block has non-trivial cohomology if and only if the centraliser of every element of order p in G is p-nilpotent.


61

Theorem 2.10 (Benson, Carlson and Rickard [6]). If YG is empty or equal to f0g then there is a one to one correspondence between the thick subcategories of stmod.B0 .kG// and the non-empty subsets of VG . Under this correspondence, V corresponds to cV \ stmod.B0 .kG//. If YG is bigger than f0g then it appears to be easy to manufacture infinite collections of thick subcategories supported on each line through the origin in YG . It appears hopeless to classify these thick subcategories. However, if we work modulo the modules supported inside the nucleus, we again obtain a classification of thick subcategories.

3 The derived category The anomalous role of the origin in the classification of thick subcategories of stmod.kG/ can be repaired by moving to a slightly bigger category, namely the derived category Db .mod.kG//. Recall that the objects in this category are the finite chain complexes of finitely generated kG-modules, and the arrows are homotopy classes of maps of complexes, with the quasi-isomorphisms inverted. A quasi-isomorphism is a map of complexes that induces an isomorphism in homology. The category Db .mod.kG// is a triangulated category, in which the triangles are formed using mapping cones. We write Kb .proj.kG// for the thick subcategory of Db .mod.kG// whose objects are the perfect complexes, i.e., the complexes quasi-isomorphic to a bounded complex of finitely generated projective kG-modules. Buchweitz [18] (see also Rickard [31]) has defined a functor from Db .mod.kG// to stmod.kG/ which is essentially surjective and has kernel Kb .proj.kG//: Kb .proj.kG// ! Db .mod.kG// ! stmod.kG/:

The thick subcategory Kb .proj.kG// is the unique minimal tensor ideal thick subcategory of Db .mod.kG//. This allows us to extend the theory of support varieties to Db .mod.kG// in such a way that an object has the origin in its support if and only if it is non-zero. As before, if V is a specialisation closed subset of VG , we write cV for the thick subcategory of Db .mod.kG// consisting of objects whose support is a finite union of elements of V . So the following theorem is an easy consequence of Theorem 2.7. Theorem 3.1. There is a one to one correspondence between tensor ideal thick subcategories of Db .mod.kG// and specialisation closed subsets of VG . Under this correspondence, V corresponds to cV . Next, we observe that G is p-nilpotent if and only if k is the only simple module in the principal block. This allows us to extend Theorem 2.10 as follows. Theorem 3.2. If YG is empty (i.e., if G is p-nilpotent) then there is a one to one correspondence between the thick subcategories of Db .mod.B0 .kG/// and subsets of VG . Under this correspondence, V corresponds to cV \ Db .mod.B0 .kG///.

62


Again, if YG is non-empty, we can work modulo objects supported inside the nucleus and obtain a classification of thick subcategories, but this returns us to quotients of stmod.B0 .kG// so nothing new is gained.

4 Rickard idempotent modules and functors The proof of Theorem 2.7 depends in an essential way on the construction of certain infinitely generated modules, and the development of a theory of support varieties in this context. The modules in question are Rickard idempotent modules. Corresponding to any specialisation closed subset V of VG , Rickard constructs two idempotent functors on StMod.kG/. He writes these as EV and FV , but for consistency with the notation of our series of papers we shall use the notation V and LV . More generally, given any thick subcategory c of stmod.kG/, we have functors c and Lc , and functorial triangles c .M / ! M ! Lc .M /: (4.1) The defining properties of these triangles are given in terms of localising subcategories of StMod.kG/. Definition 4.1. Let T be a triangulated category with small products and coproducts. A localising subcategory of T is a thick subcategory that is closed under coproducts, while a colocalising subcategory of T is a thick subcategory that is closed under products. The triangle of functors (4.1) is characterised by the following properties: (i) c .M / is in the localising subcategory Loc.c/ generated by c. (ii) If N is in c then HomkG .N; Lc .M // D 0. The functors c and Lc preserve small coproducts, see for example Corollary 6.5 in [8]. If c is a tensor ideal thick subcategory then we have c .k/ ˝k M Š c .M /;

Lc .k/ ˝k M Š Lc .M /:

If c D cV then we abbreviate cV to V and LcV to LV , so that the Rickard triangle takes the form V .M / ! M ! LV .M /: Rickard idempotent modules allow us to develop a theory of support for modules in StMod.kG/ as follows. Let p H .G; k/ be homogeneous prime ideal. We choose specialisation closed subsets V and W of VG with the property that V 6 W but V W [fpg (i.e., V XW D fpg). Then we write p for the functor LW V Š V LW . The functor p defined in this way is independent of the choice of V and W with the given properties, see [8], Theorem 6.2. Note that if p is the maximal prime m, we have m D 0.


63

Definition 4.2. If M is an object in StMod.kG/, then the support VG .M / is the subset of VG consisting of fp 2 VG j p .M / ¤ 0g: The following properties of VG .M / are proved in [6] for k algebraically closed, and in [9] for a general field k. This theorem should be compared with Theorem 2.2. Theorem 4.3. Let M , M1 , M2 , M3 be kG-modules. (i) VG .M / D ¿ if and only if M is a projective module. (ii) If M is finitely generated then VG .M / is equal to the set of closed homogeneous irreducible subvarieties of VG .M /. (iii) For a small family M˛ of kG-modules we have M [ VG . M˛ / D VG .M˛ /: ˛

˛

(iv) VG .M1 ˝k M2 / D VG .M1 / \ VG .M2 /. (v) If 0 ! M1 ! M2 ! M3 ! 0 is a short exact sequence then for i D 1; 2; 3, VG .Mi / is contained in the union of the supports of the other two modules. (vi) VG .V .k// D V X fmg, VG .LV .k// D VG X V and VG .p .k// D fpg for p ¤ m. (vii) Given a subset V VG , there L exists a module M with VG .M / D V . For example we could take M D p2V p .k/. The proof of the tensor product theorem given in [6] depends on comparison with rank varieties and the following version of Dade’s lemma for modules in StMod.kG/, which appeared as Theorem 5.2 of [5]. Theorem 4.4. Let E D hg1 ; : : : ; gr i Š .Z=p/r , let k be an algebraically closed field of characteristic p, and let K be an algebraically closed extension field of k of transcendence degree at least r 1. Then a kE-module M is projective if and only if for every choice of .1 ; : : : ; r / 2 K r with not all the i equal to zero, the restriction of K ˝k M to the cyclic subgroup of KE of order p generated by 1 C 1 .g1 1/ C C r .gr 1/ is projective. The role of this version of Dade’s lemma in the development given in [5], [6] is what necessitates the requirement that k is algebraically closed. Later we shall describe another proof avoiding rank varieties and avoiding any version of Dade’s lemma, and which works for a general field k. Remark 4.5 (Maximal versus prime ideals). The reader will have observed that when we were dealing with the stable category stmod.kG/ of finitely generated modules, we

64


used VG , the spectrum of maximal ideals in H .G; k/, whereas for the stable category StMod.kG/ of all modules we used VG , the spectrum of homogeneous prime ideals in H .G; k/. In the case of a finitely generated module, VG .M / is determined by VG .M /, see Theorem 4.3 (ii), whereas for an infinitely generated module it is not, see part (vii) of the same theorem. It would have been possible to use VG consistently throughout, but we chose not to, partly for historical reasons. The origin of the use of VG is Quillen’s work [27], [28], and much of the literature on finitely generated modules has been written in this context. A theorem of Hilbert states that for the maximal ideal spectrum and the homogeneous prime ideal spectrum of a finitely generated commutative algebras over a field, each determines the other. A graded commutative ring is commutative modulo its nil radical, so this applies here. The homogeneous prime spectrum of a nonstandardly graded ring can be quite confusing. For example a polynomial ring can give a spectrum with singularities even though it is regular in the commutative algebra sense. An explicit example of this is the ring kŒx; y; z with jxj D 2, jyj D 4, jzj D 6. The affine open patch corresponding to y ¤ 0 has coordinate ring generated by ˛ D x 2 =y, ˇ D xz=y 2 , D z 2 =y 3 , with the single relation ˛ D ˇ 2 . This has a singularity at the origin. This example arises as H .G; k/ modulo its nil radical with G D .Z=p/3 Ì †3 for p 5, and k a field of characteristic p. Finally, in the Quillen stratification theorem as described in [27], [28] (see also Section 15 of this survey), part of the statement is that NG .E/=CG .E/ acts freely on the stratum corresponding to E. This is true for inhomogeneous maximal, but not for homogeneous prime ideals. Fortunately, our use of Quillen stratification does not involve this feature.

5 Classification of tensor ideal thick subcategories The key step in the classification of tensor ideal thick subcategories of stmod.kG/ (Theorem 2.7) is the following theorem. Theorem 5.1. Let M be a finitely generated kG-module with k algebraically closed, and let V be the collection of closed homogeneous irreducible subvarieties of VG .M /. Then the tensor ideal thick subcategory generated by M is equal to cV . Proof. Write c for the tensor ideal thick subcategory generated by M . It is clear that c is contained in cV . So the natural transformation c V ! c is an isomorphism. So if N is any kG-module then we obtain a triangle c .N / ! V .N / ! Lc V .N /: The first two terms in this triangle are in Loc.cV /, and hence so is the third. So VG .Lc V .N // V :


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If S is any simple kG-module then HomkG .S; M ˝k Lc V .N // Š HomkG .S ˝k M; Lc V .N // D 0: Thus M ˝k Lc V .N / is projective, and hence ¿ D VG .M ˝k Lc V .N // D VG .M / \ VG .Lc V .N // D VG .Lc V .N //: It follows that Lc V .N / is projective, and so c .N / ! V .N / is a stable isomorphism. Thus c D cV .

6 Localising subcategories of StMod.kG / First we describe the goal. As with the thick subcategories of stmod.kG/, we only hope to classify the tensor ideal localising subcategories of StMod.kG/; this is all localising subcategories only in the case where G is a p-group. There are a lot more tensor ideal localising subcategories of StMod.kG/ than tensor ideal thick subcategories of stmod.kG/, as we can see by comparing Theorems 2.2 and 4.3. For any subset V , not just for a specialisation closed one, let CV be the full subcategory of StMod.kG/ consisting of those modules M with VG .M / V . The following lemma is the analogue of Lemma 2.4. Lemma 6.1. If V is a subset of VG then CV is a tensor ideal localising subcategory of StMod.kG/. The statement of the classification theorem is as follows. Theorem 6.2 (Benson, Iyengar and Krause [9]). There is a one to one correspondence between tensor ideal localising subcategories of StMod.kG/ and subsets of VG X f0g. Under this correspondence, V corresponds to CV . The proof of this classification theorem is much harder than the corresponding proof for tensor ideal thick subcategories of stmod.kG/. If one tries to mimic the arguments of Neeman [25], there is a basic obstruction, which is the lack of appropriate field objects. Specifically, given a prime p H .G; k/, a field object for p would be a module M such that HO .G; M / D HomkG .k; M / is isomorphic to the graded field of fractions of H .G; k/=p. Such an object does not always exist, as can be seen using the obstruction theory of Benson, Krause and Schwede [14], [15]. There is one case where there are field objects, and that is the case of an elementary abelian 2-group E. The point here is that the group algebra of an elementary abelian 2-group is the same as an exterior algebra in characteristic two. The basic property of the cohomology in this case is that it is “formal”, in the sense that the cochains and the cohomology are equivalent—there is no higher order information. This is made precise by the Bernstein–Gelfand–Gelfand (BGG) correspondence [16], which gives a correspondence between appropriate categories of modules for the exterior algebra and

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for a polynomial algebra, where the polynomial algebra is regarded as the cohomology of the exterior algebra and vice versa. In this case, one can perform the classification for the graded polynomial ring H .E; k/ and then use the BGG correspondence to prove the classification for StMod.kE/; see [12] for details. For a general finite group in characteristic two, the classification can be achieved using the Quillen stratification theorem and Chouinard’s theorem to reduce to elementary abelian subgroups. We shall describe these later. If E is an elementary abelian p-group for p odd, the problem is that the cochains are not formal. However, there is a device coming from commutative algebra that allows us to reduce to the formal case. Namely, we regard the group algebra kE as a complete intersection, and then use a suitable Koszul complex. We shall describe this construction later. Before addressing these points, we wish to set up a slightly cleaner version of the module category, where the origin f0g 2 VG is not excluded, as it is in Theorems 2.7 and 6.2. This is the category KInj.kG/ described in the next section. It bears the same relation to StMod.kG/ as the bounded derived category Db .mod.kG// does to stmod.kG/. It might be thought that D.Mod.kG// would play this role, but the compact objects in D.Mod.kG// are just the perfect complexes, namely the complexes quasiisomorphic to finite complexes of finitely generated projective modules. So we would get Kb .proj.kG// rather than the desired Db .mod.kG//.

7 The category KInj.kG / The objects of KInj.kG/ are the complexes of injective (or equivalently projective, see Lemma 1.1) kG-modules. We should emphasise that this means unbounded complexes of not necessarily finitely generated injective modules. The arrows are homotopy classes of degree preserving maps of complexes. This is a triangulated category in which the triangles come from the mapping cone construction. This category is investigated in detail in Krause [24], Benson and Krause [13]. Let Kac Inj.kG/ be the full subcategory of KInj.kG/ whose objects are the acyclic complexes. These objects can be described as Tate resolutions of modules. Definition 7.1. If M is a kG-module then a Tate resolution of M is formed by splicing together an injective resolution and a projective resolution of M :

/ P1

/ P0 / I0 CC |> CC | C! ||| = M CC CC zz z C! zz 0 0.

/ I1

Every acyclic complex of injectives ! P1 ! P0 ! P1 ! P2 !

/


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can be regarded as a Tate resolution of the image of P0 ! P1 . Furthermore, given a module homomorphism M ! M 0 , it can be extended to a map of Tate resolutions. Such extensions are homotopic if and only if the homomorphisms differ by an element of PHomkG .M; M 0 /. It follows that Tate resolutions give an equivalence of categories between StMod.kG/ and Kac Inj.kG/. We write tk for a Tate resolution of the trivial module k, as an object in KInj.kG/, ik for an injective resolution, and pk for a projective resolution. So there is a triangle in KInj.kG/ of the form pk ! i k ! t k expressing tk as the mapping cone of the map pk ! i k. We can make projective, injective and Tate resolutions of any module by tensoring: M ˝k pk ! M ˝k i k ! M ˝k t k: Tensor products in KInj.kG/ are taken to be tensor products over k of complexes of modules. Note that i k is the tensor identity of KInj.kG/. Now Kac Inj.kG/ is a localising subcategory of KInj.kG/, and the quotient category KInj.kG/= Kac Inj.kG/ is the unbounded derived category D.Mod.kG//. The inclusion Kac Inj.kG/ ! KInj.kG/

and the quotient functor KInj.kG/ ! D.Mod.kG//

each have both a left and a right adjoint, so that we get a diagram of categories and functors Homk .tk;/

Homk .pk;/

StMod.kG/ ' Kac Inj.kG/ ! KInj.kG/ ! D.Mod.kG//: ˝k tk

˝k pk

The compact objects in these categories are only preserved by the left adjoints, and give us back Rickard’s sequence stmod.kG/

Db .mod.kG//

Kb .proj.kG//:

Lemma 7.2. The only tensor ideal localising subcategories of the unbounded derived category D.Mod.kG// are zero and the whole category. Proof. Let C be a non-zero tensor ideal localising subcategory. If X is a non-zero object in C then X has non-zero homology. The homology of kG ˝k X is thus non-zero and free. Any summand isomorphic to kG of the homology splits off as a summand of kG ˝k X , and so kG is in C. But kG generates the whole of D.Mod.kG//. It follows from this lemma that corresponding to any tensor ideal localising subcategory of StMod.kG/, there are two tensor ideal localising subcategories of KInj.kG/,

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one of which is contained in Kac Inj.kG/ and the other of which is generated by this together with the image of D.Mod.kG// under ˝k pk. Correspondingly, we have a notion of support varieties for objects in KInj.kG/. The definitions involve exactly the same definitions of functors c and Lc as for StMod.kG/, and the only difference is that if X is an object in KInj.kG/ then the origin f0g is either in VG .X / or not according as the complex X has homology or not. This allows us to formulate the following version of the classification theorem for KInj.kG/, where the origin has lost its special role. Theorem 7.3. There is a one to one correspondence between tensor ideal localising subcategories of KInj.kG/ and subsets of VG . Under this correspondence, V corresponds to CV . This is the theorem whose proof we shall outline in the sections to follow.

8 Support for triangulated categories At this stage, it is appropriate to describe the general setup introduced in [8] for discussing support for objects in triangulated categories. This setup allows us to move from one category to another without too much effort. Then the game plan, which is inspired by the work of Avramov, Buchweitz, Iyengar, and Miller [1], is as follows: (i) Reduce from a finite group to its elementary abelian subgroups. (ii) Use a Koszul construction to move from complexes of modules over an elementary abelian group to differential graded modules over a graded exterior algebra. (iii) Use a version of the BGG correspondence to move from a graded exterior algebra to a graded polynomial algebra. (iv) Use a version of Neeman’s classification [25] to deal directly with differential graded modules over graded polynomial algebras. Definition 8.1. Let T be a compactly generated triangulated category with small coproducts. We write † for the shift in T and Tc for the full subcategory of compact objects in T. We write Z .T/ for the graded centre of T. Namely, Z .T/ is the graded ring whose degree n component Z n .T/ is the set of natural transformations W IdT ! †n satisfying † D .1/n †: Then Z .T/ is a graded commutative ring, in the sense that for x; y 2 Z .T/ we have yx D .1/jxjjyj xy:


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Let R be a graded commutative Noetherian ring. We say that T is an R-linear triangulated category if we are given a homomorphism of graded commutative rings W R ! Z .T/. This amounts to giving, for each object X in T, a homomorphism of graded rings X W R ! EndT .X /; such that for X and Y objects in T, the two induced actions of an element r 2 R on ˛ 2 HomT .X; Y / via X and Y are related by Y .r/˛ D .1/jrjj˛j ˛ X .r/: Example 8.2. If T D StMod.kG/ then Tc D stmod.kG/ and † D 1 . In this case, we take R D H .G; k/. Although it might seem more natural to use Tate cohomology HO .G; k/ D HomkG .k; k/, the problem is that this ring is usually not Noetherian. This is related to the awkward role of the origin in the theory of the support varieties for StMod.kG/. If T D KInj.kG/ then Tc D Db .mod.kG// and † is the usual shift. In this case, we also take R D H .G; k/. In this case, R is just the graded endomorphism ring of the tensor identity i k, and the origin no longer has an awkward role. We write Spec R for the set of homogeneous prime ideals in R. For each prime p 2 Spec R and each graded R-module M , we denote by Mp the graded localisation of M at p. If V is a specialisation closed subset of Spec R, we set TV D fX 2 T j HomT .C; X /p D 0 for all C 2 Tc ; p 2 Spec R X V g:

This is a localising subcategory of T, and there is a localisation functor LV W T ! T such that LV .X / D 0 if and only if X is in TV . We then define V .X / by completing X ! LV .X / to a triangle: V .X / ! X ! LV .X /: Example 8.3. In the case where T is either StMod.kG/ or KInj.kG/, and R D H .G; k/, this is exactly Rickard’s triangle for the subset V of Spec H .G; k/. As in Section 4, if p 2 Spec R we choose specialisation closed subsets V and W of Spec R satisfying V 6 W and V W [ p and define p D L W V D V L W : Again, this is independent of choice of V and W with the given properties. Definition 8.4. If X is an object in T then the support of X is the subset suppR .X / D fp 2 Spec R j p .X / ¤ 0g: This is a subset of suppR .T/ D fp 2 Spec R j p .T/ ¤ 0g Spec R:

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Example 8.5. Continuing Example 8.3, this notion of support agrees with VG .X / as defined in Section 4. We have suppH .G;k/ .StMod.kG// D VG X f0g; suppH .G;k/ .KInj.kG// D VG : The following theorem summarises the properties of support for R-linear triangulated categories; confer Theorems 2.2 and 4.3. Proofs can be found in [8]. Theorem 8.6. Let X , Y and Z be objects in T. Then (i) X D 0 if and only if suppR .X / D ¿. (ii) suppR .†X / D suppR .X /. (iii) For a small family of objects X˛ we have M [ suppR X˛ D suppR .X˛ /: ˛

˛

(iv) If X ! Y ! Z is a triangle in T then suppR .Y / suppR .X / [ suppR .Z/: (v) For V Spec R we have suppR .V .X // D V \ suppR .X / suppR .LV .X // D .Spec R X V / \ suppR .X /: (vi) If cl.suppR .X // \ suppR .Y / D ¿ then HomT .X; Y / D 0. Here, cl.V / denotes the specialisation closure of a subset V .

9 Tensor triangulated categories In this section we consider compactly generated triangulated categories with some extra structure. Namely, we want to consider an internal tensor product ˝ W T T ! T; exact in each variable, preserving small coproducts, and with a unit 1. It then follows (using the Brown representability theorem) that there are function objects Hom.X; Y / in T with natural isomorphisms HomT .X ˝ Y; Z/ Š HomT .X; Hom.Y; Z//: We define the dual of an object X in T to be X _ D Hom.X; 1/:


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Definition 9.1. We say that .T; ˝; 1/ is a tensor triangulated category if the following hold: (i) (ii) (iii) (iv) (v)

T is a compactly generated triangulated category with small coproducts.

The tensor product ˝ and unit 1 make T a symmetric monoidal category. The tensor product is exact in each variable and preserves small coproducts. The unit 1 is compact. Compact objects are strongly dualisable in the sense that if C is compact and X is any object in T then the canonical map C _ ˝ X ! Hom.C; X / is an isomorphism.

See [8], [11] for further discussion of this structure. Example 9.2. Both StMod.kG/ and KInj.kG/ are tensor triangulated categories. The symmetric monoidal structure ensures that the endomorphism ring of the tensor identity EndT .1/ is a graded commutative ring. Given any object X , this ring acts on EndT .X / via X˝

EndT .1/ ! EndT .X /: So if R is a graded commutative Noetherian ring, a homomorphism R ! EndT .1/ gives T a structure of an R-linear category. Definition 9.3. We say that the action of R on T is canonical if it arises from a homomorphism R ! EndT .1/ as described in the previous paragraph. Proposition 9.4. If the action of R on T is canonical then for each specialisation closed subset V Spec R and each prime p 2 Spec R there are natural isomorphisms V .X / Š X ˝ V .1/;

LV .X / Š X ˝ LV .1/;

p .X / Š X ˝ p .1/:

10 The local-global principle If X is an object or collection of objects in a triangulated category T, we write LocT .X / for the localising subcategory of T generated by X . Definition 10.1. Let T be an R-linear triangulated category with small coproducts. We say that the local-global principle holds for T if for each object X in T we have LocT .X / D LocT .fp .X / j p 2 suppR .T/g/: This condition is usually satisfied, because of the following theorem ([10], Corollary 3.5):

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Theorem 10.2. The local-global principle holds provided R has finite Krull dimension. For a tensor triangulated category T, we modify the definition slightly. If X is an object or collection of objects in T, we write Loc˝ T .X / for the tensor ideal localising subcategory of T generated by X . The following theorem says that for a tensor triangulated category the appropriate analogue of the local-global principle always holds ([10], Theorem 7.2): Theorem 10.3. Let T be an R-linear tensor triangulated category with canonical R-action. Then for each object X in T the following holds: ˝ Loc˝ T .X / D LocT .fp .X / j p 2 Spec Rg/:

When the local-global principle holds, the classification of localising subcategories can be achieved one prime at a time. The way to express this is via the following maps: μ ´ μ ´ Localising / Families .S.p//p2suppR .T/ with S.p/ a o localising subcategory of p .T/ subcategories of T where .S/ D .S \p .T// and .S.p// D LocT .S.p/ j p 2 suppR .T//. The following is Proposition 3.6 of [10]. Theorem 10.4. If the local-global principle holds then the maps and are mutually inverse bijections. In other words, specifying a localising subcategory of T is equivalent to specifying a localising subcategory of p .T/ for each prime p 2 suppR .T/. Since the tensor ideal version of the local-global principle always holds in a tensor triangulated category, we have the following theorem. Theorem 10.5. Let T be an R-linear tensor triangulated category with canonical Raction. Then the maps and above induce mutually inverse bijections between the tensor ideal localising subcategories of T and families .S.p//p2suppR .T/ with S.p/ a tensor ideal localising subcategory of p .T/. Remark 10.6. If the tensor identity 1 generates T then every localising subcategory of T is tensor ideal. In the case of StMod.kG/ and KInj.kG/ this holds if and only if G is a finite p-group.

11 Stratifying triangulated categories In the last section, we showed how to classify localising subcategories of an R-linear triangulated category T one prime at a time. The easiest case is where each p .T/ with p 2 suppR .T/, is a minimal subcategory.


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Definition 11.1. We say that a localising subcategory of T is minimal if it is non-zero and has no proper non-zero localising subcategories. We say that T is stratified by R if the local-global principle holds, and each p .T/ with p 2 suppR T is a minimal localising subcategory. Suppose that T is stratified by R, and consider the maps and of Section 10. To name a family .S.p//p2suppR .T/ with S.p/ a localising subcategory of p .T/, we just have to name the set of p 2 suppR .T/ for which S.p/ D p .T/, since the remaining primes p will satisfy S.p/ D 0. So the following is a direct consequence of Theorem 10.4. Theorem 11.2. Let T be an R-linear triangulated category. If T is stratified by R then the maps and establish a bijection between the localising subcategories of T and the subsets of suppR .T/. In the case of an R-linear tensor triangulated category T, because the corresponding local-global principle is automatic (Theorem 10.3), we say that T is stratified by R if for each p 2 suppR .T/, p .T/ is minimal as a tensor ideal localising subcategory. Theorem 11.3. Let T be an R-linear tensor triangulated category. If T is stratified by R then the maps and establish a bijection between the tensor ideal localising subcategories of T and the subsets of suppR .T/. In order to prove stratification in a given situation, we need a criterion for minimality of localising subcategories. The following can be found in Lemma 4.1 of [10] and Lemma 3.9 of [9]. Lemma 11.4. (i) Let T be an R-linear triangulated category. A non-zero localising subcategory S of T is minimal if and only if for all non-zero objects X and Y in S we have HomT .X; Y / ¤ 0. (ii) Let T be an R-linear tensor triangulated category. A non-zero tensor ideal localising subcategory S of T is minimal if and only if for all non-zero objects X and Y in S there exists an object Z in T such that HomT .X ˝ Z; Y / ¤ 0. The object Z may be taken to be a compact generator for T. We can now restate Theorems 6.2 and 7.3. Theorem 11.5. The tensor triangulated categories StMod.kG/ and KInj.kG/ are stratified by H .G; k/. In the next few sections, we outline the proof of this theorem. A lot of details will be swept under the carpet here, but are spelt out in [9].

12 Graded polynomial algebras The first step in proving Theorem 11.5 is to stratify the derived category of differential graded modules a polynomial ring. Let S be a graded polynomial algebra over the field

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k on a finite number of indeterminates. If k does not have characteristic two, we assume that the degrees of the indeterminates are even, so that S is graded commutative. We view S as a differential graded (abbreviated to dg) algebra with zero differential, and we write D.S / for the derived category of dg S -modules. The objects of this category are the dg S-modules, and the morphisms are homotopy classes of degree preserving chain maps with quasi-isomorphisms inverted. See for example Keller [23] for further details. The category D.S / is a tensor triangulated category in which the tensor product is the derived tensor product over S , X ˝LS Y and the tensor identity is S, which is a compact generator for D.S /. In particular, every localising subcategory is tensor ideal. The canonical action of S on D.S / makes D.S / into an S -linear tensor triangulated category. The following theorem is a dg analogue of the theorem of Neeman [25], and is proved in Theorem 5.2 of [9]. The existence of field objects plays a crucial role in the proof. Theorem 12.1. The category D.S / is stratified by the canonical S-action. So the maps and of Section 10 are mutually inverse bijections between the localising subcategories of D.S / and subsets of Spec S .

13 A BGG correspondence The second step in the proof of Theorem 11.5 is to use a version of the BGG correspondence to transfer the stratification from polynomial rings to exterior algebras. Let k be a field and let be an exterior algebra on indeterminates 1 ; : : : ; c of negative odd degrees. We view as a dg algebra with zero differential. Let S be a graded polynomial algebra on variables x1 ; : : : ; xc with jxi j D j i j C 1. Let J be the dg ˝k S -module with J \ D Homk . ; k/ ˝k S and d.f ˝ s/ D

X

i f ˝ xi s:

i

If M is a dg -module then Hom .J; M / is in a natural way a dg S -module. In general, for a dg algebra A, we write A\ for the underlying graded algebra of A, forgetting the differential. If M is a dg A-module, we write M \ for the underlying graded A\ -module. We say that a dg A-module I is graded-injective if I \ is injective in the category of graded A\ -module. Finally, we write KInj.A/ for the homotopy category of graded-injective dg A-modules. The following version of the BGG correspondence [16] extends the one from [1]. Theorem 13.1. The functor Hom .J; / W KInj. / ! D.S / is an equivalence of categories.


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We give a comultiplication . i / D i ˝ 1 C 1 ˝ i : This allows us to define a tensor product of dg -modules, and this makes KInj. / into a tensor triangulated category. Its tensor identity is i k, which is a compact generator. We have S Š Ext .k; k/ Š HomKInj./ .J; J / which makes KInj. / into an S-linear tensor triangulated category with canonical S action. The equivalence described in Theorem 13.1 allows us to stratify KInj. /. Theorem 13.2. The category KInj. / is stratified by the canonical action of S. So the maps and of Section 10 give a bijection between localising subcategories of KInj. / and subsets of Spec S .

14 The Koszul construction If E is an elementary abelian 2-group and k has characteristic two, then kE is an exterior algebra on generators of degree zero. So we can use Theorem 13.2 to stratify KInj.kE/. This proves Theorem 11.5 in this case; see [12] for details. Elementary abelian p-groups with p odd cannot be treated this way. In this section, we show how to use a Koszul construction to deal with this case. So let p be a prime, k be a field of characteristic p, and E D hg1 ; : : : ; gr i Š .Z=p/r be an elementary abelian p-group. Let zi D gi 1 2 kE, so that kE D kŒz1 ; : : : ; zr =.z1p ; : : : ; zrp /: We regard kE as a complete intersection, and form the Koszul construction as follows. Let A be the dg algebra A D kEhy1 ; : : : ; yr i where kE is in degree zero, and y1 ; : : : ; yr are exterior generators of degree 1 with yi2 D 0;

yi yj D yj yi ;

d.yi / D zi ;

d.zi / D 0:

Let be an exterior algebra on generators 1 ; : : : ; r of degree 1, regarded as a dg algebra with zero differential, and S be a polynomial algebra on generators x1 ; : : : ; xr of degree 2, again regarded as dg algebra with zero differential. The following is Lemma 7.1 of [9].

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Lemma 14.1. The map W ! A defined by . i / D zip1 yi is a quasi-isomorphism of dg algebras. In particular, ExtA .k; k/ Š Ext .k; k/ Š S: We give A a comultiplication .zi / D zi ˝ 1 C 1 ˝ zi ;

.yi / D yi ˝ 1 C 1 ˝ yi :

This gives KInj.A/ the structure of a tensor triangulated category with a canonical action of S . The following is a special case of Proposition 4.6 of [9]. Theorem 14.2. The map W ! A induces an equivalence of tensor triangulated categories Hom .A; / W KInj. / ! KInj.A/: As a consequence, we can stratify KInj.A/. The following is Theorem 7.2 of [9]. Theorem 14.3. The S-linear tensor triangulated category KInj.A/ is stratified by the canonical action of S. So the maps and of Section 10 give a bijection between localising subcategories of KInj.A/ and subsets of Spec S . Now the inclusion map kE ! A induces a restriction map ExtA .k; k/ ! ExtkE .k; k/: The structure of Ext kE .k; k/ is as follows. If p is odd then it is a tensor product of an exterior algebra and a polynomial algebra Ext kE .k; k/ Š .u1 ; : : : ; ur / ˝ kŒx1 ; : : : ; xr with jui j D 1 and jxi j D 2. The elements xi are the restrictions of the elements of the same name in ExtA .k; k/ Š S . If p D 2 then Ext kE .k; k/ D kŒu1 ; : : : ; ur with jui j D 1. The elements xi 2 ExtA .k; k/ restrict to u2i . In both cases, this allows us to regard S Š ExtA .k; k/ as a subring of H .E; k/ over which it is finitely generated as a module. The restriction map from ExtA .k; k/ to H .E; k/ D Ext kE .k; k/ induces a bijection Spec H .E; k/ ! Spec S: Using the induction and restriction functors, the criterion of Lemma 11.4 is essential in deducing the following theorem from Theorem 14.3. See Theorem 4.11 of [9] for details.


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Theorem 14.4. The tensor triangulated category KInj.kE/ is stratified by the canonical action of H .E; k/, or equivalently of S H .E; k/. So the maps and of Section 10 give a bijection between localising subcategories of KInj.kE/ and subsets of Spec H .E; k/. There is an issue here with the tensor structure. The usual diagonal map .gi / D gi ˝gi on kE is not compatible with the inclusion kE ! A. There is another diagonal map on kE given by .zi / D zi ˝ 1 C 1 ˝ zi coming from regarding kE as a restricted universal enveloping algebra, and this diagonal map is compatible with the inclusion. Each diagonal map gives rise to a canonical action of H .E; k/ on KInj.kE/, but the two actions are not the same. Lemma 3.10 of [9] helps us out at this point: Lemma 14.5. Let T be a triangulated category admitting two tensor triangulated structures with the same unit 1, and assume that 1 generates T. Let ; 0 W R ! Z .T/ be two actions of R on T. If the maps R ! EndT .1/ induced by and 0 agree then T is stratified by R through if and only if it is stratified by R through 0 .

15 Quillen stratification There is a general machine for understanding cohomological properties of a general finite group from its elementary abelian subgroups, called Quillen stratification. In this section, we explain how to use this machine to compare localising subcategories of KInj.kE/ and KInj.kG/. Let VG D Spec H .G; k/. If H is a subgroup of G then the restriction map H .G; k/ ! H .H; k/ induces a map resG;H W VH ! VG : Quillen [27], [28] proved the following. Given a prime p 2 VG we say that p originates in an elementary abelian p-subgroup E G if p is in the image of resG;E but not of resG;E 0 for E 0 a proper subgroup of E. Theorem 15.1. Given a prime p 2 VG , there exists a pair .E; q/ such that p originates in E and resG;E .q/ D p, and all such pairs are G-conjugate. This sets up a bijection between primes p 2 VG and conjugacy classes of pairs .E; q/ where q 2 VE originates in E. In order to be able to use this, we first need a version of the “subgroup theorem” for elementary abelian p-groups. The following is Theorem 9.5 of [9], and its proof is a fairly straightforward consequence of the Stratification Theorem 14.4 for elementary abelian p-groups.

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Theorem 15.2. Let E 0 E be elementary abelian p-groups. If X is an object in KInj.kE/ then VE 0 .X #E 0 / D .resE;E 0 /1 VE .X /: Next, we need a version of Chouinard’s theorem [19] for KInj.kG/, see Proposition 9.6 of [9]: Theorem 15.3. An object X in KInj.kG/ is zero if and only if the restriction of X to every elementary abelian p-subgroup of G is zero. We are now ready to outline the proof of Theorem 11.5. This amounts to showing that for p 2 VG , the tensor ideal localising subcategory p KInj.kG/ is minimal. For this purpose, we use the criterion of Lemma 11.4. Let X and Y be non-zero objects in p KInj.kG/. By Theorem 15.3 there exists an elementary abelian subgroup E0 such that X #E0 is non-zero. Choose a prime q0 2 VE0 .X #E0 /. Using standard properties of support under induction and restriction, we obtain resG;E0 .q0 / D p. So we can choose a pair .E; q/ with E0 E and q0 D resE0 ;E .q/, so that the conjugacy class of .E; q/ corresponds to p under the bijection of Theorem 15.1. By Theorem 15.2 we have q .X #E / ¤ 0. Since the pair .E; q/ is determined up to conjugacy by p, the object Y 2 p KInj.kG/ also has q .Y #E / ¤ 0. Let Z be an injective resolution of the permutation module k.G=E/, as an object in KInj.kG/. This has the property that X ˝k Z Š X ˝k k.G=E/ Š X #E "G : Thus using Frobenius reciprocity we have HomkG .X ˝k Z; Y / Š HomkG .X #E "G ; Y / Š HomkE .X #E ; Y #E /: Since q KInj.kE/ is minimal, using the first part of Lemma 11.4 in one direction shows that the right hand side is non-zero. Using the other part in the other direction then shows that p KInj.kG/ is minimal. Thus KInj.kG/ is stratified as a tensor triangulated category by H .G; k/.

16 Applications In this section, we give some applications of the classification of localising subcategories of StMod.kG/ and KInj.kG/ (Theorem 11.5). To illustrate the methods, we give the proof in the case of the tensor product theorem. The remaining proofs can be found in Section 11 of [9]. The tensor product theorem. The tensor product theorem states that if X and Y are objects in StMod.kG/, or more generally in the larger category KInj.kG/, then VG .X ˝k Y / D VG .X / \ VG .Y /:


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This was first proved by Benson, Carlson and Rickard in Theorem 10.8 of [5] for StMod.kG/ in the case where k is algebraically closed. The method of proof was to reduce to elementary abelian subgroups and then use the version of Dade’s lemma given in Theorem 4.4. The proof of the Stratification Theorem 11.5 does not involve any form of Dade’s lemma, and so we get a new proof of the tensor product theorem as follows. Since p .X ˝k Y / Š p .1/ ˝k X ˝k Y Š p .X / ˝k Y Š X ˝ p .Y /; if either p .X / or p .Y / is zero then so is p .X ˝k Y /. This shows that VG .X ˝k Y / VG .X / \ VG .Y /: For the reverse containment, suppose that p 2 VG .X / \ VG .Y /. Thus p .X / ¤ 0 and p .Y / ¤ 0. It follows from Theorem 11.5 and p .X / ¤ 0 that p .1/ is in Loc˝ .p .X //, and hence that p .Y / is in Loc˝ .p .X ˝k Y //. Since p .Y / ¤ 0, this implies that p .X ˝k Y / ¤ 0. The subgroup theorem. The Subgroup Theorem 15.2 for elementary abelian groups was proved using the stratification theorem in that context, and was used in order to prove the stratification theorem for general finite groups. The following general version of the subgroup theorem follows in the same way from the stratification theorem for finite groups. Theorem 16.1. Let H G be finite groups. If X is an object in KInj.kG/ then VH .X #H / D .resG;H /1 VG .X /: Thick subcategories. Theorem 11.5 also gives a new proof for the classification of tensor ideal thick subcategories of stmod.kG/ (Theorem 2.7) and Db .mod.kG//, avoiding the use of the version of Dade’s lemma given in Theorem 4.4. Localising subcategories closed under products and duality. We state the following theorem for StMod.kG/. A similar statement holds for KInj.kG/. Theorem 16.2. Under the bijection given in Theorem 6.2, the following properties of a tensor ideal localising subcategory CV of StMod.kG/ are equivalent. (i) CV is closed under products. (ii) VG X V is specialisation closed. (iii) There exists a set X of finitely generated kG-modules such that CV is the full subcategory of modules M satisfying HomkG .N; M / for all N in X. (iv) When a kG-module M is in CV so is Homk .M; k/.

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The telescope conjecture. Our final application is a statement about certain localising such categories which are defined as follows. Definition 16.3. A localising subcategory C of a triangulated category T is said to be strictly localising if there is a localisation functor L W T ! T such that an object X is in C if and only if L.X / D 0. A localising subcategory C is said to be smashing if it is strictly localising, and the localisation functor L preserves coproducts. The following is the analogue for StMod.kG/ and KInj.kG/ of the telescope conjecture of algebraic topology (Bousfield [17], Ravenel [29], [30]). Theorem 16.4. A tensor ideal localising subcategory of StMod.kG/ or of KInj.kG/ is smashing if and only if it is generated by compact objects.

17 Costratification Let T be a compactly generated triangulated category with small products and coproducts. Recall that a colocalising subcategory of T is a thick subcategory that is closed under products. If T is tensor triangulated then such a subcategory C is closed under tensor products with simple modules if and only if it is Hom closed, in the sense that for all X in T and all Y in C, the function object (see Section 9) Hom.X; Y / is in C. The main theorem of [11] classifies the Hom closed colocalising subcategories of StMod.kG/ and of KInj.kG/. Theorem 17.1. There is a one to one correspondence between Hom closed colocalising subcategories D of StMod.kG/ (respectively KInj.kG/) and tensor ideal localising subcategories C of StMod.kG/ (respectively KInj.kG/) given by C D ? D, the full subcategory of objects X satisfying Hom .X; Y / D 0 for all Y in D. The proof of this theorem goes via the notions of cosupport and costratification. We explain these concepts and fix to this end an R-linear tensor triangulated category T, as in Section 9. In addition to the axioms listed there, we assume also Hom.‹; Y / is exact for each object Y 2 T; all tensor triangulated categories encountered in this work have this property. For each prime p 2 Spec R denote by p the right adjoint of the functor p which exists by the Brown representability theorem. Definition 17.2. If X is an object in T then the cosupport of X is the subset cosuppR .X / D fp 2 Spec R j p .X / ¤ 0g: Note that p and p provide mutually inverse equivalences between p .T/ and .T/. Thus cosuppR .X / is a subset of suppR .T/. p


81

We say that the tensor triangulated category T is costratified by R if for each p 2 Spec R the category p .T/ is either zero or minimal among all Hom closed colocalising subcategories of T. As for localising subcategories, the classification of colocalising subcategories can be achieved one prime at a time. The way to express this is via the following maps: ´ μ ´ μ Colocalising / Families .S.p//p2suppR .T/ with S.p/ a o colocalising subcategory of p .T/ subcategories of T where .S/ D .S \ p .T// and .S.p// D ColocT .S.p/ j p 2 suppT .T//. The following is Corollary 9.2 of [11]. Theorem 17.3. Let T be an R-linear tensor triangulated category. If T is costratified by R then the maps and establish a bijection between the Hom closed colocalising subcategories of T and the subsets of suppR .T/. From this result one deduces Theorem 17.1 by proving costratification first for KInj.kG/ (see [11], Theorem 11.10) and then for StMod.kG/ (see [11], Theorem 11.13).

Let us include an application which justifies the study of support and cosupport; it is Corollary 9.6 of [11]. Theorem 17.4. Suppose the tensor triangulated category T is generated by its unit. Then T is stratified by R if and only if for all objects X and Y in T one has HomT .X; Y / D 0 ” suppR .X / \ cosuppR .Y / D ¿: Note that StMod.kG/ (respectively KInj.kG/) is generated by its unit if G is a p-group. We refer to [11] for more general results which do not depend on the fact that T is generated by its unit.

References [1] L. L. Avramov, R.-O. Buchweitz, S. B. Iyengar and C. Miller, Homology of perfect complexes. Advances Math. 223 (2010), 1731–1781; Corrigendum: Advances Math. 225 (2010) 3576–3578. [2] D. J. Benson, Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associative Algebras. Cambridge Studies in Advanced Mathematics 30, Cambridge University Press, Cambridge, 1991, reprinted in paperback, 1998. [3] D. J. Benson, Representations and Cohomology II: Cohomology of Groups and Modules. Cambridge Studies in Advanced Mathematics 31, Cambridge University Press, Cambridge, 1991, reprinted in paperback, 1998. [4] D. J. Benson, Cohomology of modules in the principal block of a finite group. New York Journal of Mathematics 1 (1995), 196–205. [5] D. J. Benson, J. F. Carlson and J. Rickard, Complexity and varieties for infinitely generated modules, II. Math. Proc. Cambridge Philos. Soc. 120 (1996), 597–615.

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[6] D. J. Benson, J. F. Carlson and J. Rickard, Thick subcategories of the stable module category. Fund. Math. 153 (1997), 59–80. [7] D. J. Benson, J. F. Carlson and G. R. Robinson, On the vanishing of group cohomology. J. Algebra 131 (1990), 40–73. [8] D. J. Benson, S. B. Iyengar and H. Krause, Local cohomology and support for triangulated categories. Ann. Sci. École Norm. Sup. 41 (2008), 575–621. [9] D. J. Benson, S. B. Iyengar and H. Krause, Stratifying modular representations of finite groups. Ann. of Math., to appear. [10] D. J. Benson, S. B. Iyengar and H. Krause, Stratifying triangulated categories. Journal of Topology, to appear. [11] D. J. Benson, S. B. Iyengar and H. Krause, Cosupport and colocalising subcategories for finite groups. J. Reine Angew. Math., to appear. [12] D. J. Benson, S. B. Iyengar and H. Krause, Representations of finite groups: Local cohomology and support. Oberwolfach Seminars, Birkhäuser, to appear. [13] D. J. Benson and H. Krause, Complexes of injective kG-modules. Algebra and Number Theory 2 (2008), 1–30. [14] D. J. Benson, H. Krause and S. Schwede, Realizability of modules over Tate cohomology. Trans. Amer. Math. Soc. 356 (2004), 3621–3668. [15] D. J. Benson, H. Krause and S. Schwede, Introduction to realizability of modules over Tate cohomology. Fields Inst. Comm. 45, Amer. Math. Soc., Providence, RI, 2005, 81–97. [16] I. N. Bernšte˘ın, I. M. Gel0 fand and S. I. Gel0 fand, Algebraic vector bundles on P n and problems of linear algebra. Funkc. Anal. Appl. 12 (1978), 66–67. [17] A. K. Bousfield, The localization of spectra with respect to homology. Topology 18 (1979), 257–281. [18] R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings. Preprint, Univ. Hannover 1986; http://hdl.handle.net/1807/16682. [19] L. Chouinard, Projectivity and relative projectivity over group rings. J. Pure Appl. Algebra 7 (1976), 278–302. [20] D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge, 1988. [21] M. J. Hopkins, Global methods in homotopy theory. In Homotopy Theory. London Mathematical Society Lecture Notes Series 117, Cambridge University Press, Cambridge, 1987, 73–96. [22] M. Hovey, J. H. Palmieri and N. P. Strickland, Axiomatic stable homotopy theory. Memoirs Amer. Math. Soc. 128 (1997), no. 610. [23] B. Keller, Deriving DG categories. Ann. Sci. École Norm. Sup. 27 (1994), 63–102. [24] H. Krause, The stable derived category of a noetherian scheme. Compositio Math. 141 (2005), 1128–1162. [25] A. Neeman, The chromatic tower for D.R/. Topology 31 (1992), 519–532.


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On cluster theory and quantum dilogarithm identities Bernhard Keller

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Quantum dilogarithm identities from Dynkin quivers, after Reineke . 2 Sketch of proof of Reineke’s theorem . . . . . . . . . . . . . . . . 3 Quantum dilogarithm identities from quivers with potential, after Kontsevich–Soibelman . . . . . . . . . . . . . . . . . . . . . . . . 4 DT-invariants and mutations . . . . . . . . . . . . . . . . . . . . . 5 Compositions of mutations . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . .

. . . .

85 87 92

. 94 . 96 . 104 . 113

Introduction The links between the theory of cluster algebras [19], [20], [6], [22] and functional identities for the Rogers dilogarithm first became apparent through Fomin–Zelevinsky’s proof [21] of Zamolodchikov’s periodicity conjecture for Y -systems [64] (we refer to [52] and the references given there for the latest developments in periodicity in cluster theory and its applications to T -systems, Y -systems and dilogarithm identities). This link was exploited by Fock–Goncharov [17], [18], who emphasize the central rôle of the (quantum) dilogarithm both for commutative and for quantum cluster algebras and varieties. The quantum dilogarithm is also a key ingredient in Kontsevich–Soibelman’s interpretation [50] of cluster transformations in the framework of Donaldson–Thomas theory. Indeed, Kontsevich–Soibelman show that, under suitable technical hypotheses, if two quivers with potential are related by a mutation [13], then their non commutative DT-invariants (cf. Section 4.7) are linked by the composition of a monomial transformation with the adjoint action of a quantum dilogarithm. This composition coincides with Fock–Goncharov’s cluster transformation for quantum Y -variables [17]. Therefore, Kontsevich–Soibelman’s categorical setup for DT-theory contains a ‘categorification’ of the quantum Y -seed mutations and thus, modulo passage to the ‘double torus’ [17] or to ‘principal coefficients’ [22], of the quantum X -seed mutations. By extending this idea to compositions of mutations Nagao [51] has succeeded in deducing the main theorems on the (additive) categorification of cluster algebras [14] (cf. also [53]) from Joyce’s [34], [36] and Joyce–Song’s [37] results in DT-theory (cf. also [8]). Our aim in the present survey is to give an introduction to this circle of ideas using quantum dilogarithm identities as a leitmotif. We start with the classical pentagon identity (Section 1.1). Following Reineke [55], [56] and Kontsevich–Soibelman [50], [47] we link it to the study of stability functions for a Dynkin quiver of type A2 and

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present Reineke’s generalization to arbitrary Dynkin quivers (Theorem 1.6). We sketch Reineke’s beautiful and instructive proof in Section 2. The result can be interpreted as stating that the refined DT-invariant of a Dynkin (and even an acyclic) quiver is well defined. In this form, it is conjectured to generalize to an arbitrary quiver with a potential with complex coefficients (Section 3.1). Evidence for this is given in Kontsevich–Soibelman’s deep work [50], [49]. In Section 4, we study the behaviour of the refined DT-invariant under mutations following Section 8.4 of [50]. We begin by recalling the categorical setup: The category of finite-dimensional representations of the Jacobi-algebra of the given quiver with potential is embedded as the heart of the canonical t-structure in the 3-Calabi–Yau category Df d , the full subcategory formed by the homologically finite-dimensional dg modules over the Ginzburg [29] dg algebra associated with the given quiver with potential. In Df d , the simple representations form a spherical collection (Section 4.2) in the sense of [50]. Mutation is modeled by ‘tilting’ the heart (Section 4.3), an idea going back to Bridgeland [10], cf. also [9], [11]. The comparison formula for the refined DT-invariants then becomes a simple consequence of the freedom in the choice of a stability function (Section 4.4). The refined DT-invariant is defined to be rational (Section 4.5) if its adjoint action is given by a rational transformation. This is the case in large classes of examples coming from representation theory, Lie theory and higher Teichmüller theory. In this case, the adjoint action is the ‘non commutative DT-invariant’ [62], whose behaviour under mutations is governed by Fock–Goncharov’s mutation rule for quantum Y -variables (Section 4.7). It is remarkable that this rule is involutive (Section 4.8). In Section 5, we study compositions of Fock–Goncharov mutations via functors FG W Cclop ! Sf

and

FG W Cltop ! Sf

where Sf is the groupoid of skew fields, Ccl the groupoid of cluster collections in the ambient 3-Calabi–Yau category Df d and Clt the groupoid of cluster-tilting sequences in the cluster category C . The main results of (quantum) cluster theory may be reformulated by saying that the image of a morphism ˛ W S ! S 0 of the groupoid of cluster collections, where S is the inital collection, only depends on the target S 0 (Theorem 5.2). We define an autoequivalence of Df d respectively C to be reachable if its effect on the inital cluster collection (respectively cluster tilting sequence) is given by a composition of mutations. We obtain a homomorphism F 7! .F / from the group of reachable autoequivalences of the ambient triangulated category to the group of automorphisms of the functor FG (Sections 5.3 and 5.7). If the loop functor D †1 of the ambient category is reachable, then the non commutative DT-invariant equals .†1 / (under the assumptions which ensure that it is well defined). For example, in the context of the quivers with potential associated with pairs of Dynkin diagrams [39], the loop functor is reachable and of finite order, which shows that the non commutative DT-invariant is of finite order in this case. This fact is of interest in string theory, cf. Section 8 in [12], which builds on [24], [26], [25]. Following an idea of Nagao [51] we introduce the groupoid of nearby cluster collections in Section 5.8 and show how it can

On cluster theory and quantum dilogarithm identities

87

be used to prove Theorem 5.2 (Section 5.13) and to reconstruct the refined DT-invariant (Theorem 5.12). We conclude by presenting a purely combinatorial realization of the groupoid of nearby cluster collections: the tropical groupoid. In many cases, it allows for a purely combinatorial construction of the refined DT-invariant (Section 5.14). Acknowledgment. I thank Sarah Scherotzke for providing me with her notes of the lectures and Tom Bridgeland, Changjian Fu, Christof Geiß, Pedro Nicolás, Pierre-Guy Plamondon and especially Mike Gorsky for their remarks on a preliminary version of this article. I am grateful to Kentaro Nagao, So Okada and Cumrun Vafa for stimulating (email) conversations and to Maxim Kontsevich for many inspiring lectures on the subject.

1 Quantum dilogarithm identities from Dynkin quivers, after Reineke 1.1 The pentagon identity. Let q 1=2 be an indeterminate. We denote its square by q. The quantum dilogarithm is the (logarithm of the) series 2

E.y/ D 1 C

q 1=2 q n =2 y n y C C n C q1 .q 1/.q n q/ : : : .q n q n1 /

(1.1)

considered as an element of the power series algebra Q.q 1=2 /ŒŒy. Notice that the denominator of its general coefficient is the polynomial which computes the order of the general linear group over a finite field with q elements. Let us define the quantum exponential by 1 X yn expq .y/ D ; ŒnŠ nD0 where ŒnŠ is the polynomial which computes the number of complete flags in an ndimensional vector space over a field with q elements. Then we have

E.y/ D expq

q 1=2 y ; q1

(1.2)

which explains the choice of the notation E (for the mysterious scaling factor, cf. Remark 3.3). The quantum dilogarithm has many remarkable properties (cf. e.g. [63] and the references given there), among which we single out the following. Theorem 1.2 (Schützenberger [60], Faddeev–Volkov [15], Faddeev–Kashaev [16]). For two indeterminates y1 and y2 which q-commute in the sense that y1 y2 D qy2 y1 ; we have the equality E.y1 /E.y2 / D E.y2 /E.q 1=2 y1 y2 /E.y1 /:

(1.3)

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As shown in [16], this equality implies the classical ‘pentagon identity’ for Rogers’ dilogarithm. 1.3 Reineke’s identities. Our aim in this section is to associate, following Reineke [56], an identity analogous to (1.3) with each simply laced Dynkin diagram so that the above identity corresponds to the diagram A2 . So let be a simply laced Dynkin diagram and let Q be a quiver (= oriented graph) with underlying graph . We will associate a whole family of quantum dilogarithm products with Q. All these products will be equal and among the resulting equalities, we will obtain the required generalization of the pentagon identity (1.3). The quantum dilogarithm products will be constructed from ‘stability functions’ on the category of representations of Q. We first need to introduce some notation: Let k be a field. Let A be the category of representations of the opposite quiver Qop with values in the category of finitedimensional k-vector spaces (we refer to [58], [23], [4], [3] for quiver representations). Let I D f1; : : : ; ng be the set of vertices of Q. For each vertex i , let Si be the simple representation whose value at i is k and whose value at all other vertices is zero. Let K0 .A/ be the Grothendieck group of A. It is a finitely generated free abelian group and admits the classes of the representations Si , i 2 I , as a basis. A stability function on A is a group homomorphism Z W K0 .A/ ! C to the underlying abelian group of the field of complex numbers such that, for each non zero object X of A, the number Z.X / is non zero and its argument, called the phase of X , lies in the interval Œ0; Œ, cf. Figure 1. A non zero object X of A is semistable (respectively stable) if for each non zero proper subobject Y of X , the phase of Y is less than or equal to the phase of X (respectively strictly less than the phase of X ). Sometimes, the homomorphism Z is called a central charge. Since it is a group homomorphism, it is determined by the complex numbers Z.Si /, i 2 I . Notice that each simple object of A is stable (since it has no non zero proper subobjects). Proposition 1.4 (King [46]). Let Z W K0 .A/ ! C be a stability function. For each real number , let A be the full subcategory of A whose objects are the zero object and the semistable objects of phase . a) The subcategory A of A is stable under forming extensions, kernels and cokernels in A. In particular, it is abelian and its inclusion into A is exact. Its simple objects are precisely the stable objects of phase . b) Each object X admits a unique filtration 0 D X0 X1 Xs D X whose subquotients are semistable with strictly decreasing phases. It is called the Harder–Narasimhan filtration (or HN-filtration) of X . Its subquotients are the HN-subquotients of X .

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O

Z.X / E phase of X

/

Figure 1. Image of a non zero object under the central charge.

Notice that by part a), all stable objects are indecomposable and their endomorphism algebras are (skew) fields. Under the assumptions of the proposition, let A be the full subcategory of A formed by the objects X all of whose HN-subquotients have phase greater than or equal to . Define the full subcategory A 0. (4.11) 1=2 op When q is specialized to 1 and Q replaced with Q , these formulas yield the transformation rule for Y -seeds in the sense of Fomin–Zelevinsky, cf. [22]. Notice that in deducing formula (4.9), we chose to pre-compose with †. If we post-compose with †, we obtain a variant of the intertwiner which fortunately has the same good properties and which, upon specialization of q 1=2 to 1, yields Fomin–Zelevinsky’s transformation rule for Y -seeds (without replacing Q by Qop ). 4.8 Mutation is an involution. In the setting of Section 4.3, one checks easily that . C k .A// D A: k .A/ D twSk . .A//. Thus, we obtain We also know that C k k .C .B// D twSk .B/ C k k .A/. Thus, mutation of hearts is an involution only up to the braid group for B D k action. Remarkably, the mutation intertwiner (4.9) ‘squares’ to the identity (as do its variants) by the following lemma (cf. Lemma 3.7 of [18]), where we only write ‘unprimed’ variables to avoid clutter in the notation.

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Lemma 4.9. a) We have Ad.E.yk // B Ad.E.yk1 // D tk1

where tk W K0 .A/ ! K0 .A/ is induced by the twist functor twSk . b) The composition Frac.A2 .Q/ /

'C BAd.E.yk //

k

/ Frac.AQ0 /

'C BAd.E.yk //

/ Frac.AQ /

is the identity. c) The right and left intertwiners Ad.E.yk // B 'C and ' B Ad.E.yk //1 are equal. Proof. a) If yk yi D q m yi yk , one computes, as in Lemma 4.6, that Ad.E.yk // Ad.E.yk1 //.yi / D yi E.q m yk /E.yk /1 E.q m yk1 /E.yk1 /1

D y ei Cmek D tk1 .yi /:

b) and c) are an easy consequences.

5 Compositions of mutations 5.1 The groupoid of cluster collections. To state identities between longer compositions of intertwiners (4.9), we now introduce a suitable groupoid (a category where all morphisms are invertible). We fix a quiver with potential .Q; W / and work in the setup of Section 4.2. A cluster collection [50] is a sequence of objects S10 , …, Sn0 of Df d such that a) the Si0 are spherical; b) for i ¤ j , the graded space Ext .Si0 ; Sj0 / vanishes or is concentrated either in degree 1 or in degree 2; c) the Si0 generate the triangulated category Df d . One can show [57], [43] that in this case, the closure A0 of the Si0 under iterated extensions is the heart of a non degenerate bounded t -structure on Df d and that the simples of A0 are the Si0 (up to isomorphism). On the other hand, if A0 is the heart of a non degenerate bounded t -structure on Df d , then the simples .S10 ; : : : ; Sn0 / will satisfy c) but not necessarily a) and b). If they do, we call A0 a cluster heart. In this way, we obtain a bijection between cluster hearts and permutation classes of cluster collections, cf. [43]. The groupoid of cluster collections Ccl D CclQ has as objects the cluster collections S 0 reachable from the sequence S D .S1 ; : : : ; Sn / of the simples of the initial cluster heart A by a sequence S D S .0/

/ S .1/

/ :::

/ S .N / D S 0


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of (positive and negative) mutations and permutations, where all the intermediate sequences S .i/ are cluster collections. The morphisms of Ccl are the formal compositions of mutations and permutations subject to the relations valid in the symmetric group and the relations

B "k D " .k/ B

for all permutations and mutations "k . For example, for the quiver Q W 1 ! 2, with the notations of Section 1.3, it is easy to check that we have the following two morphisms from .S1 ; S2 / to .†1 S2 ; †1 S1 / in Ccl: .S1 ; S2 /

C 1

/ .†1 S1 ; S2 /

C 2

/ .†1 S1 ; †1 S2 /

/ .†1 S2 ; †1 S1 /

(5.1)

and .S1 ; S2 /

C 2

/ .P2 ; †1 S2 /

C 1

/ .†1 P2 ; S1 /

C 2

/ .†1 S2 ; †1 S1 /:

(5.2)

Given a cluster collection S 0 D .S10 ; : : : ; Sn0 / its quiver QS 0 has the vertex set f1; : : : ; ng and the number of arrows from i to j equals the dimension of Ext1 .Sj0 ; Si0 /. Using the intertwiners (4.9) and the natural action of the permutation groups, we clearly obtain a functor FG W Cclop ! Sf; to the groupoid Sf of skew fields which takes a cluster collection S 0 to Frac.AQS 0 /. The following theorem is a corollary of the theory of cluster algebras and their (additive) categorification as developed by Fomin–Zelevinsky, Berenstein–Fomin–Zelevinsky, Fock–Goncharov, Derksen–Weyman–Zelevinsky …. Theorem 5.2. The image of a morphism ˛ W S ! S 0 of the groupoid of cluster collections under the functor FG only depends on the orbit of S 0 under the braid group Braid.Q/. We will sketch a proof in Section 5.13 below. Notice that already the statement that the image of ˛ only depends on S 0 is very strong. By the easy Lemma 4.9, it implies the statement of the theorem. As an example, consider the two morphisms (5.1) and (5.2). By the theorem, they yield the equality Ad.E.y1 //'1 Ad.E.y2 //'2 D Ad.E.y2 //'2 Ad.E.y1 //'1 Ad.E.y2 //'2

(5.3)

in the groupoid Sf. Notice that the symbols '1 and '2 denote different maps depending on the source and target fields. They are given, in the order of occurrence above, by the matrices 1 0 1 0 1 0 1 1 1 0 ; ; ; ; : 0 1 0 1 1 1 0 1 1 1

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Thus, we have Ad.E.y1 //'1 Ad.E.y2 //'2 D Ad.E.y1 /E.y2 //'1 '2 and Ad.E.y2 //'2 Ad.E.y1 //'1 Ad.E.y2 //'2 D Ad.E.y1 /E.q

1=2

(5.4)

y1 y2 /E.y1 //'2 '1 '2 : (5.5)

Since we have '1 '2 D '2 '1 '2 , the equality (5.3) is in fact a consequence of the pentagon identity (1.3) E.y1 /E.y2 / D E.y2 /E.q 1=2 y1 y2 /E.y1 /: 5.3 Action of autoequivalences of the derived category. Let F W Df d ! Df d be a triangle equivalence such that F is reachable, i.e. there is a sequence of (per-)mutations linking the initial cluster collection S D .S1 ; : : : ; Sn / to F S . Thus, the collection F S still belongs to Ccl and of course, so does F S 0 for any other cluster collection S 0 in Ccl. With F , we will associate a canonical automorphism .F / of the functor FG W Cclop ! Sf :

Indeed, for any cluster collection S 0 , we have an isomorphism of quivers QF S 0 D QS 0 given by the bijection F Si0 7! Si0 on the set of vertices. Thus, we have a canonical isomorphism of skew fields FG.F S 0 / ! FG.S 0 /:

We define .F /.S 0 / W FG.S 0 / ! FG.S 0 / to be the composition of this isomorphism with FG.˛/ for any morphism ˛ W S 0 ! F S 0 . It is immediate from Theorem 5.2, that .F / is indeed an automorphism of FG.S 0 /, and that defines a homomorphism from the group of (isomorphism classes of) reachable autoequivalences of Df d to the group of automorphisms of the functor FG. The following theorem is based on Nagao’s ideas [51]. Theorem 5.4. Suppose that the inverse suspension functor †1 W Df d ! Df d is reachable. Then .Q; W / is Jacobi-finite. If moreover Conjecture 3.2 holds for .Q; W / so that the refined DT-invariant EQ;W is well defined, then it is rational and DTQ;W equals .†1 /. C For example, consider the quiver Q W 1 ! 2. Then the morphism ˛ D C 2 1 of 1 (5.1) shows that † is reachable. Now we use the equality (5.4) and the fact that the composition '1 '2 of '1 with '2 in (5.4) equals † to deduce that, in accordance with the theorem, we have

DTQ;W D Ad.E.y1 /E.y2 // B † D FG.˛/ D .†1 /:


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5.5 The groupoid of cluster tilting sequences. In view of Theorem 5.2, it is natural to try and ‘factor out’ the braid group action on the category Df d . This is, in a certain sense, what is achieved by the passage to the cluster category. For simplicity, let us assume that .Q; W / is Jacobi-finite (the general case can be treated using Plamondon’s results [54], [53]). The cluster category CQ;W is defined [1] as the triangle quotient per./=Df d ./. It is a triangulated category with finite-dimensional morphism spaces which is 2-Calabi–Yau and admits the image T of as a cluster-tilting object, i.e. we have Ext1 .T; T / D 0 and any object X such that Ext1 .T; X / D 0 belongs to the category add.T / of direct summands of finite direct sums of copies of T . For a cluster tilting object T 0 the quiver QT 0 is the quiver of the endomorphism algebra of T 0 . A cluster tilting sequence is a sequence .T10 ; : : : ; Tn0 / of pairwise non isomorphic indecomposables of CQ;W whose direct sum is a cluster tilting object T 0 whose associated quiver QT 0 does not have loops or 2-cycles. There is a canonical mutation operation on all cluster tilting objects, cf. [32]. It yields a partially defined mutation operation on the cluster-tilting sequences. The groupoid of cluster-tilting sequences Clt D CltQ has as objects the cluster-tilting sequences of CQ;W which are reachable from the image T D .T1 ; : : : ; Tn / of the sequence of the dg modules ei , i 2 Q0 . Its morphisms are defined as formal compositions of mutations and permutations as for the groupoid of cluster collections. The following theorem proved in [42] yields a link between the groupoids Clt and Ccl. Theorem 5.6 (Keller–Nicolás [42]). There is a canonical bijection from the set of Braid.Q/-orbits of cluster collections in Df d to the set of cluster tilting sequences in CQ;W . It is compatible with mutations and permutations and preserves the quivers. The bijection is based on the exact sequence of triangulated categories 0

/ Df d

/ per./

/ CQ;W

/ 0:

Namely, we define a silting sequence to be a sequence .P10 ; : : : ; Pn0 / of objects in per./ such that a) Hom.Pi0 ; †p Pj0 / vanishes for all p > 0, b) the Pi0 generate per./ as a triangulated category and c) the quiver of the subcategory whose objects are the Pi0 does not have loops nor 2-cycles. For such a sequence, the subcategory A0 of Df d formed by the objects X such that Hom.Pi0 ; †p X / D 0 for all i and all p ¤ 0 is the heart A0 of a non degenerate bounded t-structure whose simples form a cluster collection. The map from silting sequences to cluster collections thus defined is a bijection. We obtain the bijection of the theorem by composing its inverse with the map taking a silting sequence to its image in the cluster category, which one shows to be a cluster tilting sequence using Theorem 2.1 of [1]. Thanks to the theorem, we can define a functor FG W Cltop ! Sf

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by sending a cluster-tilting sequence T 0 to the image under FG W Cclop ! Sf of the corresponding cluster collection S 0 . 5.7 Action of autoequivalences of the cluster category. In analogy with Section 5.3, if .Q; W / is Jacobi-finite, one obtains a homomorphism, still denoted by , from the group of reachable autoequivalences of the cluster category CQ;W to the automorphism group of the induced functor FG W CltQ ! Sf : Again, if the suspension functor †1 of the cluster category is reachable, we find that .†1 / equals DTQ;W . In particular, if † is of finite order N as an autoequivalence of the cluster category, then the automorphism DTQ;W of Frac.AQ / is of finite order dividing N . Applications of these ideas include the (quantum version of the) periodicity theorem for the Y -systems (and T -systems) associated with pairs of simply E and E 0 be alternating orientations of laced Dynkin diagrams [38], [39]. Indeed, let E E 0 and W the canonical simply laced Dynkin diagrams, Q the triangle product potential on Q defined in Proposition 5.12 of [39]. Then the cluster category CQ;W is equivalent to the cluster category CA˝k A0 associated in [1] with the tensor product of E and A0 D E 0 , by Proposition 5.12 of [39]. Let be the the path algebras A D k sequence of mutations of Q defined in (3.6.1) of [39]. Then by Section 7.4 of [39], the Zamolodchikov autoequivalence Za D 1 ˝ 1 W CA˝k A0 ! CA˝k A0

(5.6)

is reachable and its image under is . Moreover, if h and h0 are the Coxeter numbers of and 0 , then we have Zah D h ˝ 1 D †2 ˝ 1 D †2 :

(5.7)

Since we also have (cf. the proof of Theorem 8.4 in [loc. cit.]) Za D 1 ˝ ;

we find that

0

0

Zah D 1 ˝ h D 1 ˝ †2 D †2 : 2

(5.8) (5.9)

Notice that this implies in particular that † is reachable and yields a sequence of mutations whose composition is the square of the non commutative DT-invariant DTQ;W . Equation (5.9) confirms equation (8.19) of [12]. In fact, it is not hard to show that †1 is also reachable. By combining equations (5.7) and (5.9) we obtain 0 0 ZahCh D 1 and thus hCh D 1. By applying the functor FG to this last equation, we obtain a statement equivalent to the quantum version of the periodicity for the Y system associated with .; 0 /. Notice that in the course of this reasoning, we have also found that 0 †2 D Zah D Zah :


109

This implies that the non commutative DT-invariant DTQ;W is of order dividing 2

h C h0 ; gcd.h; h0 /

a fact already present in Section 8.3.2 of [12]. 5.8 Nearby cluster collections. Let S 0 be a cluster collection and A0 the associated heart. We define A0 to be a nearby heart and S 0 to be a nearby cluster collection if there is a torsion pair .U; V / in A (cf. Section 1.3) such that A0 is the full subcategory formed by the objects X of Df d such that the object H 0 .X / lies in V , the object H 1 .X / in U and H p .X / vanishes for all p ¤ 0; 1. Theorem 5.9 (Nagao [51]). a) Let A be the initial heart and A0 a nearby heart. Then each simple Sk0 of A0 either lies in A or in †1 A. b) Each nearby cluster collection S 0 is determined by the classes ŒSi0 of its objects in K0 .Df d /. Part a) of the theorem is equivalent to the ‘sign-coherence’ of the classes ŒPi0 2 K0 .per /, where the Pi0 form the silting sequence associated with S 0 . The ‘signcoherence’ is also proved in [54] and, in another language, in [14]. Part b) is an easy consequence. The following theorem goes back to the insight of Nagao [51]. In this form, it follows [40] from the proof of Theorem 2.18 in [54] and the generalization of Theorem 5.6 to quivers with potential which are not necessarily Jacobi-finite, cf. [42]. Theorem 5.10. a) Each cluster collection S 0 belongs to the Braid.Q/-orbit of a unique nearby cluster collection .S 0 /. b) If S 0 and S 00 are cluster collections related by a mutation, then .S 0 / and .S 00 / are related by a mutation. More precisely, if S 00 D ˙ .S 0 / for some sign ˙ and some k " 00 0 1 k n, then .S / D k . .S //, where " D C1 if the object Sk0 of S 0 lies in A and " D 1 if Sk0 lies in †1 A. Clearly, a cluster collection is reachable iff each cluster collection in its Braid.Q/orbit is reachable. Thus, the reachable nearby cluster collections form a system of representatives for the Braid.Q/-orbits in Ccl. Let Ncc denote the full subgroupoid of Ccl formed by the reachable nearby cluster collections. Then the projection restricts to a functor / / Ccl = Braid.Q/ Ncc which is full and yields a bijection between the sets of objects. The map S 0 7! .S 0 / yields a (non functorial!) section. We have the following diagram of groupoids and functors / Ccl Ncc SSS NNN SSS NNN SSS NNN SSS NNN SSS) ) & / Sfop : Ccl = Braid.Q/

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The resulting functor Nccop ! Sf can be refined so as to yield identities between y Q (rather than just in the automorphism group of Frac.AQ /): products of series in A Let U A be a torsion subcategory associated with a reachable nearby cluster heart A0 (so that an object X of Df d belongs to A0 iff the object H 1 .X / belongs to U, the object H 0 .X / belongs to U? and H p .X / vanishes for all p ¤ 0). Let ˛ W S ! S 0 be a morphism of Ncc, where S is the initial cluster collection and S 0 a cluster collection whose associated heart is A0 . After permuting the elements of S 0 we may assume that no permutations occur in ˛ so that ˛ is a composition of mutations of nearby cluster collections k1 kN / S .1/ k2 / / S .N / : S D S .0/ For 1 i N , let ˇi be the class ki -th object of S .i/ in K0 .Df d /. By Theorem 5.9, either ˇi belongs to the positive cone determined by A or to its opposite. We put "i D C1 in the first case and "i D 1 in the second. Then "i ˇi is a positive integer linear combination of the classes ŒSj . We write E."i ˇi / for E.y v /, where v 2 N n is the vector of the coefficients of the decomposition of "i ˇi in the basis given by the ŒSj . Define the invertible element EU;˛ of AQ by EU;˛ D E."1 ˇ1 /"1 E."2 ˇ2 /"2 : : : E."N ˇN /"N : Theorem 5.11 ([40]). The element EU;˛ does not depend on the choice of ˛. The proof of the theorem is independent of Conjecture 3.2. It is based on the work of Nagao [51], Plamondon [54], [53], Derksen-Weyman-Zelevinsky [13], [14], Berenstein-Zelevinsky [7], …. We put EU D EU;˛ for any ˛. Notice that if the cluster heart †1 A is reachable, the corresponding torsion subcategory is U D A. Theorem 5.12 ([40]). Suppose that the ground field equals C and that Conjecture 3.2 holds for .Q; W / so that the refined DT-invariant EQ;W of (3.1) is well defined. If the cluster heart †1 A is reachable, then EA equals EQ;W . 5.13 On the proof of the main theorem. To prove Theorem 5.2, by Lemma 4.9, it suffices to show that if ˛ W S ! S 0 is a morphism of Ccl to a reachable nearby cluster collection, then FG.˛/ only depends on S 0 . The proof [40] of this fact uses 1) the technique of the ‘double torus’ (cf. e.g. [17]) to reduce the statement to a statement about seeds in quantum cluster algebras; 2) Berenstein–Zelevinsky’s theorem which states that the exchange graph of the quantum cluster algebra of a quiver is canonically isomorphic to the exchange graph of its cluster algebra (Theorem 6.18 of [7]); 3) the expression of the (classical) cluster variables in terms of quiver Grassmannians first obtained in this generality by Derksen–Weyman–Zelevinsky [14], cf. also [51], [54].

111


Among these three ingredients, the third one is perhaps the deepest. Let us make it explicit: Let S 0 be a reachable nearby cluster collection. After permuting its objects, we may assume that there is a sequence of mutations transforming the initial cluster collection S to the given cluster collection S 0 . This sequence determines a vertex t in the n-regular tree and thus a cluster .Xi .t /; 1 i n/ in the cluster algebra associated with Q, cf. [22]. Following [51] and [53], we can express the cluster variables Xj .t / in terms of the cluster collection S 0 as follows: Let .T10 ; : : : ; Tn0 / be the silting sequence associated with S 0 and .T1 ; : : : ; Tn / the initial silting sequence. Define the integers gij , 1 i; j n, by the equality ŒTj0

D

n X

gij ŒTi

iD1

in K0 .per /. Then we have Xj .t / D

n Y iD1

g xi ij

X

.Gre .H

1

.Tj0 ///

e

n Y

hSi ;ei

xi

;

iD1

where Gre is the Grassmannian of submodules of dimension vector e and the Euler characteristic (for singular cohomology with rational coefficients of the underlying topological space). 5.14 The tropical groupoid. We will exhibit a groupoid defined in purely combinatorial terms which, if the potential W is generic, is isomorphic to the groupoid of nearby cluster collections (and thus admits a full surjective functor to the groupoid of reachable cluster collections modulo the braid group action and to the groupoid of tiltz be the quiver obtained from Q by adding ing sequences in the cluster category). Let Q 0 0 a new vertex i and a new arrow i ! i for each vertex i of Q. The new vertices i 0 are called frozen because we never mutate at them. The tropical groupoid Trp D TrpQ z by mutating at the non frozen vertices has as objects all the quivers obtained from Q 1; : : : ; n. Its morphisms are formal compositions of mutations at the non frozen vertices and permutations of these vertices as in the definition of the groupoid of cluster tilting sequences in Section 5.7. We construct a morphism of groupoids Ncc ! Trp as follows: For a reachable z nearby cluster collection S 0 , define the quiver q.S 0 / to have the same vertices as Q, such that the full subquiver on 1; : : : ; n is the Ext-quiver of S 0 , there are no arrows between frozen vertices, and for each old vertex i , the number of arrows from i to a frozen vertex j 0 is the integer bij defined by the equality ŒSi0

D

n X

bij ŒSj

j D1

in K0 .Df d /. Here, if bij is negative, we draw bij arrows from j 0 to i . From Theorem 5.9, we deduce the following corollary.

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Corollary 5.15. a) The quiver q.S 0 / uniquely determines S 0 . ! Trp. b) The map S 0 7! q.S 0 / underlies a unique isomorphism of groupoids Ncc Now we can give a purely combinatorial version of Theorem 5.11: Let k D .k1 ; : : : ; kN / be a sequence of vertices of Q (no frozen vertices are allowed to ocz be the quiver cur). Let k .Q/ z kN kN 1 : : : k1 .Q/; z s/ be the quiver and, more generally, for each 1 s N , let Q.k; z ks1 ks2 : : : k1 .Q/: z

Let B Q.k;s/ be the antisymmetric matrix associated with this quiver and let ˇs be the vector n X z bkQ.k;s/ ˇs D 0 ej s ;j j D1

in Zn . We know from Theorem 5.9 that either all components of ˇs are non negative or all are non positive. We put "s D C1 in the first case and "s D 1 in the second. Now we define E.k/ D E."1 ˇ1 /"1 E."2 ˇ2 /"2 : : : E."N ˇN /"N : Let k0 be another sequence of vertices of Q. Theorem 5.16. If there is an isomorphism of quivers z z k .Q/ ! k0 .Q/

which is the identity on the frozen vertices j 0 , 1 j n, then we have E.k/ D E.k0 / y Q. in A For example, if we apply the theorem to Q W 1 ! 2 and the sequences k D .1; 2/ and k0 D .2; 1; 2/, cf. Figure 5, then all the ˇs are positive and we find the pentagon identity (1.3). If we use k D .1; 2; 1/ and k0 D .2; 1/ instead, then for k, the vector ˇ3 is negative and we find the identity E.y1 /E.y2 /E.y1 /1 D E.y2 /E.q 1=2 y1 y2 /; which is of course equivalent to (1.3). z 0 be a quiver of the tropical groupoid TrpQ . Define a non frozen vertex i Let Q z 0 to be green if there are no arrows from frozen vertices to i and red otherwise. of Q z 0 to be maximal if all non frozen vertices of Define a sequence k of green vertices of Q 0 z / are red. In many cases, the following proposition allows one to construct the k .Q refined DT-invariant combinatorially (cf. also Section 5.2, page 49 of [25]).

113


z admits a maximal green sequence k. Then the Proposition 5.17. Suppose the Q 1 cluster heart † A is reachable and, if the refined DT-invariant EQ;W is well defined, it equals E.k/. /.-, ()*+ 1

/.-, ()*+ /2 2

1

1O o

/.-, ()*+ 2

10

20

10

20

12

?? ? 2 ?

1O

/2 O

10

20

/.-, ()*+ 1 ?o ? 2O ?? ? / ?? ? 10 20

66 66 66 66 66 66212 66 66 66 66 66 isom

1 o_?? ?2 ?? ? ?? ? 10 20

?? 1 ?? // .-, ()*+ 1O _?? 2 ?? ? ?? ? 10 20 2

Figure 5. The two maximal green sequences for A2 .

In Figure 5, the two maximal green sequences for the quiver AE2 are given (green vertices are encircled). Examples of classes of quivers Q to which the proposition applies include those enumerated in Section 4.5. The quiver mutation applet [41] makes it easy to search for maximal green sequences. They exist for acyclic quivers, for square products of acyclic quivers and also for the quivers associated in [5] with each pair consisting of an acyclic quiver and a reduced expression of an element in the Coxeter group associated with its underlying graph. For this case, a maximal green sequence is constructed in Section 12 of [27].

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[45] B. Keller, D. Yang and G. Zhou, The Hall algebra of a spherical object. J. London Math. Soc. 80 (2009), 771–784. [46] A. D. King, Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. 45 (1994), 515–530. [47] M. Kontsevich, Master class on wall-crossing. Lectures at the Centre for quantum geometry of moduli spaces, Aarhus University, August 2010, available at qgm.au.dk/calendar/videorecordings. [48] M. Kontsevich, Private communication. February 2, 2011. [49] M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants. Preprint, arXiv:1006.2706 [math.AG]. [50] M. Kontsevich and Y. Soibelman, Stability structures, Donaldson-Thomas invariants and cluster transformations. Preprint, arXiv:0811.2435 [math.AG]. [51] K. Nagao, Donaldson-Thomas theory and cluster algebras. Preprint, arXiv:1002.4884 [math.AG]. [52] T. Nakanishi, Periodicities in cluster algebras and dilogarithm identities. In Representations of Algebras and Related Topics. European Mathematical Society Series of Congress Reports, European Math. Soc. Publishing House, Zürich, 2011, 407–443. [53] P.-G. Plamondon, Cluster algebras via cluster categories with infinite-dimensional morphism spaces. Compositio Math., to appear, arXiv:1004.0830 [math.RT]. [54] P.-G. Plamondon, Cluster characters for cluster categories with infinite-dimensional morphism spaces. Advances Math. 227 (2011), 1–39. [55] M. Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli. Invent. Math. 152 (2003), 349–368. [56] M. Reineke, Poisson automorphisms and quiver moduli. J. Inst. Math. Jussieu 9 (2010), 653–667. [57] J. Rickard and R. Rouquier, Stable categories and reconstruction. Preprint, arXiv:1008.1976 [math.RT]. [58] C. M. Ringel, Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics 1099, Springer-Verlag, Berlin, 1984. [59] O. Schiffmann, Lectures on Hall algebras. Preprint, arXiv:math/0611617 [math.RT] [60] M. P. Schützenberger, Une interprétation de certaines solutions de l’équation fonctionnelle: F .x C y/ D F .x/F .y/. C. R. Acad. Sci. Paris 236 (1953), 352–353. [61] P. Seidel and R. Thomas, Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108 (2001), 37–108. [62] B. Szendr˝oi, Non-commutative Donaldson-Thomas invariants and the conifold. Geom. Topol. 12 (2008), 1171–1202. [63] D. Zagier, The dilogarithm function. In Frontiers in Number Theory, Physics, and Geometry. II, Springer-Verlag, Berlin, 2007, 3–65. [64] Al. B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories. Phys. Lett. B 253 (1991), 391–394.

Quantum loop algebras, quiver varieties, and cluster algebras Bernard Leclerc

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1 Representations of quantum loop algebras . . . . . . . 2 Nakajima quiver varieties and irreducible q-characters . 3 Tensor structure . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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117 118 125 139 150

Introduction This article is an extended version of a minicourse given at the workshop of the 14th International Conference on Representations of Algebras (ICRA XIV), held in August 2010 in Tokyo. The aim of the minicourse, consisting of three lectures, was to explain some new connections between cluster algebras and the representation theory of quantum affine algebras. At the very origin of the theory of quantum groups is the search of an algebraic procedure for constructing solutions of the quantum Yang–Baxter equation R12 .u/R13 .uv/R23 .v/ D R23 .v/R13 .uv/R12 .u/:

(1)

The unknown R.u/ of this equation is an endomorphism of V ˝ V for some finitedimensional vector space V . This endomorphism depends on a parameter u 2 C , and we denote by Rij .u/ the endomorphism of V ˝ V ˝ V acting via R.u/ on the product of the i th and j th factors, and via IdV on the remaining factor. The Yang–Baxter equation appeared in several different guises in the literature on integrable systems (see [25] for a nice selection of early papers on the subject). As is well known, Drinfeld and Jimbo showed how to associate a solution of (1) with every irreducible finite-dimensional representation V of a quantum loop algebra Uq .Lg/. Here g is a simple complex Lie algebra, Lg D g ˝ CŒt; t 1 is its loop algebra, and Uq .Lg/ denotes the quantum analogue of the enveloping algebra of Lg with parameter q 2 C not a root of unity. This gives a strong motivation for studying the category C of finite-dimensional representations of Uq .Lg/, and many authors have brought important contributions (see e.g. [2], [6], [7], [10], [20], [32]). In particular the simple objects of C have been classified by Chari and Pressley, an appropriate notion of q-character has been introduced by Frenkel and Reshetikhin, and, when g is of simply laced type, Nakajima

118

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has calculated the irreducible q-characters in terms of the cohomology of certain quiver varieties. After reviewing these results in the first two lectures, we will turn to more recent attempts to understand the tensor structure of C . In the case of g D sl2 , Chari and Pressley [6] proved that every simple object is a tensor product of Kirillov–Reshetikin modules. These irreducible representations (which are defined for every g) are very particular. They also come from the theory of integrable systems [27], [28] where they were first studied in relation with row-to-row transfer matrices. However, if g is different from sl2 , the Kirillov–Reshetikin modules are not the only prime simple objects and the situation is far more complicated. Thus, already for g D sl3 we do not know a factorization theorem for simple objects. In the last lecture, we will explain a general conjecture (for g of simply-laced type) which would imply that the prime tensor factorization of many simple objects can be described as the factorization of a cluster monomial into a product of cluster variables. In retrospect, this should not be too surprising, since one of the first applications of cluster combinatorics given by Fomin and Zelevinsky was the proof of Zamolodchikov’s periodicity conjecture for Y -systems [13], also intimately related with the representation theory of Uq .Lg/ [28]. Our conjecture, which involves a positive integer `, is now proved when ` D 1 [24], [35], but the general case remains open.

1 Representations of quantum loop algebras In this first lecture, we review the definition of quantum loop algebras, the classification of their finite-dimensional irreducible representations, and the notion of qcharacter. For simplicity, we only consider quantum loop algebras of simply laced type A; D; E. We also formulate the T -systems satisfied by the q-characters of the Kirillov–Reshetikhin modules. All this is illustrated in the case of Uq .Lsl2 /. We briefly explain the connection with the Yang–Baxter equation. Finally, we introduce some interesting tensor subcategories of the category of finite-dimensional representations, which will be used in §3. For a recent and more complete survey on these topics, see [5]. 1.1 The quantum loop algebra Uq .Lg/. Let g be a simple Lie algebra over C of type An ; Dn or En . We denote by I D Œ1; n the set of vertices of the Dynkin diagram, by A D Œaij i;j 2I the Cartan matrix, and by … D f˛i j i 2 I g the set of simple roots. Let Lg D g ˝ CŒt; t 1 be the loop algebra of g. This is a Lie algebra with bracket Œx ˝ t k ; y ˝ t l D Œx; y ˝ t kCl

.x; y 2 g; k; l 2 Z/:

(2)

Following Drinfeld [8], the enveloping algebra U.Lg/ has a quantum deformation Uq .Lg/. This is an algebra over C defined by a presentation with infinitely many generators C xi;r ; xi;r ; hi;m ; ki ; ki1

.i 2 I; r 2 Z; m 2 Z n f0g/;

(3)

Quantum loop algebras, quiver varieties, and cluster algebras

119

and a list of relations which we will not repeat (see e.g. [10]). These relations depend on q 2 C which we assume is not a root of unity. If xiC ; xi ; hi .i 2 I / denote the ˙ Chevalley generators of g then xi;r is a q-analogue of xi˙ ˝ t r , hi;m is a q-analogue m of hi ˝ t , and ki stands for the q-exponential of hi hi ˝ 1. In fact Uq .Lg/ is isomorphic to a quotient of the quantum enveloping algebra Uq .g/ O attached by Drinfeld and Jimbo to the affine Kac–Moody algebra g. O It thus inherits from Uq .g/ O the structure of a Hopf algebra. Our main object of study is the category C of finite-dimensional Uq .Lg/-modules1 . Since Uq .Lg/ is a Hopf algebra, C is an abelian monoidal category. It is well-known that C is not semisimple. We denote by R its Grothendieck ring. Given two objects M and N of C , the tensor products M ˝ N and N ˝ M are in general not isomorphic. However, they have the same composition factors with the same multiplicities, so R is a commutative ring. For every a 2 C there exists an automorphism a of Uq .Lg/ given by ˙ ˙ a .xi;r / D ar xi;r ;

a .hi;m / D am hi;m ;

a .ki˙1 / D ki˙1 :

(This is a quantum analogue of the automorphism x ˝ t k 7! ak .x ˝ t k / of Lg.) Each automorphism a induces an auto-equivalence a of C , which maps an object M to its pullback M.a/ under a . 1.2 q-characters. By Drinfeld’s presentation, the generators ki˙1 and hi;m are pairwise commutative. So every object M of C can be written as a finite direct sum of common generalized eigenspaces for the simultaneous action of the ki and of the hi;m . These common generalized eigenspaces are called the l-weight-spaces of M . (Here l stands for “loop”). The q-character of M , introduced by Frenkel and Reshetikhin [27], is a Laurent polynomial with positive integer coefficients in some indeterminates Yi;a .i 2 I; a 2 C /, which encodes the decomposition of M as the direct sum of its l-weight-spaces. More precisely, the eigenvalues of the hi;m .m > 0/ in an l-weightspace V of M are always of the form ki li X q m q m X m m .a / .b / i;r i;s m.q q 1 / rD1 sD1

(4)

for some nonzero complex numbers ai;r ; bi;s . Moreover, they completely determine the eigenvalues of the hi;m .m < 0/ and of the ki on V . We encode this collection of eigenvalues with the Laurent monomial mV D

ki Y Y i2I

1

rD1

Yi;ai;r

li Y

1 : Yi;b i;s

sD1

We only consider modules of type 1, a mild technical condition, see e.g. [7], §12.2 B.

(5)

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B. Leclerc

The collection of eigenvalues (4), or equivalently the monomial (5), will be called the ˙1 l-weight of V . Let Y D ZŒYi;a I i 2 I; a 2 C . One then defines the q-character of M 2 C by X dim V mV 2 Y; (6) q .M / D V

where the sum is over all l-weight spaces V of M (see [9], Proposition 2.4). Theorem 1.1 ([10]). The Laurent polynomial q .M / depends only on the class of M in R, and the induced map q W R ! Y is an injective ring homomorphism. The subalgebra of Uq .Lg/ generated by C ; xi;0

xi;0 ;

ki ;

ki1

.i 2 I /

is isomorphic to Uq .g/. Hence every M 2 C can be regarded as a Uq .g/-module by restriction. The l-weight-space decomposition of M is a refinement of its decomposition as a direct sum of Uq .g/-weight-spaces. Let P be the weight lattice of g, with basis given by the fundamental weights $i .i 2 I /. Let ZŒP be the group ring of P . As usual, 2 P is written in ZŒP as a formal exponential e to allow multiplicative notation. We denote by ! the ring homomorphism from Y to ZŒP defined by ! .Yi;a / D e $i :

(7)

If V is an l-weight-space of M with l-weight the Laurent monomial m 2 Y, then V is a subspace of the Uq .g/-weight-space with weight such that e D !.m/. Hence, the image !.q .M // of the q-character of M is the ordinary character of the underlying Uq .g/-module. For i 2 I and a 2 C define Y a Yj;aij : (8) Ai;a D Yi;aq Yi;aq 1 j 6Di

Thus !.Ai;a / D e ˛i , and the Ai;a .a 2 C / should be viewed as affine analogues of the simple root ˛i 2 . Following [10], we define a partial order on the set M of Laurent monomials in the variables Yi;a by setting m m0 ”

m0 m

is a monomial in the Ai;a with exponents 0.

(9)

This is an affine analogue of the usual partial order on P , defined by 0 if and only if 0 is a sum of simple roots ˛i . A monomial m 2 M is called dominant if it does not contain negative powers of the variables Yi;a . We will denote by MC the set of dominant monomials. They parametrize simple objects of C, as was first shown by Chari and Pressley [7]3 . More precisely, we have 2

In the non-simply-laced case, the definition of Ai;a is more complicated, see [10]. The original parametrization of [7] is in terms of Drinfeld polynomials, but in these notes we will rather use the equivalent parametrization by dominant monomials. 3


121

Theorem 1.2 ([9]). Let S be a simple object of C . The q-character of S is of the form X (10) Mp ; q .S / D mS 1 C p

where mS 2 MC , and all the Mp 6D 1 are monomials in the variables A1 i;a with nonnegative exponents. Moreover the map S 7! mS induces a bijection from the set of isoclasses of irreducible modules in C to MC . The dominant monomial mS is called the highest l-weight of q .S /, since every other monomial mS Mp of (10) is less than mS in the partial order (9). The onedimensional l-weight-space of S with l-weight mS consists of the highest-weight vecC tors of S, that is, the l-weight vectors v 2 S such that xi;r v D 0 for every i 2 I and r 2 Z. For m 2 MC , we denote by L.m/ the corresponding simple object of C . In particular, the modules L.Yi;a / .i 2 I; a 2 C / are called the fundamental modules. It is known ([10], Corollary 2), that the Grothendieck ring R is the polynomial ring over Z in the classes of the fundamental modules. 1.3 Kirillov–Reshetikhin modules. For i 2 I , k 2 N and a 2 C , the simple .i/ object Wk;a with highest l-weight m.i/ k;a

D

k1 Y

Yi; aq 2j

j D0

.i/ is called a Kirillov–Reshetikhin module. Thus Wk;a is an affine analogue of the irreducible representation of Uq .g/ with highest weight k$i . In particular for k D 1, .i/ .i/ W1;a coincides with the fundamental module L.Yi;a /. By convention, W0;a is the trivial representation for every i and a. .i/ .i/ The classes ŒWk;a in R, or equivalently the q-characters q .Wk;a /, satisfy the following system of equations indexed by i 2 I , k 2 N , and a 2 C , called the T -system4 : Y .j / .i/ .i/ .i/ .i/ ŒWk;aq ŒWk;aq aij : ŒWk;a (11) 2 D ŒWkC1;a ŒWk1;aq 2 C j 6Di

This was conjectured in [28] and proved in [34], [21]. Using these equations, one .i/ can calculate inductively the expression of any ŒWk;a as a polynomial in the classes .i/ ŒW1;a of the fundamental modules. Thus, one can obtain the q-characters of all the Kirillov–Reshetikhin modules once the q-characters of the fundamental modules are known. 4

In the non-simply laced case, the T -systems are more complicated, see [28], [21].

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B. Leclerc

1.4 The case of sl 2 . Let us illustrate the previous statements for g D sl2 . Here .i/ I D f1g, and we may drop the index i in ˛i , $i , Yi;a , Ai;a , ŒWk;a . We have Aa D Yaq Yaq 1 . The fundamental modules W1;a .a 2 C / are the affine analogues of the vector representation C 2 of Uq .sl2 /, whose character is equal to e $ C e $ D e $ .1 C e ˛ /: Since the l-weight spaces are subspaces of the one-dimensional Uq .sl2 /-weight spaces, W1;a also decomposes as a sum of two l-weight spaces, and the two l-weights are easily 1 checked to be Ya and Yaq 2 . Hence 1 1 q .W1;a / D Ya C Yaq 2 D Ya .1 C Aaq /:

The T -system (11) reads ŒWk;a ŒWk;aq 2 D ŒWkC1;a ŒWk1;aq 2 C 1

.a 2 C ; k 2 N /:

From the identity ŒW1;a ŒW1;aq 2 D ŒW2;a ŒW0;aq 2 C 1 D ŒW2;a C 1

(12)

one deduces that 1 1 1 1 1 1 q .W2;a / D Ya Yaq 2 C Ya Yaq 4 C Yaq 2 Yaq 4 D Ya Yaq 2 .1 C Aaq 3 C Aaq Aaq 3 /:

More generally, we have q .Wk;a / D

k1 Y j D0

Yaq 2j 1 C A1 1 C A1 .1 C .1 C A1 aq / / : (13) aq 2k1 aq 2k3

Following Chari and Pressley [6], we now describe the q-characters of all the simple objects of C . We call q-segment of origin a and length k the string of complex numbers †.k; a/ D fa; aq 2 ; : : : ; aq 2k2 g: Two q-segments are said to be in special position if one does not contain the other, and their union is a q-segment. Otherwise we say that they are in general position. It is easy to check that every finite multi-set fb1 ; : : : ; bs g of elements of C can be written uniquely as a union of q-segments †.ki ; ai / in such a way that every pair .†.ki ; ai /; †.kj ; aj // is in general position. Then, Chari and Pressley have proved that the simple module S with highest l-weight mS D

s Y

Ybj

j D1

is isomorphic to the tensor product of Kirillov–Reshetikhin modules q .S/ can be calculated using (13).

N i

Wki ;ai . Hence


123

1.5 Trigonometric solutions of the Yang–Baxter equation. Let us briefly indicate how quantum loop algebras give rise to families of solutions of the quantum Yang– Baxter equation. A nice introduction to these ideas is given by Jimbo in [26]. Consider the tensor product Wk;a ˝ Wk;b of Kirillov–Reshetikhin modules for Uq .Lsl2 /. For a generic choice of u D a=b 2 C , the q-segments †.k; a/ and †.k; b/ are in general position, and therefore Wk;a ˝ Wk;b is irreducible. Moreover, since the Grothendieck group is commutative, when the tensor product is irreducible it is isomorphic to Wk;b ˝ Wk;a . Therefore, there exists up to normalization a unique isomorphism I.a; b/ W Wk;a ˝ Wk;b ! Wk;b ˝ Wk;a : Now, if †.k; c/ is another q-segment in general position with †.k; a/ and †.k; b/, then I12 .b; c/I23 .a; c/I12 .a; b/ and I23 .a; b/I12 .a; c/I23 .b; c/ are two isomorphisms between the irreducible modules Wk;a ˝ Wk;b ˝ Wk;c and Wk;c ˝ Wk;b ˝ Wk;a , hence they are proportional. These intertwiners can be normalized in such a way that I12 .b; c/I23 .a; c/I12 .a; b/ D I23 .a; b/I12 .a; c/I23 .b; c/: Putting R.a; b/ D P I.a; b/ where P is the linear map from Wk;b ˝ Wk;a to Wk;a ˝ Wk;b defined by P .w ˝ w 0 / D w 0 ˝ w, it follows that R23 .b; c/R13 .a; c/R12 .a; b/ D R12 .a; b/R13 .a; c/R23 .b; c/: Moreover it can be seen that R.a; b/ only depends on a=b, thus setting u D a=b and v D b=c, we obtain that R23 .v/R13 .uv/R12 .u/ D R12 .u/R13 .uv/R23 .v/; that is, R.u/ is a solution of the Yang–Baxter equation (1). These solutions were obtained by Tarasov [39], [40]. For example, the solution coming from the 2-dimensional representation W1;a can be written in matrix form as 0 1 1 0 0 0 B C 1q 2 B0 q.u1/ C 0 2 2 B C uq uq B C: R.u/ D B C 2 B0 u.1q2 / q.u1/ 0C uq uq 2 @ A 0

0

0

1

This is the R-matrix associated with two famous integrable models: the spin 1/2 XXZ chain, and the six-vertex model. Note that it is well-defined and invertible if and only if u 6D q ˙2 . In fact, for u D q ˙2 , Eq. (12) shows that the tensor product W1;au ˝ W1;a is not irreducible, and one can check that W1;au ˝W1;a is not isomorphic to W1;a ˝W1;au . More generally, the same method can be applied to any finite-dimensional irreducible representation W of Uq .Lg/, using the general fact that W .u/ ˝ W is irreducible except for a finite number of values of u 2 C . We shall return to this special feature of quantum loop algebras in §3.

124

B. Leclerc

1.6 Subcategories. Since the Dynkin diagram of g is a bipartite graph, we have a partition I D I0 t I1 such that every edge connects a vertex of I0 with a vertex of I1 . For i 2 I we set ´ 0 if i 2 I0 , (14) i D 1 if i 2 I1 : Let MZ be the subset of M consisting of all monomials in the variables Yi;q i C2k

.i 2 I; k 2 Z/:

Let CZ be the full subcategory of C whose objects V have all their composition factors of the form L.m/ with m 2 MZ . One can show that CZ is an abelian subcategory, stable under tensor products. Its Grothendieck ring RZ is the subring of R generated by the classes of the fundamental modules L.Yi;q 2kCi /

.i 2 I; k 2 Z/:

It is known that every simple object S of C can be written as a tensor product S1 .a1 / ˝ ˝ Sk .ak / for some simple objects S1 ; : : : ; Sk of CZ and some complex numbers a1 ; : : : ; ak such that ai 62 q 2Z .1 i < j k/: aj (Here Sj .aj / denotes the image of Sj under the auto-equivalence aj , see §1.1.) Therefore, the description of the simple objects of C essentially reduces to the description of the simple objects of CZ . We will now introduce, following [24], an increasing sequence of subcategories of CZ . Let ` 2 N. Let M` be the subset of MZ consisting of all monomials in the variables Yi;q i C2k .i 2 I; 0 k `/: Define C` to be the full subcategory of C whose objects V have all their composition factors of the form L.m/ with m 2 M` . Proposition 1.3 ([24]). C` is an abelian monoidal category, with Grothendieck ring the polynomial ring R` D Z ŒL.Yi;q 2kCi /I i 2 I; 0 k ` : The simple objects of the category C0 are easy to describe. Indeed, it follows from Proposition 6.15 of [9] that every simple object of C0 is a product of fundamental modules of C0 , and conversely any tensor product of fundamental modules of C0 is simple. We will see in §3 that the simple objects of the subcategory C1 are already non trivial, and that they have a nice description involving cluster algebras. Clearly, every simple object of CZ is of the form S.q k / for some k 2 Z and some simple object S in C` with ` large enough. Therefore, the description of the simple objects of C eventually reduces to the description of the simple objects of C` for arbitrary ` 2 N.


125

2 Nakajima quiver varieties and irreducible q-characters The characters of the irreducible finite-dimensional Uq .g/-modules are identical to those of the corresponding g-modules, and are thus given by the classical Weyl character formula. Moreover, Kashiwara’s theory of crystal bases gave rise to a uniform combinatorial description of these characters, generalizing the Young tableaux descriptions available for g D sln . The situation is much more complicated for Uq .Lg/. Indeed, there is no analogue of Weyl’s formula in this case, and it is believed that only Kirillov– Reshetikhin modules (and their irreducible tensor products) have a crystal basis (see [37] for the existence of crystals of KR-modules for g of classical type). However, inspired by earlier work on Springer theory for affine Hecke algebras, Ginzburg and Vasserot [20] gave a geometric description of the irreducible q-characters of Uq .Lg/ for g D sln in terms of intersection cohomology of closures of graded nilpotent orbits. This was extended to all simply-laced types by Nakajima [32], using a graded version of his quiver varieties. In this second lecture, we shall review Nakajima’s geometric approach. 2.1 Graded vector spaces. Recall the partition I D I0 t I1 of §1.6. Define the sets of ordered pairs: Iy D I Z; Iy0 D .I0 2Z/ t .I1 .2Z C 1//; Iy1 D .I0 .2Z C 1// t .I1 2Z/: We will consider finite-dimensional Iy-graded C-vector spaces. More precisely, we will use the letters V; V 0 ; : : : for Iy1 -graded vector spaces, and the letters W; W 0 ; : : : for Iy0 -graded vector spaces. We shall write M M V D Vi .r/; W D Wi .r/; .i;r/2Iy1

.i;r/2Iy0

where the spaces Vi .r/, Wi .r/ are finite-dimensional, and nonzero only for a finite number of .i; r/. We write V V 0 if and only if dim Vi .r/ dim Vi0 .r/ for every .i; r/ 2 Iy1 . Consider a pair .V; W / where V is Iy1 -graded and W is Iy0 -graded. We say that .V; W / is l-dominant if di .r; V; W / ´ dim Wi .r/ dim Vi .r C 1/ X dim Vi .r 1/ aij dim Vj .r/ 0

(15)

j 6Di

for every .i; r/ 2 Iy0 . The pair .0; W / is always l-dominant, and for a given W there are finitely many isoclasses of Iy1 -graded spaces V such that .V; W / is l-dominant.

126

B. Leclerc

2.2 ADHM equations. Let .V; W / be a pair of vector spaces, where V is Iy1 -graded and W is Iy0 -graded. Define

L .V; W / D

M

Hom.Vi .r/; Wi .r 1//;

.i;r/2Iy1

L .W; V / D

M

Hom.Wi .r/; Vi .r 1//;

.i;r/2Iy0

M

E .V / D

Hom.Vi .r/; Vj .r 1//:

.i;r/2Iy1

j W aij D1

Put M .V; W / D E .V /˚L .W; V /˚L .V; W /. An element of M .V; W / is written .B; ˛; ˇ/, and its components are denoted by Bij .r/ 2 Hom.Vi .r/; Vj .r 1//; ˛i .r/ 2 Hom.Wi .r/; Vi .r 1//; ˇi .r/ 2 Hom.Vi .r/; Wi .r 1//: We define a map W M .V; W / !

L

.i;r/2Iy1

.i;r/ .B; ˛; ˇ/ D ˛i .r 1/ˇi .r/ C .1/i

Hom.Vi .r/; Vi .r 2// by

X

Bj i .r 1/Bij .r/

..i; r/ 2 Iy1 /:

j W aij D1

We can then introduce ƒ .V; W / ´ 1 .0/ M .V; W /. In other words, ƒ .V; W / is the subvariety of the affine space M .V; W / defined by the so-called complexAtiyah– Drinfeld–Hitchin–Manin equations (or ADHM, in short): ˛i .r 1/ˇi .r/ C .1/i

X

Bj i .r 1/Bij .r/ D 0

..i; r/ 2 Iy1 /:

(16)

j W aij D1

2.3 Graded quiver varieties. A point .B; ˛; ˇ/ of ƒ .V; W / is called stable if the following condition holds: for every Iy1 -graded subspace V 0 of V , if V 0 is B-invariant and contained in Ker ˇ then V 0 D 0. The stable points form an open subset of ƒ .V; W / denoted by ƒs .V; W /. Let GV ´

Y

GL.Vi .r//:

.i;r/2Iy1

This reductive group acts on M .V; W / by base change in V : g .B; ˛; ˇ/ D .gj .r 1/Bij .r/gi .r/1 /; .gi .r 1/˛i .r//; .ˇi .r/gi .r/1 / :


127

Note that there is no action on the space W . This action preserves the subvariety ƒ .V; W / and the open subset ƒs .V; W /. Moreover, the action on ƒs .V; W / is free. One can then define, following Nakajima, M .V; W / ´ ƒs .V; W /=GV :

This set-theoretic quotient coincides with a quotient in the geometric invariant theory sense. The GV -orbit through .B; ˛; ˇ/, considered as a point of M .V; W /, will be denoted by ŒB; ˛; ˇ. Note that M .V; W / may be empty (if there is no stable point). One also defines the affine quotient M0 .V; W / ´ ƒ .V; W /==GV :

By definition, the coordinate ring of M0 .V; W / is the ring of GV -invariant functions on ƒ .V; W /, and M0 .V; W / parametrizes the closed GV -orbits. Since the orbit f0g is always closed, M0 .V; W / is never empty. We have a projective morphism

V W M .V; W / ! M0 .V; W /; mapping the orbit ŒB; ˛; ˇ to the unique closed GV -orbit in its closure. Finally, the third quiver variety is L .V; W / ´ V1 .0/: 2.4 Properties. If it is not empty, the variety M .V; W / is smooth of dimension X dim M .V; W / D dim Vi .r C 1/di .r; V; W / C dim Wi .r/ dim Vi .r 1/ .i;r/2Iy0

where di .r; V; W / is defined by (15). The coordinate ring of M0 .V; W / is generated by the following GV -invariant functions on ƒ .V; W /: .B; ˛; ˇ/ 7! h ; ˇj .r n1/Bjn1 j .r n/ : : : Bj1 j2 .r 2/Bij1 .r 1/˛i .r/ i; (17) where .i; r/ 2 Iy0 , .i; j1 ; j2 ; : : : ; jn1 ; j / is a path (possibly of length 0) on the (unoriented) Dynkin diagram, and is a linear form on Hom.Wi .r/; Wj .r n 2// [31]. When the path is the trivial path at vertex i , the function (17) is .B; ˛; ˇ/ 7! h

; ˇi .r 2/˛i .r/ i;

where is a linear form on Hom.Wi .r/; Wi .r 2//. In particular, if W D Wi .r/ is supported on a single vertex .i; r/ 2 Iy0 then every function of the form (17) is equal to 0, so for every V the coordinate ring of M0 .V; W / is C, and M0 .V; W / D f0g. It follows that M .V; W / D L .V; W / in this case. reg Let M0 .V; W / be the subset of M0 .V; W / consisting of the closed free GV reg orbits. This is open, but possibly empty. In fact, M0 .V; W / 6D ; if and only if

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B. Leclerc

M .V; W / 6D ; and the pair .V; W / is l-dominant. In this case, the restriction of V to reg reg V1 .M0 .V; W // is an isomorphism, and in particular M0 .V; W / is non singular of dimension equal to dim M .V; W /. If V V 0 , that is, if Vi .r/ Vi0 .r/ for every .i; r/ 2 Iy1 , then we have a natural closed embedding M0 .V; W / M0 .V 0 ; W /. One defines [ M0 .W / D M0 .V; W /: V

In fact, one has a stratification

M0 .W / D

G

reg

M0

.V; W /;

ŒV

where V runs through the Iy1 -graded spaces such that .V; W / is l-dominant, and ŒV denotes the isomorphism class of V as a graded space. It follows from §2.1 that M0 .W / has finitely many strata. 2.5 Examples. 1. Take g D sl2 of type A1 . Assume that I0 D I D f1g. Since I is a singleton, we can drop indices i in the notation and write Iy0 D 2Z, Iy1 D 2Z C 1. Hence M M W D W .r/; V D V .s/; r22Z

s22ZC1

and M .V; W / D L .W; V / ˚ L .V; W / consists of pairs .˛; ˇ/: the B-component is zero in this case. In particular, any subspace V 0 of V is B-stable, so .˛; ˇ/ is stable if and only ˇ.s/ is injective for every s 2 2Z C 1. The ADHM equations reduce to

˛.s 1/ˇ.s/ D 0

.s 2 2Z C 1/:

With any pair .˛; ˇ/ we associate x D ˇ˛ 2 End.W /, and E D ˇ.V / W . Clearly, x and E depend only on the GV -orbit of .˛; ˇ/. Let N .W / denote the subvariety of End.W / consisting of degree -2 endomorphisms of W (that is, x.W .r// W .r 2/ for every r) satisfying x 2 D 0. Let F .V; W / denote the variety of pairs .x; E/, where E is a graded subspace of W with dim E.r/ D dim V .r C 1/, and x 2 N .W / is such that Im x E Ker x: Then one can check that the map Œ˛; ˇ 7! .x; E/ establishes an isomorphism ! F .V; W /: M .V; W /

Moreover, the affine variety M0 .W / is isomorphic to N .W /, and the projective morphism V W M .V; W / ! M0 .V; W / M0 .W / is the first projection V .x; E/ D x:


129

Thus the fiber L .V; W / D V1 .0/ is the Grassmannian of graded subspaces E of W with dim E.r/ D dim V .r C 1/. Finally, M0 .V; W / is isomorphic to the subvariety of N .W / defined by the rank conditions: dim x.W .r// dim V .r 1/

.r 2 2Z/:

2. Take Q of type A3 , with sink set I0 D f1; 3g and source set I1 D f2g. Choose first W D W1 .0/ D C: One can check that the only choices of V such that M .V; W / 6D ; are (a) V D 0. (b) V D V1 .1/ D C. (c) V D V1 .1/ ˚ V2 .2/ D C ˚ C. (d) V D V1 .1/ ˚ V2 .2/ ˚ V3 .3/ D C ˚ C ˚ C. For instance, this can be checked by using the dimension formula of §2.4: in all other cases, this formula produces a negative number. Case (d) is illustrated in Figure 1. Let us determine ƒs .V; W / in each case. In cases (b), (c), (d), the map ˇ1 .1/ W V1 .1/ ! W1 .0/ has to be injective. Indeed its kernel is B-invariant, and so the stability condition V3 .3/ w w w ww wwB32 .3/ w {w V2 .2/ ww ww w ww {ww B21 .2/ V1 .1/

ˇ1 .1/

W1 .0/

V2 .4/ GG w GGB23 .4/ w w GG w GG wwB .4/ w G# w 21 {w V1 .3/ V3 .3/ GG w GGB12 .3/ w w GG ww GG wwB32 .3/ G# w {w V2 .2/

ˇ2 .2/

W2 .1/

Figure 1. The cases W D W1 .0/ D C, and W D W2 .1/ D C, in type A3 .

forces it to be trivial. Similarly, in cases (c), (d), the map B21 .2/ W V2 .2/ ! V1 .1/ has to be injective. Indeed its kernel is B-invariant and contained in Ker ˇ2 .2/ D V2 .2/, and so the stability condition forces it again to be trivial. Finally, in case (d), for the same reasons, the map B32 .3/ W V3 .3/ ! V2 .2/ has to be injective. Thus we have (b) ƒs .V; W / D C and M .V; W / D C =C D fpointg. (c) ƒs .V; W / D C C and M .V; W / D .C C /=.C C / D fptg.

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B. Leclerc

(d) ƒs .V; W / D C C C , M .V; W / D .C C C /=.C C C / D fptg. As a second choice, take W D W2 .1/ D C: One can check that the only choices of V such that M .V; W / 6D ; are (a) V D 0, (b) V D V2 .2/ D C, (c) V D V2 .2/ ˚ V1 .3/ D C ˚ C, (d) V D V2 .2/ ˚ V3 .3/ D C ˚ C, (e) V D V2 .2/ ˚ V1 .3/ ˚ V3 .3/ D C ˚ C ˚ C, (f) V D V2 .2/ ˚ V1 .3/ ˚ V3 .3/ ˚ V2 .4/ D C ˚ C ˚ C ˚ C. Case (f) is illustrated in Figure 1. Let us determine ƒs .V; W / in each case. In cases (b), (c), (d), (e) the map ˇ2 .2/ W V2 .2/ ! W2 .1/ has to be injective. Indeed its kernel is B-invariant, and so the stability condition forces it to be trivial. Similarly, in cases (c), (e), (f) the map B12 .3/ has to be injective. Indeed its kernel is B-invariant and contained in Ker ˇ1 .3/ D V1 .3/, and so the stability condition forces it again to be trivial. Similarly, in cases (d), (e), (f) the map B32 .3/ has to be injective. Finally, in case (f), the ADHM equations imply the relation B12 .3/B21 .4/ C B32 .3/B23 .4/ D 0:

(18)

Since B12 .3/ and B32 .3/ are injective, this implies that B21 .4/ and B23 .4/ are both injective or both equal to 0. If they are both equal to 0 then V2 .4/ is B-invariant and contained in Ker ˇ2 .4/, so .B; ˛; ˇ/ is not stable. Hence B21 .4/ and B23 .4/ are both injective. Thus we have (b) ƒs .V; W / D C and M .V; W / D C =C D fptg, (c) ƒs .V; W / D C C and M .V; W / D .C C /=.C C / D fptg, (d) ƒs .V; W / D C C and M .V; W / D .C C /=.C C / D fptg, (e) ƒs .V; W / D C C C , M .V; W / D .C C C /=.C C C / D fptg, (f) ƒs .V; W / D C C C C , because B23 .4/ can be expressed in terms of B12 .3/; B21 .4/, and B32 .3/ in view of (18). Hence, we find again that M .V; W / D .C C C C /=.C C C C / D fptg. 3. Assume that Q is of type An and 1 2 I0 . Take W of the form W D W1 .r/ ˚ W1 .r C 2/ ˚ ˚ W1 .r C 2k/

.r 2 2Z; k 2 N/;


131

where dim W1 .r C 2j / D d .j / .0 j k/. Thus, W can be regarded as a 2Z-graded vector space. One can check that the map .B; ˛; ˇ/ 7! x ´ .ˇ1 .r C1/˛1 .r/; ˇ1 .r C3/˛1 .r C2/ : : : ; ˇ1 .r C2k 1/˛1 .r C2k// induces an isomorphism from M0 .W / to the variety of degree -2 endomorphisms x of W satisfying x nC1 D 0. In other words, M0 .W / is the affine space of representations in W of the quiver r

1

2

3

k

r C 2 r C 4 r C 2k

bound by the relations i iC1 : : : iCn D 0

.1 i k n/:

Let us now determine ƒs .V; W / for a given Iy1 -graded space V . First, the stability condition implies that Vi .j / D 0 for j < r C i . Next, it is easy to show by induction that if .B; ˛; ˇ/ is stable then the following maps are injective: ˇ1 .r C 1 C 2j /

.0 j k/;

Bi;i 1 .r C i C 2j /

.2 i n; 0 j k/:

Moreover, we have Vi .j / D 0 for j > r C i C 2k. Therefore a typical example of .B; ˛; ˇ/ in ƒs .V; W / looks like in Figure 2 with all maps ˇ1 .j / and Bi;i1 .j / injective. Put E1.j / ´ ˇ1 .r C 1 C 2j /.V1 .r C 1 C 2j // W1 .r C 2j /

.0 j k/;

and for i D 2; : : : ; n, Ei.j / ´ ˇ1 .r C 1 C 2j /B21 .r C 2 C 2j / : : : Bi;i1 .r C i C 2j /.Vi .r C i C 2j //: The vector spaces Ei D

L j

Ei.j / form an n-step flag

F D .E0 D W E1 En 0 D EnC1 / of graded subspaces of W D E0 , with graded dimension di D .dim Vi .r C i /; dim Vi .r C i C 2/; : : :/

.i D 1; : : : n/:

We get a well-defined map .B; ˛; ˇ/ 7! .x; F / from ƒs .V; W / to the variety F .V; W / of pairs consisting of a graded flag F of dimension .d1 ; : : : ; dn / in W , together with a graded nilpotent endomorphism x preserving this flag (that is, such that x.Ei.j / / .j 1/ EiC1 ). This map is GV -equivariant, hence induces a map M .V; W / ! F .V; W /, and one can check that this is an isomorphism. Moreover, in this identification the map V W M .V; W / ! M0 .V; W / becomes the projection .x; F / 7! x. Finally, the

132

B. Leclerc

V3 .7/ w w w ww wwB32 .7/ w {w w V2 .6/ GG GG B .6/ vv GG23 vv v GG v GG vv B21 .6/ v {v # V1 .5/ V3 .5/ HH w HH B .5/ ww H 12 ww ˇ1 .5/ HH w HH ww B32 .5/ H# {ww W1 .4/ V2 .4/ GG GG B .4/ vv v GG23 v ˛1 .4/ vv GG v B .4/ GG v 21 {vv # V1 .3/ V3 .3/ HH HH B .3/ ww H 12 ww w ˇ1 .3/ HH w HH ww B32 .3/ H# {ww W1 .2/ V2 .2/ v v vv ˛1 .2/ vv v B21 .2/ v {vv V1 .1/

ˇ1 .1/

W1 .0/ Figure 2. The case W D W1 .0/ ˚ W1 .2/ ˚ W1 .4/ in type A3 .

zero fibre L .V; W / D V1 .0/ is isomorphic to the variety of graded flags of W of dimension .d1 ; : : : ; dn /. More generally, the fiber V1 .x/ is the variety of graded flags of W of dimension .d1 ; : : : ; dn / which are preserved by x. Ginzburg and Vasserot [20] have shown that the Borel–Moore homologies of the varieties G Mx D V1 .x/ .x 2 M0 .W //; V

y nC1 /(where V runs over isoclasses of Iy1 -graded spaces) have natural structures of Uq .sl modules, called standard modules. These modules are not simple, but can be decomposed into simple ones using the decomposition theorem for perverse sheaves. In the next sections, we review Nakajima’s extension of these results to other root systems.


133

2.6 Quiver varieties and standard modules for Uq .Lg/. To a pair .V; W / of graded spaces as in §2.1 we attach two monomials in Y given by Y Y dim V .s/ dim W .r/ YW D Yi;q r i ; AV D Aj;q s j : (19) .i; r/2Iy0

.j; s/2Iy1

One can check that the monomial Y W AV is dominant if and only if the pair .V; W / is l-dominant in the sense of §2.1. Let us associate with W the simple Uq .Lg/-module L.W / ´ L.Y W /, which belongs to the subcategory CZ . We can also attach to W the tensor product O M.W / D L.Yi;q r /˝ dim Wi .r/ : .i; r/2Iy0

This product is not simple in general. Moreover, its isomorphism class may depend on the chosen ordering of the factors. However, we will only be interested in its qcharacter (or in its class in RZ ) which is independent of this ordering. The modules M.W / are called standard modules. The morphism V W M .V; W / ! M0 .V; W / being projective, its zero fiber L .V; W / is a complex projective variety. Let .L .V; W // denote its Euler characteristic. Note that L .V; W / has no odd cohomology [32], §7, so .L .V; W // is equal to the total dimension of the cohomology. Theorem 2.1 (Nakajima [32]). The q-character of the standard module M.W / is given by X q .M.W // D Y W .L .V; W // AV ; ŒV

where the sum runs over all isomorphism classes ŒV of Iy1 -graded spaces V . If W has dimension 1, M.W / D L.W / is a fundamental module, hence Theorem 2.1 describes in particular the q-characters of all fundamental modules. On the other hand, q .M.W // is the product of the q-characters of the factors of M.W /, so it can also be expressed as a product of q-characters of fundamental modules. 2.7 Examples. We illustrate Theorem 2.1 with the examples of §2.5. 1. We take g D sl2 . The variety L .V; W / is isomorphic to the product of ordinary Grassmannians Y Gr .dim V .r C 1/; dim W .r// ; r22Z

so its Euler characteristic is

!

Y

dim W .r/

r22Z

dim V .r C 1/

:

134

B. Leclerc

Hence q .M.W // D Y

W

X Y ŒV r22Z

dim W .r/

!

dim V .r C 1/

AV :

/. We can check that the On the other hand, recall that q .L.Yq r // D Yq r .1 C A1 q rC1 above value of q .M.W // is equal to Y q .L.Yq r //dim W .r/ ; r22Z

as it should. 2. Take g of type A3 . It follows from the calculations of §2.5.2 that 1 1 1 1 1 q .L.Y1;q 0 // D Y1;q 0 1 C A1 1;q C A1;q A2;q 2 C A1;q A2;q 2 A3;q 3 ; C A1 A1 C A1 A1 C A1 A1 A1 : q .L.Y2;q 1 // D Y2;q 1 1 C A1 2;q 2 2;q 2 1;q 3 2;q 2 3;q 3 2;q 2 1;q 3 3;q 3 C A1 A1 A1 A1 : 2;q 2 1;q 3 3;q 3 2;q 4 3. Assume that g is of type An . Choosing W of dimension 1, it follows from §2.5.3 that q .L.Y1;q r // D Y1;q r 1 C A1 C A1 A1 1;q rC1 1;q rC1 2;q rC2 1 1 C C A1 1;q r A2;q rC2 : : : An;q rCn : 2.8 Standard modules and the graded preprojective algebra. Let Q be the sinksource orientation of the Dynkin diagram of g, with set of sinks I0 and set of sources I1 . Define the repetition quiver ZQ as the infinite quiver with set of vertices Iy0 , and two types of arrows: (i) for every arrow i ! j in Q we have arrows .i; 2m C 1/ ! .j; 2m/ in ZQ for all m 2 Z; (ii) for every arrow i ! j in Q we have arrows .j; 2m/ ! .i; 2m 1/ in ZQ for all m 2 Z. As an example, the quiver ZQ for Q of type A3 is shown in Figure 3. We then introduce a set of degree two elements in the path algebra of ZQ: for every .i; r/ 2 Iy0 , let i;r be the sum of all paths from .i; r/ to .i; r 2/. The graded y of the path algebra of ZQ by preprojective algebra of Q is by definition the quotient ƒ y y is well known the two-sided ideal generated by the i;r ..i; r/ 2 I0 /. The algebra ƒ to be the universal cover of the preprojective algebra ƒ of Q, in the sense of [17]. It turns out that the quiver variety L .V; W / is homeomorphic to a quiver Grassy mannian of an injective ƒ-module. To state this precisely, let us denote by Si;r the y one-dimensional simple ƒ-module supported on vertex .i; r/ of ZQ. Let i;r be the


.1;4/

.1;2/

KKK KK% ss y ss s

:: : .2;3/

KKK KK%

ss sy ss

KKK KK%

135

.3;4/

.3;2/

ss sy ss .2;1/ KKK ss KK% s s ys .1;0/ .3;0/ KKK s s KK% s yss .2;1/

KKK KK%

ss y ss s

.1;2/

:: :

.3;2/

Figure 3. The quiver ZQ in type A3 .

injective hull of Si;r . (This is a finite-dimensional module.) To the Iy0 -graded vector space W we attach the injective module M ˚ dim Wi .r/ i;r : W D .i;r/2Iy0

To the Iy1 -graded vector space V we attach the dimension vector dV D .dim Vi .r C 1/I .i; r/ 2 Iy0 /: We then have the following result, due to Lusztig [31] in the ungraded case, and extended to the graded case by Savage and Tingley [38]. Proposition 2.2. The complex variety L .V; W / is homeomorphic to the Grassmannian y Gr.dV ; W / of ƒ-submodules of W with dimension vector dV . It follows that we can rewrite Nakajima’s formula for standard modules of CZ as X .Gr.dV ; W // AV : (20) q .M.W // D Y W ŒV

In particular, for the fundamental modules of CZ we get X .Gr.dV ; i;r // AV : q .L.Yi;q r // D Yi;q r ŒV

(21)

136

B. Leclerc

2.9 Examples. 1. One can easily recover the formulas of §2.7 using the well-known y description of the indecomposable injective ƒ-modules for a Dynkin quiver Q of type An . Thus, §2.7.3 follows immediately from the fact that the injective module 1;r is an n-dimensional module with a unique composition series. More generally, all the Grassmannians of submodules of all indecomposable injecy tive ƒ-modules in type An are reduced to a point. This implies that all fundamental Uq .LslnC1 /-modules have multiplicity-free q-characters, that is, all their l-weight spaces have dimension 1. 2. Let g D so8 be of type D4 . We label by 3 the central node of the Dynkin diagram, and we set I0 D f3g, I1 D f1; 2; 4g. The injective module 3;0 has total dimension 10, and its dimension vector is supported on the finite strip of ZQ displayed in Figure 4. More precisely, every vertex of the picture carries a 1-dimensional vector .3;4/ HHH jj jjjvvvv HH j j j z $ v j uj .1;3/ T .2;3/ .4;3/ TTT HH v TTT HHH v v TTT $ zvv ) .3;2/ j H HHH jjj v H$ jjj zvvvv ujjj .1;1/ T .2;1/ .4;1/ TTT HH TTT HHH vv v TTT $ v zv ) .3;0/

Figure 4. The skeleton of the injective module 3;0 in type D4 .

space, except .3; 2/ which has a 2-dimensional vector space. There are 28 non-trivial quiver Grassmannians Gr.d; 3;0 /, and it easy to check that all of them are points, except for the following d : d3;0 D d1;1 D d2;1 D d4;1 D d3;2 D 1;

d1;3 D d2;3 D d4;3 D d3;4 D 0;

for which Gr.d; 3;0 / ' P .C/. It follows that the fundamental module L.Y3;q 0 / has dimension 29. More precisely, writing for short vi;s ´ A1 i;q s , we have 1

q .L.Y3;0 // D Y3;q 0 .1 C v3;1 C v3;1 v1;2 C v3;1 v2;2 C v3;1 v4;2 C v3;1 v1;2 v2;2 C v3;1 v1;2 v4;2 C v3;1 v2;2 v4;2 C v3;1 v1;2 v2;2 v4;2 C v3;1 v1;2 v2;2 v3;3 C v3;1 v1;2 v4;2 v3;3 C v3;1 v2;2 v4;2 v3;3 C 2 v3;1 v1;2 v2;2 v4;2 v3;3 C v3;1 v1;2 v2;2 v3;3 v4;4 C v3;1 v1;2 v4;2 v3;3 v2;4 C v3;1 v2;2 v4;2 v3;3 v1;4 2 C v3;1 v1;2 v2;2 v4;2 v3;3 C v3;1 v1;2 v2;2 v4;2 v3;3 v1;4

C v3;1 v1;2 v2;2 v4;2 v3;3 v2;4 C v3;1 v1;2 v2;2 v4;2 v3;3 v4;4 2 2 C v3;1 v1;2 v2;2 v4;2 v3;3 v1;4 C v3;1 v1;2 v2;2 v4;2 v3;3 v2;4 2 2 C v3;1 v1;2 v2;2 v4;2 v3;3 v4;4 C v3;1 v1;2 v2;2 v4;2 v3;3 v1;4 v2;4


137

2 2 C v3;1 v1;2 v2;2 v4;2 v3;3 v1;4 v4;4 C v3;1 v1;2 v2;2 v4;2 v3;3 v2;4 v4;4 2 C v3;1 v1;2 v2;2 v4;2 v3;3 v1;4 v2;4 v4;4 2 C v3;1 v1;2 v2;2 v4;2 v3;3 v1;4 v2;4 v4;4 v3;5 /:

The restriction of L.Y3;0 / to Uq .so8 / decomposes into the direct sum of the fundamental module with fundamental weight $3 , of dimension 28, and of a copy of the trivial representation. 2.10 Perverse sheaves. Recall from §2.4 the stratification G reg M0 .V 0 ; W /; M0 .W / D ŒV 0

where V 0 runs through the Iy1 -graded spaces such that the pair .V 0 ; W / is l-dominant. We denote by ICW .V 0 / the intersection cohomology complex associated with the reg trivial local system on the stratum M0 .V 0 ; W /. Consider an arbitrary Iy1 -graded space V such that M .V; W / 6D ;. The map

V W M .V; W / ! M0 .V; W / is projective, and the variety M .V; W / is smooth. Hence, by the decomposition theorem, the push-down .V /Š .1M .V;W / / of the constant sheaf on M .V; W / is a direct sum of shifts of simple perverse sheaves in the derived category D.M0 .V; W //. We can regard these perverse sheaves as objects of D.M0 .W // by extending them by 0 on the complement M0 .W / n M0 .V; W /. Nakajima has shown that all these perverse sheaves are of the form ICW .V 0 / for some l-dominant pair .V 0 ; W / with V 0 V . So we can write in the Grothendieck group of D.M0 .W // X Œ.V /Š .1M .V;W / Œdim M .V; W // D aV;V 0 IW .t /ŒICW .V 0 /: (22) V 0 V

Here t is a formal variable implementing the action of the shift functor t j ŒL D ŒLŒj ; and aV;V 0 IW .t/ 2 NŒt ˙1 is the graded multiplicity. (The additional shift in degree by dim M .V; W / makes the left-hand side invariant under Verdier duality.) Note that in (22), the pair .V; W / is not necessarily l-dominant. We can now state the main result of this lecture. Theorem 2.3 (Nakajima [32]). Let W be an Iy0 -graded space, and let L.W / be the corresponding simple module in CZ . The coefficient of the monomial Y W AV in q .L.W // is equal to aV;0IW .1/. In other words, the l-weight multiplicities of L.W / are calculated by the (ungraded) multiplicities of the skyscraper sheaf ICW .0/ D 1f0g in the expansions of the pushdowns Œ.V /Š .1M .V;W / / on the basis fŒICW .V 0 /g.

138

B. Leclerc

2.11 Examples. 1. Let L.W / D L.Yi;q r / be a fundamental module. Then, by §2.4, M0 .W / D f0g, and M .V; W / D L .V; W /. So aV;0IW .t / is the Poincaré polynomial of the cohomology of L .V; W / (up to some shift). Since the odd cohomology groups of L .V; W / vanish, we recover that the coefficient of Y W AV in q .L.W // is the Euler characteristic of L .V; W /, in agreement with Theorem 2.1. 2. Take g of type A1 , and W D W .r/ ˚ W .r C 2/ ˚ ˚ W .r C 2k/; with dim W .r C 2i / D 1 for every i D 0; 1; : : : ; k. The corresponding simple Uq .g/O module is the Kirillov–Reshetikhin module L.W / D WkC1;q r . Recall from §2.5.1 the description of the quiver varieties M .V; W / and M .W /. The variety M .V; W / is non-empty if and only if dim V .r C 2j C 1/ dim W .r C 2j / for every j D 0; : : : ; k. For any such choice, since dim V .r C 2j C 1/ is equal to 0 or 1, there is a unique graded subspace E of W satisfying dim E.r C 2j / D dim V .r C 2j C 1/. Moreover, the set of x’s such that Im x E Ker x is isomorphic to the vector space of linear maps of degree -2 from W =E to E. We have two cases: (i) If V is such that there exists s 2 f0; : : : ; kg with ´ 0 if j < s; dim V .r C 2j C 1/ D 1 if j s; then this space is reduced to f0g, and .V /Š .1M .V;W / / D 1f0g ;

aV;0;W .1/ D 1:

(ii) Otherwise, if there is s 2 f0; : : : ; k 1g with dim V .r C 2s C 1/ D 1;

dim V .r C 2s C 3/ D 0;

then this space has positive dimension, and .V /Š .1M .V;W / / is a simple perverse sheaf 6D 1f0g . So aV;0;W D 0. In conclusion, the q-character of the Kirillov–Reshetikhin module WkC1;q r is given by q .WkC1;q r / D Yq r Yq rC2 : : : Yq rC2k .1 C A1 C A1 A1 q rC2kC1 q rC2k1 q rC2kC1 C C A1 A1 : : : A1 /; q rC1 q rC3 q rC2kC1 in agreement with Eq. (13). 2.12 Algorithms. Let m 2 MC . We say that the simple module L.m/ is minuscule if m is the only dominant monomial of q .L.m//. (In [33] these modules are called special.) There exists an algorithm due to Frenkel and Mukhin [9] which attaches to any m 2 MC a polynomial FM.m/ 2 Y, and in case L.m/ is minuscule it is


139

proved that FM.m/ D q .L.m//. Moreover, all fundamental modules L.Yi;a / are minuscule, so this algorithm allows to calculate their q-characters. It was proved in [34] that Kirillov–Reshetikhin modules are also minuscule. But there also exist simple modules for which the Frenkel–Mukhin algorithm fails. For example in type A2 , 2 2 q .L.Y1;1 Y2;q 3 // 6D FM.Y1;1 Y2;q 3 /; [24], Example 5.6. (For an earlier example in type C3 see [36].) In [33], Nakajima has introduced a t -analogue q;t of the q-character q . This is obtained by keeping the t -grading in the graded multiplicities aV;0IW .t / of Theorem 2.3. Imitating the Kazhdan–Lusztig algorithm for calculating the intersection cohomology of a Schubert variety, he has described an algorithm for computing the .q; t /-character of an arbitrary simple module in terms of the .q; t /-characters of the fundamental modules. The .q; t /-characters of the fundamental modules can in turn be obtained using a t -version of the Frenkel–Mukhin algorithm. We therefore have, in principle, a way of calculating q .L.m// for every m 2 MC .

3 Tensor structure In the category of finite-dimensional Uq .g/-modules, tensor products of irreducible modules are almost never irreducible. This is in sharp contrast with what happens for tensor products of finite-dimensional Uq .Lg/-modules. Indeed, if M and N are simple objects of C , the tensor product M ˝ N.a/ is simple for all but a finite number of a 2 C . (Here N.a/ is the image of N under the auto-equivalence a of §1.1.) Hence many tensor products of simple Uq .Lg/-modules are simple, or equivalently, many simple modules can be factored as tensor products of smaller simple modules. The following questions are therefore natural: (i) what are the prime simple modules, i.e., the simple modules which have no factorization as a tensor product of smaller modules ? (ii) which tensor products of prime simples are simple ? We have seen in §1.4 that these questions have a simple answer when g D sl2 , namely, the prime simples are the Kirillov–Reshetikhin modules, and a tensor product of Kirillov–Reshetikhin modules is simple if and only if the corresponding q-segments are pairwise in general position. In this third lecture, we will report on some recent progress in trying to extend these results to an arbitrary simply-laced g. 3.1 The cluster algebra A` . We will assume the reader has some familiarity with cluster algebras. Nice introductions to this theory with pointers to the literature have been written by Zelevinsky [42] and Fomin [11]. All the necessary material for understanding this lecture can also be found in [30], §2. For ` 2 N, we define a new quiver ` . Put Iy0 .`/ D f.i; i C 2k/ j i 2 I; 0 k `g:

140

B. Leclerc

The graph ` is obtained by taking the full subgraph of ZQ with vertex set Iy0 .`/, and by adding to it new vertical up-arrows corresponding to the natural translation .i; r/ 7! .i; r C 2/. For example, if g has type A3 and I0 D f1; 3g, the quiver 2 is shown in Figure 5.

.1;4/

.2;5/ v O HHHH v v H$ zv v

.3;4/ v O v v zv v .2;3/ O HHH vv HH zvvv $ .1;2/ .3;2/ O HHH O vv HH v v $ zv .2;1/ HHH v v HH v zvv $

O HHH HH $

.1;0/

.3;0/

Figure 5. The quiver 2 in type A3 .

Let z D fz.i;r/ j .i; r/ 2 Iy0 .`/g be a set of indeterminates corresponding to the vertices of ` , and consider the seed .z; ` / in which the variables z.i;i / .i 2 I / are frozen. This is the initial seed of a cluster algebra A` Q.z/. It follows easily from [14] that A` has in general infinitely many cluster variables. The exceptional pairs .g; `/ for which A` has finite cluster type are listed in Table 1. Table 1. Algebras A` of finite cluster type.

Type of g

`

Type of A`

A1

`

A`

Xn

1

Xn

A2

2

D4

A2

3

E6

A2

4

E8

A3

2

E6

A4

2

E8

3.2 Conjectural relation between A` and C` . Recall the subcategory C` from §1.6, and its Grothendieck ring R` . We say that a simple object S of C` is real if S ˝ S is simple.


141

Conjecture 3.1 (Hernandez–Leclerc [24]). The assignment z.i;i C2k/ 7! W .i/

`C1k; q i C2k

extends to a ring isomorphism ` W A` ! R` . The map ` induces a bijection between cluster monomials and classes of real simple objects of C` , and between cluster variables and classes of real prime simple objects of C` . Note that in [24] we have chosen a different initial seed for defining A` , so the Kirillov–Reshetikhin modules assigned to the initial cluster variables have different spectral parameters5 . For g D sl2 , Conjecture 3.1 holds, as it is just a reformulation of the classical results of §1.4. If true in general, Conjecture 3.1 will give a combinatorial description in terms of cluster algebras of the prime tensor factorization of every real simple module of C . Note that, by definition, the square of a cluster monomial is again a cluster monomial. This explains why cluster monomials can only correspond to real simple modules. For g D sl2 , all simple Uq .Lg/-modules are real. However for g 6D sl2 there exist imaginary simple Uq .Lg/-modules (i.e., simple modules whose tensor square is not simple), as shown in [29]. This is consistent with the expectation that a cluster algebra with infinitely many cluster variables is not spanned by its set of cluster monomials. We arrived at Conjecture 3.1 by noting that the T -system equations satisfied by Kirillov–Reshetikhin modules (see §1.3) are of the same form as the exchange relations of a cluster algebra. This was inspired by the seminal work [12], in which cluster algebra combinatorics is used to prove Zamolodchikov’s periodicity conjecture for Y -systems attached to Dynkin diagrams. 3.3 The case ` D 1. Our main evidence for Conjecture 3.1 is the following: Theorem 3.2 ([24], [33]). Conjecture 3.1 holds for g of type A, D, E and ` D 1. In this case, all simple modules are real. This was first proved in [24] for type A and D4 by combinatorial and representation-theoretic methods, and soon after, by Nakajima [33] in the general case, by using the geometric description of the irreducible q-characters explained in §2. These two different proofs will be explained in §3.4 and §3.5. Let us illustrate Theorem 3.2 for g D sl4 . As the cluster algebra A1 has finite cluster type A3 , the cluster variables, and therefore the non frozen prime simple modules, are in bijection with the almost positive roots of A3 [14]. Of course, there can be several 5 We take this opportunity to correct a typo in [30]: in the statement of Conjecture 9.1, one should replace .i/ .i/ by Wk; . Wk; q i C2.`C1k/ q i

142

B. Leclerc

such bijections. The bijection chosen in [24] is as follows: S.˛1 / D L.Y1;q 2 /;

S.˛2 / D L.Y2;q /;

S.˛3 / D L.Y3;q 2 /;

S.˛1 / D L.Y1;q 0 /;

S.˛2 / D L.Y2;q 3 /;

S.˛3 / D L.Y3;q 0 /;

S.˛1 C ˛2 / D L.Y1;q 0 Y2;q 3 /;

S.˛2 C ˛3 / D L.Y2;q 3 Y3;q 0 /;

S.˛1 C ˛2 C ˛3 / D L.Y1;q 0 Y2;q 3 Y3;q 0 /: Note that the last three modules are not Kirillov–Reshetikhin modules. There are three more prime simples corresponding to the three frozen variables of A1 , namely F1 D L.Y1;q 0 Y1;q 2 /; F2 D L.Y2;q Y2;q 3 /; F3 D L.Y3;q 0 Y3;q 2 /: The cluster algebra A1 has fourteen clusters, in bijection with the vertices of the associahedron shown in Figure 6 [14]. The faces of the associahedron are naturally labeled by the almost positive roots (the rear, bottom, and leftmost faces are labeled by rp r pp ˛2 @ p @ r pp r p @ pp B B @ ppp pp B @ pp B ˛2 C ˛3 @ pp ˛ Cpp ˛ Br @r 1 pp 2 pp @ @ p @ pp pp @r pp ˛ C˛ C˛ pp 1 2 3 ppp r r ˛ pp @ p p p p p p 3p p p p p p r p p p p p p@ p p p p p p p p p p p p p r p @ ppp p p @r p p ˛1 p p p pp ppp p p rp p r Figure 6. The associahedron of type A3 .

˛1 , ˛2 , and ˛3 , respectively). Each vertex corresponds to the cluster consisting in its three adjacent faces: f˛1 ; ˛2 ; ˛3 g;

f˛1 ; ˛2 ; ˛3 g;

f˛1 ; ˛2 ; ˛3 g; f˛1 ; ˛2 ; ˛3 g

f˛1 ; ˛2 ; ˛3 g; f˛1 ; ˛2 ; ˛2 C ˛3 g; f˛1 ; ˛3 ; ˛2 C ˛3 g; f˛3 ; ˛2 ; ˛1 C ˛2 g; f˛3 ; ˛1 ; ˛1 C ˛2 g;

f˛1 C ˛2 ; ˛2 ; ˛2 C ˛3 g;

f˛1 ; ˛1 C ˛2 ; ˛1 C ˛2 C ˛3 g;

f˛1 ; ˛3 ; ˛1 C ˛2 C ˛3 g;

f˛3 ; ˛2 C ˛3 ; ˛1 C ˛2 C ˛3 g;

f˛1 C ˛2 ; ˛2 C ˛3 ; ˛1 C ˛2 C ˛3 g:


143

The simple modules of C1 are exactly all tensor products of the form S.ˇ1 /˝k1 ˝S.ˇ2 /˝k2 ˝S.ˇ3 /˝k3 ˝F1˝l1 ˝F2˝l2 ˝F3˝l3 ; .k1 ; k2 ; k3 ; l1 ; l2 ; l3 / 2 N 6 ; in which fˇ1 ; ˇ2 ; ˇ3 g runs over the 14 clusters listed above. Note that by Gabriel’s theorem, positive roots are in one-to-one correspondence with indecomposable representations of Q. By inspection of the above list of clusters, one can check that two roots belong to a common cluster if and only if the corresponding representations of Q have no extension between them. This is true in general, as will be explained below (see Corollary 3.3, and the end of §3.5). 3.4 Proof of Theorem 3.2: approach of [24]. Write for short zi D z.i;i C2/ for the cluster variables of the initial cluster z. We know that R1 is the polynomial ring in the classes of the fundamental modules L.Yi;q i / and L.Yi;q i C2 /. On the other hand, it is not difficult to show that, because of the presence of the frozen variables, A1 is the polynomial ring in the variables zi , together with the variables zi0 of the cluster z0 obtained from z by applying the product of mutations Y Y i i z: z0 D i2I0

i2I1

Therefore, the assignment zi 7! ŒL.Yi;q i C2 /;

zi0 7! ŒL.Yi;q i /

.i 2 I /;

extends to a ring isomorphism from A1 to R1 . To calculate the images under of the remaining cluster variables, we use the fact [15] that every cluster variable is entirely determined by its F -polynomial (and its g-vector) with respect to the reference cluster z. For an almost positive root ˇ D P b ˛ , let zŒˇ be the cluster variable whose cluster expansion with respect to z has i i i Q b denominator i zi i . In particular zi D zŒ˛i . Denote by Fˇ the F -polynomial of zŒˇ. By convention F˛i D 1. On the other hand, the q-character of an object M of C1 is uniquely determined by its truncation obtained by specializing A1 to 0 for k > i C 1. The truncated i;q k q-character of a simple object L.m/ of C1 is of the form q .L.m//2 D m Pm .v1 ; : : : ; vn /

(23)

where P is a polynomial in the variables vi ´ A1i C1 .i 2 I / with constant term 1. i;q Moreover, the map W ŒL.m/ 7! q .L.m//2 is an injective ring homomorphism from R1 to its image in Y. The injectivity comes from the fact that the truncated q-character of a module of C1 already contains all its dominant monomials. It is thus enough to determine the images of the cluster variables of A1 under 0 ´ .

144

B. Leclerc

Let si .i 2 I / be the Coxeter generators of the Weyl group. In [24], it is proved that for ˇ > 0,

0 .zŒˇ/ D Y ˛ Fˇ .v1 ; : : : ; vn /; (24) where ˛D

X

ai ˛i ´

i

and Y˛ D

8Q a i < Yi;3 i :

Y

si ˇ

if ˛ > 0;

i2I

Yi;2i

(25)

i2I1

if ˛ D ˛i :

(26)

Thus, setting m D Y ˛ and comparing (24) with (23), we see that an important step in proving Theorem 3.2 is to show that the two polynomials Pm and Fˇ coincide. This last statement is verified in [24] for every root ˇ in types An and Dn , and for every multiplicity-free root ˇ in type En . The proof uses the Frenkel–Mukhin algorithm for evaluating Pm , and the combinatorial description of the Fibonacci polynomials of Fomin and Zelevinsky [13] for evaluating Fˇ . Thus, except for these missing roots in type En , this shows that all cluster variables of A1 are mapped by to the classes of some simple modules in R1 . The second main step is the following tensor product theorem, proved for all types An , Dn , En . Let S1 ; : : : ; Sk be simple modules of C1 , and suppose that for every 1 i < j k the tensor product Si ˝Sj is simple. Then it is shown [24], Theorem 8.1, that S1 ˝ ˝ Sk is simple6 . Thus, to show that the image of a cluster monomial by

is the class of a simple module, it is enough to prove it when the monomial is the product of two cluster variables. Finally, the third step consists in proving that if zŒˇ and zŒ are two compatible cluster variables of A1 , that is, if zŒˇzŒ is a cluster monomial, then the tensor product of the corresponding simple modules of C1 is simple. Since, for a given g, there are only finitely many cluster variables in A1 , and so finitely many compatible pairs, this is in principle only a “finite check”. Unfortunately it is not easy in general to decide if a product of (truncated) irreducible q-characters is simple, and in [24] this was only settled completely in types An and D4 . Although this (partial) proof is combinatorial and representation-theoretic, it has an interesting geometric consequence. Indeed, it shows that the truncated q-characters of the prime simple objects of C1 coincide, after dividing out the highest l-weight monomial, with the F -polynomials of the cluster variables of A1 . But the F -polynomials have a geometric description due to Fu and Keller [16] in terms of quiver Grassmannians, inspired from a similar formula of Caldero and Chapoton for cluster expansions of cluster variables [4]. Therefore we get the following geometric description of the truncated q-characters. 6

This theorem was later extended by Hernandez [23] to the whole category C .


145

Let M Œˇ be the indecomposable representation of the Dynkin quiver Q attached to a positive root ˇ, and denote by Gr .M Œˇ/ the quiver Grassmannian of subrepresentations of M Œˇ with dimension vector . Let ˛ and Y ˛ be related to ˇ as in Eq.(25), (26). Finally, recall the notation vi ´ A1i C1 . i;q

Corollary 3.3 ([24]). Conjecture 3.1 for C1 implies that X q .L.Y ˛ //2 D Y ˛ .Gr .M Œˇ// v11 : : : vnn :

(27)

More generally, we have a similar truncated q-character formula for every simple module of C1 , in which the indecomposable representation M Œˇ of the right-hand side is replaced by a generic representation of Q, i.e., a representation without selfextension. Corollary 3.3 should be compared to Eq.(20) and (21) for q-characters of standard modules. What is remarkable here is that we obtain a similar formula for simple modules of C1 : for all these modules, we do not need to use the decomposition theorem for perverse sheaves, as was done in §2.10. 3.5 Proof of Theorem 3.2: approach of [35]. In [35], Nakajima reverses the logic of [24], and first proves the formula of Corollary 3.3 for all simple modules of C1 and for all Dynkin types, by means of his description in terms of perverse sheaves. This is made possible because of the following simple description of the quiver varieties M .V; W / and M0 .W / when W corresponds to the highest l-weight of a simple object of C1 , and the monomial Y W AV contributes to its truncated q-character. One first notes that L.W / is in C1 if and only if the Iy0 -graded space W satisfies Wi .r/ 6D 0

only if

r 2 fi ; i C 2g:

(28)

Moreover, if Y W AV appears in q .L.W //2 then Vi .r/ 6D 0

only if

r D i C 1:

(29)

Thus W and V are supported on a zig-zag strip of height 2, as shown in Figure 7. Therefore the ADHM equations are trivially satisfied in this case, and we have M .V; W / D ƒ .V; W /. Since every dominant monomial appears in the truncated q-character, and since M0 .W / is equal to M0 .V; W / for some l-dominant pair .V; W /, we see that M0 .W / D M .V; W /==GV for some V satisfying (29). Define the following GV -invariant maps on M .V; W /: xi D ˇi .i C 1/˛i .i C 2/ .i 2 I /; .i 2 I1 ; j 2 I0 ; aij D 1/: yij D ˇj .1/Bij .2/˛i .3/

(30) (31)

146

B. Leclerc

W2 .3/ ˛2 .3/

W1 .2/ V2 .2/ W3 .2/ HH HH B .2/ vv v H 23 v ˛1 .2/vvv ˇ .2/ HHH ˛3 .2/ HH v B21 .2/ 2 v {v # V1 .1/ W2 .1/ V3 .1/

ˇ1 .1/

W1 .0/

ˇ3 .1/

W3 .0/

Figure 7. The graded spaces W and V associated with a simple object of C1 in type A3 .

z obtained by The data .xi ; yij / amount to a representation of a decorated quiver Q attaching to every vertex i of Q a new vertex i 0 and an arrow i 0 ! i (resp. i ! i 0 ) if i 2 I0 (resp. i 2 I1 ) (see Figure 8). Let M M EW D Hom.Wi .i C 2/; Wi .i // ˚ Hom.Wi .3/; Wj .0// i2I

i2I1 ; j 2I0 ; aij D1

z based on W . Nakajima shows that the map be the space of representations of Q .B; ˛; ˇ/ 7! .xi ; yij / induces an isomorphism from M0 .W / to EW . Hence, for L.W / in C1 the affine variety M0 .W / is isomorphic to a vector space. 25 555 5 0 x2 55 10 55y23 3 55 y21 55 0 x1 x 2 55 3 5 1 3 z in type A3 . Figure 8. The decorated quiver Q

Put i D dim Vi and D .i / 2 N I . Let F .; W / be the variety of n-tuples X D .Xi / of subspaces of W satisfying dim Xi D i and M Xi Wi .1/ ˚ Xj .i 2 I1 /: Xi Wi .0/ .i 2 I0 /; j W aij D1

Define Fz .; W / as the closed subvariety of EW F .; W / consisting of all elements


147

..xi ; yij /; X / such that Im xi Xi .i 2 I0 /;

Im xi ˚

M

yij Xi .i 2 I1 /:

(32)

j W aij D1 Nakajima shows that .B; ˛; ˇ/ L2 M .V; W / is stable if and only if all maps ˇi .1/ .i 2 I0 / and i .2/ ´ ˇi .2/ ˚ j W aij D1 Bij .2/ .i 2 I1 / are injective. Clearly the collection X of spaces

Xi D ˇi .1/.Vi .1// .i 2 I0 /; Xi D i .2/.Vi .2// .i 2 I1 /;

(33) (34)

is GV -invariant, and dim Xi D i if .B; ˛; ˇ/ is stable. Therefore, the map .B; ˛; ˇ/ 7! ..xi ; yij /; X / defined by (30), (31), (33) and (34) induces a map from M .V; W / to Fz .; W /, and Nakajima shows that this is an isomorphism [35], Proposition 4.6. Moreover, when M .V; W / and M0 .W / are realized as Fz .; W / and EW , respectively, then the projective morphism V W M .V; W / ! M0 .W / becomes the first projection. (Compare this description with the prototypical example of §2.5.1.) By Theorem 2.3, to calculate the truncated q-character of a simple module of C1 , one must now compute the multiplicity aV;0IW .1/ of the skyscraper sheaf ICW .0/ in the expansion of Œ.V /Š .1M .V;W / / on the basis fŒICW .V 0 /g. Since M0 .W / ' EW is a denote the dual space, vector space, one can use for that a Fourier transform. Let EW and let be the Fourier–Sato–Deligne functor from the derived category D.EW / to D.EW /. The functor maps every simple perverse sheaf ICW .V / on EW to a simple . In particular, the image of the skyscraper sheaf is perverse sheaf on EW Œdim EW ; .ICW .0// D 1EW

the constant sheaf on EW , with degree shifted by dim EW . We can regard the product EW F .; W / as a trivial vector bundle on F .; W / with fiber EW . By (32), the fibers of the restriction of the second projection to Fz .; W / are vector spaces of constant dimension, and Fz .; W / can be seen as a subbundle of EW F .; W /. Denote by Fz .; W /? the annihilator of Fz .; W / in the dual trivial bundle EW F .; W /. We also have a Fourier–Sato–Deligne functor 0 from the derived category of the trivial bundle EW F .; W / to that of EW F .; W /. It satisfies 0

1Fz .;W / Œdim Fz .; W / D 1Fz .;W /? Œdim Fz .; W /? :

the bundle Moreover, denoting by W Fz .; W / ! EW and ? W Fz .; W /? ! EW maps, we have the commutation relation

Š? B

0

D

B Š :

148

B. Leclerc

It follows that the required (ungraded) multiplicity aV;0IW .1/ is equal to the multi? in the expansion of .1 z plicity of the constant sheaf 1EW / in terms of the Š F .;W /? f .ICW .V //g. The advantage of this Fourier transformation is that we can now evaluate this new multiplicity by looking at the stalk of Š? .1Fz .;W /? / over a generic point . of EW At this point we remark that we can without loss of generality assume that for every i 2 I one has Wi .2 i / D 0. In other words, we suppose that EW is a space of representations of the quiver Q without decoration. (One can easily reduce the general case to this one by factoring out from L.W / a tensor product of frozen Kirillov– Reshetikhin modules L.Yi;i Yi;i C2 /, as in [24], §9.2, or [35], §6.3.) So EW is a space of representations of the quiver Q obtained from Q by changing the orientation of every arrow. Let Y GW ´ GL.Wi .3i //: i2I

EW

has an open dense GW -orbit corresponding to the Since Q is a Dynkin quiver, generic representation of dimension vector .dim Wi .3i //, and all other GW -orbits have strictly smaller dimension. Now, all the simple perverse sheaves .ICW .V // are GW -equivariant, hence they are supported on a union of GW -orbits, so the only one Œdim EW . having a nonzero stalk over a generic point of EW is .ICW .0// D 1EW ? Therefore, by definition of the pushdown functor Š , the multiplicity aV;0IW .1/ is nothing else than the dimension of the total cohomology of a generic fiber of ? . It remains to describe this generic fiber. Because of our simplifying assumption, a point of EW is now just a collection of maps yij 2 Hom.Wi .3/; Wj .0// .aij D 1/, and a point in F .; W / is a collection of subspaces X D .Xi / of W such that M Xi Wi .0/ .i 2 I0 /; Xi Xj .i 2 I1 /: j Waij D1

The pair ..yij /; X / belongs to Fz .; W / if and only if Im.˚j yij / Xi for all i 2 I1 . Clearly, the annihilator Fz .; W /? consists of pairs ..yij /; X / in EW F .; W / such P that Xi Ker j yij for every i 2 I1 . To get a nicer description of the fibers of ? we consider the product of Gelfand–Ponomarev reflection functors at every sink i 2 I1 of Q . The functor sends .yij / 2 EW to .yij / 2 EW defined by Wi .0/ D Wi .0/ .i 2 I0 /;

Wi .3/ D Ker

X

yij

.i 2 I1 /;

j

P and, for i 2 I1 , yik is the composition of the embedding of Ker j yij in ˚j Wj .0/ followed by the projection onto Wk .0/. The collection of linear maps y D .yij / is a representation of the original quiver Q. By construction, Xi Wi .3i / for every i 2 I . Moreover, one can easily check that ..yij /; X / 2 Fz .; W /? if and only if


149

yij .Xi / Xj for every i 2 I1 . In other words, X belongs to the fiber of ? above .yij / if and only if X is a point of the quiver Grassmannian Gr .y /. It follows that the multiplicity aV;0IW .1/ of the monomial Y W AV in q .L.Y W //2 is the total dimension of the cohomology of Gr .y / for a generic representation y of Q Q in EW . Note that the product of reflection functors categorifies the product s in the Weyl group, so if we denote by ˇ the graded dimension of W , and i i2I1 if we assume that ˇ is a positive root, then the graded dimension ˛ of W is related to ˇ by (25), in perfect agreement with (27). Moreover, Nakajima explains that the vanishing of the odd cohomology of L .V; W / implies that this generic fiber has no odd cohomology, therefore aV;0IW .1/ is also equal to the Euler characteristic of the quiver Grassmannian Gr .y /. Thus, Corollary 3.3 follows in full generality. After this q-character formula is established, Nakajima proceeds to show that the tensor product factorization of the simple modules L.W / of C1 is given by the canonical direct sum decomposition of the corresponding generic quiver representation y of EW into indecomposable summands. The proof uses the geometric realization given by Varagnolo and Vasserot [41] of the t-deformed product of .q; t /-characters in terms of convolution of perverse sheaves. Finally, to relate the q-character formula with cluster algebras, Nakajima makes use of the cluster category of Buan, Marsh, Reineke, Reiten and Todorov [3], and of the Caldero-Chapoton formula for cluster variables [4]. It is worth noting that Nakajima’s approach is more general: most of his results work for the quantum affinization Uq .Lg/ of a possibly infinite-dimensional symmetric Kac– Moody algebra g. This yields some important positivity results for all cluster algebras attached to an arbitrary bipartite quiver. However, when g is infinite-dimensional Uq .Lg/ is no longer a Hopf algebra, and the meaning of the multiplicative structure of the Grothendieck group is less clear (see [22]). 3.6 The case ` > 1. If g D sl2 , Conjecture 3.1 holds for every `. Otherwise, Conjecture 3.1 has only been proved for g D sl3 and ` D 2 [24], §13. In that small rank case, A2 still has finite cluster type D4 (see Table 1), and this implies that C2 has only real objects. There are 18 explicit prime simple objects with respective dimensions 3; 3; 3; 3; 3; 3; 6; 6; 6; 6; 8; 8; 8; 10; 10; 15; 15; 35; and 50 factorization patterns (corresponding to the 50 vertices of a generalized associahedron of type D4 [14]). Our proof in this case is quite indirect and uses a lot of ingredients: the quantum affine Schur–Weyl duality, Ariki’s theorem for type A affine Hecke algebras [1], the coincidence of Lusztig’s dual canonical and dual semicanonical bases of CŒN in type A4 [18], and the results of [19] on cluster algebras and dual semicanonical bases. This proof could be extended to g D sln and every ` if the general conjecture of [19] about the relationship between Lusztig’s dual canonical and dual semicanonical bases was established.

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Acknowledgements. The author is very grateful to the organizing committee of ICRA XIV for the invitation to give a minicourse in Tokyo, and to the editors for their efforts in preparing this volume. Figure 6 was borrowed from a paper of Fomin and Zelevinsky.

References [1] S. Ariki, On the decomposition numbers of the Hecke algebra of G.n; 1; m/. J. Math. Kyoto Univ. 36 (1996), 789–808. [2] T. Akasaka and M. Kashiwara, Finite-dimensional representations of quantum affine algebras. Publ. RIMS 33 (1997), 839–867. [3] A. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov, Tilting theory and cluster combinatorics. Advances Math. 204 (2006), 572–618. [4] P. Caldero and F. Chapoton, Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv. 81 (2006), 595–616. [5] V. Chari and D. Hernandez, Beyond Kirillov-Reshetikhin modules. In Quantum Affine Algebras, Extended Affine Lie Algebras, and Their Applications. Contemporary Mathematics 506, Amer. Math. Soc., Providence, 2010, 49–81. [6] V. Chari and A. Pressley, Quantum affine algebras. Comm. Math. Phys. 142 (1991), 261– 283. [7] V. Chari and A. Pressley, A Guide to Quantum Groups. Cambridge University Press, Cambridge, 1994. [8] V. G. Drinfeld, A new realization of Yangians and quantized affine algebras. Soviet Math. Doklady 36 (1988), 212–216. [9] E. Frenkel and E. Mukhin, Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras. Comm. Math. Phys. 216 (2001), 23–57. [10] E. Frenkel and N. Reshetikhin, The q-characters of representations of quantum affine algebras. In Recent Developments in Quantum Affine Algebras and Related Topics. Contemporary Mathematics 248, Amer. Math. Soc., Providence, 1999, 163–205. [11] S. Fomin, Total positivity and cluster algebras. In Proceedings of the International Congress of Mathematicians 2010, vol. 2, Hyderabad, 2010, 125–145. [12] S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations. J. Amer. Math. Soc. 15 (2002), 497–529. [13] S. Fomin and A. Zelevinsky, Y -systems and generalized associahedra. Ann. of Math. 158 (2003), 977–1018. [14] S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification. Invent. Math. 154 (2003), 63–121. [15] S. Fomin andA. Zelevinsky, Cluster algebras IV: coefficients. Compositio Math. 143 (2007), 112–164. [16] C. Fu and B. Keller, On cluster algebras with coefficients and 2-Calabi-Yau categories. Trans. Amer. Math. Soc. 362, (2010), 859–895.


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[17] P. Gabriel, The universal cover of a representation-finite algebra. In Representations of Algebras. Lecture Notes in Mathematics 903, Springer-Verlag Berlin, 1981, 68–105. [18] C. Geiss, B. Leclerc and J. Schröer, Semicanonical bases and preprojective algebras. Ann. Sci. École Norm. Sup. 38 (2005), 193–253. [19] C. Geiss, B. Leclerc and J. Schröer, Rigid modules over preprojective algebras. Invent. Math. 165 (2006), 589–632. [20] V. Ginzburg and E. Vasserot, Langlands reciprocity for affine quantum groups of type An . Internat. Math. Res. Notices 3 (1993), 67–85. [21] D. Hernandez, The Kirillov-Reshetikhin conjecture and solutions of T -systems. J. Reine Angew. Math. 596 (2006), 63–87. [22] D. Hernandez, Drinfeld coproduct, quantum fusion tensor category and applications. Proc. London Math. Soc. 95 (2007), 567–608. [23] D. Hernandez, Simple tensor products. Invent. Math. 181 (2010), 649–675. [24] D. Hernandez and B. Leclerc, Cluster algebras and quantum affine algebras. Duke Math. J. 154 (2010), 265–341. [25] M. Jimbo (Editor), Yang-Baxter Equation in Integrable Systems. Advanced Series in Mathematical Physics 10, World Scientific, Singapore, 1990. [26] M. Jimbo, Topics from Representations of Uq .g/ – An Introductory Guide to Physicists. In Quantum Groups and Quantum Integrable Systems. Nankai Lectures on Mathematical Physics, World Scientific, Singapore, 1992, 1–61. [27] A. N. Kirillov and N. Reshetikhin, Representations of Yangians and multiplicities of the inclusion of the irreducible components of the tensor product of representations of simple Lie algebras. J. Sov. Math. 52 (1990), 3156–3164. [28] A. Kuniba, T. Nakanishi and J. Suzuki, Functional relations in solvable lattice models. I. Functional relations and representation theory. Internat. J. Modern Phys. A 9 (1994), 5215–5266, [29] B. Leclerc, Imaginary vectors in the dual canonical basis of Uq .n/. Transformation Groups 8 (2003), 95–104. [30] B. Leclerc, Cluster algebras and representation theory. In Proceedings of the International Congress of Mathematicians 2010. Hyderabad, vol. 4, 2471–2488. [31] G. Lusztig, On quiver varieties. Advances Math. 136 (1998), 141–182. [32] H. Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Amer. Math. Soc. 14 (2001), 145–238. [33] H. Nakajima, Quiver varieties and t -analogs of q-characters of quantum affine algebras. Ann. of Math. 160 (2004), 1057–1097. [34] H. Nakajima, t -analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras. Representation Theory 7 (2003), 259–274. [35] H. Nakajima, Quiver varieties and cluster algebras. Kyoto J. Math. 51 (2011), 71–126. [36] W. Nakai and T. Nakanishi, On Frenkel-Mukhin algorithm for q-character of quantum affine algebras. In Exploration of New Structures and Natural Constructions in Mathematical Physics, Adv. Stud. in Pure Math., Amer. Math. Soc., to appear; arXiv:0801.2239 [math.QA].

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[37] M. Okado and A.Schilling, Existence of Kirillov-Reshetikhin crystals for nonexceptional types. Representation Theory 12 (2008), 186–207. [38] A. Savage and P. Tingley, Quiver Grassmanianns, quiver varieties and preprojective algebras. Pacific J. Math. 251 (2011), 393–429. [39] V. O. Tarasov, On the structure of quantum L-operators for the R-matrix of XXZ-model. Theoret. Math. Phys. 61 (1984), 163–173. [40] V. O. Tarasov, Irreducible monodromy matrices for the R-matrix of XXZ-model and local lattice quantum Hamiltonian. Theoret. Math. Phys. 63 (1985), 440–454. [41] M. Varagnolo and E. Vasserot, Perverse sheaves and quantum Grothendieck rings. In Studies in Memory of Issai Schur. Progress in Mathematics 210, Birkhäuser 2003, 345–365. [42] A. Zelevinsky, What is a cluster algebra? Notices of the Amer. Math. Soc. 54 (2007), 1494– 1495.

Weighted projective lines and applications Helmut Lenzing Dedicated to Idun Reiten on the occasion of her 70th birthday

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . 2 Generalities . . . . . . . . . . . . . . . . . . . . . 3 Euler characteristic and stability . . . . . . . . . . 4 Triangle singularities . . . . . . . . . . . . . . . . 5 Kleinian and Fuchsian singularities . . . . . . . . . 6 Flags of invariant subspaces for nilpotent operators 7 Comments . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction This survey covers the topics of my mini-course on “Weighted projective lines” given at the Workshop of ICRA XIV. Weighted projective lines, and their defining equations of shape x a ˙ y b ˙ z c D 0 have a long history going back to F. Klein [24] and H. Poincaré [49]. Accordingly their study has a high contact surface with many mathematical subjects, classical and modern. Among the many related subjects we mention representation theory of algebras and groups, invariant theory, function theory, orbifolds, 3-manifolds, singularities and the study of nilpotent operators. Since the formal definition of the category of coherent sheaves by W. Geigle and the author in 1987, see [14], substantial progress has been made by a number of authors where, however, many results are difficult to locate. So to some extent this survey also serves as a guide to later developments. As a recent application, the analysis of the singularity category of triangle singularities, and Kleinian and Fuchsian singularities is covered in some detail. In the center of this analysis is the structure of the corresponding stable categories of vector bundles. The analysis is in the spirit of Buchweitz [3] and Orlov [48], and concerns work in progress with Kussin and Meltzer [26], [27], [28], recent work by Kajiura, Saito, and Takahashi [21], [22] and joint work with J. A. de la Peña [36]. These methods are further applied to the study by C. M. Ringel and M. Schmidmeier [55] on the invariant subspace problem for nilpotent operators.

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2 Generalities Throughout we work over an algebraically closed base field k. Roughly speaking, a weighted projective line X is given by marking a finite number of points 1 ; : : : ; t from the projective line P 1 .k/ by positive integers p1 ; : : : ; p t , where pi is attached to i . Without loss of generality we may assume 1 D 1, 2 D 0 and 3 D 1. We call p D .p1 ; : : : ; p t / the weight sequence and D .1 ; : : : ; t / the parameter sequence of X. Thus 3 ; : : : ; t are pairwise distinct, non-zero elements from k. In the spirit of non-commutative algebraic geometry we are going to define an abelian category coh-X, to be thought of as the category of coherent sheaves on X, without specifying what X is. The upshot is that all information we get on X is already covered by the category coh-X. 2.1 The category of coherent sheaves. To the data p D .p1 ; : : : ; p t / and D .1 ; : : : ; t / we are associating the following objects: (i) The rank one abelian group L D L.p/ on generators xE1 ; : : : ; xE t subject to the relations p1 xE1 D p2 xE2 D D p t xE t DW cE: The element cE will be called the canonical element ofP L. The group L is an ordered group whose cone of positive elements is given as tiD1 N xEi . Due to the above relations, each element xE of L has a unique expression xE D

t X

ì xEi C È c;

where 0 ì < pi and ` 2 Z:

iD1

It follows that L is almost linearly ordered, namely for each xE from L we either have P xE 0 or xE !E C cE, where !E D .t 2/E c tiD1 xEi . (ii) The k-algebra S D S.p; /kŒx1 ; : : : ; x t =I , where I D .f3 ; : : : ; f t / is the p ideal generated by all so-called canonical relations fi D xi i .x2p2 i x1p1 /, i D 3; : : : ; t. S carries a natural L-grading by declaring the xi to be homogeneous of degree xEi . In this way M SD SxE ; where SxE SyE SxC E yE ; and S0 D k: x0 E

By modL -S we denote the category of all finitely generated L-graded S-modules with morphisms being the graded S-linear maps of degree zero. Note that L acts on modL -S by grading shift: for an L-graded module M , and xE from L we denote by M.x/ E the S -module M with the new grading M.x/ E yE D MxC . E yE We are further interested in the full subcategory modL 0 -S consisting of all graded modules of finite length (= finite k-dimension). This subcategory is a Serre subcategory of modL -S, that is, is closed under forming submodules, factor modules and extensions.

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(iii) This allows to apply Serre’s construction, see [56], forming the quotient category H WD modL -S=modL 0 -S in the sense of Serre–Grothendieck–Gabriel [13]. H has the same objects as modL -S. We further keep the morphisms of modL -S but formally invert all morphisms having both kernel and cokernel in modL 0 -S . The category H is again an abelian category; moreover the natural quotient functor q W modL -S ! H is exact with kernel modL 0 -S. We say, that H is the category of coherent sheaves on X, z for the image of notation: H D coh-X. For emphasis, we are using the notation M z sheafification. Note that M under the quotient functor, and call the process M 7! M z .x/ the L-action on modL -S induces an L-action on coh-X such that M E WD .M.x//Q. E z We call O D S the structure sheaf of X, and call the O.x/ E the shifted (or twisted) structure sheaves. Remark 2.1. The above procedure is the fastest introduction of the category coh-X. Readers who are not happy with this virtual, and highly implicit definition of the weighted projective line X itself, are advised to consult [14] which contains a definition of X as a ringed space by means of a graded sheaf theory. Theorem 2.2 ([14]). The category coh-X is connected, Hom-finite, k-linear with the following properties: (i) coh-X is abelian and also noetherian, that is, ascending chains of subobjects are stationary. (ii) coh-X has Serre duality in the form D Ext1 .X; Y / D Hom.Y; X /, where the k-equivalence W coh-XP ! coh-X is the shift X 7! X.!/ E with the dualizing element !E D .t 2/E c tiD1 xEi . (iii) coh-X is hereditary, that is, Exti .; / D 0 for each i 2. (iv) Each X from coh-X has a decomposition X D X0 ˚ XC , where X0 has finite length in coh-X and XC has no non-zero subobjects of finite length. (v) The simple objects from coh-X are naturally parametrized by the projective line P 1 .k/, where each … f1 ; : : : ; t g is associated with a unique simple S , called ordinary simple, and each D i comes associated with pi simple objects, called exceptional simple. Moreover, each ordinary simple S has Ext1 .S; S / D k, while each exceptional simple S has Ext1 .S; S / D 0. (vi) For any x; E yE from L, the quotient functor q induces isomorphisms Sy E x E D Hom.O.x/; E O.y//. E In particular, Hom.O.x/; E O.y// E ¤ 0 if and only if xE y. E By coh0 -X we denote the full subcategory of finite length (= torsion) objects of coh-X. The objects without non-zero subobjects of finite length form a category vect-X that we call the category of vector bundles on X. 2.2 Grothendieck group and Euler form. We denote by K0 .X/ D K0 .coh-X/ the Grothendieck group of coh-X. The Grothendieck group is equipped with the Euler

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form which is the Z-bilinear form h; i W K0 .X/ K0 .X/ ! Z, given on classes of objects X; Y of coh-X by hŒX ; ŒY i D dim Hom.X; Y / dim Ext1 .X; Y /: There are two important linear forms rk and deg on K0 .X/, called rank and degree. The rank rk W K0 .X/ ! Z is characterized by the fact that rk.O.x// E D 1 for each xE from L. The rank is zero on torsion sheaves and strictly positive on non-zero vector bundles. Let ı W L ! Z be the homomorphism sending xEi to p=p N i for i D 1; : : : ; t, where pN is the least common multiple of p1 ; : : : ; p t . Then deg W K0 .X/ ! Z is characterized by the property that deg.O.x// E D ı.x/. E Another characterization is that deg.O/ D 0 and the degree deg.S / of a simple sheaf S equals pN if S is ordinary simple, and deg.S/ D p=p N i if S is an exceptional simple attached to the point i . Vector bundles of rank one are called line bundles. It follows from the graded factoriality of S that each line bundle L has the form L Š O.x/ E for a uniquely defined xE 2 L. Moreover, line bundles serve as building blocks for vector bundles in general: Proposition 2.3 (Line bundle filtration). Each vector bundle E has filtration by line bundles 0 D E0 E1 E2 Er D E with factors Li D Ei =Ei1 which are line bundles. Moreover, the number r of line bundles factors of E is an invariant of E, the rank of E. Note that morphisms and by Serre duality also extensions between line bundles are explicitly known. An object E in an abelian Hom-finite k-category is called exceptional if End.E/ D k and Extn .E; E/ D 0 for each n > 0. Similarly, an object E from a Hom-finite triangulated k-category is called exceptional if End.E/ D k and Hom.E; EŒn/ D 0 for each integer n ¤ 0.1 Exceptional objects have a somewhat rigid structure. In the case of weighted projective lines they are already determined by their class in the Grothendieck group, see [18], Lemma 4.2, or [44], Proposition 4.4.1. For the convenience of the reader, we reproduce the proof which is based on the fact that the Grothendieck group K0 .X/ exerts a tight control on coh-X: Lemma 2.4. Assume X from coh-X has rank and degree zero. Then X D 0. In particular, from ŒX D 0 in K0 .X/ it follows that X D 0. Proof. Since X has rank zero, it has finite length. Since each simple sheaf has strictly positive degree, the degree of X from coh0 -X then equals a (positively) weighted length of X . So if zero, X has to be zero. 1

If k is not algebraically closed, one allows End.E / to be a finite skew field extension of k.


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x D Q [ f1g By Lemma 2.4 each non-zero object X has a well-defined slope in Q given by .X / D deg.X /= rk.X /. The slope belongs to Q for non-zero vector bundles. It is 1 for each non-zero sheaf of finite length. The following property of the slope will be frequently used: Lemma 2.5. For each vector bundle F from coh-X and xE from L we have F .x/ E F D ı.x/. E In particular, F F D ı.!/. E Proof. The formula is obviously true for line bundles. By Proposition 2.3, F has a finite filtration with line bundle factors ŒL1 ; : : : ; ŒLr implying that F .x/ EPhas such a filtration with factors ŒL1 .x/; E : : : ; ŒLr .x/. E It follows that ŒL.xE1 /ŒL D riD1 .ŒL.x/ŒL E i /. Passing to degrees, we obtain deg.F .x// E deg.F / D rk.F /ı.x/. E The claim follows. The following lemma does not extend to a smooth projective curve of genus g > 0 (or Euler characteristic 0). Indeed, also there a line bundle L has trivial endomorphism ring but the space Ext 1 .L; L/ has dimension g. Hence exceptionality of line bundles implies genus zero, a result also true in a weighted setting. Exceptionality of line bundles on a weighted projective line is therefore indicating that the genus of the smooth projective curve C underlying X is zero, that is, C is the projective line. (Recall that we work over an algebraically closed field k.) Lemma 2.6. For a weighted projective line each line bundle L is exceptional. Proof. We know that L D O.x/ E for some xE from L. Since Hom.O.x/; E O.x// E D S0 D k, we get End.L/ D k. Further D Ext 1 .L; L/ D Hom.O.x/; E O.xE C !// E D S!E . Since !E 6 0 this expression is zero, and L is exceptional. Proposition 2.7 ([18], Lemma 4.2). Let E and F be exceptional sheaves on a weighted projective line having the same class ŒE D ŒF in the Grothendieck group K0 .X/. Then E is isomorphic to F . Proof. From the assumptions we obtain 1 D hŒE; ŒEi D hŒE; ŒF i and hence get a non-zero morphism u W E ! F . Applying Hom.E; / to the resulting exact sequence u

.E;u/

0 ! K ! E ! F ! C ! 0, we obtain exactness of 0 ! .E; K/ ! .E; E/ ! .E; F /, where .E; u/ is a non-zero map defined on a one-dimensional k-space, implying Hom.E; K/ D 0. By heredity of coh-X we further deduce Ext 1 .E; C / D 0 from Ext1 .E; E/ D 0. Our assumption ŒE D ŒF then implies ŒK D ŒC and further dimExt 1 .E; K/ D hŒE; ŒKi D hŒF ; ŒC i D dimHom.F; C /: This yields, in particular, Hom.F; C / D 0, thus C D 0, and then also ŒK D ŒC D 0 which finally yields C D 0. We have shown that u is an isomorphism. We will later need a refinement of the degree function det W K0 .X/ ! L, called the determinant. For construction and proof compare [14] and [39].

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Lemma 2.8. There is a group homomorphism det W K0 .X/ ! L, called determinant which is characterized by the following equivalent properties. (i) det O D 0, for each ordinary simple sheaf S we have det S D cE, and for each exceptional simple sheaf concentrated in i we have det S D xEi , i D 1; : : : ; t. (ii) det O.x/ E D xE for each xE 2 L. 2.3 Canonical tilting object. An object T in coh-X is called a tilting object if it has no self-extensions, that is, Ext 1 .T; T / D 0 and, moreover, T generates coh-X homologically, that is, for any object X from coh-X the condition Hom.T; X / D 0 D Ext1 .T; X / implies that X D 0. The next theorem links weighted projective lines to the representation theory of finite dimensional algebras. Theorem L 2.9 ([14]). Let L be a line bundle on X. Then the category coh-X has a tilting object TL 0xE E consisting of line bundles. Moreover, the endomorphism ring E c L.x/ End.T / is (isomorphic to) a canonical algebra. We call TL the canonical tilting bundle associated with L. E O.y// E D Proof. Let 0 x; E yE cE, then from Serre duality we get D Ext 1 .O.x/; Hom.O.y/; E O.xE C !// E D SxC . From x E C ! E y E c E C ! E we conclude that E ! E yE xE C !E yE 6 0, hence SxC D 0. This shows that T has no self-extensions. E ! E yE Analyzing the relationship between line bundles and simple sheaves, it is not difficult to prove that T generates coh-X. Canonical algebras were introduced and studied by C. M. Ringel in [52]. For instance, the canonical algebra of type .2; 3; 7/ is given by the quiver hh4 xE1 VVVVVVV hhhh VVVV x1 VVVV h h VVVV h h VVVV hhh h h h VVV+ x2 x2 x2 hh h / xE2 / 2xE2 / cE E0 7 B 77 7 x x3 77 3 / 2xE3 / 3xE3 / 4xE3 / 5xE3 / 6xE3 xE3 x1 hhhhh

x3

x3

x3

x3

x3

and the canonical relation x12 C x23 C x37 D 0. (If X has t weights we have t 3 relations, the canonical relations from the definition of the coordinate algebra S.) Corollary 2.10. The categories Db .coh-X/ and Db .mod-ƒ/ are equivalent as triangulated categories. In particular, they have naturally isomorphic Grothendieck groups Pt of rank 2 C iD1 .pi 1/ Proposition 2.7 applies, in particular, to the indecomposable summands T1 ; : : : ; Tn of a tilting object T in coh-X, implying that essential information on End.T / is kept by the classes ŒTi . For instance, the information on these classes is sufficient to recover the weight type of X.


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L Proposition 2.11. Let T D tiD1 Ti be a tilting object in coh-X, where X has weight type p D .p1 ; : : : ; p t /. Let CT D hŒTi ; ŒTj i, 1 i; j n, be the Cartan matrix of T . Then the characteristic polynomial T of the Coxeter matrix ˆT D CT1 CTt r is Q given as T D .x 1/2 tiD1 vpi , where vn D .x n 1/=.x 1/. In particular, the Coxeter polynomial T determines the weight type of X. Proof. One first checks that the matrix ˆT represents the k-linear automorphism N of K0 .X/ induced by the Auslander–Reiten translation in the Z-basis ŒT1 ; : : : ; ŒTn of K0 .X/. Its characteristic polynomial T thus has an invariant meaning, hence can be calculated with respect to any Z-basis of K0 .X/. Choosing such a basis appropriately, then allows to determine the specific form of the Coxeter polynomial T D X . For details we refer to [37], [34], and [31]. The proposition is a very basic instance for the interplay between coh-X and its ‘Ktheoretic shadow’ .K0 .X/; h; i/. For more sophisticated applications of this method of spectral analysis we refer to the survey article [38]. 2.4 Perpendicular categories. We call a full subcategory H 0 of an abelian category H exact if it is closed under kernels and cokernels. Lemma 2.12. Assume H is an Ext-finite hereditary abelian category, and E is exceptional in H . Then the (right) perpendicular category E ? , that is, the full subcategory consisting of all objects X from H satisfying Hom.E; X / D 0 D Ext1 .E; X / is an exact subcategory of H which is closed under extensions. In particular E ? with the induced exact structure is again abelian and hereditary. Proof. It is straightforward to show that H 0 D E ? is exact and extension-closed in H . This immediately implies that H 0 is an abelian category in its own right. Concerning heredity, just note that for any two objects X; Y from H 0 the first extensions, taken in Yoneda’s sense, will be the same when taken in H 0 and H , respectively. Accordingly, for any X from H 0 the functor Ext 1 .X; / is right exact on H and on H 0 implying that H 0 is hereditary. The next result establishes formation of perpendicular categories as a natural tool for induction arguments. Proposition 2.13 ([15]). Assume in the above setting that E is additionally exceptional. Then the inclusion from E ? to H has a left adjoint. Moreover, one gets a splitting K0 .H / D ZŒE ˚ K0 .E ? /. The next proposition yields another link between weighted projective lines and hereditary algebras. Proposition 2.14 ([37]). Assume X has weight type .p1 ; : : : ; p t / and L is a line bundle from coh-X. Then the full subcategory L? of coh-X right perpendicular category to a line bundle L is an abelian hereditary category which is equivalent to the category of right modules over the path algebra ƒ0 of the equioriented star Œp1 ; : : : ; p t .

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Proof. Since the group L acts transitively on the isomorphism classes of line bundles, we may assume that L D O.E c /. By Lemma 2.12 the category H 0 WD O.E c /? is a hereditary abelian category with the exact structure inherited from H D coh-X. By the very definition of H 0 we have Ext 1 .O.E c /; X / D 0 for each X from H 0 . Here, the Ext-spaces, taken in the sense of Yoneda, are the same when taken in H 0 or H since H 0 is extension-closed in H . Moreover, by heredity of H the inclusions O.x/ E ,! O.E c/ yield Ext 1 .O.x/; E X / D 0 for each 0 xE cE implying that the line bundles O.x/ E with 0 xE < cE all belong to H 0 and also are projective in H 0 . As the direct sum of all O.x/ E 0 with 0 xE cE is tilting in H it further follows that the direct sum T of all O.x/ E with 0 xE < cE is a projective generator for H 0 which then implies the claim. The proposition is a special instance of a more general unpublished result obtained jointly with T. Hübner [20] which states that the (right) perpendicular category E ? to any exceptional vector bundle in coh-X is already (equivalent to) the module category mod-H over a finite dimensional hereditary algebra H . For a proof we refer to the paper [11] by Crawley-Boevey. Note that, in general, H will not be connected. For examples of hereditary algebras H arising that way we refer to [19]. It is an open and difficult question to determine the class of hereditary algebras H where there exists a full embedding from mod-H into coh-X for some weighted projective line X such that, moreover, mod-H becomes the (right) perpendicular category with respect to an exceptional sequence of vector bundles in coh-X. By contrast, it is a straightforward task to describe the perpendicular category E ? E n, for an exceptional sheaf of finite length. Up to a possible factor of type mod-k A ? 0 E where An is equioriented, the category E is always of type coh-X where the weight type of X0 is dominated by the weight type of X. For a detailed treatment of that question, we refer to Section 9 of [15].

3 Euler characteristic and stability The Euler characteristic P of a weighted projective line X of weight type .p1 ; : : : ; p t / is defined as X D 2 tiD1 .1 1=pi /. Here, the number 2 stands for the Euler characteristic of the projective line P 1 .k/ over k, and each inserted weight pi yields the correction term .1 1=pi /. Hence for a small number of small weights the Euler characteristic is positive while for a large number of weights or for large weights (at least three are needed) the Euler characteristic is negative.PA closely related concept is the degree of the dualizing element !E D .t 2/E c tiD1 xEi which is given by Pt ı.!/ E D p..t N 2/ iD1 1=pi /, where pN D lcm.p1 ; : : : ; p t /. Indeed, we have N !/. E X D 1=pı. 3.1 Stability and semistability. A non-zero object X is called semistable (resp. stable) if for each non-zero subobject X 0 of X we have X 0 X (resp. X 0 < X ). By convention, the zero bundle is semistable for any slope. The definition implies


161

that coh0 -X consists of all objects of infinite slope. If X and Y are semistable, and f W X ! Y is a non-zero morphism, the definition implies that X Y . The slope thus indicates roughly the position of an object X in the category coh-X. It is easy to see that each line bundle is stable. For the general treatment of stability and semistability for categories of coherent sheaves on projective curves we refer to [57]. Lemma 3.1. Assume q belongs to Q [ f1g. The full subcategory C .q/ of coh-X consisting of all semistable objects of slope q is an exact and extension-closed subcategory of coh-X. It is therefore an abelian and hereditary category. Moreover, each object in coh-X has finite length. Proof. If q D 1 then C .q/ D coh0 -X, where the claim is known. For q 2 Q it is straightforward to show that the C .q/ is exact and extension-closed in coh-X. Moreover, it follows from Lemma 2.4 that each proper chain of subobjects E1 E2 Es E in C .q/ has at most s D rk E members which shows the last claim. Proposition 3.2 ([14], Proposition 5.5). If X 0, then each indecomposable vector bundle F is semistable. If, moreover, X > 0, then F is stable. In more detail: (i) For X > 0 each indecomposable vector bundle is exceptional. (ii) For X D 0 and q 2 Q the category Cq of all semistable vector bundles of slope q is uniserial with an Auslander–Reiten quiver consisting of a family of tubes. Moreover Hom.Cq ; Cq 0 / ¤ 0 if and only if q q 0 . Also for negative Euler characteristic stability arguments are a useful tool. This is largely due to the existence of a Harder–Narasimhan filtration which exists for each vector bundle F . This is based on the next result. Lemma 3.3 ([14], Lemma 5.3). Each non-zero bundle F on X has a non-zero subbundle F1 such that each non-zero sub-bundle (sub-sheaf ) F 0 of F satisfies .F 0 / .F1 /. F1 is uniquely determined if we assume additionally that F1 has maximal rank. For the next result we refer to [14] for a definition of the Hom-sheaf Hom.E; F /, the tensor product F ˝ G, and for the fact that Hom.O; Hom.F; G// D Hom.F; G/. Theorem 3.4 ([37], Theorem 2.7). Let F and G be non-zero bundles on X with G F > ı.E c C !/ E D pN C ı.!/, E then Hom.F; G/ ¤ 0. Proof. We first note that the vector bundle H D Hom.F; G/ D F _ ˝ G, where F _ is the dual vector bundle Hom.F; O/, has slope H D G F . It thus suffices to prove that we obtain Hom.O; H / ¤ 0 if H is a bundle with H > ı.E c C !/. E Let ss H ss denote the maximal semistable subbundle of H . Notice that H H > L O.E c / and assume ı.E c C !/. E We now invoke the canonical tilting bundle T D 0xE E c for contradiction that Hom.O; H ss / D 0. Then we also obtain Hom.O.x/; E H ss / D 0 since any non-zero morphism from a line bundle to a vector bundle is a monomorphism.

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Since T is tilting, this now implies Hom.T; H ss / D 0 and then Ext 1 .T; H ss / ¤ 0. Serre c /; H ss / D Hom.H ss ; O.E c C !//, E therefore duality finally implies 0 ¤ D Ext1 .O.E ss H .O.E c C !// E D ı.E c C !/, E contradicting our assumption. 3.2 Positive Euler characteristic. The following result is due to T. Hübner [17], see [40] for a slightly more general version. Theorem 3.5 (Hübner, 1989). Assume X > 0. Let T be the direct sum of (a representative system of ) all indecomposable vector bundles X of slope ı.!/ E < X 0. Then T is a tilting bundle in coh-X. Moreover, the endomorphism ring of End.T / is isomorphic to the path algebra kQ of a quiver Q whose underlying graph is extended Dynkin. Moreover, the AR-quiver of vect-X has shape ZQ. Proof. Let X and Y be indecomposable bundles with slope q in the range ı.!/ E < 1 q 0. We claim that D Ext .X; Y / D Hom.Y; X / D 0. By Lemma 2.5 we obtain that X D X C ı.!/, E hence X Y < 0. By stability this implies that Hom.Y; X / D 0, proving the above claim. It remains to show that T is tilting. Let U be the smallest subcategory of coh-X which contains T and is closed under direct summands, kernels of epimorphisms, cokernels of monomorphisms and extensions. We have to show that U D coh-X, and first treat the case of an indecomposable vector bundle F . Assume that F does not belong to U. Then either F > 0 or F ı.!/. E We are going to deal with the case F > 0, the treatment of the other case is similar. Invoking Lemma 2.5 we may assume that F has minimal slope. We then consider the almost-split sequence 0 ! F .!/ E ! Fx ! F ! 0. Because X > 0 each indecomposable vector bundle is stable implying that each indecomposable summand and also F .!/ E has a slope in the range ı.!/ E < q < F . Hence F .!/ E and Fx belong to U. The properties of U then imply that also F is in U, a contradiction. Finally, each simple sheaf S is the cokernel term of an exact sequence 0 ! L0 ! L ! S ! 0, where L0 and L are line bundles. It follows from this that U also contains all finite length sheaves. We have shown that U D coh-X. Then Lemma 2.5 implies that the Auslander–Reiten quiver of vect-X has the shape Z, where is the underlying graph of the quiver of End.T /. It is further easy to see that the rank function on vect-X induces an additive function on such that is extended Dynkin. We say that the algebra HX D End.T / from the theorem is the hereditary algebra associated with X. Note that this algebra only exists if X has positive Euler characteristic. Corollary 3.6. With the above assumptions, the following holds: (1) The categories Db .coh-X/ and Db .mod-kQ/ are triangle-equivalent. (2) The category vect-X is equivalent to the mesh category k.ZQ/. Next we illustrate how – for positive Euler characteristic – the canonical tilting bundle sits in the category of vector bundles.

163


We have marked the positions of the members O.x/ E of the canonical configuration by fat dots . The figure also visualizes the action of the Picard group L.2; 3; 4/ on the category vect-X. In particular, xE1 and xE3 act by glide-reflection, whereas xE2 acts by translation. For further illustrations we refer to [52], p. 196, where the canonical algebras of domestic type .2; a; b/, 1=a C 1=b > 1=2, are realized by tilting objects in the preprojective component of the hereditary algebra HX associated with X. x E3

x E1

3x E3

B B B : : ? ? ? ? ? ? ? ? ? ? ? ? }? > A?A A ? ? ? ? ? ? ? ? ? ? ? ? } ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? }}?? ? AA?? ? ?? ? ??.-? ????? ? ?? ? ?? ? .-BB } A ? ? ? ? ? ? ? ? B ? ? ?? }? }}?? ? ??? ? AA?A? IJ ? ? ? ? ? ???? ? ?? ? ?? ? ? / ???/ / BB?B/ IJ }/ / >= / / A/ / ?/ / ?/ ./ ??/ / / / / ? / /? ? / /? ?. ? ? ? ? BB? / /? } }? ? ? / ? ? ? ? ? ??? ? } ? ? ? ???? ? ?? ? ?? ? ?? ? ????? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ???? ? ? ? ?? ? ? ? ? ???? ? ? ? ? ? ? ? ? ? ? ??? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ??? ? B

B

B

B

B

B

B

B

B

B

B

B

2x E3

x E2

0

B

B

2x E2

B

cE

Figure 1. The canonical tilting bundle, weight type .2; 3; 4/.

Let be an extended Dynkin graph, and W ! N its (unique normalized positive) additive function. We thus request that for each vertex v we have 2.v/ D P .u/. For instance, to weight type .2; 3; 4/ corresponds the extended Dynkin u-v z graph E7 . Its additive function is displayed below. Note that here assumes the value 1 twice. 2 1

2

3

4

3

2

1

Generally, the number ˛./ of vertices v of with .v/ D 1 and the corresponding weight types are given by the table below. Table 1. The invariant ˛./ for an extended Dynkin diagram.

z p;q A

zn D

z6 E

z7 E

z8 E

.p; q/

.2; 2; n 2/

.2; 3; 3/

.2; 3; 4/

.2; 3; 5/

pCq

4

3

2

1

Corollary 3.7. Assume X > 0. The invariant ˛./ equals the index ŒL W Z!, E that is, the number of -orbits of line bundles in vect-X. Moreover, each tilting bundle T contains at least ˛./ pairwise nonisomorphic line bundle summands.

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Proof. In view of the proposition, the first claim follows directly from the definitions. The last assertion is a refinement of [43], Proposition 4.2, whose proof goes in two steps. We first show that each line bundle L satisfies Hom.T; L/ D 0 or Ext 1 .T; L/ D 0. Indeed, if both expressions are non-zero then we obtain a monomorphism L ,! T by Serre duality using that T is a bundle. There results a non-zero morphism T ! L ,! T , contradicting the fact that T is tilting. (This part of the proof works for arbitrary weight type). Next we show that each -orbit Z L of a line bundle L contains a direct summand of T : Since ı.!/ E < 0 we obtain an integer n such that Hom.T; n L/ ¤ 0 nC1 and Hom.T; L D 0/. The preceding step then implies that Ext 1 .T; n L/ D 0. By Serre duality, the second expression states that further Ext 1 . n L; T / D 0. Since T is tilting, this yields that n L is a direct summand of T , thus proving our claim. z n such that For t 2 weights, the underlying graph of the quiver Q has shape A ˛./ D n C 1; we hence obtain: Corollary 3.8. If X has at most two weighted points, then each indecomposable bundle is a line bundle. In particular, the middle term of the almost-split sequence 0 ! L.!/ E ! E ! L ! 0 splits. We note that also the converse is true. Remark 3.9. We will see in Section 4 that the above invariant ˛./ D ŒL W Z! E has also a meaning for arbitrary weight triples .p1 ; p2 ; p3 /, where it appears (up to sign) as the Gorenstein parameter or Gorenstein index of the triangle singularity x1p1 C x2p2 C x3p3 . The treatment there will also provide the explicit formula ˛./ D

3 Y

.pi 1/

iD1

3 X

pi 1 D .p1 p2 C p2 p3 C p1 p3 / p1 p2 p3

iD1

in terms of the associated weight triple. 3.3 Euler characteristic zero. Here we have an explicit classification running as follows. Theorem 3.10 ([39]). Assume X has Euler characteristic zero. For q 2 Q [ f1g let C .q/ denote the full subcategory consisting of all semistable sheaves of slope q. Then the following holds: (i) We have C .1/ D coh0 -X. W W (ii) We have coh-X D q2Q[f1g C .q/ , where the symbol means ‘additive closure of the union’ and additionally indicates that non-zero morphisms exist from C .q/ 0 to C .q / only for q q 0 . (iii) For each q there is an auto-equivalence of Db .coh-X/ inducing an equivalence from coh0 -X to C .q/ .


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Proof. Assertions (i) and (ii) follow directly from Proposition 3.2. The difficult part is assertion (iii), where different proofs exist, see [39], [42], [35], see also [25]. Corollary 3.11. Each component of the Auslander–Reiten quiver of coh-X is a tube. 3.4 Negative Euler characteristic. For the general properties of coh-X for X < 0 we refer to the joint paper [37] with de la Peña. We are presenting here only the most basic ones. Proposition 3.12. Assume X < 0. Then each component of the Auslander–Reiten quiver of vect-X has shape ZA1 . Assume X has wild weight type .p1 ; : : : ; p t / such that X < 0. Then X is said to z is a have Dynkin label 2 fD4 ; E6 ; E7 ; E8 g in case the extended Dynkin diagram subtree of the star Œp1 ; : : : ; p t , and further the number of vertices of is chosen to be minimal. We further say that X has Dynkin index 2, 3, 4, or 6, if equals D4 , E6 , E7 or E8 , respectively. For instance, X.2; 3; 7/ has Dynkin label E8 and X.2; 2; 2; 2I / has Dynkin label D4 . In the first case we have Dynkin index 6 in the second case Dynkin index 2. Theorem 3.13 ([37]). Assume X has negative Euler characteristic and Dynkin index . Then each exceptional vector bundle has quasi-length 1. Moreover, this bound is attained in each Auslander–Reiten component containing a line bundle.

4 Triangle singularities Let tX denote the number of weights pi 2. Let L be a line bundle on a weighted projective line X. Then the middle term E of the almost-split sequence L W 0 ! L.!/ E ! EL ! L ! 0 is called the Auslander bundle associated with L. Note that EL decomposes if and only if tX 2: For tX 2 each indecomposable bundle is a line bundle, hence EL decomposes. Conversely, assume that EL D L.xE1 / ˚ L.xE2 /. It follows !E xE1 0 which only happens for tX 2. The next result indicates that the case of three weights behaves special. Proposition 4.1. Let t D tX denote the number of weights of X. Then the Auslander bundle EL has trivial endomorphism ring End.E/ D k if and only if t 3. Moreover, EL is exceptional if and only if t D 3. Proof. From the exact Hom-Ext sequence .L; / we first obtain Hom.L; E/ D 0 D Ext1 .L; E/ since the connecting homomorphism is an isomorphism. As a next step, from the exact Hom-Ext sequence .L.!/; E / we deduce Hom.L.!/; E L/ D k 4t and 1 t3 n Ext .L.!/; E L/ D k with the convention that k D 0 for n < P 0. The above uses P the normal form expressions !E D tiD1 xEi C .2 t /E c and 2!E D tiD1 .pi 2/xEi C .t 4/E c . Finally, application of .; E/ to yields exactness of 0 ! .L.!/; E E/ ! .E; E/ ! .L; E/ ! 1 .L.!/; E E/ ! 1 .E; E/ ! 1 .L; E/ ! 0, hence End.E/ D k 4t and Ext 1 .E; E/ D k t3 .

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This section deals with hypersurface singularities of the shape f D x1p1 Cx2p2 Cx3p3 where p1 , p2 , p3 are integers 3. More specifically we intend to study the singularity category associated with f . As we have seen before, the mathematics of f is encoded in the weighted projective line X given by the weight triple .p1 ; p2 ; p3 /. Compared to weighted projective lines in general, those with exactly three weights enjoy a number of mathematical properties not present for weighted projective lines in general. Since S D kŒx1 ; x2 ; x3 =.f / is L-graded local with m D .x1 ; x2 ; x3 / being its (unique) graded maximal ideal, each L-graded simple module U has the form U D k.x/ E for some xE from L. Here, k D A=m is concentrated in degree 0. A finitely generated L-graded S-module M is called (maximal) Cohen–Macaulay (CM for short) if Hom.U; M / D 0 D Ext1 .U; M / holds for all simple graded S -modules. By CML -S we denote the category of all L-graded CM-modules. Proposition 4.2 ([14],Theorem 5.1). Sheafification q W modL -S ! coh-X, M 7! L L z M L, induces an equivalence q W CM -S ! vect-X with inverse LW vect-X, X 7! Hom.O.x/; E X /. Moreover, q restricts to an equivalence proj -S ! add.L/, x2L E where L denotes the full subcategory fO.x/ E j xE 2 Lg of line bundles on X. We use the above equivalence CML -S D vect-X as an identification and thus arrive at the following situation. The category vect-X of vector bundles on X is simultaneously fully embedded as an extension-closed subcategory into two different abelian categories modL -S

- CML -S D vect-X ,! coh-X:

Each of these embeddings induces a “natural” exact structure (in Quillen’s sense [50], §2, [23], Appendix A) on vect-X. These two exact structures are different; more precisely each sequence in vect-X, induced from an exact sequence in modL -S , is exact in coh-X; the converse is not true. The exact sequences in vect-X, induced from modL -S will be called distinguished exact. Proposition 4.3. A sequence W 0 ! X 0 ! X ! X 00 ! 0 in vect-X is distinguished exact if and only if Hom.L; / is exact for each L 2 L. This is if and only if Hom. ; L/ is exact for each L 2 L. Moreover, each distinguished exact sequence is exact in the abelian category coh-X. As a complete intersection the algebra S is L-graded Gorenstein hence the exact structure on vect-X induced from CML -S is Frobenius with the indecomposable projective-injectives just forming the system L of line bundles. Recall in this context that a Frobenius category is an exact category with enough (relative) projectives and injectives such that, moreover, the notions projective and injective coincide. A general result, due to Happel [16], Chapter I, then states that the attached stable category vect-X D

vect-X ŒL

is triangulated. Here ŒL denotes the two-sided ideal of morphisms in vect-X factoring through add.L/. We have the following result. As shown in [16] the distinguished


167

triangles in vect-X are induced by the distinguished exact sequences from vect-X. Moreover, the suspension functor Œ12 is induced by means of the formation of injective hulls. In more detail, let X be a vector bundle without direct summands from L and let 0 ! X ! =.X / ! X 00 ! 0 denote the injective hull of X in the Frobenius category vect-X. Then in vect-X we have X Œ1 D X 00 . Proposition 4.4. The category vect-X is a triangulated, Hom-finite k-category which is Krull–Schmidt. The Picard group L acts on vect-X by shift X ! X.x/, E xE 2 L. The category vect-X inherits Serre duality Hom.X; Y Œ1/ D D Hom.Y; X / from coh-X where is denotes the Picard-shift X 7! X.!/. E Accordingly, vect-X has Auslander– Reiten triangles. Moreover, the functor Œ2 is induced by the Picard-shift with the canonical element cE. Further vect-X is homologically finite, that is, Hom.X; Y Œn/ D 0 for jnj 0. Proof. We have already seen that vect-X is triangulated. Hom-finiteness follows from the Hom-finiteness of vect-X. Further, each E 2 vect-X is a finite direct sum E D E1 ˚ ˚ En of indecomposable objects with local endomorphism rings, and this property is inherited by vect-X. Hence vect-X is Krull–Schmidt. Since the Picard group action sends line bundles to line bundles, it preserves the ideal ŒL, hence induces an action on vect-X. Next, we are dealing with Serre duality: Let X , Y be objects of ˛

ˇ

vect-X, and let 0 ! Y ! =.Y / ! Y Œ1 ! 0 be an injective hull of Y . Invoking Serre duality of coh-X there results an exact sequence ˇ

˛

Hom.X; =.Y // ! Hom.X; Y Œ1/ ! D Hom.Y; X.!// E ! D Hom.=.Y /; X.!//: E This implies that cokerˇ is isomorphic to ker ˛ . But cokerˇ is just the morphism space Hom.X; Y Œ1/ in the stable category, and ker ˛ is isomorphic to the k-dual of coker.Hom.=.Y /; X.!// E ! Hom.Y; X.!/// E D Hom.Y; X.!//, E proving the claim on Serre duality. By general theory, see [51] this implies that vect-X has Auslander– Reiten triangles and that the corresponding Auslander–Reiten translation on vect-X is induced by the Picard-shift with !, E that is, the Auslander–Reiten translation of vect-X. That the two-fold suspension Œ2 equals the Picard-shift by the canonical element cE uses the fact that we are dealing with three weights, implying that S D kŒx1 ; x2 ; x3 =.f / is a (graded) hypersurface singularity and relies on a graded version on a general result on matrix factorizations, see [60], Proposition 7.2, using that cE is the degree of f . Finally, homological finiteness of vect-X is deduced from D Hom.X; Y Œ2n C 1/ D Hom.Y Œ2n; X.!// E D Hom.Y .nE c /; X.!//: E Just note that the left hand side is zero for n 0 and the right hand side is zero for n 0. 2 We avoid the name shift for the suspension functor Œ1 because of possible confusion with the Picard shifts.

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4.1 Tilting objects. It is much harder to show that there exists a tilting object in vect-X. Since Auslander-bundles serve as a kind of replacement in vect-X for the no longer existing line bundles, it is most natural to search first for tilting objects composed from Auslander bundles. It seems that this is only successful for weight triples of the form .2; 3; p/. Indeed, Auslander bundles are exceptional in vect-X, and we have the following result. By An .r/ we denote the path algebra of the equioriented quiver An where we request that each composition of r consecutive arrows is zero. Theorem 4.5 ([28]). Assume X has weight type .2; 3; p/ with p 2. We put xN i D xEi C !E for i D 1; 2; 3, and M D faxN 1 C b xN 3 j a D 0; : : : ; p 2; b D 0; 1g. Let E D EL be an Auslander-bundle. Then M T D E.x/ E x2M E

is a tilting object in vect-X with endomorphism ring End.T / D A2.p1/ .3/. Corollary 4.6. Assume weight type .2; 3; p/ and let E D EL be an Auslander bundle. Then the right perpendicular category E ? formed in vect-X is triangle-equivalent to the derived category Db .mod-A2p3 .3//. P We assume tX D 3 and put xEmax WD 2!E C cE D 3iD1 .pi 2/xEi . For any line E L.!// E D bundle L and Picard group member 0 xE xEmax we have Ext 1 .L.x/; D Hom.L; L.x// E D k. Hence the middle term E of the non-split exact sequence W 0 ! L.!/ E ! E ! L.x/ E ! 0 is uniquely determined up to isomorphism. We call E the extension bundle with the data .L; x/, E and use the notation E D EL hxi. E Our next theorem highlights the special role of weighted projective lines with three weights. Its proof needs some preparation. Let E be an indecomposable bundle and =.E/ be its injective hull in the Frobenius category vect-X. There results a distinguished exact sequence 0 ! E ! =.E/ ! EŒ1 ! 0, where EŒ1 is again indecomposable, and EŒ1 takes the role of the suspension in vect-X. Lemma 4.7. Assume tX D 3. Then the following holds. (1) For each xE in L and vector bundle E we obtain det.E.x// E D det.E/ C rk.E/x. E E is the extension bundle given by the line bundle L and 0 xE (2) If E D EL hxi xEmax , then det.EŒ1/ D det.E/ C cE. (3) Assume E; F are indecomposable bundles of rank two, then Hom.E; F / ¤ 0 implies det.F / det.E/ 0 with equality if and only if E Š F . Proof. The first assertion follows by using a line bundle L filtration for E. For the second assertion one uses the expression =.E/ D L.x/ E ˚ 3iD1 L.!E C .ì C 1/xEi / for the injective hull of E, see [28]. Concerning the last assertion we note that any u W E ! F yielding a non-zero member from Hom.E; F / is a monomorphism, since otherwise u would factor through a line bundle, the image of u. This implies that the cokernel C of


169

u has finite length, such that the determinant det.C / of C , the sum of the determinants of the simple composition factors of C is 0 in L and, moreover, equal to zero if and only if C has length zero. The claim now follows from det.F /det.E/ D det.C /. Theorem 4.8 ([28]). We assume that X has triple weight type. Then each indecomposable vector bundle E of rank two is an extension bundle E D EL hxi E for some line bundle L and Picard group member xE where 0 xE xEmax . Moreover, E is exceptional in coh-X, hence in Db .coh-X/, and also exceptional in vect-X. Proof. Step 1. We show that E is an extension bundle. For this we choose a line bundle L such that (a) Hom.L.!/; E E/ ¤ 0 and (b) Hom.L.!E C xEi /; E/ D 0 for i D 1; 2; 3. For instance, we may choose L to be of maximal degree such that (a) is satisfied. Since E is indecomposable, we obtain a non-split exact sequence W 0 ! L.!/ E !E ! C ! 0. The cokernel term C has rank one and because of (b) has no torsion, hence is a line bundle L.x/ E for some xE from L. Since by assumption 0 ¤ Ext 1 .L.x/; E L.!// E D D Hom.L; L.x// E we obtain xE 0. Applying Hom.L.!E C xEi /; / to and invoking (b) we obtain exactness of the sequence 0 D .L.!E C xEi /; E/.L.!E C xEi /; L.x// E ! 1 .L.!E C xEi /; L.!// E D D .L; L.!E C xEi // D 0. This yields Hom.L.!E C xEi /; L.x// E D 0, hence xE !E xEi cE C !E and then xE .2!E C cE/ C xEi for i D 1; 2; 3, implying xE xEmax . Step 2. We prove a slightly more general claim and show that the middle term E of the non-split exact sequence W 0 ! L.!/ E ! E ! L.x/ E ! 0 is exceptional P in coh-X for each xE D 3iD1 ì xEi , where 0 ì pi 1 for i D 1; 2; 3.3 We first apply .L.x/; E / to obtain an exact sequence 0 D .L.x/; E L.!// E ! .L.x/; E E/ ! .L.x/; E L.x// E ! 1 .L.x/; E L.!// E ! 1 .L.x/; E E/ ! 1 .L.x/; E L.x// E D 0. Since does not split, the boundary morphism of the Hom-Ext sequence is an isomorphism, yielding Hom.L.x/; E E/ D 0 and Ext 1 .L.x/; E E/ D 0:

(4.1)

Next, we form .L.!/; E / and obtain exactness of the sequence 0 ! .L.!/; E L.!// E ! 1 1 1 E L.!// E ! .L.!/; E E/ ! .L.!/; E L.x// E D .L.!/; E E/ ! .L.!/; E L.x// E ! .L.!/; D .L.x/; E L.2!// E D 0. This yields Hom.L.!/; E E/ D k

and Ext 1 .L.!/; E E/ D 0:

(4.2)

Finally, we form . ; E/ to obtain exactness of the sequence 0 ! .L.x/; E E/ ! .E; E/ ! .L.!; E E// ! 1 .L.x/; E E/ ! 1 .E; E/ ! 1 .L.!; E E// ! 0. By means of (4.1) and (4.2) we obtain End.E/ D k and Ext 1 .E; E/ D 0, that is, exceptionality of E E with 0 xE xEmax . in coh-X. In particular this concerns any extension bundle EL hxi Step 3. For the last assertion we assume that E is an indecomposable vector bundle of rank two. We know already that End.E/ D k such that End.E/ D k follows. By Serre duality we further have Hom.E; EŒn/ D D Hom.EŒn 1; E.!//, E and we have to prove that this expression is zero for each non-zero integer n. Assume for 3

Note, that we reserve the term ‘extension bundle’ for the more restricted situation 0 ì pi 2.

170

H. Lenzing

contradiction that it is non-zero for some n ¤ 0. By Lemma 4.7 it follows that for such an n the inequalities (a) nE c 0 and (b) .n 1/E c 2!E hold. Now, (a) is violated for n < 0 and (b) is violated for n > 0, thus proving our claim. The next result is a far-reaching improvement of Theorem 4.5, proceeding along similar lines but technically substantially more involved. A key step in its proof is Theorem 4.8. For details we refer to [28]. Theorem 4.9 ([28]). Assume X has triple weight type .p1 ; p2 ; p3 / with pi 2, and fix a line bundle L. Then the direct sum T of all extension bundles EL hxi E with E 0 xE xEmax is a tilting object in vect-X with endomorphism ring k Ap1 1 ˝k E p 1 ˝k k A E p 1 . kA 2 3 E n denotes the path algebra of the equioriented quiver of type An or, Here, k A equivalently, the ring of all upper triangular matrices Tn .k/ over k. Tensor products of these algebras have recently attracted much attention. There is recent unpublished work by S. Ladkani on this topic dealing with the aspect of Calabi–Yau dimension, see also [29]. Further [41] deals with tensor products of path algebras of Dynkin E p 1 ˝k k A E p 1 is naturally isomorphic to E p 1 ˝k k A quivers. Note further that k A 1 2 3 the incidence algebra kP of the poset P D Œ0; p1 2 Œ0; p2 2 Œp3 2, where interval-notation Œ0; n refers to the linear poset f0; 1; : : : ; ng. Corollary 4.10. Assume triple weight type .p1 ; p2 ; p3 / for X. Then the Grothendieck group K0 .vect-X/ of the triangulated category vect-X is finitely generated free of rank .p1 1/.p2 1/.p3 1/. E of the extension bundles EL hxi, E with Proof. By generalPtheory the classes ŒEL hxi 0 xE xEmax D 3iD1 .pi 2/xEi , form a Z-basis of K0 .vect-X/. 4.2 Calabi–Yau property. Despite its simple proof, the next result is also quite important since it produces an infinite family, indexed by weight triples, of triangulated categories which all are fractionally Calabi–Yau. Recall that a triangulated category T with Serre duality is called m=n-Calabi–Yau if Sn D Œm, where m, n are integers with n > 0. In our context, the Serre functor is given by S D B Œ1 D Œ1 B , where is the Auslander–Reiten translation. For the possible smallest values of n, where vect-X is m=n-Calabi–Yau yielding the exact Calabi–Yau dimension m=n, to be thought of as the pair .m; n/, of vect-X, we refer to [28]. Theorem 4.11 ([28]). Assume X has weight type .p1 ; p2 ; p3 /. Then vect-X is m=nCalabi–Yau where m=n D 1 2X . Proof. Let pN be the least common multiple of .p1 ; p2 ; p3 /. Then pN !E D ı.!/E E c;

3 X where ı.!/ E D pN 1 1=pi / : iD1

171


E hence SpN D ŒpC2ı. Passing to the attached Picard shifts, we pN D Œ2ı.!/, N !/. E Pobtain 3 Now the quotient .pN C 2ı.!//= E pN D iD1 .1 1=pi / equals 1 2X , and the claim follows. Note that the Auslander–Reiten translation for vect-X induces a k-linear automorphism of K0 .vect-X/, called the Coxeter transformation, whose characteristic polynomial is called the Coxeter polynomial. Corollary 4.12. Assume X has three weights. Then the Coxeter transformation of E p 1 ˝k k A E p 1 ˝k k A E p 1 is periodic. In particular, vect-X or of the algebra k A 1 2 3 the Coxeter polynomial factors into cyclotomic polynomials. E Now, the suspension Proof. Recall from the proof of the theorem that pN D Œ2ı.!/. Œ1 induces multiplication by 1 on K0 .vect-X/. Hence ˆpN D 1 if ˆ denotes the Coxeter transformation. The claim follows. 4.3 Positive EulerL characteristic. We present some examples illustrating how the tilting object T D 0x E of vect-X sits in the AR-component of vect-X. E x Emax EL hxi For weight type .2; 2; p/ it forms an equioriented subquiver of type Ap1 . We have marked the positions of the extension bundles EL hj xE3 i, 0 j p 2, by the symbol , and the position of the involved line bundles L.!/ E and L.j xE3 /, 0 j p 2 by the symbol , the remaining line bundles are marked by the symbol B. We also display the situation for weight types .2; 3; 3/, .2; 3; 4/ and .2; 3; 5/. We observe that in these examples the extension bundle EL hxEmax i is always an Auslander bundle. This is, indeed, a general fact for weight type .2; a; b/. By contrast, in the preceding examples the extension bundle EL hxE3 i is not an Auslander bundle, the only exception being weight type .2; 3; 3/ where all indecomposable vector bundles of rank two are Auslander bundles.

h4x E3 i h3x E3 i h2x E3 i hx E3 i h0i

! E

?B? ?B? ?B? ?B? ?B ??? ??? ??? ??? / / / / / / /? ? / B ? /B B B B ?? ? ??? ? ??? ? ??? ?? ? ? ? ? ? ?? ?? ?? ? ? ? ??? ??? ??? ??? ? ? ?? ? ?? ?? ? ?? ?? ?? ?? ??? ?? ? ? ??? ? ??? ? ??? ? ? ??? ? ? ? ? ? ?? / B /? ?? / /? ?? / B /? ?? / /? ?? / B ?? ?? ?? ?? ??

B

B

0

x E3

2x E3

3x E3

4x E3

Figure 2. Standard tilting object for vect-X, weight type .2; 2; 6/.

172

H. Lenzing

hx E3 i B

x E2 ??

B

?? ?B? ?B? ? ? ?B ?? ??? ??? ??? ??? ? ? ? ? ? B OO ? ? o7 B OO ?8 ? o7 OO ? ? o7 B OO ? ? o7 B OO ? ? o7 B OO' o?oo OO' o?o?o OO' o?o?o OO' o?o?o OO' o?o?o oo7 OO?O?? oo7 C OO?O?? oo7 OO?O?? oo7 OO?O?? oo7 OO?O?? o o O O O o O O' o o o 'o 'o 'o 'o ? ? ?? ? ? ?? ? ?? ? ?? ?? ?? ?? ?? ?? ?? ? ? ?? ? ? ? ) ? ?? ? ?? ? ?? ? ?? ? ??? ? ? ? ? ?? ? ? ? ?? ? ? ?

B

h0i

! E

0

hx E2 i

B

B

E2 C x E3 i x E3 hx

E3 x E2 C x


hx E2 i

x E3

hx E2 C 2x E3 i

x E2 C x E3

B

?? ?B? ?B? ?B? ?B? ?B? ?B? ?? ? ? ?? ?? ?? ?? ?? ?? ?? A ? ???? ? ???? ?: ??? ? ???? ? ???? ? D ??? ? ???? ? ???? ? ??? ? ???? ? ???? ? ???? ? ???? ? ???? ? ???? ? ??? ? / / / / / / / / * / / / / / / / / ? ???? ? ????4 ? ???? ? ???? ? ???? ? ???? ? ???? ? ??? ? ?? ? ???? ? ???? ? ??? ? ???? ? ???? ? ???? ? ???? ? ? ??? ? ??? ? ??? ? ??? ? ??? ? ??? ? ??? ? ???

! E

h0i

B

0

hx E3 i

B

B

h2x E3 i

x E2

B

B

2x E3

Figure 4. Standard tilting object for vect-X, weight type (2,3,4).

Lemma 4.13. Assume weight type .2; a; b/. Then the following holds: E D EL hxi. (i) We have EL hxEmax xi E xN 1 x/ E for any 0 xE xEmax . (ii) The extension bundle EL hxEmax i is isomorphic to the Auslander bundle EL .xN 1 /. (iii) Assume weight type .2; 3; b/. Then the extension bundle EL hxE2 i is isomorphic to the Auslander bundle EL .xN 3 /. Proof. To prove (i), it suffices to show that the two objects have the same class in the

! E

h0i

0

B

B

B

B hx E2 i

B

h2x E3 i

x E3

B

B

hx E2 C x E3 i

h3x E3 i

B


hx E3 i

x E2

B

hx E2 C 2x E3 i

2x E3

B hx E 2 C 3x E3 i

B


173

174

H. Lenzing

Grothendieck group. Observe that we have short exact sequences x1

0 ! L.!/ E ! L.xE1 C !/ E ! S1 ! 0; x1

0 ! L.!E C xN 1 x/ E ! L.E c C 2!E x/ E ! S1 ! 0; where S1 denotes the unique simple sheaf concentrated in x1 with Hom.L; S1 / D k. This uses that p1 D 2, and that xE1 does not occur in the expression xE D `2 xE2 C `3 xE3 . Now, ŒEL .xEmax x/ E D ŒL.!/ E C L.xE1 C !/ and ŒEL .xN 1 x/ E D ŒL.xE1 C ! x/ E C E The claim follows. ŒL.xE1 C 2!/. Assertion (ii) is a special case of (i). The proof of (iii) is similar to the proof of (i) and uses that for weight type .2; 3; b/ we have xE2 .xN 3 C !/ E D xE3 . 4.4 Euler characteristic zero Proposition 4.14. Assume X has tubular weight type .2; 3; 6/, .2; 4; 4/ or .3; 3; 3/. Then there exists a tilting object T in the stable category vect-X whose endomorphism ring is the canonical algebra ƒ of the same weight type. In particular, we have triangle equivalences vect-X Š Db .coh-X/ (depending on the choice of T ). Proof. We sketch the argument, leaving details to [28]. As shown in [14], the direct sum T of all line bundles O.xE3 C x/ E with xE in the range 0 xE cE is a tilting object for coh-X and Db .coh-X/. By [39] there is an auto-equivalence of Db .coh-X/ acting on slopes q by q 7! 1=.1 C q/. It follows that T is a bundle whose indecomposable summands have slopes q in the range 1=2 < q < 1. It follows from this property that T is a tilting object for vect-X having all the claimed properties. Recall in this context that the category H D coh-X is hereditary, yielding the very W concrete description of Db .coh-X/ as the repetitive category n2Z H Œn, where each H Œn is a copy of H (objects written X Œn with X 2 H ) and where morphisms are given by Hom.X Œn; Y Œm/ D Extmn H .X; Y / and composition is given by the Yoneda product. Remark 4.15. The classification of indecomposable bundles over the weighted projective line X D X.2; 3; 6/ is very similar to Atiyah’s classification of vector bundles on a smooth elliptic curve, compare [1] and [39]. Indeed the relationship is very close: Assume the base field is algebraically closed of characteristic different from 2 and 3. If E is a smooth elliptic curve of j -invariant 0, it admits an action of the cyclic group G of order 6 such that the category cohG .E/ of G-equivariant coherent sheaves on E is equivalent to coh-X.2; 3; 6/. Thus vect-X.2; 3; 6/ has the additional description as stable category vect G -E of G-equivariant vector bundles on E with respect to a suitable Frobenius structure. E of a standard tilting It is interesting to analyze the position of the summands EL hxi object in vect-X. By way of example we consider weight type .2; 3; 6/. The analysis is similar for the two remaining weight triples .2; 4; 4/ and .3; 3; 3/.


175

Lemma 4.16. Assume X is given by a tubular weight triple. Let pN be the largest weight. Then each indecomposable bundle F of rank two either has either integral or half-integral slope. In the case of integral slope it either has quasi-length 2 in a tube of -period p, N and then is an Auslander bundle, or else it is quasi-simple in a tube of -period p=2 N (if such a tube exists). In the case of half-integral slope F is quasi-simple in a tube of -period p. N In particular F is quasi-simple in vect-X. Proof. This follows from the description of coh-X given in [39]. By way of example, we discuss weight type .2; 3; 6/ in more detail. Let F be an indecomposable vector bundle of rank two. Then only the following cases appear: (1) F has integral slope, and then either has quasi-length two in the tube of -period 6, and then is an Auslander bundle, or else F is quasi-simple in the tube of -period 3. (2) F has half-integral slope, and then is quasi-simple in the tube of -period 6. By Theorem 4.8 any of these possibilities occurs for the indecomposable summands E xE D axE3 Cb xE2 , a D 0; : : : ; 4, b D 0; 1 of the standard tilting object for vect-X. EL hxi, E is an Auslander For xE 2 f0; xE2 ; xEmax ; xEmax xE2 g the extension bundle F D EL hxi bundle, for xE 2 fxE3 ; xE2 C xE3 ; xEmax xE3 ; xEmax .xE2 C xE3 /g the bundle F has halfintegral slope and then -period 6. For the two remaining cases xE 2 f2xE3 ; xE2 C 2xE3 g the bundle F has integral slope and is quasi-simple of -period 3. It follows that all extension bundles EL hxi E are quasi-simple in vect-X. 4.5 Negative Euler characteristic Proposition 4.17. For X < 0 each Auslander–Reiten component of vect-X is of shape ZA1 . Proof. First it follows from Proposition 3.12 that each Auslander–Reiten component of vect-X has shape ZA1 . Invoking stability of line bundles, one deduces that line bundles are quasi-simple in their components. Passing to the stable category vect-X then yields the claim. 4.6 The Orlov context. Let R be a positively Z-graded affine k-algebra R with R0 D k which is commutative noetherian of Krull dimension d . Then R is called graded Gorenstein if R has finite graded self-injective dimension d . Then d is the Krull dimension of R, and R has a minimal graded injective resolution 0 ! R ! E 0 ! E 1 ! ! E d ! 0 where E d is the graded injective hull of the simple module k.a/. The uniquely defined integer a is then called the Gorenstein parameter of R. The algebraic analysis of singularities has made big progress through Orlov’s analysis [48] between two triangulated categories associated with R. One category is the derived category of the Serre quotient modZ -R=modZ 0 -R, a category

176

H. Lenzing

which is to be thought of as the category of coherent sheaves on some (virtual) noncommutative space X . The other triangulated category is the singularity category Z Z b b Db;Z Sg .R/ D D .mod -R/=D .proj -R/ of R which is known to be equivalent to the stable category of graded Cohen–Macaulay modules CMZ -R, see [48] or [3]. The comparison of the two categories depends on the signature of the Gorenstein parameter a, and Orlov’s theorem [48] roughly states that the two categories are equivalent if a D 0 and that otherwise one gets a full embedded as triangulated categories from one of the two categories into the other, where the direction of this embedding depends on the sign of the Gorenstein parameter. More precisely, even, one can identify one of the two categories as a full triangulated subcategory of the other one being the perpendicular category to a finite exceptional sequence with jaj members. When going to apply Orlov’s theorem to the projective coordinate algebra S of a weighted projective line X of weight type .p1 ; : : : ; p t /, we need an extension of Orlov’s theorem to the L-graded case in order to obtain the same type of relationship between the derived category Db .coh-X/ of coherent sheaves on X, respectively the L categories Db;L Sg .S / Š CM -S Š vect-X. A key point for this extension is to define the Gorenstein parameter properly. By construction S has Krull dimension two, and it is not difficult to show that the minimal L-graded injective resolution of S has the form 0 ! S ! E 0 ! E 1 ! E 2 ! 0, where E 2 is the graded injective hull of the graded simple module k.!/. E (As an aside we mention that E 0 is the graded quotient field 1 of S and that E is the direct sum of the injective hulls of the S -modules S=p, where p runs through the height-one prime ideals of S.) The conclusion at this point is that !E carries all the information to derive the numerical Gorenstein parameter a for the L-graded setting. The first guess, here, would be to take the degree ı.!/ E which turns out to have already the right sign, but still needs to be renormalized. A more careful, and technical, analysis of Orlov’s theorem then yields the following definition for the Gorenstein parameter of the L-graded algebra S . Definition 4.18. The Gorenstein parameter a of the L-graded projective coordinate algebra S of weight type .p1 ; : : : ; p t / is defined as a D ı.!/ E

Qt

t t t X Y pi 1 Y pi D X pi : D .t 2/ lcm.p1 ; : : : ; p t / pi iD1

iD1

iD1

iD1

Note that the Gorenstein parameter a of S and the Euler characteristic of X have the same sign. Moreover, if the weights p1 ; : : : ; p t are pairwise coprime then the degree map ı establishes an isomorphism L D Z. Further the correction term Qt E In the next lemma iD1 pi = lcm.p1 ; : : : ; p t / equals one, such that a D ı.!/. we give alternative descriptions of the Gorenstein parameter in terms of coh-X. Lemma 4.19. (a) Assume arbitrary weight type and non-zero Euler characteristic of X, then the Gorenstein parameter a of S equals the Index ŒL W Z! E up to sign. Therefore, jaj counts the number of -orbits of line bundles in vect-X.


177

(b) Assume tX D 3. Then the Gorenstein parameter a equals a D rk K0 .coh-X/ rk K0 .vect-X/ D .p1 C p2 C p3 1/ .p1 1/.p2 1/.p3 1/: We have a D 0 for X D 0. For X ¤ 0, the subgroup Z!E has finite non-zero index jaj D ŒL W Z!, E and then a and X have the same sign. E c . Moreover pN Proof. (a) Let pN D lcm.p1 ; : : : ; p t /. As is easily checked pN !E D ı.!/E is the smallest positive integer n such that n!E belongs to ZE c . With our assumption on the Euler characteristic n D jı.!/j E is the smallest positive integer such that nE c belongs to Z!. E The claim then follows using the diagram L? ?? ?? ?? ?? Z!E ZE c? ?? ?? ? pN ??? jı.!/j E ZE c \ Z!E

Qt

i D1 pi

where pN D lcm.p1 ; : : : ; p t /. (b) The assertion on the ranks of the groups K0 .X/ and K0 .vect-X/ is covered by Corollary 2.10 and Theorem 4.9. The L-graded version of Orlov’s theorem, applied to the present context, now yields the following trichotomy. Theorem 4.20. The following assertions hold: (i) For X > 0 the category vect-X is a triangulated subcategory of Db .coh-X/ obtained as the (right) perpendicular subcategory with respect to an exceptional sequence E1 ; : : : ; Ejj in Db .coh-X/. (ii) For X D 0 the triangulated categories vect-X and Db .coh-X/ are equivalent. (iii) For X < 0, the category Db .coh-X/ is a triangulated subcategory of vect-X obtained as the (right) perpendicular subcategory with respect to an exceptional sequence E1 ; : : : ; E in vect-X. In the preceding sections 4.3, 4.4 and 4.5 we have seen many instances for properties (i), (ii) and (iii).

5 Kleinian and Fuchsian singularities For the base field C of complex numbers the material of this section deals with classical material. The Kleinian singularities arise as rings of invariants of binary polyhedral

178

H. Lenzing

group actions on the polynomial ring CŒx; y in two variables, see [24]. Fuchsian singularities deal with (graded) rings of entire automorphic forms with respect to certain discrete group actions on the upper half plane, see [45] and [47]. In our context, and then for an algebraically closed base field k of any characteristic, these singularities arise as follows. Let X be aL weighted projective line of non-zero Euler L characteristic. For X > 0, we put R D S . For > 0 we put R D X E E . We call R algebra n0 n! n0 Sn! obtained from the L-graded coordinate algebra S by restriction of the grading to the subgroup Z!. E Note that for X D 0 the subgroup Z!E is finite, implying that the restriction of the grading to Z!E behaves mathematically quite different. Proposition 5.1 ([15], [30]). Assume non-zero Euler characteristic and let R be the Z-graded algebra obtained from S by restricting the L-grading to the subgroup Z!. E Then the following holds: (i) R is Z-graded Gorenstein. (ii) The restriction functor modL -S ! modZ -R; M 7! MjZ!E induces equivZ Z alences coh-X D modL -S=modL 0 -S ! mod -R=mod0 -R and vect-X D L Z CM -S ! CM -R. Moreover, under the equivalence vect-X D CML -S ! CMZ -R the -orbit Z of the structure sheaf corresponds to the category of indecomposable projective Z-graded R-modules. Each of the two full embeddings modZ -R

- vect-X ,! coh-X

of vect-X as an extension-closed subcategories into the abelian categories modZ -R and coh-X, respectively, imposes on vect-X the an exact structure. The exact structure obtained from the embedding vect-X ,! modZ -R induces on vect-X the structure of a Frobenius category. For distinction, the induced exact sequences on vect-X will be called Z O-exact. Proposition 5.2. (i) A sequence W 0 ! X 0 ! X ! X 00 ! 0 in vect-X is Z -exact if and only if Hom.L; / is exact for each L 2 Z O. This is if and only if Hom. ; L/ is exact for each L 2 Z O. Moreover, each Z O-exact sequence is exact in the abelian category coh-X. (ii) The Z O-exact sequences define on vect-X the structure of a Frobenius category where the members from Z O are the indecomposable projective-injective objects. Proof. The proof follows from the preceding proposition using that the algebra R is Z-graded Gorenstein. We note that the setting is analogous to, but different from, the setting of Section 4. Only for weight types .2; 3; 5/ and .2; 3; 7/ the two concepts agree. From the setting we obtain another stable category of vector bundles vect Z -X D vect-X=Œ Z O:

179


Using different methods these categories were investigated by Kajiura, Saito, Takahashi in [21], [22] and in joint work with de la Peña [36]. Theorem 5.3. Assume X has weight type .p1 ; p2 ; p3 / with X > 0. In this case the hereditary star D Œp1 ; p2 ; p3 is a Dynkin diagram. Then the algebra R obtained from S by restricting the grading from L to Z!E yields a Z-graded algebra having a (minimal) system of three homogeneous generators x, y, z, all being monomials in x1 , x2 , x3 . With this choice of generators we have R WD SjZ!E D kŒx; y; z=.f / where f is the simple graded singularity from the table below. Moreover, with the above assumptions, the singularity f can be chosen as a sum of monomials in x, y, z and then is unique. Dynkin diagram ApCq D Œp; q

generators .x; y; z/ .x1 x2 ; x2pCq ; x1pCq /

D2lC2 D Œ2; 2; 2l

.x32 ; x12 ; x1 x2 x3 /

D2lC3 D Œ2; 2; 2l C 1

.x32 ; x1

x2 ; x12

x3 /

E6 D Œ2; 3; 3

.x1 ; x2 x3 ; x23 /

E7 D Œ2; 3; 4

.x2 ; x32 ; x1

E8 D Œ2; 3; 5

x3 /

.x3 ; x2 ; x1 /

deg.x; y; z/

relation f pCq

deg.f /

yz

pCq

.1; p; q/

x

.2; 2l; 2l C 1/

z 2 C x.y 2 C y x l / 2

2

l

4l C 2

.2; 2l C 1; 2l C 2/

z C x.y C z x /

4l C 4

.3; 4; 6/

z2 C y3 C x2 z

12

2

3

.4; 6; 9/

3

z Cy Cx y

18

.6; 10; 15/

z2 C y3 C x5

30

The simple graded surface singularities

We are now going to discuss what happens with the restriction procedure if we apply it to weight triples with X < 0. In general the algebra R is graded Gorenstein but not always a hypersurface singularity or, more generally, a graded complete intersection. The following result is taken from [30] and [36] where additional information is available. Proposition 5.4. Let k be a field and assume .p1 ; p2 ; p3 / is a weight triple with ı.!/ E > 0. Let R D SjZ!E be the Z-graded restriction of the L-graded triangle singularity S the subgroup Z!E which we identify with Z by the correspondence !E $ 1. Then the following holds: Exactly for the weight triples .p1 ; p2 ; p3 / of Arnold’s strange duality list the algebra R is generated by three homogeneous elements x, y, z and then has the form R D kŒx; y; z=.f / where the generators x, y, z, the relation f and their degrees is given by the list below.

180

H. Lenzing

.p1 ; p2 ; p3 /

generators .x; y; z/

deg.x; y; z/

relation f

deg f

N

42

12

30

13

.6; 8; 9/

y C xz C x

4

24

14

.4; 10; 15/

z2 C y3 C x5y

30

11

3

22

12

18

13

2

3

7

.2; 3; 7/

.x3 ; x2 ; x1 /

.6; 14; 21/

z Cy Cx

.2; 3; 8/

.x32 ; x2 ; x1 x3 /

.6; 8; 15/

z 2 C x 5 C xy 3

.2; 3; 9/

.x33 ; x2 x3 ; x1 / .x3 ; x22 ; x1 x2 /

.2; 4; 5/

3

2

.4; 6; 11/

z C x y C xy

.4; 6; 7/

y 3 C x 3 y C xz 2

.3; 4; 5/

.x32 ; x22 ; x1 x2 x3 / .x33 ; x22 x3 ; x1 x2 / .x2 x3 ; x1 ; x25 / .x2 x32 ; x1 x3 ; x24 / .x3 ; x1 x2 ; x13 / .x32 ; x1 x2 ; x3 x12 / .x23 ; x1 x2 x3 ; x23 / .x2 x3 ; x12 ; x1 x24 / .x2 x33 ; x12 x3 ; x1 x23 /

.4; 4; 4/

.x1 x2 x3 ; x14 ; x24 /

.2; 4; 6/ .2; 4; 7/ .2; 5; 5/ .2; 5; 6/ .3; 3; 4/ .3; 3; 5/ .3; 3; 6/ .3; 4; 4/

2

2

4

2

5

.4; 5; 10/

z Cy zCx

20

12

.4; 5; 6/

xz 2 C y 2 z C x 4

16

13

.3; 8; 12/

z2 C y3 C x4z

24

10

.3; 5; 9/

z 2 C xy 3 C x 3 z

18

11

.3; 5; 6/

y C x z C xz

2

15

12

.3; 4; 8/

z2 y2z C x4y

16

11

.3; 4; 5/

2

x y C xz C y z

13

12

.3; 4; 4/

x 4 yz 2 C y 2 z

12

12

3

3

3

2

Arnold’s strange duality list

Here, the bullet marks the cases where one has a choice for the monomial generators. Further, N denotes the sum of the three weights. For a recent discussion of Arnold’s strange duality we refer to [12]. As to the general qualitative properties of vect Z -X, like Serre duality andAuslander– Reiten components, the situation is analogous to the properties of vect-X for triangle singularities. From a quantitative point of view, however, the triangle case and the Kleinian/Fuchsian case differ sensibly: For X > 0, the Gorenstein parameter L L of R D E equals C1, while for X < 0 the Gorenstein parameter of R D E n0 Rn! n0 Sn! equals 1. This implies that the two triangulated categories Db .coh-X/ and vect Z -X are much closer than in the comparable case for triangle singularities. In particular, the ranks of the two Grothendieck groups K0 .X/ and K0 .vect Z -X/ just differ by ˙1. For a detailed discussion we refer to [36]. Here, we only mention the following result concerning tilting objects. Theorem 5.5 ([36]). (a) Assume X has weight type .p1 ; p2 ; p3 / with X > 0. Let T be the direct sum of (a representative system of ) all indecomposable vector bundles X 6Š O of slope ı.!/ E < X 0. Then T is a tilting bundle in vect Z -X. Moreover, the endomorphism ring of End.T / is isomorphic to the path algebra kQ of a quiver Q whose underlying graph is the Dynkin graph Œp1 ; p2 ; p3 . Further, vect Z -X is equivalent to the mesh category of the translation quiver ZQ. (b) Assume X has weight type p D .p1 ; : : : ; p t / with X < 0. Then the triangulated category vectZ -X has a tilting object T whose endomorphism algebra is an extended x of type p, that is, a one-point extension of the canonical algebra canonical algebra ƒ ƒ of type p by an indecomposable projective module.

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It does not matter which indecomposable projective one takes for the one-point extension (or coextension). These algebras all happen to be derived equivalent. In x we keep quiver and more detail, in order to construct the extended canonical algebra ƒ relations for ƒ, but add a single new arrow with a new vertex (without introducing any new relations). Below we display the quiver of the extended canonical algebra of type .2; 3; 7/ subject to the single relation x12 C x23 C x37 D 0. ii4 xE1 VVVVVVV iiii VVVV x1 i i i VVVV iii i VVVV i i iix VVVV i i x x2 i 2 2 i V/+ i / / E x E2 2x E2 cE 0 77 B 77 x x3 7 7 3 / 2xE3 / 3xE3 / 4xE3 / 5xE3 / 6xE3 x E3 x1

x3

x3

x3

x3

/

?

x3

Figure 6. Extended canonical algebra of type .2; 3; 7/.

6 Flags of invariant subspaces for nilpotent operators In recent work of C. M. Ringel and M. Schmidmeier, see [54] and[55, 53], categories of invariant subspaces for nilpotent k-linear operators are treated. The resulting categories may be considered as mono-representations of the two element poset (quiver) 1 ! 0 in the category mod-kŒx=.x b / consisting of all monomorphic k-linear maps X1 ,! X0 . More generally, one can investigate monorepresentations of the poset P D n ! .n 1/ ! ! 0 in the categories mod-kŒx=.x b / resp. the category modZ -kŒx=.x b / of Z-graded kŒx=.x b /-modules. In the graded case the resulting Z b. This category inherits from the category of monorepresentations is denoted a1 category of all representations of P the structure of a Frobenius category. Quite surprisingly, this category is related to the category of vector bundles on the weighted projective line of weight type .2; a; b/, as we are going to describe now.The following result is taken from unpublished work with D. Kussin and H. Meltzer [26], see also [27] where the case of invariant subspaces of nilpotent operators is treated. Note that for this application we deal with the Frobenius structure on vect-X coming from the triangle singularity x12 C x2a C x3b . Theorem 6.1 ([26]). Assume X of weight type .2; a; b/. Then there is a partition L D P t F of the class of L of line bundles such that the following holds: (i) The factor category vect-X=ŒF is a Frobenius category with the class P D P =ŒF as the indecomposable projective-injectives.

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(ii) There is a naturally defined functor Z ˆ W vect-X ! a1 .b/;

E 7! Hom.; E/jP

with kernel ŒF . (iii) ˆ yields equivalences of Frobenius resp. triangulated categories Š Z Š Z vect-X=ŒF ! a1 .b/ and vect-X ! a1 .b/:

Note that for the above theorem the order .2; a; b/ of the weights matters, since the last weight, here b, corresponds to the nilpotency degree while a corresponds to the flag length a 1. Assume X > 0. From the structure of vect-X and vect-X, described Z in Section 4 we obtain the structure of the Auslander–Reiten quiver of a1 .b/ in a straightforward fashion. We first illustrate this for weight type .2; 4; 3/ which by the theorem is related to the flag problem 3Z .3/. B? @ > / @ > ?

? B > ? B ? = @ > @ @ = @ > / / / / = @ > @ @ = @ > > ? ? ? B

? ? > @ @ > / / > @ @ > ? ? B

B ? > @ / > @ ?

@

= > @ / / @ = > @ ?

@ = / @ =

? ? > @ > @ > @ > @ > @ / / / / / > @ > @ > @ > @ > @ ? ? ?

3Z .3/

2Z .4/

cylindric gluing

Möbius gluing

Figure 7. The categories vect-X.2; 4; 3/ and 3Z .3/.

The two Auslander–Reiten quivers of the figure should be thought to periodically extend to the left and the right with gluing schemes of a cylindric respectively Möbius type. The quiver on the left shows the category vect-X, where X has weight type .2; 4; 3/. The line bundles are either denoted by fat dots (corresponding to line bundles of P ) or else by circles B which correspond to line bundles from F . The Auslander Reiten quiver on the right shows the factor category vect-XŒF , hence the category 3Z .3/. Since the weighted projective line X is not changed by a permutation of the weights, we obtain the following application: Theorem 6.2. For all integers a; b 2 we have Z Z a1 .b/ Š b1 .a/

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as triangulated categories. Proof. Let X D X.2; a; b/ and Y D X.2; b; a/. Then vect-X D vect-Y as triangulated categories. Hence Z Z a1 .b/ Š vect-X D vect-Y Š b1 .a/:

@

== >> @ / / @ == >> @ ?

@ == / @ ==

? ? ? >> @ >> @ >> @ >> @ >> @ / / / / / >> @ >> @ >> @ >> @ >> @ ?? ? ??

? ? ? @ >> @ >> @ >> / / / @ >> @ >> @ >> ?

== @ / == @

@ == / @ ==

? >> @ / >> @

3Z .3/

2Z .4/

cylindric identification

Möbius identification

Figure 8. A surprising symmetry, .2; 4; 3/ versus .2; 3; 4/.

We remark that there is further work on the invariant flag problem for nilpotent operators by M. Schmidmeier and his student A. Moore, see [46]. With different methods Z E a1 ˝k A E b1 was .b/ with endomorphism algebra k A the standard tilting object for a1 also established by X.-W. Chen [5]. The representation types for categories of monorepresentations for finite one-peak posets were determined by D. Simson in a number of papers, among them [58] and [59]. We also point to the paper of P. Zhang [61]. The whole problem has its roots in the theory of finite abelian groups because Birkhoff’s problem [2] from 1934 already asks for the classification of the possible positions of subgroups in p-groups, p a prime, whose elements have bounded order p n . In different terms Birkhoff raised the question to classify all monorepresentations of the poset 1 ! 0 in the category mod-Z=Zp n . This problem corresponds largely to the ungraded version of the problem addressed in this section.

7 Comments Many important aspects of weighted projective lines and their applications are not touched upon in this survey for reasons of time and space. There is first and foremost the impressive work of W. Crawley-Boevey [8], [9], [10], [11] yielding, among other subjects, a proof of Kac’s theorem for weighted projective lines.

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Another aspect, neglected here, is the relationship between weighted projective lines and Lie theory through the Ringel–Hall-algebra approach, we refer to [4] and the literature quoted there. Further, there is an axiomatic approach to weighted projective lines and, more generally, to weighted projective curves. Here, we quote work by X.-W. Chen and H. Krause [7], and [6], two papers by the author [32], [35] another one jointly with I. Reiten [40], and finally a paper by I. Reiten and M. Van den Bergh [51]. This last paper describes a larger world in characterizing (in different language, however) categories of coherent sheaves on weighted smooth projective curves. We further note that the paper [7] describes a construction which is related to, but different from, the author’s construction of categories of p-cycles in [33].

References [1] M. F. Atiyah, Vector bundles over an elliptic curve. Proc. London Math. Soc. 7 (1957), 414–452. [2] G. Birkhoff, Subgroups of Abelian groups. Proc. London Math. Soc., II. Ser. 38 (1934), 385–401. [3] R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-Cohomology over Gorenstein rings. Preprint, 1986. [4] I. Burban and O. Schiffmann, Composition algebra of a weighted projective line. Preprint 2010, arXiv:1003.4412 [math.RT]. [5] X.-W. Chen, The stable monomorphism category of a Frobenius category. Math. Res. Letters 18 (2011), 125–137. [6] X.-W. Chen and H. Krause, Expansions of abelian categories. J. Pure Appl. Algebra 215 (2011), 2873–2883. [7] X.-W. Chen and H. Krause, Introduction to coherent sheaves on weighted projective lines. Preprint 2011, arXiv:0911.4473 [math.RT]. [8] W. Crawley-Boevey, Quiver algebras, weighted projective lines, and the Deligne-Simpson problem. In International Congress of Mathematicians. Vol. II. European Math. Soc. Publishing House, Zürich, 2006, 117–129. [9] W. Crawley-Boevey, General sheaves over weighted projective lines. Colloq. Math. 113 (2008), 119–149. [10] W. Crawley-Boevey, Kac’s theorem for weighted projective lines. J. Eur. Math. Soc. (JEMS) 12 (2010), 1331–1345. [11] W. Crawley-Boevey, Connections for weighted projective lines. J. Pure Appl. Algebra 215 (2011), 35–43. [12] W. Ebeling and A. Takahashi, Strange duality of weighted homogeneous polynomials. Compositio Math., to appear; arXiv:1003.1590 [math.AG]. [13] P. Gabriel, Des catégories abéliennes. Bull. Soc. Math. France 90 (1962), 323–448.


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[14] W. Geigle and H. Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional algebras. In Singularities, Representation of Algebras, and Vector Bundles. Lecture Notes in Mathematics 1273, Springer-Verlag, Berlin, 1987, 265–297. [15] W. Geigle and H. Lenzing, Perpendicular categories with applications to representations and sheaves. J. Algebra 144 (1991), 273–343. [16] D. Happel, Triangulated Categories in the Representation Theory of Finite-dimensional Algebras, London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge, 1988. [17] T. Hübner, Classification of indecomposable vector bundles on weighted curves. Diplomarbeit, Paderborn, 1989. [18] T. Hübner, Exzeptionelle Vektorbündel und Reflektionen an Kippgarben über projektiven gewichteten Kurven. Dissertation, Paderborn, 1996. [19] T. Hübner, Hereditary module categories arising as categories perpendicular to exceptional vector bundles. In Algebras and Modules II. Canad. Math. Soc. Conf. Proc. 24, Amer. Math. Soc., Providence, RI, 1998, 327–336. [20] T. Hübner and H. Lenzing, Categories perpendicular to exceptional bundles. Preprint, Paderborn, 1993. [21] H. Kajiura, K. Saito and A. Takahashi, Matrix factorization and representations of quivers. II. Type ADE case. Advances Math. 211 (2007), 327–362. [22] H. Kajiura, K. Saito and A. Takahashi, Triangulated categories of matrix factorizations for regular systems of weights with D 1. Advances Math. 220 (2009), 1602–1654. [23] B. Keller, Chain complexes and stable categories. Manuscripta Math. 67 (1990), 379–417. [24] F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. Birkhäuser Verlag, Basel, 1993. Reprint of the 1884 original, Edited, with an introduction and commentary by Peter Slodowy. [25] D. Kussin, Noncommutative curves of genus zero: related to finite dimensional algebras. Memoirs Amer. Math. Soc. 201 (2009), no. 942. [26] D. Kussin, H. Lenzing and H. Meltzer, Invariant flags for nilpotent operators and weighted projective lines. In preparation. [27] D. Kussin, H. Lenzing and H. Meltzer, Nilpotent operators and weighted projective lines. Preprint 2010, arXiv:1002.3797 [math.RT]. [28] D. Kussin, H. Lenzing and H. Meltzer, Triangle singularities, ADE-chains and weighted projective lines. In preparation. [29] S. Ladkani, On derived equivalences of lines, rectangles and triangles. Preprint 2009, arXiv:0911.5137 [math.RT]. [30] H. Lenzing, Wild canonical algebras and rings of automorphic forms. In Finite-dimensional Algebras and Related Topics. NATO ASI Series C: Mathematical and Physical Sciences 424, Kluwer Academic Publishers, Dordrecht, 1994, 191–212. [31] H. Lenzing, A K-theoretic study of canonical algebras. In Representation Theory of Algebras. Canad. Math. Soc. Conf. Proc. 18, Amer. Math. Soc., Providence, RI, 1996, 433–454.

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[32] H. Lenzing, Hereditary Noetherian categories with a tilting complex. Proc. Amer. Math. Soc. 125 (1997), 1893–1901. [33] H. Lenzing, Representations of finite-dimensional algebras and singularity theory. In Trends in Ring Theory. Canad. Math. Soc. Conf. Proc. 22, Amer. Math. Soc., Providence, RI, 1998, 71–97. [34] H. Lenzing, Coxeter transformations associated with finite-dimensional algebras. In Computational methods for representations of groups and algebras. Progress in Mathematics 173, Birkhäuser, Basel, 1999, 287–308. [35] H. Lenzing, Hereditary categories. In Handbook of Tilting Theory. London Mathematical Society Lecture Notes Series 332, Cambridge University Press, Cambridge, 2007, 105–146. [36] H. Lenzing and J. A. de la Peña, Extended canonical algebras and Fuchsian singularities. Math. Z. 268 (2011), 143–167. [37] H. Lenzing and J. A. de la Peña, Wild canonical algebras. Math. Z. 224 (1997), 403–425. [38] H. Lenzing and J. A. de la Peña, Spectral analysis of finite dimensional algebras and singularities. In Trends in Representation Theory of Algebras and Related Topics. European Mathematical Society Series of Congress Reports, European Math. Soc. Publishing House, Zürich, 2008, 541–588. [39] H. Lenzing and H. Meltzer, Sheaves on a weighted projective line of genus one, and representations of a tubular algebra. In Representations of Algebras. Canad. Math. Soc. Conf. Proc. 14, Amer. Math. Soc., Providence, RI, 1993, 313–337. [40] H. Lenzing and I. Reiten, Hereditary Noetherian categories of positive Euler characteristic. Math. Z. 254 (2006), 133–171. [41] Z. Leszczyński, On the representation type of tensor product algebras. Fund. Math. 144 (1994), 143–161. [42] H. Meltzer, Tubular mutations. Colloq. Math. 74 (1997), 267–274. [43] H. Meltzer, Derived tubular algebras and APR-tilts. Colloq. Math. 87 (2001), 171–179. [44] H. Meltzer, Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projective lines. Memoirs Amer. Math. Soc. 171 (2004), no. 808. [45] J. Milnor, On the 3-dimensional Brieskorn manifolds M.p; q; r/. In Knots, Groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox). Ann. of Math. Studies 84. Princeton University Press, Princeton, N. J., 1975, 175–225. [46] A. Moore, Auslander-Reiten theory for systems of submodule embeddings. PhD thesis, Florida Atlantic University, 2009. [47] W. D. Neumann, Brieskorn complete intersections and automorphic forms. Invent. Math. 42 (1977), 285–293. [48] D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities. In Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin, Vol. II. Progress in Mathematics 270, Birkhäuser, Boston, Inc., Boston, MA, 2009, 503–531. [49] H. Poincaré, Mémoire sur les fonctions fuchsiennes. Acta Math. 1 (1882), 193–294. [50] D. Quillen, Higher algebraic K-theory. I. In Algebraic K-theory, I: Higher K-theories. Lecture Notes in Mathematics 341, Springer-Verlag, Berlin, 1973, 85–147.


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Cohomology of block algebras of finite groups Markus Linckelmann

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . 2 Some special cases and examples . . . . . . . . 3 Defect groups . . . . . . . . . . . . . . . . . . 4 Relative projectivity . . . . . . . . . . . . . . . 5 Brauer’s First Main Theorem . . . . . . . . . . 6 Source algebras of blocks . . . . . . . . . . . . 7 Fusion systems of blocks . . . . . . . . . . . . 8 Conjectures . . . . . . . . . . . . . . . . . . . 9 Symmetric algebras and transfer . . . . . . . . 10 Separably equivalent algebras . . . . . . . . . . 11 Block cohomology . . . . . . . . . . . . . . . 12 Block cohomology and Hochschild cohomology 13 Further remarks and questions . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction One of the main questions serving as a guideline for the present notes is this: which finite-dimensional algebras occur as block algebras of finite groups? Block algebras of finite groups are precisely the class of finite-dimensional indecomposable algebras which arise as indecomposable direct factors of finite group algebras. Block algebras are always symmetric but not every symmetric algebra arises as a block algebra of some finite group. In fact, the prominent finiteness conjectures suggest that there are only ‘few’ algebras which do occur in this way. Block algebras have all the usual invariants associated with algebras – module categories, derived and stable categories, as well as cohomological invariants such as Hochschild cohomology, for instance. But beyond that, block algebras have also invariants related to their nature as direct factors of finite groups algebras, such as defect groups, fusion systems, block cohomology and associated cohomology varieties. The second main question underlying much of the material in these notes is this: to what extent are block algebras determined by their defect groups and fusion systems? Throughout these notes we denote by p a prime and by k an algebraically closed field of characteristic p. For G a finite group, a block algebra of the group algebra kG is an indecomposable direct factor B of kG, as an algebra. The block algebras of kG correspond bijectively to primitive idempotents in the center Z.kG/ via the map

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sending a block algebra B to its unit element 1B . In other words, if B is a block algebra of kG then B D kGb for a unique primitive idempotent b 2 Z.kG/, also called block of kG or the block idempotent of the block algebra B. In particular, a block algebra B is also an indecomposable direct summand of kG as a kG-kG-bimodule. The category of finite-dimensional k-algebras has finite direct products but not direct sums. The category of finitely generated kG-kG-bimodules has both (and they coincide, as they do in any additive category). The decomposition of kG as a direct product of its block algebras kG D B1 B2 Br is equal, set-theoretically speaking, to the (unique) decomposition of kG as a direct sum of indecomposable kG-kG-bimodules, kG D B1 ˚ B2 ˚ ˚ Br : Setting bi D 1Bi , the set fbi j 1 i rg is the set of all primitive idempotents in Z.kG/; any two different idempotents in this set are orthogonal, and their sum is 1kG . Thus, for any kG-module U we have a direct sum decomposition of kG-modules L U D riD1 bi U: Each summand bi U can be viewed as a module over the block algebra Bi ; in this way, the category of finitely generated kG-modules mod.kG/ decomposes as the direct sum of the categories of finitely generated Bi -modules, with i running from 1 to r. In particular, if U is indecomposable to begin with then U D bi U for a unique i , 1 i r, and bj U D f0g for j ¤ i , in which case we say that the module U belongs to the block Bi . The choice of an algebraically closed field k as a base ring is somewhat restrictive – group algebras can be defined over any commutative ring, and any finite group algebra over a commutative Noetherian ring admits a block decomposition as above. Many of the results formulated below remain true with k replaced by complete discrete valuation ring O having k as residue field and a quotient field K of characteristic zero. In that case, the block decomposition above ‘lifts’ uniquely to a block decomposition OG D By1 By2 Byr of OG as a direct product of indecomposable O-algebras, and this is as before also the unique decomposition of OG as a direct sum of indecomposable OG-OG-bimodules. Extending coefficients to K yields a decomposition of (no longer necessarily indecomposable) K-algebras KG D .K ˝O By1 / .K ˝O By2 / .K ˝O Byr /: The algebra KG is semi-simple; in particular, if K is ‘large enough’, KG is a direct product of matrix algebras. By a theorem of Brauer, K is ‘large enough’ for any

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subgroup of G if K contains a primitive jGj-th root of unity. Our initial question may as well be formulated over O: which are the O-algebras arising as block algebras of finite group algebras over O? Any finitely generated KG-module X is determined by its character X , which is defined as the function G ! K sending x 2 G to the trace of the endomorphism of X given by multiplication with x on X . Since conjugate matrices have the same trace, this is well-defined, and X is a class function; that is, the value X .x/ depends only on the conjugacy class of x in G. Moreover, since any element in G has finite order, the eigenvalues of any endomorphism of X induced by multiplication on X with a group element x 2 G are roots of unity, hence contained in O, and therefore the character X of X is in fact a class function from G to O. If X is a simple KG-module, its character X is called an irreducible character of G with coefficients in K. The set of irreducible characters of G with coefficients in K is denoted by IrrK .G/. If X is a simple K ˝O Byi module, or equivalently, if X .bi / D X .1/, we say that X and X belong to the block Bi . Denoting by IrrK .Bi / the set of irreducible characters belonging to Bi , the set IrrK .G/ is the disjoint union of the subsets IrrK .Bi /, with 1 i r. If K is large enough, the set IrrK .G/ is an orthonormal basis of the space ClK .G/ of class functions from G to K, with respect to the scalar product defined 1 P 1 by h˛; ˇiG D jGj / for any two K-valued class functions on G. x2G ˛.x/ˇ.x Denoting by RK .G/ the Grothendieck group of finitely generated KG-modules, the map sending X to X induces an isomorphism of abelian groups RK .G/ Š Z IrrK .G/. The interplay between kG and KG via OG is a rich source of structural invariants – but for the sake of expository convenience, this aspect of the theory will be developed rather sporadically. The first six sections are a brief introduction to standard material on block theory. Many background results are stated with proofs, however terse. From Section 7 onwards, including proofs would have been beyond the intended scope of these notes as a survey.

2 Some special cases and examples Our first fundamental question – which k-algebras occur as block algebras? – can be answered completely in a few ‘extreme’ cases. There is a surprisingly simple answer, due to Okuyama and Tsushima, for when a block algebra is Morita equivalent to a commutative algebra: Theorem 2.1 ( [114]). Let B be a block algebra of kG for some finite group G. If B is Morita equivalent to a commutative k-algebra A then A Š kP for some finite abelian p-group P . In that case, P is uniquely determined as the defect group of the block B, a concept, due to Brauer, which we will define and investigate in more detail in the next section. The uniqueness of P , up to isomorphism, requires the fact, due to Deskins [37], that finite abelian p-groups with isomorphic group algebras over a field of characteristic

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p are isomorphic. This is a special case of the – in general still open – modular isomorphism problem. The proof of this theorem in [114] amounts to characterising block algebras whose center is symmetric as precisely those blocks which are nilpotent (in the sense of Broué and Puig [25]) with abelian defect groups. By a result of Roggenkamp and Zimmermann, derived equivalent symmetric local k-algebras are Morita equivalent, and hence the above theorem can be formulated more generally using derived equivalences instead of Morita equivalences. Any abelian finite p-group arises in the above theorem, since if P is a finite abelian p-group then kP is in particular local. This holds more generally for arbitrary finite p-groups, and is another case in which one has a complete answer regarding the block decomposition of a finite group algebra: Theorem 2.2. Let P be a finite p-group. Then the algebra kP is local; that is, its unit element is its unique idempotent. In particular, kP is indecomposable as a k-algebra. Moreover, the augmentation ideal I.kP / D ker.kP ! k/ is the unique maximal ideal, the unique maximal left ideal, and the unique maximal right ideal of kP , and we have I.kP /jP j D f0g. Proof. One of the standard proofs uses induction over the order of P , using the fact that if P is non-trivial, then so is its center, and hence there is an element z in Z.P / of order p. Setting Z D hzi and Px D P =Z, one verifies that the kernel of the canonical map from kP onto k Px is equal to the ideal I.kZ/kP generated by the augmentation ideal I.kZ/ of kZ and that the I.kP / is the inverse image of I.k Px /. Every element in Z is of the form z s for some integer s such that 1 s p. Thus z s 1 D .z 1/.1 C z C C z s1 /, which shows that I.kZ/ D .z 1/kZ. Since char.k/ D p we have .z 1/p D z p 1p D 0, hence I.kZ/p D f0g. By induction, x x I.k Px /jP j D f0g, hence I.kP /jP j is contained in I.kZ/kP . Since the p-th power of this ideal is zero by the previous remarks, we get that I.kP /jP j is zero. In particular, I.kP / is nilpotent, hence contained in the Jacobson radical of kP . Since I.kP / has codimension one, it is equal to the Jacobson radical. Since the Jacobson radical is of a finite-dimensional algebra is equal to any of the intersections of the maximal ideals, or maximal left ideals, or maximal right ideals, the result follows. One of the immediate consequences of this theorem is that for P a finite p-group and Q a subgroup of P , the transitive permutation kP -modules kP =Q, having as a basis the set P =Q of cosets xQ, x 2 P , is indecomposable. Indeed, the canonical map kP ! kP =Q is a surjective homomorphism of left kP -modules, and hence the image of I.kP / is the unique maximal submodule of kP =Q – whence the indecomposability of kP =Q as a left kP -module. Theorem 2.2 shows that finite p-groups provide examples of group algebras with a unique block. In general, for a finite group G to have only one block has strong structural implications – if p is odd this happens exactly if G has a normal p-subgroup Q satisfying CG .Q/ D Z.Q/, while for p D 2 the situation is slightly more involved (the Mathieu groups M22 and M24 are the unique simple groups with exactly one block if p D 2). One direction involves the classification of finite


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simple groups; see Harris [52]. If p does not divide the group order, block algebras are simply matrix algebras: Theorem 2.3 (Maschke/Dickson). If p does not divide the order of the finite group G then every block algebra of kG is a matrix algebra; equivalently, kG is isomorphic to a direct product of matrix algebras. Proof. The proof of this theorem shows more precisely that kG is separable; that is, kG is isomorphic, as a kG-kG-bimodule, to kG ˝k kG. Indeed, the canonical surjective bimodule homomorphism W kG ˝k kG ! kG induced by multiplication 1 P 1 in kG splits, having as section the map sending x 2 G to jGj . This y2G xy ˝ y expression makes sense thanks to the fact that jGj is invertible in k, and is easily seen to define a bimodule homomorphism W kG ! kG ˝k kG satisfying B D IdkG . If K is a field of characteristic zero as above, then jGj is invertible in K and so this proof shows that KG is separable, hence semi-simple. In general, KG need not be a direct product of matrix algebras, but by a theorem of Brauer, if K contains a primitive jGj-th root of unity, then KG is indeed also a direct product of matrix algebras. The converse of the above theorem holds as well: if p divides jGj, then kG is not semisimple. P One way to see this consists of considering the sum of all group elements z D x2G x in kG. Clearly z 2 Z.kG/, hence zkG D kGz is a two-sided ideal in kG, and we have z 2 D jGjz, which is zero if p divides jGj. Thus zkG is an ideal which squares to zero, hence contained in the Jacobson radical J.kG/. It follows that at least one block algebra of kG has a non-zero radical, hence is not isomorphic to a matrix algebra. Example 2.4. Let G D S3 , the symmetric group on three letters. The block decomposition of kG is as follows: (i) If p D 2 then kG Š kC2 M2 .k/, where C2 is a cyclic group of order 2. (ii) If p D 3 then kG is indecomposable as an algebra, hence has a unique block. (iii) If p 5 then kG Š k k M2 .k/. Determining the block decomposition of kG is by no means an easy exercise – for the following example the ordinary character tables from the Atlas are required (the presentation follows that given in [76]): Example 2.5. Let G D 2:M22 , the non-split central extension of the simple Mathieu group M22 by an involution and let p D 3. Then kG has nine blocks B1 , B2 , …,B9 . Of these, five are matrix algebras. The dimensions of these blocks can be read off the character table; up to relabelling, we get B1 Š B2 Š M45 .k/; B3 Š M99 .k/; B4 Š B5 Š M126 .k/; mod.B6 / Š mod.B7 / Š mod.kS3 /:

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Danz and Külshammer showed in Theorem 4.3 of [36] that there is a Morita equivalence mod.B8 / Š mod.k.C3 C3 / Ì Q8 / for a faithful action of the quaternion group Q8 on C3 C3 . For the block B9 there is no such Morita equivalence, but there is a derived equivalence, due to Okuyama, D b .mod.B9 // Š D b .mod.k.C3 C3 / Ì Q8 //: What this shows is that for G D 2:M22 and p D 3 all blocks of kG are Morita equivalent, or at least derived equivalent, to blocks of much smaller groups. This is a typical phenomenon, reflected by some of the conjectures we will discuss in Section 8 below. What both examples illustrate is that even if p divides the order of G, some of the block algebras of kG may still be matrix algebras. The number of blocks of a group algebra which are matrix algebras will turn out to be of particular interest – whence the following notation: Definition 2.6. Let G be a finite group. We denote by `0 .kG/ the number of block algebras of kG which are matrix algebras. Since kG is symmetric, the number `0 .kG/ is equal to the number of isomorphism classes of modules which are both simple and projective. One of the great numerical mysteries of this subject is Alperin’s weight conjecture, predicting that the number `.kG/ of isomorphism classes of simple kG-modules can be expressed as a sum of numbers `0 .kH /, for certain quotients H of local subgroups of G; we present here the group theoretic version of this conjecture. Conjecture 2.7 (Alperin’s weightPconjecture, group theoretic version). For any finite group G we have `.kG/ D Q `0 .kNG .Q/=Q/, where Q runs over a set of representatives of the G-conjugacy classes of p-subgroups of G. The term for Q D 1 in this sum is equal to `0 .kG/. If p does not divide jGj, this is the only term in this sum, and one sees easily that the conjecture holds in that case. Alperin’s weight conjecture has a stronger block theoretic version which we will describe later. As mentioned above, we will ignore for the most part the aspect of block theory over the ring O. It is not known whether the Morita equivalence class of a block B of a finite group algebra kG determines that of the block By of OG lifting B. Algebras over k may lift in more than one way, but not every k-algebra lifts to an O-algebra whose coefficient extension to K is semi-simple, providing a potential strategy to rule out certain algebras as block algebras. We conclude this section with two examples, illustrating this aspect.


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Examples 2.8. The ’obvious’ candidate for lifting the algebra B D kŒx=.x 2 / of dual numbers over k to an algebra over the ring O is the algebra OŒx=.x 2 /. Extending this to the quotient field of characteristic zero K yields the algebra of dual numbers KŒx=.x 2 / over K, which is not semi-simple. But there are other ways to construct O-algebras lifting kŒx=.x 2 /. Consider the algebra By with O-basis f1; yg, where the multiplication is given by y 2 D p 2 1. Since p 2 J.O/, the image of y in k ˝O By squares to zero, hence k ˝O By Š B. Moreover, we have K ˝O By Š K K. 1 Indeed, one verifies that this algebra has two idempotents, namely 12 ˙ 2p y.

If B is a symmetric finite-dimensional k-algebra which lifts to a symmetric Oalgebra By such that K ˝O By is a direct product of matrix algebras, then the Cartan matrix C of B is of the form C D D t D, where D is the decomposition matrix; in particular, det.C / is not negative. It is, however, easy to construct symmetric algebras whose Cartan matrices have negative determinant. The following example arose in discussions with J. Chuang and B. Külshammer. Let A be the path algebra of the quiver with two vertices 0, 1 and exactly three arrows from 0 to 1. In particular, dimk .A/ D 5, and the Cartan matrix of A is 1 3 CA D : 0 1 The trivial extension algebra B D T .A/ is equal to A ˚ A as a k-vector space, with multiplication given by .a; /.b; / D .ab; a C b/, where a, b 2 A, ; 2 A D Homk .A; k/, considered as an A-A-bimodule as usual. This algebra is well known to be symmetric (with symmetrising form sending .a; / to .1/; see Section 9 below for more details and background material on symmetric algebras). The Cartan matrix of B is equal to 2 3 t CA C .CA / D ; 3 2 hence has determinant 5.

3 Defect groups We introduce Brauer’s concept of defect groups of blocks and describe its main properties. Defect groups of blocks are a generalisation of Sylow p-subgroups of finite groups: the defect groups of a block of kG form a G-conjugacy class of p-subgroups of G, and there is always at least one block having the Sylow p-subgroups of G as defect groups. Definition 3.1. Let G be a finite group and B a block algebra of kG. A defect group of B is a minimal subgroup P of G such that B is isomorphic to a direct summand of B ˝kP B as a B-B-bimodule.

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Brauer [15] originally defined defect groups as maximal p-subgroups P for which 1B is not in the kernel of the canonical linear projection kG ! kCG .P / (sending group elements in G but not in CG .P / to 0). Green [50] characterised defect groups as minimal p-subgroups P for which 1B can be written as a relative trace TrPG .c/, for some element c in the subalgebra .kG/P of P -fixed points (with respect to the conjugation action of P on kG). The above version, which is well known to be equivalent to those of Brauer and Green, has been used in [99] and [101], 3.2, for instance. Any B-module U is in particular relatively kP -projective; that is, U is isomorphic to a direct summand of a module of the form B ˝kP V for some kP -module V . More precisely, since B is a summand of B ˝kP B, tensoring with ˝B U implies that U is isomorphic to a direct summand of B ˝kP U . This leads to one of the key concepts of the subject, namely relative projectivity, which we consider in more detail in Section 4. Thus kP controls to a degree the representation theory of B. For instance, the block algebra B and its defect group algebra kP have the same representation type – which is either finite, tame or wild. Example 3.2. A block algebra B of a finite group algebra kG has the trivial group f1g as a defect group if and only if B is isomorphic to a direct summand of B ˝k B as a BB-bimodule; in other words, if and only if B is separable. Since B is indecomposable as an algebra and since k is algebraically closed, this is the case if and only if B is isomorphic to a matrix algebra. We show now that the defect groups of a block form a conjugacy class of psubgroups: Theorem 3.3. Let G be a finite group and B a block of kG. The defect groups of B form a G-conjugacy class of p-subgroups of G. Moreover, if P is a defect group of B then kP is isomorphic to a direct summand of B as a kP -kP -bimodule. This theorem shows that if p does not divide the order of G then the defect groups of all blocks of kG are trivial, and hence, by 3.2, this proves again that kG is a direct product of matrix algebras. We collect some technicalities, for future reference, in a separate statement: Lemma 3.4. Let P , Q be p-subgroups of a finite group G and let x 2 G. Denote by kŒP xQ the kP -kQ-bimodule having the double coset P xQ as a k-basis. Then kŒP xQ is indecomposable, and there are isomorphisms of kP -kQ-bimodules kŒP xQ Š kP ˝kR kŒxQ Š kŒP x ˝kT kQ where R D P \ xQx 1 and T D x 1 P x \ Q D x 1 Rx. Proof. View kŒP xQ as a k.P Q/-module, with u 2 P acting by left multiplication, and v 2 Q by right multiplication with v 1 ; this is a transitive permutation module for the group P Q, hence indecomposable by the remarks following 2.2. One verifies that the maps sending uxv to u ˝ xv and ux ˝ v yield the bimodule isomorphisms as stated.


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Proof of Theorem 3.3. We show first that any defect group of any block of B is a psubgroup of G. For any Sylow p-subgroup S of G, the canonical map W kG ˝kS kG ! kG sending x ˝ y to xy is a surjective kG-kG-bimodule homomorphism, and since char.k/ D p, the index jG W S j is invertible in k. This homomorphism is split 1 P as a bimodule homomorphism, with section sending 1 to jGWSj x ˝ x 1 , x2ŒG=S where ŒG=S a set of representatives of the cosets G=S in G. To see this, one first checks that this expression does not depend on the choice of ŒG=S . Next, if y 2 G then yŒG=S is again a set of representatives of G=S in G, and hence the above sum 1 P 1 1 is equal to jGWSj y , which implies that there is indeed a unique x2ŒG=S yx ˝ x bimodule homomorphism with the above value for 1. Finally, the fact that is a section of follows from the trivial verification ..1// D 1. Thus, if D is a defect group of B and if P is a Sylow p-subgroup of D then kD is isomorphic to a direct summand of kD ˝kP kD, hence B ˝kD B Š B ˝kD kD ˝kD B is isomorphic to a direct summand of B ˝kP B. Thus B is isomorphic to a direct summand of B ˝kP B, whence D D P by the minimality of D with this property. Thus the defect groups of B are p-subgroups of G. Suppose next that P and Q are defect groups of B. Let R G be a set Sof representatives of the P -Q-double cosets in G; that is, G is a disjoint union G D x2R P xQ. For x 2 R denote as before by kŒP xQ the k-subspace of kG spanned by the double coset P xQ. This yields a decomposition L kG D x2R kŒP xQ of kG as a direct sum of kP -kQ-bimodules. Any such summand can be viewed as a k.P Q/-module, with .u; v/ 2 P Q acting by left multiplication with u and by right multiplication with v 1 . Transitive permutation modules of finite p-groups over k are indecomposable, and hence each summand kŒP xQ is an indecomposable kP kQ-bimodule. By the Krull–Schmidt theorem, this decomposition of kG as a direct sum of indecomposable kP -kQ-bimodules is unique, up to isomorphism and order of summation. Now B is a direct summand of kG as a B-B-bimodule, hence also as a kP -kQ-bimodule. Therefore, L B D x2 kŒP xQ for some subset of R. Since B is isomorphic to a direct summand of B ˝kP B, it is also isomorphic to a direct summand of L B ˝kP B ˝kP B Š x2 B ˝kP kŒP xQ ˝kQ B: The right side is a direct sum of (not necessarily indecomposable) B-B-bimodules. Since moreover B is indecomposable, it follows from the Krull–Schmidt theorem again that B is isomorphic to a direct summand of B ˝kP kŒP xQ ˝kQ B for some x 2 . By 3.4 we have an isomorphism of kP -kQ-bimodules kŒP xQ Š kP ˝kR kŒxQ

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where R D P \ x Q. This shows that B is isomorphic to a direct summand of B ˝kR B. The minimality of P implies R D P , hence P is conjugate to a subgroup of Q. Exchanging the roles of P and Q implies that P and Q are conjugate. This argument applied to P D Q shows that kŒxP is, for some x 2 NG .P /, isomorphic to a direct summand of B as a kP -kP -bimodule. But x B D B, so multiplication by x 1 shows that kP is isomorphic to a direct summand of B as a kP -kP -bimodule. Theorem 3.3 shows that all defect groups of a block B have the same order. This leads to the following definition, due to Brauer: Definition 3.5. Let G be a finite group and B a block of kG. The defect of B is the non-negative integer d.B/ such that p d.B/ is the order of the defect groups of B. Example 3.6. A block algebra B of a finite group algebra kG has defect zero if and only if the defect groups of B are trivial. Thus, by 3.2, B has defect zero if and only if B is isomorphic to a matrix algebra over k. Theorem 3.3 can be used to obtain the following characterisation of defect groups: Theorem 3.7. A p-subgroup P of a finite group G is a defect group of a block algebra B of kG if and only if the following two statements hold: (i) B is isomorphic to a direct summand of B ˝kP B as a B-B-bimodule. (ii) kP is isomorphic to a direct summand of B as a kP -kP -bimodule. Proof. If P is a defect group of B then (i) holds by definition and (ii) holds by the last statement in 3.3. The converse is an immediate consequence of the two lemmas 3.8 and 3.9 below. Lemma 3.8. A p-subgroup P of a finite group G contains a defect group of the block algebra B of kG if and only if B is isomorphic to a direct summand of B ˝kP B as a B-B-bimodule. Proof. This is just a reformulation of the definition of defect groups. Lemma 3.9. A p-subgroup P of a finite group G is contained in a defect group of the block algebra B of kG if and only if kP is isomorphic to a direct summand of B as a kP -kP -bimodule. Proof. Let D be a defect group of B and P a p-subgroup of G. Suppose that kP is isomorphic to a direct summand of B as a kP -kP -bimodule. Since B is isomorphic to a direct summand of B ˝kD B as a B-B-bimodule, it follows that kP is isomorphic to a direct summand of B ˝kD B as a kP -kP -bimodule. Now, as a kP -kD-bimodule, B is isomorphic to a direct sum of bimodules of the form kŒP xD, with x running over a suitable subset of G. Similarly, as a kD-kP -bimodule, B is isomorphic to a direct sum of bimodules of the form kŒDyP , with y running over some subset of G. Thus


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kP is isomorphic to a direct summand of kŒP xD ˝kD kŒDyP , for some x; y 2 G. Using 3.4 one easily sees that this is only possible if D contains x 1 P x and yP y 1 ; in particular, P is contained in a conjugate of D. Conversely, if P is contained in D then kP is a direct summand of kD as a kP -kP -bimodule (having as a complement the kP -kP -submodule kŒD P of kD). Since kD is isomorphic to a direct summand of B as a kD-kD-bimodule, by 3.3, it follows that kP is isomorphic to a direct summand of B as a kP -kP -bimodule. Question 3.10. Do Morita equivalent block algebras (of possibly different) finite groups have isomorphic defect groups? Note that this question contains as a subquestion the modular isomorphism problem mentioned above, which asks whether the isomorphism class of a finite p-group P is uniquely determined by the isomorphism class of its group algebra kP . See [54] for an overview. If one wanted to be more prudent and avoid getting this close to the modular isomorphism problem one could ask the above question with k replaced by a complete discrete valuation ring O of characteristic zero having k as its residue field – by a result of Roggenkamp and Scott [133] it is known that the isomorphism class of a finite p-group is indeed determined by its group algebra over O. At least the order of a defect group of a block B is an invariant of the Morita equivalence class of that block – in fact, even an invariant of its stable (and hence also derived) module category. To state this more precisely, we denote by Rk .B/ the Grothendieck group of mod.B/ – a free abelian group with basis the set of isomorphism classes of simple B-modules – and we denote by Pr k .B/ the subgroup generated by the images in Rk .B/ of the finitely generated projective B-modules. The quotient group Rk .B/= Pr k .B/ is a finite abelian group of order equal to j det.CB /j, where CB is the Cartan matrix of B. By a classical result of Brauer, all elementary divisors of CB divide jP j, and there is exactly one elementary divisor equal to jP j. This yields immediately the following observation: Proposition 3.11. Let G be a finite group and B a block of kG with a defect group P . Then the exponent of the abelian group Rk .B/= Pr k .B/ is equal to jP j. In particular, jP j is invariant under stable equivalences of Morita type, hence under derived and Morita equivalences. We will show next that there is always a distinguished block having the Sylow p-subgroups as a defect groups. Definition 3.12. Let G be a finite group and denote by I.kG/ the kernel of the augmentation homomorphism kG ! k sending all group elements to 1k . The unique block B0 of kG not contained in I.kG/ is called the principal block of kG. This makes sense: since k has exactly one idempotent, the augmentation homomorphism kG ! k must send all block idempotents but one to zero. Equivalently, the principal block of kG is the unique block B0 which does not annihilate the trivial kG-module k.

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Theorem 3.13. Let G be a finite group and B0 the principal block of kG. Then the defect groups of B0 are the Sylow p-subgroups of G. Proof. The trivial kG-module k, viewed as a B0 -module, is isomorphic to a direct summand of B0 ˝kP k, where P is a defect group of B0 . Thus k is isomorphic to a direct summand of kG ˝kP k. An easy verification shows that the composition of any two kG-homomorphisms k ! kG ˝kP k ! k is a scalar multiple of jG W P j, and so invertible only if P is a Sylow p-subgroup, whence the result. Example 3.14. The group 2:M22 considered in 2.5 above has an elementary abelian Sylow 3-subgroup C3 C3 of order 9. Thus, for p D 3, any block with a non-trivial defect group has defect groups isomorphic to either C3 or to C3 C3 . With the blocks of 2:M22 labeled as in 2.5, the blocks B1 , B2 , B3 , B4 , B5 have the trivial group as defect group, B6 and B7 have defect groups isomorphic to C3 , and the remaining two blocks B8 , B9 have defect groups isomorphic to C3 C3 . Moreover, B9 is the principal block. Remark 3.15. The definitions and results on defect groups in this section make sense for block algebras over O. More precisely, let B be a block algebra of a finite group algebra kG and let By be the block of O which lifts B; that is, k ˝O By Š B. Then a defect group of B is also one of By in the sense that P is minimal with the property y B-bimodule. y that By is isomorphic to a direct summand of By ˝OP By as a B-

4 Relative projectivity The notion of an object in a category being relatively projective or injective with respect to a functor from that category to a possibly different category is one of the key concepts in representation theory. We review this in cases where the categories under consideration are module categories and where the functors are induced by tensoring with bimodules. Definition 4.1. Let A be a k-algebra and B a subalgebra of A. An A-module U is called relatively B-projective if there is a B-module V such that U is isomorphic to a direct summand of A ˝B V . Remarks 4.2. 1. Let B be a block of a finite group algebra with a defect group P . Every B-module U is relatively kP -projective, where kP is identified to its image kP 1B in B. Indeed, since B is isomorphic to a direct summand of B ˝kP B as a B-B-bimodule, tensoring by ˝kP U shows that U is isomorphic to a direct summand of B ˝kP U . 2. Let G be a finite group and S a Sylow p-subgroup. Then every kG-module U is relatively kS -projective. This follows from the previous remark and the fact that S contains a defect group of every block of kG, by 3.3. One can see this also directly, using a variation of the argument at the beginning of the proof of 3.3: the canonical


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surjective kG-homomorphism kG ˝kS U ! U , sending a ˝ u to au, where a 2 kG and u 2 U , is split. IsP has as a section the kG-homomorphism U ! kG ˝kS U 1 1 sending u 2 U to jGWSj u. Here ŒG=S is a set of representatives of x2ŒG=S x ˝ x the cosets G=S in G; one checks first that the previous expression does not depend on the choice of ŒG=S . Using the fact that for any y 2 G the set yŒG=S is again a set of representatives of G=S in G we get that this map is a kG-homomorphism. A trivial verification shows that this is a section as required. If U is relatively B-projective as in 4.1 then V can be chosen to be the restriction of U to B: Proposition 4.3. Let A be a k-algebra, B a subalgebra of A and U an A-module. If U is relatively B-projective then the canonical surjective A-homomorphism A ˝B U ! U sending a ˝ u to au, for a 2 A and u 2 U , is split. In particular, if U is relatively B-projective then U is isomorphic to a direct summand of A ˝B U . Proof. This is a consequence of the slightly more general result 4.4 below, applied in the case where C D k. Proposition 4.4. Let A, B, C be k-algebras such that B is a subalgebra of A. Let M be an A-C -bimodule. The following are equivalent. (i) There is a B-C -bimodule N such that M is isomorphic to a direct summand of the A-C -bimodule A ˝B N . (ii) The canonical surjective A-C -bimodule homomorphism A˝B M ! M sending a ˝ m to am, where a 2 A, m 2 M , is split. Proof. If (ii) holds then (i) holds with N D M , viewed as a B-C -bimodule. Suppose that (i) holds. Let N be a B-C -bimodule such that A ˝B N D M ˚ M 0 for some AC -bimodule M 0 . Define a A-C -bimodule homomorphism ˇ W A ˝B N ! A ˝B M by ˇ.m/ D 1 ˝ m for m 2 M and ˇ.m0 / D 0 for m0 2 M 0 . Denote by W A ˝B N ! M the canonical projection of A ˝B N onto M with kernel M 0 . Note that B ˇ D . Indeed, both are the identity on M and zero on M 0 . Define an A-C -bimodule homomorphism ˛ W A ˝B N ! A ˝B M by setting ˛.a ˝ n/ D aˇ.1 ˝ n/ for all a 2 A and n 2 N . Then, for a 2 A and m 2 M we have . B ˛/.a ˝ n/ D .aˇ.1 ˝ n// D a.ˇ.1 ˝ n// D a .1 ˝ n/ D .a ˝ n/. Since restricts to the identity on M we get that B ˛jM D IdM , whence the result. The notion of relative projectivity as defined in 4.1 is a special case of more general concepts. For instance, if A, B are k-algebras and M is an A-B-bimodule, then an A-module U is relatively M -projective if it is a direct summand of M ˝B V for some B-module V . The functor M ˝B is left adjoint to the functor HomA .M; /. If U is relatively M -projective, one can adapt the arguments in 4.4 to show that the adjunction counit M ˝B HomA .M; U / ! U , sending m ˝ ' to '.m/, is split. This is in turn a special case of yet more general statements on relative projectivity with respect to

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functors between arbitrary categories, in terms of splitting of adjunction units and counits – see Chouinard [29] and Broué [24]. There is, of course, also a dual notion of relative injectivity; we spell out a few useful and well-known special cases: Proposition 4.5. Let A, B, C be k-algebras and V a B-C -bimodule. Let ˛ W B ! A be a k-algebra homomorphism. For any A-C -bimodule U denote by ˛ U the BC -bimodule obtained by restriction via ˛. If V is isomorphic to a direct summand of ˛ U for some A-C -bimodule U then the canonical B-C -bimodule homomorphism W V ! ˛ A ˝B V sending v 2 V to 1A ˝ v is split injective. Proof. Let W V ! ˛ U and W ˛ U ! V be B-C -bimodule homomorphisms such that B D IdV . Let ˇ W A ˝B V ! U be the A-C -bimodule homomorphism sending a ˝ v to a.v/, where a 2 A and v 2 V ; that is, ˇ corresponds to through the usual adjunction. Clearly ˇ B D , where ˇ is considered as a B-C -bimodule homomorphism. Thus . B ˇ/ B D B D IdV , which shows that is split injective with retraction B ˇ. Corollary 4.6. Let ˛ W B ! A be a homomorphism of k-algebras. Suppose that B is isomorphic to a direct summand of A as a B-B-bimodule. Then ˛ is injective and Im.˛/ is a direct summand of A as a B-B-bimodule. Proof. This follows from 4.5 applied to B D C and A instead of U , B instead of V . Let G be a finite group and H a subgroup of G. For V a kH -module, its induced G kG-module is defined as IndH .V / D kG ˝kH V . For U a kG-module, its restriction G to kH is the module U viewed as a kH -module, denoted by ResH .U /. For x 2 G x 1 x x we set H D xH x and denote by V the k. H /-module which is equal to V as a k-module, with xhx 1 acting as x on V . Induction and restriction are covariant exact G functors; more precisely, IndH is the functor kG ˝kH from mod.kH / to mod.kG/, G and ResH is isomorphic to the functor kG ˝kG from mod.kG/ to mod.kH /, where G G here kG is considered as a kH -kG-bimodule. The functors IndH and ResH are left and right adjoint to each other; this fact, known as Frobenius reciprocity, is a special case of a more general adjunction of exact functors between module categories of symmetric algebras, which we review in 9.5. With this notation, a kG-module U is relatively kH projective if there is a kH -module V such that U is isomorphic to a direct summand G G G of IndH .V /. In that case, 4.3 implies that the canonical map IndH ResH .U / ! U sending x ˝ u to xu, for x 2 G, u 2 U , is split surjective. One of the standard ingredients for dealing with relative projectivity in the context of finite group algebras is Mackey’s formula, which is essentially a functorial reinterpretation of the bimodule decomposition of kG induced by the double coset partition of two subgroups of a finite group G. Theorem 4.7 (Mackey’s formula). Let G be a finite group and let H , L be subgroups of G. For any kL-module W there is a natural isomorphism of kH -modules L xL G G H x ResH IndL .W / Š x2ŒH nG=L IndH \x L ResH \x L . W /:


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G G Proof. We have ResH IndL .W / D kG ˝kL W , where kG is considered as a kH -kLbimodule. Writing kGL as a disjoint union of H -L-double cosets yields a direct sum decomposition kG D x2ŒH nG=L kŒH xL as kH -kL-bimodules. For any x 2 G there is a unique isomorphism of kH -kL-bimodules kŒH xL Š kH ˝k.H \x L/ kŒxL, mapping yxz to y ˝ xz, where y 2 H and z 2 L. Moreover, since xL D x Lx, the space kŒxL is a k.x L/-kL-bimodule, and we have an isomorphism of k.x L/-modules kŒxL ˝kL W Š x V . Combining these isomorphisms yields the result.

The results of this section up to and including Mackey’s formula hold over any commutative ring as ring of coefficients instead of the field k (except for the second remark in 4.2 which requires ŒG W S to be invertible in k). Definition 4.8. Let G be a finite group and let M be a finitely generated indecomposable kG-module. A subgroup Q of G is called a vertex of M if Q is a minimal with the property that M is relatively Q-projective. If Q is a vertex of M , a kQ-source of M is an indecomposable kQ-module V such that M is isomorphic to a direct summand of IndG Q .V /. The following theorem states that every indecomposable kG-module M has a vertex Q, every vertex is a p-group and for every such vertex there is a kQ-source V . The theorem then tells us where to look for sources, namely in the restriction to a vertex Q of M , and concludes that pairs .Q; V / consisting of a vertex and a source are unique up to conjugation by elements in G. Its proof combines relative projectivity, Mackey’s formula and the Krull–Schmidt Theorem. Theorem 4.9. Let G be a finite group and let M be a finitely generated indecomposable kG-module. (i) M has a vertex, and every vertex of M is a p-subgroup of G. (ii) For every vertex Q of M there is a kQ-source V of M . (iii) Given a vertex Q every kQ-source V of M is isomorphic to a direct summand of ResG Q .M /. (iv) Given a vertex Q of M , an indecomposable summand V of ResG Q .M / is a source of M if and only if V has vertex Q. (v) Given a vertex Q of M , a kQ-source V , a p-subgroup R of G and an indecomposable kR-module W such that W has R as vertex and is isomorphic to a direct x x summand of ResG R .W /, there is x 2 G such that R Q and such that W is Q isomorphic to a direct summand of Resx R .V /. (vi) Given two vertices Q, R of M , a kQ-source V and kR-source W of M , there is x 2 G such that x R D Q and x W Š V . In particular, the set of vertices of M is a conjugacy class of p-subgroups of G. Proof. The existence of a vertex Q is, of course, trivial: just take a minimal subgroup

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Q for which one can find a kQ-module V such that M is a summand of IndG Q .V /; since G is finite this is clearly possible. It follows from the second remark in 4.2 that any vertex Q is a p-group. ThisL proves (i). We next show that V can then be chosen indecomposable: write V D i are indecomposable 1in Vi , where the VL OQ-modules for 1 i n. Thus M is a direct summand of 1in IndG Q .Vi /. It follows from the Krull–Schmidt Theorem that in fact M is the summand of one of the IndG Q .Vi /, for some i. Thus we may replace V by Vi , or equivalently, we may assume that V is a kQ-source of M . This completes the proof of (ii). By 4.3, if Q is a vertex of G M then in fact M is a summand of IndG Q ResQ .M /. Thus, as before, we may choose an indecomposable direct summand V of ResG Q .M / such that M is a summand of IndG .V /. This shows that at least some source V of M is a summand of ResG Q Q .M /. Note that a source must have vertex Q, because otherwise Q would not be minimal with the property that M is relatively Q-projective. Let W be another kQ-source of G M . Then M is a summand of IndG Q .W /. Thus V is a summand of ResQ .M /, which is a summand of L xQ Q G x ResG Q IndQ .W / D x2ŒQnG=Q IndQ\x Q ResQ\x Q . W / by Mackey’s formula. Since V is indecomposable, V is a summand of the kQ-module xQ x IndQ Q\x Q ResQ\x Q . W / for some x 2 G. But V must have vertex Q, so this forces x Q D Q, and hence V Š x W , as W is indecomposable. Since x M Š M via the map sending m 2 M to xm, we get that W itself is a summand of M restricted to Q. This proves (iii) and (iv). Note that this also proves a special case of (v), namely when Q D R. Let now Q, V , R, W be as in (v). Then, by assumption, W is a summand of G ResG R .M /, and M is a summand of Ind Q .V /. Thus W is a summand of G ResG R Ind Q .V / D

L y2ŒRnG=Q

y

Q IndR R\y Q ResR\y Q .V /

by Mackey’s formula. Since W is indecomposable, the Krull–Schmidt Theorem imyQ y plies that W is a summand of IndR R\y Q ResR\y Q . V / for some y 2 G. Since R is y a vertex of W , it follows that R Q and hence that W is isomorphic to a direct y summand of ResRQ .y V /. Conjugating this back with x D y 1 yields (v). In the situation of (vi), applying (v) twice implies that actually x R D Q, and hence x W Š V for some x 2 G. This concludes the proof of (vi), hence of the theorem. The above theorem says that every indecomposable kG-module M determines a pair .Q; V / consisting of a vertex Q and a kQ-source V , uniquely up to G-conjugacy. In general, the pair .Q; V / does not determine the isomorphism class of M . One can show that given .Q; V / there are at most finitely many isomorphism classes of indecomposable kG-modules M having Q as vertex and V as source. These can be parametrised by a third invariant, the multiplicity module, defined by Puig; see [147] for a broader exposition of this material, and also Thévenaz [146] for a generalisation to interior algebras.


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Examples 4.10. Let G be a finite group.

The vertices of the trivial kG-module are the Sylow p-subgroups of G and the trivial module for any of these is then a source. One can use this to provide another proof of the fact that the Sylow p-subgroups of G are the defect groups of the principal block of kG. Using that k is a field one checks that an indecomposable kG-module is projective if and only of it has the trivial subgroup of G as its vertex. Given a finite p-group P and a subgroup Q of P , the transitive permutation module kP =Q is indecomposable with Q as a vertex and the trivial kQ-module as a source. Indeed, kP =Q Š IndP Q .k/, hence Q contains a vertex of kP =Q. The restriction P ResP Ind .k/ has a direct summand k, and hence, by the first example, Q is a Q Q vertex.

The Green correspondence, due to J. A. Green, is a vertex and source preserving correspondence between indecomposable modules of a finite group G over a p-local ring and modules of normalisers of p-subgroups of G. Theorem 4.11 (Green Correspondence). Let G be a finite group, let Q be a p-subgroup of G and let H be a subgroup of G containing NG .Q/. There is a bijection between the sets of isomorphism classes of indecomposable kG-modules with Q as vertex and indecomposable kH -modules with Q as vertex given as follows. (i) If U is an indecomposable kG-module having Q as a vertex, then there is, up G .U / to isomorphism, a unique indecomposable direct summand f .U / of ResH having Q as vertex. Moreover, every kQ-source of f .U / is a source of U , and G .U / has a vertex contained in x Q \ H for every other direct summand of ResH some x 2 G H and not H -conjugate to Q. (ii) If V is an indecomposable kH -module having Q as a vertex, then there is, up G .V / to isomorphism, a unique indecomposable direct summand g.V / of IndH having Q as vertex. Moreover, every kQ-source of V is a source of g.V /, and G every other indecomposable direct summand of IndH .V / has a vertex contained x in Q \ Q for some x 2 G H . (iii) We have g.f .U // Š U and f .g.V // Š V . The Green correspondence does not say anything about the structural connections between U and f .U /. For instance, if one of U , f .U / is simple this does not imply that the other is simple as well. We formulate some technical parts of the proof of the Green correspondence separately. Lemma 4.12. Let G be a finite group, let Q be a p-subgroup of G and let H be a subgroup of G containing NG .Q/. Let V be a relatively Q-projective indecomposable kH -module.

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G G (i) We have ResH IndH .V / D V ˚V 0 for some kH -module V 0 , and every indecomposable summand of V 0 has a vertex contained in H \ x Q for some x 2 G H . In particular, no indecomposable direct summand of V 0 has Q as vertex. G .V / D U ˚ U 0 for some indecomposable kG-module U such (ii) We have IndH G .U /, and then every indecomposable summand of that V is a summand of ResH 0 U has a vertex contained in Q \ x Q for some x 2 G H . In particular, no indecomposable direct summand of U 0 has Q as vertex.

Proof. By Mackey’s formula we have L xH G G H x 0 IndH .V / D x2ŒH nG=H IndH ResH \x H ResH \x H . V / Š V ˚ V : Now V is relatively kQ-projective, and V is indecomposable. Thus V is a direct H summand of IndH Q .S / for some indecomposable kQ-module S. Write Ind Q .S / D G G V ˚ V0 for some kH -module V0 . As for V we may write ResH IndH .V0 / D V0 ˚ V00 G G 0 for some kH -module V0 . We compute ResH IndQ .S / in two different ways. On one hand, the Mackey formula yields that L xQ G H H x 00 ResH IndG Q .S / D x2ŒH nG=Q IndH \x Q ResH \x Q . S / D Ind Q .S / ˚ V ; where V 00 has the property that all indecomposable summands have a vertex contained in H \ x Q for some x 2 G H . On the other hand, G G G G G IndG ResH S .S / D ResH IndH .V / ˚ ResH IndH .V0 / 0 0 D V ˚ V 0 ˚ V0 ˚ V00 D IndH Q .S / ˚ V ˚ V0 :

The Krull–Schmidt Theorem implies that V 00 D V 0 ˚ V00 . In particular, all indecomposable summands of V 0 have a vertex contained in H \ x Q for some x 2 G H . Thus any indecomposable direct summand of V 0 has a vertex R contained in H \ x Q for some x 2 G H . If Q were a vertex of that summand as well, then h R D Q for some h 2 H . But then Q H \ hx Q, hence Q Q\ hx Q, which is impossible since hx 62 H does not normalise Q. This proves (i). Note that since V is a summand of G G ResH .U / it follows from (i) that ResH .U 0 / is a summand of V 0 , by the Krull–Schmidt Theorem. Clearly every indecomposable summand of U 0 has a vertex contained in Q 0 because U 0 is a summand of IndG Q .S /. Let U1 be an indecomposable summand of U and let R Q be a vertex of U1 . Let T be a kR-source of U1 . Then T is a summand G of ResG R .U1 /, by 4.9. Thus, for some indecomposable summand V1 of ResH .U1 /, the H module T is a summand of ResR .V1 /. By (i), V1 has a vertex contained in H \ x Q for some x 2 G H . Since all vertices of V1 are H -conjugate, there is y 2 H and 1 1 x 2 G H such that y R H \ x Q. Thus R H \ y x Q, hence R Q \ y x Q as required. This completes the proof. Proof of Theorem 4.11. (i) Let S be a kQ-source of U . Then U is a summand of G H IndG Q .S/ D IndH Ind Q .S /. Since U is indecomposable, there is an indecomposable


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G H summand V of IndH Q .S / such that U is a summand of IndH .V /. Write Ind Q .S / D V ˚ V0 for some kH -module V0 . Note that then Q is a vertex of V and S a kQ-source G G of V . By Mackey’s formula we can write ResH IndH .V / D V ˚ V 0 for some kH G G G G 0 module V . Note that we also have IndQ .S / D IndH .V / ˚ IndH .V0 /. Now ResH .U / G G G is a summand of ResH IndH .V /, thus, by 4.12, ResH .U / has at most one summand with vertex Q, namely V , and all other summands have a vertex contained in H \ x Q for some x 2 G H . Since U has vertex Q, which is contained in H , U is a summand G G of IndH ResH .U /, by Higman’s criterion. Since all vertices of U are G-conjugate, no G vertex can be contained in H \ x Q with x 2 G H , and hence ResH .U / must have V as direct summand. Thus setting f .U / D V shows the result. This will also be useful in the proof of (iii), because by construction of V we have that U is a summand G of IndH .V /. For (ii), note that by 4.12 there is an indecomposable summand U of G G G IndH .V / such that V is a summand of ResH .U /, and then IndH .V / D U ˚ U 0 , where 0 U is an OG-module all of whose indecomposable summands have a vertex contained in Q \ x Q for some x 2 G H . Setting g.V / D U proves (ii). Statement (iii) follows from the construction of the correspondences in (i) and (ii).

Remark 4.13. Definition 4.8, Theorem 4.9 and the Green correspondence 4.11 remain true for bounded complexes of finitely generated modules of finite group algebras over k and over the ring O, because the Krull–Schmidt theorem holds for O-algebras which are finitely generated as O-modules. Inducing an indecomposable module from a subgroup does not, in general, yield an indecomposable module. There is one situation where it does: Theorem 4.14 (Green’s Indecomposability Theorem). Let G be a finite group, H a subnormal subgroup of p-power index and V an indecomposable kH -module. Then G the kG-module IndH .V / is indecomposable. This result holds for modules over OG and also when k is not algebraically closed provided one assumes that V is absolutely indecomposable; that is, for any field extension k 0 =k, the k 0 H -module k 0 ˝k V is still indecomposable. Corollary 4.15. Let G be a finite group and B a block algebra of kG. Let a and d be the non-negative integers such that p a is the order of a Sylow p-subgroup of G and such that p d is the order of a defect group of B. Let U be a finitely generated B-module. Then p ad divides dimk .U /. Proof. Let P be a defect group of B and S a Sylow p-subgroup of G. Since U is relaG G tively P -projective it follows that ResG S .U / is a direct summand of ResS IndP .U / Š L x S P x x IndS \x P ResS\x P . U /. By 4.14, every indecomposable direct summand of any term in this sum is of the form IndSQ .V / for some subgroup Q of order at most jP j and some indecomposable kQ-module V , and thus has dimension divisible by p ad .

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5 Brauer’s First Main Theorem How does one detect whether a finite group algebra kG has blocks with a given psubgroup P of G as a defect group? The following theorem provides a useful criterion: Theorem 5.1 (Brauer’s First Main Theorem). Let G be a finite group and P a psubgroup of G. For any block B of kG with P as a defect group there is a unique block C of kNG .P / with P as a defect group such that C is isomorphic to a direct summand of B as a kNG .P /-kNG .P /-bimodule. This correspondence is a bijection between the set of blocks of kG with P as a defect group and the set of blocks of kNG .P / with P as a defect group. Proof. This is a special case of the Green correspondence. We view kG as a k.G G/-module. There is an isomorphism IndGG P .k/ Š kG ˝kP kG, where P D f.u; u/ j u 2 P g. Thus the blocks of kG with P as a defect group are the indecomposable direct summands of kG having P as a vertex. The group NG .P / NG .P / contains NGG . P /, and hence every block B has, up to isomorphism, a unique direct summand C as a kNG .P /-kNG .P /-bimodule, with P as a vertex. Any such C must be a direct summand of kNG .P / because B is a direct summand of B ˝kNG .P / B, hence in particular, any B has a Green correspondent which is a summand of kNG .P /. There are other proofs of this theorem, notably a more algebra theoretic proof using the Brauer homomorphism. The block C as in the theorem above is called the Brauer correspondent of B. The mere existence of blocks with P as a defect group is thus played back to the potentially much smaller group NG .P /. Theorem 5.2. Let G be a finite group, B a block of kG and P a defect group of B. Then P contains every normal p-subgroup of G; equivalently, P contains the largest normal p-subgroup Op .G/ of G. Proof. Let Q be a normal p-subgroup of G. Then every Q-Q-double coset is in fact a left or right coset. Thus, as a Q-Q-bimodule, every indecomposable direct summand of B is isomorphic to kŒQx, for some x 2 G. Right multiplication by x 1 on B is an automorphism of B as a left kQ-module which therefore sends a decomposition of B as a direct sum of indecomposable kQ-kQ-bimodules to a decomposition of B as a direct sum of (not necessarily indecomposable) left kQ-modules. Since x 1 normalises Q, this decomposition is again a decomposition of B as a direct sum of indecomposable kQ-kQ-bimodules. Under this map, a direct summand isomorphic to kŒQx will be sent to a direct summand isomorphic to kQ. Thus, by 3.9, Q is contained in a defect group of B. Since the defect groups of B are G-conjugate (by 3.3) and since Q is normal in G it follows that Q is contained in any defect group of B, whence the result. The previous two theorems imply the following necessary group theoretic criterion for G to have any blocks with P as a defect group:


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Corollary 5.3. Let G be a finite group and P a p-subgroup of G. If kG has a block with P as a defect group then P D Op .NG .P //, or equivalently, NG .P /=P has no non trivial normal p-subgroup. In general there is no obvious structural connection between a block B and its Brauer correspondent C , but the structure of C is well understood, thanks to the following result of Külshammer in [90] on blocks with a normal defect group: Theorem 5.4. Let G be a finite group and B a block of kG with a defect group P such that P is normal in G. Then there is a p 0 -subgroup E of Aut.P / and an element ˛ 2 H 2 .EI k / such that B is Morita equivalent to the twisted group algebra k˛ .P Ì E/, where ˛ is extended trivially to P Ì E. We conclude this section with a list of examples in which the structure of block algebras with a given defect group are known. Examples 5.5. Let G be a finite group, B a block of kG and P a defect group of B.

If P is cyclic then B is a Brauer tree algebra, and this is the only case in which B has finite representation type. The long list of authors having contributed to the theory of cyclic blocks includes Brauer [14], Dade [33], Thompson [148], Janusz [65], Kupisch [93], Green [51], Gabriel and Riedtmann [49], to name a few. It is not known which Brauer trees actually occur in block algebras. Using the classification of finite simple groups it has been shown by Feit [47] that ‘most’ trees do not occur. Brauer trees of cyclic blocks of general linear groups have been determined by Fong and Srinivasan in [48], and, with few exceptions, Brauer trees of cyclic blocks of sporadic finite simple groups have been determined by Hiss and Lux [58]. There are other symmetric algebras of finite representation type – but these cannot occur as block algebras. The reason for that is that if P is cyclic then B is stably equivalent to its Brauer correspondent C , and C in turn is Morita equivalent to the algebra k.P Ì E/, for some cyclic automorphism group E of P of order dividing p 1, and this is a Nakayama algebra; that is, k.P ÌE/ is symmetric and every indecomposable k.P ÌE/-module has a unique composition series. A symmetric Brauer tree algebra can be described in terms of generators and relations as follows. Let I be set of edges of the Brauer tree. Label the vertices of the Brauer tree by either or in such a way that vertices connected by an edge have different labels. The Brauer tree has one exceptional vertex with an exceptional multiplicity m; choose the labeling of the vertices in such a way that the exceptional vertex has as its label (if the exceptional multiplicity m is equal to 1, choose any -labeled vertex as exceptional vertex). The Brauer tree is considered as a tree in the plane; this is equivalent to fixing a cyclic order on each set of edges emanating from a common vertex, corresponding to the counterclockwise order of the edges in the plane. The product of the cycles around each -labeled edge yields a permutation of I , again denoted , and similarly, the product of the cycles around the -labeled edges yields a permutation of I . Using the fact that the Brauer tree is connected one verifies that B is a transitive cycle on I .

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Using the fact that the Brauer tree is a tree one sees that for any i 2 I the intersection of the -orbit i of i and of the -orbit i of i is exactly fi g. In this way, the Brauer tree is encoded in terms of the combinatorial information .I; ; /, together with a designated exceptional -orbit and its exceptional multiplicity m 1. The set of vertices of the Brauer tree corresponds bijectively to the disjoint union of the set of

-orbits and the set of -orbits in I . The symmetric Brauer tree algebra is generated by the disjoint union I [ fr; sg subject to the following relations: P – i 2 D i for all i 2 I , i i 0 D 0 for all i , i 0 2 I such that i ¤ i 0 , and i2I D 1; – rs D sr D 0; – ir D r .i / and i s D s .i / for any i 2 I ; – ir a.i/ D i s b.i/ , where a.i / D ji j and where b.i / D ji j if the vertex i is non exceptional and b.i / D m ji j if the vertex i is exceptional.

One can ‘lift’ these relations over O in order to obtain a description of block algebras with cyclic defect groups over O; see [98], 3.10. This, and the next example, are due to Erdmann as part of a systematic investigation of blocks of tame representation type – the main reference, which points to background and further references is Erdmann’s book [44]. Suppose that p D 2 and that P is a Klein four group. Then B is Morita equivalent to either kP , kA4 or the principal block algebra of kA5 . Here A4 and A5 are the alternating groups on four and five letters, respectively. Both have a Klein four group as a Sylow 2-subgroup. The double covers AQ4 , AQ5 of A4 , A5 , respectively, have a quaternion group Q8 as a Sylow 2-subgroup. Blocks with a quaternion defect group turn out to be, up to Morita equivalence, exactly central extensions of blocks with a Klein four defect group by an involution: if p D 2 and P is a quaternion group of order 8 then B is Morita equivalent to either kP , k AQ4 or the principal block algebra of k AQ5 . Suppose p is an odd prime, let P D Cp Cp be an elementary abelian p-group of order p 2 , and consider the semi-direct product G D P Ì Q8 , with Q8 acting on P such that Z D Z.Q8 / acts trivially on P and such that in a decomposition of the Klein four group V4 D Q8 =Z D C2 C2 each factor C2 acts on one of the factors Cp of P by inversion. Then kG has two blocks, namely the principal block B0 , having as block idempotent the central idempotent 12 .1 C z/, where Z D f1; zg and a non-principal block B1 whose block idempotent is 12 .1 z/. Both blocks have the (normal) Sylow p-subgroup P as their defect group. We have B0 Š k.P Ì V4 / and B1 Š k˛ .P Ì V4 /, where ˛ is the image in H 2 .V4 I k /, under the unique injective group homomorphism Z ! k , of the element in H 2 .V4 I Z/, inflated to P Ì V4 , representing the non-split exact sequence 1

/Z

/ Q8

/ V4

/ 1:

It has been shown by Alperin and Benson that k˛ .P Ì V4 / is Morita equivalent to the symmetric local algebra khx; yi=.x p ; y p ; xy C yx/. One of the main ingredients is

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the fact that k˛ V4 is isomorphic to the matrix algebra M2 .k/. A finite p 0 -group H is called of central type if H has a cyclic center Z and there exists ˛ 2 H 2 .H=ZI k / such that k˛ H=Z is a matrix algebra. Using the classification of finite simple groups it has been shown by Howlett and Lehrer [62] that finite groups of central type are solvable. The previous example is a special case of certain blocks with a single isomorphism class of simple modules which are isomorphic to what is known as quantum complete intersections – the main references are Benson–Green [10], Benson–Kessar [11] and Holloway–Kessar [59]. If P D hy1 i hy2 i hyr i is an abelian p-group of rank r, written as a direct product of cyclic p-groups hyi i of order p ni for some positive integers ni , then there is an isomorphism of k-algebras n

n

nr

kP Š kŒx1 ; x2 ; : : : ; xr =..x1 /p 1 ; .x2 /p 2 ; : : : ; .xr /p / mapping yi to the image of 1 C xi , for 1 i r. If we replace the commutativity of the variables xi by a ‘commutativity up to a scalar’ we obtain other examples of local symmetric algebras which arise as block algebras. Let .qij / be a matrix in Mr .Fp /, where Fp is identified to its image in k, such that qij qj i D 1 and qi i D 1 for 1 i; j r. Denote by A the k-algebra generated by x1 , x2 ; : : : ; xr with n relations xi xj D qij xj xi and .xi /p i D 0, for 1 i; j r. Then A is a local symmetric k-algebra. By [59], Theorem 1.1, the algebra A is Morita equivalent to a block B of a finite group G with P as a defect group; moreover, one can choose G such that P is normal in G. Note that if all entries qij are 1 then A Š kP ; in general, we have dimk .A/ D jP j. Little is known about symmetric local algebras A Morita equivalent to blocks with non-abelian defect groups. It is still true that the dimension of A is a power of p; more precisely, the last statement in the previous example holds in general for blocks with a single isomorphism class of simple modules: Proposition 5.6. Let A be a symmetric local k-algebra. If A is Morita equivalent to a block algebra B of kG for some finite group G then dimk .A/ D jP j, where P is a defect group of B. The proof uses the fact, due to Brauer, that the elementary divisors of the Cartan matrix of B, hence of A, divide jP j and exactly one of them is equal to jP j. Thus if A is local, its Cartan matrix consists of the single entry jP j, which then also is the dimension of A.

6 Source algebras of blocks The fact that we do not know in general whether the Morita equivalence class of a block determines its defect groups means that we cannot simply replace a block algebra by its basic algebra, because this might result in losing too much information. This

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consideration is at the heart of Puig’s concept [121] of a source algebra of a block – which, although potentially ‘much smaller’ than the block algebra itself, is still ‘big enough’ to contain a defect group and related invariants. Here is a way to define a source algebra of a block algebra B with a defect group P , using only the concepts developed so far. We denote by B P the subalgebra of B consisting of all elements in B which are invariant under the conjugation action by P on B. If i is an idempotent in B P then Bi is a direct summand of B as a B-kP -module, having as a complement the bimodule B.1 i /. The k-dual of Bi is isomorphic to iB as a kP -B-bimodule, using the symmetry of B. As a B-B-bimodule, B is isomorphic to a direct summand of B ˝kP B. More precisely, 4.4 implies that the map B ˝kP B ! B induced by multiplication in B is split surjective. Now B is indecomposable as a BB-bimodule, hence there is a primitive idempotent i in the P -fixed point algebra B P such that the map Bi ˝kP iB ! B is still split surjective as a homomorphism of B-B-bimodules. Definition 6.1. Let B be a block algebra of a finite group algebra kG and P a defect group of B. A primitive idempotent i in B P is called a source idempotent of B, and the algebra iBi is called a source algebra of B, if B is isomorphic to a direct summand of Bi ˝kP iB, as a B-B-bimodule. We view iBi as interior P -algebra; that is, we consider iBi together with the group homomorphism P ! .iBi / sending y 2 P to iyi . Since i belongs to the block algebra B, multiplication by i annihilates all blocks other that B, so iBi D i kGi . The idempotent i commutes with the elements in P , hence iyi D iy D yi for all y 2 P , which implies that the map sending y 2 P to iyi is indeed a group homomorphism from P to .i kGi / . The next result shows that iBi is essentially unique, not just up to isomorphism of k-algebras, but in fact up to isomorphism of interior P -algebras, modulo allowing twists by automorphisms of P . Theorem 6.2 (Puig). Let B be a block of a finite group algebra and let i , j be two source idempotents in B P . Then i and j are conjugate by an element in the subgroup .B P / NG .P / of B . In particular, there is an algebra isomorphism ˛ W iBi Š jBj and a group automorphism ' of P induced by conjugation with an element in NG .P / such that ˛.iui / D j'.u/j for all u 2 P . The proof uses the following two lemmas, the first of which implies that any .B P / conjugate of a source idempotent is again a source idempotent. Lemma 6.3. Let G be a finite group, P a subgroup of G and i , j idempotents in .kG/P . The kG-kP -bimodules kGi and kGj are isomorphic if and only if the idempotents i and j are conjugate by an element in the group ..kG/P / .


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Proof. We have kG D kGi ˚ kG.1 i / D kGj ˚ kG.1 j /. Thus, by the Krull– Schmidt theorem, if there is an isomorphism of kG-kP -bimodules kGi Š kGj then there is also an isomorphism kG.1 i / Š kG.1 j /. The direct sum of these isomorphisms is an automorphism of kG as a kG-kP -bimodule, hence given by right multiplication with an invertible element a in .kG P / . Then kGi a D kGj and kG.1 i/a D kG.1 j /. This shows that j D ci a and 1 j D d.1 i /a for some c, d in kG. Thus 1 D j C .1 j / D .ci C d.1 i //a, which shows that ci C d.1 i/ D a1 and hence ci D a1 i , whence a1 i a D j as required. The converse is easy. Lemma 6.4. Let B be a block of a finite group algebra kG with a defect group P and source idempotent i 2 B P . As kP -kP -bimodules, Bi and iB have a direct summand isomorphic to kP . Proof. Since Bi is a direct summand of B, hence of kG, every direct summand of Bi as a kP -kP -bimodule is isomorphic to kŒP yP for some y 2 P , by the Krull–Schmidt theorem. By 3.4 we have kŒP yP Š kP ˝kQ kŒyP , where Q D P \ yP y 1 . Now B is isomorphic to a summand of Bi ˝kP iB, hence Bi ˝kP iB has a summand isomorphic to kP as a kP -kP -bimodule. But that implies the existence of an element y as before with Q D P \ yP y 1 D P , hence y 2 NG .P /. It follows that P yP D yP , and thus kŒyP is isomorphic to a direct summand of Bi as a kP -kP -bimodule. Left multiplication by y 1 implies that kP is isomorphic to a direct summand of Bi . The same argument applied to iB completes the proof. Proof of 6.2. As a B-B-bimodule, B is isomorphic to a direct summand of Bi ˝kP iB and of Bj ˝kP jB. Thus B Š B ˝B B is also isomorphic to a direct summand of their tensor product Bj ˝kP jBi ˝kP iB. Since jBi is a direct summand of B, hence of kG as a kP -kP -bimodule, every indecomposable direct summand of jBi is of the form kŒP yP for some y 2 G. The indecomposability of B and the Krull–Schmidt theorem imply that B is isomorphic to a direct summand of Bj ˝kP kŒP yP ˝kP iB for some y. Using 3.4 again, this B-B-bimodule is a direct summand of B ˝kQ B, where Q D P \ yP y 1 . The minimality of P with this property implies that y 2 NG .P /, hence B is isomorphic to a direct summand of Bjy ˝kP iB. Thus Bi is isomorphic, as a B-kP -bimodule, to a direct summand of Bjy ˝kP iBi . Again, as every direct summand of iBi is isomorphic to kŒP zP for some z 2 G, the Krull–Schmidt theorem implies that Bi is isomorphic to a direct summand of Bjy ˝kP kŒP zP . The fact that, by 6.4, Bi has a summand kP forces z 2 NG .P /, and hence Bi is isomorphic to a direct summand of Bjyz. But Bjyz D B.yz/1 jyzB is indecomposable as a BkP -bimodule, and hence Bi Š B.yz/1 jyz. Since yz 2 NG .P /, it it follows from 6.3 that there is an element c in .B P / NG .P / such that j D ci c 1 . The map sending a 2 iBi to cac 1 is an algebra isomorphism ˛ W iBi Š jBj . Since .B P / acts trivially on the image iP i of P in jBj it follows that ˛ sends i ui to j'.u/j for some automorphism ' of P induced by conjugation with an element of NG .P /. This completes the proof of 6.2.

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Any source algebra iBi is Morita equivalent to B – thus no module theoretic information is lost by passing from B to any of its source algebras: Theorem 6.5 (Puig). Let B be a block of a finite group algebra kG with a defect group P and source idempotent i 2 B P . We have BiB D B; equivalently, the algebras B and iBi are Morita equivalent via the B-iBi -bimodule Bi and the iBi -B-bimodule iB. Proof. The equality BiB D B is equivalent to the surjectivity of the map Bi ˝kP iB ! B. The obvious map iB ˝B Bi ! iBi is an isomorphism. Thus, by standard properties of Morita equivalences, this implies the result. Any .B P / -conjugate of a source idempotent is a source idempotent; indeed, two idempotents i , j in B P are conjugate in .B P / if and only if the B-kP -bimodules Bi and Bj are isomorphic. Examples 6.6. Let G be a finite group, B a block of kG and P a defect group of B.

If P is trivial – that is, if B is a block of defect zero – then the source algebras are isomorphic to the trivial algebra k. Indeed, in that case, B is isomorphic to a matrix algebra Mn .k/. All primitive idempotents in a matrix algebra are conjugate, so any one of them is a source idempotent, and if i is a primitive idempotent in Mn .k/ then iMn .k/i Š k. If P is cyclic then, by [98], the isomorphism class of a source algebra iBi is determined by the Brauer tree of B (or equivalently, the Morita equivalence class of B) together with an indecomposable endo-permutation kP -module having P as a vertex. If p D 2 and P is a Klein four group then, by [32], Theorem 1.1, the source algebras of B are isomorphic, as interior P -algebras, to either kP , kA4 , or the principal block algebra of kA5 . This makes sense as P is isomorphic to a Sylow 2-subgroup of A4 and of A5 , hence we can consider kA4 and the principal block algebra of kA5 as interior P -algebras, through any identification. Since any group automorphism of P can be extended to a group automorphism of A4 and A5 , it does not matter which identification we choose. This result uses the classification of finite simple groups – without using the classification this has been shown to hold ‘up to Heller translates’ in [96]

Remarks 6.7. 1. The concepts of source idempotents and source algebras make sense over O, without any change. Let B be a block of a finite group algebra kG, and let By be the block of OG lifting B; that is, k ˝O By D B. Since By and B have stable bases with respect to the conjugation action of a defect group on By and on B, the canonical map By P ! B P is surjective (and its kernel J.O/By P is contained in J.By P /). By standard lifting theorems, any primitive idempotent i in B P lifts uniquely, up to conjugation, to a primitive idempotent iO in By P . Moreover, one can show that if i is a source idempotent then iO is also a source idempotent in the analogous sense – that is, By is isomorphic to


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y Setting A D iBi , the source algebra AO D iO By iO of a direct summand of By iO ˝OP iO B. y O B satisfies k ˝O A Š A. 2. By a result of Puig, source algebras lift uniquely to source algebras over O – in contrast to finite-dimensional k-algebras, which can lift in many different ways. To be O AO0 be source algebras of blocks of finite group algebras OG, OG 0 more precise, let A, with a common defect group P such that k ˝ AO Š k ˝O A0 , as interior P -algebras. Then AO Š AO0 as interior P -algebras.

7 Fusion systems of blocks The fusion system of a finite group G on one of its Sylow p-subgroups P is a category, denoted FP .G/, defined as follows. The objects of FP .G/ are the subgroups of P . The morphism sets HomFP .G/ .Q; R/ between two subgroups Q, R of P consist of the group homomorphisms induced by conjugation by elements in G; that is, ' W Q ! R is a morphism in FP .G/ if there exists an element x 2 G such that '.u/ D x 1 ux for all u 2 Q; in particular, Qx R, where Qx D x 1 Qx. Since the Sylow p-subgroups of G are all G-conjugate, this determines FP .G/ up to isomorphism. In order to extend this to blocks we follow essentially the approach of Alperin and Broué [2] and of Broué and Puig [26]. For Q a p-subgroup of G denote by BrQ W .kG/Q ! kCG .Q/ the restriction to the subalgebra of Q-fixed elements .kG/Q in kG of the linear projection map kG ! kCG .Q/ sending x 2 CG .Q/ to x and x 2 G CG .Q/ to zero. It is easy to see that this map is in fact a k-algebra homomorphism; this is called the Brauer homomorphism. Fix a block B D kGb of kG, where b is the block idempotent in Z.kG/ of B; that is, b D 1B . Then, for Q a p-subgroup of G, either Br Q .b/ D 0, or Br Q .b/ is a sum of block idempotents of kCG .Q/. A B-Brauer pair is a pair .Q; e/ consisting of a p-subgroup Q of G and a block idempotent e of kCG .Q/ satisfying BrP .b/e D e. The set of B-Brauer pairs is partially ordered, with a partial order defined by .R; f / .Q; e/ if R Q and there exists an idempotent j 2 .kG/Q such that Br Q .j /e D Br Q .j / ¤ 0 and Br R .j /f D Br R .j / ¤ 0. Note that this makes sense as .kG/Q .kG/R . We collect the main properties of this partial order: Theorem 7.1 ([2]). Let B be a block of a finite group algebra kG. The partial order on the set of B-Brauer pairs defined above has the following properties. (i) The partial order is invariant under G-conjugation; that is, the set of B-Brauer pairs is a G-poset. (ii) Given a B-Brauer pair .Q; e/ and a subgroup R of Q there is a unique block idempotent f of kCG .R/ such that .R; f / .Q; e/.

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(iii) The maximal B-Brauer pairs, with respect to this partial order, are all G-conjugate, and if .P; e/ is a maximal B-Brauer pair then P is a defect group of B. (iv) If B is the principal block of kG and .Q; e/ a B-Brauer pair then e is the block idempotent of the principal block of kCG .Q/. Statement (iv) is a variation of what is referred to as Brauer’s Third Main Theorem. Combining the properties (ii) and (iii) of this theorem implies that once we fix a maximal B-Brauer pair .P; eP /, for every subgroup Q of P there is a unique block idempotent eQ of kCG .Q/ such that .Q; eQ / .P; eP /. In other words, the partially ordered set of B-Brauer pairs contained in .P; eP / is isomorphic to the partially ordered set of subgroups of P . This observation is used in the definition of fusion systems of blocks. Definition 7.2. Let B be a block of a finite group algebra kG, fix a maximal B-Brauer pair .P; eP / and, for any subgroup Q of P denote by eQ the unique block idempotent of kCG .Q/ satisfying .Q; eQ / .P; eP /. The fusion system F D FP .B/ of B on P is the category defined as follows. The objects of F are the subgroups of P . For any two subgroups Q, R of P the morphism set HomF .Q; R/ consists of all group homomorphisms ' W Q ! R for which there exists an element x 2 G satisfying

'.u/ D x 1 ux for all u 2 Q; in particular, Qx R; .Q; eQ /x .R; eR /.

The fusion system FP .B/ depends on P and eP , but since the maximal B-Brauer pairs are G-conjugate, eP , and hence FP .B/, is unique up to conjugation by elements in NG .P /. The first of the two conditions on morphisms implies that FP .B/ is a subcategory of FS .G/, where S is a Sylow p-subgroup of G containing P . The second condition is equivalent to x 1 eQ x D ex 1 Qx , by the uniqueness property of the partial order on B-Brauer pairs. For instance, we have Aut F .Q/ Š NG .Q; e/=CG .Q/, for any subgroup Q of P . An immediate consequence of statement (iv) in 7.1 is the following: Corollary 7.3. Let B be the principal block of a finite group algebra kG and let P be a Sylow p-subgroup of G (hence a defect group of B). Then FP .B/ D FP .G/. This shows that every fusion system of a finite group is also a fusion system of some block. It is sometimes useful to consider the related orbit category Fx ; this is the category having the same objects as F (that is, the subgroups of P ), and the morphism sets in Fx are obtained from those by factoring out inner automorphisms; that is, HomFx .Q; R/ D Inn.R/n HomF .Q; R/ for any two subgroups Q, R of P . For instance, we have AutFx .Q/ Š NG .Q; e/=QCG .Q/ for any subgroup Q of P . The group E D AutFx .P / D NG .P; eP /=P CG .P /


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is called inertial quotient of the block B, and this is known to be a p 0 -group. Prior to the work of Alperin and Broué in [2], Brauer had considered special cases of Brauer pairs, called centric – these are precisely the B-Brauer pairs .Q; e/ such that Z.Q/ is a defect group of the block kCG .Q/e. The property of being centric can be characterised within F : the Brauer pair .Q; e/ is centric if and only if every subgroup R of P which is isomorphic to Q in F satisfies CP .R/ D Z.R/. The subgroups of P obtained in this way are called F -centric, and the full subcategory of F -centric subgroups of P is denoted by F c . One can show that this category is ‘upwardly closed’; that is, if Q R P and Q is F -centric then so is R. As a consequence of work of Alperin and Broué [2], the categories FP .G/, for P a Sylow p-subgroup of a finite group G, and FP .B/, for P a defect group of a block B, enjoy strikingly similar properties. Puig introduced in the early 1990s an axiomatic version of abstract fusion systems, frequently now called saturated fusion systems, on finite p-groups. We skip the precise definition here and point to the rapidly growing literature on the subject, both from a group theoretic and a homotopy theoretic perspective. See for instance [19], [20], [25], [38], [39], [40], [41], [83], [78], [79], [80], [105], [108], [115], [127], [128], [137], [143]. There are ‘exotic’ abstract fusion systems which do not arise as fusion system of a finite group – the Solomon fusion systems [94] are a case in point, as are the Ruiz–Viruel systems in [137]. These cannot show up as fusion systems of blocks, either, by results of Kessar [75], and Kessar– Stancu [84], providing evidence that the following question should have a positive answer: Question 7.4. Is every fusion system FP .B/ of a block B of a finite group algebra kG with defect group P isomorphic to a fusion system FP .H / for some finite group H having P as a Sylow p-subgroup? Equivalently, is no fusion system of a block exotic? There are some cases where this is known to be true: Examples 7.5. Let G be a finite group.

Let B be the principal block of kG and P a defect group of B, hence a Sylow p-subgroup of G. By 7.3 we have FP .B/ D FP .G/. In particular, every fusion system of a finite group is a fusion system of a block. Let B be a block of kG with an abelian defect group P . Then there is a p 0 -subgroup E of Aut.P / such that FP .B/ D FP .P Ì E/; the group E is the inertial quotient of B mentioned above. Since E is a p 0 -group, P is a Sylow p-subgroup of P Ì E, so this yields a positive answer to the above question if P is abelian. This is a consequence of work of Alperin and Broué [2]. Let B be a block algebra of kG. Suppose that G is p-solvable. By a result of Puig [120], there is a finite p-solvable group L containing P as a Sylow p-subgroup such that FP .B/ D FP .L/. More precisely, L can be chosen to have a non-trivial normal p-subgroup Q satisfying CL .Q/ D Z.Q/.

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Question 7.6. What is the fusion system of a block with a single isomorphism class of simple modules? More precisely, is the fusion system of a block B with defect group P and a single isomorphism class of simple modules isomorphic to that of a solvable finite group having P as a Sylow p-subgroup? One reason for asking this question is that the known constructions of blocks with a single isomorphism class of simple modules tend to involve finite groups of central type – and those are solvable, as mentioned in one of the examples in 5.5. Many group theoretic results involving fusion systems carry over to blocks and abstract fusion systems – see the sample list of work in that direction above. We describe one case – which is one of the corner stones of block theory – in more detail: the theory of nilpotent blocks, due to Broué and Puig. If P is a Sylow p-subgroup of a finite group G, then for any subgroup H of G containing P , the fusion system FP .H / is a subcategory of FP .G/. If FP .H / D FP .G/ we say that H controls G-fusion in P . In particular, FP .G/ contains the ‘trivial’ fusion system FP .P /, induced by the inner automorphisms of P . By a theorem of Frobenius, if FP .G/ D FP .P / then G D K ÌP for some normal p 0 -subgroup K of G; that is, G is p-nilpotent. Frobenius’ theorem is a case where the p-local structure of G determines the structure of G to the extent that this is possible with p-local information (it does not specify anything about the structure of the p 0 -group K or the action of P on K). Broué and Puig introduced in [25] the block theoretic analogue: Definition 7.7. A block B of a finite group algebra is called nilpotent if for some (and hence any) choice of a maximal B-Brauer pair .P; e/ we have FP .B/ D FP .P /, in other words, if B has a trivial fusion system. The source algebras of nilpotent blocks are as follows: Theorem 7.8 (Puig [123]). Let B be a nilpotent block of a finite group algebra B. The source algebras of B are of the form S ˝k kP , where P is a defect group of B and S D Endk .V / for some indecomposable endo-permutation kP -module V having P as a vertex. In particular, mod.B/ Š mod.kP /. Puig has shown further a converse to the last statement, namely that nilpotent blocks are precisely the blocks which are Morita equivalent to a finite p-group algebra. An elegant proof of the above theorem has been given by Külshammer, Okuyama and Watanabe in [92]. There is a different approach to fusion systems of finite groups and of blocks, using the bimodule structure of group algebras and source algebras. We motivate this by first describing the group case. Let P be a Sylow p-subgroup of a finite group G, and let Q, R be subgroups of P . Write GD

[ x2ŒQnG=R

QxR


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as disjoint union of its Q-R-double cosets, where ŒQnG=R is a set of representatives in G of said double cosets. If x 1 Qx R then QxR D xR is a Q-R-double coset with jRj elements and corresponds to the group homomorphism W Q ! R induced by conjugation with x 1 , hence is a morphism in FP .G/, and every morphism in FP .G/ arises that way. In other words, the category FP .G/ can be read off the P -P -biset structure of G. This idea can be carried over to blocks. Let B a block of kG with a defect group P . Instead of considering bisets, we consider bimodules, using arguments similar to those in the proof of 3.3: for Q, R subgroups of P we have a direct sum decomposition L kG D x2ŒQnG=R kŒQxR Note that each kŒQxR is indecomposable as a kQ-kR-bimodule. Now B is a direct summand of kG as a kG-kG-bimodule, hence as a kQ-kR-bimodule, and thus we have an isomorphism L B Š x2 kŒQxR of kQ-kR-bimodules, for some subset of ŒQnG=R, thanks to the Krull–Schmidt theorem. It is tempting to try and define FP .B/ by taking for morphisms the group homomorphism ' W Q ! R for which there is an element x 2 satisfying QxR D xR and '.u/ D x 1 ux, for all u 2 Q. One rapidly sees that this yields categories which are ‘too big’ in that they may contain information which relates to the group G but which goes beyond what should be considered as a structural part of the block B. This is where one needs to replace the algebra B by a potentially much smaller source algebra. Let i be a source idempotent in B P ; that is, i is a primitive idempotent in B P such that B is isomorphic to a direct summand of Bi ˝kP iB as a B-B-bimodule. Since i is an idempotent which commutes with all elements of P , hence of Q and R as above, the source algebra iBi is a direct summand of B, as a kQ-kR-bimodule. Thus, the Krull–Schmidt theorem implies that there is an isomorphism of kQ-kR-bimodules L iBi Š x2T kŒQxR for some subset T of . And this is what we use to define the fusion system FP .B/ of the block B on its defect group P . As before, the objects of FP .B/ are the subgroups of P . If x 2 T such that QxR D xR, then the group homomorphism ' W Q ! R defined by '.u/ D x 1 ux for u 2 Q is a morphism in HomFP .B/ .Q; R/. In order to make sure that this yields a category, we need to impose in addition that FP .B/ contains any composition and any restriction to a subgroup of morphisms obtained in this way. It is follows from results in [122] that this definition of FP .B/ coincides with that given in 7.2 above.

8 Conjectures The prominent conjectures in block theory can be divided into three classes: finiteness conjectures, which state that once a defect group is fixed there should be only finitely

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many ‘different ’ block algebras with that defect group; counting conjectures, which express certain numerical invariants of a block algebra in terms of invariants associated with the defect groups, and structural conjectures, trying to predict the structure of block algebras, their module and derived module categories in terms of ‘local’ information. The line between the latter two classes of conjectures is fluid in that, whenever possible, one seeks to give structural explanations for the numerical coincidences observed. Finiteness conjectures Conjecture 8.1 (Donovan). For a fixed finite p-group P there are only finitely many Morita equivalence classes of block algebras of finite groups with defect groups isomorphic to P . Donovan’s conjecture is known to hold for the following p-groups:

P cyclic (Brauer [14], Dade [33], Janusz [65], Kupisch [93]); p D 2 and kP of tame representation type (that is, P is generalised dihedral, semi-dihedral or quaternion) in almost all cases (Erdmann [44]); finite p-groups which admit only the trivial fusion system (as a consequence of Broué–Puig), such as C2 C4 , for p D 2.

One of the difficulties of classifying blocks of tame representation type up to Morita equivalence, in terms of quivers and relations, lies in gaining control over certain scalars in socle relations. Some of those scalars in [44] have been determined by Holm in [61], at least up to finitely many possibilities (which is sufficient for the purpose of Donovan’s conjecture) but there is still one open case, namely where P is generalised quaternion and B has two isomorphism classes of simple modules. Donovan’s conjecture is also known in some cases where one restricts attention to blocks of certain classes of finite groups G; that is, for a fixed finite p-group P , if G runs over any of the classes below then there are only finitely many Morita equivalence classes of blocks of kG with defect groups isomorphic to P :

blocks of finite p-solvable groups (Külshammer [89]); blocks of symmetric groups (Scopes [140]); blocks of alternating groups (Hiss [55]); blocks of Weyl groups of type B and D (Kessar [71]); blocks of double covers of symmetric and alternating groups (Kessar [70]); unipotent blocks of general linear groups GLn .q/ for a fixed prime power q (Jost [66]); unipotent blocks of classical groups in the unitary prime case (Hiss–Kessar [56], [57]); principal 3-blocks with an abelian defect group (Koshitani [87]).


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The last item in this list extends earlier work of Koshitani and Miyachi [88] on principal 3-blocks with an elementary abelian defect group of order 9; unlike the other items in this list, this requires the classification of finite simple groups. One of the philosophical reasons for Donovan’s conjecture is that there aren’t that many finite simple groups. Besides the 26 sporadic simple groups, the finite simple groups of Lie type of a given Dynkin type are expected to behave ‘generically’ in the same way. This would suggest that using the classification of finite simple groups might provide an access route to Donovan’s conjecture. It is not known, however, whether Donovan’s conjecture can actually be reduced to the classification in general. There is a weaker form of Donovan’s conjecture, predicting that there are only finitely many Cartan matrices of blocks with defect group isomorphic to a fixed finite p-group. This weaker version has been reduced to blocks of quasi-simple finite groups by Düvel [42]. Kessar showed in [74] that the ‘gap’ between the weak form and the original form of Donovan’s conjecture can be formulated as a rationality conjecture. If is an automorphism of the field k then extends to a ringP automorphism,Pabusively still denoted by , of the finite group algebra kG, sending x2G x x to x2G . x /x. If B is a block algebra of kG then so is its image .B/ under this ring automorphism. But unless is trivial, it is not a k-algebra automorphism of kG (precisely because the coefficients get ‘twisted’ by ). Thus, although B and .B/ are isomorphic as rings, they may not be isomorphic as k-algebras – they need not even be Morita equivalent. We denote by m.G; B/ the number of pairwise Morita inequivalent blocks of kG of the form d .B/, where is the Frobenius automorphism 7! p of k and where d runs over the set of non negative integers . Quantum complete intersections provided the first example of ring isomorphic but not Morita equivalent block algebras over k, due to Benson and Kessar [11]. Conjecture 8.2 (Kessar). For any finite p-group P there is a positive integer m.P /, depending only on the isomorphism class of P , such that for any finite group G and any block B of kG with defect groups isomorphic to P , we have m.G; B/ m.P /. It is shown in [74], Theorem 1.4, that this conjecture, together with the weak form of Donovan’s conjecture, is indeed equivalent to Donovan’s conjecture 8.1. Kessar’s conjecture holds trivially for principal blocks, and hence Donovan’s conjecture for principal blocks is indeed reduced to principal blocks of finite simple groups by Düvel’s results in [42]. Since a source algebra of a block B is Morita equivalent to B, the following conjecture, stated by Puig in 1982, would imply Donovan’s conjecture: Conjecture 8.3 (Puig). Let P be a finite p-group. Up to isomorphism of interior P -algebras, there are only finitely many source algebras of blocks with defect groups isomorphic to P . ‘Up to isomorphism of interior P -algebras’ means that if A, A0 are source algebras of blocks with defect groups Q, Q0 isomorphic to P , we have an algebra isomorphism A Š A0 mapping the image of Q in A to the image of Q0 in A0 . Equivalently, for some

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identifications P Š Q Š Q0 the isomorphism A Š A0 preserves the images of P in these algebras elementwise. Puig’s conjecture is known to hold in the following cases:

P cyclic (Linckelmann [98]); p D 2 and P Š C2 C2 a Klein four group (Craven, Eaton, Kessar and Linckelmann [32]).

The proof of Puig’s conjecture for P a Klein four group in [32] requires the classification of finite simple groups – the main problem is to show that a block with a Klein four defect group has at least one algebraic simple module (in the sense of Alperin). It was previously known, as a consequence of work of Erdmann, that at least some Heller translate of a simple module would be algebraic – and this led to the classification of the source algebras of blocks with a Klein four defect group ‘up to a Heller translate’in [96], as mentioned in example 6.6. As in the case of Donovan’s conjecture, Puig’s conjecture is known to hold for blocks of certain classes of groups with a given defect group:

blocks of symmetric groups (Puig [124]); blocks of alternating groups (Kessar [73]); blocks of Weyl groups of type B and D (Kessar [71]); blocks of double covers of symmetric groups (Kessar [70]); unipotent blocks of series of finite general linear groups over a fixed finite field (Kessar [72]); unipotent blocks of classical groups in the unitary prime case (Hiss–Kessar [56], [57]); principal 3-blocks with an elementary abelian defect group of order 9 (Koshitani [87]).

As before, the last item on this list requires the classification of finite simple groups. Counting conjectures. Counting conjectures attempt to predict numerical invariants of a block algebras in terms of local information – that is, in terms of invariants associated with the defect groups and fusion systems. One of the most prominent counting conjectures is Alperin’s weight conjecture, announced in [1], of which we mentioned the group theoretic version earlier in 2.7. In order to describe the general block theoretic version we need the following notation. Given a block algebra B of kG, we denote, for any p-subgroup Q of G, by BQ the (possibly empty) product of all block algebras of kNG .Q/ which are isomorphic to a direct summand of B as a kNG .Q/kNG .Q/-bimodule. We then denote by BxQ the image of BQ in kNG .Q/=Q under the canonical algebra homomorphism kNG .Q/ ! kNG .Q/=Q. This is again a (possibly empty) product of block algebras of kNG .Q/=Q. Conjecture 8.4 (Alperin’s weight conjecture, block Ptheoretic version). For any finite group G and any block B of kG we have `.B/ D Q `0 .BxQ /, where Q runs over a set of representatives of the G-conjugacy classes of p-subgroups of G.


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Unlike in the group theoretic version, the term for Q D 1 in this sum is zero whenever P has a non-trivial defect group. The term for Q D P is equal to `.C /, where C is the Brauer correspondent of B. If B has an abelian defect group P one can show that this is the only non-zero term, and hence, for blocks with abelian defect groups, Alperin’s weight conjecture is equivalent to `.B/ D `.C / The block theoretic version of Alperin’s weight conjecture implies the group theoretic version stated earlier in 2.7 but it is not known whether the converse implication holds. Alperin’s weight conjecture holds in the following cases:

blocks with cyclic defect groups (Dade [33]); blocks with Klein four, generalised dihedral, quaternion, semi-dihedral defect groups (Brauer [16], [17], Olsson [116]); nilpotent blocks (Broué–Puig [25]); the remaining cases of 2-blocks with metacyclic 2-groups and certain minimal abelian 2-groups as defect groups (Sambale [138]); blocks of symmetric groups and general linear groups (Alperin [1], Alperin and Fong [3], An [4]); blocks of finite groups of Lie type in defining characteristic (Cabanes); blocks of finite p-solvable groups (Okuyama); blocks of some sporadic groups (An); blocks with an elementary abelian defect group of order p 2 with a single isomorphism class of simple modules (Kessar–Linckelmann [81]); the proof uses a stable equivalence due to Rouquier [135] and classical isometry arguments for character groups; 2-blocks with an elementary abelian defect group of order 8 (Kessar, Koshitani and Linckelmann [77]); the proof uses the classification of finite simple groups.

Alperin’s weight conjecture can be reformulated in a slightly different way, closer to the concept of fusion systems of blocks. For .Q; e/ a B-Brauer pair, denote by eN the image of e in kCG .Q/=Z.Q/. Modulo the identification CG .Q/=Z.Q/ Š QCG .Q/=Q, this is a central idempotent in kNG .Q; e/=Q, hence determines a product of blocks of kNG .Q; e/=Q. Alperin’s weight conjecture is equivalent to X `.B/ D `0 .kNG .Q; e/=QeN .Q;e/

where now .Q; e/ runs over a set of representatives of the G-conjugacy classes of B-Brauer pairs. By a result of Knörr [85], only summands with .Q; e/ centric contribute to this sum. By work of Külshammer and Puig [91], the algebra kNG .Q; e/=QeN is Morita equivalent to a twisted group algebra k˛.Q/ AutFx .Q/, where Aut Fx .Q/ D

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NG .Q; e/=QCG .Q/; this is the automorphism group of Q in the orbit category Fx of the fusion system F of B, where we now assume that we have made a choice of a maximal B-Brauer pair .P; eP /. One of the open problems in this context is the gluing problem ([103], Conjecture 4.2): is there a class ˛ in H 2 .F c I k / whose restriction to Aut F .Q/ yields the class ˛.Q/ (inflated to Aut F .Q/ via the canonical map AutF .Q/ ! AutFx .Q/), for any F -centric subgroup Q of P ? It has been shown by Park [119] that even if ˛ exists it need not be unique. Knörr and Robinson [86] reformulated Alperin’s weight conjecture as an alternating sum of the form X dimk .Z.B// D .1/jj dimk .Z.B //

where runs over a set of representatives of the G-conjugacy classes of chains of non-trivial p-subgroups D Q0 < Q1 < < Qn , where j j D n, and where B is a product of blocks of kNG . / related to B as above. As before, one can rewrite this alternating sum in a slightly different way, so that the summation is over chains of B-Brauer pairs rather than p-subgroups. This reformulation is equivalent to Alperin’s weight conjecture in the sense that if one holds for all blocks then so does the other – but it is not clear whether the two equalities are equivalent for a fixed block. Note that this reformulation counts dimensions of centers, or equivalently, numbers of ordinary irreducible characters, rather than isomorphism classes of simple modules. This reformulation by Knörr and Robinson has paved the way for refinements and further reformulations of Alperin’s weight conjecture, due to Dade [34], [35], and Robinson [132] – with a growing number of cases being proved by many authors including An, Eaton, Okuyama, O’Brien, Olsson, Satoshi, Sukizaki, Uno, Wilson, to name a few. It also led to a structural interpretation, due to Symonds [144], of the group theoretic version of Alperin’s weight conjecture in terms of Bredon cohomology, which then has been extended to the block theoretic version in [104] – modulo a hypothetical solution of the gluing problem (Conjecture 4.2) in [103] mentioned above. Conjecture 8.5 (McKay). If G is a finite group with Sylow p-subgroup P then G and NG .P / have the same number of irreducible characters of degree prime to p. This conjecture is known to hold for instance if G is p-solvable (Wolf, Dade, Okuyama, Wajima), symmetric (Fong), sporadic simple (Wilson), GLn .q/ with p dividing q (Fong), or if G has cyclic Sylow p-subgroups (as a consequence of Dade’s results on blocks with cyclic defect groups). The McKay conjecture has a blocktheoretic version, known as the Alperin–McKay conjecture. To state it, we need the following concept. If B is a block algebra of a finite group algebra kG then one can show that (essentially as a consequence of 4.15) the degree .1/ of every irreducible character belonging to the block B is divisible by p ad , where p a is the order of a Sylow p-subgroup of G and p d the order of a defect group of B. We say that has height zero if p ad is the exact power of p dividing .1/. By a result of Brauer, to every block of kG belongs at least one irreducible character of height zero.


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Conjecture 8.6 (Alperin–McKay). Let G be a finite group and B a block of G with defect group P . Let C be the block of NG .P / corresponding to B. Then B and C have the same number of irreducible characters of height zero. The McKay conjecture follows from the Alperin–McKay conjecture by running over all blocks of kG having a Sylow p-subgroup as a defect group. This conjecture has been further refined by Isaacs and Navarro [64], taking into account certain congruences of character degrees. Another refinement, due to Navarro [111], takes into account Galois actions on irreducible characters (a principle which can be applied as well to the conjectures of Alperin and Dade above). Yet another refinement due to Turull [149] takes into account fields of values of characters and local Schur indices over the p-adic number field. The Alperin–McKay conjecture (and in many cases some of its refinements) holds for blocks with cyclic defect groups (as before a consequence of Dade’s work), blocks of p-solvable groups, blocks of symmetric groups and their covers (Fong, Olsson, Michler), blocks with a TI defect group (An, Eaton), certain finite groups of Lie type (Lehrer, Michler, Olsson, Späth). Uno [150] has formulated a generalisations of the McKay conjecture, combining conjectures of Dade and Isaacs– Navarro. See also [5], §2. Structural conjectures Conjecture 8.7 (Broué’s Abelian Defect Group Conjecture, [21], 6.1). Let B be a block of a finite group algebra kG with an abelian defect group P . Denote by C the block of kNG .P / which is the Brauer correspondent of B. Then there is an equivalence of bounded derived categories D b .mod.B// Š D b .mod.C //. Since the number of isomorphism classes of simple modules is invariant under derived equivalences, this conjecture would imply `.B/ D `.C /; in other words, for blocks with abelian defect groups, Alperin’s weight conjecture would be a consequence of Broués Abelian Defect Group Conjecture. This conjecture has been strengthened in various ways such as for block algebras over a complete discrete valuation ring, for instance. At the level of characters, a derived equivalence between block algebras over O induces a perfect isometry between the blocks; see [21], [22]. It is expected that such a derived equivalence can be realised by a ‘splendid’ two-sided tilting complex consisting of p-permutation bimodules (see e.g. [131]). Broué’s Abelian Defect Group Conjecture is known to hold in a growing list of cases (in many cases in its ‘splendid’ version):

P cyclic; this was the first case of a derived equivalence between block algebras, due to Rickard [129], subsequently lifted over O in [95]; a splendid equivalence was constructed by Rouquier [134]; p D 2 and P Š C2 C2 a Klein four group (this follows from combining results of Rickard [131] and Erdmann’s description, up to Morita equivalence, of blocks with a Klein four defect group in [43]; over O, including the splendid version,

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this follows from combining [131] with the description in [96] of blocks with a Klein four defect group over O, lifting Erdmann’s results). As with the earlier conjectures, Broué’s Abelian Defect Group Conjecture holds for certain blocks with an abelian defect group of the following finite groups G:

all blocks with abelian defect groups of symmetric groups (Chuang–Rouquier [31], using earlier work of Chuang–Kessar [30]); all blocks with abelian defect groups of alternating groups (Marcu¸s [110], using [31] and Clifford theory [109]); 3-blocks B with defect group P Š C3 C3 , and `.C / D 1, where C is the Brauer correspondent of B (Kessar); in that case mod.B/ Š mod.C /; the proof calculates low degree Hochschild cohomology and outer automorphism groups of 9-dimensional symmetric local algebras, and yields an explicit description of the basic algebras of B as one of the quantum complete intersections khx; yi=.x 3 ; y 3 ; xy ˙ yx/; a long – and in recent years rapidly growing – list of blocks of sporadic groups, some of their central extensions, and some series of finite groups of Lie type in low ranks (Danz–Külshammer, Koshitani, Kunugi, Miyachi, Müller, Okuyama, Waki, …)

By a result of Roggenkamp and Zimmermann, two derived equivalent symmetric local algebras are in fact isomorphic. Thus Broué’s Abelian Defect Group Conjecture predicts that a block B with an abelian defect group and a unique isomorphism class of simple modules should be Morita equivalent to its Brauer correspondent C . The structure of C is well understood – in particular, if also the inertial quotient of C is abelian then C , hence B, is Morita equivalent to a quantum complete intersection. Although there is no immediately obvious generalisation of Broué’s Abelian Defect Group Conjecture to blocks with arbitrary defect groups, there are many examples of such equivalences. For instance, by [31], Theorem 7.2, any two blocks of symmetric groups with isomorphic defect groups are derived equivalent, and by a result in [97], any two block algebras with dihedral defect groups and three isomorphism classes of simple modules are derived equivalent. Holm has classified in [60] blocks of tame representation type up to derived equivalence (making use of Erdmann’s work [44]). Except for the Klein four case, we do not have explicit two-sided complexes realising the derived equivalences between tame blocks. Last but not least, Brauer conjectured the following: Conjecture 8.8 (Brauer’s height zero conjecture). Let G be a finite group, B a block of kG and P be a defect group of B. All irreducible characters belonging to B have height zero if and only if P is abelian. Since a perfect isometry preserves heights of characters (cf. [21], 1.5), Broué’s Abelian Defect Group conjecture would imply the Alperin–McKay conjecture for blocks with abelian defect groups. It would also prove one implication of Brauer’s


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height zero conjecture: if Broué’s Abelian Defect Group Conjecture holds then any irreducible character in a block B with an abelian defect group P has height zero, because every irreducible character in the Brauer correspondent C of B has height zero. For the group theoretic versions of some of the above conjectures there has recently been significant progress towards reductions to quasi-simple groups; see for instance [63], [142], [112].

9 Symmetric algebras and transfer There are few general statements one can make about block algebras – and one of them is that they are symmetric. We denote, for any finite-dimensional k-algebra A, as before by mod.A/ the category of finitely generated left A-modules. For A a finite-dimensional k-algebra, a right A-module can be viewed, if necessary, as a left module over the opposite algebra Aop ; we identify the category of finitely generated right A-modules with mod.Aop /. If U , U 0 are left A-modules, we will write HomA .U; U 0 / for the kvector space of A-homomorphisms from U to U 0 , whereas if V , V 0 are right A-modules, we will denote the space of A-homomorphisms from V to V 0 by HomAop .V; V 0 /. For A, B two finite-dimensional k-algebras, an A-B-bimodule is assumed to have the same k-vector space structure induced by the left action of A and the right action of B; equivalently, an A-B-bimodule is the same as an A ˝k B op -module. There are three – possibly non isomorphic – duality functors from the category of finitely generated A-B-bimodules mod.A ˝k B op / to the category of finitely generated B-A-bimodules mod.B ˝k Aop /, defined as follows: for M a finitely generated A-B-bimodule, the k-dual M D Homk .M; k/ becomes a B-A-bimodule via .b a/.m/ D .amb/, where 2 M , m 2 M , a 2 A and b 2 B. Since we restrict attention to finitely generated modules, we have a canonical isomorphism M Š M ; that is, applying the k-duality functor Homk .; k/ is a contravariant equivalence with inverse equivalence given by applying k-duality again. The A-duality functor HomA .; A/ sends M to the B-A-bimodule HomA .M; A/, with bimodule structure given by .b ' a/.m/ D '.mb/a, where ' 2 HomA .M; A/, m 2 M , a 2 A and b 2 B. This is also a contravariant functor but not necessarily an equivalence because it need not be exact (it is exact if and only if A is injective as an A-module, hence if and only if A is a self-injective algebra). Similarly, the B-duality functor HomB op .; B/ sends M to the B-A-bimodule HomB op .M; B/, with bimodule structure given by .b a/.m/ D b .am/, where 2 HomB op .M; B/, m 2 M , a 2 A and b 2 B. Again this functor need not be an equivalence since it is not exact unless B is self-injective. One of the main specialties of symmetric algebras is that all three duality functors are isomorphic. Definition 9.1. A finite-dimensional k-algebra is symmetric if A Š A as A-A-bimodules.

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We have not specified any isomorphism A Š A in particular, so for uniqueness considerations, it will be useful to know what the set of all such isomorphisms is. If ˛, ˇ are two bimodule isomorphisms from A to A then ˇ 1 B ˛ is an automorphism of A as an A-A-bimodule, hence induced by left or right multiplication on A with an invertible element in Z.A/. In other words, ˛ and ˇ ‘differ’ at most by multiplication by an element in Z.A/ . Since 1A generates A as a left and right A-module and since a 1A D 1A a for any a 2 A, the image s D ˛.1A / 2 A under a bimodule isomorphism ˛ W A Š A generates A as a left and right A-module, and we have a s D s a for all a 2 A, which is equivalent to s.ab/ D s.ba/ a for all a, b 2 A. Any such s is called a symmetrising form of A. Choosing a bimodule isomorphism A Š A is equivalent to choosing a symmetrising form of A. If s D ˛.1A / and t D ˇ.1A / are two symmetrising forms, the above considerations imply that t D z s for a uniquely determined z 2 Z.A/ , or equivalently, t .a/ D s.az/ for all a 2 A. Examples 9.2. If G is a finite group then kG is symmetric. More precisely, there is a canonical isomorphism .kG/ Š kG sending 2 .kG/ to the element P bimodule 1 /x in kG. The symmetrising form of kG corresponding to this isox2G .x morphism is the linear map s W kG ! k defined by s.1G / D 1k and s.x/ D 0 for x 2 G f1g. Any matrix algebra Mn .k/ is symmetric, with symmetrising form the trace map tr W Mn .k/ ! k, sending a matrix to the sum of its diagonal entries. Hecke algebras of finite Coxeter groups are symmetric. Finite direct products and tensor products of symmetric algebras are symmetric, direct factors of symmetric algebras are symmetric, and any algebra Morita equivalent to a symmetric algebra is symmetric. Since the k-dual U of a finitely generated projective A-module is an injective Aop -module and vice versa, a symmetric k-algebra is in particular self-injective – that is, the classes of projective and injective modules coincide. In particular, every projective indecomposable module U over a symmetric k-algebra A is also injective indecomposable, hence has a simple socle – and being symmetric implies moreover that soc.U / Š U=rad.U /. Proposition 9.3. A finite-dimensional k-algebra A is symmetric if and only if there is an isomorphism of duality functors HomA .; A/ Š Homk .; k/ from mod.A/ to mod.Aop /. Proof. Suppose A is symmetric. Choose a symmetrising form s W A ! k. This induces a natural transformation of functors HomA .; A/ ! Homk .; k/ sending an A-homomorphism ' W U ! A to the k-linear map s B ' W U ! k, for any finitely generated A-module U . This natural transformation is clearly k-linear. This is in fact an isomorphism of functors: for U D A this is an isomorphism because A is symmetric, hence by k-linearity and naturality, applied first to direct sums of copies of A and then to direct summands thereof, this is an isomorphism for any finitely generated projective


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A-module U . Both functors are exact (for k-duality this is trivial and for A-duality this holds as A is self-injective), hence, the natural transformation applied to a projective presentation of an A-module, yields isomorphisms for all finitely generated A-modules. Conversely, if there is an isomorphism of functors HomA .; A/ Š Homk .; k/, evaluating this isomorphism at A yields an isomorphism HomA .A; A/ Š A . By naturality, this is an isomorphism of A-A-bimodules, and the left side is, as an A-A-bimodule, isomorphic to A. This shows that A is symmetric. Corollary 9.4. Let A, B be symmetric k-algebras. We have isomorphisms of duality functors HomA .; A/ Š Homk .; k/ Š HomB op .; B/ from mod.A ˝k B op / to mod.B ˝k Aop /. Proof. The isomorphism of functors HomA .; A/ Š Homk .; k/ from mod.A/ to mod.Aop / from 9.3 induces by naturality a functor, hence an equivalence, from mod.A˝k B op / to mod.B ˝k Aop /. The second isomorphism is obtained similarly. One of the key properties of symmetric algebras is the following statement on biadjoint functors: Theorem 9.5. Let A, B be symmetric k-algebras and M an A-B-bimodule which is finitely generated projective as a left A-module and as a right B-module. Then M is finitely generated projective as a left B-module and as a right A-module. The functors M ˝B and M ˝A between mod.B/ and mod.A/ are left and right adjoint to each other, and any choice of symmetrising forms on A, B determines a choice of adjunction isomorphisms HomA .M ˝B V; U / Š HomB .V; M ˝A U / HomB .M ˝A U; V / Š HomA .U; M ˝B V / for any A-module U and any B-module V . Proof. This is based on the canonical adjunction isomorphism HomA .M ˝B V; U / Š HomB .V; HomA .M; U // for any A-module U and any B-module V , sending an A-homomorphism ' W M ˝B V ! U to the B-homomorphism W V ! HomA .M; U / defined by .v/.m/ D '.m ˝ v/, where v 2 V , m 2 M . This isomorphism, also known as ‘isomorphisme cher à Cartan’, means that M ˝A is left adjoint to HomA .M; /. Since M is finitely generated projective as a left A-module, there is a canonical isomorphism of functors HomA .M; A/ ˝A Š HomA .M; /, given, for any A-module U , by the map sending ' ˝ u to the map m 7! '.m/u, where u 2 U , m 2 M and ' 2 HomA .M; A/. By 9.4, any choice of a symmetrising form on A determines an isomorphism of functors HomA .M; A/˝A Š M ˝A , which shows that M ˝B is left adjoint to M ˝A . Exchanging the roles of M and M shows that M ˝B is also right adjoint to M ˝A , with an adjunction isomorphism determined by a choice of a symmetrising form on B.

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With the notation and assumptions of this theorem, any choice of symmetrising forms determines the units and counits of a left and right adjunction isomorphism of M ˝B and M ˝A ; that is, we get bimodule homomorphisms B ! M ˝A M;

M ˝B M ! A

representing the unit and counit of the left adjunction of M ˝B to M ˝A , as well as A ! M ˝B M ; M ˝A M ! B representing the unit and counit of the right adjunction of M ˝B to M ˝A . These are used in [100], §2, to define transfer maps trM W HH .B/ ! HH .A/; trM W HH .A/ ! HH .B/: If M and M induce a Morita equivalence, these transfer maps are ring isomorphisms; if M and M induce a stable equivalence of Morita type, they induce an isomorphism on the Tate analogue of Hochschild cohomology HH .A/ Š HH .B/, in particular, they induce isomorphisms HH n .A/ Š HH n .B/ in any positive degree n; see e.g. [100], Remark 2.13.

b

b

Example 9.6 (cf. [100], Example 2.6). Let G be a finite group and H as subgroup. We consider kG and kH as symmetric algebras endowed with their canonical symmetrising forms (sending the unit element of the group to 1k and any non-trivial group element to zero). Set M D kG, considered as kG-kH -bimodule. Then M is free as left kG-module (of rank 1) and as a right kH -module (of rank jG W H j). The adjunction isomorphisms for the functors M ˝kH and M ˝kG are known as Frobenius reciprocity. By the symmetry of kG, we have an isomorphism M Š kG, now considered as a kH -kG-bimodule. Note that this isomorphism is determined by the choice of symmetrising forms. Thus we have an isomorphism M ˝kH M Š kG ˝kH kG as kG-kG-bimodules, and an isomorphism M ˝kG M Š kG ˝kG kG Š kG as kH -kH -bimodules. Modulo identifications through these isomorphisms, the above adjunction units and counits can be calculated explicitly (the verifications are elementary and left to the reader as an exercise): for the left adjunction of M ˝kH to M ˝kG , the adjunction unit is represented by the inclusion map kH ! kG viewed as homomorphism of kH -kH -bimodules, and the corresponding counit is represented by the map kG ˝kH kG ! kG


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induced by multiplication in kG. It is worth noting that after replacing M by kG, the symmetrising forms do not seem to play a role anymore: this is related to the fact that an ‘induction functor’ of the form A ˝B from a subalgebra B of an algebra A is always left adjoint to the ‘restriction’ from A to B, via the inclusion B ! A as unit and the multiplication map A ˝B A ! A as counit (this is just a special case of the ‘isomorphisme cher à Cartan’). For the right adjunction of M ˝kH to M ˝kG , the symmetrising forms do matter. The unit is represented by the unique kG-kG-bimodule homomorphism kG ! kG ˝kH kG P 1 sending 1 to x2ŒG=H x ˝ x where ŒG=H is a set of representatives of the right H -cosets in G, and the counit is represented by the ‘projection’ kG ! kH sending x 2 H to x and x 2 G H to 0; this is clearly a homomorphism of kH -kH bimodules. The unit is what gives rise to relative trace maps, and the counit is what gives rise to the Brauer homomorphism.

10 Separably equivalent algebras Definition 10.1 ([107], 3.1). Two k-algebras A and B are called separably equivalent if there is an A-B-bimodule M which is finitely generated projective as a left A-module and as a right B-module and a B-A-bimodule N which is finitely generated projective as a left B-module and as a right A-module, such that A is isomorphic to a direct summand of M ˝B N as an A-A-bimodule and such that B is isomorphic to a direct summand of N ˝B M as a B-B-bimodule. Remarks 10.2. 1. Separable equivalence is an equivalence relation on the class of finite-dimensional k-algebras. 2. A finite-dimensional k-algebra A is separably equivalent to the field k if and only A is separable (that is, A is projective as an A ˝k Aop -module). 3. Morita equivalent algebras are separably equivalent. If A and B are separably equivalent symmetric indecomposable k-algebras then a separable equivalence can be realised by some indecomposable bimodule and its dual: Proposition 10.3 ([107], 3.2). Let A, B be indecomposable symmetric separably equivalent k-algebras. There is an indecomposable A-B-bimodule M which is finitely generated projective as a left A-module and as a right B-module such that A is isomorphic to a direct summand of M ˝B M and B is isomorphic to a direct summand of M ˝A M . In the situation of this proposition we will loosely say that the bimodule M realises a separable equivalence between A and B. The proof given in [107], 3.2, is based on some general abstract nonsense on the splitting of adjunction maps.

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Examples 10.4. If A, B are derived equivalent symmetric k-algebras then A and B are separably equivalent. Indeed, by a result of Rickard [130], a derived equivalence between two symmetric algebras given by a bounded two-sided tilting complex induces a stable equivalence of Morita type (cf. Broué [23]) which is trivially a separable equivalence. If G is a finite group and B a block algebra of kG with a defect group P , then B and kP are separably equivalent. More precisely, we can take as bimodule realising a separable equivalence the algebra B itself, here viewed as a B-kP -bimodule. Since B is symmetric, its k-dual is isomorphic to B again, now viewed as a kP -Bbimodule. By the definition of defect groups, B is isomorphic to a direct summand of B ˝kP B, and we noted in 3.3 that kP is isomorphic to a direct summand of B Š B ˝B B as a kP -kP -bimodule. The fact that any two block algebras with a common defect group are separably equivalent suggests that separable equivalence is not a very strong connection between two algebras – but it is just strong enough for preserving the representation type and various cohomological invariants: Proposition 10.5 ([107], 3.5). If A, B are separably equivalent symmetric k-algebras then A and B have the same representation type. This is proved using a result of Erdmann and Nakano in [46]. The stable category stmod.A/ and the derived bounded category D b .mod.A// of finitely generated modules over a symmetric k-algebra are finite-dimensional as triangulated categories, in the sense of Rouquier [136], Definition 3.6. These dimensions are invariant under separable equivalences: Proposition 10.6 ([107], 3.6, 3.7). Let A, B be separably equivalent symmetric kalgebras. We have dim.stmod.A// D dim.stmod.B// and dim.D b .mod.A/// D dim.D b .mod.B///. As noted above, block algebras are always separably equivalent to some finite pgroup algebra. This is not the case for arbitrary symmetric algebras – so this is one of the few general properties distinguishing block algebras. Example 10.7. Suppose that p is odd, and set A D kŒx=.x 2 /. Thus A is a twodimensional local symmetric algebra; it has exactly two isomorphism classes of indecomposable modules represented by the regular A-module, denoted again A, and the one-dimensional A-module, denoted by k. In order to show that A is not separably equivalent to a finite p-group algebra we argue by contradiction. Let P be a finite p-group such that kP and A are separably equivalent. Then P is cyclic as kP must have finite representation type by 10.5. Let V be a 2-dimensional kP -module; that is, we have a non-split exact sequence of kP -modules 0

/k

/V

/k

/ 0:


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Note that V is not projective since p is odd. Suppose the A-kP -bimodule M realises a separable equivalence between A and kP . Applying M ˝kP to the previous exact sequence yields an exact sequence of A-modules, which, when stripped off all projective summands occurring at its left or right end, will take the form 0

/W

/X

/ W0

/0

where W , W 0 are direct sums of one-dimensional A-modules. This sequence is not split because applying M ˝A yields an exact sequence which has as a non-split direct summand the sequence we started out with, thanks to the fact that kP is a direct summand of M ˝A M as a kP -kP -bimodule. Thus X must have a summand isomorphic to A. It is an easy exercise to see that this sequence is the direct sum of an exact sequence involving only direct sums of one-dimensional modules and an exact sequence whose middle term is projective. But then applying M ˝A yields an exact sequence which is a direct sum of a split exact sequence and an exact sequence with a projective middle term, in contradiction to the fact that the sequence we started out with has no projective summands. This shows that A cannot be separably equivalent to a finite p-group algebra. As far as cohomological invariants go, separable equivalence preserves under suitable hypotheses the Krull dimension of Hochschild cohomology – by which we mean the Krull dimension of the even part of the Hochschild cohomology if p is odd (which is commutative since Hochschild cohomology is graded commutative, by a result of Gerstenhaber). This is one of the main motivations for considering the notion of separably equivalent algebras. Theorem 10.8 ([107], 4.1). Let A, B be separably equivalent symmetric k-algebras. Then ExtA˝k A0 .A; U / is Noetherian as an HH .A/-module for any finitely generated A-A-bimodule U if and only if ExtB˝k B 0 .B; V / is Noetherian as an HH .B/-module for any finitely generated B-B-bimodule V . In that case, the Krull dimensions of HH .A/ and of HH .B/ are equal. Applied to a block algebra B with defect group P this yields a proof of the wellknown fact that the Krull dimension of HH .B/ is equal to the p-rank of P (that is, the rank of an elementary abelian subgroup of P of maximal order). Other applications include the following result on finite generation of Hochschild cohomology of Hecke algebras: Theorem 10.9 ([107], 1.1). Let H be a Hecke algebra of a finite Coxeter group .W; S / over a field K of characteristic zero with non-zero parameter q in K. Suppose that all irreducible components of W are of type A, B, D, and suppose in addition that if W involves a component of type B or D then the order of q in k is not even. Then, for any finitely generated H -H -bimodule M , the HH .H /-module Ext H ˝ H 0 .H ; M / k is Noetherian. In particular, HH .H / is finitely generated as a K-algebra.

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The defect group algebras of two Morita or even derived equivalent block algebras are separably equivalent, by the above examples, hence have defect groups of the same p-rank. They also have the same order – and hence if one of them has elementary abelian defect groups, so has the other. Question 10.10. Let P , Q be finite p-groups with separably equivalent group algebras kP , kQ. Does this imply P Š Q? There is little evidence to back this up at this point, but a positive answer to this question would in particular yield a positive answer to 3.10, showing that block algebras which are stably equivalent of Morita type have isomorphic defect groups. The charm of phrasing it this way is that, although a positive answer would have strong implications in block theory, it is a question entirely within the universe of local Noetherian rings, requiring no block theory. Even assuming P , Q abelian, this remains an interesting and open problem. Perhaps more realistically, one could ask for which finite p-groups P does 10.10 have a positive answer. And as in the case of 3.10, it may be wise to replace k by the ring O mentioned earlier, in order to circumvent the modular isomorphism problem. Remarks 10.11. 1. The notion of separable equivalence makes sense over arbitrary commutative base rings; in particular, it makes sense for symmetric algebras over a complete discrete valuation ring O having k as residue field and a quotient field K of characteristic zero. If A, B are separably equivalent O-algebras then the k-algebras k ˝O A, k ˝O B are separably equivalent, and the K-algebras K ˝O A, K ˝O B are separably equivalent. In that situation, K ˝O A is a separable K-algebra if and only if K ˝O B is so, too. Block algebras of finite group algebras over O have this property. 2. There is an obvious generalisation of the notion of separable equivalence, with M , N as in 10.1 replaced by bounded complexes X , Y of bimodules whose restrictions to A and B are finitely generated projective. For symmetric algebras this makes no difference: one observes first that, as in 10.3, one can choose Y Š X , and second that one can replace X , X by their canonical images M , M , in the appropriate stable categories of bimodules under Rickard’s canonical functors describing stable categories as Verdier quotients of derived module categories. 3. There is another obvious generalisation of this concept to additive (in particular triangulated) categories: two additive categories C , D are separably equivalent if there are additive functors F W C ! D, G W D ! C such that IdC is a direct summand of G B F and IdD is a direct summand of F B G ; if the categories C , G are abelian or triangulated, the functors F , G are in addition required to be exact in the appropriate sense.

11 Block cohomology By a classical result of Cartan and Eilenberg, the cohomology algebra H .GI k/ is isomorphic to a certain subalgebra of the cohomology H .P I k/ of a Sylow p-subgroup


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P of G; this subalgebra consists precisely of those 2 H .P I k/ which are stable with respect to G-fusion in P ; that is, which have the property that for any subgroup Q of P and any element x 2 G such that x 1 Qx P , the element ResP Q . / is mapped to the P 1 element Resx 1 Qx . / under the isomorphism H .QI k/ Š H .x QxI k/ induced by conjugation with x. This is equivalent to the following statement: Theorem 11.1 ([28], Theorem 10.1). Let G be a finite group, P a Sylow p-subgroup of G and denote by FP .G/ the fusion system of G on P . The restriction map ResPG induces an isomorphism of graded k-algebras H .GI k/ Š lim H .QI k/: FP .G/

Here H .QI k/ is understood to be the contravariant functor on FP .G/ sending a subgroup Q of P to the graded k-algebra H .QI k/. By a theorem of Evens and Venkov, the graded commutative algebra H .GI k/ is finitely generated, hence its maximal ideal spectrum is an affine variety, denoted VG .k/. One of the milestones towards understanding the structure of this variety is Quillen’s work in [125], [126]. Theorem 11.2 (Quillen [126], Stratification Theorem 10.2)). Let G be a finite group, P a Sylow p-subgroup of G and denote by EP .G/ the full subcategory of the fusion system FP .G/ consisting of all elementary abelian subgroups of P . The graded algebra homomorphism qG W H .GI k/ ! lim H .EI k/ EP .G/

induced by the product of the restriction maps ResG E , with E running over the elementary abelian subgroups of P , is an F -isomorphism; that is,

ker.qG / is a nilpotent ideal in H .GI k/, and Im.qG / contains a p a -th power of every element in limEP .G/ H .EI k/, for some non negative integer a.

In terms of varieties this means that the map induced by qG from the maximal ideal spectrum of limEP .G/ H .EI k/ to the maximal ideal spectrum VG .k/ of H .GI k/ is a morphism of varieties and a homeomorphism, but the inverse homeomorphism need not be a morphism of varieties. It implies that VG .k/ can be written as a disjoint union [ C VG;E VG .k/ D E C of locally closed subvarieties VG;E , indexed by a set of representatives of the Gconjugacy classes of elementary abelian subgroups of P . This is known as the Quillen stratification. This has been generalised by Avrunin and Scott [6] to Carlson’s cohomology varieties of modules. For a finitely generated module M over a finite group

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algebra kG, define the cohomology variety of M , denoted VG .M /, as the maximal ideal spectrum of the quotient H .G/=.ker. W H .G/ ! ExtkG .M; M // where is the algebra homomorphism induced by ˝k M . Then, by [6], the variety VG .M / can be written as a disjoint union [ C VG .M / D VG;E .M / E C .M /, as before indexed by a set of representatives of locally closed subvarieties VG;E of the G-conjugacy classes of elementary abelian subgroups of P . In other words, the variety VG .M / has a Quillen type stratification, for any finitely generated kGmodule M , which coincides with Quillen’s original stratification if M is the trivial kG-module. See §5 in [7] for an exposition of this material. Although blocks need not have an augmentation (because the trivial module belongs to the principal block), the description of Cartan–Eilenberg in terms of stable elements carries over to blocks by simply replacing the fusion system of G by that of a block.

Definition 11.3 ([100], Definition 5.1). Let G be a finite group, B a block algebra of kG with defect group P and fusion system F . Set H .B/ D lim H .QI k/ F

the inverse limit of the contravariant functor from F to the category of graded kalgebras, sending a subgroup Q of P to its cohomology H .QI k/. More explicitly, H .B/ can be identified with the subalgebra of ‘stable elements’ in H .P I k/, consisting of all 2 H .P I K/ satisfying ResP Q . / D Res' . / for all P Q P and all ' 2 HomF .Q; P //. Here ResQ W H .P I k/ ! H .QI k/ is the algebra homomorphism induced by restriction from Q to P , and Res' W H .P I k/ ! H .QI k/ is the composition of the restriction ResP '.Q/ followed by the isomorphism H .'.Q/I k/ Š H .QI k/ induced by '. If B is the principal block of kG then P is a Sylow p-subgroup of G and F is equal to the fusion system FP .G/ of G on P . Thus, by the theorem of Cartan–Eilenberg above, the following holds. Theorem 11.4. Let G be a finite group, B the principal block of kG and P a Sylow p-subgroup of G. Restriction from G to P induces an isomorphism of graded algebras H .GI k/ Š H .B/. In other words, ordinary group cohomology is an invariant of the fusion system of the principal block B of kG, a fact that can, of course, also be observed simply by noting that every non principal block annihilates the trivial kG-module k, and hence H .GI k/ D Ext kG .k; k/ D ExtB .k; k/. In general, for B a non principal block, there


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need not be a module M whose Ext-algebra is isomorphic to H .B/; see [81], §6, for an example. Block cohomology has been calculated for 2-blocks with defect groups of rank 2 in [68], and the connection between the block cohomology of a block and that of its Brauer correspondent has been investigated in [69], [139].

12 Block cohomology and Hochschild cohomology Group cohomology is related to Hochschild cohomology as follows. Let G be a finite group. Set G D f.x; x/ j x 2 Gg. There is an isomorphism of k.G G/-modules IndGG G .k/ Š kG sending .x; 1/ ˝ 1 to x, for x 2 G. In other words, modulo identifying G and the diagonal subgroup G, the ‘diagonal induction functor’ sends the trivial kG-module to the k.G G/-module kG, which we can then consider as a kG-kG-bimodule. Since this ‘diagonal induction functor’ is exact, it sends a projective resolution of the trivial kG-module k to a projective resolution of the kG-kG-bimodule kG, hence induces an algebra homomorphism ıG W H .GI k/ ! HH .kG/ This homomorphism is split injective: the functor ˝kG k induces a retraction. The trivial kG-module is annihilated by all non-principal blocks, and hence, multiplying by the block idempotent of the principal block B of kG yields a split injective algebra homomorphism H .GI k/ ! HH .B/ For non-principal blocks, we have a slightly weaker statement: Theorem 12.1 ([100], Theorem 5.6; [101], Corollary 4.3). Let G be a finite group and B a block of kG. There is an injective graded algebra homomorphism ıB W H .B/ ! HH .B/ Moreover, HH .B/ is finitely generated as a module over H .B/; in particular, the Krull dimensions of H .B/ and of HH .B/ are both equal to the rank of P . The algebras H .B/ and HH .B/ are graded commutative, hence commutative if p D 2 while if p is odd then their even parts are commutative. The Krull dimensions are understood to be those of the commutative even parts in case p is odd. The fact that HH .B/ has the rank of P as Krull dimension has also been proved by S. F. Siegel, using a short more direct argument which does not require the map ıB . It is not known whether ıB is split injective. The proof of 12.1 is rather technical; it is based on a commutative diagram consisting of two pullbacks of the following form. As before, P

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is a defect group of B, and i 2 B P is a source idempotent. HH .B/ O

˛

/ Ext kP ˝ B op .iB; iB/ k O ˇ

HHiB O

/ HH .kP / O ıP

/ H .P I k/

HiB O

H .B/ All maps in this diagram are injective, and they are obtained as follows. The map ˛ is induced by the functor iB ˝B , with iB viewed as a kP -B-bimodule. The map ˇ is induced by the functor ˝kP iB. The upper rectangle is the pullback of ˛ and ˇ; this defines HHiB . Since ˛ and ˇ are injective, so are the two maps starting at HHiB , and their images in HH .kP / and HH .B/ are the subalgebras of stable elements in the sense of [100], Definition 3.1, with respect to the bimodules iB and Bi . The lower rectangle is again a pullback, defining the subalgebra HBi of H .P I k/. One can show that this subalgebra contains the subalgebra H .B/, whence the diagram. Composing the vertical maps of the left column yields the map in 12.1. Let M and M 0 be finitely generated modules over a non principal blocks B and 0 B of finite group algebras kG and kG 0 . Suppose that there is a Morita equivalence mod.B/ Š mod.B 0 / sending M to M 0 . Then the cohomology varieties VG .M / and VG 0 .M 0 / need not be isomorphic, in general – not even if this Morita equivalence is induced by a source algebra isomorphism. The reason for this is that the definition of VG .M /, VG 0 .M 0 / involves the cohomology algebras H .GI k/ and H .G 0 I k/, which are invariants of the principal blocks B0 and B00 of kG and kG 0 , respectively. It may well happen that B, B 0 have isomorphic source algebras but G, G 0 have different fusion systems, even non isomorphic Sylow p-subgroups. We define the block variety VB of the block B of kG as the maximal ideal spectrum of H .B/. If B is the principal block of kG then VB Š VG .k/, by 11.4. One of the consequences of the existence of the map ıB is that we can use it to define more generally for any finitely generated B-module M a graded algebra homomorphism M W H .B/ ! ExtB .M; M /

by composing with the canonical map

ıB W H .B/ ! HH .B/ .M; M / HH .B/ ! ExtB


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induced by the functor ˝B M . We then define the block variety of M as the maximal ideal spectrum of the quotient H .B/= ker.M /: The connection between VB .M / and VG .M / is as follows: Theorem 12.2 ([101], Corollary 4.4). Let B be a block of a finite group algebra kG having a defect group P and let M be a finitely generated B-module. Restriction from G to P induces an algebra homomorphism H .G; k/ ! H .B/, which in turn induces a finite surjective morphism of varieties VB .M / ! VG .M /. If B is the principal block, this is an isomorphism of varieties. The variety VB .M / is a subvariety of VB . Although B need not have an augmentation and despite the fact that H .B/ need not be the Ext-algebra of a B-module, the variety VB is isomorphic to VB .M / for suitable modules M . More precisely, we have: Proposition 12.3. Let B be a block of a finite group algebra kG with a defect group P and M a finitely generated indecomposable B-module with P as a vertex and a kP -source V of dimension prime to p. Then M is an isomorphism; in particular, we have an isomorphism of varieties VB .M / Š VB . This can be proved either directly, or as a consequence of a more general statement which characterises VB .M / in terms of vertices and sources of M . Theorem 12.4 ([12], Theorem 1.1). Let B be a block of a finite group algebra kG with a defect group P and a source idempotent i in B P . Let M be an indecomposable B-module. Then M has a vertex Q contained in P and a kQ-source V which is isomorphic to a direct summand of iM as a kQ-module. Restriction from P to Q induces an algebra homomorphism rQ W H .B/ ! H .QI k/, and then VB .M / is equal to the image of VQ .U / under the map rQ W VQ .k/ ! VB induced by rQ . The proof is based on a result due independently to Kawai [67] (Theorem 1.1) and Linckelmann [102] (Theorem 2.1), showing that VB .M / is the image of VP .iM / under the map induced by the ‘inclusion’ H .B/ ! H .P I k/, where the notation is as in the above theorem. Block varieties admit Quillen stratifications: Theorem 12.5 ([102]). Let B be a block of a finite group algebra kG with a defect group P and fusion system FP .B/. Denote by EP .B/ the full subcategory of FP .B/ consisting of all elementary abelian subgroups of P . The canonical graded algebra homomorphism qB W H .B/ ! lim H .EI k/ EP .B/

is an F -isomorphism. In particular, VB admits a Quillen stratification; more generally, for any finitely generated B-module M the variety VB .M / is a disjoint union [ C VB .M / D VB;E .M / E

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C for some locally closed subvarieties VB;E .M / indexed by a set of representatives of the isomorphism classes in EP .B/.

The proof is essentially a variation of that of Quillen’s theorem as presented in [7], §5, except for one additional ingredient, namely a certain fusion stable biset constructed by Broto, Levi and Oliver in [20], Proposition 5.5. One of the consequences of the existence of such a biset is that H .B/ is the cohomology of a splitting summand of the classifying space of P as a p-complete spectrum, and in particular, H .B/ is a module over the mod-p Steenrod algebra. The Hochschild cohomology of a finite-dimensional algebra is graded commutative (by a result of Gerstenhaber), and hence, whenever it is also finitely generated, its maximal ideal spectrum is again an affine variety – see [141], [45] for a development of this general approach. This applies in particular to the Hochschild cohomology of block algebras (cf. 12.1). For B a block algebra of a finite group algebra we denote by XB the maximal ideal spectrum of HH .B/. Pakianathan and Witherspoon showed in [118], Theorem 2.5, that for any finite group G with a Sylow p-subgroup P there is a graded algebra homomorphism Y H .EI k/ ˝k H .CG .k/I k/ HH .kG/ ! E

with E running over a set of representatives of the G-conjugacy classes of elementary abelian p-subgroups of P , which has a nilpotent kernel. As a consequence they showed in [118], Theorem 2.10, that XB is a union of subvarieties XB;E indexed by a set of representatives of the G-conjugacy classes of elementary abelian p-subgroups of P . This is pushed further to get a Quillen stratification for the Hochschild cohomology of block algebras: Theorem 12.6 ([118]. Theorems 4.2, 4.3). Let B be a block of a finite group algebra kG with a defect group P and fusion system FP .B/. The variety XB is a disjoint union [ C XB D XB;E E C of locally closed subvarieties XB;E , with E running over a set of representatives of the FP .B/-isomorphism classes of elementary abelian subgroups of P . Moreover, if B is or principal type (that is, FP .B/ is a full subcategory of FS .G/ for some Sylow C C p-subgroup S of G containing P ) then XB;E Š VB;E for all such E; in particular, the block cohomology variety VB and the Hochschild cohomology variety XB are homeomorphic.

Pakianathan and Witherspoon raised in [118] the question whether these varieties are actually isomorphic; more precisely, they asked whether the map ıB W H .B/ ! HH .B/ becomes an isomorphism upon taking quotients by suitable nilpotent ideals. They showed this to be true for certain special cases in [117]. The interest of


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this question lies in the fact that H .B/ is an invariant of the fusion system FP .B/, whereas HH .B/ is an invariant of the Morita equivalence class – and as we have seen, conjecture such as Alperin’s are precisely about the relationship between FP .B/ and mod.B/. It was shown in [106] that the answer is positive in general: Theorem 12.7 ([106], Theorem 1.1). Let G be a finite group and B be a block algebra of kG with defect group P . The canonical graded algebra homomorphism ıB from block cohomology H .B/ to the Hochschild cohomology HH .B/ induces an isomorphism modulo nilpotent ideals, or equivalently, an isomorphism of varieties XB Š VB . For principal blocks one can be slightly more precise because the map ıB is split injective in that case: Theorem 12.8. Let B be the principal block of a finite group algebra. There is a nilpotent ideal N in HH .B/ such that HH .B/ Š H .B/ ˚ N . Take for N the kernel of a retraction HH .B/ ! H .B/ of ıB ; this is a nilpotent ideal by 12.7. It is not known whether theorem 12.8 holds for all blocks because we do not know whether the map ıB is split injective. It would possibly suffice to find a B-module M such that the graded center of ExtB .M; M / is isomorphic to H .B/, because the canonical map HH .B/ ! ExtB .M; M / might then provide a retraction for ıB . The existence of such a module M is an open problem.

13 Further remarks and questions To what extent can one use the Hochschild cohomology of a symmetric algebra in order to decide whether that algebra arises as a block algebra or not? In other words, can one use Hochschild cohomology to rule out certain algebras? As in the principal block case, dealt with in a more general way in work of Benson, Carlson and Robinson [9], one can make some simple statements in that direction: Theorem 13.1. Let P be a non-trivial finite p-group. There is a positive integer s such that for any block B of a finite group with defect group P , there are no s consecutive indices n 0 for which HH n .B/ is zero. In other words, the number of consecutive vanishing Hochschild cohomology spaces of a block is bounded in terms of its defect group. In particular, HH .B/ is non-zero in infinitely many degrees. Proof. Since H .B/ has as Krull-dimension the rank of P , there is a homogeneous element in some positive degree H t .B/ which is not nilpotent. The number t depends on the fusion system of B, but there are only finitely many fusion systems on P , and hence for some sufficiently large s, the existence of a non-nilpotent element in H s .B/ will hold for all blocks with a defect group P . Since H .B/ embeds into HH .B/, the theorem follows.

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M. Linckelmann

b

Using Tate duality applied to Hochschild cohomology one can formulate the above theorem with HH .B/ replaced by its Tate analogue HH .B/. This theorem is to be seen in the context of an intriguing one-parameter family, due to Buchweitz, Green, Madsen and Solberg [27], of self-injective algebras whose Hochschild cohomology is finite-dimensional if the parameter is transcendental. If the parameter is a root of unity in k, the ‘gaps’ in Hochschild cohomology depend on the order of this root of unity, and they can become arbitrarily large. The fact that a transcendental parameter yields finite-dimensional Hochschild cohomology has been extended to quantum complete intersections of codimension 2 by Bergh and Erdmann [13]. The algebras in [27], [13] are not symmetric – one may want to speculate whether any symmetric algebra defined over a finite field has infinite Hochschild cohomology, with ‘gaps’ of bounded lengths. Since gaps in Hochschild cohomology of block algebras are bounded by their defect groups, this invites further speculations as to whether this could be exploited for detecting bounds on the size of the field of definition of a block algebra. In other words: can Hochschild cohomology be used to approach Kessar’s rationality conjecture 8.2? Block cohomology H .B/ is an invariant of the fusion system F of B on a defect group of P , and hence in particular, for a fixed finite p-group P , there are only finitely many isomorphism classes of graded k-algebras which arise as block cohomology algebras of blocks with P as a defect group. It is not known whether an analogous statement holds for Hochschild cohomology – but at least the dimensions of HH n .B/ are bounded in terms of n and P : Theorem 13.2 ([82], Theorem 1). There is a function f W N0 N0 ! N0 such that for any integer n 0 and any block algebra B of a finite group algebra kG of defect d we have dimk .HH n .B// f .n; d /: For n D 0 this follows from a theorem of Brauer and Feit [18], since HH 0 .B/ Š Z.B/. This theorem of Brauer and Feit is used in the proof of theorem 13.2. Since Hochschild cohomology is invariant under Morita equivalences, Donovan’s conjecture would imply that for a fixed defect group P there should indeed be only finitely many isomorphism classes of graded algebras which arise as Hochschild cohomology of block algebras with defect groups isomorphic to P . The next question is related to results of Henn, Lannes and Schwartz in [53]. Let E be an elementary abelian p-subgroup of a finite group G. The inclusion maps induce a group homomorphism E CG .E/ ! G. This group homomorphism induces a graded algebra homomorphism H .GI k/ ! H .EI k/ ˝k H .CG .E/I k/. Dividing this by the ideal generated by all elements of H .CG .E/I k/ of degree at least n yields an algebra homomorphism H .G/ ! H .EI k/ ˝k H 0. Then, we have W yi .ui / 7! Yi .ui /. Furthermore, one can prove that W yi .u/ 7! Yi .u/ for any .i; u/ 2 PC by induction on the forward mutations for u > ui and on the backward mutations for u < ui , using the common Y-systems for the both sides. By the restriction of to Gi .B; y/, we obtain a group homomorphism ' W Gi .B; y/ ! YB .B; i /, which is the inverse of . Aside from the direct connection between T- andY-systems in Proposition 5.6, there is an algebraic connection, which has been noticed since the inception of the original T- and Y-systems [36], [34]. Proposition 5.11. Let T .B; i / be the ring in Definition 5.9. For each .i; u/ 2 PC , we set Y Tj .v/HC .j;vIi;u/ .j;v/2PzC Yi .u/ WD Y : (5.28) Tj .v/H .j;vIi;u/ z .j;v/2PC

Then, Yi .u/ satisfies the Y-system (5.10) in T .B; i / by replacing yi .u/ with Yi .u/.

430

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Note that (5.28) is the ratio of the first and second terms in (5.16). Proof. Thanks to the isomorphism in Theorem 5.10, one can work in the localization of Ai .B; x/ by generators ŒxQ i .u/1 (.i; u/ 2 PzC ), which is a subring of the ambient field Q.x/. The claim is translated therein as follows: For each .i; u/ 2 PC , we set Y Y H 0 .j;vIi;u/ H .j;vIi;u/ ŒxQj .v/1 C Œxj .v/1 C z .j;v/2PC .j;v/2PC yNi .u/ WD Y D Y H .j;vIi;u/ H 0 .j;vIi;u/ ŒxQj .v/1 Œxj .v/1 .j;v/2PzC .j;v/2PC Y b .u/ D Œxj .u/1j i : j 2I

Then yNi .u/ satisfies the Y-system (5.10) by replacing yi .u/ with yNi .u/. In fact, this claim is an immediate consequence of Proposition 3.9 in [16]. Remark 5.12. Some classic examples of T- andY-systems are not always in the ‘straight form’ presented here, but represented by generators TiN .u/ and YiN .u/ whose indices iN belong to the orbit space I = of I by . For example, the T- and Y-systems for type .X; `/ D .B4 ; 4/ in Section 5.3 [24], and the sine-Gordon T- and Y-systems for Example 3.6 [46] are such cases. In these examples, it is just a ‘change of notation’ for generators. However, this makes the reconstruction of the initial exchange matrix B from given T- or Y-systems nontrivial, because a priori we only know I =, and we have to find out true index set I and with some guesswork. 5.5 T- and Y-systems for general period. Conceptually, the notions of T- and Ysystems can be straightforwardly extended to general -periods of B, though they become a little apart from the ‘classic’T- andY-systems. We will use them in Section 6. Let i D i .0/ j j i .t 1/ be a slice of any (not necessarily regular) -period i of B. One can still define the sequence of seeds .B.u/; x.u/; y.u// (u 2 Z) and the forward mutation points .i; u/ 2 PC as in the regular case. Fix i 2 I , and let : : : ; .i; u/; .i; u0 /; .i; u00 /; : : :

. < u < u0 < u00 < /

(5.29)

be the sequence of the forward mutation points. (It may be empty for some i .) In general, if it is not empty, the sequence : : : ; u; u0 ; u00 ; : : : is periodic for u ! u C tg, but it does not necessarily have the common difference. For each .i; u/ 2 PC , let .i; uC C .i; u// and .i; u .i; u// be the nearest ones to .i; u/ in the sequence (5.29) in the forward and backward directions, respectively; in other words, .i; u .i; u//, .i; u/, .i; u C C .i; u// are three consecutive forward mutation points in (5.29). If i is regular, then ˙ .i; u/ D tgi , which is the common difference (therefore, called regular). In general, we have 0 < ˙ .i; u/ < tg, ˙ .i; u C tg/ D ˙ .i; u/, and C .i; u .i; u// D .i; u/:

(5.30)

431

Periodicities in cluster algebras and dilogarithm identities

Let J.i ; / be the subset of I consisting of all the components of j .i ; /. Note that the condition (A1) in Definition 5.1 means that J.i ; / D I . Using these notations, the relations (5.8) and (5.12) are generalized as follows. For .i; u/ 2 PC , Y 0 .1 C yj .v//GC .j;vIi;u/ .j;v/2PC ; (5.31) yi .u/ yi .u C C .i; u// D Y 0 .1 C yj .v/1 /G .j;vIi;u/ .j;v/2PC ´ bj i .v/ v 2 .u; u C C .i; u//; bj i .v/ 7 0 0 G˙ .j; vI i; u/ D (5.32) 0 otherwise; and xi .u/xi .u C C .i; u// yi .u/ D 1 C yi .u/ C

Y j 2I nJ.i ;/W bj i >0

1 1 C yi .u/

Y

b .u/

xj j i

0

xj .v/HC .j;vIi;u/

.j;v/2PC

Y

bj i .u/

xj

j 2I nJ.i ;/W bj i 0 defined by ' t .yi / D t for any i 2 I . Then, thanks to (2.6) and the parts (a) and (c) of Conjecture 2.1/Theorem 2.2, the limit lim t!0 ' t is indeed a 0=1 limit, and the total number of .i; u/ 2 SC such that the value '.yi .u/=.1 C yi .u/// goes to 1 is exactly N .

435


Remark 6.5. In the above proof, Step 1 (constancy) is due to [12], and Step 2 (evaluation at 0=1 limit) is due to [4]. 6.3 First proof of Proposition 6.3. Now we only have to prove Proposition 6.3. We give two proofs in this and the next subsections. The first one here is a generalization of the former proofs for special cases [44], [24], which is a somewhat brute force proof with ‘change of indices’. To start, we claim that one can assume all the components of i exhaust I . In fact, if the condition does not hold, one may reduce the index set I so that the condition holds, since doing that does not affect the Y-system (5.31). Note that the condition is nothing but the condition (A1) in Definition 5.1 with D id. Thus, the accompanying T-system has the simplified form (5.35). Furthermore, by the periodicity assumption and Theorem 4.3 (a), we have xi .u C / D xi .u/:

(6.15)

Now, let Fi .u/ be the F -polynomials at .i; u/. Lemma 6.6. The following properties hold. (a) Periodicity: Fi .u C / D Fi .u/. (b) For .i; u/ 2 SC , Fi .u/Fi .u C C .i; u// D

yi .u/ 1 C yi .u/

T

Y

0

Fj .v/HC .j;vIi;u/

.j;v/2PC

1 C 1 C yi .u/

Y

T

Fj .v/

0 .j;vIi;u/ H

(6.16) :

.j;v/2PC

(c) For .i; u/ 2 SC , Q

0

.j;v/2PC

Fj .v/HC .j;vIi;u/

.j;v/2PC

Fj .v/H .j;vIi;u/

yi .u/ D Œyi .u/T Q

0

;

Fi .u/Fi .u C C .i; u// 1 C yi .u/ D Œ1 C yi .u/T Q 0 .j;vIi;u/ : H .j;v/2PC Fj .v/

(6.17)

(6.18)

Proof. (a) This is obtained from the specialization of (6.15). (b) This is obtained from the specialization of (5.35). (c) The first equality is obtained by rewriting (2.6) with (5.34). The second one is obtained from the first one and (b). Proof of Proposition 6.3. We prove (6.12). We put (6.17) and (6.18) into the left hand side of (6.12), expand it, then, sum them up into three parts as follows.

436

T. Nakanishi

The first part consists of the terms only involving tropical coefficients, i.e., X

Œyi .u/T ^ Œ1 C yi .u/T :

(6.19)

.i;u/2SC

By Theorem 2.2, each monomial Œyi .u/T is either positive or negative. If it is positive, then Œyi .u/T ^ Œ1 C yi .u/T D Œyi .u/T ^ 1 D 0. If it is negative, then, Œyi .u/T ^ Œ1 C yi .u/T D Œyi .u/T ^ Œyi .u/T D 0. Therefore, the sum (6.19) vanishes. The second part consists of the terms involving both tropical coefficients and F polynomials. We separate them into five parts, X

Œyi .u/T ^ Fi .u/;

.i;u/2SC

X .i;u/2SC

X

Œyi .u/T ^ Fi .u C C .i; u// D

X

Y

Œyi .u/T ^

.i;u/2SC

D

X

Œyi .u .i; u//T ^ Fi .u/;

.i;u/2SC 0

Fj .v/H .j;vIi;u/

.j;v/2PC

Y

0 .i;uIj;v/ H

Œyj .v/T

^ Fi .u/;

(6.20)

.i;u/2SC .j;v/2PC

X

Y

Œ1 C yi .u/T ^

.i;u/2SC

D

X

0

Fj .v/HC .j;vIi;u/

.j;v/2PC

Y

0 .i;uIj;v/ HC

Œ1 C yj .v/T

^ Fi .u/;

.i;u/2SC .j;v/2PC

X

Œ1 C yi .u/T ^

.i;u/2SC

D

X

Y

0

Fj .v/H .j;vIi;u/

.j;v/2PC

Y

H 0 .i;uIj;v/

Œ1 C yj .v/T

^ Fi .u/;

.i;u/2SC .j;v/2PC

where we changed indices and also used the periodicity of Œyi .u/T , Fi .u/, .i; u/, 0 0 0 and H˙ .j; vI i; u/. Recall the relation H˙ .i; uI j; v/ D G˙ .j; vI i; u .i; u// in (5.37). Then, the sum of the above five terms vanishes due to the ‘tropical Y-system’ Y Œyi .u .i; u//T Œyi .u/T D Y

.j;v/2PC .j;v/2PC

which is a specialization of (5.31).

G 0 .j;vIi;u .i;u//

Œ1 C yj .v/T C

G 0 .j;vIi;u .i;u//

Œ1 C yj .v/1 T

; (6.21)


437

The third part consists of the terms involving only F -polynomials. It turns out that this part requires the most elaborated treatment. We separate them into three parts, Y 0 Fj .v/HC .j;vIi;u/ X .j;v/2PC ^ Fi .u/; (6.22) .A/ D Y 0 Fj .v/H .j;vIi;u/ .i;u/2SC .j;v/2PC Y 0 Fj .v/HC .j;vIi;u/ X .j;v/2PC .B/ D ^ Fi .u C C .i; u//; (6.23) Y 0 .j;vIi;u/ H F .v/ j .i;u/2SC .j;v/2PC X Y Y 0 0 .C/ D Fj .v/HC .j;vIi;u/ ^ Fj .v/H .j;vIi;u/ : (6.24) .j;v/2PC

.i;u/2SC

.j;v/2PC

Let us rewrite each term so that their cancellation becomes manifest. The first term (A) is rewritten as follows. Y X Fj .v/bj i .u/ ^ Fi .u/ .A/ D .i;u/2SC

.j;v/2PC u2.v .j;v/;v/

X

D

bj i .u/Fj .v/ ^ Fi .u/

.i;u/2SC ; .j;v/2PC u2.v .j;v/;v/

X

D

bj i .v/Fj .v/ ^ Fi .u/

.i;u/2SC ; .j;v/2PC v2.u .i;u/;u/

D

1 2

X

bj i .min.u; v//Fj .v/ ^ Fi .u/:

.i;u/2SC ; .j;v/2PC .u .i;u/;u/\.v .j;v/;v/¤;

Here the third line is obtained from the second one by the exchange .i; u/ $ .j; v/ of indices, the skew symmetric property bj i .u/ D bij .u/, and the periodicity; the last line is obtained by averaging the second and the third ones; therefore, there is the factor 1=2 in the front. We also mention that, in the last line, the pair .i; u/; .j; v/ with u D v does not contribute to the sum, because in that case we have bj i .u/ D 0 due to the condition (5.4). Similarly, the second term (B) is rewritten as follows. Y X Fj .v/bj i .u/ ^ Fi .u C C .i; u// .B/ D .i;u/2SC

D

.j;v/2PC u2.v .j;v/;v/

X

bj i .u .i; u//Fj .v/ ^ Fi .u/

.i;u/2SC ; .j;v/2PC u .i;u/2.v .j;v/;v/

438

T. Nakanishi

D

X

1 2

bj i .max.u .i; u/; v .j; v///Fj .v/ ^ Fi .u/:

.i;u/2SC ; .j;v/2PC .u .i;u/;u/\.v .j;v/;v/¤;

The third term (C) is written as follows. X X .C/ D .i;u/;.j;v/2PC

.k;w/2SC k2.u .i;u/;u/\.v .j;v/;v/ bj k .w/>0; bi k .w/0

j W bj k 0

j W bj0 i 0. An integral quadratic form q W Zn ! Z is said to be • • • • •

positive if q.x/ > 0 for all x ¤ 0 in Zn ; nonnegative if q.x/ 0 for all x 2 Zn ; negative if q.x/ < 0 for some x 2 Zn ; weakly positive if q.x/ > 0 for all x > 0 in Zn ; weakly nonnegative if q.x/ 0 for all x 0 in Zn .

The Tits forms of tame algebras and their roots

451

For a nonnegative form q, the set rad q D fx 2 Zn j q.x/ D 0g is a subgroup of Zn called the radical of q, and its rank is called the corank of q, denoted by corank q. For a natural number m, a vector x of Zn with q.x/ D m is said to be an m-root of q. An 1-root x of Zn will be simply called a root of q. A vector x 2 Zn with all coordinates nonzero is said to be omnipresent. Let q W Zn ! Z be an integral quadratic form. For a nonempty subset I D fi1 ; : : : ; im g of f1; : : : ; ng, we denote by q I W Zm ! Z the integral quadratic form qdI , where dI W Zm ! Zn maps the canonical basis vectors e1 ; : : : ; em of Zm onto the basis vectors ei1 ; : : : ; eim of Zn , called the restriction of q with respect to I . For each i 2 f1; : : : ; ng, we denote by q .i/ the restriction of q with respect to I D fi g. Following S. A. Ovsienko [64], an integral quadratic form q W Zn ! Z is said to be critical provided q is not weakly positive but all restrictions q .i/ W Zn1 ! Z, with i 2 f1; : : : ; ng, are weakly positive. Similarly, following H.-J. von Höhne [49], q is said to be hypercritical if q is not weakly nonnegative but all restrictions q .i/ W Zn1 ! Z, with i 2 f1; : : : ; ng, are weakly nonnegative. We note that an integral quadratic form q W Zn ! Z is not weakly positive (respectively, is not weakly nonnegative) if there is a subset I of f1; : : : ; ng such that the restriction q I of q with respect to I is critical (respectively, hypercritical). We start with a theorem from [64]. Theorem 2.1 (Ovsienko). Let q W Zn ! Z be a weakly positive quadratic form and x D .xi / 2 N n a positive root of q. Then xi 6 for any i 2 f1; : : : ; ng. We exhibit a combinatorial criterion for an integral quadratic form to be weakly positive, established by S. A. Ovsienko [64] and H.-J. von Höhne [49]. Theorem 2.2 (Ovsienko, von Höhne). Let q W Zn ! Z be an integral quadratic form. Then q is weakly positive if and only if q.x/ > 0 for every nonzero vector x 2 Œ0; 6n . Moreover, we have the following combination of results proved by Yu. A. Drozd [37] and D. Happel [45]. Theorem 2.3 (Drozd, Happel). Let q W Zn ! Z be an integral quadratic form. Then q is weakly positive if and only if q has only finitely many positive roots. The next theorem proved by S. A. Ovsienko [64] is fundamental for the representation theory of tame algebras. Theorem 2.4 (Ovsienko). Let q W Zn ! Z be a critical integral quadratic form. Then one of the statements holds. (1) n D 2. (2) q is nonnegative of corank 1.

452

J. A. de la Peña and A. Skowroński

Moreover, if n 3, then rad q D Zhq for a positive omnipresent vector hq 2 Zn . We have also the following fact from [103]. Theorem 2.5 (Zeldich). Let q W Zn ! Z be a hypercritical integral quadratic form. Then q .i/ is nonnegative for any i 2 f1; : : : ; ng. The following characterization of hypercritical integral quadratic forms have been established by J. A. de la Peña in [67] and M. V. Zeldich [103]. Theorem 2.6 (de la Peña, Zeldich). Let q W Zn ! Z be an integral quadratic form. The following conditions are equivalent. (1) q is hypercritical. (2) There is a vector x 2 Zn with q.x/ < 0 and for every vector y 2 Zn with q.y/ 0 we have y 0 or y 0. We end this section with the following combinatorial criterion for an integral quadratic form to be weakly nonnegative, proved by S. A. Ovsienko [64], J. A. de la Peña in [67], and H.-J. von Höhne [49]. Theorem 2.7 (Ovsienko, de la Peña, von Höhne). Let q W Zn ! Z be an integral quadratic form. Then q is weakly nonnegative if and only if q.x/ 0 for any vector x 2 Œ0; 12n . We refer also to the paper by A. Dean and J. A. de la Peña [27] for algorithms to decide when a given integral quadratic form is weakly nonnegative.

3 Tits form of an algebra Let A D KQ=I be a triangular algebra. The Tits form of A is an integral quadratic form qA W ZQ0 ! Z, defined, for x D .xi / 2 ZQ0 , by X X X qA .x/ D xi2 xs.˛/ x t.˛/ C r.i; j /xi xj i2Q0

˛2Q1

i;j 2Q0

where Q0 is the set of vertices of Q, Q1 is the set Sof arrows of Q, and r.i; j / D jR \ I.i; j /j for a minimal set of generators R i;j 2Q0 I.i; j / of the admissible ideal I . K. Bongartz proved in [11], using the triangularity of A, that r.i; j / D dimK ExtA2 .Si ; Sj / for all i; j 2 f1; : : : ; ng, and hence the coefficients r.i; j / of qA do not depend on the choice of a minimal set of generators R. In order to exhibit the geometric nature of the Tits form, we need a geometric context. Let A D KQ=I be a bound quiver algebra and d D .di / 2 N Q0 a dimension vector. Denote by modA .d/ the set of all representations V D .Vi ; '˛ /i2Q0 ;˛2Q1


453

in the category repK .Q; I / of finite dimensional K-linear representations of the bound quiver .Q; I / with Vi D K di for all i 2 Q0 . Then a representation V in modA .d/ is given by d t.˛/ ds.˛/ -matrices V .˛/ determining the K-linear maps '˛ W K ds.˛/ ! K d t .˛/ , in the canonical bases of K di , i 2 Q0 . Moreover, the matrices V .˛/, ˛ 2 Q1 , satisfy the relations m X i V ˛1.i/ : : : V ˛n.i/i D 0 Pm

iD1

for all relations iD1 i ˛1.i/ : : : ˛n.i/i 2 I , which are K-linear combinations of paths of length 2 in Q with a common source and a common target. Hence, we may view modA .d/ as a subset of the affine space Y K d t .˛/ ds.˛/ A.d/ D ˛2Q1

defined by the vanishing of a finite number of polynomials, given by the matrix relations described above. Therefore, modA .d/ is a closed subset of the affine space A.d/ in the Zariski topology. We call modA .d/ the affine variety of A-modules of dimension vector d. Further, we have a natural action W G.d/ modA .d/ ! modA .d/ of the algebraic group G.d/ D

Y

GLdi .K/

i2Q0

on the variety modA .d/ by the conjugation formula 1 .g V /.˛/ D g t.˛/ V .˛/gs.˛/

for g D .gi / 2 G.d/, V 2 modA .d/, ˛ 2 Q1 . Then two representations M and N in modA .d/ are isomorphic if and only if M and N belong to the same G.d/-orbit in modA .d/, or equivalently, G.d/ M D G.d/ N . Moreover, if A D KQ=I is a triangular algebra, then it follows from the Krull’s Principal Ideal Theorem that qA .d/ dim G.d/ dim modA .d/ for any positive vector d in N Q0 , and hence qA is a geometric form. The following theorem has been proved by K. Bongartz in [11]. Theorem 3.1 (Bongartz). Let A D KQ=I be a triangular representation-finite bound quiver algebra. Then dim G.d/ > dim modA .d/: In particular, qA is a weakly positive quadratic form.

454


The corresponding theorem for tame algebras has been proved by J. A. de la Peña in [68]. Theorem 3.2 (de la Peña). Let A D KQ=I be a triangular tame bound quiver algebra. Then dim G.d/ dim modA .d/: In particular, qA is a weakly nonnegative quadratic form. Unfortunately, in general, the weak positivity (respectively, weak nonnegativity) of the Tits form qA of a triangular bound quiver algebra A D KQ=I does not force A to be representation-finite (respectively, tame). One has to impose some nondegeneracy conditions on a triangular algebra A to recover its representation type from the weak positivity (respectively, weak nonnegativity) of the Tits form qA of A. The following example from Section 2 of [11] illustrates the problem. Example 3.3 (Bongartz). Let Q be the quiver 1 zz z zz z }z 2 DD DD DD D! " 4 ˛

DD DDˇ DD D! z3 ı zz z z z }z

5 DD 6 DD zz D zz DD! }zzz 7.

Consider the sets of relations in KQ R1 D f˛ ˇı; " ; ı g and R2 D f˛ ˇı; " ; g ; the ideals I1 and I2 in KQ generated by R1 and R2 , and the associated bound quiver algebras A1 D KQ=I1 and A2 D KQ=I2 , respectively. Then A1 and A2 are triangular algebras whose Tits forms qA1 and qA2 are equal to the integral quadratic form q W Z7 ! Z given for x D .xi / 2 Z7 by q.x/ D

7 X

xi2 x1 x2 x1 x3 x2 x4 x2 x5 x3 x4 x3 x6

iD1

x 4 x 7 x5 x 7 x 6 x 7 C x 1 x 4 C x 2 x 7 C x 3 x 7 :


455

Then q, considered as a rational form Q7 ! Q, can be written as follows:

2 2 1 1 q.x/ D x5 .x2 C x7 / C x6 .x3 C x7 / 2 2 2 1 1 C x4 C .x1 x7 x2 x3 / C .x1 C x7 /2 2 4 2 1 3 C x1 .x2 C x3 / C .x2 x3 /2 ; 2 8

for any x D .xi / 2 Q7 . Hence q is a weakly positive integral quadratic form. Observe that q is not positive. The algebra A1 is a representation-finite algebra, which is a strongly simply connected algebra without convex critical subcategory (see Section 7). On the other hand, A2 is not simply connected and admits a (simply connected) Galois covering Az2 containing as a convex subcategory the path algebra K of the wild quiver of the form y 2 EEE y 3 EEE " yy ı yy EE EE y y EE yy EE y |yy " |y " 5 4 6 50 DD 60 DD zz z D z DD zz ! }z 7,

and hence A2 is wild. We may associate to a triangular algebra A D KQ=I also a nonsymmetric bilinear form h; iA W ZQ0 ZQ0 ! Z defined, for x D .xi /; y D .yi / 2 ZQ0 , by X X X xi yj xs.˛/ y t.˛/ C r.i; j /xi yj ; hx; yiA D i;j 2Q0

˛2Q1

i;j 2Q0

which found many important applications. Clearly, we have qA .x/ D hx; xiA for any x 2 ZQ0 . The following useful inequality has been established recently in [78]. Theorem 3.4 (de la Peña–Skowroński). Let A D KQ=I be a triangular algebra. Then for any modules M and N in mod A, we have the inequality hdim M; dim N iA dimK HomA .M; N / dimK ExtA1 .M; N /: It has been proved by W. Crawley-Boevey in [25] that, for any tame algebra A and a positive vector d 2 K0 .A/, all but finitely many modules X in ind A of dimension vector d lie in stable tubes of rank 1 of the Auslander–Reiten quiver A of A. The following result has been proved recently in [78], using Theorem 3.4.

456


Theorem 3.5 (de la Peña–Skowroński). Let A be a tame triangular algebra and w 2 K0 .A/ a connected positive vector such that qA .w/ D 0. Then there is a decomposition w D w1 C C ws of w into direct sum of nonnegative vectors in K0 .A/ such that the following statements hold: (1) qA .wi / D 0 for any i 2 f1; : : : ; sg. (2) For each i 2 f1; : : : ; sg, there is an infinite family Xi; , 2 Xi , of pairwise nonisomorphic modules in ind A lying in stable tubes of A of rank 1 with dim Xi; D wi for any 2 Xi . (3) HomA .Xi; ; Xj; / D 0 and ExtA1 .Xi; ; Xj; / D 0 for all i ¤ j in f1; : : : ; sg and 2 Xi , 2 Xj . Besides the Tits form there is another quadratic form for algebras of finite global dimension, which is analogous to the Euler characteristic of a topological space. This form was introduced by C. M. Ringel in [81], and plays a fundamental role in the representation theory of algebras of finite global dimension. Let A D KQ=I be a triangular algebra. The Euler form of A is the integral quadratic form A W ZQ0 ! Z such that, for any module M in mod A, we have A .dim M / D

1 X

.1/i dimK ExtAi .M; M /:

iD0

The following result of K. Bongartz ([11], Proposition 2.1) gives a connection between the Tits and Euler forms of a triangular algebra. Theorem 3.6 (Bongartz). Let A D KQ=I be a triangular algebra of global dimension at most 2. Then qA D A . We end this section with an example showing that, for triangular algebras A of global dimension bigger than 2, the Tits form qA and the Euler form A may have completely different behaviour. Example 3.7 (de la Peña–Skowroński). Let Q be the quiver 5

4

/3o

ı

1o

˛

'

ˇ

2o

7o

%

6o

9o

11

"

8o

10 ,

I the ideal in the path algebra KQ generated by the elements ıˇ %; ' ";

; ıˇ˛; "%; ";


457

and A D KQ=I the associated bound quiver algebra. It has been shown in Example 5.6 of [74] that A is a tame strongly simply connected algebra of global dimension 3, however not of polynomial growth. Moreover, for any positive integer n, we constructed an indecomposable right A-module Yn with the dimension vector dim Yn of the form n n 2n C 1 1 1 1 n 2n n 0 1; qA .dim Yn / D 2n C 1 and A .dim Yn / D 1 3n (negative!).

4 Simply connected algebras P Let .Q; I / be a connected bound quiver. A relation % D jmD1 j wj 2 I.x; y/ is said to be minimal if m 2 and, for each nonempty proper subset J of f1; : : : ; mg, we have P j 2J j wj … I . We denote by m.I / the set of all minimal relations of the ideal I . Further, let …1 .Q; x0 / be the fundamental group of the quiver Q at a fixed vertex x0 of Q. Let N.Q; m.I /; x0 / be the normal subgroup of …1 .Q; x0 / generated by all elements of the form Œw 1 u1 vw, where w isP a walk from x0 to x and u; v are paths from x to y in Q such that there is a relation jmD1 j wj 2 m.I / with u D wi and v D wk for some i; k 2 f1; : : : ; mg. Then, following R. Martinez and J. A. de la Peña [58], the fundamental group …1 .Q; I / of .Q; I / is defined as the quotient group …1 .Q; I / D …1 .Q; x0 /=N.Q; m.I /; x0 / (see also [44]). Following I. Assem and A. Skowroński [2], a triangular algebra A is said to be simply connected if, for any presentation A Š KQ=I of A as a bound quiver algebra, the fundamental group …1 .Q; I / is trivial. Example 4.1 (Assem–Skowroński). Let A D KQ=I , where Q is the quiver 4 2 ^= === == = 1o 3o 5 ˛ ˇ

ı

and I is the ideal of KQ generated by ı˛ ıˇ. Then …1 .Q; I / is trivial. On the other hand, A Š KQ=I 0 , where I 0 is the ideal in KQ generated by ı˛, and clearly …1 .Q; I 0 / Š Z. Hence, A is not simply connected. On the other hand, the bound quiver algebra B D KQ=J , where J is the ideal in KQ generated by ı˛ ıˇ and ˛, is a simply connected algebra. We note also the following intrinsic characterization of simply connected algebras proved in [88], Lemma 4.2.

458


Proposition 4.2 (Skowroński). Let A be a triangular algebra. Then A is a simply connected algebra if and only if A does not admit a proper Galois covering (in the sense of [58]). Let A D KQ=I be a triangular algebra. For each vertex x of Q, denote by Q.x/ the subquiver of Q obtained by deleting all those vertices of Q being a source of a path in Q with target x (including the trivial path from x to x). We shall denote by A.x/ the full subcategory of A whose objects are the vertices of Q.x/. Moreover, for each vertex x of Q, denote by P .x/ the indecomposable projective A-module at x, and by R.x/ the radical of P .x/. Then R.x/ is said to be separated if R.x/ is a direct sum of pairwise nonisomorphic indecomposable modules whose supports are contained in pairwise different connected components of Q.x/. Then, generalizing definition proposed by R. Bautista, F. Larrion and L. Salmeron in [9], we say that A has the separation property if R.x/ is separated for any vertex x of Q. The following result has been proved in [89] (Proposition 2.3). Proposition 4.3 (Skowroński). Let A be a triangular algebra with the separation property. Then A is a simply connected algebra. Following [89], Definition 2.2, an algebra A is said to be strongly simply connected if every convex subcategory of A is simply connected. Clearly, if A is strongly simply connected then A is simply connected. The algebra B D KQ=J considered in Example 4.1 is simply connected but not strongly simply connected, because B contains a convex subcategory which is the path algebra K of the quiver of the form 2 ^= === == == ˇ

1o

˛

3,

whose fundamental group is Z. For an algebra A, we denote by H 1 .A/ the first Hochschild cohomology space of A with respect to the A-A-bimodule A. Then H 1 .A/ Š Der.A; A/=Der 0 .A; A/; where DerK .A; A/ D fı 2 HomK .A; A/ j ı.ab/ D aı.b/ C ı.a/bg ; is the space of K-linear derivations from A to A, and 0 DerK .A; A/ D fıx 2 HomK .A; A/ j ıx .a/ D ax xag ;

is the subspace of inner derivations from A to A. Hence H 1 .A/ is isomorphic to the K-vector space of external derivations from A to A. The following characterization of strongly simply connected algebras has been established in Theorem 4.1 of [89].


459

Theorem 4.4 (Skowroński). Let A be a triangular algebra. The following conditions are equivalent. (1) (2) (3) (4)

A is strongly simply connected. Every convex subcategory of A has the separation property. Every convex subcategory of Aop has the separation property. The first Hochschild cohomology space H 1 .C / of any convex subcategory C of A vanishes.

In [15] K. Bongartz and P. Gabriel proposed the concept of simple connectedness of a represention-finite algebra A invoking the Auslander–Reiten quiver A . Namely, a triangular representation-finite algebra A is called in [15] simply connected if the fundamental group …1 .jA j/ of the geometric realization jA j of the Auslander–Reiten quiver A of A is trivial. The following combination of results from [19] and [58] shows that for representation-finite algebras all concepts of simple connectedness coincide. Theorem 4.5 (Bretscher–Gabriel, Martinez–de la Peña). Let A be a triangular representation-finite algebra. The following conditions are equivalent. (1) A is a simply connected algebra. (2) A is a strongly simply connected algebra. (3) …1 .jA j/ is trivial. We also exhibit the following deep result proved by R.-O. Buchweitz and S. Liu in [23]. Theorem 4.6 (Buchweitz–Liu). Let A be a representation-finite algebra. Then the following conditions are equivalent. (1) A is a simply connected algebra. (2) H 1 .A/ D 0. We note that there are wild simply connected algebras A (even with the separation property) such that H 1 .A/ ¤ 0 (see [89], Example 3.4). On the other hand, the following problem posed in [89] (Problem 1) seems to be still open: “Let A be a tame triangular algebra. Is it true that A is simply connected if and only if H 1 .A/ D 0?” We end this section with an example from [22], Example 1.7, showing that, for representation-infinite triangular algebras, the simple connectedness is not sufficiently strong assumption to recover the tame representation type from the weak nonnegativity of the Tits form.

460


Example 4.7 (Brüstle–de la Peña–Skowroński). Let A D KQ=I be the bound quiver algebra given by the quiver Q 7 ^= 2o == ˇ == == 8o 1o

3o

˛

4 o^= 6 == ı == == 5 !

and the ideal I of KQ generated by the relations ˛ , ˛ ı ˇ, ! ˇ. Denote by B (respectively, H ) the full subcategory of A formed by the vertices 1, 2, 3, 4, 5 and 6 (respectively, 1, 2, 3, 4 and 5). Then B is a one-point extension of the hereditary algebra z 4 by an indecomposable regular module of regular length 3 lying H of Euclidean type A in the unique stable tube of rank 4 of the Auslander–Reiten quiver H , and hence B is wild, by [82], Theorem 3. Therefore, A is also wild. Further, A is a triangular algebra with the separation property, and hence is simply connected. Clearly, A is not strongly simply connected, because the full convex subcategory H of A is not simply connected. On the other hand, the Tits form qA of A coincides with the Tits form qƒ of the bound quiver algebra ƒ D K =J given by the quiver 7 aBB 2o BB ˇ BB BB B 8o 1o

˛

5o

3 Y3 33 33 3 ı

4o

!

6

and the ideal J of K generated by the relations ˛ , ˛, !ı˛ ! ˇ. Denote by R z 4 of ƒ formed by the vertices 1, 2, 3, the hereditary full subcategory of Euclidean type A 4 and 5. Then ƒ can be obtained from R by two one-point coextensions of R (with the coextension vertices 7 and 8) by the same simple regular R-module lying in the unique stable tube of rank 2 of R , and the one-point extension (with extension vertex 6) by the simple regular R-module lying in a stable tube of rank 1 of R . Invoking again Theorem 3 of [82], we conclude that ƒ is tame (even one-parametric). In particular, we obtain that qA D qƒ is weakly nonnegative. Finally, we also note that ƒ is simply connected but clearly not strongly simply connected.

5 Critical, pg-critical and hypercritical algebras Let be a finite connected acyclic quiver and H D K the associated hereditary algebra. Then we may visualize the shape of the Auslander–Reiten quiver H of H as

461


follows 55 55 P .H /

:::

y x ~ } z { R.H / |

55 55

: : : Q.H /

where P .H / is the preprojective component containing all indecomposable projective H -modules, Q.H / is the preinjective component containing all indecomposable injective H -modules, and R.H / is the family of all regular components. More precisely, we have the following facts: (1) If is a Dynkin quiver, then R.H / is empty and P .H / D Q.H /. (2) If is a Euclidean quiver, then P .H / Š .N/ op , Q.H / Š N op , and R.H / is a P1 .K/-family of stable tubes. (3) If is a wild quiver, then P .H / Š .N/ op , Q.H / Š N op , and R.H / is a union of jKj components of the form ZA1 . 1 A module T in mod H is called a tilting module if ExtH .T; T / D 0 and T is a direct sum of n pairwise nonisomorphic modules in ind H , with n equal the rank of the Grothendieck group K0 .H / of H . Then the associated endomorphism algebra EndH .T / is called a tilted algebra of type . In the special case, when T is a direct sum of modules from P .H /, B D EndH .T / is called a concealed algebra of type . An algebra A is said to be a critical algebra if A is a concealed algebra of type N one of the following Euclidean graphs. with the underlying graph

zn D n4

8 88

z6 E

:::

8 88

z8 E

z6 E

The critical algebras are strongly simply connected and have been classified completely by quiver and relations by K. Bongartz [12] and D. Happel and D. Vossieck [48]. Moreover, we have the following theorem from [12] and [48]. Theorem 5.1 (Bongartz, Happel–Vossieck). Let A be a strongly simply connected algebra. The following statements are equivalent. (1) A is a critical algebra. (2) The Tits form qA is critical.

462


An algebra A is said to be hypercritical if A is a concealed algebra of type with N of one of the trees the underlying graph T5 8 88 88 8 z z6 E

z zn 8 D 88

:::

z z7 E

z z8 E

8 88

z z n the number of vertices is n C 2, 4 n 8. The hypercritical where in the case of D algebras are strongly simply connected and have been classified completely by quivers and relations by M. Lersch [54], L. Unger [99], and J. Wittman [101]). The following theorem from [67] justifies the name hypercritical. Theorem 5.2 (de la Peña). Let A be a strongly simply connected algebra. The following statements are equivalent. (1) A is a hypercritical algebra. (2) The Tits form qA is hypercritical. Following R. Nörenberg and A. Skowroński ([63], (3.2)) by a pg-critical algebra (polynomial growth critical algebra) we mean here a bound quiver algebra obtained from one of the frames .1/–.16/ below by operations of the following forms: (a) Replacing each subgraph ~~ ~~ @ @@ @

by ~~ ~~ @ @@ @

or

p RRRR) ppp : p p p :: : NxpNN NNN NN& ullll

463


(b) Choosing arbitrary orientations in nonoriented edges. (c) Constructing the opposite algebra. .1/ @ @

~~

:::

~~

:::

.5/ @@

~~

~~ @ @

:::

2 ~~~~ 222 : 2 22 22 0@@ : 0 0 :. :E EE"

.3/ @@

0 00 00 0

~~ @ @

:::

:::

.7/ @@

~~

* @@ * @ * @ * ~ ~~

.2/ @@

~~

:::

: : : E E |yyy E" EE E" |yyy : ::

~~ @ @

.4/

:::

3EE E yy : : : 3 " |y 3 :::

~~ @@

~~ @@

~~ @ @

2 ~~~~ 222 *0 : 0*0 . 222 : 22 0 0@* 00 2 @ 0* 0* :. : D DD "

.6/ * @ @ ~ * ~~ * @@ * * ~~~

:::

~~ @ @

:::

.8/ @@ y 6 |yy: : : ~C {ENEE ~~p "

~~ @ @

~~ @ @

464


.9/ ~ ~

...

q~ ~ ~~ ~~~ 00 0 000 :. @ : D @ .. DD " .

.11/

.13/

/ ~~ // / // // // / ;;;

. |zzz .. :. .. : .. * @ .. @ * .. * . * * :. : D DD "

@ @ / ' // '' // ' // '' // '' // '' //'' /' ~~

.:

:::

~~ @ @

.12/

:::

~~ @ @

.14/ :.

:

, ~~ ,, , : ,, z ,, |zz ,, 0 ,, 000 ,, 00 0 00 00 0 DD D" :. : @ @

.10/

: D DD " 2 2 222 2 22 22 z |zz : .

@@ - & -- & -- && -- && -- && -- && -- && --&&

;;; 2 ) 22 ) 22 )) 22 )) 22 )) 22)) ~~

..

:

.

:.

:

~~ @ @

:::

@@

@M @ @ 3 @@ 2 :. ~~ 222 : . 2 |zzz ...

~~~ .:

:::

~~ @ @

465


.15/

@ @ : .: n ~~wn~nnn~n~ 2 ) ~ 22 ) ~ ) 22 ) ) 22 ) : 22 )) . : 22)) ~~

.16/

;;;

2 ) 22 ) 22 )) 22 )) 22 )) 22)) ~~

~~ ~ ~ ~~ .:

:.

:

:

where any dashed line indicates a relation being the sum of all paths from the starting point to the end point. We note that the pg-critical algebras introduced above are strongly simply connected algebras of global dimension 2. The following special case of a more general result from [63] justifies the name pg-critical. Theorem 5.3 (Nörenberg–Skowroński). Let A be a pg-critical algebra. Then A is tame of nonpolynomial growth but every proper convex subcategory of A is of polynomial growth.

6 Coil algebras The aim of this section is to recall the coil algebras, which play a fundamental role in the representation theory of tame strongly simply connected algebras. Given a standard component of A ˇand an indecomposable module X in , the support .X / of the functor HomA .X; /ˇ is the K-linear category defined as follows (see [4]). Let HX denote the full subcategory of mod A formed by the indecomposable modules M in such that HomA .X; M / ¤ 0, and JX denote the ideal of HX consisting of the morphisms f W M ! N (with M , N in HX ) such that HomA .X; f / D 0. We define .X / to be the quotient category HX =JX . Let A be an algebra and be a standard component of A . For an indecomposable module X in , called the pivot, three admissible ˇ operations are defined, depending on the support .X / of the functor HomA .X; /ˇ . These admissible operations yield in each case a modified algebra A0 , and a modified component 0 of (see [3] for more details): (ad 1) If .X / is the path category of the infinite linear quiver X D X0 ! X1 ! X2 ! X is called an (ad 1)-pivot, and we set A0 D .A D/ŒX ˚ Y1 , where D is the full t t upper triangular matrix algebra (with t 1), and Y1 is the unique indecomposable 0 projective-injective D-module. In this case, is obtained by inserting in a rectangle consisting of the modules Zij D K; Xi ˚ Yj ; 11 for i 0, 1 j t , and

466


Xi0 D .K; Xi ; 1/ for i 0, where Yj , 1 j t , denote the indecomposable injective D-modules. If t D 0, we set A0 D AŒX and the rectangle reduces to the ray formed by modules of the form Xi0 . (ad 2) If .X / is of the form Yt

Y1

X D X0 ! X1 ! X2 !

with t 1 (so that X is injective), X is called an (ad 2)-pivot, and we set A0 D AŒX . In this case, 0 is obtained by inserting in a rectangle consisting of the modules Zij D K; Xi ˚ Yj ; 11 for i 0, 1 j t , and Xi0 D .K; Xi ; 1/ for i 0. (ad 3) If .X / is the bound quiver category of a quiver of the form / Y2 / ::: / Y t1 / Yt YO 1 O O O X D X0

/ X1

/ :::

/ X t2

/ X t1

/ Xt

/ X tC1

/ :::

with t 2 (so that X t1 is injective), bound by the mesh relations of the squares, X is called an (ad 3)-pivot, and we set A0 D AŒX . In this case, 0 is obtained by inserting in a rectangle consisting of the modules Zij D K; Xi ˚ Yj ; 11 for i 1, 1 j i , and Xi0 D .K; Xi ; 1/ for i 0. It was shown in [3] that 0 is a standard component of A0 containing the module X . The dual coextension operations (ad 1 ), (ad 2 ), (ad 3 ) are also called admissible. A translation quiver C is called a coil if there exists a sequence of translation quivers 0 ; 1 ; : : : ; n D C such that 0 is a stable tube and, for each 0 i < n, iC1 is obtained from i by an admissible operation [3]. Let C be a critical algebra and T be the P1 .K/-family of standard stable tubes in C . Following [5] an algebra B is called a coil enlargement of C if there is a finite sequence of algebras C D A0 ; A1 ; : : : ; Am D B such that, for each 0 j < m, Aj C1 is obtained from Aj by an admissible operation with pivot or copivot in a stable tube of T or in a coil of Aj , obtained from a stable tube of T by means of the sequence of admissible operations done so far. A distinguished property of a coil enlargement of a critical algebra is the existence of a P1 .K/-family of standard coils. We also note that every coil enlargement B of a critical algebra C is a strongly simply connected algebra. Recall also that a tubular extension (respectively, tubular coextension) of C in the sense of C. M. Ringel [83], (4.7), is a coil enlargement B of C such that each admissible operation in the sequence defining it is of type (ad 1) (respectively, (ad 1 )). An essential role in our considerations will be played by the following structure result proved in Theorem 3.5 of [5]. Theorem 6.1 (Assem–Skowroński–Tomé). Let B be a coil enlargement of a critical algebra C . Then: (1) There is a unique maximal tubular coextension B of C which is a convex subcategory of B, and B is obtained from B by a sequence of admissible operations of types (ad 1), (ad 2), (ad 3).

467


(2) There is a unique maximal tubular extension B C of C which is a convex subcategory of B, and B is obtained from B C by a sequence of admissible operations of types (ad 1 ), (ad 2 ), (ad 3 ). (3) Every object of B belongs to B or B C . We note that the bound quiver of a tubular extension (respectively, tubular coextension) B of a critical algebra C is obtained from the bound quiver of C by adding a finite family of branches at the extension vertices of one-point extensions (respectively, at the coextension vertices of one-point coextensions) of C by simple regular modules. Recall that a branch [83], (4.4), is a finite connected bound subquiver of the following infinite bound quiver, containing the root b, :::

:::

:::

:::

55 55 55 55 55 55 55 55 9 9 9 9 B B B B 99 9 9 9 99_ _ _ 99_ _ _ 99_ _ _ 9_9 _ _ 9 9 9 9 99 99 B B 9 99 99 99 99 99 9 99_ _ 99_ _ 9 KK s9 KK ss KK s s KK ss KK ss KK s s KK _ _ _ _ ss KK KK ssss % s b

where the dashed lines denote the zero-relations of length 2. We also note that the class of bound quiver algebras of branches coincides with the class of tilted algebras of the hereditary algebras given by the equioriented quivers ! ! ! ! of types Am , m 1 (see [83], Proposition 4.4 (2)). Finally, we point out that the bound quiver algebra of a branch is a strongly simply connected representation-finite special biserial algebra, and hence the support of any of its indecomposable modules is the path algebra of a linear quiver (usually with many sources and sinks) of type An , n 1 (see [97]). We have the following characterization of tame coil enlargement of critical algebras (see [5], Corollary 4.2). Theorem 6.2 (Assem–Skowroński–Tomé). Let B be a coil enlargement of a critical algebra C . The following statements are equivalent. (1) B is tame. (2) B is of polynomial growth. (3) qB is weakly nonnegative.

468


A tame coil enlargement of a critical algebra is said to be a coil algebra. The Auslander–Reiten quiver B of a coil algebra B consists of a preprojective component, a preinjective component and infinitely many standard coils, all but finitely many of them being stable tubes (see [5], Theorem 3.5 and Corollary 4.2). In the article, by a tubular algebra we mean a tubular extension (equivalently, a tubular coextension) of a critical algebra of tubular type .2; 2; 2; 2/, .3; 3; 3/, .2; 4; 4/, or .2; 3; 6/ (see Section 5 in [83] for a more general concept of a tubular algebra). We note that every tubular algebra is a strongly simply connected algebra of global dimension 2. We end this section with the following characterization of tubular algebras in the class of coil algebras, which follows from [83], (5.2), [88], Lemma 3.6, and Theorems 6.1 and 6.2. Theorem 6.3. Let B be a coil algebra. The following statements are equivalent. (1) B is a tubular algebra. (2) B is of infinite growth but every proper convex subcategory of B is of finite growth.

7 Strongly simply connected algebras of polynomial growth We start with the following characterizations of representation-finite simply connected algebras established by K. Bongartz in [11], [12], [13]. Theorem 7.1 (Bongartz). Let A be a strongly simply connected algebra. The following statements are equivalent. (1) A is representation-finite. (2) The Tits form qA is weakly positive. (3) A does not contain a convex subcategory which is critical. Theorem 7.2 (Bongartz). Let A be a representation-finite strongly simply connected algebra. Then the dimension vector function X 7! dim X induces a bijection between the set of isomorphism classes of modules in ind A and the set of positive roots of the Tits form qA of A. The following characterization of strongly simply connected algebras of polynomial growth has been established in [93], Theorem 4.1 and Corollary 4.2. Theorem 7.3 (Skowroński). Let A be a strongly simply connected algebra. The following conditions are equivalent. (1) A is of polynomial growth. (2) A does not contain a convex subcategory which is pg-critical or hypercritical. (3) The Tits form qA of A is weakly nonnegative and A does not contain a convex subcategory which is pg-critical.


469

The crucial role in the proof of the above theorem is played by a complete understanding the structure of the module category mod A as well as the shapes of components of A of a strongly connected algebra of polynomial growth. Following I. Assem and A. Skowroński [3], [4], a multicoil of an Auslander–Reiten quiver A is a component obtained from a finite number of coils glued together by some acyclic translation quivers, and a multicoil algebra is an algebra A having the property that every cycle in ind A consists of modules of a standard coil of a multicoil of A . Then we have the following structure theorem from [93], Theorem 4.1. Theorem 7.4 (Skowroński). Let A be a strongly simply connected algebra. The following conditions are equivalent. (1) (2) (3) (4) (5)

A is of polynomial growth. The component quiver †A is acyclic. Every component of A is standard. Every cycle in ind A is finite. A is a multicoil algebra.

The following consequence of Theorem 4.1 and Corollary 4.7 of [93] describes the one-parameter families of indecomposable modules over strongly simply connected algebras of polynomial growth and their dimension vectors. Theorem 7.5 (Skowroński). Let A be a strongly simply connected algebra of polynomial growth and M a module in ind A. The following statements are equivalent. (1) qA .dim M / D 0. (2) There are infinitely many pairwise nonisomorphic modules N in ind A with dim N D dim M . (3) B D supp M is a critical or tubular convex subcategory of A and M lies in a stable tube of B . The following theorem on the structure of the category of indecomposable modules over a strongly simply connected algebra of polynomial growth is also a consequence of [93], Theorem 1 and its proof. Theorem 7.6 (Skowroński). Let A be a strongly simply connected algebra of polynomial growth. Then there exist convex coil subcategories B1 ; : : : ; Bm of A whose indecomposable modules exhaust all but finitely many isomorphism classes of modules in ind A. Moreover, if the support D D supp X of a module X in ind A is not contained in one of the categories B1 ; : : : ; Bm , then X is a directing module and D is a tame tilted algebra. We mention that the module categories of coil algebras are rather well understood. Moreover, a complete classification of coil algebras with sincere nondirecting indecomposable modules lying in nonstable coils has been established in the paper by P. Malicki,

470


A. Skowroński and B. Tomé [57]. Hence, in order to classify the indecomposable finite dimensional modules over strongly simply connected algebras of polynomial growth, it remains to describe the supports of directing modules over such algebras. Further, if X is a directing module over an algebra A, then supp X is a convex subcategory of A [11], Proposition 3.2, and is a tilted algebra [83], p. 376. Hence, we need a classification of tame tilted strongly simply connected algebras with sincere directing modules. Those of finite representation type are classified in [10], [31], [84] (see also Section 6 in [83]). It was also shown in [69] that the representation-infinite tame tilted algebras A with sincere directing modules are at most 2-parametric (A .d / 2 for any d 1). The families of 2-parametric tame algebras with sincere directing modules and having at least 20 vertices in the Gabriel quiver have been classified in [70]. A strongly simply connected algebra A is said to be extremal (see [10]) if there is an indecomposable (finite dimensional) A-module M whose support supp M contains all extreme vertices (sinks and sources) of the Gabriel quiver QA of A. Observe that the convex hull of the support of an indecomposable module over a strongly simply connected algebra is an extremal strongly simply connected algebra. The following fact proved in [77], Theorem (extending [75], Theorem, and [76], Theorem 1) is also essential for our considerations. Theorem 7.7 (de la Peña–Skowroński). Let A be a strongly simply connected algebra satisfying the following conditions: (1) A is extremal; (2) qA is weakly nonnegative; (3) A contains a convex subcategory which is either a representation-infinite tilted zp , p D 6; 7; 8, or a tubular algebra. algebra of type E Then A is of polynomial growth. As a direct consequence of Theorems 7.3 and 7.7 we obtain the following fact. Corollary 7.8. Let A be an extremal strongly simply connected algebra with weakly nonnegative Tits form qA and containing a pg-critical convex subcategory. Then every z m , for some m 4. critical convex subcategory of A is of type D The following geometric and homological characterizations of strongly simply connected algebras have been established in [73]. Theorem 7.9 (de la Peña–Skowroński). Let A be a strongly simply connected algebra. Then the following statements are equivalent. (1) A is of polynomial growth. (2) For every module X in ind A and d D dim X , we have A .d/ D dim G.d/ dimX modA .d/ 0:


471

Theorem 7.10 (de la Peña–Skowroński). Let A be a strongly simply connected algebra. The following statements are equivalent. (1) A is of polynomial growth. (2) qA is weakly positive and ExtA2 .X; X / D 0 for any module X in ind A. (3) dimK EndA .X / dimK ExtA1 .X; X / and ExtAr .X; X / D 0 for r 2 and any module X in ind A. We note that the condition ExtA2 .X; X / D 0 means that X is a nonsingular point of the variety modA .dim X /. The following result from [98] gives another characterization of strongly simply connected algebras of polynomial growth, invoking the Tits and Euler forms. Theorem 7.11 (Skowroński–Zwara). Let A be a strongly simply connected algebra. The following statements are equivalent. (1) A is of polynomial growth. (2) qA is weakly nonnegative and there exists a positive integer m such that, for each positive vector x 2 K0 .A/, there are at most m isomorphism classes of modules X in ind A such that x D dim X and qA .x/ ¤ 0. (3) A is tame and there exists a positive integer m such that, for each positive vector x 2 K0 .A/, there are at most m isomorphism classes of modules X in ind A such that x D dim X and A .x/ ¤ 0. Moreover, we have the following consequence of the above theorem. Corollary 7.12. Let A be a nonsimple strongly simply connected algebra of polynomial growth, n be the rank of K0 .A/, and x be a positive vector in K0 .A/. Then the following statements hold. (1) The number of isomorphism classes of modules X in ind A with x D dim X and qA .x/ ¤ 0 is bounded by n 1. (2) The number of isomorphism classes of modules X in ind A with x D dim X and A .x/ ¤ 0 is bounded by n 1. For an algebra A, denote by n.A/ the number of vertices of the Gabriel quiver QA of A and by e.A/ the number of isomorphism classes of projective-injective modules in ind A. The following theorem proved in [74] provides common bounds on the values of the Tits and Euler forms on the dimension vectors of indecomposable modules over strongly simply connected algebras of polynomial growth. Theorem 7.13 (de la Peña–Skowroński). Let A be a strongly simply connected algebra of polynomial growth, and X a module in ind A. Then the following inequalities hold. (1) 0 qA .dim X / 18n.A/.

472


(2) 0 A .dim X / 2 C e.A/. (3) A .dim X / qA .dim X /. (4) If X is a faithful A-module, then qA .dim X / 2 and A .dim X / 2. We note that the assumption on A to be of polynomial growth is essential for the validity of the above theorem (see Example 3.7). We present also the following result proved in [72]. Theorem 7.14 (de la Peña–Skowroński). Let A be a strongly simply connected algebra such that qA .dim X / 2 f0; 1g for any module X in ind A. Then A is of polynomial growth. We present now some examples from [74] illustrating properties of the Tits and Euler forms of strongly simply connected algebras of polynomial growth. Example 7.15 (de la Peña–Skowroński). Let s be a positive integer, Q.s/ the quiver

s0 o

.s 1/0 o

::: o

a0 b 00 0 ˛ 00 ˇ 0 /o 1o 10 o / // / c d

::: o

s1o

s;

I .s/ the ideal in the path algebra KQ.s/ generated by ˛ , ˛ , ˇ , ˇ and all the paths of length s C 1 of the form i ! i 1 ! ! 1 ! 0 ! 10 ! ! .s i 1/0 ; and A.s/ D KQ.s/ =I .s/ the associated bound quiver algebra. Then A.s/ is a representation-infinite strongly simply connected algebra of polynomial growth (even with A.s/ .d / 1 for any d 1). Let X .s/ be the indecomposable module in mod A.s/ D repK .Q.s/ ; I .s/ / of the form

Ko

1

Ko

1

::: o

K: K :: 2 1 3 :: :: 4 1 5 0 K o 1 1 0 K 3:o 2 0 3 Ko : 4 5 0 :: :: 1 : 0 1 0 1 0 0 K K 2 3 1 415 1

1

1

::: o

1

Ko

1

K:

Then we have qA.s/ .dim X .s/ / D s C 1 2 D A.s/ .dim X .s/ /: We also note that for s D 1, the A.1/ -module X .1/ is faithful and qA.1/ .dim X .1/ / D 2 D A.1/ .dim X .1/ /.

473


Example 7.16 (de la Peña–Skowroński). Let Q be the quiver 1= 4 == ==˛ == @3 ˇ 2

@7 /5 == == == ı == 8; 6

I the ideal of KQ generated by and ı, and A D KQ=I the associated bound quiver algebra. Consider the indecomposable module in mod A D repK .Q; I / of the form K B 1 K K BB 1 |> 11 BB || 1 | BB || B 0 || / 2 2 > K 0 1 K BB | 00 BB 1 0 | BB || 01 BB || 0 | | 1 K K K: Then A is a representation-infinite strongly simply connected algebra of polynomial growth (again with A .d / 1 for any d 1), gl: dim A D 3, and X is a faithful A-module with qA .dim X / D 2 > 1 D A .dim X /: Example 7.17 (de la Peña–Skowroński). Let r be a positive integer, .r/ the quiver of the form 333r1 ˛r 333 333 ˛ 333 ˛r1 1 2 2 33 33 33 r 44 DD < 333

3 3 33 4 DD zzz 3 3

3 4 " z ! 3 3DD % 3 4 : : : 3 o < zz 3 D

33 33 44 33 D z

3 z 4 33 " 33 33 33

3 3 3 33 r ˇ1 3 1 ˇ2 3 2 ˇr1 3 r1 ˇr ˛1

'

o

;

J .r/ the ideal of K .r/ generated by ˛1 !, ˛1 %, ˇ1 , ˇ1 , ˛i ˇi i i for i 2 f1; : : : ; rg, and i1 ˇi , ˛i i1 for i 2 f2; : : : ; rg (if r 2), and ƒ.r/ D K .r/ =J .r/ the associated bound quiver algebra. Then ƒ.r/ is a representation-infinite strongly simply connected algebra of polynomial growth (even with ƒ.r/ .d / 1 for any d 1). Consider the

474


indecomposable module M .r/ in mod ƒ.r/ D repK . .r/ ; J .r/ / of the form K K

444 1 1

444 1

0 0 44 0

K 444

K FF 44

; 44

x FF 4

#

xxx0 1 0 K3 K2 K2 x; 44F4FF1F0 0

444

333 x x

x 3 44 # 44

K213 44 K

44

0 1 415 4

0 1 0 1 4

1 1 0 1 K K 2 3 1 415 0

2 3 0 405 1

K

444 1

44 0 33

44

33

4 1 0 10

01 01 o o K2 K2 K2 K 2:

444

44

44

01 4

0 1 K 1 0

:::

Then gl: dim ƒ.r/ D 2, r D e.ƒ.r/ / the number of indecomposable projectiveinjective modules in mod ƒ.r/ , and qƒ.r/ .dim M .r/ / D ƒ.r/ .dim M .r/ / D r C 2: Example 7.18 (de la Peña–Skowroński). Let m and r be a positive integers. Denote by Q.m; r/ the quiver of the form

11 21 / / ' '' ' 12 ' 22 ' ' ' ' ' .. ˛2 .. ˛1 . .' '' ' MM '' '' MMM qqqq8 1m ' 2m ' M q ! q & ; 7 qq8 ;M;MM% q ;;MMM ;;; 777 qqq & ;; ;; ; ˇ1 ;; ; 1 ˇ2 ;; 2

r1 / '' r2 ' ' ' .. ˛r .. .' .' ' ' ' ' ' '' rm ' , ;;; ;; ; r ˇr ;; ;

I.m; r/ the ideal of KQ.m; r/ generated by ˛1 !, ˛1 %, ˇ1 , ˇ1 , ˛i ˇi i1 : : : im i for i 2 f1; : : : ; rg, and i1m ˇi , ˛i i1 for i 2 f2; : : : ; rg (if r 2), and A.m; r/ D KQ.m; r/=I.m; r/ the associated bound quiver algebra. Then A.m; r/ is a representation-infinite strongly simply connected algebra of polynomial growth (even with A.m;r/ .d / 1 for any d 1) and with gl: dim A.m; r/ D 2. For any d 2 f1; : : : ; rg and i 2 f1; : : : ; mg, consider the indecomposable module


475

Xi.d / in mod A.m; r/ D repK .Q.m; r/; I.m; r// of the form d1 D1 1 / 1 / /K K K' K K' K ' d 2 D1 ' ' . ' ' 1 1 .. 2 3 ' K' K' 0 '' '' 1 405 . 0 . K' 1 . . . . 1 d i D0 .' .' .' ˛d D 0 ' ' ' 2 3 ' ' ' K' 1 415 . K EE0 K K K K 8 .. ' ' ' EE qqq qq 1 ' 1 ' 1 ' ' d m D1 " q 0 1 0 0 0 0 3 2 2 2 K K K K K ;;MMMM1 0 0 ; / ;; y< ;;ˇr D 0 1 ;; MM& ;;; // y2yy3 ; ; ; 1 ; ; ; K 415 D1 ;; ; K ; ; ; 1 ; d ; 0 1 ; 0 1 1 1 0 01 K, K K

having in the remaining vertices (if d < m) the zero spaces. Then we have that .d / • X1.d / ; : : : ; Xm are pairwise nonisomorphic; .d / • dim X1.d / D D dim Xm D x .d / ; • qA.m;r/ .x .d / / D d .

In connection to Theorem 7.13 we present now a surprising example exhibited by T. Brüstle in [20]. Example 7.19 (Brüstle). Let Q be the quiver 1 [6 66 ˛ 66

2

ˇ

ı

3o

o D 4 Z66 66 6 8 9

6

5 Z6 66 % 66

7;

I the ideal of KQ generated by , and A D KQ=I the associated bound quiver algebra. Then A is a pg-critical algebra, and hence a tame strongly simply connected algebra of nonpolynomial growth. Moreover, gl: dim A D 2, and so qA D A , by Theorem 3.6. It has been proved in [20] that for any module X in ind A we have qA .dim X / 2. We end this section with characterizations of strongly simply connected algebras of polynomial growth (respectively, finite growth) via isotropic corank of their Tits forms. Let A D KQ=I be a triangular algebra and consider the Tits form qA of A as a quadratic form QQ0 ! Q. Denote by QC the set of nonnegative rational numbers. Then ˚

rad0 .qA / D x 2 N Q0 j qA .x/ D 0 ;

476


is called the set of isotropic roots of qA , and ˚

rad0Q .qA / D x 2 .QC /Q0 j qA .x/ D 0 ; the set of rational isotropic roots of qA . A subset V of rad0Q .qA / is said to be a halfspace if ax C by 2 V for any x; y 2 V and a; b 2 QC . The dimension of V is the maximal number of Q-linearly independent vectors of V . The isotropic corank of qA , denoted by corank 0 qA , is the maximal dimension of a connected half-space contained in rad0Q .qA /. The following characterization of strongly simply connected algebras of finite growth has been established in Theorem 3.3 of [71]. Theorem 7.20 (de la Peña). Let A be a strongly simply connected algebra. The following statements are equivalent. (1) A is of finite growth. (2) The Tits form qA is weakly nonnegative with corank 0 qA 1. The following theorem follows from Theorem 2.5 of [71]. Theorem 7.21 (de la Peña). Let A be a strongly simply connected algebra. The following statements are equivalent. (1) A is of polynomial growth. (2) The Tits form qA is weakly nonnegative with corank 0 qA 2 and every convex subcategory B of A such that qB accepts an omnipresent isotropic root is either critical or tubular.

8 Tame strongly simply connected algebras The following main result proved in [22] is a natural generalization of the Bongartz Theorem 7.1 to the tame algebras, and solves the problem raised by S. Brenner more than 30 years ago [18]. Theorem 8.1 (Brüstle–de la Peña–Skowroński). Let A be a strongly simply connected algebra. The following statements are equivalent. (1) A is a tame algebra. (2) The Tits form qA of A is weakly nonnegative. For the very special class of strongly simply connected algebras formed by the tree algebras (the Gabriel quiver is a tree) the above theorem has been proved by T. Brüstle in [21]. The following direct consequence of Theorems 5.2 and 8.1 gives another handy criterion for a strongly simply connected algebra A to be tame.


477

Corollary 8.2. Let A be a strongly simply connected algebra. The following statements are equivalent. (1) A is a tame algebra. (2) A does not contain a convex hypercritical subcategory. Since the Gabriel quivers of hypercritical algebras have at most 10 vertices, we obtain also the following result. Corollary 8.3. Let A be a strongly simply connected algebra. The following statements are equivalent. (1) A is a tame. (2) Every convex subcategory of A with at most 10 objects is tame. It follows from the result proved by O. Kerner [51] that every concealed algebra of wild type is strictly wild. Then we obtain the stronger version of Drozd’s Theorem 1.2 for strongly simply connected algebras. Corollary 8.4. Every strongly simply connected algebra is either tame or strictly wild, and not both. For a positive integer d , we denote by algd .K/ the affine variety of associative algebra structures with identity on the affine space K d . Then the general linear group GLd .K/ acts on algd .K/ by transport of the structure, and the GLd .K/-orbits in algd .K/ correspond to the isomorphism classes of d -dimensional algebras (we refer to [53] for more details). We identify a d -dimensional algebra A with the point of algd .K/ corresponding to it. For two d -dimensional algebras A and B, we say that B is a degeneration of A (A is a deformation of B) if B belongs to the closure of the GLd .K/-orbit of A in the Zariski topology of algd .K/. The following important result is due to P. Gabriel [41]. Theorem 8.5 (Gabriel). For any positive integer d , the class of representation-finite algebras in algd .K/ forms an open subset. Applying Corollary 8.3, S. Kasjan established in [50] the following fact. Theorem 8.6 (Kasjan). For any positive integer d , the class of tame strongly simply connected algebras in algd .K/ forms an open subset. In the remaining part of this section we will outline the main steps of the proof of sufficiency part of Theorem 8.1, given in [22]. We start with some general results. The following important result has been proved by C. Geiss in [43]. Theorem 8.7 (Geiss). Let d be a positive integer, A and B two d -dimensional Kalgebras such that A degenerates to B in algd .K/ and B is a tame algebra. Then A is a tame algebra.

478


Following A. Skowroński and J. Waschbüsch [97] an algebra A is said to be special biserial if A is isomorphic to a bound quiver algebra KQ=I , where the bound quiver satisfies the conditions: (a) each vertex of Q is a source and target of at most two arrows, (b) for any arrow ˛ of Q there are at most one arrow ˇ and at most one arrow with ˛ˇ … I and ˛ … I . The following fact has been proved in [100] (see also [24], [36]). Theorem 8.8 (Wald–Waschbüsch). Every special biserial algebra is tame. In the study of nonpolynomial growth tame strongly simply connected algebras z n, a fundamental role is played by some enlargements of critical algebras of types D n 4. Recall from [12] and [48] that there are only four families of critical algebras z n , n 4, given by the following bound quivers of types D 2 22 22 2

~ 22 ~~ 22 : 22 . :. @@ @

:::

:::

2 22 22 2

2 22 22 2

@@@ : . 2 22 : . 22 2 ~~~

:::

@@@ : . 2 22 :. 22 2 ~~~

/ ::: / @ @@@ ~? ~ ~ ~ _ _ _ _ _ _ _ _ _m6/ QBQQ = m BB QQQ | m BB QQ( mmmmm||| BB | BB || | B! ||

means / or o . It where the number of vertices is equal n C 1 and is well known (see [83], (4.3)) that the Auslander–Reiten quiver C of a critical algebra z n consists of a preprojective component, a preinjective component, and a C of type D P1 .K/-family of standard stable tubes, two of them of rank 2, one of rank n 2, and the remaining ones of rank 1. Observe that, for n D 4, C has 3 stable tubes of rank n 2 D 2. In this paper, by a D-coil algebra is meant a coil enlargement B of a critical algebra z n using modules from a fixed stable tube of rank n2 in C . It follows from C of type D [5] (Theorem 4.1, Corollary 4.2) and results of [83], (4.9), that the Auslander–Reiten quiver B of a D-coil algebra B consists of a preprojective component, a preinjective component, a K-family of standard stable tubes (two tubes of rank 2, the remaining ones of rank 1), and a standard coil having at least n 2 rays and at least n 2 corays, and usually many projective modules and many injective modules. This coil will be

479


called the large coil of B . Observe that the large coil of B is uniquely defined, except z 4 . In this case, by the large coil we mean the case B D C is a critical algebra of type D a fixed stable tube of rank 2 of C . Clearly, a D-coil algebra is a coil algebra. We also note that every D-coil algebra contains exactly one critical convex subcategory, and is zr, a glueing of two representation-infinite tilted algebras of (usually different) types D r 4. In order to define the class of D-algebras we need also the concepts of D-extensions and D-coextensions of D-coil algebras. Suppose A and A0 are two algebras (considered as K-categories) containing a common convex subcategory B. Then we denote by ` ƒ D A A0 the pushout A and A0 along the embeddings of B into A and A0 . Observe B

that the quiver Qƒ of ƒ is obtained by glueing the quivers QA and QA0 along the quiver QB , and the ideal defining ƒ is the ideal in the path algebra KQƒ generated by the ideals defining the algebras A and A0 . Let B be a D-coil algebra and a large coil of B . By a D-extension of B we mean a strongly simply connected algebra of one of the forms: ` H , where X is an indecomposable module in such that the support (d1) BŒX K! ˇ .X / of the functor HomB .X; /ˇ is the path category of the linear quiver X D X0 ! X1 ! X2 ! ; H is the path algebra of a quiver .m/ of the form

! D a1

/ a2

/ :::

w; b ww ww / am HHH HH # c;

m 1, and K! D K is the simple algebra given by the extension vertex ! of BŒX and the unique source ! D a1 of .m/; (d2) BŒX , ˇ where X is an indecomposable module in and the support .X / of HomB .X; /ˇ is the bound quiver category of the quiver

X D X0

/ X1

/ :::

YO 1

/ Y2 O

/ Y3 O

/ :::

/ Xt

/ X tC1

/ X tC2

/ :::

with t 0, bound by the mesh relations of the squares; (d3) BŒX , where X is an indecomposable module in and the support .X / of

480


ˇ HomB .X; /ˇ is the bound quiver category of the quiver YO 2 YO 1

/ Z1 O

X D X0

/ X1

/ X2

/ X3

/ :::

bound by the mesh relation of the unique square; (d4) BŒX , ˇ where X is an indecomposable module in and the support .X / of HomB .X; /ˇ is the bound quiver category of the quiver YO 2

/ Z2 O

YO 1

/ Z1 O

X D X0

/ X1

/ X2

/ X3

/ :::

bound by the mesh relations of the two squares. A D-coextension of B is defined dually invoking the dual coextension constructions (d1 ), (d2 ), (d3 ), (d4 ). Since the class of D-coil algebras is closed under making the opposite algebras, we conclude that the class of D-coextensions of D-coil algebras coincides with the class of opposite algebras of D-extensions of D-coil algebras. We would like to mention that D-extensions of types (d1) and (d2) were applied in [63] to define the pg-critical algebras. In fact, it is rather easy to see that every D-extension (respectively, D-coextension) A of a D-coil algebra B creates a new critical algebra of z n , which can be used to create new D-coil algebras and their D-extensions or Dtype D coextensions. Finally, we mention that in general the one-point extensions of type (d2) (respectively, the one-point coextensions of type (d2 )) may contain convex hereditary z m , and hence they are not strongly simply connected (see the subcategories of type A algebras of types (17)–(31) in [63], Theorem 3.2). Therefore, the assumption that a D-extension (respectively, D-coextension) is strongly simply connected is essential for our considerations. We need also the concept of a blowup of an algebra. Let A D KQ=I be a bound quiver algebra. A vertex a of Q is said to be narrow if the quiver Q of A contains a convex subquiver of the form x

˛

/a

ˇ

/y

with ˛ˇ … I , and ˛ (respectively, ˇ) is the unique arrow of Q ending (respectively, starting) at a. For a narrow vertex a of Q, we define the blowup Ahai D KQhai=I hai

481


of A at the vertex a as follows. The quiver Qhai is obtained from the quiver Q by replacing the subquiver by the subquiver hai of the form a t9 1 tt t tt x JJ JJ J ˛2 J% a2 ˛1

JJ ˇ1 JJ JJ % 9y t t tt t t ˇ2

and keeping the remaining vertices and arrows of Q unchanged. Then the ideal I hai of KQhai is obtained from the ideal I of KQ by adding the generator ˛1 ˇ1 ˛2 ˇ2 , replacing any generator of the form u˛ by two generators u˛1 and u˛2 , any generator of the form ˇv by two generators ˇ1 v and ˇ2 v, any generator containing ˛ˇ by the generator with ˛ˇ replaced by ˛1 ˇ1 , and keeping the remaining generators of I unchanged. Further, a set S of narrow vertices of Q is said to be orthogonal if Q does not admit an arrow connecting two vertices of S. By a blowup of A we mean an iterated blowup Aha1 ; : : : ; ar i D Aha1 iha2 i : : : har i of A with respect to an orthogonal set a1 ; : : : ; ar of narrow vertices of Q. We are now in position to give a recursive definition of a D-algebra: (i) All D-coil algebras are D-algebras. (ii) All D-extensions and D-coextensions of D-coil algebras are D-algebras. (iii) Suppose A is a D-algebra and contains a D-coil algebra B as a convex subcateD-coil algebra congory. Let A0 be a D-extension or a D-coextension of B, or a ` taining B as a convex subcategory. Then the pushout ƒ D A A0 is a D-algebra B

provided it does not contain a hypercritical convex subcategory (equivalently, the Tits form qƒ of ƒ is weakly nonnegative). (iv) All blowups of D-algebras are D-algebras. We would like to mention that there is a complete local understanding of the bound quiver presentations of D-algebras. Namely, by Theorem 6.1, every D-coil algebra B is a suitable glueing of a tubular extension B C and a tubular coextension B of the z n . Moreover, by [83], (4.7), the tubular extensions same critical algebra C of type D (respectively, coextensions) of the critical algebras C are obtained from C by adding branches (in the sense of [83], (4.4)) at the extension (respectively, coextension) vertices of the one-point extensions (respectively, coextensions) of C by the applied simple regular C -modules. Further, a complete description of all simple regular modules and all indecomposable regular modules of regular length 2 (applied in the D-extensions z n is given in Section 2 of [62]. and D-coextensions) over the critical algebras of types D Finally, the forbidden hypercritical algebras are described by quivers and relations in [54], [99], [101]. We exhibit the following properties of D-algebras which will be essential in further considerations.

482


Proposition 8.9. Let A be a D-algebra. Then (1) A is a strongly simply connected algebra. (2) Aop is a D-algebra. (3) Every object a of A is an object of a convex subcategory ƒ of A which is a tubular extension or a tubular coextension of a critical convex subcategory C . (4) Every object a of A is an object of a convex D-coil subcategory B of A. The statement (4) says that every D-algebra A admits a finite family of convex D-coil subcategories which together exhaust all objects of A (an atlas of A by Dcoil algebras). We also note that the class of algebras which are tubular extensions or tubular coextensions of critical algebras and occur as convex subcategories of Dalgebras coincides with the class of all strongly simply connected representation-infinite z n , n 4 (see also [83], (4.9)). Therefore, the statement (3) tilted algebras of types D of Proposition 8.9 can be reformulated as follows: every D-algebra A admits an atlas formed by convex subcategories which are representation-infinite tilted algebras of z n and together exhaust all objects of A (an atlas of A by representation-infinite type D z n ). tilted algebras of types D The following example illustrates the above considerations. Example 8.10. Let B be the algebra given by the bound quiver 13 L L 11 1 EE 10 EE E7E EE G G 7" 9 73 s s ssLs L sss 2 14 15

/ 12

5 II II II I$ 6 19 @@ ysss /4 @@@ 7 18 >> @@ s s >> s @@ >> ysFs4ss >> 8 17 H ' >> z x HHH >> zzm H z HH >> zzz HH }z HH HH 9 H# 16 .

We claim that B is a D-coil algebra. Denote by C the critical convex subcategory of z 8 given by the objects 1, 2, 3, 4, 5, 6, 7, 8, 9. Then B is a coil enlargement B of type D of C by four admissible operations of type (ad 1), creating the sets of vertices f10g, f11; 12g, f13g, f18g, three admissible operations of type (ad 1 ), creating the sets of vertices f14g, f15g, f16; 17g, and one admissible operation of type (ad 2), creating the vertex 19. Moreover, B is a D-coil algebra because only the simple regular C -modules (the simple modules SC .3/ and SC .8/ at the vertices 3 and 8) from the unique (large)


483

stable tube of rank 6 of C are used. In the notation of Theorem 6.1, the maximal tubular coextension B of C is the convex subcategory of B given by the objects of C and the objects 14, 15, 16 and 17, while the maximal tubular extension B C of C is the convex subcategory of B given by the objects of C and the objects 10, 11, 12, 13, 17, 18 and 19. We also note that the object 17 belongs to B and B C . Consider now the algebra A given by the bound quiver 13 L L / 12 11 5 II t 20 rr MMMMM r IIII tt r t MM& r t r I$ ytt xr 6 100 G 1000 GG v GGS vv 19 GGS S vvv G# {vSvS rr 2BBB x r /4 L r/8 3 1 7 t 18 + BBB ?? BB t rr L ? t r K % ?? B! Ettt rr K r y 3 t ?? r 0 ?? 2 8 17 1700 H { x HHH' ?? { ' HH {{ n ?? 14 HH / ? }{{{{ HH H H 8 9 HH $ 15 16 . Then A is a D-algebra, obtained from the D-coil algebra B by D-extension, creating the vertex 20, and two blowups at the vertices 10 and 17, creating the sets of vertices f100 ; 1000 g and f170 ; 1700 g. Observe that A contains five pairwise different critical convex subcategories: the category C D C1 , the category C2 given by the objects 1, 2, 3, 100 and 1000 , the category C3 given by the objects 1, 2, 3, 4, 5, 6, 7 and 20, the category C4 given by the objects 1, 2, 3, 4, 5, 6, 7, 8, 170 , 1700 , 18 and 19, and the category C5 given by the objects 1, 2, 3, 4, 8, 9, 16, 170 and 1700 . We note that in A the vertices 11, 12, 13 form the branch of a tubular extension of the critical category C2 , and do not belong to a tubular extension of the critical subcategory C . Further, the convex subcategory B1 of A given by the objects 1, 2, 3, 100 , 1000 , 11, 12, 13, 14 and 15 is a D-coil algebra, which is the coil enlargement of the critical algebra C2 by two admissible operations of type (ad 1), creating the sets of vertices f11; 12g, f13g, and two admissible operations of type (ad 1 ), creating the sets of vertices f14g, f15g. Clearly, the objects 14, 15 and the objects of C form another convex D-coil subcategory of A. Finally, observe that if we take the blowup ƒ D Bh6; 10; 17i of B at the pairwise orthogonal narrow vertices 6, 10, 17, then ƒ is a D-algebra which does not contain the unique critical subcategory C of B as a convex subcategory. Theorem 8.11. Let A be a D-algebra. Then A is a tame algebra. The proof of the above theorem is divided into three main steps. A D-algebra A is said to be mild if, in the D-extensions and D-coextensions of

484


D-coil algebras applied to obtain A, the procedures (d3), (d4), (d3 ) and (d4 ) are not involved. The following proposition ([22], Proposition 4.5) is the first reduction step. Proposition 8.12. Let A be a D-algebra. Then there are a mild D-algebra A0 , canonically associated to A, two full cofinite subcategories X of ind A0 and Y of ind A, and a functor F W mod A0 ! mod A such that: (1) F is exact and preserves indecomposable modules; (2) F defines a functor X ! Y which is dense and reflects isomorphisms. Moreover, if A0 is tame then A is tame. Let B be a D-coil algebra, the large coil of B , and X an indecomposable module in . Assume that X is the pivot of an admissible operation of type (ad 1), (ad 2), or (ad 3). We say that the pivot X is maximal if does not contain a pivot X 0 of an admissible operation of the same type (ad 1), (ad 2), or (ad 3), such that .X / is a proper convex subcategory of .X 0 /. Observe that if X is the pivot of an admissible operation of type (ad 2) then X is maximal. Similarly, if X is the pivot of a D-extension of B of type (d1) then X is said to be maximal provided X is maximal as the pivot of an admissible operation of type (ad 1). Further, if X is the pivot of a D-extension of B of type (d2) then X is said to be maximal if t D 0 and does not contain a pivot X 0 of a D-extension of B of type (d2) such that .X / is a proper convex subcategory of .X 0 /. We also note that, if X is the pivot of a D-extension of B of type (d3) or (d4), then X is maximal, that is, .X / is not a proper convex subcategory of .X 0 / for a pivot X 0 2 of a D-extension of B of type (d3) or (d4). Dually, one defines maximal copivots of the dual operations (ad 1 ), (ad 2 ), (ad 3 ), (d1 ), (d2 ), (d3 ), (d4 ). A D-coil algebra B is said to be smooth if B is a coil enlargement of a critical z n invoking only admissible operations with maximal pivots and algebra C of type D maximal copivots. A D-extension (respectively, D-coextension) A of a D-coil algebra B is said to be smooth provided the pivot of the D-extension operation (respectively, the copivot of the D-coextension operation) is maximal. Finally, a D-algebra A is said to be smooth if all D-coil algebras, D-extensions and D-coextensions, occurring in the recursive definition of A, are smooth. The second step in the proof of Theorem 8.11 is the following proposition ([22], Proposition 4.7). Proposition 8.13. Let A be a D-algebra. Then there is a smooth D-algebra A# , canonically associated to A, such that A is a factor algebra of A# . In particular, if A# is tame then A is tame. Combining the procedures presented in the proofs of Propositions 4.5 and 4.7 of [22], we may associate (in a canonical way) to an arbitrary D-algebra A the mild and smooth D-algebra A D .A0 /# D .A# /0 . Moreover, if A is tame then A is also tame. The following proposition ([22], Proposition 4.9) is the third step of the proof of Theorem 8.11. Proposition 8.14. Let A be a mild and smooth D-algebra. Then A degenerates to a special biserial algebra.


485

Summing up, Theorem 8.11 is a direct consequence of Theorems 8.7 and 8.8 and Propositions 8.12, 8.13, and 8.14. The following theorem from [22] (Theorem 6.1) completes the proof of the sufficiency part of Theorem 8.1. Theorem 8.15. Let A be a strongly simply connected algebra satisfying the following conditions: (1) A is extremal. (2) qA is weakly nonnegative. (3) A contains a pg-critical convex subcategory. Then A is a D-algebra. We end this section with the following characterization of tame strongly simply connected algebra established in Corollary 5 of [22]. Theorem 8.16 (Brüstle–de la Peña–Skowroński). Let A be a strongly simply connected algebra. The following statements are equivalent. (1) A is tame algebra. (2) The convex hull of the support of any module M in ind A inside A is an algebra of one of the forms: a tame tilted algebra, a coil algebra, or a D-algebra.

9 Tame algebras and partially ordered sets In this section we provide a link between the representation theory of a wide class of strongly simply connected algebras and the representation theory of partially ordered sets, initiated in 1972 by L. A. Nazarova and A. V. Roiter [61] in connection with the study of lattices over orders and indecomposable modules over finite dimensional algebras. We refer to the book [85] for a general theory of linear representations of partially ordered sets and vector space categories. Instead of the matrix representations of partially ordered sets proposed by L. A. Nazarova and A. V. Roiter we will use the filtered linear representations of partially ordered sets proposed by P. Gabriel in [39], [40]. Let I be a finite partially ordered set with its partial order relation denoted by . For elements i; j 2 I , we write i j if i j and i ¤ j . Without loss of generality we may assume that I D f1; : : : ; ng and that i j implies i j . The Hasse diagram of I is the quiver whose vertices are the elements of I and there is an arrow i ! j provided i j and there is no t 2 I with i t j . The Tits form qI W ZnC1 ! Z of I is defined, for x D .xi / 2 ZnC1 , by qI .x/ D

nC1 X iD1

xi2

C

X ij in I

xi xj

nC1 X iD1

xi xnC1 :

486


For a finite partially ordered set I D .I; / we denote by KI the incidence algebra of I over K. Recall that KI has a basis over K all pairs .y; x/ of elements x; y 2 I with y x, and the multiplication of two basis elements .y; x/ and .y 0 ; x 0 / of KI is given by ´ .y; x 0 /; x D y 0 ; 0 0 .y; x/.y ; x / D 0; x ¤ y0: Then ex D .x; x/, x 2 I , form a set of pairwise orthogonal primitive idempotents of KI whose sum is the identity 1KI of KI . Moreover, the radical rad KI of KI is generated as K-space by all pairs .y; x/ with y x. Let I D .I; / be a finite partially ordered set. Following P. Gabriel [39], [40], an I -space ( filtered representation of I ) over K is a system V D .V; Vi /i2I , where V is a finite dimensional K-vector space, Vi is a K-vector subspace of V for any i 2 I , and Vi Vj provided i j . A map f W V ! W of I -spaces V D .V; Vi /i2I and W D .W; Wi /i2I is a K-linear map f W V ! W such that f .Vi / Wi for all i 2 I . The direct sum of V and W is the system V ˚ W D .V ˚ W; Vi ˚ Wi /i2I . An I -space V is said to be indecomposable if V is nonzero and is not a direct sum of two nonzero I -spaces. We denote by I -spK the category of (finite dimensional) I -spaces over K. We provide now a module theoretical interpretation of the category of I -spaces of a finite partially ordered set I over K. Let I D .I; /, with I D f1; : : : ; ng, be a finite partially ordered set. We denote by I the enlargement I D I [ f g of I by a unique maximal element D n C 1. Consider the incidence algebra KI of I over K and the full subcategory modsp KI of mod KI consisting of modules with projective socle. Then there is a canonical equivalence categories % W I - spK ! modsp KI (see [85], Lemma 5.1, for details). A finite partially ordered set I is said to be representation-finite if I -spK admits only finitely isomorphism classes of indecomposable objects. The following characterizations of representation-finite partially ordered sets has been proved by M. Kleiner [52]. Theorem 9.1 (Kleiner). Let I D f1; : : : ; ng be a partially ordered set. The following conditions are equivalent. (1) I is representation-finite. (2) I does not contain a full partially ordered subset whose Hasse diagram is one of the forms

K1 D .1; 1; 1; 1/ D ,

K2 D .2; 2; 2/ D

,

487


K3 D .1; 3; 3/ D

,

K5 D .1; 2; 5/ D

K4 D .N; 4/ D

7 77 7

,

.

(3) The Tits form qI W ZnC1 ! Z of I is weakly positive. The partially ordered sets K1 , K2 , K3 , K4 , K5 occurring in the above theorem are called the critical partially ordered sets of Kleiner. Let I be a finite partially ordered set. Following L. A. Nazarova [60] (and D. Simson [85]), we say that I is wild if there is a Khx; yi-KI -bimodule M such that M is a finite rank free left Khx; yi-module and the functor ˝Khx;yi M W mod Khx; yi ! modsp KI preserves the indecomposability and isomorphism classes of modules. Further, the poset I is said to be tame if, for any dimension d , there exists a finite number of KŒxKI -bimodules Mi , 1 i nd , which are free finite rank left KŒx-modules and all but finitely many isomorphism classes of indecomposable modules in modsp KI of dimension d are of the form KŒx=.x / ˝KŒx Mi for some 2 K and some i 2 f1; : : : ; nd g. It follows from the general result of Yu. A. Drozd [38] that a finite partially ordered set is either wild or tame, and not both. The following characterizations of tame partially ordered sets has been established by L. A. Nazarova [60]. Theorem 9.2 (Nazarova). Let I D f1; : : : ; ng be a partially ordered set. The following conditions are equivalent. (1) I is tame. (2) I does not contain a full partially ordered subset whose Hasse diagram is one

488


of the forms

N1 D .1; 1; 1; 1; 1/ D ,

N3 D .2; 2; 3/ D

N5 D .N; 5/ D

7 77 7

,

,

N2 D .1; 1; 1; 2/ D ,

N4 D .1; 3; 4/ D

N6 D .1; 2; 6/ D

, .

(3) The Tits form qI W ZnC1 ! Z of I is weakly nonnegative. The partially ordered sets N1 , N2 , N3 , N4 , N5 , N6 are called the critical partially ordered sets of Nazarova. An algebra A is said to be schurian if dimK HomA .P; P 0 / 1 for any indecomposable projective modules P and P 0 in mod A. It follows from Theorem 4.4 that the class of schurian strongly simply connected algebras coincides with the class of completely separating algebras introduced by P. Dräxler in [32]. It has been proved in [32] that a completely separating algebra A can be written as A D KS=I where KS is the incidence algebra of a finite partially ordered set S over K and I is an ideal of KS contained in .rad KS/2 . Then the Hasse diagram of S is the Gabriel quiver QA of A. Following P. Dräxler [32], a module V in mod A is said to be thin if the dimension vector dim V of V has coordinates 0 or 1. Further, for a vertex i of S (equivalently, of QA ), a module V in ind A is said to be a start module for i if HomA .Pi ; V / ¤ 0 and ExtA1 .V; fac Pi / D 0, where Pi is the indecomposable projective right A-module ei A at i and fac Pi denotes the family of all factor modules of Pi . Following [30], we denote by P .A; i / the set of all indecomposable thin start modules V for a given vertex i of S with the partial order defined as follows: for two modules V and W in P .A; i /, we have V W if there is a nonzero homomorphism ' 2 HomA .W; V / with 'i ¤ 0.


489

The following theorem from [30], [31], [102] gives a link between Bongartz’s Theorem 7.1 and Kleiner’s Theorem 9.1. Theorem 9.3 (Dräxler, Xi). Let A be a schurian strongly simply connected algebra. The following conditions are equivalent. (1) A is representation-finite. (2) For each vertex i of the Gabriel quiver QA of A, the partially ordered set P .A; i/ is representation-finite. The above theorem extends the characterization of representation-finite tree algebras proved by K. Bongartz and C. M. Ringel in [16]. In [34] P. Dräxler and R. Nörenberg proved the following theorem. Theorem 9.4 (Dräxler–Nörenberg). Let A be a schurian strongly simply connected algebra. The following conditions are equivalent. (1) The Tits form qA of A is weakly nonnegative. (2) For each vertex i of the Gabriel quiver QA of A, the partially ordered set P .A; i/ is tame. Therefore, we obtain the following consequence of Theorems 8.1 and 9.4. Theorem 9.5 (Brüstle–de la Peña–Skowroński). Let A be a schurian strongly simply connected algebra. The following statements are equivalent. (1) A is tame. (2) For each vertex i of the Gabriel quiver QA of A, the partially ordered set P .A; i/ is tame.

10 Tits forms with maximal roots In this section we discuss properties of triangular algebras whose Tits form admits a maximal omnipresent positive root. Let q W Zn ! Z be an integral quadratic form, say q.x/ D

n X iD1

xi2 C

n X

aij xi xj

1i<j n

for x D .xi / 2 Zn . We denote by q.; / W Zn Zn ! Z the symmetric Zbilinear form such that q.x C y/ D q.x/ C q.x; y/ C q.y/ for x; y 2 Zn . Hence, aij D q.ei ; ej / D q.ej ; ei / for i < j and the canonical basis vectors e1 ; : : : ; en of Zn . Observe that, if q is weakly positive then, by Theorem 2.3, q admits only finitely many positive roots, and hence a maximal positive root. A weakly nonnegative form which is not weakly positive usually has infinitely many positive roots. For example, it is the case for the Tits forms of critical algebras, because the dimension vectors of

490


all indecomposable modules lying in the preprojective and preinjective components of their Auslander–Reiten quivers are positive roots. On the other hand, it frequently happens that the set of positive roots of a weakly nonnegative integral quadratic form has maximal elements. In particular, it is the case for 2-parametric tame tilted algebras with sincere directing modules (see [69]). The following theorem has been proved in [33]. Theorem 10.1 (Dräxler–Golovachtchuk–Ovsienko–de la Peña). Let q W Zn ! Z be a weakly nonnegative integral quadratic form and x D .xi / 2 N n a maximal positive root of q. Then xi 12 for any i 2 f1; : : : ; ng. The bound 12 in the above theorem is the best possible (as the following example from [33] shows). Example 10.2 (Dräxler–Golovachtchuk–Ovsienko–de la Peña). Let q W Z10 ! Z be the integral quadratic form q.x1 ; : : : ; x10 / D

10 X

xi2 x1 x3 x2 x3 x3 x4 x5 x6 x6 x7 x7 x8 x7 x9

iD1

x9 x10 C 3x1 x2 and x D .1; 1; 4; 6; 8; 10; 12; 6; 8; 4/ 2 Z10 . Then q.x/ D 1. Moreover, a simple checking shows that x is a maximal omnipresent positive root of q. Clearly, in study the positive roots of integral quadratic forms we may restrict to the omnipresent roots, taking the suitable restrictions of forms. We have also the following result proved in [46]. Theorem 10.3 (Happel–de la Peña). Let q W Zn ! Z be an integral quadratic form with aij 5 for all i < j in f1; : : : ; ng. Assume that q has a maximal omnipresent positive root. Then q is weakly nonnegative. We have the following facts (see [79], Lemma 1.2 and Proposition 1.3). Theorem 10.4. Let q W Zn ! Z be a weakly nonnegative integral quadratic form. Then the following statements hold. (1) For an omnipresent root v 2 N n , we have 2 q.v; ei / 2 for any i 2 f1; : : : ; ng. (2) An omnipresent root v 2 N n is maximal if and only if 0 q.v; ei / 2 for any i 2 f1; : : : ; ng. (3) If q admits a maximal omnipresent positive root, then q has only finitely many omnipresent positive roots. Proposition 10.5. Let q W Zn ! Z be a weakly nonnegative integral quadratic form having a maximal omnipresent positive root v. Then one of the following two situations occur.


491

(1) There exists a unique index a 2 f1; : : : ; ng with q.v; ea / > 0. Moreover, then q.v; ea / D 1 and va D 2. (2) There are two indices a ¤ b in f1; : : : ; ng with q.v; ea / > 0 and q.v; eb / > 0. Moreover, then q.v; ea / D 1, q.v; eb / D 1 and va D 1, vb D 1. The vertices a (respectively, a and b) satisfying the condition (1) (respectively, (2)) of the above proposition are called exceptional indices of q. We are concerned now with the problem of realization of a positive root x of the Tits form qA of a triangular algebra A as the dimension vector of a module X in ind A. In general, this problem seems to be difficult. Example 10.6. Let A D KQ=I be the bound quiver algebra given by the quiver Q of the form 3 BB |4 BB || BB | B! }||| 5 T kkk 2 TTTTTT ˇ ˛ kkkk TTTT k TTTT kkk k k T) kk 1= = 11 == || == | | = || / / / 5 6 ı |= 7 bDD 10 DD | DD! % ||| | D | 8 9 and the ideal I of KQ generated by ˛ˇ ı . Then A is a strongly simply connected algebra and the Tits form qA D Z11 ! Z is as follows q.x1 ; : : : ; x11 / D

11 X

xi2 x1 x2 x1 x5 x2 x3 x2 x4 x2 x11

iD1

x5 x6 x6 x7 x7 x8 x7 x9 x7 x10 x10 x11 C x1 x11 : Consider the positive vector x of ZQ0 D Z11 of the form 1 1

3

1

2 1 2 1 1 1

1.

Then q.x/ D 1. On the other hand, a simple analysis shows that there is no indecomposable module X in mod A D repK .Q; I / with dim X D x. Observe that A is a wild algebra with qA negative, because the full subquiver of Q given by the vertices z z 7. 2; 3; 4; 6; 7; 8; 9; 10; 11 is a wild quiver of hypercritical type D For triangular algebras with weakly positive Tits forms, we have the following general answer (see [79], Proposition 2.3).

492


Theorem 10.7 (de la Peña–Skowroński). Let A be a triangular algebra whose Tits form is weakly positive. Then for any positive root x 2 K0 .A/ there exists a module X in ind A such that x D dim X . The following theorem has been proved in [79] (Theorem 2) using essentially Theorem 8.1. Theorem 10.8 (de la Peña–Skowroński). Let A be a strongly simply connected algebra such that the Tits form qA of A admits a maximal omnipresent positive root. Then the following statements hold. (1) A is a tame tilted algebra. (2) The dimension vector function X ! dim X establishes a bijection between the isomorphism classes of sincere modules in ind A and the omnipresent positive roots of qA . The following examples from [70], Theorem 1, illustrate possible situations. Example 10.9 (de la Peña). Let A D KQ=I be the bound quiver algebra given by the quiver Q of the form ?3@ ~~ @@@ ~ ~ ?5> @ 2 @@@ ~ @ ~~~ >>> >> ˛ >> 4 >> 1< 10 @ 4p.ƒ/s.ƒ/ and let M the (unique) since representation of B. Note that the indecomposable B-modules are string modules, they are given by words using as letters the arrows of the quiver of B. Since n > 4l.ƒ/s.ƒ/, it follows easily that there is a simple ƒ-module S such that Œsoc .M W S / 3. But this implies that there is an indecomposable B-module N which is given by a word of the form w D l1 l1 : : : l t , with arrows l1 and l t1 such that .l1 / and .l t1 / are different arrows of Q and end in the same vertex of Q (namely the support vertex of S). We denote by A the support algebra of N , it is given by a quiver of type A tC1 without any relation, and .N / has a one-parameter family of simple submodules U such that the ƒ-modules .N /=U are indecomposable and pairwise non-isomorphic. Since ƒ is minimal representation-infinite, it follows that .N / is faithful, thus we obtain all the vertices and all the arrows of Q by applying to the vertices and arrows of A, respectively. Since we now know that the vertices and the arrows of Q are images under of vertices and arrows in the support of A, it follows that the support of A contains a z In particular, we see that fundamental domain for the action of the Galois group on Q. 0 we can compose the word w with a word w such that .w/ D .w 0 / and so on. Now assume that there is a vertex x of Q with 3 arrows ending in x, say ˛; ˇ; . Q Q ending in the same vertex x. Looking at the universal cover, we obtain arrows ˛; Q ˇ; Q 0 0 0 0 But the arrow ˛Q is the first letter of a word w D l1 l2 l3 of length 3 which yields a string z Similarly, the arrow ˇQ is the first letter of a word w 00 D l 00 l 00 l 00 of length module for ƒ. 1 2 3 z But the union of the support of the words 3 which also yields a string module for ƒ. z 7 . This contradicts z of type E w 0 , w 00 and is a subquiver (without any relation) of Q z is representation-finite. the assumption that any finite subcategory of ƒ The dual argument shows that any vertex of Q is starting point of at most two arrows. Now assume that the vertex x of Q is endpoint of the arrows ˛ ¤ ˇ and starting point of the arrow and that neither ˛ nor ˇ is a zero relation. We looking again at the universal cover, and obtain arrows ˛, Q ˇQ ending in a vertex x, Q as well as Q starting in the vertex x. Q Since ˛ is not a zero relation, we see that Q ˛Q must be a subword of w or w 1 , in particular we can prolong it to a word w 0 D l10 l20 l30 l40 of length 4 so that we have a corresponding string module M.w 0 / Similarly, we prolong Q ˇQ to a word w 00 D l100 l200 l300 l400 of length 4 with string module M.w 00 /. We consider the union of the z again of type support of the words w 0 , w 00 ; it is a subquiver (without any relation) of Q, z E7 , thus again we obtain a contradiction to the assumption that any finite subcategory z is representation-finite. of ƒ

The minimal representation-infinite algebras which are special biserial

517

10 Further examples First, we present a second example of a barbell algebras with non-serial bar. Example 3. Start with D .C/, D .C/, 0 D .C C/. Then H. 0 1 / has the following shape:

˛1

˛7 ˛8 6 a0 ..................................... a7 ...................................... a6 ................˛ .................... .. a

. .......... . .. ..... ... ... ..

............... .......................

a1 ..................................... a2 ...................................... a3 ˛2 ˛3

˛4

5.........................

a4

...... ... .. ..... . . . . . . . ............

˛5

and B. ; ; 0 / will be .................... ....... ... ..... ............. ... .. .... . ... ... .. 1 .................................. ... .. .. ... .. ... ..... ........ ............ ...........

a

a2 ...................................... a3

Again, the bar (given by the arrows a1

. .......... ...... ...... ...... . ...... ... . .......... ... ....... . ...... ...... ...... ..

a5 ............................ a4

... ... ... .. .. .. . . . . . . . ..... ............

a2 ! a3 / is not serial.

The next examples are wind wheel algebras. Example 4. The wind wheel algebra for the word w

3.. . . . . . .

...... ...... ...... ...... ...... . . . . . .

6 ...........

u1

2.. ........... . . . . . . . . . .

1.. . . . . . .

3

.... . ...... .. . . . . . . 2 ...

v1 u

4

. ..... ..... ..... .. ........... ...... . ...... . ...... . .. . . . . . . . 2 ... 3

v

u

5

1

.... . ...... .. . . . . . . . . .

2

. ..... ..... ..... .. ...... . . . . . . . . . . 3 ... 4

v

u

The permutation is D .13/.24/. The short zero relations are u1;! u3C1;1 u2;! u4C1;1 u3;! u1C1;1 u4;! u2C1;1

D 6 2 4; D 1 3 6; D 5 1 3; D 2 4 5:

The long zero relations are u1;! v1 u1 3;!to D 6 2 1 5; u2;! v2 u1 4;!to D 1 3 4 2; 1 1 u3;! v3 u1 1;!to D .u1;! v1 u3;!to / ; 1 1 u4;! v4 u1 2;!to D .u2;! v2 u4;!to / :

4.. ........... . . . . . . . . . .

3.. . .

v4 ....

518

C. M. Ringel

The corresponding wind wheel algebra W D W .w/ is 6

.. ........ ....... ....... ....... ....... ....... ....... ....... ....... . . . . . . ....... . ............ .... . . . ...... . ... ......... . .. .. . . ... ........ . ...... ....... . ........ . . . ... . ........ ..... ... .... ........ ........ ........ ... ........ ... ........ ........ . . ... . . . ... . . . . ............. ... . . . . . . ... . . ... ..... ............. . . . . . . . ... ........ ... ..... . . . . . . . . . . . ... ........ ... ..... . . . . . . . . . . . ......... ... ....... ......... . . . . . ......... . . . . .. .. ... .. ....... .. .. ........ ....... . ....... ....... ....... ....... ....... ....... . . . . . ....... . . ........... ........ . . . ...... ......

2

3

˛

"

1

4

5

with the further relations 6 ! 2 ! 1 ! 5 and 2 ! 4 ! 3 ! 1. There are two bars, they are given by the arrows ˛ and . In order to construct W , one starts with the following orientation sequence: . . . . . . . . . . .

. . . . . . . . . . .

C

1

1

. . . . . . . . . . .

C

2

. . . . . . . . . . .

. . . . . . . . . . .

C 2

C

3

. . . . . . . . . . .

C 3

. . . . . . . . . . .

4

. . . . . . . . . . .

4

. . . . . . . . . . .

and constructs the hereditary algebra H. 1 1 : : : 4 4 /:

4...

..... ..... ..... ............ . . . ...

2 ........................... 1 ........................... 5....... ..... .... ..

4...

... . ....... ..

... . ....... ..

a0 D 3. ...... .... ... . ..

a1 D 6 ........................... 2 ........................... 1

.... ..... ..... ..... . . . . . . . .....

3

Thus, we start with a quiver which can be drawn either as a zigzag (with arrows pointing downwards), where the left end and the right end have to be identified, or else as a proper cycle: .... .... .... .... .... .... .... .. . .. . ..................... ...................... ... . . . ............. .... .... .... .... .... ...... ........ . . . ........................... ...................... ............. ...... ...... ....... ....... .................... .................... .............. .............. ....... ....... ........................ ........................ ............. ............. ............. ............. ....... ....... ......................... ................. .... .... .... .... .... .... ................ . . . . . ... ......... . . ...... . ................... ..................... . .. ... .... .... .... .... .... .... ....

.... .... .... ..... ......... .... .... ................. ... ..... .. . ............................... . . .... . . . . . . . .......... ......................... ... .............................. .... ..... ..... .... .... .............................................................. ................................ .. ..... ...... .... ..... . . . . . . . . . . . . ... . . . . ...... ......................... ..... ... .... ...... .. .. .................................................... ....................... .. ........... ...... .... ......... ...... . . .. ........................... . . ...... ..... .... .. ......................... .......... . .... ... .......................... .... ... ... .......... .......... ... .... . .................. ... . .... ... ......... . . . .... . . ...... .. ... .

and we barify on the one hand the two subquivers which are enclosed in rectangular boxes, on the other hand also the two subquivers with shaded background. In both cases, the barification yields an identification of a projective serial module of length 2 with an injective serial module of length 2. Example 5. We start with the following orientation sequence: . . . . . . . . . . .

1

. . . . . . . . . . .

1

. . . . . . . . . . .

C

2

. . . . . . . . . . .

C 2

. . . . . . . . . . .

3

. . . . . . . . . . .

3

. . . . . . . . . . .

C

4

. . . . . . . . . . .

C 4

. . . . . . . . . . .


519

and construct the hereditary algebra H. 1 1 : : : 4 4 /:

3.......

...... ....... ..... .... . . . . ..... .....

3 ........................... 4 ...........

....... ..... ..... ..... .

.... ... ....

2... ... . ....... ...

a0 D 4 .... ...

... ..... ..... ..... ....... ........

..... ..... ...... ..........

a1 D 2 ........................... 1

1

The quiver which we obtain is ................................ . ............ . ....... .. .............. . ..... ... ... ............... .......... ... ... . ............................................ ... ... ... ......... ......... . .

2

4

1

3

...... .......... . ............... ..... .. .. ..... ... ... ... .... ........ ........... .......

...... .......... . ............... ..... .. .. ..... ... ... ... .... ........ ........... .......

with the additional relations: 4 ! 2 ! 1 ! 1 and 2 ! 4 ! 3 ! 3. The bars are given by the arrows 2 ! 1 and 4 ! 3. Example 6. As in Example 4, consider again . . . . . . . . . . .

C

1

. . . . . . . . . . .

. . . . . . . . . . .

1

C

2

. . . . . . . . . . .

C 2

. . . . . . . . . . .

. . . . . . . . . . .

C

3

. . . . . . . . . . .

C 3

4

. . . . . . . . . . .

4

. . . . . . . . . . .

But now we take D .12/.34/:

4...

.. ..... ..... ..... ....... ........

4 ........................... 3 ........................... 5...... ..... .. ... .

2...

... . ....... .

... . ....... .

3...... .... .. .. ...

6 ........................... 2 ........................... 1 Here is the quiver:

....... ... . ... ....... .. ................. ......... . . ....... .. ........ ......... .....

2

1

........ ....... ........... .... ... ... ... .... . ... .. .. ...... ......... ...... ........ .....

........ ....... ........... .... ... ... ... .... . ... .. .. ...... ......... ...... ........ .....

1.....

.. ..... ..... ..... ....... ........

5

4...

... ... ......... . . . .............. ... . . .......... .......... ...........

6 ................ .

3

with the additional relations 6 ! 2 ! 1 ! 1 and 5 again the arrows 2 ! 1 and 4 ! 3.

3

4

4. The bars are

520

C. M. Ringel

Part II The module categories 11 The cycle algebras We consider the algebras H D H. / where is not constant, so that H is finite dimensional. 11.1 The Auslander–Reiten quiver. The structure of the Auslander–Reiten quiver of H is well-known: there is the preprojective and the preinjective component, the remaining components are regular tubes. The string modules form four components of the Auslander–Reiten quiver, namely the preprojective component, the preinjective component, and two tubes (we call them string tubes); the remaining components (the band tubes) are homogeneous. Note that the string tubes may be homogeneous or exceptional! The Auslander–Reiten quiver of H looks as follows:

...................................................................... .... .... .... .... ... ... ... ... ... ... ... ... ..................................................................... .... .... .... ....

P

... ... .. .. ... ... ... ... . . ..... .... ... ... ... ... ... . . ... .... ... ... ... ... ... ... ... ... ... ... 0 ..... ... ... .... ... ... ... ... ... ... ... ... ... ................................................

R

... ... .. .. ... ... ... ... . . ..... .... ... ... ... ... ... . . ... .... ... ... ... ... ... ... ... ... ... ... 1 ..... ... ... .... ... ... ... ... ... ... ... ... ... ................................................

... ... .. .. ... ... ... ... . . ..... .... ... ... ... ... ... .. ... ... ... ... ... ... ... .. ... ... ... ... ... .. ... ... ... ... ... .. ... ... ... ... ... ... ... .. ...............

R

string tubes

... ... ... ... ... ... .

... ... .. .. ... ... ... ... . . ..... .... ... ... ... ... ... .. ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... .. ...............

.............

.............

R

.............

... ... .. .. ... ... ... ... . . ..... .... ... ... ... ... ... .. ... ... ... ... ... ... ... .. ... ... ... ... ... .. ... ... ... ... ... .. ... ... ... ... ... ... ... .. ...............

.... .... .... .... ...................................................................... .. .. ... ... ... ... ... . .... .... .... .... ......................................................................

Q

band tubes

... .... ... ... ... ........................................................................................................................................................................................................................

preprojectives

the regular modules (tubes indexed by P1 .k/)

preinjectives

One should stress that in this case all the Auslander–Reiten components are (considered as simplicial complexes, thus as topological spaces) surfaces with boundary; all are homeomorphic to Œ0; 1ŒS 1 . 11.2 An example. We consider D .C C C C / and depict here, for later reference, the four components which contain string modules, always we draw two fundamental domains inside the universal cover of the component, they are separated by dashed lines (horizontal ones for the preprojective and the preinjective component, vertical ones for the regular components). Note that instead of arrows, we show edges, the orientation is from left to right.


the preprojective component P

........... ............... ............... ............ ..... .......... .......... .......... ........ ........ ........ ........ ..... .......... .......... .......... ..... ............ ............... .............. .............. ....... . . . ... .... .... .... .... .......... .......... .......... .......... .......... ......... ................. ................. ................. ................. ......... .......... .......... .......... .......... .......... ..... ......... ......... ........ ......... .......... .......... .......... .......... ..... ..... ........ ........ ......... ..... .......... .......... .......... .......... ..... ........ ........ ......... ......... .......... .......... ..... ......... ............... .............. ........... ..... ..... ..... .......... ..... ..... ............ ................. ................ ................. . . . . ......... .............. ............. .............. ....... . . . . . . . . . . . ..... ........ ........ ........ ........ .......... ......... ......... .......... ......... ........ .............. ............... ............... ............... ........ .......... .......... .......... .......... .......... ..... ........ ........ ........ ........ .......... .......... .......... .......... ..... .... ........ ....... ....... .... .......... .......... .......... .......... ..... ........ ........ ........ ......... ......... ......... ..... ..... .... ..... .... ..... .... .....

521

the preinjective component Q

.... ........ ............... ............... ............... ........ .......... .......... .......... .......... . . . . ......... ......... ......... .... ........ ............... ............... ............ .......... .......... .......... ...... ......... ..... . . . . . . . . ......... ................ ................ .......... .......... ..... . . . .......... .......... .......... ......... ................. ................. ............. .......... .......... .......... ..... ......... ............. ............. ............. . . . . .... ........... ...................... ...................... ...................... ............ .......... .......... .......... .......... . .. .. .. ......... ......... ......... ..... ........ ............... ............... ........... .......... .......... .......... . . . . . . . . .... .... .... .... .... ........ ............... ............... .......... .......... ..... . . . ......... ......... ......... ........ ............... ............... ........... .......... .......... .......... ..... ...... ......... ......... ......... . . . . . . . . . . . .... ........ ............... ............... ............... .........

the two exceptional tubes: ..... ...... ...... ...... ........ ...... ...... ...... ..... ................ ............. ............. ............................. ............. ............. ................ . . . . ......... .......... .......... .......... . .......... .......... .......... .......... ........... ............... ............... ............... ................ ................. ................. ................. ........... .. ........... .......... .......... ........... .. ........... ............ ............ ........... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . ..... ......... ......... ......... .......... ........... ................. ................ ................ ................. ................ ............... ................ ........... ... ............... ............... ............... ............... ... ............... ............... ............... ............... ... ......... .............. ............. ............. ............... ............. ............. .............. ......... ..... ......... ......... ......... ......... ......... ......... ......... ..... . ... ................. ................ ................ ................ ... ................ ................ ................ ................. ... ......... ............. ............. ............. ............... ............. ............. ............. ......... ..... .......... .......... .......... .......... ......... ......... ......... .... ... ................ ................ ................ ................ ... ................ ................ ................ ................ ... ......... . ............. . ............. . ............. . ............... . ............. . ............. . ............. . .........

R0

..... ...... ...... ...... ........ ...... ...... ...... ..... ................ ............. ............. ............................. ............. ............. ................ . . . . ......... .......... .......... .......... . .......... .......... .......... .......... ........... ............... ............... ............... ................ ................. ................. ................. ........... .. ........... .......... .......... ........... .. ........... ............ ............ ........... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . ..... ......... ......... ......... .......... ........... ................. ................ ................ ................. ................ ............... ................ ........... ... ............... ............... ............... ............... ... ............... ............... ............... ............... ... ......... .............. ............. ............. ............... ............. ............. .............. ......... ..... ......... ......... ......... ......... ......... ......... ......... ..... . ... ................. ................ ................ ................ ... ................ ................ ................ ................. ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ........... ................. ................. ................. ................... ................ ................ ................ ........... . . . . . . . . . . . . . . . . ... ................ ................ ................ ................ ... ................ ................ ................ ................ ... . .. . .. . .. . .. . . .. . .. . .. . .. . . ...... . .......... . .......... . .......... . ........... . .......... . .......... . .......... . .......

R1

11.3 The serial modules for a cycle algebra. Recall that a module is called serial provided it has a unique composition series. We consider the serial H -modules, where H is a cycle algebra. The following assertions are easy to verify: Lemma. Any serial H -module M is projective or regular or injective. If M is a serial H -module of length at least two, and M is projective, then M= soc belongs to R0 or R1 . If M is a serial H -module of length at least two, and M is injective, then rad M belongs to R0 or R1 . In the barification process, we barify a projective serial module M.b/ of length at least 2 with M.b/= soc say in R0 and an injective serial module M.b 0 / of the same length with rad M.b 0 / in R1 . Our convention for distinguishing R0 and R1 will be the following: given an indecomposable projective H -module P with radical rad P D X ˚ X 0 , where X and X 0 are serial modules, we fix the order X , X 0 and assume that the module P =X 0 as well as the composition factors of X 0 = soc are simple regular objects of R0 , whereas the module P =X as well as the composition factors of X= soc are simple regular objects of R1 : .. ........... .............. ..... . . ... .. . ...... ... ... .. ..... ........ ..... ..... .. .......... .... .. . . . . . ............... ..... ..... ..... .. .... .... ............... . . . . . ... ... . ... .. ... ... .. . R0 R1 ............. .... .............. ... . .. ... ..... ..... ... ... ... ... ... ... ... ... ...... ........ ... ... ... ... ... ... ............. .... ......... ... ..... .... . . . ... . . . . .... ... ......... .. .... ... ... ..... ......... . . . ... .. .. .. . . . ... . .. ... . ... ... . ... .. ..... . . .. . . ... .. ... ... .... .... ... ... ... ... .... .... . . . . ..... ... ..... .........

. ..

.. .

......... .. .. ..... ..... ... ............ ... ... ..

. ............ ..... .... .... ........... ... . . ...

... . ... ... . ... ... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ...........

522

C. M. Ringel

12 The wind wheel algebras z and W H a corresponding wind wheel algebra. We Let H be hereditary of type A, consider the restriction functor W mod H ! mod W . First, let us recall the shape of the category mod H . preprojectives

the regular modules

string tubes

........................................................... .... .... .... ... ... ... ... ... ... ... ........................................................ .... .... ....

P

preinjectives

...................................................................................................................................................................................... . . .... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ......... ....................................... .. ... ... .............................................................................................. ... ... .. ................................................................................................................. .. .. .. ... ....................................................................................................... ... ... ... .. ......... ....................................... .. .... ..... .... .................................................................................................... ..... .. ... ... ... ............................................................................................ . .. . ............................................................. ... . . .... .... .... .......................................................................................................................... .... .... .... ......................................................... ... ... ... ... ... ... ........................................................................................... ... ... ... .... ... ............................................................................... .. ... ... .. ... ......................................................................................... . ... . ................................................................ ... ... . . . .. . ................................................................ . . ... . ... . 0 ... .. 1 ... ........................................................................................ ... ... . . . . . . . .. . . . . . . . . . . . . . . . . ... . . ... .. .... .... .... ................................................................................................ . .. . ............................................................. .... .... .... ........................................................... .. . . . .... ................... . .... .... .... .... ............................................................................................................ ... ... ... ... ............................................................................................ ...................................... ...................................... ............................................................................... .......................................................

R

band tubes

R

R

Q

We have shaded the homogeneous tubes: they remain untouched; whereas the other four components are cut (between rays or corays) into pieces and these pieces are embedded (with some overlap) into a component which contain in addition so-called quarters. This cut-and-paste process will now be explained. 12.1 Example. We consider the wind wheel algebra W .w/ for the word w D ˛ˇ1 ˇ2 ˇ3 1 ˇ31 ˇ21 ˇ11 ; thus we start with the quiver H D H. .w// 10

ˇ10

ˇ20

20

ˇ30

30

B .......................... B .......................... B .................. ..... .......

... ............

B ...................

....... .....

0

˛

B ........................... B ........................... B 1

ˇ1

2

ˇ2

....... ..... ... ....... . . . . . . . ...........

3

B 4

ˇ3

and barify the subquivers 1 2 3 4 and 0 10 20 We obtain in this way the wind wheel algebra W D W .w/ ˛

........ ......... ........ ........... ....... ........... ..... .... ... ........... ... ... .. ... 1 2 3 .. . .... . . . . . . . ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... . ... ... ..... ... .. .. ... ............................. . . . . ... ... . . .. . . . .. . . .. . . ...... ...... ...... .... ...................... ...............................

B

ˇ

B

ˇ

B

ˇ

B

30 .

with ˛ 2 D 2 D ˛ˇ1 ˇ2 ˇ3 D 0. Recall that we know: There is precisely one non-periodic (but biperiodic) Z-word, namely r.b/ D 1 .w 00 / ˛ 1 ˇ1 ˇ2 ˇ3 1 .w 0 /1 where w 0 D ˇ31 ˇ21 ˇ11 ˛ˇ1 ˇ2 ˇ3 1 , and w 00 D ˛ 1 ˇ1 ˇ2 ˇ3 ˇ31 ˇ21 ˇ11 .

523


w 00

........................ .. .. .... ˇ ........ . .... . . . 3 ....... ......... ..................... ..... ..... ...... ............... ................ ..... ........... .... ..... . . ..... .............. ..... . . ... ˇ3 ..... ˇ3 ........ ................... ˇ3 .................. . ..... . . . . . . . . ˇ . . . . . . . . . . . ..... ... 2 ...... ..... ..... ..... ............... ..... ..... ..... . ........ ..... ..... ................... . . . . . . ..... . . . . . ..... . . . . . . . . . . . . . .. ˇ2 . ...... . ˇ1 .......................... ......................... ˇ2 ˇ.. 3 ........... ..... ˇ2 ..... . ..... . ..... . . ......... ......... . .................. ............. . . . . . .. . ..... ..... . ......................... . . ..... . ˛ ... ˇ1 . ˛ ........ ˇ1 . . ˇ2 ............. .... ˇ1 . . . . . . . . ..... ....... . . . . .. .. . . . . . . . . ˇ1 .......... ˛ . . . . . . . . . . . . . . 00 . . 0 . . . . . . . . . . . . . .

bN

w

w

w0

with b D ˇ1 ˇ2 ˇ3 and bN D 1 ˇ1 ˇ2 ˇ3 ˛ 1 (the word w 0 is obtained from w by rotation, the word w 00 by rotation and inversion). Here we see the reason why we call these algebras the wind wheels: We consider the word r.b/ as a pair of opposite “rotor blades”. 12.2 Proposition. The restriction functor W mod H ! mod W has the following properties: (1) Indecomposable modules are sent to indecomposable modules. (2) Corresponding modules on the two A4 -quivers which yield the bar become isomorphic, otherwise non-isomorphy is preserved. (3) The indecomposable W -modules which are not in the image of the functor are the string modules for words which contain ˛ 1 ˇ1 ˇ2 ˇ3 1 as a subword. Note that bN D ˛ 1 ˇ1 ˇ2 ˇ3 1 is the closure of the bar b D ˇ1 ˇ2 ˇ3 , as defined in Section 6. In order to outline the cut-and-paste process, we start with the Auslander–Reiten components containing string modules, as shown above. We assume that R0 is the tube which contains at the boundary the simple H -modules 1, 2, 3 as well as the serial module with composition factors 0, 10 , 20 , 30 , 4, whereas R1 is the tube which contains at the boundary the simple H -modules 10 , 20 , 30 and the serial module with composition factors 0, 1, 2, 3, 4. The following two pictures show the full subcategories of mod H with modules with support in f0; 1; 2; 3g on the one hand (see the left picture) as well as those with support in f10 ; 20 ; 30 ; 4g on the other hand (the right picture) and describe the role of the various modules inside mod H : M.b/

........ ...... ........ . ... .... ........ .... ......... ......... . . ........................... . . . ..... ............ ....... .... ..... ..... ...... .......... .... ..... ............. ......... ..... . . .... ............ ......... ..... ................. .... . . ..... . . . . . . . . . . . . ....... . . ..... ......... ... ....... ........... . . . . . ... . ... . . . . . . . . . . . ..... ..... ..... .................................. ........................ ........ ... .... ........ ........ ................... ..... . . . . . . . . . . . . . . . . . . ... ....... ....... ... ..................................... ........... ......... . . . . . . . . ..... .......... . ...... . . ....... . .......... ......... ..... ..... ..... ..... ......... ................. ..... ..................................................................................... ......... . . . . . . . . .. .. .. .. .. .. ... .. .. .. .. .. ..... ..... ..... .................. ............................................................................................... ......... . . . ... .... . . ..... ......... ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ..... ..... ...... . . . . . . . ....... ............ . . . ..... . . ..... ... .... ..... ......... . . . . . . . . ............ ............ . . ......... . ..... ..... .... ..... ........ . . . . ..... .. .. .. ... .. ..... ............................................................................................................ ....... .......... ................. ......

M.b/= soc

0

1

2

3

M.b 0 /

........ ...... ........ . ... .... ........ .... ......... ......... . . .......................... . . . . ..... ......... ........ ...... 0 .................................. ..... ......... ... ....... .... .... ............ ........ ..... . .... ................... .... . . . . .......... ........ ........ . ... ........................ ..... . . . . . . . ..... . . . . . . .. .. .. .. .. . .. ..... ................................................................... ......... ......... ........ .. .. . . ..... ..... . .. ..... ......... ........ ..... ...................................................................... ......... . . . . .... ..... ......... ... .......................................................... ....... . . . . . . . . ........ ....... ........ ... ................................................................. ....... . . .... . . . . .. . . . ..... ..... ..... .. .. .. .. .. .. .. .. .. ..... ......... ........ ..... ......................................................... .. . ..... ..... ................................................................................. ................ . . ..... ......... ...... . . .... . .... ......................................................................................... ......... ..... . ..... .. .. .. ... .. ..... .............................................................................................................. ..... .... . . . . . . . . . . .................... ................

rad M.b /

10

20

30

4

524

C. M. Ringel

In the left picture, the shaded area marks those modules which belong to R0 , whereas the remaining modules (those which form the left boundary) are projective, thus in P . In the right picture, the shaded area marks those modules which belong to R1 , the remaining ones are injective, thus in Q. According to property (2), any module of the left triangle is identified under with the corresponding module of the right triangle. Let us look at the various components of mod H which contain string modules. We will add bullets in order to mark the position of the indecomposables with support contained either in f0; 1; 2; 3g or else in f10 ; 20 ; 30 ; 4g. We are going to cut these components into suitable pieces: these are the dashed areas seen in the pictures. First, we exhibit the preprojective component (left) and the preinjective component (right): P

..... .... .... .......... .... .... ..... .... .... ........ .... .. . ................................................... ....................................................... ..... ................................................................................................................... . . . . . . . ... .................................................................................................... . . . . ................................................................ ..... .. . .. ........ .............................................................. ............. ..... ........................................................................................................... .......... . . . ........... . . . .......... . ................. ..... ............................................. ......................................... . . . . . . . . . . .... ........................................................................ .......................... ........................................... .................................... ..... ......................................................................... . . . . . . . .. . . . . . . . ............................................. ............................................ ............................................. ........ ........................................................... . . ........................................... .................................. . .. . . . .. . . ............................................. ............................................. ......... ............................................. ................................................. .......................... ........................................... ....... ............................................. ...................... ............................................. ......... ............................ ............................................... .... ............................................. ............. ............................................. ......... ............. .................................................. ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... .... . . . ......... . . . .... ... ................ .... .. ..................................................... . ..................................................... .... ....................................................................................................................... . . . . ..... ........................................................................................................... ............................................................... ..... . . . . . . . . . . . . . . . .. . . . . . . ....... ..... .......................................................... ............. .......................... ........ ........................................................................................... . . . . ..................................... . . ...................................................... . . .................................................... ......... ........................................................................................ . . . ........................................ ....................................................................... .................................................................. ............................................. . . . . ................................... ............................................. ..... ................................. ............................................. .... . . . . . .. .. . . ............................................. ............................................. ......... ............................................ .................................................. ......................... ........................................... . . . . . . .. ............................................. ...................... ............................................. ......... ............................ ............................................. .... ............................................. ............. ............................................. ......... ............. ................................................. . .... .... .... .... .... .... .... ..... .... .... ........ .... ..

.... .............. ...... ..................... .... .................. ...... ....................... ...... .......... ......................... ....... ................................................................. .................................................. ............... .. . . . . . . .. . . . . . . .. ............. ........ ...... ............................................. ..... ..... ........................................................................... ............................................ ......... ........................................... . ..... . . . ..... ................................................................. .... . . . . . .... . . . . . .... ......................................... .......................................... ............................................ .......................................................................... .................................................................... ................................................ ...................................................... .......... .. ..... . ..................................................................... ......... . ......................................................................... ....... ........................................................ ....... ....................... ........................ ................................... ................................. .......................... . . . . . . .. . .. . . .. . . . . . . .. . . ...................................... ....... ...................................................... .... ................................................................ .... ...................................................................................................... ......................... ....... ................................................................... ................................................ ............... ............. .... . . . . . ... . . . . . . .. ........ ...... ........................................... ..... ..... .......................................................................... .... . . . . . .... . . . . . ...... ......... ................................................................................ ..... . . . . ..... .................................................. ............................................ .. . . . . . . . . . . . ............................................. .............................................. ......................................................................... ...................................................................... ................................................ ........... ...... . ......................................................................................... ....................................... ..... . ....................................................................... ....... ....................................................... ....... ....................... ....................... ................................... ................................ ..................................... . . . ............................ ..... . .... ..................................................... ....................... .... .............. ...... ..................... .... ................. ...... ....................... ...... .........

Q

Next, the two regular components (the boundary of any of the two components contains three simple modules; they are labeled): R0

.. .. .. .. .. .. ...... ................................................................................................. ............................................................................................................................ . . ....... . . . . . . .......................................................... .................................................................. ..... .. ....................................................... ..................................................................... ............................ . . . ........................ ...................................................................... .................................................... ................. ........ ............................................................................................................ ......... ............................................................................... .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .......................................... ............................................. .................................................................... ............................................ ......... ............................................................................................................ ......... ................................................................ ....................... ... .................................................................................... ............................................ ... ... . ... ... ... . .... ........................................................... ..... ................................................................ .... ............................. .................................................... ...... ....................................................................................................................... ........ .................................................. .. . ................................................................... ............................................................................ ........................................... .. ........................................................................ . ............................................... . . . . . . . .................................................................. ..... .................. . . . . . . . . . . . . . . . . .. ................................................................. .............. .......................................................................................... ............ ................................................................. . . . . . . ................................................................................... .................................................. ....... ....................................................................................................... ......... ............ ...... . . . . . ...... . . . . . ..... . . . . . .... .... .............................................................................................................. ............................................................. .... ... .................................................................................................. ........ ... ........................................................................................................... ........ ... .. ................................................................................................... .. .................................................................................................. .. . ..... .............................................................. . .... ................................................................ . . . . . . . . . . . . . . . . . ... ..... .................................................................................................................................. .......................................................................................................................... .. . . . . . ..... . . . . ... . . . . . . . ... . . . . . ... . . . . . ... . . . . .... .......... ................................................................. ...................................................................... ....... . . ..... ........................................................................................................... ......... .............................................................................................. ................................................................... ..................................................... .......................... .... ........................................................... .... .............................................. ........................... ............... ........ .................................................................................................. ........ ........................................................................ ........................... . . . . .................................................................... . . . . ............................................

1

2

3

1

2

3

R1

.. .. .. .. .. .. ....... . . . .......... ........................................................................ .............................................................................................................. . . ................... ..... . . . . . . ..... .................................................. ..... ......................................................................... ..... .................................................... ................................................................. ........................... ......... . . . .......... . . . .......... . .. ... .... . . . .... .... . . . .... .... . .. . .... ... ............... ....... ..... ........................................................................................................................... ......... ........................................................................................................... ..... .................................................................. ...... . . . . . ...... . . . . . ..... . . . . . ... ........... ..... . . . . . ..... . . . . . ..... . . . . . ..... ............................................................. ... ........ .......................................................................................................... ............ ........................................................................................................... ... .. .................................................................................................. .. ................................................................................................. .. . .............................................................. ..... . .............................................................. ..... . . . . . . . . . . . ... .................................................................................... ...................................................................................................................................... ............ . . . . . . . . . . . . . ... . . . . . . . ......................................................................................................... ....... .................................................................... .......................................................... ..... ..... ............................................................................................................. ........ ............... . . . . . . . . . . . . . . . . . . .. .............. ..................................................................... ........................................................................................... ........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................................................................... ........ . . . . . . . . . . . . . . . . . . . . . . . . . . ..... ......................................................................................................... ......... .............................. ....................................... ..... . . . . . ...... . . . . . ..... . . . . . .... ..... . . . . . ... ........................................................................... ........................ .................................................................. ....................................................................... ........ ........................................................................................................................... ......... ................................................... ................................................................. .................................. ................................................... ................................... ..................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . .. . . . ... ... . . ... ... . . . .. ...... ........................................... . .. . . .. . . . .. .. ..... .................................... ............................. ...................................................................................................................... .... . . . . .... . . . . . . .. ........................................... . .............. ....... ................... ..... ....................................................................... ............................................... ..... ............................................................................................................. ......... ................................................................................. ..... ................................................................. ........................... .................................................... ................ ........ ...................................................................................................... ........ ......................................................................................... ... ................................................... ... ............................................... .......... . . . . . . ........................................................................ . . . . ...................................................................

10

20

30

10

20

30

As we have mentioned, under the restriction functor W mod H ! mod W , some serial H -modules become isomorphic, namely, the ten H -modules with support in the subquiver with vertices f0; 1; 2; 3g are identified with the corresponding ten H -modules


525

with support in the subquiver with vertices f10 ; 20 ; 30 ; 4g. In a first step, we make the identification of the nine pairs consisting of modules of length at most 3. We obtain the following partial translation quiver: R00

0 R1 ..... .. .. .... ........ ........ ..... ................ ................ ..... ......... ........ ......... ........ .......... ..... .......... ... ... ..... . . ............. ................. ................. ............. ............... ............... . . . . . . . . ..... .... ..... ..... ..... ..... ..... ..... ..... ..... .. ..... .... ... ... ......... ......... ......... ........ ..... ..... .................. ................. ................. ..... ......... ......... ......... ......... ......... ......... .......... .......... ......... .... ..... .. .. .. ..... . . . ............ .......... ............ ........... . . . . . . ..... .................. .................. .................. . . . .......... ......... ......... .... .... ............... ............... ............... ..... . . . ..... . . ..... .................. .................. .................. .......... .......... .......... ..... .......... ......... ......... ..... ..... ................. ................. .................. ..... ..... ..... ..... ......... . . . . . . ..... ..... . .. . . . . . . . . . ..... .... ..... .... ..... .... ..... ... ....... ......... ......... ......... ......... ......... . . .......... .......... .......... ..... . . ...... ...... ...... ..... . . . .... ..... ................ ................ ................ .......... .......... .......... ..... ..... .... ..... .... ..... .......... ..... .................. ................ ................. ... ... .. ..... . .. .. ..... ..... ................ ................ ................ .............. .................. .................. ......... . . . ..... .... ..... .... ..... . .......... ......... ......... ......... .... . .. . ......... ........ ....... ....... ........ ........ ........ ......... ..... ......... ........ ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........ .. .. .. .. ..... .. .. .. .. ............ ................. ................. ................ ................ ................ ................ ................ ............... . . . . ..... ..... ..... ..... .... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... . . ........... ............... .......... . . . . . ... . . . . . ... . . . . . ... . . . . ...... . . . ........... ................ ............... . . . . ..... .... .... .... ..... . ..... ..... ..... ..... ........ ...... ....... ....... ..... ........ ..... ....... .. .......... . . . . . . . . . . . . . . . . . . . . ..... ..... .... ......... ......... ..... ................ ................. ....... ....... ...... ...... .......... ...... . . . . . . . . . . . . . . . . . . . . . ..... ..... .... ......... ......... ... ....... ....... ....... ....... . . . . . . ..... . . . . . ...... ..... ....... ......... ....... . . . . . . . . . . . . . . . . . . . ..... ..... .... .................. . . . . . ..... .... ......... ..... ..... .................. ................. ...... .... . ......... ..... . . . . . . . . . . . . . . . . .... . ..... ..... ..... ...... ...... ..... ......... ......... ........ ........ ..... ......... .......... ..... ........ ... . 0 . . . . . . . . . . . . . . . . . . . . . M.b/ ....... ....... ...... ....... ....... .... ........ ....... ........ ....... M.b / . . . . ......... ..... ......... ......... ......... . ..... ......... . ..... ................. ........ ..... ................ ........ .......... .......... .......... .......... ..... ............ ........... ........ . . . . . . . . . . . ..... .... ..... .. ............ ...... . .. ........ ..... ... ..... ........ ..... ........ .......... ....... ..... .... . . . . ..... .... .... .....

Q0

P0

Here we denote by P 0 the rays coming from P , and so on. Note that we did not yet identify the points labeled M.b/ and M.b 0 /, these are H -modules of length 4 which are identified under the restriction functor. Now let us make this last identification, and insert the W -modules which are not in the image of W mod H ! mod W . II

III

. . . . . ..... .. .. .. .. .. .. .. .. ..... .................. .................. .. .......................................................................................................................................................................... ......... .................. .................. ......... .......... .......... ........ .......... .......... . ...... . ...................................................................................... .. .................................................................................. ..... ..... ..... ..... ..... ..... ..... ......... ......... ......... ......... ......... .... . ................................................................................................................................. . .... ......... ......... ......... ......... ......... ......... . . . . ....... ....... ..... ....... .... ....... ..... ......................................................................... ........ ..... .................. .................. .................. .... ... ................................................................................................................... ... .... ................ ................. ................. ........ .......... .......... .......... ....... ......... .......... .......... ............................................................................................. ........... .... ..... ...... .. ..... ..... ..... ..... ...................... ... ........ ........ ......... ........ ......... ........ ......... .... ................................................................................ .... ......... ........ ......... ........ ......... ......... ........ . .... . ........ ....... ....... ........................................................... ....... ........ ........ ........ ......... ....... ... .... .... .... ... . .. .... .... .... ................ ...... .... ......... ......... ......... ......... ......... ......... ......... ...... ................................................................... .... .. ......... ......... ......... ......... ......... ......... ......... .... ........ ........ . .. . .. . .. .... ..... .. . .. ................................... .. .. .. .......... ........ ........ ........ . . .. .. ....... ....... ....... ....... ..... ..... . .................. .... ......... ......... ......... ........ ......... ........ ......... .. .... ... ......................................... ... ... ......... ......... ........ ......... ........ ......... ........ ..... .................. ........ . .. . .. . .. ..... . .. . .. .. .. ...... .. .......................... . ............. ... .. ........ ....... ................... ........ ....... ....... ....... ....... .................. . . ......... ........ ......... ......... ......... ........ ......... ... ... .. ............ ... .. ... ......... ......... ......... ......... ......... ........ ......... ..... ............................. ..... . .. ....... ....... ......... ........ ........ ........ .................................. ... ....... .... ...... ... .......................................... . ..... . ..... ..... . ..... . . ..................... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................ ... ..... ...... ..... ..... ..... ..... ..... .... .... ...... ....... ...... ....... ...... ....... ...... .... .................................. .. .. .. .. .. ....... . ... .. .. ......................................... ................................................. .. .. .. ......... ......... ........ ........ ........ ........ ....... ....................................... ... .. ......... ........ ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ......... ........ ......... .... ....................................... ......... ......... ......... ......... ......... ........ ........ . . .. . . . .. ........ .......... ................................. . . . . . . . . . . . . . . . .................................. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................. . . ....... ........ ....... ....... ........ ....... ....... ....... ....... ....... ....... ....... ....... ........ ....... ........ ....... ........ ..... ............................. ........ . . . ......... . . . ......... . . . ......... . . . ......... . . . ........ ........ ................... ........ ........ ......... ........................... ..... .. . .. . .. .. ..... ......... ........ ......... ......... .... ...................... ................. .... ......... ........ ......... ......... ........ ....... ........... ........ ........ ......... ........ ....... .......... ..... .... .. . .. . .. . .. . ..... ........ ........ ........ ......... .... ......... ...... .... ......... ........ ......... ........ ......... ........ .... ........ ........ ....... ........ . ....... ... .. ..... .. . ........ ..... ......... ........ ......... ......... .... .... ........ ........ ........ ......... ........ ........ ...... ....... ...... ...... .... ..... .......... ............... ........... ................ ........ . . . . . . . . . . ..... ..... ..... ..... ..... ..... ..... ..... .... .. ... .... .... .... .... ........ M.b/ ..... ................ ................................................................ ............................ ........ ................ ........ ............................. ............................ .......... .......... .......... ... ... .................................................... ........ .................... ........................................ . . . . . . . .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................. . ..... .... ..... .... ............... .................................... ............................................................... .... . ......... .......... .... ......... ............................. ......... ......... ..... ........ ......................................................... ........ ............................. ........ .... .... ........ ........................................................ .......... .......... .............. ............ .... . ............................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ............................. ..... . ............................. ........ .... .................. . .... ......... ......................................................... ..... ..... ............... .............. . ..... .................... . .... .... .. ... . .... . .... . ...

I

IV What are the additional modules? These are the string modules for words which contain a completed bar as a subword. These modules form quarters (as introduced

526

C. M. Ringel

in [20]), namely the four shaded areas in the picture above. The four quarters can be rearranged in order to be parts of a tile, similar to those exhibited in [20], p 54f, this will be explained in the next section. Of special interest seem to be the four encircled module, the corner modules for the quarters. 12.3 The corner modules. It may be worthwhile to identify explicitly the four corner modules (in the presentation of this component given above, we have encircled these modules): quarter I

quarter II

........... . ......... ........ .......... .......... ............ ˇ3 ........... . . . . . . .... ........ . . . . . . ............ ˇ3............. ..... ˇ2 .... ............ .. ˛............................ˇ1 ˇ2............. .. ˇ1.....

...........

quarter III

......... ..... ............... ˇ3.. ..... ....... .. ..... . ... ˇ3.. .......... ............... ................. ..... ..... ...... ...... ............ˇ3 .. ..... . ˇ2.. ..... . ......... ............. . . . . . . . ..... ..... ..... .. ............ˇ3 .. ........ . ˇ . . . . . . . . 2 . . . . . . . . . ..... ..... ........ ... ......... ˇ2 ˇ1 ˇ3 ..... ......... .......... .......... ....... ... .. . ......... ˇ1.. ........... ˇ2 ......... ........ ..... .........ˇ1 ˛........ ........ ........... ˇ2.. ..... ......... ˇ1 ˛ . ... ... ˇ1..

.rad M.b//

...

N

.M.b/= soc/

quarter IV

........ .................. .......... .......... ............ ˇ3 ....... . . . . . . ........ .......... ........ ........ ............... ˇ2 ˛.............................ˇ

1

x .b/ M

In our example both the socle and the top of all corner modules are of length 2. In general, the corner modules for the quarters I and IV may have a socle of length 3, and dually, the corner modules for the quarter III and IV may have a top of length 3, as the following description shows: For the quarter I, the corner module .rad M.b// is obtained from rad M.b/ by adding hooks on the left and on the right. Dually, for the quarter III, the corner module .M.b/= soc/ is obtained from M.b/= soc by adding cohooks on the left and on the right. x .b/ by adding to M.b/ a hook For the quarter IV we obtain the corner module M N where bN is the x .b/ D M.b/, on the left, a cohook on the right. In our case we have M completion of b. The corner module for the quarter II has been denoted here by N D N.b/, in our example, we start with rad M.b/= soc and add a cohook on the left and a hook on the right, in order to obtain N — however, this rule makes sense only in case rad M.b/= soc is non-zero, thus in case the bar module M.b/ is of length at least 3. In general, let N0 be the boundary module in R0 which has the same socle as M.b/ and N1 the boundary module in R1 which has the same top as M.b/. Then N has a filtration 0 N 00 N 0 N with N 00 D .N0 /;

N 0 =N0 D rad M.b/= soc;

N=N 0 D .N1 /:

In case M.b/ is of length 2, say b D ˇ where ˇ is an arrow, then we deal with an exact sequence 0 ! .N0 / ! N ! .N1 / ! 0: This is one of the Auslander–Reiten sequences involving string modules and having an indecomposable middle term, namely that corresponding to the arrow ˇ, see [7].


527

This description of the corner modules shows that all of them are related to the following Auslander–Reiten sequence 0 ! rad M.b/ ! M.b/ ˚ rad M.b/= soc ! M.b/= soc ! 0 for W =I , where I is the annihilator of M.b/. Recall that we have used the Auslander– Reiten quiver of W =I as our gluing device, let us mark the Auslander–Reiten sequence in question: M.b/ ........... . ...... ..... ... ... . .... ........ ... ... ............... .... ........ .............. . . ................................ . . . . ...................................................................... ..... ..... ...... . . . . . . . ..... ..... ..... ..... .... . . . . . . ... ...... ..... ..... ....................................... ..... ..... . . ..... ...... ... . . . . . . ... . ..... ..... . . ..... ... ..... .................................................. ......... . . . . . ..... . . . ..... ... .................. ....... ....... ... .... ..... . . . . . . . . . . . . . ..... ..... ...................... ..... ..... ... ... ... . . . . . . . . . . . . . . . . . . . ..... . . . . . . . . ..... ..... ... ... ........... ............ ... . . . . ..... . . . . . . . . . . . . . . . . ..... ..... ..... ... ... ............ ....... ... ... . . . . . . . . . . . . . . . . . . . . ..... ..... ......... ..... ... ... ........... ....... ........... . . . . . . . . . . ..... . . . . . . . . . . . . . . . . ....... ..... ..... ..... ........ .. .. ... . . ..... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... ..... ..... ..... .. ... ....... ....... ... ....... ... . ... . . . . . . . . . . . . . . . . . . ..... . . . . ..... . . . ..... ..... .... ..... ..... ..... ... ... ... . . . .... ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... ..... ..... ... ........ ........ ....... ... .. . .... . . . . . . . . . . . . . . . . . . . . . . . .......... ..... ..... ... ..... . ... ........... ... . . . . . . . . . . . ..... ..... . . . . ..... ..... ... ... . . . . ... ... . . . . . . . . ..... ........................................................................................................................ ... .. ..... . . . . . . . . . . . . ............... ..............

rad M.b/

M.b/= soc

rad M.b/= soc

In case M.b/ is of length at least 3 we deal with a square, if it is of length 2, then with a triangle. In Section 13.2 we will see in which way this square or triangle is enlarged in mod W . 12.4 Wind wheels with several bars. Now we consider the general case of t bars. Any of the components P , R0 , R1 , Q will be cut into t pieces, and always the pieces will be indexed the set B of the direct bars. Write w D w1 : : : w t where all the words wi start with a direct letter and end with an inverse letter, and such that any wi ends with an inverse bar, say .bi /1 . We denote by W B ! B the cyclic permutation with .bi / D biC1 . Similar to the case t D 1, we remove arrows from the preprojective component, but now we want to retain t connected pieces P .b/ with b 2 B. The piece P .b/ is supposed to contain the projective modules starting with rad M.b/ and ending with M. b/. Here is a picture of P .b/: ......

......

......

...

M.b/........................ ..................................... .................................... .................................... rad M.b/...................... ............................... ............................. ............................. ............... ... ....... ... ........ ........ ........ ........ ........ ........ .... .... .................. ..... . .. ........................... ......... ........ ........ . .................................................... ......... ......... ......... ......... ........ ......... . . . ......... ........ ....................................................... . . . . . .................................................................................... ......... .......... .......... .......... ...............

....... ............................................... ......... ... ............................................ .. ............................................ ......... ......... ........ ......... .............................................. ..... ........ ........................................... .. . ............................ ............. ......... ......... ......... ................................................ .......... ........................................... ..... . . . . ..................... .................. ..... ..... ...... .................. .................. ..... . . ..... ... ... .. .... ............................................. .......................... .......................... ..... ............ .............. .............................. ..... ............................................................................................................................. . . . . ..... ................................................................................................... ...... .. . . . . . ... . . . . . .. . . . . .. ..... ..... ..... ......................................................... ..... .................. ....................................................................................... ....... . . .......................................... . ....... . .... ........ ........ ......... ......... .......................................................................... . . . ............................ .............. ........ ....... ......... ..... ................ .................. ............................................................................ ..... ..... ..... ..... .................................... ........... ..... . .. ............ .......... ..... . . ............................ . . . . . . ..... ...... ...... ...... ...... ......... ........................................... ....................... ........ ....... ........ . . . . .. . ..... . . . . . . . . . . . . . . . . . . . ..... ...... ....... ....... ....... ...... ......................... .............. ........ ....... ........ ..... .... .... . . .... ..... ........ ........ ........ ........ ........ ............ .......... .......... ..... .......... . . .

M. b/

P .b/

528

C. M. Ringel

Next, consider the regular component R0 ; according to our convention, this is the component which contains the simple modules T .b/ D top M.b/. Again, we remove arrows in order to obtain t pieces consisting of full corays; the piece R.b/ with index b shall contain the modules T . b/; 1 T . b/; 2 T . b/; : : : up to T .b/.

R.b/

..... ...... ...... ... ...... ...... ...... ...... ...... ......... ........ ......... ......... ......... ......... ......... ......... ......... ........ ......... ......... ......... ........ ......... ......... .......... .......... .......... .......... .......... .......... .......... . . .. .. ...................... ........... ........... ........... ........... ........... ........... ............ . . . . . . . ................ . . . . . . . . . . . . . . . . . . . . . ..... ..... ...... ..... ...... ..... ...... ..... ...... ..... ...... ..... ...... ............... ..... ........ ........ ........ ........ ........ .... ......... .......................................... .... ..... ..... ..... ..... ..... ... ...................................... ........ ......... ......... ......... ........ ......... ........ ......... ........ ......... ........ ......... ........ .......... ......... .......... .......... .......... .......... . . .. . .. . . . . . . . . ..................................................... . . . . . . . . . .......... .......... .......... .......... .......... ........... . . . . . . ..................................... . . . . . . . . . . . . . . . . . . ..... .... ..... .... ..... ..... ..... ..... ..... ..... ..... ................................. ..... .... ......... ......... ........ ......... ........ .......................................................................... ..... .... ....... ...... ..... ....... ................................................................... ........ ........ ........ ........ ........ ........ ........ ......... ......... ......... ........ ......... ......... ......... .......... .......... ...................................................... . . . . . . . . ...................................................... . . . . . . . . ........... ........... ... ....... ... ....... ... ....... . . . . . ........................................................ . . . . . . . . . . . . . . . ..... ..... ..... ..... ..... ..... ..... ..... ..... ................................................. ..... ........ ........ ........ .... ........ ......................................................................................................... . ................................................................. .................. .................. .......... ..... ......... ................................................................ .... ..... ........ ......... ........ ......... ......... ..... .... . . ................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. . . . . . .. .. . . . . . .. .. . . . . . .. .. .. .. .. .. .. .. ..... ............................................................................................................ ......... ......... ......... ......... ......... ......... ......... ......... ......... ................................................................. ..... ..... ......... ......... ...... ...... ....... ........ . ......... .... ......... ......... ..................................................................... ..... ..... ......... .......................................................................................................... ......... ......... ......... ......... ......... ......... ........ .............................................................. ......... ....... ........ . . . . . . . . . . . . . . .. . .. .. . .. . . . . . . .. . . . . . . .. . . . . . . . . .. ..... ......... ......... ......................................................................................................... ......... ........ ......... ......... ......... ........ .. . . . . . . .. . . . . . . .. . . . . . . . . .. .. . . . ..... . . ...... . . . . ........... . . . . .................................................................... . . . . .......... . . . . ........... . . . . .....

T . b/

T .b/

Similarly, we consider a word w 0 obtained from w by cyclic rotation, such that w D w10 : : : w t0 where any wi0 starts with an inverse letter and ends with a direct letter, and ends in a (direct) bar, say the bar b .i/ and we denote by W B ! B the permutation which sends b.i/ to b .iC1/ . 0

Now we cut the preinjective component in order to get pieces made up of corays. the piece Q.b/ has to contain M.b/= soc up to M.b/. Here is a picture of Q.b/.

..... . . . . ..... ..... .......... .......... .......... ..... ........ ......... ........ ......... ........ ......... ........ ...... ...... ...... ...... .............. ..... .............. .............. . . . . . . . . . . . . . . . . ..... ..... ..... ..... ..... ..... .... ...... ...... ...... ........ ..... .......... ..... .......... ..... ........ ......... ........ ........ ......... ....... ........ ........ . . . . . .. .. . ..... ......... ......... ......... ......... ......... ..... ........ ........ . .......... .......... ..... ..... ......... ......... ......... ......... .. . ......... ....... ........ . ..... ......... ......... ......... ......... ......... .......... ..... ..... .......... ......... . ........ ......... . . . . . ..... ........... . . . ..... ........ ........ ........ ........ ........................... ........ ......................... ........ . . . . . . . . .... ........ ........ ........ ......................................................... . . . ..... ..................................... ..... ..... .. .... .... .. . ........ ......... ..................................................... ..... ..... ......... ......... ........................................................................... ............................................... ......... . . . . . . . . . . . . . . . . . . . . . . . . .... .... ......................................... ..... ................................................................................... .. . . . . . . .. . . . . . . .. ........ ..... ......................................... ..... .......................................................................... ................................................ .. .. . . . . . .. .. . . . . . .. .......................................... ............................................ .......................................................................... . . . . . . ................................. ............................................................................. .................................................... . . .. . .. . . . . . . . . . . . .. . .................................................................... ......... .. .......................................................................... ....... .................................. ....... ........ ......... ..... ......... ....................................................... ................................. . ...... . . . . . . . . .. . . . . . . .......... . . ...................................... ......... ......... ......... ......... ......... ...... ...... ... ...... . . . ..........

Q.b/

M.b= soc/

M.b/

Finally, we consider the regular component which contains the simple modules S.b/ D soc M.b/. Again, we remove arrows in order to obtain t pieces consisting now of full rays. The piece R1 .b/ indexed by b contains the rays starting at


529

1 S.b/; 2 S.b/; : : : , up to S.b/. . . . . ..... .. .. .. .. .......... ............................................. ..... ......... ......... ......... ......... ..... ......... ........ ......... ........ ......... ........ ......... ........ ......... ......... ............................................................................... ...... ...... ...... ...... ........ ................................................... ........ ................................................... ........ ........ . ..... ........ . . . . . . . . . . . . . . . . . . . . . . . . . ..... ..... ..... ..... ..... ..... ..... ..... ..... ......................................................... .......... .......... .......... .......... ................................................................ ..... . ......... . .. . . . . . . .. . . . . . . .. . . . . . . . .. .. .......... ..... .......... .............................................................. ......... ......... ..... ......... ......... ........ ......... ......... ......... ........ ......... ........................................................................................................ ........ .................................................................. ........ ........ ........ ..... . ..... . ..... ............................................................. ..... . ..... . . . . . . . . . . . . . . . . . . . . . . ..... ..... ..... ..... ..... ..... ..... ................................................................ ..... ................................................................... ......... ......... .......... ..... ..... . ......... .. .. . . . . . . .. . . . . . . .. . . . . . . . . .. .................................................................... ......... ..... .......... .......... ..... ..... ......... ......... ........ ......... ......... ......... ........................................................................................................... ......... ........ . . ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ ........................................................................... ........ .. ...... .......... . . . . . . . . . . . . . . . . . . . . . ..... ..... ..... ..... ...... .................................................................... ..... ..... ..... ..... ..... ......... ......... ........ .................................................................................... . ........ ........... .......... ......... ................................................................... ..... ..... ..... ......... ......... ......... ......... ............................................................................................................ ......... ......... ........ ........ . . . . . . . . ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... .......... ................................................................................................. .......... . ... ...... . . . . . . . . . . . . . . . . . . ..... ..... ..... ................................................................. ..... ..... ..... ..... ..... ..... .... .................................................................... ......... ........ ......... . ......... . .. . .. . ..... .......... ..... ................................................................ ......... ......... ..... ......... ......... ......................................................................................................... ......... ......... ......... ......... ......... ......... ....... ................................................................... ........ . . . . . . . . . . . . .......... .......... .. ...... ................................................................................... ...... ......... . . . . . . . . . . . . . . . . . . . . ..... ................................................................. ..... ..... ..... ..... ..... ..... ..... ..... . . . ...... . . . . ..................................................................... . . . . ........... . . . . .......... . . . . ........... . . . . ......

S.b/

S.b/

R1 .b//

Altogether we have cut the four string components of mod H into flat pieces: any such component yields t pieces. The gluing of these pieces is done by identifying H -modules which become isomorphic under the restriction functor mod H ! mod W (and finally we will have to add various quarters). As in the case t D 1, we first will look at the proper subfactors of the bars. Identifying the corresponding H -modules, we obtain partial translation quivers which are planar: Any of the components P , Q, R0 , R1 has been cut into t pieces, and the identification process will use one piece of each kind, in order to obtain t planar partial translation quivers of the following form (): R0 .b/

..... . . ...... ... ...... ..... .......... .......... ..... ......... ......... ......... ......... ..... ........ ......... ........ ......... ..... .......... .......... ...... ...... ... . . ..... .............. .............. .............. .............. .............. .............. . . . . . . . . . . . . . . . . . . . . . . . . ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .... .... ...... ...... .... ..... .. ..... .... ........ ..... ......... ......... ......... ..... .......... .......... .......... ..... ......... ........ ......... ........ ......... ......... ..... ........ ......... ........ ......... ........ ......... ......... ......... ........ ..... ....... ...... ......... ......... . . . . . . . . . . . . . . . ..... . . . ..... ..... ..... ..... ..... ..... ..... ........ ......... ........ ......... ........ ......... ..... ..... .................. .................. .................. .......... ..... .......... .......... ..... . ..... . . . ..... ......... ......... ......... ..... .......... .......... .......... .... ........ ......... ......... ........ ........ ......... ..... ......... ......... ......... ......... ......... ......... . . . . ....... ........ ....... ....... . .. .. .... ..... ....... ........ ........ ..... ......... ......... ......... ..... ......... ......... ......... ......... ......... ......... ..... ......... ......... ........ ......... ......... ......... .......... ..... .......... .......... ........ ..... ........ ....... .... ....... ........... ....... ...... ........ . ......... . . . . . . . . . . . . . . ..... . . . . . . . . . . . . . . . . ..... ....... ....... ....... ........ ....... ....... .... ........ ........ ......... ........ ........ ........ ..... ....... ........ ........ .. .. ..... .. .. ..... .. ... ...... ....... ........ ........ ..... ......... ......... ......... ......... ......... ......... ..... ......... ......... ......... ......... ......... ......... ..... .... ..... .... ..... .......... .......... .......... .......... ..... . . . . . . . . . . . . . ....... ......... ......... ......... ......... ........ ........ .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ........ ........ ........ ........ ........ ........ ........ ....... ........ ........ ........ ....... ........ ........ ........ ... ... .. .. .. . ..... ... ... ........ ......... ......... ........ ......... ......... ........ ........ ........ ..... ......... ......... ......... ........ ......... ......... ......... ......... ......... ......... ......... ......... ........ ......... ......... ......... ......... .... ... ... ... .......... ... ... ... ... ... ... .......... ..... ..... ......... . . . . ..... ........ ........ .. . . . . . ...... . . . . ...... . . . . ...... . . . . ...... . . . . . ......... ......... . . . . . . . . . . . . . . . . ..... ..... ..... ......... ......... ..... ......... ......... ......... ........ .......... .......... ..... ... ...... ........ . . . . . . . . . . . .... . . . . . . . . . . ..... ... ....... ....... ....... ....... ... ....... ........ ......... . . . . . . . . . . ..... .... ..... ..... ..... .... ..... .... ..... ... . . . . . . . . . . . . . . . . . . . . . ..... ........ ......... ...... .. .......... . . . . . . . . . . . . . . . . . . ..... .... .... ........ ......... ..... .................. ................. .......... .......... ..... .. ........ .... ..... ..... . . . . . . . . . . . . . . . . . ......... ..... .... ........ ....... ........ ........ ..... ......... ......... ......... ......... .......... .......... ..... . . ... . . . . . . . . . ......... . . . ......... .......... .......... .......... . . . . ..... . . . . . ..... . . . . . . . ..... ....... ....... ....... ....... .... ........ ....... ........ ........ ........ ........ .... ....... 0/ 00 / ......... ....... . . . . . . . . . . . . M.b M.b . . . . . . . . ..... . .. . .. .... ........ ........ ........ ........ ..... ......... ......... ......... ......... ......... ......... .... ....... . .. . . ......... ..... ........ .... ......... ..... ..... ..... ......... ......... ......... ..... ......... ......... ......... . . .......... . . . . . ...... ....... ....... ..... . . . . . . . . . . . . . . . ..... . . .... ........ ........ ..... ......... ......... .... ......... ....... . ......... ..... .... ..... ..... ..... ........ ..... ......... .......... ....... ..... . . . . ..... ... ..... ..... ... .....

M. b/= soc

Q. b/

R1 .b/

rad M.b/

P .b/

530

C. M. Ringel

It is important to observe that in contrast to the case t D 1, the modules labeled M.b 0 / and M.b 00 / (corresponding to bars b 0 and b 00 ) now may be different! We obtain in this way a permutation of the bars such that b 00 D .b 0 /. We know that b 0 D .b/ and b 00 D .b/. Now b 0 D .b/ means that b D 1 1 .b 0 /, thus

.b 0 / D b 00 D .b/ D 1 1 .b 0 / D Œ ; .b 0 /: This shows:

D Œ ; : Of course, as we have mentioned already, we still have to add the indecomposable W -modules which do not belong to the image of . We know that there are precisely t non-periodic (but biperiodic) Z-words; they give rise to 4 t quarters which have to be inserted as in the case t D 1. Before making the final identifications, let us attach the quarters of type IV to the pieces of the form Q.b/. In this way, we obtain t partial translation quivers of the form .. .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .. ... ... ... .. ... .......... ............ ................ ..... ... ... ... . ... . . .. ... . ..... ..... ........... ... ................................... .. ................ .. ................ ... ................ ... ................. ... .... . .... .... .... .... .... .... .... .... .... .... .... .... .... .... .............

or better

.... .... .... .... .... .... .... ... .... .. . . . .... ... . .... . ... .... . . . .... ........................... . . .. . . . . . . . . .... ..... ... .. . . . . . .... . .. ... ........ . ... ..... .. .... . . .. . . . . . . . ................................. .... . .. . . . . . . . . . .... .............. .............. .... .... .... ......... .... .. .... ... .... .. . . . .... .... .. .... .... ..

(The visualization on the right hand side takes into account the embedding of these Auslander–Reiten components into the corresponding “Auslander–Reiten quilt” which we will discuss in the next section.) The partial translation quivers are sewn together in the same way as one constructs the Riemann surfaces of the n-the root functions in complex analysis (taking into account the permutation ). For example, we may obtain a 3-ramified component which roughly will have the following shape: ... .... ........ .... .... ... .... .... ... ... ..... .... ...... ...... ...... ........ . . . . . ... ... ..... .... ... ... ... ... ..... .... ... .. ... ... .... .... .. ... ... ... ..... .... ... ... ... ... .... .... ...... ..... . . ... ... ..... . . . . ... .... ..... . . . . . . . . . . ... ... .... . . . . . . . . . . . . . . . . . . . . . . . ........ ... ... ..... ...... ... .... .... . . . . . . . ... ... .... . .. ............................................................ ...... ... .... .... . . ... ... ..... . . . ..... .. ...... .... .... .......... ... ... .... . . . . . . . . . . . . . . . . . . . . . . . . ................. ..... ... ......... . . .......... ... ......... . .... ... .......................... ........ .............. . ... . . . . . . . . . . . . . . .... ........ . . . . . . . . . . . . . .. .... ............. ........ ................... ................................. . ......... ........ ........ . . .. ......... ......... .. ........................................ .... .... ..... . . . . . . . . .... .... .... . ... .... .... .... ...................... ........... ...... ....... ...... .... .... .... ............................................... ......... ......... .... .... .... ... .............. .......... ....... .... .... .... . .... .... .... ...................................... . . .... .... .... . ... ... ..... .. .... .... .... .... .... .... .... .... .... .... .... .... ....... ....... ........ .... .... ... .... .... .... .... .... .... ....... ....... .... .. .... .... .... .......

We will call such a component with r leaves an r-ramified component of type A1 1. Proposition. Let W be a wind wheel with t bars. Then: Any non-regular Auslander– Reiten component is an r-ramified component of type A1 1 with 1 r t . If C1 ; : : : ; Cc are the non-regular Auslander–Reiten components of W and Ci is ri -ramified, for P 1 i c, then ciD1 ri D t .

531


We may assume that r1 r2 rc , thus we deal with a partition and we call this partition .r1 ; r2 ; : : : ; rc / the ramification sequence of W . 12.5 Wind wheels with arbitrarily many non-regular components. We are going to present a wind wheel with t bars which has t non-regular Auslander–Reiten components (all being necessarily 1-ramified: the ramification sequence is .1; 1; : : : ; 1/). Here is the quiver: ........ ....................... ....................... ........ ............ ... ... ...... ...... .... ... ... ... ... ... ... ... .. .. ... . . . . .... . . ... ... ... ... .. ... .. . . ............... .. ..................... .... ... .......... ... . ... . . .... . . . . . . . . . . . . .......... .......... ...... ...... . . .. .. .. .. .. .. .. .. . .. .... .. ... .. ... .. ... .. ... .. .. .. ... .. ... .. ........ . . . . . . . . . . . . . . .. . . .. . . .. .. . .. . ....................... ................................................ .......................... ................................ .................. ... ..... .. .. ... . . . ... .... ........... ........... ............... .. ... .. ... ... ...... .............................................................................................................................................................................................................................................................................................

1

3

2t 1

0

2

2t 2

For example, for t D 5, the primitive cyclic word is

0

8

9

9

8

6

7

7

6

4

5

5

4

2

3

3

2

0

1

1

0:

All the non-regular Auslander–Reiten components of this algebra look similar, here is one of these components: ..... ..... .... .... ..... ..... 67 ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... . ..... ..... . . . . . . . ..... ..... .... 5 .................... ..... ..... ..... 3 7 5 5 .... .... . .... .... 4....... .7 .... 2 5 7 4 5 4 5 ........... .... ..... .... ... ...6 .... .. 4 .. ....... 7 ..... 6 4 ..... .4 ..... ..... .....6......... ..... . . 6 ..... ..... . . . . . . . . . . . . . . . . . . . . . . . . ............ . . . . ....... . . . .... ........ ...... ...... ...... ........ ... ...... ...... ... ... ... ....... .... .... ...... ...... ...... .................................................... .......... .......... .......... .......... .......... ........... . .... ..... .. ..... ..... ..... . ..... ..... .. . .................... . . .. 2 . . 3 .................... ...... 5 7 5 . . . .. .......4...... 5 .......... .......... .......... . . 2 5 4 7 . . . . . .. ..... 45 .. ...... .......5... 6 6 ......... ....4 ..... ..... ........ ..... ........... ........... ........... ..4 ..... ..... ...... .. ....... ......... ..... ....... ..... ..... ......... .......... ....... ... .... .... .... .... ....... .... ....... . . . .... ... . . . . . 5 2 ..... ... ..... ..... 5 4 .. ..... ..... ..... ........ ..... ..... ...... ........... ............ ...... ...... ....... ....... .... . ...... ................. .... . .......... . . . . . . . . . . . . . ..... . ..... ........ ... . . . . . . . ..... 5 3 3 ... . . . . . . ..... .......... 5 2 3 2 .................. ... . . . . . . . . . . . . . . . . . . . . . . . . .................. ..... .. ...... 2 ..... ....... ....4 .................. ............. .......... 4 ................. ..... .......... .................. ................. ........ .. ... ......... ...... .................. ................ .... ..... ....... ..... ......... ..... .................. . ................. ................. ..... ..... ................. . . . . . . . . . . . . . . ... ... 3 . . . . . .................. . 2 .. .................. .................... ..... .................. ..... ... ................. . . . . . . . . ............ . . . . . . . . . . . . . . . . . . . . . .................. ....... ....... .. .......... . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................. .. .. .. .. ............. ...................... ................ ...... 3 0 3......... ..... .................. 2 3.. 2 .. .................. ..................... . . . . . . . . . . . . . . . . . . .................. . . . . ..... ..... ...................... ....... ....... .............. .................. ........... ....... ........... ................. ...... .................. ....... ................. .................. ..... ............. ............ .......... ................. ................... .... . . .... ................. ... .... .....3........ .... . . . . 0 . . . . ..... ......2...... 3 ..... ..... . ................ ...... .. ..... ..... ......... ................ ..... ....... ................ ..... .......... ....... ................ ...... .......... ................ .......... ....... ....... ................ ................ ...... ....... .......... ................ ............... ..... ..... ......... ................ . . . . . . . . . ..... . . . . . . . . . . . . . ................... . ..... . ..... ..... ..... ................ ......... ..... ..... ..... ..... ..... ..... .... . . . . . . .. .

II

B

B

III

4

5

B

I45

B

B

B

B

B

B

B

B

B

B

B

B

B

IV23

The inserted quarters are labeled I, II, III, IV, with a bar b as an index: all the modules in such a quarter are of the form M.v/, where v is a word which contains bN as a subword (such a quarter will later be seen as part of the tile T .b/).

532

C. M. Ringel

12.6 Wind wheels with non-regular Auslander–Reiten components with arbitrary ramification. First, let us present an example with a 3-ramified component. Here is the quiver with the zero relations of length 2 (in addition all paths of length 3 are zero relations): ......................... ......................... ..... ..... ... ... ... ... ... ... ... ... ... ... .... .... . ............... ... ............... .... . . ... ... . . . . . . ....... ....... . . ....... ....... ...... ...... ... ... ... ... ... ... ... ... ... ........ ........ ........ . . . ................................................ ................................................ .. . . .... . .. ........... .... .. . ..... ... ...... . . ... ..... ...... . . ............ ....... ...... ........ . . . . . . .......... .. .............. .......... .......................................................

1

3

5

0

2

4

....................................... ............ . . . . .. ... ..... .. .. . ........... ...... ... .......... ........................

7... ... ... . ....... .

6

... ...... ........ .... .... . ............... ..... ..... . .. ... ... ... . . ..... .........................

The primitive cyclic word is

4

0

1

1

0

2

3

3

2

4

5

7

6

6

7

5

4:

Let us exhibit one part of the 3-ramified Auslander–Reiten component (as before, the inserted quarters are labeled I, II, III, IV, with a bar b as an index):

..... ..... ..... ..... ..... ..... ..... ..... 3 .... 2 1 .... .... ... 0 .. . . . . . ..... ..... ... ...... ........ . . . . ....... . . . . . . ........ .. .. ....... .... ..... ..... ... 3 .................... ..... 2....... .1 ... ... . 01 .. . . ........ ...1.. ..... ...... .......... .......... ..0 ....... ........ ...... ...... ....... ..... ..... .... ....... .... . . . 1 .... ..... 1 ..... ......... ....... ....... .......... ...... ...... . . . . . . . . . . . . . . . . . . . . ... .. ... ... . . . . . 1 3 ..... ..... 1 2 .... ..... ........ ........... 0 .... ......... . ...... . .. ......... ....... ..... ..... ..... ..... ..... 3 2 .. ..... ...... ............ ..... ....... ....... . . . . . . . . . . ......... ....... ..... .....

II

B

III

..... .. ..... ..... 45 ..... ..... ..... ..... ..... ..... . . . . ..... .... ..... 1 .................... ..... ..... 5 1 ..... . .... 0....... .5 7 6 1 0 ..... .... .... ...4 ... 7 .... . . . . . . . 4 . . . . . . . . . . . . .... ... ....6 ..... ........... 6 ........ ........ ... . . . . . . . . . . . . . . . ...... . . . . . ........ .. .. .... ...... ...... ..... ....... .... ..... ..... ..... ............. ....... . . . . . . . . . . 10 . . . . . . . . . . 57 . . . . . . . . . .

1

4

B

B

3 2 3..

B

4 3 2

B

B

I67

4

..... ..... ...................... .......... ...... .......... ...... .......... .............. ... ... ..... .. .............. ........ . ..... .... .... . . . . . . . 4...... .3 ..... .. ... . . . . . . . . ..... ...2 . ...... 3 ..... ....... ..... ..... ..... ......... .......... .......... .......... ...... ...... ......... ...... ...... .... .......... .......... ............. ..... ..... ..... ......... ..... .. . . . . . . . ..... ... . . . . . . ..... ... . . . . . . ..... .. ..... ..... . .... .......... ..... .....

6

B

B

..... .......... ..

B

B

B

B

6

.. ..... ..... ..... ..... . . . . .. ..... 7 ..... 5 ........... . 4 6 . ..... . . . . . . ..... .. ..... ...... ...... ........ ....... ...... ..... .......... ..... ..... .. .... .......... ... ... 6........7......... .. ......6..... 5 .. ...... .. .. ....4 ..... .......... .......... ..... ...... ........ ...... ........ .......... .... ..... ..... ....... .... ..... 6 ..... 6 .. ..... . ..... ..... ....... ..... ...... . .. ....... ......... .......... ....... ..... ......... ..... 7 ..... ..... 5 6 ..... . 4 ....6 ..... ....... ..... .. ............ ........ ......... .... ..... ..... ........ ..... .... 5

B

..... ...... ..........

B

........... .....

..... ...... ..........

..... ....... .......

.......... ..... .....

0 5 4 ..

B

B

..... ...... ..........

B

B

.......... ...... .....

IV23

1

3

This concerns the part of the component containing the module I.0/ D 102 : The leaves containing the modules I.6/ and I.2/ look similar. These three leaves together


533

form a component, namely a 3-ramified component of type A1 1. ............................................................................................................................................................................................................. ......................................................................................................................................... .............................. .......................................................1 ......................5 .......... . 0 . . . . . . . . . . .7. . ... .. ..... .. ... .. . . .................................................. ................................. ..................................... ................................ 6 4 ........ . . . . . . . . . . . . . ..................................... ................................. ......................... .. .6...................... . ..................... 1 .......................... 1 6................................. ........................ ................................. ..... 3 .......1 7 . ..........1.. 2 5.....6................. ...... ............0 .4 ... ..........6 ............................... ......................... . . . .......... ...................... .5 ............................... ....................... 3 . 4...................... . ......................2 ....................... ................................. ....................... ...........

1

6

............................................................................................................................................................................................................. ......................................................................................................................................... .............................. .......................................................7 ......................7 .......... . 6 . . . . . . . . . . .5. . ... .. ..... .. ... .. . . .................................................. ................................. ..................................... ................................ 4 6 ........ . . . . . . . . . . . . . ..................................... ................................. ......................... .. .2...................... . ..................... 5 ......................... 7 0................................. ........................ ................................. . . . .......5 7 3 ............7... 6 3.....2................. ...... ............6 ... ... .......0 ......2 ............................... ............................... . ...................... .3. . . . . . . . . . ....................... 7 2............................................ . ......................6 ....................... ................................. ....................... ...........

5

0

............................................................................................................................................................................................................. ......................................................................................................................................... .......................................................3 .............................. ......................1 .......... . 2 . . . . . . . . . . .1. . ... .. ..... .. ... .. . . .................................................. ................................. .................................... ................................ 0 0 ........ . . . . . . . . . . . . . ..................................... ................................. . . ......................... .4...................... . ..................... 3 .......................... 3 2................................. . . . .................... ................................. ..... 5 .......3 5 . ..........3.. 4 7.....4................. ............2 ...... .6 ... .. .......2 . . . ............................ .......................... . . . .......... ...................... .7 ............................... ....................... 5 . 6 4 ....................... ................................. ....................... ...................... ....................... ...........

3

2

In addition there is a second non-regular Auslander–Reiten component, namely the component containing the module I.4/; it is 1-ramified. The boundary looks as follows: ............................................................................................................................................................................................................. ......................................................................................................................................... .............................. ......................3 .......................................................5 .......... . 4 . . . . . . . . . . .3. . ... .. ..... .. ... .. . . .................................................. ................................. ............................... ..................................... 2 2 ......... . . . . . . . . . . . . . ..................................... ................................. . . ......................... .0...................... . ..................... 7 .......................... 5 4................................. . . . .................... ................................. ..... 1 .......7 1 ......... . ......0 ..........5....0. .............. ........ ................ ...1 .................................................. ............4 ...................................0 . . . . . . . ... .......4 . . . . . . . . . . . . . . . . . . . . . ...................................... ............................. ............... ...................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................................... ................................. ................ .................................................. . . . . . . . . . . . . . . . . . . . . 1 . . ........................................................................................................................................... 0 ......................................................................................................................................... ......................................................................................................................................... .........................................................................................................................................

7

4

We use Galois coverings of this wind wheel in order to exhibit wind wheels with arbitrary ramification. Let us consider the Galois covering obtained by an s-covering of the cycle of length 2, thus we deal with a wind wheel of the following shape (for s D 3, we have to identify the upper left hand arrow with the upper right hand arrow

534

C. M. Ringel

in order to have an arrow 70 ! 0 5): .......... .......... .......... .......... .......... .......... 0 ....................................... 0 ....................................... ...................... ....................................... ....................................... 0 ........................................ 0 ...................... . ... ... ... ... ... ..... ... ... ... ... ... ... ... ... ... ... ... ........ ........ ........ ........ ........ ........ .. .. .. .. .. .. 0 0 0 0 . . . . . . . . . . . . . . . . . . . . . . ........ ........ ........ ...... ...... ...... . . . . . . . . . . . . . . .. .. .. .. .. .. .. ...... .. ... ... .. .......... ..... .. .......... ..... .. .......... ..... .... .......................... .......................... .... ................... ..... ..... ..... ...... . ........ ........ .. .. .. ........ ......... ......... ... ... ... .. .. .. ........... ........... ........... . . . . . . . . . . . ... . . . . . .... .... . ... . ... . ... .. ..... ... .... ...... .... .. . ... . ... . ... ......................... .................... 0 ........ 0 ........................... 0 ........ .................................. ........................................ .................................. 0 . . . . . . ......... ......... ......... ......... ......... ......... ... ... ... ... ... ... .. .. .. .. .. .. ... ... ... ... ... ...

7

5

7

5

7

5

6

4

6

4

6

4

2

0

2

0............ ... . .... ..... ... ........ ..... ... .... .. ... . .. ... ... .... . . . . ...... ....................

0............ ... . .... ..... ... ........ ..... ... .... .. ... . .. ... ... .... . . . . ...... ....................

............ ... . .... ..... ... ........ ..... ... .... .. ... . .. ... ... .... . . . . ...... ....................

1

1

3

0

2

0

3

0 ............ ... . .... ..... ... ........ ..... ... .... .. ... . .. ... ... .... . . . . ...... ....................

0 ............ ... . .... ..... ... ........ ..... ... .... .. ... . .. ... ... .... . . . . ...... ....................

1

............ ... . .... ..... ... ........ ..... ... .... .. ... . .. ... ... .... . . . . ...... ....................

3

The corresponding primitive cyclic word is 0

0 0

2

4

5

0 0

7

0 0

6

6

7

5

4

0

1

1

0

2

3

3

2

4

5

7

6

6

7

0

5

4

0

0

0

10

1

0

0

0

2

0

30

3

0

2

0

4

0

50

70 60

Let us show the leaves which contain the modules I.0/; I.6/; I.2/, they are quite similar to those seen above — the only difference occurs on the right hand side of the upper leaf and in the upper row of the middle leaf: ............................................................................................................................................................................................................. ......................................................................................................................................... .............................. ......................5 .......................................................1 .......... . 0 . . . . . . . . . . .7. . ... .. ..... .. ... .. . . .................................................. ................................. ............................... ..................................... 6 4 ......... . . . . . . . . . . . . . ..................................... ................................. ......................... .. .6...................... . ..................... 1 ......................... 1 6................................. ........................ ................................. . . . .......1 3 .......... .0.7 ............1... 2 6 05 ................ ...... ............0 ...............6 ....4 . .............................. . . . ...................... . ...................... 0 ...................... 5 ....................... 3 40................................. . ......................2 ....................... ................................. ....................... ...........

1

6

............................................................................................................................................................................................................. ......................................................................................................................................... . . . . . . . . . . . 0........................................................... ......................................................0.7 ................................. . . . . . . . . . . . 0. . .. .. ... .. ..... .. .. . 7 ................................. 5 .......... 60 ............................... ..................................... ......... . . . . . . . . . . . . . 4 6 .................................... ................................. . . ......................... .2...................... 5 ..................... . ......................... 7 0................................. . . . . . ................... ................................. ..... 7 .......5 3 . ..........7.. 6 3.....2................. ...... ............6 .2 ... .. .......0 . . . . . .......................... .......................... . . . . . . . ...................... .3. . . . ............ . . . . . . . . . . . .7 2............................................ . .................................6 ....................... ................................. ....................... ...........

5

0

............................................................................................................................................................................................................. ......................................................................................................................................... .............................. ......................1 .......................................................3 .......... . 2 . . . . . . . . . . .1. . ... .. ..... .. ... .. . . .................................................. ................................. ............................... ..................................... 0 0 ......... . . . . . . . . . . . . . ..................................... ................................. ......................... .. .4...................... . ..................... 3 ......................... 3 2................................. ........................ ................................. ..... 5 .......3 5 . ..........3.. 4 7.....4................. ...... ............2 .6 ... ..........2 ............................... ......................... . . . .......... ...................... .7 ............................... ....................... 5 . 6...................... . ......................4 ....................... ................................. ....................... ...........

3

2


535

As before, the upper leaf and the middle leaf are sewn together (both contain the module M.23/), similarly, the middle leaf and the lower leaf are sewn together (both contain the module M.67/). But the change of the right hand side of the upper leaf is important, since it means that the upper leave and the lower leave no longer are sewn together (the lower leaf contains the module M.45/, the upper one the shifted module M.40 50 /). It follows that for the s-fold covering 3s leaves are sewn together and form a 3s-ramified component (this is the Auslander–Reiten component which contains the modules I.0/, I.6/, I.4/ and their shifts under the Galois group). What happens with the remaining non-regular component (the 1-ramified one)? Thus, let us start to calculate the Auslander–Reiten component which contains the module P .1/. Here is the relevant part of the boundary: ............................................................................................................................................................................................................. ......................................................................................................................................... .............................. .......................................................5 ......................3 ................................. .......... . 4 . . . . . . . . . . .3. . ... .. ..... .. ... .. . . .................................................. ..................................... ............................... 2 2 ......... . . . . . . . . . . . . . ..................................... ................................. ......................... .. .0...................... . 0 ..................... 7 ......................... 5 4................................. .......................0. ................................. ......7 ....... 1 1 ..0.. 00 ........5 1.....0................. .............0..... ... ... .......4 ......0 . . .......... ...........4 ............................... . .............................. 0 .......... .1 .......................01 ................................. 0 . .....................0 ...................... ........................ ...................... ....................... ...........

7

4

It follows that the Galois shifts of the module P .1/ all lie in one component, and this is a component of type A1 1 which is s-ramified. This shows that any ramification does occur. 12.7 The ramification sequence of a wind wheel. We have seen above, that the sewing of the leaves is accomplished via the permutation D Œ ; . Thus, we see: All the non-regular components are 1-ramified if and only if the permutations and commute (in particular, this will be the case if these permutations coincide, as in Example 12.5). In general, we see that we do not get all the possible permutations for . The mathematics behind it, is as follows: In the symmetric group † t , we fix one t -cycle as and form for any t -cycle the commutator D Œ ; : these are the permutations which arise for the sewing procedure. Proposition. A partition of t is the ramification sequence of a wind wheel if and only if it is the cycle partition for the commutator of two t -cycles. (By definition, the cycle partition of a permutation has as parts the lengths of the cycles when written as a product of disjoint cycles.) For t D 2, the group † t is commutative, thus we get as only the identity. This means: For t D 2, we always get two non-regular components, both being 1-ramified. For t D 3, the group † t is no longer commutative, however the 3-cycles commute, thus again the only commutator D Œ ; is the identity, thus again we see that we only get 1-ramified components.

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C. M. Ringel

The first case where one can obtain an r-ramified component with r > 1 is t D 4I an explicit example has been discussed in 12.6. For t D 4 one checks easily that the possible ramification sequences are .3; 1/ and .1; 1; 1; 1/.. For t D 5, they are .5/, .3; 1; 1/ and .1; 1; 1; 1; 1/ (for example, the commutator of the permutations .12345/ and .12354/ has the cycle partition .3; 1; 1/, that of .12345/ and .12453/ has the cycle partition .5/). In particular, we see that for t 5, there are no 2-ramified components. For t D 6, the commutator of .123456/ and .124653/ has the cycle partition .4; 2/. It seems that a description of the structure of the commutators Œ ; , where and are t-cycles in † t is not known (we are indebted to G. Malle for drawing our attention to the related investigations [3], [12]).

13 The Auslander–Reiten quilt of a wind wheel Auslander–Reiten quilts have been considered until now only for suitable special biserial algebras ƒ. A general definition can be given in case ƒ is a 1-domestic special biserial algebra, see [20]: The vertices are (finite or infinite) words, and there are arrows, meshes, but also a convergence relation. The Auslander–Reiten quilt considers not only the indecomposable ƒ-modules of finite length, but also related indecomposable ƒ-modules of infinite length which are algebraically compact. The main objective is to sew together Auslander–Reiten components which contain string modules, using N-words and Z-words, the Z-words yield “tiles”. We denote by rad the radical of mod ƒ, it is the ideal generated by the non-invertible homomorphisms between indecomposable ƒ-modules. Using transfinite induction, one defines rad for any ordinal number as follows (see [16], [24]): for any ideal I , and n 2 N, let I n be the n-fold product ofTcopies of I , in this way, radn is defined for n 2 N. If is a limit ordinal, let rad D ˛ 0, for all vertices z of „ (one may say that such a function with values in Z is “clusteradditive”, see [23]). Other similarities with cluster tilted algebras (see [14]) should be mentioned: Proposition 14.2. The barbell algebras are Gorenstein algebras of Gorenstein dimension 1 and the stable category of Cohen–Macaulay modules is Calabi–Yau of CY-dimension 3. For our Example 2, here are the minimal injective resolutions of the indecomposable projective modules: 0 ! P .1/ ! I.2/ ˚ I.2/ ! I.1/ ˚ I.3/ ˚ I.3/ ! 0; 0 ! P .2/ ! I.2/ ! I.1/ ˚ I.3/ ! 0; 0 ! P .3/ ! I.2/ ˚ I.2/ ! I.1/ ˚ I.1/ ˚ I.3/ ! 0:

552

C. M. Ringel

Let L be the full subcategory of all torsionless modules (by definition, a module is torsionless if it can be embedded into a projective module) and P the full subcategory of all projective modules. We have to calculate the factor category L D L=P . Since we deal with a 1-Gorenstein algebra, L is a triangulated category with Auslander–Reiten translation. It is not difficult to check that the only indecomposable modules which are torsionless and not projective are the two serial modules of length 2 with socle P .2/, we denote them by L.1/ D 12 and L.3/ D 32 . Thus, L has the following Auslander–Reiten ... ... quiver: .. .. ..

..

L.1/

L.1/

............................................................................. .................................................................................. .................................................. ............................................. ... ... ..................................... ................................. ... ... ............................. .. .. ......... ... ...

P .1/

P .2/

.. . ...

.. .......... ...... . ...... ...... ......

...... ...... ...... ...... ........ .....

P .3/

... .. ......... .. ............................... ................................. ........................................................... . . . . . ................. .................................................... .................................................................................... .............................................................................

.. . ...

L.3/

L.3/

... ...

... ...

The dashed line indicate that we have to identify vertices of the triangles exhibited: Note that both serial modules L.1/ and L.3/ are shown twice, these are the vertices which have to be identified. It follows that the (triangulated) category L has just two indecomposable objects, both being fixed under the suspension functor as well as under the Auslander–Reiten translation functor (so that L is the product of two copies of the stable module category of the algebra kŒ D kŒT =hT 2 i of dual numbers), the Auslander–Reiten quiver of L looks as follows: .. .. .. ... .. ... .. . . . . . . ... . .. .. .. ....

L.1/

.. .. .. ... .. ... .. . . . . . . ... . .. .. .. ....

L.3/

Thus, we deal with a triangulated category for which both the suspension functor as well as the Auslander–Reiten translation functor are the identity functor. This means that L is 3-Calabi–Yau, and indeed n-Calabi–Yau for any n. Since the module category of a barbell algebra shares so many properties with the module category of a cluster tilted algebra, one may wonder whether also for a barbell algebra ƒ the module category mod ƒ is obtained from a triangulated category C by forming C =hT i for some object T in C . As Idun Reiten has pointed out, this is indeed the case if we deal with a barbell algebra ƒ with two loops (as in our running Example 2) provided we assume that the characteristic of k is different from 3: such an algebra is 2-CY-tilted (this means: the endomorphism ring of some cluster tilting


553

object of a 2-Calabi–Yau category, [17]). Namely, if ƒ is a barbell algebra with two loops ˛, ı in its quiver Q and if the characteristic of k is different from 3, then ƒ is the Jacobian algebra J.Q; W / D kQ=h3˛ 2 ; 3ı 2 i, where W is the potential W D ˛ 3 C ı 3 , see [9], thus one can apply Theorem 3.6 of Amiot [1].

15 Sectional paths Recall that a (finite or infinite) path . ! Xi ! XiC1 ! / in the Auslander– Reiten quiver of a finite dimensional algebra is called sectional provided XiC1 is not isomorphic to Xi1 for all possible i . Such a path will be called maximal provided it is not a proper subpath of some sectional path. An infinite sectional path involving only monomorphisms will be called a mono ray, an infinite path involving only epimorphisms will be called an epi coray; of course, mono rays start with some module, epi corays end in a module. Note that for Auslander–Reiten components of the form ZA1 as well as for stable tubes, all maximal sectional paths are mono rays and epi corays. Theorem 15.1. Let ƒ be a k-algebra which is minimal representation-infinite and special biserial. Then any maximal sectional path is a mono ray, an epi coray or the concatenation of an epi coray with a mono ray. In the special case of ƒ being a wind wheel, we will see that any maximal sectional path is a mono ray or an epi coray: there are concatenations only for the barbells, as well in case there are nodes. Corollary. Assume that ƒ is minimal representation-infinite and special biserial. Let X , Y , Z be indecomposable ƒ-modules with an irreducible monomorphism X ! Y and an irreducible epimorphism Y ! Z. Then X D Z. Proof. We may assume that there are no nodes: Namely, if all the sectional paths of nn.ƒ/ are as mentioned, the same has to be true for ƒ: the only maximal sectional paths for ƒ to be looked at are those passing through the node. Resolving the node we will obtain sectional paths which are not double infinite paths, thus by the assumption on nn.ƒ/, we will deal with an epi coray ending in the node and a mono ray starting in the node, thus with a concatenation as listed. z n , thus a cycle algebra, then any maximal sectional path If ƒ is hereditary of type A is a mono ray, an epi coray. We only have to look at the preprojective component and the preinjective component. But if f W X ! Y is a non-zero map between indecomposable preprojective modules, then f has to be always a monomorphism: otherwise, the kernel of f would have negative defect, and since the defect of X is 1, it would follow that the image of f is a non-zero submodule of Y with non-negative defect, a contradiction. The dual argument shows that the maximal sectional paths in the preinjective component are epi corays. Next, let us look at the wind wheels: again, only the non-regular components have to be considered. But we know how to construct these components: we use mono rays

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from the preprojective component and the tube R1 as well as epi corays from the tube R0 and from the preinjective component, and in addition rays and corays in the tiles. But all the maximal sectional paths in the four quarters of a tile are mono rays and epi corays (in the quarter I we have only mono rays, in III only epi corays, whereas II and IV have both mono rays and epi corays. Finally, let us look at the barbells. There is only one non-regular component which has to be treated separately. The band modules lie in homogeneous tubes and there will be an additional regular tube containing string modules. What really is of interest are the remaining components C , they are of the form ZA1 1 . Let us look at the example 2 (the general case is similar). Let M D M.v/ be the Geiß module ([10]) for C (it is the unique module in C of minimal length) and one easily observes that v is a word of the form 1 ! 2 2 3. It is easy to see that all the modules t1 M for t 0 t are obtained from M by adding hooks both on the left and on the right; similarly, all the modules tC1 M for t 0 are obtained from t M by adding cohooks both on the left and on the right. But this implies that all the maximal sectional paths in C are concatenation of an epi coray with a mono ray. It may be helpful to call an indecomposable ƒ-module a valley module if it is the concatenation vertex for a sectional path which is the concatenation of an epi coray with a mono ray, and to exhibit corresponding pictures: always we encircle the “valleys”. First, we present a non-regular component of a wind wheel:

B

B

B

...... ....... .... ..... ..... ..... ........ .... .... ........ ..... .....

B

..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... . . . . . . . . ..... ..... .... .... ..... ..... ..... ..... ..... 5 ..... ..... 7 3...... 5 ..... ..... 55 ......... 7 4 ..... 5 2 ......... 4 7 4 5 . ..... . . . . ..... 4 ..... 7 6 . . . . .. 4 .......4.... .... ..... .... ..... ..... 6 .......... ...... ......6 ...... 6 ........ ..... ....... ..... ..... ..... ...... ....... ....... ......... ......... ......... .......... ....... ....... ........ .......... ........ ........... ........ ........ ........ ..... .... .... .... ..... ........ . ..... ...... ....... ............. .... .... .... . ..... ........ ..... ..... ..... ..... ..... .... ... . . . . . . . . . . . ..... 3 5 5 7 .. ... . . . . . . . . . . ..... 5 2 5 2 4 . . . . . . . . . . . 7 . . . . . . . . . . ........ ............. .. ..... .. .................... 5 4 ..... 4 4..... 6 6 .... ..... ........... ........... .............. ........... ..... ..... . . . . . . . . . . . . . ........ . . . . . .. .. .. .......... .......... .... ..... ..... .....

4

5

B

..... ..... ........ ....

3

.... ........ ..... .....

5

B

5

4

5 2 5 ....................... .................. 4 ............

2

..... ..... ........ ....

3

.... ........ ..... .....

B

..... ..... ........ .... .... ........ ..... ..... ..... ..... ........ ....

................. 42 32 ................. . ..... .... ................ ..... .................. ........ ................. .................. ........ ..... ................. . . . . . .................. . .... . . . . . . . . . . . . . . . . . . . . . . . . ..... . . ..... .................. ......... .......... . . . . . . . . . . . . . . ............ . . . . . ......... ..... .. 3 . ..... .................. ................ ..... ....... 2...... ........ .................. ................. . ..... .. ..... .................. .. . . .. ............ ................ ..... .................. ..... ............. ................ ........ ........ . . . ................. . . . . ........ .................. .. ................. . ... ....... ....... ....... ............ ........ . . . . . . . .... .................. . . . . . . . . . . . . ..... ..... ..... ................... ........................... ... . . . . . . . . . . . . . . .......... . ..... ........ ... . . . 3 3 0 . . . . . . . . ..... ..... .......... ... 3 2....... .................. . . . . . . . . . . . . 2 . . . . . . . . .................. ... ... .... .... ................. .................. ........... ......... ........ ......... ..... ............. ........ ......... ................. .................. . ..... . ...... ..... ................. . .. .................. ..... ..... ............. ......... ....... ................ .................. ..... ....... ......... ..... ..... ..... ................................ ............................. . . . . . . . . . . . . . ......... ..... .......... ..... . ..... 3 0 ..... ..... ..... ..... ..... 3 2 .. ................ ..... ..... . ... ..... .. ................ . ....... ..... ................ ..... ............ ............ ......... ........ ........ ................ ................ ..... .............. ........ .......... . . . . . . . . . . . ................ . . . . ........ ................ ..... ........... ........ ....... ... ... ... ...... ....... ................ . . . . . . . . . ................ . . . . ..... . . . . . . . . ..... .. ................ . ..... ................ ... . . . . . . . . . . . . . .................. . . . . . . . . . . . . ..... .......... ... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .... .... .. . ..... ..... ..... ..... ........ ....

B

.... ........ ..... .....

B

B

B

B

B

B

B

B

B

B

The valleys may be considered as the natural places where to cut such a component into pieces. Of course, in our cut-and-paste process, we followed this rule. The second example is the non-regular component of the barbell given as Example 2:


..... ....... ...........

B

B

B

........... ....... ..... ..... ..... ..... ....... .......... ........... ....... ..... ..... ..... ..... ....... ........... .............. .. ..... ..... ..... ..... ..... .

B

..... ..... .... .... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ........... .... ......... .... .... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... ..... ..... . .. .. .. ............ ............ ............ ....... ..... ....... ..... ........... ..... ..... ..... ... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .... ..... . ..... ..... . . . . . . .. ..... .... .... .. ..... ..... ......... . ..... ..... ...... ........... ..... ........... .. ..... ..... ..... . ..... ............... ..... ......... ....... ......... ....... ....... ........ ....... ....... ......... ....... ..... . . ......... ..... ........ ..... ...... ..... ...... ..... ..... ..... . ........... . . ........... . . . . . . . . . . . . . . . . . . . ...... ..... . . ..... ..... . ... . . ..... . . . . . ... ..... . . . . . . . . . . . . . ..... . 1 ............ ..... ... ..... ... .. .. ..... ....2 ........ ................ .... ..... . .................... ..... . . . . . . ..... . ................ . ............ . ..... ..... ...... . ....... . . . . ..... . . ....... . . . . . .... ..... .... .............. ........... .... .... ....

B

B

B

B

B

B

S.1/

I.1/

P .1/

........... ....... ..... .....

..... ..... ....... ...........

..... ..... ....... ...........

.......... ....... ..... ....

..... ..... ....... ...........

........... ....... ..... .....

.......... ....... ..... ....

..... ..... ....... ...........

..... ..... ....... ...........

..... ..... .............. ..... ..... .. ....... ....... ..... ........... .............. ........... ........................ . ..... . . . . . . . . ..... ... . ... . . . . . . . . ..... .. . . . . . . . . . . . ... 3 ... . . . . . . . . ..... ... .. . ..... .... 2 ..... ... ..... . ..... . ..... ........ ..... .............. ............. ......... ......... ..... ..... ............... ..... ......... ..... ..... .. ..... ..... . ..... .... ...... ..... . . ..... ...... . . . . . . ......... . ..... . . . ..... ..... .... . . . . . . . . . ..... ..... .... .... . . . . . . . . . . . . .... .... .... ....

I.2/

B

555

.......... ....... .....

P .2/

I.3/

B

.............. .. ..... .....

B

. ..... ..... ..... ..... ..... ..... .

P .3/

S.3/

B

..... ..... ..... ..... ..... ..... .

Of course, when dealing with a barbell and look at a regular component C of string modules, say with Geiß-module M , then the valley modules are precisely those which lie on the sectional paths which contain M . In all these components, the “valleys” provide a clear division into regions with common growth pattern. For example, in the regions on the left, all irreducible maps are epimorphisms, whereas in the regions on the right, all are monomorphisms.

Part III Appendix The appendix collects some remarks related to the investigations presented above. First, we show an example of an algebra which may be considered as a twisted version of a barbell.

16 Further minimal representation-infinite algebras Consider the following algebra: ˛

....................... ..... ............. ... .. .. . ... .... ................................. .. ... ... ...... ...... . ....................

1

ˇ

2

3

30

....... ....... ....... ........ .......... .......... ........ ....... ....... 0 .......

(or, more generally, the corresponding algebras where ˛ and ˇ are replaced by longer paths). Note that the universal covering are the “dancing girls” of Brenner–Butler. This is a Gorenstein algebra of Gorenstein dimension 1, the minimal injective resolutions of the indecomposable projective modules are as follows:

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0 ! P .1/ ! I.1/ ! I.2/ ˚ I.2/ ! 0; 0 ! P .2/ ! I.1/ ! I.2/ ˚ I.3/ ˚ 30 ! 0; 0 ! P .3/ ! I.1/ ! I.2/ ˚ I.30 / ! 0; 0 ! P .30 / ! I.1/ ! I.2/ ˚ I.3/ ! 0; and here is the central part of the non-regular component: ................................................................................................................................................................................................. ................................................................................................................................................................................................... .................................................................................................................................... ............................................................................................................................ ............................................................................................................................ ............................................................................................................................ ............................................................................................................................................................................ ............................................................................... ............................................................................................................. ...................................................................................................................................................................................................... ......................................................... ................................................................................ ............................................................................................................................ ................... ............................1 ........................... . . . . . . . . . . . . . . . ...................................... .......................2 ......................... ..... ......................... ....... ..... ............................................................................................................... ... .... . . . . . . . .......... . . . . ....... . . . . . . ......... . . . . ...... . . ......................................... . . ......................... ..................... ..................................... . ..... .................................... ..... ............ . . . ........ . . . . . . ............. . . . ........ . . . . . . . .. . ... .................... ... .... ..................................................................................... ............... ..... .... ....................................................... .... .............. ..................................................... ................ . ..... .... ................................................... .... .......................... ..... .. ..... . . . . . . . . .......... . . . ....... . . . . . . . . . ..... ............................ .................................... . ........................................ . . . . . . ..... ........ . . . . ....... . . . . . . ........ . . . . . ..... . ... . ........................................... . . . . . .......................................................................... ..................................... . ..... ..... . . ........................................... ... ..... .................................. . ......................................................... .................................... ........ ............ ........ ........ ........ ........ ................................................................................................... .......................................... ................................................... .... .... ... ....................... ................ ......................... ........................ . . . . . . . ..... . ..... . . . . . . . ............ . . . . ....... . ...................................................................... ..... . ........... . . . . ...... . .................. .. . ............................ ......................................... .. ... . . . . .... . . .. . . . . . ..... . ................................................................ ............................................................... ..... . .............................. ............................................. ... ....... .......0................................ . ...............................................................0 .......................... .............................................

B

B

B

B

B

B

S.1/

I.1/

B

B

B

P .1/

B

I.2/ I.3/

P .2/ P .3/

B

I.3 /

P .3 /

B

17 Barification may change the representation type Consider the path algebra of the quiver 0 ˛ 10 ˇ 0 20 100 ˇ 00 200 ı 3 B .......................... B .......................... B ........................... B .......................... B .......................... B

and barify the arrows ˛2 and ˛4 . The we obtain the quiver 0 ˛ 1 .............ˇ .............. 2 ı 3 . ....... B .......................... B : B ............................ ......................................B .............. ... ...

Here, starting with a representation-finite algebra, we obtain a tame one. Similarly, if we start with the following tame quiver, the barification of b 0 and b 00 yields a wild algebra: 4 . ..................... 5 ..... B .... B 0 ˛ 10 ˇ 0 20 100 ˇ 00 200 ı 3 ........................ B ........................... B ........................... B ........................... B ........................... B ........................... B .................. ....... ..... 6 B 0

B

˛ .

1

ˇ

2

ı

. ...... ............. ......... .......................... ........................ ........................ ........................... .... ........ ... . .

B

B

3

B

.. ....... . ....... ........... ............ . ......... ....... .

4

5

B .......................... B 6

B


557

18 Accessible representations We have mentioned in the introduction that the recent paper [5] of Bongartz has drawn the attention to the minimal representation-infinite algebras which have a good cover z such that all finite convex subcategories of ƒ z are representation-finite. As we show ƒ, above, these algebras are special biserial and can be completely classified. The title of the Bongartz paper [5] indicates that his main concern was to proof the following theorem: Let ƒ be a finite dimensional k-algebra where k is an algebraically closed field. If there exists an indecomposable ƒ-module of length n > 1, then there exists an indecomposable ƒ-module of length n1. Unfortunately, the statement does not assert any relationship between the modules of length n and those of length n 1. There is the following open problem: Given an indecomposable ƒ-module M of length n 2. Is there an indecomposable submodule or factor module of length n 1‹ The three subspace quiver shows that this may not be true in case the field k is not algebraically closed, say if it is finite field with few elements. In [22] we slightly modified the arguments of Bongartz in order to strengthen his assertion. Using induction, one may define accessible modules: First, the simple modules are accessible. Second, a module of length n 2 is accessible provided it is indecomposable and there is a submodule or a factor module of length n 1 which is accessible. The open problem mentioned above can be reformulated as follows: Are all indecomposable representations of a k-algebra ƒ, where k is algebraically closed, accessible? This is known to hold in case ƒ is representation-finite and the aim of [22] was to show that any representation-infinite algebra over an algebraically closed field has at least accessible modules of arbitrarily large length. In dealing with special biserial algebras, we do not have to worry about the size of the base field k. The following assertion is valid for k-algebras with k an arbitrary field. Proposition 18.1. Any indecomposable representation of a special biserial algebra is accessible. Proof. It is obvious that string modules are accessible, thus we only have to consider band modules. It will be sufficient to show the following: any band module has a maximal submodule which is a string module. Thus, let M be a band module. First, let us consider the special case of dealing with the Kronecker algebra, thus M D .M1 ; M2 I ˛; ˇ/ with vector spaces M1 and M2 , and invertible linear maps ˛; ˇ W M1 ! M2 . Let M 0 be a submodule of M which is a band module and of smallest possible dimension. Note that M 0 is uniquely determined and is contained in any non-zero regular submodule of M . Let 0 ¤ x 2 M10 and choose a direct complement U M1 for kx. Then N D .U; M2 I ˛jU; ˇjU / is a submodule of M , and of course a maximal one. We claim that N is a string module. As a submodule of a regular Kronecker module, we can write N D N 0 ˚ N 00 with N 0 preprojective and N 00 regular. But N 00 has to be zero, since otherwise M 0 N 00 , thus x 2 N100 U , a contradiction. This shows that N is a direct sum of say t indecomposable preprojective

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Kronecker modules. Since dim N1 dim N2 D 1, it follows that t D 1. This shows that N is an indecomposable preprojective Kronecker module and thus a string module. Now consider an arbitrary special biserial algebra ƒ with quiver Q. There is a primitive cyclic word w 2 .ƒ/ and an indecomposable vector space automorphism W V ! V such that M D M.w; /. Let w D l1 : : : ln with letters li ; we can assume that ln1 is a direct letter, whereas ln is an inverse letter. Denote by xi1 the terminal point of li , for 1 i n. Then M is given by t copies Vi of V , indexed by 0 i n 1, such that the arrows of Q operate as follows: if li D ˛ is a direct letter (thus an arrow), then ˛ is the identity map Vi ! Vi1 , if li is an inverse letter, say li D ˛ 1 for some arrow ˛, then ˛ is the identity map Vi1 ! Vi for i ¤ n and the map W Vn1 ! V0 for i D n. Note that .V; V I 1; / is a band module for the Kronecker quiver, thus, as we have seen already, it has a maximal submodule L .U; V I 1jU; jU / which is a string module. We obtain a submodule N of M D n1 iD0 Vi by taking the L V ˚ U , where U is considered as a subspace of Vn1 . Since subspace N D n2 iD0 i .U; V I 1jU; jU / is a string module for the Kronecker algebra, it follows that N is a string ƒ-module.

19 Semigroup algebras It should be mentioned that algebras defined by a quiver, commutativity relations and zero relations can be considered as factor algebras of a semigroup algebra kŒS modulo a one-dimensional ideal generated by a central idempotent e, thus the paper may be seen as dealing with a class of minimal representation-infinite semigroups. Let S be a semigroup (a set with an associative binary operation). An element z of S is called a zero element provided sz D z D zs for all s 2 S. Of course, if there is a zero element, then it is uniquely determined. Let S be a semigroup with zero element z, we want to consider the semigroup algebra kŒS . Obviously, the element z considered as an element of kŒS is a central idempotent and the ideal hsi generated by z is one-dimensional, thus z is a primitive idempotent. With z also 1 z is a central idempotent, and we obtain a direct decomposition of kŒS as a product of k-algebras kŒS D hzi h1 zi D kz kŒS .1 z/: One may call kŒS .1 z/ D kŒS=hzi the reduced semigroup algebra of S . It follows that the modules for the reduced semigroup algebra of S are precisely the kŒS modules M with zM D 0. The product decomposition of the semigroup algebra kŒS shows that there is a unique simple (one-dimensional) kŒS-module which is not annihilated by z, all other indecomposable kŒS-modules are annihilated by z and thus are modules over the reduced semigroup-algebra. Given a quiver Q, let S.Q/ be obtained from the set of all paths (including the paths of length 0) by adding an element z (it will become the zero element). As in the definition of the path algebra kQ of a quiver, define the product of two paths to


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be the concatenation, if it exist, and to be z otherwise. In this way, S.Q/ becomes a semigroup with zero element z, and the reduced semigroup algebra of S.Q/ can be identified with the path algebra kŒQ of the quiver Q. Of course, if we deal with a set of commutativity relations and zero relations, then we may consider the factor semigroup S.Q; / D S.Q/=hi, this is again a semigroup with zero, and its reduced semigroup algebra is just the algebra defined by the quiver Q and the relations .

References [1] C. Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential. Annales de l’Institut Fourier 59 (2009), 2525–2590. [2] R. Bautista, P. Gabriel, A. V. Roiter and L. Sameron, Representation-finite algebras and multiplicative bases. Invent. Math. 81 (1985), 217–285. [3] E. Bertram, Even permutations as a product of two conjugate cycles. J. Combin. Theory Ser. A 12 (1972), 368–380. [4] K. Bongartz, Treue einfach zusammenhängende Algebren I. Comment. Math. Helv. 57 (1982), 282–330. [5] K. Bongartz, Indecomposables live in all smaller lengths. Preprint 209, arXiv:0904.4609 [math.RT]. [6] A. Buan, R. Marsh and I. Reiten, Cluster-tilted algebras. Trans. Amer. Math. Soc. 359 (2007), 323–332. [7] M. C. R. Butler and C. M. Ringel, Auslander-Reiten sequences with few middle terms and applications to string algebras. Comm. Algebra 15 (1987), 145–179. [8] P. Dowbor and A. Skowroński, Galois coverings of representation-infinite algebras. Comment. Math. Helv. 62 (1987), 311–337. [9] H. Derksen, J. Weyman and A. Zelevinsky, Quivers with potentials and their representations I: Mutations. Selecta Math. 14 (2008), 59–119. [10] C. Geiss, On components of type ZA1 1 for string algebras. Comm. Algebra 26 (1998), 749–758. [11] I. M. Gelfand and V. A. Ponomarev, Indecomposable representations of the Lorentz group. Russian Math. Surveys 23 (1968), 1–58. [12] D. H. Husemoller, Ramified coverings of Riemann surfaces. Duke Math. J. 29 (1962), 167–174. [13] J. P. Jans, On the indecomposable representations of algebras. Ann. of Math. 66 (1957), 418–429. [14] B. Keller and I. Reiten, Cluster-tilted algebras are Gorenstein and stably Calabi-Yau. Advances Math. 213 (2007), 140–164. [15] R. Martinez-Villa, Algebras stably equivalent to l-hereditary. In Representation Theory II. Lecture Notes in Mathematics 832, Springer-Verlag, Berlin, 1980, 396–431.

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[16] M. Prest, Model Theory and Modules. London Mathematical Society Lecture Note Series 130. Cambridge University Press, Cambridge, 1988. [17] I. Reiten, Homological properties of cluster tilted algebras. Talk at the workshop: Cluster Algebras and Cluster Tilted Algebras. Bielefeld 2006. [18] C. M. Ringel, Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics 1099, Springer-Verlag, Berlin, 1984. [19] C. M. Ringel, Some algebraically compact modules I. In Abelian Groups and Modules. Kluwer Academic Publishers, Dordrecht, 1995, 419–439. [20] C. M. Ringel, Infinite length modules. Some examples as introduction. In Infinite Length Modules. Birkhäuser Verlag. Basel, 2000, 1–73. [21] C. M. Ringel, On generic modules for string algebras. Bol. Soc. Mat. Mexicana 7 (2001), 85–97. [22] C. M. Ringel, Indecomposables live in all smaller lengths. Bull. London Math. Soc., to appear. [23] C. M. Ringel, Cluster-additive functions on stable translation quivers. Preprint 2011, arXiv:1105.1529 [math.RT]. [24] J. Schröer, On the infinite radical of a module category. Proc. London Math. Soc. 81 (2000), 651–674. [25] A. Skowroński and J. Waschbüsch, Representation-finite biserial algebras. J. Reine Angew. Math. 345 (1983), 172–181. [26] C. Thiele, The topological structure of Auslander-Reiten quivers of special string algebras. Comm. Algebra 21 (1993), 2507–2526. [27] B. Wald and J. Waschbüsch, Tame biserial algebras. J. Algebra 95 (1985), 480–500.

Coalgebras of tame comodule type, comodule categories, and a tame-wild dichotomy problem Daniel Simson

Contents 1 2 3 4 5 6 7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries, basic facts and notation . . . . . . . . . . . . . . . . . . . The Grothendieck group and the composition length vector of a comodule The left-valued Gabriel quiver of a basic coalgebra . . . . . . . . . . . . Irreducible morphisms and the Auslander–Reiten quiver of a coalgebra . . Two concepts of tameness and wildness for a K-coalgebra . . . . . . . . Quivers, profinite bound quivers, path coalgebras and locally nilpotent representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Euler coalgebras, Cartan matrices and Euler characteristic . . . . . . . . . 9 Almost split sequences in comodule categories . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction Throughout we fix an arbitrary field K. Given a finite-dimensional K-algebra R, we denote by Mod R the category of right R-modules, and by mod R the full subcategory of Mod R whose objects are modules of finite dimension. The representation theory splits the finite-dimensional K-algebras over an algebraically closed field K into two classes: (a) the K-algebras R of wild (module) type, for which the category mod R is Kwild, that is, there is a representation embedding mod R0 ! mod R, for any finite-dimensional K-algebra R0 , and (b) the K-algebras R of tame (module) type, for which the category mod R is Ktame, that is, the isomorphism classes of the indecomposable modules in mod R of any fixed K-dimension d 0 form at most finitely many one-parameter families; see [80], Chapter 14, and [99], Chapter XIX, for details. A famous tame-wild dichotomy theorem of Drozd [29] asserts that every algebra over an algebraically closed field K is either tame or wild, and the two types are mutually exclusive. The class of tame algebras contains the representation-finite algebras, that is, the algebras R with finitely many isomorphism classes of indecomposable modules

Supported by Polish Research Grant N.C.N. 2011/STat/2011-2015.

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in mod R. We recall from [4] that the class of representation-finite algebras coincides with the class of pure semisimple algebras [77], that is, such algebras R that every module in Mod R is a direct sum of modules in mod R; see [75]–[77]. In this survey article, we study K-coalgebras C and their comodule categories. This includes the study of modules of finite-dimensional K-algebras R, because any such K-algebra has the form R D C D HomK .C; K/, where C is a finite-dimensional K-coalgebra such that the category mod R is equivalent to the category of finite-dimensional C -comodules. We discuss the concepts of tameness, wildness, discrete comodule type, polynomial growth, and the pure semisimplicity, introduced by the author in [83], [84], [93], for basic K-coalgebras C over an algebraically closed field K and intensively studied during the last decade. In particular, we show that the tame-wild dichotomy holds for a wide class of coalgebras C of infinite K-dimension. Moreover, we prove a weak version of tame-wild dichotomy for coalgebras asserting that, over an algebraically closed field K, any basic K-coalgebra of tame comodule type is not of wild comodule type. To formulate the main definitions and results, we recall that a (unitary) K-coalgebra C is a K-vector space C together with two K-linear maps W C ! C ˝ C (the comultiplication) and " W C ! K (the counity) satisfying the coassociativity condition . ˝ idC / D .idC ˝ /, and the counity conditions ." ˝ idC / D idC , .idC ˝ "/ D idC , under the identification C ˝ K Š C Š K ˝ C , where we set ˝ D ˝K . A subcoalgebra of a K-coalgebra C is a K-vector subspace D of C such that .D/ D ˝ D C ˝ C . A coalgebra C is said to be simple if C has no non-zero subcoalgebras. We denote by C op the coalgebra opposite to C . A K-linear map f W C ! C 0 between K-coalgebras C and C 0 is defined to be a coalgebra homomorphism if 0 f D .f ˝ f / and "0 f D ". Given a K-coalgebra C , a left C -comodule (or a left corepresentation of C ) is a vector space M together with a K-linear map ıM W M ! C ˝ M such that . ˝ idM /ıM D .idC ˝ıM /ıM and ."˝idM /ıM is the canonical isomorphism M Š K˝M . Left C -subcomodules of the C -comodule C are called left coideals of C . Given a left C -comodule M we denote by soc M the socle of M , that is, the sum of all simple C -subcomodules of M . A K-linear map f W M ! M 0 between left C -comodules M and M 0 is a C -comodule homomorphism if ıM 0 f D .idC ˝ f /ıM . The vector space of all C -comodule homomorphisms f W M ! M 0 is denoted by HomC .M; M 0 /. We set EndC M D HomC .M; M /. Given a K-coalgebra C , we denote by C -Comod and C -comod the categories of left C -comodules and left C -comodules of finite K-dimension, respectively. The right C -comodules will be identified with the left C op -comodules. We set C op -Comod D Comod-C and C op -comod D comod-C . In general, the category C -Comod has not enough projectives, and sometimes it has no non-zero projective object, see [66], [83], and [84]. One of the aims of the representation theory of coalgebras (and algebras of finite dimension) is to give, for a K-coalgebra C , a detailed description of the category

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C -comod, for any K-coalgebra C , where K is a field. In particular, the main aims are the following. (A1) Construct a complete list of the indecomposable comodules in the category C -comod, up to isomorphism, and parametrize them in a suitable way. (A2) Given a comodule M in C -comod (resp. in the category C -Comodfc of finitely copresented comodules, see Section 2.1, find a decomposition of M in a direct sum of indecomposable comodules. Note that C -comod is a Krull–Schmidt category. (A3) For any pair of indecomposable comodules M and N in the category C -comod, give an explicit description of the endomorphism K-algebras EndC M and EndC N by means of generators and relations, and a description of the vector spaces HomC .M; N / and Ext m C .M; N /, for m 1, together with their End C N EndC M -bimodule structures. m (A4) In case dimK Extm C .M; N / is finite for all m 0, and Ext C .M; N / vanishes for m sufficiently large, compute the integer C .M; N / D

1 P

.1/j dimK ExtjC .M; N / 2 Z;

j D0

called the Euler characteristic of the pair M , N . (A5) Describe the connected components of the valued Auslander–Reiten quiver .C -comod/ of the category C -comod in the sense of Definition 5.4. The question whether or not a K-coalgebra C admits a “suitable parametrization” of indecomposable comodules in C -comod (required in (A1)), leads us in [84] to a definition of K-tameness and K-wildness for coalgebras, in an analogy to Drozd0 s tameness and wildness of bocses and finite-dimensional K-algebras given in [29]. In Section 6, we discuss two different concepts of tameness for a K-coalgebra C . One of them is the tame comodule type (resp. wild comodule type) of C defined by the requirement that the category C -comod is tame (resp. wild). We show that the definition of wild comodule type is left-right symmetric; see [83], [84], and Sections 6.1–6.3 for details. Unfortunately, the tame-wild dichotomy remains an open problem, that is, we are not able to prove that every K-coalgebra over an algebraically closed field K is either of tame comodule type or of wild comodule type, and the two types are mutually exclusive. To get a solution, at least for a relatively large class of coalgebras C , we have introduced in [93] and [95] a stronger definition of tameness and wildness; namely we define the fc-tame comodule type and the fc-wild comodule type of C by means of the tameness and wildness of the full subcategory C -Comodfc of C -Comod whose objects are the (socle) finitely copresented C -comodules; see Section 2. We show that the definition is left-right symmetric if C is computable, that is, dimK EndC E < 1, for all socle-finite direct summands of C C .

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The class of coalgebras C of tame comodule type (resp. of wild comodule type) can differ from the class of coalgebras C of fc-tame comodule type (resp. of fc-wild comodule type) in case the simple C -comodules are not finitely copresented, that is, if the category C -comod is not contained in C -Comodfc . However, if C -comod C -Comodfc and C is of fc-tame comodule type (resp. of fc-wild comodule type) then C is also of tame comodule type (resp. of wild comodule type). In particular, this is the case if C is a left locally artinian coalgebra; see Sections 6.3 and 6.4. Under the assumption that K is algebraically closed, we show in Section 6.4 that the f c-tame-wild dichotomy holds for computable K-coalgebras by a reduction of the problem to the tame-wild dichotomy theorem, for free triangular Roiter bocses and finite-dimensional K-algebras, proved by Drozd [29], see also Crawley-Boevey [22], [23], and Drozd–Greuel [30]. Hence we deduce that the usual tame-wild dichotomy holds for a wide class of K-coalgebras C of infinite dimension, containing the hereditary coalgebras, left semiperfect coalgebras, right semiperfect coalgebras, and the incidence coalgebras K I of interval finite posets I . Recall that a coalgebra C is said to be left (resp. right) semiperfect if every simple left C -comodule (resp. right C -comodule) has a finite-dimensional projective cover in C -comod; see [57]. To see a relationship between the representation theory of K-coalgebras and the representation theory of finite-dimensional K-algebras, note that if C is a coalgebra of finite dimension then the K-dual vector space C ´ HomK .C; K/ to C admits a K-algebra structure (induced by and ") and there is an equivalence of categories C -comod Š mod C . If C is of infinite dimension then C is a directed union of a directed family of finite-dimensional subcoalgebras Cˇ C , the category C -comod is a directed union of the full subcategories CS C -comod ˇ -comod of C -comod S and therefore has a directed union form C -comod D ˇ Cˇ -comod Š ˇ mod Cˇ : Consequently, the category C -comod can be studied “locally” by means of the categories of finite-dimensional modules over the finite-dimensional algebras Cˇ . One of the main tools in the study of comodule categories are quivers and their locally nilpotent K-linear representations, and a categorical duality between C -Comod and the category C -PC of pseudocompact left modules over the pseudocompact algebra C (in the sense of Gabriel [34] and [36]) described in Section 2.2. One of the motivations for our study of comodule categories is the fact that, for every locally finite Grothendieck K-category A, there exist a K-coalgebra C and an equivalence of K-categories A Š C -Comod, see [34], [40], [79], [84], and [104] for more details. Important examples of such categories are hereditary Ext-finite length K-categories with Serre functor studied by Reiten and Van den Bergh [71], Section 3; see also [50]. Another motivation for the study of comodule categories and pseudocompact modules over pseudocompact K-algebras is the role they play in a description of the Jacobian algebras of quivers with potentials in the sense of [27] and the Jacobi-finite quivers with potentials studied by Amiot [1], and their extension by Keller and Yang in [48] to c see [88] and Sections 7.2 and 7.5. complete pseudocompact path algebras KQ;


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Let us also mention another sources, where the study of coalgebras and comodules play a key role. The first one comes form the fact that in the framework of “noncommutative geometry” the notion of principal bundles generalizes by means of the so-called entwining structures .A; C /' in the sense of Brzeziński and Majid [8], where A is an algebra, C a coalgebra, and ' W A ˝ C ! C ˝ A a linear map verifying some compatibility conditions. Here the coalgebra C plays the role of a principal bundle. These structures were used by several authors in order to unify different kind of modules as Doi–Koppinen Hopf modules, Hopf modules, or Yetter–Drinfeld modules; see [10] and references therein. Recall also that Kontsevich and Soibelman [51] have associated to each noncommutative thin scheme (or non-commutative formal manifold) M , a huge coalgebra CM , called the coalgebra of distributions on M , such that M 7! CM induces is an equivalence between the category of coalgebras and the category of noncommutative thin schemes. The comodule category CM -Comod is equivalent to the category K Q-Comod, where K Q is the path coalgebra of an infinite quiver Q associated to the coordinate ring of the non-commutative formal manifold M ; see also [62] and [102]. The main tools we use in the study of K-coalgebras C , their comodule categories, and the representation types are: the Gabriel quiver CQ and the Auslander–Reiten quiver of C -comod, path coalgebras of quivers, Dynkin diagrams, Euclidean diagrams, profinite bound quivers and their locally nilpotent K-linear representations, a categorical duality between C -Comod and the category C -PC of pseudocompact left modules over the pseudocompact algebra C in the sense of Gabriel [34] and [36], (e) almost split sequences in categories of C -comodules, (f) an Euler Z-bilinear form bC , the Euler characteristic C , and a Coxeter transformation ˆC , and (g) a cotilting technique in categories of comodules developed in [92], [106]–[107]. (a) (b) (c) (d)

The reader is referred to [32], [59] and [103] for the coalgebra and comodule terminology, to [34] and [68] for the category theory terminology, and to [2], [5] and [80] for the representation theory terminology.

2 Preliminaries, basic facts and notation To keep the article self-contained, we collect in this section the terminology, notation, definitions, and basic facts we need in the study of comodule categories C -Comod. The following theorem contains the main properties of comodule categories we use in the article. The reader is referred to [14], [15], [32], [59], [84], [85], [89], and [103] for details.

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Theorem 2.1. Let C be a K-coalgebra, where K is a field. (a) The coalgebra C is a directed union of finite-dimensional subcoalgebras and every left C -comodule M is a directed union of finite-dimensional subcomodules. In particular, every simple comodule is finite-dimensional and it is isomorphic to a subcomodule of the left comodule C C . (b) C -Comod is a locally finite Grothendieck K-category and C -comod is the full subcategory of C -Comod consisting of objects of finite length (see [34] and [68]). The category C -comod is a skeletally small abelian Krull–Schmidt K-category. (c) The coalgebra C , viewed as a left C -comodule, is an injective cogenerator in the category C -Comod. If M is a left C -comodule, then the left comultiplication map ıM W M ! C ˝ M is a C -comodule embedding and the left C -comodule C ˝ M is injective. (d) The category C -Comod has enough injective objects. Every injective object in C -Comod is a direct sum of injective envelopes E.S/ of simple C -comodules S . L (e) The direct sum ˇ Eˇ of a family of comodules Eˇ is injective in C -Comod if and only if all summands Eˇ are injective. ! HomK .M; K/ D M ; (f) The Yoneda K-linear map "QM W HomC .M; C / ' 7! "', is an isomorphism of right EndC M -modules, for any left C -comodule M . L (g) If socC C D j 2IC S.j /tj is a direct sum decomposition of the left comodule socC C , where IC is a set, tj 1, and S.j / are pairwise non-isomorphic simple comodules, then tj is finite and tj D

dimK S.j / : dimK EndC S.j /

(2.2)

(h) The global homological dimensions gl.dim C and gl.dim C op of C and C op coincide [61]. 2.1 Basic notation and definitions. Throughout, K is a field. We denote by N the set of non-negative integers, and by Z the abelian group of integers. Given a set I , we denote by ZI the direct product of I copies of the group Z, and by Z.I / ZI the direct sum of I copies of the group Z. We view Z.I / ZI as abelian groups. By a ring, we always mean an associative ring R with an identity element. The Jacobson radical of R is denoted by J.R/. Let C be a K-coalgebra. For m 0 and a pair of left C -comodules M and M 0 , 0 we denote by Ext m C .M; M / the m-th extension group in the Grothendieck category C -Comod (see [28], [68]), E.M / denotes an injective envelope of M in C -Comod, and soc M is the socle of M , that is, the sum of all simple C -subcomodules of M . In particular, socC C is the socle of the left C -comodule C C . A C -comodule M is said to be socle-finite if the K-dimension of the socle soc M is finite. We denote by C -inj the full subcategory of C -Comod which objects are the soclefinite injective comodules.


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We say that a comodule M is (socle) finitely copresented if there is an exact sequence 0 ! M ! E0 ! E1 in C -Comod with E0 and E1 in C -inj, called a socle-finite injective copresentation of M . If E0 ; E1 2 add.E/ for some socle-finite injective C -comodule E, the comodule M is called finitely E-copresented. We denote by C -Comodfc C -ComodE fc the full subcategories of C -Comod the objects of which are the finitely copresented comodules and finitely E-copresented comodules, respectively. Here add.M / means the full subcategory of C -Comod which objects are summands of direct sums of finitely many copies of M . If N is a right C -comodule and M is a left C -comodule, the cotensor product of M and N is the K-vector space (see [59], [103]) ıN ˝idM idN ˝ıM

N M D Ker.N ˝ M ! N ˝ C ˝ M /:

(2.3)

Given a simple C -comodule S and M in C -Comod, the S-socle socS .M / soc M of M is the sum of all simple subcomodules of M isomorphic to S . A C -comodule M is defined to be piecewise socle-finite [92] (or quasi-finite [104]) if socS .M / is finite-dimensional, for any simple comodule S , or equivalently, dimK HomC .Z; M / is finite, for any comodule Z in C -comod. This means that the simple summands of soc M have finite (but perhaps unbounded) multiplicities [104]. Following the standard notation, we denote by C0 C1 Cm C the coradical filtration of the coalgebra C , where C0 D socC C D soc CC is the coradical of C (i.e., the sum of all simple subcoalgebras of C ), C1 D C0 ^ C0 is the wedge of two copies of C0 , and CmC1 D C0 ^ Cm for m 1; see [59], Corollary S1 5.1.8. 1 We recall that Cm D .C ˝ Cm1 C C0 ˝ C / for any m 1, and C D mD0 Cm , see [32], [59], [103]. We recall that c 2 C is said to be a group-like element of C if .c/ D c ˝ c and ".c/ D 1. The set of all group-like elements of C is denoted by G.C /. Note that, given c 2 G.C /, H D Kc is a simple subcoalgebra of C . Moreover, a simple subcoalgebra H of C is of dimension one if and only if H D Kc for some c 2 G.C /. A K-coalgebra C is defined to be: • connected (or indecomposable) if C is not a direct sum of two non-zero coalgebras; • pointed if every simple subcoalgebra H of C is one-dimensional, or equivalently, H D Kc, for some c 2 G.C /; • hereditary if gl.dim C D gl.dim C op D 1, that is, the left C -comodule E=M is injective, for any injective left C -comodule E and any subcomodule M E [61]; • left locally artinian if every direct summands E, with simple socles soc.E/, of the left C -comodule C C is an artinian object in C -Comod [50]; • left cocoherent if every socle-finite epimorphic image M of a socle-finite injective C -comodule E is finitely copresented in C -Comod [50],

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• Hom-computable (or computable, in short) if dimK HomC .E1 ; E2 / is finite, for every pair of direct summands E1 and E2 , with simple socles soc.E1 / and soc.E2 /, of the left C -comodule C C [90]; • Ext-finite if the category C -comod is Ext-finite, that is, dimK Extm C .S1 ; S2 / is finite for all m 0 and all pairs of simple subcomodules S1 and S2 of C C ; equivalently, dimK Ext m C .M1 ; M2 / is finite for all m 0 and all pairs M1 and M2 in C -comod. Remarks 2.4. (i) The definitions of Hom-computable coalgebras and Ext-finite coalgebras are left-right symmetric, because there are dualities C -inj Š C op -inj and C -comod Š C op -comod, see Corollary 6.27. (ii) The class of left cocoherent coalgebras contains right semiperfect coalgebras, hereditary coalgebras, and left locally artinian coalgebras; see [50]. (iii) If the coalgebra C is computable and left cocoherent, then the K-category C -Comodfc is abelian, it has enough injective objects, and it is Ext-finite, that is, dimK Extm C .M; N / is finite for all m 0 and all comodules M , N in C -Comod fc . (iv) A pointed socle-finite K-coalgebra C is computable if and only if C is of finite K-dimension [90]. 2.2 Pseudocompact algebras and pseudocompact modules. For a convenience of the reader we collect from [6], [7], [33], [34], [55] and [56] some facts on linear topological rings and modules, and pseudocompact K-algebras and their pseudocompact modules, see also [42] and [110]. By a topological ring we mean a ring R equipped with a topology such that the addition and the multiplication are continuous. A topological ring is said to be right linear topological if R has a basis (of neighborhoods of zero) consisting of right ideals. We state without proof the following useful results. Lemma 2.5. Let R be a right linear topological ring. Then the open right ideals of R satisfy the following conditions. (a) If I1 , I2 are open right ideals, then I1 \ I2 is open. (b) If I1 I2 are right ideals and I1 is open then I2 is open. (c) If I1 is an open right ideal of R, then for each element r 2 R the right ideal .I1 W r/ D fs 2 R j rs 2 I1 g is open. Lemma 2.6. Let R be any ring and let F be a set of right ideals of R satisfying the following two conditions: (i) For any pair I1 ; I2 of right ideals in F there is I3 2 F such that I3 I1 \ I2 . (ii) If I1 2 F and r 2 R, there exists a right ideal I2 2 F such that I2 .I1 W r/. Then there exists a unique right linear topology on R having F as a basis. Let R be a right topological ring. By a topological right R-module we mean a right R-module M equipped with a topology such that the addition and the multiplication


569

M R ! M are continuous. If the topology on R is right linear, a topological right R-module M is said to be right linear topological if M has a basis (of neighborhoods of zero) consisting of right R-submodules. If R is a right linear topological ring, we denote by Dis.R/ the full subcategory of Mod.R/ formed by the discrete R-modules, that is, linear topological right R-modules with the discrete topology. The following result is easily verified. Lemma 2.7. Let R be a right linear topological ring with a basis F . (a) A right R-module M is discrete if and only if for each element m 2 M the annihilator .0 W m/ D fs 2 R j ms D 0g is open in R. (b) The category Dis.R/ is closed under subobjects, factor objects and arbitrary direct sums in Mod.R/. (c) Dis.R/ is a Grothendieck category and the modules R=I , with I 2 F , form a set of generators of Dis.R/. In particular, the category Dis.R/ has enough injective objects. Definition 2.8 (Gabriel [34], [36], see also [55]). Let K be a field. (a) A pseudocompact K-algebra is a Hausdorff linear topological K-algebra R that admits a basis F consisting of two-sided ideals I such that dimK R=I is finite for all I 2 F and the natural K-algebra homomorphism R ! lim R=I is an I 2F isomorphism. (b) A pseudocompact K-algebra R is said to be basic, the if the quotient algebra R=J.R/ of R modulo its Jacobson radical J.R/ is a (topological) product of division rings. (c) A right linear topological R-module M is called a pseudocompact R-module if M has a basis consisting of right R-submodules N such that dimK M=N is finite and the canonical R-homomorphism M ! lim M=N is an isomorphism. N The following lemma shows that it is sufficient to assume in Definition 2.8 (a) that R admits a basis F consisting of right (or left) ideals I such that dimK R=I is finite, for all I 2 F . Lemma 2.9. Assume that R is a linear topological K-algebra that admits a basis F consisting of right ideals I such that dimK R=I is finite for all I 2 F . For each I 2 F , the set IO D fr 2 R j R r I g is an open two-sided ideal of R such that IO I and dimK R=IO is finite. Proof. We fix a right ideal I 2 F . It is clear that IO is a two-sided ideal of R and IO I . Since dimK R=I is finite, there exist elements f1 ; : : : ; fn 2 R such that R D Kf1 C C Kfn C I . Note that the intersection A D .I W f1 / \ \ .I W fn / is an open right ideal of R, because I is open. In particular, dimK R=A is finite. Now we show that IO is open, by showing that A IO. For this purpose, we note that every element f 2 R has the form f D 1 f1 C Cn fn Ch, where 1 ; : : : ; n 2 K and h 2 I . Then, given g 2 A, we get f g D 1 .f1 g/ C C n .fn g/ C h g 2 I:

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It follows that R A I and, hence, A IO. We denote by P C K the category of all pseudocompact K-algebras together with continuous K-algebra homomorphisms. Given an algebra R in P C K , we denote by R-PC and PC-R the category of pseudocompact left R-modules and right R-modules, respectively. Given a pair M , N of R-modules in R-PC, we denote by homR .M; N / the K-vector space of all continuous R-module homomorphisms from M to N . A proof of the following result can be found in [34], see also [7], Proposition 2.3. Proposition 2.10. Let R be a pseudocompact K-algebra. (a) The category R-PC of pseudocompact left R-modules is abelian with exact inverse limits and enough projective objects. (b) A pseudocompact R-module P is projective in R-PC if and only if P is a direct summand of a direct product of copies of R with the product topology. (c) Every finitely generated discrete R-module is of finite K-dimension. Following Gabriel [34] (see also Brumer [7], p. 448, and [68]) we define a pair of duality functors D1 D./

! Dis.R/ R-PC ı D2 D./

(2.11)

as follows. Given M in Dis.R/, we set D1 .M / D M D HomK .M; K/; and we view D1 .M / as a left R-module in a natural way. A linear topology on D1 .M / is defined by taking for the basis the submodules N? D f' 2 HomK .M; K/ j '.N / D 0g; where N runs through all finitely generated submodules of M . It follows from Lemma 2.5 (a) that N is annihilated by an open ideal I of R, that is, N is a finite-dimensional module over the finite-dimensional K-algebra R=I . Therefore dimK N < 1 and so dimK M =N? D dimK N < 1. Since obviously M Š lim N Š lim M =N? , the R-module D1 .M / D M is pseudocompact. N N Given a pseudocompact left R-module L Š lim L=Lˇ in R-PC, with open subˇ modules Lˇ of L, we set D2 .L/ D Lı ´ homK .L; K/; that is, D2 .L/ consists of all ' 2 HomK .L; K/ such that Ker ' contains an open submodule Lˇ of L. It follows from Lemma 2.7 (a) that the right R-module D2 .L/ Š lim .L=Lˇ / is discrete. The functors D1 and D2 are defined on morphisms in a !ˇ natural way.

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Proposition 2.12 ([34]). The contravariant functors D1 D ./ and D2 D ./ı in (2.11) are dualities of categories that are inverse to each other, and restrict to a duality dis.R/ Š .dis.Rop //op D .R- dis/op . An interesting application of the pseudocompact algebras and their pseudocompact modules in the study of hereditary abelian categories with Serre duality can be found in [71]. Another useful application can be found in a recent paper by Keller and Yang in [48] on the Jacobian algebras of quivers with potentials in the sense of [27], the Jacobifinite quivers with potentials studied by Amiot [1], and their extension to complete path c [88]; see also [69] and [105]. algebras KQ 2.3 Coalgebras and comodules versus pseudocompact algebras and modules. We denote by CoalgK the category of K-coalgebras together with coalgebra homomorphisms. It is well known that CoalgK has arbitrary coproducts and direct limits. An important role in our study of coalgebras and comodule categories is playing by the pair of duality functors D1 D./ ! CoalgK (2.13) P C K D2 D./ı

defined as follows, see [79], p. 404, [84], [110]. Given a K-coalgebra C we equip the K-vector space C D HomK .C; K/

with the K-algebra structure given by the induced maps C ˝C ! .C ˝C / ! C (the convolution product [32], [59], [103]) and " W K ! C . Obviously, " W C ! K is the identity element of C . A linear profinite topology on the algebra C is defined by taking for its basis the two-sided ideals H ? D f' 2 HomK .C; K/ j '.H / D 0g;

(2.14)

where H runs through all finite-dimensional subcoalgebras of C . It follows from Lemma 2.6 that we have defined a linear topology on the K-algebra C such that C is a pseudocompact K-algebra. For this purpose we note that, according to Theorem 2.1, the coalgebra C is a directed union of its finite-dimensional subcoalgebras Hˇ and therefore C Š lim Hˇ Š lim C =Hˇ? : (2.15) Hˇ Hˇ Note also that dimK C =Hˇ? D dimK Hˇ is finite. We call C the convolution pseudocompact K-algebra associated to C . It is well known that all open one-sided ideals of C are closed, see [55], [56], [108] and [110], Section 1. We define the functor D1 in (2.13) by associating to any K-coalgebra C the Kalgebra D1 .C / D C equipped with the profinite topology defined above. The functor D1 is defined on coalgebra homomorphisms in an obvious way.

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Now let R Š lim R=I be a pseudocompact K-algebra with a basis F of I 2F neighborhoods of zero consisting of two-sided ideals I such that dimK R=I < 1. Consider the topologically K-dual space Rı D homK .R; K/ to R, where homK .R; K/ consists of all K-linear functionals ' W R ! K such that Ker ' contains an open ideal from F . It follows that Rı D homK .R; K/ Š homK .lim R=I; K/ Š lim .R=I / ; I 2F !I 2F

(2.16)

where .R=I / D HomK .R=I; K/ D homK .R=I; K/ because dimK R=I is finite. The K-algebra structure on the finite-dimensional K-algebra R=I induces a dual K-coalgebra structure on .R=I / in such a way that the K-linear map .R=I2 / ! .R=I1 / induced by the algebra surjection R=I1 ! R=I2 is a coalgebra embedding for all ideals I1 I2 in F . This defines a unique coalgebra structure on Rı such that .R=I / is a subcoalgebra of Rı for all I in F (compare with [40], Section 3.3). It follows from the definition of the profinite linear topology on C applied to C D Rı that there are functorial isomorphisms .Rı / D .lim .R=I / / Š lim .R=I / Š lim R=I Š R !I 2F I 2F I 2F and the composite isomorphism .Rı / Š R is an isomorphism of pseudocompact K-algebras. Similarly we show that there is a functorial coalgebra isomorphism C Š .C /ı for any K-coalgebra C . We define the functor D2 in (2.13) by associating to any pseudocompact K-algebra R Š lim R=I the K-vector space D2 .R/ D Rı , (2.16), equipped with the coalgeI 2F bra structure defined above. The functor D2 is defined on morphisms in the category P C K in an obvious way. By a standard argument we get the following important observation made in [49], see also [32], [84] and [110], Section 2. Theorem 2.17. (a) Given a K-coalgebra C , the vector space D1 .C / D C D HomK .C; K/ is a pseudocompact K-algebra with respect to the induced linearly topological K-algebra structure. Given a pseudocompact K-algebra R, the vector space D2 .R/ D Rı D homK .R; K/ is a K-coalgebra with respect to the induced coalgebra structure. D1 D./ ! (b) The contravariant functors CoalgK P C K in (2.13) are dualities of D2 D./ı categories that are inverse to each other. (c) The map C 7! C defines a duality between the category of finite-dimensional K-coalgebras and the category of finite-dimensional K-algebras. It follows from [34] and [36] that the algebra C is left (and right) topologically semiperfect, that is, every simple left C -module admits a projective cover

573


in C -PC Q (see also [55], [56] [68], [84]); equivalently, C admits a decomposition C Š j 2I C ej in C -PC, where fej gj 2I is a topologically complete set of pairwise orthogonal primitive idempotents such that ej C ej is a local algebra for every i 2 I . The decomposition is unique up to isomorphism and permutation. Assume that C is a K-colagebra and C is the pseudocompact K-algebra of C . Together with the duality (2.13), a key role in the study of the comodule category C -Comod is played by the pair of duality functors D1 D./

! C -Comod C -PC ı D2 D./

(2.18)

defined by applying the duality (2.11) to the category of rational C -modules in the sense of [103], see Theorem 2.22 (d) and [32], [49], [84], [110], Section 2. To define the functors (2.18), we recall from [103] that, given a left C -comodule M D .M; ıM /, the composed K-linear map ıM ˝id

id˝

ev˝"

M ˝ C ! C ˝ M ˝ C ! C ˝ C ˝ M ! K ˝ M Š M

(2.19)

defines a right C -module structure on M (called the rational C -module structure (see [32], [59], [103]), where W M ˝ C ! C ˝ M is the twist isomorphism and ev W C ˝ C ! K is the evaluation map c ˝ ' 7! '.c/. It is shown in [103] that this correspondence defines a categorical isomorphism C -Comod Š Rat.C /

(2.20)

of C -Comod with the category Rat.C / of rational right C -modules; see [103]. The following observation is very useful; see [49] and [110]. Lemma 2.21. Given a K-coalgebra C , we have Rat.C / D Dis.C /. Proof. Let M be a left C -comodule, viewed as a rational C -module. Given m 2 M , we have ıM .m/ D c1 ˝m1 C Ccr ˝mr , where c1 ; : : : ; cr 2 C and m1 ; : : : ; mr 2 M . By Theorem 2.1 (a), there is a finite-dimensional subcoalgebra H of C containing the elements c1 ; : : : ; cr . It follows from formula (2.19), defining the rational right C module structure on M , that the two-sided ideal H ? , (2.14), of C annihilates the element m of the right C -module M , that is, the annihilator .0 W m/ contains H ? . Hence, according to Lemma 2.5, the right ideal .0 W m/ is open; hence, the right C -module M is discrete, by Lemma 2.7. Conversely, assume M is a discrete right C -module and let N be a finite-dimensional submodule of M . It follows from Lemma 2.7 that N is annihilated by an open ideal of C , that is, there exists a finite-dimensional subcoalgebra H of C such that N H ? D 0. Then N is a finite-dimensional right module over the finite-dimensional K-algebra C =H ? isomorphic to the algebra H . Then N is a left H -module and the multiplication map H ˝ N ! N induces N Š N ! .H ˝ N / Š H ˝ N Š H ˝ N defining a H -comodule structure on N such that the induced

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rational right module structure of N over the algebra C =H ? Š H coincides with the original one. Since N is arbitrary, then we have defined a unique left C -comodule structure on M such that the induced rational right C -module structure on M is the original one. This finishes the proof. In the study of comodule categories we frequently use the following useful observation (see [49], [84], [110], Section 2). Theorem 2.22. Let C be a K-coalgebra and C D HomK .C; K/ the associated pseudocompact K-algebra (2.15). (a) The map associating to any left C -comodule M the underlying vector space M endowed with rational right C -module structure defines category isomorphisms C -Comod Š Rat.C / D Dis.C / and C -comod Š rat.C / D dis.C /; (2.23) where Dis.C / is the category of discrete right C -modules and dis.C / is the full subcategory of Dis.C / formed by finite-dimensional modules. (b) A coalgebra embedding H C induces a fully faithful exact embedding H -Comod C -Comod such that a comodule M in C -Comod lies in H -Comod if and only if M , viewed as a discrete right C -module, is annihilated by the ideal H ? of C . (c) For any finite-dimensional left C -comodule M there is a finite-dimensional subcoalgebra H of C such that M lies in H -Comod C -Comod. (d) The contravariant functors (2.18) defined by the formulae M 7! D1 .M / D M and L 7! D2 .L/ D Lı D homC .L; K/ are mutually inverse K-dualities of categories. The following corollary is a consequence of Theorem 2.22 and Proposition 2.10. Corollary 2.24. Let C be a K-coalgebra, C -inj the category of socle-finite injective C -comodules, and C -pr the category of top-finite projective modules in C -PC. (a) The composite functors C -comod Š rat.C / ./

! C - dis D C - rat D C op -comod D comod-C D dis.C / ./

define the pair of mutually inverse K-linear duality functors D1 D./

! C op -comod D comod-C: C -comod ı D2 D./

(2.25)

(b) The composite functors D D./

D D./

./C

1 1 op ! ! ! C -inj pr-C C -inj D inj-C; C -pr C D D./ı D D./ı ./ 2

with ./ functors

C

D hom

C

2

.; C /, define the pair of mutually inverse K-linear duality rC

! C -inj inj-C: rC

(2.26)


575

2.4 Basic coalgebras. The following notion is of importance, because allows us to reduce the study of C -Comod to the study of C b -Comod over a basic coalgebra in the following sense. Definition 2.27 ([79], p. 404). A K-coalgebra C is called basicL if the left C -subcomodule socC C of C has a direct sum decomposition socC C D j 2IC S.j /, where IC is a set, S.j / are simple comodules and S.i / 6Š S.j / for all i ¤ j . The following lemma shows that the definition is left-right symmetric and the notion of a basic coalgebra introduced in [20] (see also [15], [60]) is equivalent to the above one. Lemma 2.28. Assume that K is an arbitrary field and that C is a K-coalgebra. The following conditions are equivalent. (a) The coalgebra C is basic. L (b) The left C -comodule C has a direct sum decomposition C D i2I Ei , where every left C -comodule Ei is indecomposable, soc Ei is simple and Ei 6Š Ej for i ¤ j. (c) If D is a simple subcoalgebra of C then D is a division K-algebra. (d) Every simple left C -subcomodule of C is a simple subcoalgebra of C . (e) dimK S D dimK EndC S, for any simple left C -comodule S . (f) The pseudocompact K-algebra C is basic. For a proof, we refer to [20], Theorem 2.4, [84], Lemma 5.3, [89], Lemma 2.1. Corollary 2.29. Assume that K is a field and C a K-coalgebra. (a) The coalgebra C is basic if it is pointed, i.e., every simple subcoalgebra of C is one-dimensional. (b) If K is algebraically closed, the following conditions are equivalent: (b1) The coalgebra C is basic. (b2) The coalgebra C is pointed. (b3) Every simple subcoalgebra H of C has the form H D Kc, where c 2 G.C / is a group-like element of C . (b4) Every simple left C -subcomodule of C is one-dimensional. Following [79], p. 404, we associate to any K-coalgebra C a basic coalgebra simple C b as follows. We fix a complete set fSj gj 2J of pairwise non-isomorphic L subcomodules Sj of C C and we form the injective C -comodule E D j 2J E.Sj /. Fix a S directed family fEˇ g of finite-dimensional subcomodules Eˇ of E such that E D ˇ Eˇ . By Gabriel [34] and Lemma 2.9, the induced directed family of right ideals HomC .E=Eˇ ; E/ define a profinite linear topology on the K-algebra ƒE D EndC .E/ Š lim HomC .Eˇ ; E/ Š lim ƒE =HomC .E=Eˇ ; E/ ˇ ˇ

(2.30)

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such that ƒE is a pseudocompact K-algebra. We take for C b D ƒıE D homK .ƒE ; K/

(2.31)

the topological dual K-coalgebra of ƒE (see (2.15)). In the notation of [15] and [104], we have C b D coend.E/. The following useful fact was proved in [79], p. 404, [84] by applying an idea of Gabriel [34], [36], 7.2, see also [20] and [25]. Theorem 2.32. Assume that K is an arbitrary field. (a) For every K-coalgebra C , the K-coalgebra C b D ƒıE , (2.15), is basic and there is an equivalence of categories C -Comod Š C b -Comod; the conditions determine C b uniquely, up to a K-coalgebra isomorphism. (b) If C and H are K-coalgebras then C -Comod Š H -Comod if and only if C b Š H b.

3 The Grothendieck group and the composition length vector of a comodule Throughout, we assume that C is a basic K-coalgebra, with comultiplication W C ! C ˝ C and counit " W C ! K. We fix a left comodule decompositions L L socC C D S.j / and C C D E.j /; (3.1) j 2IC

j 2IC

where IC is a set and S.j /, with j 2 IC , are pairwise non-isomorphic simple comodules in C -comod and E.j / is the injective envelope of S.j / in C -Comod, for each j 2 IC . It follows from Lemma 2.26 that S.j / is a simple subcoalgebra of C , for any j 2 IC . We have observed in Sections 2.2 and 2.3 that the algebra C D HomK .C; K/ is left (and right) topologically semiperfect, that is, every simple left C -module admits a projective cover in C -PC; equivalently, C admits a decomposition Q C Š C ej (3.2) j 2IC

in C -PC, where fej gj 2IC is a topologically complete set of pairwise orthogonal primitive idempotents such that ej C ej is a local algebra, for every i 2 IC . The decomposition is unique up to isomorphism and permutation. Remarks 3.3. (a) Since we assume that C is basic, we can apply the formula (2.30) to the injective comodule E D C C and, by Theorem 2.1 (f) applied to M D C , there is an isomorphism ƒC D EndC .C; C / Š C given by ' 7! "'. One can show that it is an (anti)isomorphism of pseudocompact K-algebras such that the idempotent ej in (3.2), with j 2 IC , corresponds to the composite C -comodule homomorphism j

uj

C ! E.j / ! C , where j is the canonical projection and uj is the canonical embedding.


577

(b) The coalgebra C (or more generally, any C -C -bicomodule) can be viewed as a bimodule over the algebra C with respect to the right and the left hit actions of C on C , usually denoted by the symbols (, * as in [32] and [59]. Here we omit these symbols and simply use juxtaposition, e.g., eC D e * C and C e D C ( e for any e 2 C . Notice that eC is an injective right C -comodule and C e is an injective left C -comodule for any idempotent e 2 C ; see also [26]. The following simple lemma is often used later. Lemma 3.4. Assume that C is a coalgebra, e D e 2 is an idempotent in C , and D./ D HomK .; K/. (a) There is an isomorphism D.eC / Š C e of left C -modules. (b) If C is of finite dimension, then there is an isomorphism D.C e/ Š eC of right C -modules. Proof. See [19] and [25]. To any comodule M in C -comod we associate the composition length vector lgth M D Œ`j .M /j 2IC 2 Z.IC / ; where Z.IC / is the direct sum of IC copies of the free abelian group Z and `j .M / 2 N is the number of simple composition factors of M isomorphic to the simple comodule S.j /. We recall from [84] that the map M 7! lgth M extends to the group isomorphism (see [84], [85], and [90]) ' lgth W K0 .C / ! Z.IC / ; (3.5) where K0 .C / D K0 .C -comod/ D F =F0 is the Grothendieck group of the coalgebra C (or of the category C -comod). Here F is the free abelian group having as a basis z of comodules M in C -comod and F0 is a the set of the isomorphism classes M z L z Nz corresponding to all exact subgroup of F generated by the elements M sequences 0 ! L ! M ! N ! 0 in C -comod. We denote by ŒM the image of z of the module M under the canonical group epimorphism the isomorphism class M F ! F =F0 . Following [90], Section 2, we extend the definition of lgth M to a class of infinitedimensional C -comodules M . We recall that given a left C -comodule M the socle filtration of M is the chain soc0 M soc1 M socm M M;

(3.6)

where soc0 M D soc M and socmC1 M is the preimage of soc.M= socm M / under the canonical comodule projection M ! M= socm M for any m 0. It is well known that for M D C the socle filtration of C coincides with the coradical filtration C0 C1 Cm C of C , that is, socm C D Cm (see [59], p. 64). Note x m D socm M= socm1 M is semisimple for also that, by definition, the comodule M x 0 D soc M . each m 0, where we set M

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Definition 3.7 ([90]). Assume that K is a field and C a basic K-coalgebra (3.1). (a) Given a left comodule M in C -Comod, the composition length multiplicity of a simple C -comodule S.j / in M is the cardinal (finite or infinite) `j .M / D

1 sj

dimK HomC .M; E.j //;

where sj D dimK S.j / D dimK EndC S.j /. (b) The composition length vector of M is the vector lgth M D Œ`j .M /j 2IC :

(3.8)

(c) Given a left C -comodule M (viewed as a rational right C -module), we define its dimension vector dim M D ŒdimK M ej j 2IC ; (3.9) where dimK M ej has values in Z [ f1g. (d) A comodule M in C -Comod is defined to be computable (or composition length computable) if, for each j 2 IC , the cardinal `j .M / is finite; or equivalently, lgth M 2 ZIC (the direct product of IC copies of the free abelian group Z). We denote by C -Comp the full subcategory of C -Comod whose objects are computable comodules. P x x It is shown in [90] that `j .M / D 1 mD0 `j .Mm /, where `j .Mm / is the number of times the simple comodule S.j / appears as a summand in a semisimple decomposition x m . In other words, `j .M / is the multiplicity the simple comodule S.j / appears of M as a composition factor in the socle filtration (3.6) of M . The category C -Comp of computable left C -comodules is abelian and contains the category C -comod. Moreover, C -Comp is closed under subobjects, finite direct sums and extensions in the category C -Comod. The map M 7! lgth M induces a group monomorphism lgth W K0 .C -Comp/ ! ZIC . Remark 3.10. If the coalgebra C is pointed, we have sj D 1, E.j / D C ej , and dimK M ej D dimK HomC .M; C ej / D dimK HomC .M; E.j // D `j .M / for any j 2 IC . Hence the dimension vector dim M coincides with the composition length vector lgth M of M .

4 The left-valued Gabriel quiver of a basic coalgebra Following the concept of a quiver, introduced implicitly by Gabriel in [33] (and explicitly in [35]), we study K-coalgebras and their comodule categories by means of quivers and their K-linear representations in a similar way the quiver representation technique is applied in the study of categories of modules over finite-dimensional algebras over an algebraically closed field. We recall from [35] that a quiver Q D .Q0 ; Q1 / is an oriented graph, with the set of vertices Q0 and the set Q1 of arrows.


579

The left-valued Gabriel quiver (or the valued Ext-quiver) of a coalgebra is defined as follows, compare with [36], Section 7. Definition 4.1 ([52], 4.3 (c), [84], 8.6). Let K be a field and C a basic K-coalgebra with fixed decompositions (3.1). (a) The left-valued (Gabriel) quiver of C is the valued quiver .CQ; C d/, where Q C 0 D IC is the set of vertices of .CQ; Cd/ and, given two vertices i; j 2 CQ0 , there exists a unique valued arrow 0 ; d 00 / .C dij C ij

i ! j from i to j in CQ1 if and only if Ext 1C .S.i /; S.j // ¤ 0 and 0 C dij

D dim Ext 1C .S.i /; S.j //Fi

and

00 C dij

D dimFj Ext 1C .S.i /; S.j //;

where Fi D EndC S.i /. If .C dij0 ;C dij00 / D .1; 1/, we omit .C dij0 ;C dij00 / over the arrow, and we simply write the simply-laced arrow i ! j . (b) The right-valued (Gabriel) quiver of C is the valued quiver .QC ; d C / D .CxQ; Cx d/, where Cx D C op , that is, .QC ; d C / is the left-valued Gabriel quiver of the coalgebra Cx D C op opposite to C . (c) Assume that the coalgebra C in (3.1) is pointed. The left (Gabriel) quiver of C is the quiver CQ D .CQ0 ; CQ1 /, where CQ0 D IC is the set of vertices of CQ, and the arrows from i to j are elements of a fixed basis of the K-vector space 1 jEi D Ext C .S.i /; S.j //, viewed as an Fj -Fi -bimodule. In other words, the vertices of CQ are just the elements j of the set IC (identified with the simple left C -comodules S.j /); and, in case (c) where the coalgebra C is pointed, the quiver CQ D .CQ0 ; CQ1 / is obtained from the valued quiver .CQ; Cd/ 0 ; d 00 / .C dij C ij

by interchanging each of the valued arrows i ! j with the set ˇ1 i◦

ˇ2

.. .

◦j

ˇdij

of dij arrows, where dij D dimK Ext 1C .S.i /; S.j //. Note that dimK Fi D dimK EndC S.i / D 1 if the coalgebra C in (3.1) is pointed; compare with [20], [60], and [111]. Following [53], [87], and [111], we give two alternative descriptions of the valued quiver .CQ; Cd/: the first one by means of irreducible morphisms, and the second one by means of the of the C0 -C0 -bicomodule C1 =C0 defined by the first to terms of the radical filtration of C . To present the first description, denote by D C -inj C -Comod the category of socle-finite injective comodules in C -Comod. Given two comodules E 0 and E 00 in D

580

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C -inj, we define the radical of HomC .E 0 ; E 00 / to be the K-subspace rad.E 0 ; E 00 / D rad .E 0 ; E 00 / of HomC .E 0 ; E 00 / generated by all non-isomorphisms ' W E.i/ ! E.j / between indecomposable summands E.i / of E 0 and E.j / of E 00 , respectively. The square rad2 .E 0 ; E 00 / D rad2 .E 0 ; E 00 / of rad.E 0 ; E 00 / is defined to be the K-subspace rad2 .E 0 ; E 00 / rad.E 0 ; E 00 / of rad.E 0 ; E 00 / generated by all composite comodule homomorphisms of the form fj0

fj00

E 0 ! E.j / ! E 00 ; where j 2 IC , fj0 2 rad.E 0 ; E.j // and fj00 2 rad.E.j /; E 00 /. For any m 3, the 0 00 0 00 m-th power radm .E ; E / of rad .E ; E / is defined in a similar way, see the following section. A C -comodule homomorphism f W E.j / ! E.i/ is said to be irreducible in C -inj if f 2 rad .E.j /; E.i // n rad2 .E.j /; E.i //. In describing the valued Gabriel quiver of a basic coalgebra the following fact proved in [87], Theorem 2.3, is useful. Proposition 4.2. Let C be a basic K-coalgebra as in (3.1). T1 m (a) For every pair i; j 2 IC , rad1 mD1 rad .E.j /; E.i // D 0, .E.j /; E.i // ´ and for any non-invertible non-zero map f 2 HomC .E.j /; E.i // there is m 1 such mC1 that f 2 radm .E.j /; E.i //. Moreover, there exits a path .E.j /; E.i // n rad '1

'2

'3

'm

E.j / ! E.j1 / ! E.j2 / ! : : : ! E.i/ of irreducible morphisms '1 ; : : : ; 'm in C -inj such that the composition 'm : : : '2 '1 is non-zero. (b) If dimK HomC .E.j /; E.i // is finite and rad .E.j /; E.i // ¤ 0, then m

(b1) there exist an integer mij 1 such that rad ij .E.j /; E.i // ¤ 0 and radm .E.j /; E.i // D 0 for all m 1 C mij , and (b2) every non-invertible non-zero homomorphism f 2 HomC .E.j /; E.i // is a P finite K-linear combination f D tsD1 s fsrs : : : fs2 fs1 of compositions fs1

fs2

fs3

fsrs

E.j / ! E.js1 / ! E.js2 / ! ! E.jsrs / D E.i/ of irreducible morphisms fsa in C -inj. For each pair of indices a; b 2 IC , we define the bimodule of irreducible morphisms in to be the K-vector space Irr .E.b/; E.a// D rad .E.b/; E.a//= rad2 .E.b/; E.a//

(4.3)

viewed as an Fa -Fb -bimodule; see [52], [83], [87]. To present the second description of .CQ; Cd/, note that the simple left C -comodules in the decomposition (3.1) are simple subcoalgebras of C and C0 D socC C D


L i2IC

581

S.i / is a coalgebra decomposition inducing the C0 -C0 -bicomodule decompo-

sition

C0 .C1 =C0 /C0

D

L

a .C1 =C0 /b ;

(4.4)

a;b2IC

where the S.a/-S.b/-bicomodule a .C1 =C0 /b ´ S.a/.C1 =C0 /b S.b/ is viewed as a rational Fa -Fb -bimodule, along the division algebra isomorphisms EndC S.a/ Š Fa and EndH S.b/ Š Fb , compare with [11], [60], and [111]. Given a 2 IC , we denote by E.a/ E1 .a/ the injective envelope of S.a/ in the category C -Comod and C1 -Comod, respectively. For any pair a; b 2 IC , we define the Fa -Fb -bimodule homomorphism resab W Irr .E.b/; E.a// ! Irr I 1 .E1 .b/; E1 .a//

(4.5)

by associating to any non-isomorphism f W E.b/ ! E.a/ the restriction resab .f / W E1 .b/ ! E1 .a/ of f to the subcomodule E1 .b/ of E.b/. The following theorem presents three equivalent descriptions of the left Gabriel quiver of a basic coalgebra. Theorem 4.6 ([53], [87]). L Assume that C be a basic K-coalgebra, with a left comodule decomposition socC C D i2IC S.i / as in (3.1). Let C1 D C0 ^C0 and let .CQ; C d/ be the left-valued Gabriel quiver of C . (a) Given a; b 2 IC , the Fa -Fb -bimodule homomorphism resab in (4.5) is an isomorphism. (b) For any pair a; b 2 IC , there exist Fa -Fb -bimodule isomorphisms ' ' ! Irr .E.b/; E.a// ! a .C1 =C0 /b : HomFa .Ext 1C .S.a/; S.b//; Fa /

(4.7)

0 ;d 00 / .dab ab

(c) There exists a unique valued arrow a ! b in .CQ; C d/ if and only if the Fa -Fb -bimodule Irr.E.b/; E.a// is non-zero and 0 D dim Irr .E.b/; E.a//Fb ; dab

00 dab D dimFa Irr .E.b/; E.a//:

(4.8)

0 ;d 00 / .dab ab

(d) There exists a unique valued arrow a ! b in the quiver .CQ; C d/ if and only if the Fa -Fb -bicomodule a .C1 =C0 /b D S.a/ .C1 =C0 /b S.b/ is non-zero and 0 D dim.a .C1 =C0 /b /Fb ; dab

00 dab D dimFa .a .C1 =C0 /b /:

(4.9)

For the proof the reader is referred to [60], Theorem 1.7, [52], Proposition 4.10, [87], Theorem 2.5; see also [11] and [111]. Corollary 4.10. Let C be a basic K-coalgebra. (a) The left-valued Gabriel quiver of C and the right-valued Gabriel quiver of C are dual to each other. (b) The left-valued Gabriel quivers of C and C1 coincide.

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Proof. Apply the duality functor rC W C -inj ! C op -inj D inj-C (2.26) and Theorem 4.6. Details can be found in [53], Section 3; see also [39], [62] and [63]. Corollary 4.11. A basic K-coalgebra is connected (indecomposable) if and only if the left-valued Gabriel quiver .CQ; C d/ of C is connected. Proof. Apply Theorem 4.6 (c) and Proposition 4.2 (a) as in the proof of [87], Corollary 2.4; see also [20] and [60]. CQ

Following [111], we have the following useful description of the left Gabriel quiver D CQ0 ; CQ1 / of a pointed coalgebra C , see also [11] and [12].

Description 4.12. Assume that the coalgebra C in (3.1) is pointed. By Corollary 2.29, simple subcoalgebras of C are simple subcomodules S.j / of C C . On the other hand, simple subcoalgebras of C can be identified with non-zero group-like elements of C , that is, with the elements of the set G.C / D fc 2 C j .c/ D c ˝ c and ".c/ D 1g:

(4.13)

Therefore, the left Gabriel quiver CQ D .CQ0 ; CQ1 / can be described as follows. The set CQ0 D IC of vertices j of CQ can be identified with the set G.C /. Given two vertices ci ; cj 2 G.C /, with i; j 2 IC , the set ˇ1

CQ1 . ci ; cj /W

ci ◦

ˇ2

.. .

◦ cj

ˇdij

of arrows with a source ci and the terminus cj can be identified with a K-basis of the x i ; cj / D P.ci ; cj /=K.ci cj /, where quotient vector space P.c P.ci ; cj / D fc 2 C j .c/ D cj ˝ c C c ˝ ci g C

(4.14)

x i ; cj /; see [111] is the set of .ci ; cj /-primitive elements of C . Here dij D dimK P.c for details (see also [11], [12], [15], [20], [60], and [63]). Alternatively, by Theorem 4.6, the arrows ˇ1 ; : : : ; ˇdij in CQ1 .ci ; cj / can be identified with elements of a fixed K-basis of the space Irr .E.cj /; E.ci //, or with elements of a K-basis of the space ci .C1 =C0 /cj ; see Theorem (4.6).

5 Irreducible morphisms and the Auslander–Reiten quiver of a coalgebra One of the aims of the representation theory of coalgebras (and finite-dimensional algebras) is to give, for a K-coalgebra C , a detailed description of the categories C -comod and C -Comodfc , for any connected K-coalgebra C , where K is a field. In particular, our aims are the following.


583

• Construct a complete list of the indecomposable comodules in the categories C -comod and C -Comodfc , up to isomorphism, and parametrize the indecomposable comodules in a suitable way. • Given a comodule M in C -comod (resp. in C -Comodfc ), find a decomposition of M in a direct sum of indecomposable subcomodules. Note that C -comod is a Krull–Schmidt category and that if C is computable, C -Comodfc is a Krull–Schmidt category, too. • Given a pair of indecomposable comodules M and N in the categories C -comod (resp. in C -Comodfc ), describe the endomorphism K-algebras EndC M and EndC N by means of generators and relations, and describe the vector spaces HomC .M; N / and Ext m C .M; N / together with their End C N -EndC M -bimodule structures for m 1. m • In case dimK Ext m C .M; N / is finite, for all m 0, and Ext C .M; N / vanishes, for m sufficiently large, describe the Euler characteristic

C .M; N / D

1 P

.1/j dimK Ext jC .M; N /;

j D0

of the pair M , N . • Describe the connected components of the valued Auslander–Reiten quivers .C -comod/ and .C -Comodfc / of the categories C -comod and C -Comodfc in the sense of Definition 5.4 presented later. Following the Auslander–Reiten theory for finite-dimensional algebras, first we introduce the notion of an irreducible morphism between left C -comodules as follows, see [2], [5], Section 5.5, and [80], Section 11.1. Definition 5.1. Let C be a K-coalgebra over a field K. (a) A C -comodule homomorphism f W M 0 ! M 00 in C -comod (resp. in C -Comodfc ) is an irreducible morphism if f is not an isomorphism and given a factorisation f / M 00 M 0B BB {= { BB {{ B {{ 00 f 0 BB {{ f Z of f with Z in C -comod (resp. in C -Comodfc ), f 0 is a section or f 00 is a retraction, that is, f 0 has a left inverse or f 00 has a right inverse, see [2]. Irreducible morphisms in any full subcategory of C -Comod are defined analogously. (b) Given two comodules M 0 and M 00 in C -comod (resp. in C -Comodfc ), we define the radical of HomC .M 0 ; M 00 / to be the K-subspace rad.M 0 ; M 00 / HomC .M 0 ; M 00 /;

584

D. Simson

of HomC .M 0 ; M 00 / generated by all non-isomorphisms ' W X ! Y between indecomposable direct summands X of M 0 and Y of M 00 , respectively. In particular, rad.M 0 ; M 00 / is the set of all non-isomorphisms ' W M 0 ! M 00 if M 0 and M 00 are indecomposable. (c) The square of rad.M 0 ; M 00 /, with M 0 ; M 00 2 C -comod (resp. the fc-square of rad.M 0 ; M 00 /, with M 0 ; M 00 2 C -Comodfc ) are defined to be the K-subspaces rad2 .M 0 ; M 00 /; rad2fc .M 0 ; M 00 / rad.M 0 ; M 00 / HomC .M 0 ; M 00 / of rad.M 0 ; M 00 / generated by all composite C -comodule homomorphisms of the form 0

fj0

fj00

M ! Z ! M 00 , where Z is a comodule in C -comod (resp. in C -Comodfc ), fj0 2 rad.M 0 ; Z/ and fj00 2 rad.Z; M 00 /. 0 00 0 00 (d) The m-th powers radm .M 0 ; M 00 / and radm fc .M ; M / of rad.M ; M / are defined analogously for each m 2. For m D 1, we set rad1 .M 0 ; M 00 / D

1 T mD1

0 00 radm .M 0 ; M 00 / and rad1 fc .M ; M / D

1 T mD1

0 00 radm fc .M ; M /

if M 0 and M 00 lie in C -comod and C -Comodfc , respectively. 0 00 We call the vector spaces rad1 .M 0 ; M 00 / and rad1 fc .M ; M / the infinite radical of HomC .M 0 ; M 00 / and the infinite fc-radical of HomC .M 0 ; M 00 /, respectively. We have then defined two chain of vector spaces HomC .M 0 ; M 00 / rad.M 0 ; M 00 radm .M 0 ; M 00 / rad1 .M 0 ; M 00 /; 0 00 HomC .M 0 ; M 00 / radfc .M 0 ; M 00 .M 0 ; M 00 / radm fc .M ; M / 0 00 rad1 fc .M ; M / if M 0 and M 00 lie in C -comod and C -Comodfc , respectively. The following simple lemma is very useful in application. Lemma 5.2. Assume that C is a K-coalgebra. (a) If M 0 ; M 00 is a pair of indecomposable comodules in C -comod, the radical rad.M 0 ; M 00 / consists of all non-isomorphisms f 2 HomC .M 0 ; M 00 /. Moreover, a C -comodule homomorphism f W M 0 ! M 00 is an irreducible morphism in C -comod if and only if f 2 rad.M 0 ; M 00 / n rad2 .M 0 ; M 00 /. (b) If M 0 ; M 00 is a pair of indecomposable comodules in C -Comodfc such that EndC M 0 and EndC M 00 are local algebras then the radical rad.M 0 ; M 00 / consists of all non-isomorphisms f 2 HomC .M 0 ; M 00 /. Moreover, a C -comodule homomorphism f W M 0 ! M 00 is an irreducible morphism in C -Comodfc if and only if f 2 rad.M 0 ; M 00 / n rad2fc .M 0 ; M 00 /. Proof. Apply the standard Auslander–Reiten theory arguments, see [2], [5], or [80], p. 174. The reader can also consult the proof of [87], Lemma 2.2.


585

Assume that M 0 , M 00 is a pair of indecomposable comodules in C -comod (resp. in C -Comodfc ). Following the finite-dimensional algebras terminology [73], we call the K-vector spaces Irr C .M 0 ; M 00 / D rad.M 0 ; M 00 /=rad2 .M 0 ; M 00 /; Irr fc .M 0 ; M 00 / D rad.M 0 ; M 00 /=rad2fc .M 0 ; M 00 /

(5.3)

the bimodule of irreducible morphisms in C -comod and the bimodule of fc-irreducible morphisms in C -Comodfc , respectively (see [83], [84], [66], [52]). Note that the K-vector spaces Irr C .M 0 ; M 00 /, Irr fc .M 0 ; M 00 / are F .M 00 /-F .M 0 /bimodules over the division K-algebras F .M 0 / D EndC M 0 =J.EndC M 0 /

and

F .M 00 / D EndC M 00 =J.EndC M 00 /:

Following Ringel [73] (and [52], [83], [66]), given a coalgebra C , we define the valued AR-quiver of C -comod and of C -Comodfc as follows, see also [2], Chapter IV, [5], Chapter V, and [80], Chapter 11. Definition 5.4. Let C be a K-coalgebra over a field K. (a) The Auslander–Reiten valued quiver .C -comod/ of C -comod is the valued quiver whose vertices are the isomorphism classes ŒX of indecomposable comodules in C -comod. There is a unique valued arrow 0 00 / .dXY ;dXY

ŒX ! ŒY in .C -comod/ if there exists an irreducible morphism X ! Y in C -comod and 0 00 dXY D dimF .Y / Irr C .X; Y /, dXY D dim Irr C .X; Y /F .X/ . (b) Assume that C is a computable coalgebra. The Auslander–Reiten valued quiver .C -Comodfc / of C -Comodfc is the valued quiver whose vertices are the isomorphism classes ŒM of indecomposable comodule in C -Comodfc . There is a unique valued arrow 0 00 .dMN ;dMN /

ŒM ! ŒN in .C -Comodfc / if there exists an irreducible morphism M ! N in C -Comodfc and 0 00 dMN D dimF .N / Irr C .M; N /, dMN D dim Irr C .M; N /F .M / . 0 00 If dMN D dMN D 1, we write the simply-laced arrow ŒM ! ŒN instead of the 0 00 .dMN ;dMN /

valued arrow ŒM ! ŒN . The quivers .C -comod/ and .C -Comodfc / will be called in short the valued Auslander–Reiten quiver of C and the valued Auslander–Reiten fc-quiver of C , respectively. It is clear that the valued quivers .C -comod/ and .C -Comodfc / are disjoint unions of connected components; called the Auslander–Reiten components of C and the Auslander–Reiten fc-components of C , respectively.

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Although the shape of the valued quivers .C -comod/ and .C -Comodfc / do not determine the categories C -comod and C -Comodfc , they carry a lot of important information. Therefore we are interested in describing .C -comod/ and .C -Comodfc /, at least for coalgebras of tame comodule type and for coalgebras of fc-tame comodule type studied in Section 6. A description of .C -comod/, for a class of coalgebras C of infinite dimension, is given in [52], [66], [83], and [91].

6 Two concepts of tameness and wildness for a K -coalgebra We recall from Section 5 that, in analogy to Drozd’s tameness for finite-dimensional K-algebras [29], our concepts of a tame K-coalgebra C given in [84] and [93], is inspired by the question if C admits a “suitable parametrization” of the isoclasses of the indecomposable comodules in C -comod or in C -Comodfc . This leads us to the definitions of a tame comodule type and an fc-tame comodule type, discussed in the following subsections, together with corresponding definitions of a wild comodule type and fc-wild comodule type. For coalgebras C of finite K-dimension, these two definitions coincide; and therefore they coincide with Drozd’s definition of tameness, for finite-dimensional K-algebras, given in [29]. 6.1 Tame comodule type and wild comodule type. In this subsection we discuss the concepts of a tame comodule type and of a wild comodule type, for a K-coalgebra. In particular, we prove the following weak version of tame-wild dichotomy for coalgebras: If K is an algebraically closed field then there is no basic K-coalgebra of tame comodule type and of wild comodule type. Let K be an arbitrary field. We recall from [84] and Section 3 that the map M 7! lgth M , associating to any comodule M in C -comod the composition length vector (3.8) lgth M D Œ`j .M /j 2IC 2 Z.IC / , extends to the group isomorphism ' ! Z.IC / ; lgth W K0 .C /

(6.1)

where K0 .C / D K0 .C -comod/ is the Grothendieck group of the coalgebra C (or of the category C -comod). We recall that lgth M D dim M (the dimension vector (3.9) of a comodule M ) if the coalgebra C is pointed or, more generally, basic. Given v 2 K0 .C / we denote by indv .C -comod/ the full subcategory of C -comod formed by the indecomposable comodules M with lgth M D v. Following Drozd [29], we introduce tame comodule type and wild comodule type for a coalgebra as follows (see also a motivation given in Section 5). Definition 6.2 ([84]). Assume that K is an arbitrary field. (a) Let R be a K-algebra. By a C -R-bimodule C LR we mean a K-vector space L equipped with a left C -comodule structure and a right R-module structure satisfying the condition ıL .xr/ D ıL .x/r for all x 2 L and r 2 R.


587

(b) A K-coalgebra C is defined to be of wild comodule type (or wild, in short) if the category C -comod of finite-dimensional left C -comodules is of K-wild representation type in the sense that there exists an exact K-linear representation embedding (see [80], Chapter 14, and [99], Section XIX.1) 3 T W mod 3 .K/ ! C -comod; where 3 .K/ D K0 KK : If, in addition, the functor T is fully faithful, we call C -comod of fully K-wild comodule type, or strictly K-wild comodule type. (c) A K-coalgebra C is defined to be of tame comodule type (or tame, in short) if the category C -comod of finite-dimensional left C -comodules is of K-tame representation type; that is, for every vector v 2 K0 .C / Š Z.IC / there exist C -KŒt -bimodules L.1/ ; : : : ; L.rv / , which are finitely generated free KŒt -modules, such that all but finitely many indecomposable left C -comodules M , with lgth M D v, are of the form M Š L.s/ ˝ K1 , where s rv , 2 K and K1 D KŒt =.t /: In this case, we say that L.1/ ; : : : ; L.rv / is a an almost parametrising family for the category indv .C -comod/ of all indecomposable C -comodules N with lgth N D v (d) If there is a common bound for the numbers rv of such C -KŒt -bimodules L.1/ ; : : : ; L.rv / in each vector v, the tame coalgebra C is called domestic (see [74], [101], (2.1), [80], Section 14.4, and [99], Section XIX.3). In other words, C is of tame comodule type if the indecomposable left C -comodules of any fixed composition length vector v 2 Z.IC / form a finite set of at most oneparameter families. The reader is referred to [29], [80], Sections 14.2–14.4, and [99], Section XIX.3, for an introduction and a discussion of tameness for finite-dimensional K-algebras. A discussion of representation K-types of module categories can be also find in [86]. Remark 6.3. In Definition 6.2 of tame coalgebra, the polynomial K-algebra KŒt in one indeterminate t can be replaced by the rational K-algebra KŒt h D KŒt; h.t /1 , with a non-zero polynomial h.t / 2 KŒt; see [22], [29], [80], Section 14.4. The discrete comodule type, the polynomial growth, and the linear growth, for coalgebras of tame comodule type, are defined as follows (compare with [99], Section XIX.4, and [100]). Definition 6.4 ([84]). Assume that K is a arbitrary field and C is a K-coalgebra of tame comodule type. (a) We define two growth functions 1C ; 0C W K0 .C / ! N

(6.5)

as follows. Given a vector v 2 K0 .C / Š Z.IC / , we define 1C .v/ D rv to be the minimal number rv 1 of non-zero C -KŒth -bicomodules L.1/ ; : : : ; L.rv / satisfying

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the conditions in Definition 6.2 of tame comodule type of C . We set 1C .v/ D rv D 0 if there is no such a family, that is, there is only a finite number of the isoclasses of indecomposable C -comodules N in C -comod with lgth N D v. We define 0C .v/ to be the minimal number of isoclasses of indecomposable discrete C -comodules N in C -comod, with lgth N D v, that is, C -comodules N in C -comod, that are not of the form N Š L.s/ ˝KŒth K1 , where s 1C .v/, K1 D KŒt =.t /, 1 2 K, and L.1/ ; : : : ; L.C .v// is a minimal family of C -KŒt h -bicomodules satisfying the conditions in the definition of tame comodule type. (b) A K-coalgebra C of tame comodule type is defined to be of K-discrete comodule type if the growth function 1C W K0 .C / ! N of C in (3.5) is zero, that is, for every vector v 2 K0 .C / Š Z.IC / , the number of the isomorphism classes of indecomposable C -comodules M in C -comod, with lgth M D v, is finite. (c) A K-coalgebra C of tame comodule type is said to be of finite growth if there exists an integer m 0 such that 1C .v/ m, for each v 2 K0 .C / Š Z.IC / . (d) A K-coalgebra C of tame comodule type is said to be of polynomial growth if there exist a formal power series G.z/ D

1 P

P

P

mD1 j1 ;:::;jm 2IC sj ;:::;sjm 2N 1

.sj ;:::;sjm / sj1 sj2 zj1 zj2

1 gj1 ;:::;j m

s

jm : : : zjm 2 KŒŒfzj gj 2I

in the set fzj gj 2IC of indeterminates zj such that 1C .v/ G.v/ for all vectors P v D .vj /j 2IC 2 K0 .C / Š Z.IC / with kvk ´ j 2IC vj 2, where z D .zj /j 2IC , .sj ;:::;sj /

m 1 2 Z are non-negative coefficients, and sj1 0; sj2 0; : : : ; sjm 0, gj1 ;:::;j m j1 ; : : : ; jm are pairwise different. (e) A K-coalgebra C of tame comodule type is said to be of linear growth if C is of polynomial growth Pand the power series G.z/ 2 KŒŒfzj gj 2I in part (d) is of the linear form G.z/ D j 2IC gj zj , where gj 2 N D f0; 1; 2; : : : g.

We note that G.v/ is a polynomial of the coordinates vj of the vector v 2 K0 .C / Š Z.IC / because vj D 0 for all but a finite number of the indices j 2 IC . Note also that tame domestic coalgebras are of linear growth. A discussion of the growth for finite-dimensional tame K-algebras can be found in [80], Section 14.4, and [99], Section XIX.3. We start with the following useful lemma ([84], p. 147, and [85], Corollary 5.5). Lemma 6.6. Assume that K is an algebraically closed field and C is a basic Kcoalgebra as in (3.1). If dimK Ext1C .S.i /; S.j // 3 for some i; j 2 IC , then the coalgebra C is of wild comodule type and contains a wild subcoalgebra H such that 4 dimK H 5. Proof. Here we apply the arguments used in the proof of Theorem 5.4 in [85]. Since the K-coalgebra C in (3.1) is basic and the field K is algebraically closed, it follows that C is pointed, dimK S.a/ D dimK EndC S.a/ D 1 for all a 2 IC , and the left-valued Gabriel quiver .CQ; C d/ of C is the left Gabriel quiver CQ D .CQ0 ; CQ1 / of C , with


589

CQ0

D IC . By Corollary 4.10, CQ D .CQ0 ; CQ1 / coincides with the left Gabriel quiver C1 Q of the first term C1 of the coradical filtration of C . Assume that i; j 2 IC are such that dimK Ext 1C .S.i /; S.j // 3. It follows that there are at least three arrows with the source i and the terminus j in the left Gabriel quiver CQ of C . Case 1ı : i ¤ j . Then the three arrow quiver ˇ1

Q 0W

ˇ2

i◦

◦j

ˇ3

is a subquiver of CQ D C1 Q. It follows that the subcoalgebra C1 of C contains the coalgebra H 0 D .KQ0 / D Kei ˚ Kej ˚ Kˇ1 ˚ Kˇ2 ˚ Kˇ3 of dimension 5 dual to the path K-algebra KQ0 Š .K/ of the quiver Q0 , where ei and ej are the stationary paths at i and j , respectively, see [85], p. 483, and Section 6.7 for details. Since H 0 -comod Š mod KQ0 Š mod .K/ and KQ0 Š .K/ is a self-dual wild K-algebra (see [80], Section 14.3, [99], Corollary XIX.1.9) then the subcoalgebra H 0 D .KQ0 / of C1 C of dimension five is of wild comodule type. Case 2ı : i D j . Then the three loop quiver Q00 :

:ıd

at vertex i is a subquiver od C Q D C1 Q. If 1 ; 2 ; 3 W i ! i are the loops of Q00 at i , 00 D Kei ˚ K 1 ˚ then the subcoalgebra C1 of C contains the coalgebra H 00 D KQ1 K 2 ˚ K 3 of dimension 4 (see Section 6.7), which is the dual K-coalgebra A to the self-dual K-algebra A D KŒt1 ; t2 ; t3 =.t1 ; t2 ; t3 /2 of dimension 4. Let modsp 3 .K/ be the full subcategory of mod 3 .K/ formed by modules having no simple injective direct summands. It is clear that modsp 3 .K/ is equivalent to the full subcategory of repK .Q0 / consisting of all K-linear representations V D .Vi ; Vj ; 'ˇ1 ; 'ˇ2 ; 'ˇ3 /, where 'ˇ1 ; 'ˇ2 ; 'ˇ3 W Vi ! Vj are K-linear maps such that the intersection Ker '1 \ Ker '2 \ Ker '3 is zero. Define the K-linear functor F W modsp 3 .K/ ! mod A Š comod-H 00 by attaching to any K-linear representation V D .Vi ; Vj ; 'ˇ1 ; 'ˇ2 ; 'ˇ3 / of the quiver Q0 (viewed as a 3 .K/-module) the A-module F .V / D Vi ˚ Vj equipped with the multiplication 0 ' ˇs N .x; y/ ts D Œx; y D Œ0; 's .x/ 0 0 of elements .x; y/ 2 Vi ˚ Vj by the cosets tN1 ; tN2 ; tN3 2 A of the indeterminates t1 , t2 and t3 , for s D 1; 2; 3. It is easy to check that F is exact, carries indecomposable modules to indecomposable ones, and F .V / Š F .V 0 / implies V Š V 0 for any pair of

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modules V and V 0 of modsp 3 .K/. This means that the functor F is a representation embedding and therefore the K-algebra A of dimension 4 is of wild representation type. Since H 00 -comod Š mod .K/, then the subcoalgebra H 00 of C1 C is of wild comodule type. The following theorem collects main properties of tame and wild coalgebras. Theorem 6.7. Assume that K is an arbitrary field and C is a K-coalgebra. (a) The wild comodule type and the tame comodule type are Morita-invariant. In particular, a coalgebra C is of wild comodule type (resp. of tame comodule type) if and only if the basic coalgebra C b in (2.31) associated to C is of wild comodule type (resp. of tame comodule type). (b) If dimK C is finite and C D HomK .C; K/ is the convolution K-dual algebra in (2.15) to C , then C is of wild comodule type (resp. of tame comodule type) if and only if the finite-dimensional K-algebra C is of wild comodule type (resp. of tame comodule type). (c) The definition of wild comodule type for K-coalgebras is left-right symmetric, that is, C is of wild comodule type if and only if the coalgebra C op opposite to C of wild comodule type. (d) Assume that K is an algebraically closed field K. The following three conditions are equivalent. (d1) C is of wild comodule type. (d2) C contains a finite-dimensional K-coalgebra H of wild comodule type. (d3) C is a directed union of a family of finite-dimensional K-subcoalgebras Hˇ of wild comodule type. (e) Assume that K is an algebraically closed field K. If C is a basic coalgebra of tame comodule type and H is a subcoalgebra of C then H is of tame comodule type. In particular, any K-coalgebra C is of tame comodule type is a directed union of a family of finite-dimensional K-subcoalgebra Hˇ of tame comodule type. Proof. (a) Apply definitions. (b) If dimK C is finite, we have dis.C / D mod C . Then, in view of the equivalence C -comod Š rat.C / D dis.C /, see (2.23), we get C -comod Š mod C . Hence (b) follows. (c) Apply the K-duality .C -comod/op Š C op -comod D comod-C given in (2.25) and note that there exits a K-duality mod 3 .K/ Š .mod 3 .K//op , see [84], Lemma 6.9. (d) The implications (d2) H) (d3) H) (d1) are obvious. To prove the implication (d1) H) (d2), we assume that C is a basic coalgebra as in (3.1). In the case dimK Ext1C .S.i /; S.j // 3 for some i; j 2 IC , by Lemma 6.6, the coalgebra C is of wild comodule type and contains a wild subcoalgebra H such that 4 dimK H 5, and we are done. Then, without loss of generality, we may assume that dimK Ext1C .S.i /; S.j // 2 for all i; j 2 IC . It follows that dimK Ext1C .S; S 0 / 2

591


for any pair of simple C -comodules S and S 0 in C -comod. In this case, a finite-dimensional K-subcoalgebra H of C of wild comodule type can be constructed by applying rather lengthy arguments used in the proof of [84], Theorem 6.10. (e)Assume that the coalgebra C is of tame comodule type and has the decomposition (3.1). Let H be a finite-dimensional L subcoalgebra of C . Then there exists a finite subset J of IC such that soc H H D j 2J S.j / and K0 .H / Š Z.J / . It follows that H -Comod is a full exact subcategory of C -Comod and, for every vector v 2 K0 .H / Š Z.J / Z.IC / , the left H -comodules M such that lgth M D v coincide with the left C -comodules M such that lgth M D v. Fix a vector v 2 K0 .H / Š Z.J / Z.IC / . Since C is of tame comodule type, then there exist C -KŒt-bimodules L.1/ ; : : : ; L.rv / , which are finitely generated free KŒt-modules, such that all but finitely many indecomposable left H -comodules M , with lgth M D v, are of the form M Š L.s/ ˝ K1 , where s rv , K1 D KŒt =.t / and 2 K. If we view the C -KŒt-bimodules L.1/ ; : : : ; L.rv / as H -KŒt -bimodules, we get an isomorphism M Š L.s/ ˝ K1 of H -comodules. This shows that H is of tame comodule type and finishes the proof. As a consequence we get the following weak version of tame-wild dichotomy for coalgebras over an algebraically closed field K (compare with [29]). Corollary 6.8 ([85], Corollary 5.6). Let K be an algebraically closed field. If C is a basic K-coalgebra of tame comodule type then C is not of wild comodule type. Proof. Assume, to the contrary, that C is of wild comodule type. By Theorem 6.7 (d), there exists a finite-dimensional subcoalgebra H of C of wild comodule type. On the other hand, by Theorem 6.7 (e), the coalgebra H is of tame comodule type because C is assumed to be of tame comodule type. Since the category isomorphism H -comod Š dis.H / D mod.H / presented in the proof of (b) preserves tame representation type and wild representation type, we get a contradiction, by applying the tame-wild dichotomy theorem of Drozd [29] to the finite-dimensional algebra H . Corollary 6.9 ([85], Corollary 5.6). Let K be an algebraically closed field and assume that C is a basic K-coalgebra of tame comodule type as in (3.1). Then: (a) dimK Ext 1C .S; S 0 / 2 for any pair of simple left C -comodules S and S 0 , (b) the left Gabriel quiver CQ of C does not contain a subquiver of one of the following four forms Q0: ◦

◦

Q00:

:◦d

L0 :

s

◦←−− ◦

L00 :

s

◦−−→◦

where s in the circle of the quivers L0 and L00 means that this is a one-way oriented cycle of length s 1.

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Proof. (a) is a consequence of the proof of Theorem 6.7 (d), because C is basic of the form (3.1) and, hence, any simple left C -comodule has the form S.i / for some i 2 IC . (b) Since C is of tame comodule type, it is not of wild comodule type by Corollary 6.8. Then [84], Lemma 8.18, and the proof of Lemma 6.6 yield (b). We hope that the following tame-wild dichotomy holds, see [84] and [85]. Tame-Wild Dichotomy 6.10. Any K-coalgebra over an algebraically closed field K is either of tame comodule type, or of wild comodule type, and these types are mutually exclusive. This is a coalgebra analogue of the well-known tame-wild dichotomy theorem for finite-dimensional K-algebras proved by Drozd [29]. Note that, in view of Corollary 6.8, to solve problem 6.10, it is sufficient to prove that, over an algebraically closed field K, any K-coalgebra that is not of tame comodule type is of wild comodule type. By applying [72], Lemma 1.5, or [81], Theorem 3.12 (b), we get: Corollary 6.11. Let K be an algebraically closed field and C a K-coalgebra. The coalgebra C is of fully wild comodule type if and only if there exists a pair of finite-dimensional left C -modules U1 , U2 satisfying the following three conditions: (i) EndC .U1 / Š EndC .U2 / Š K. (ii) HomC .U1 ; U2 / D 0 and HomC .U2 ; U1 / D 0. (iii) dimK Ext1C .U1 ; U2 / 3. The polynomial growth and the linear growth behavior of coalgebras are discussed in [84]. The problem whether or not a localisation of coalgebras preserves the tame comodule type is discussed in [45], [62], [63], [89]. 6.2 Examples. One can easily construct a lot of examples of coalgebras of wild comodule type from wild finite-dimensional K-algebras R (see [80] and [99]), because any such finite-dimensional K-algebra R produces the dual wild coalgebra H D R such that mod R Š H -comod. Recall from Theorem 6.7 (d) that any wild coalgebra C of infinite K-dimension always contains a finite-dimensional wild K-coalgebra H D R . Examples of coalgebras of fully wild comodule type can also be produced by applying Corollary 6.11. Now we present two examples of coalgebras of tame comodule type. Example 6.12. Consider the cocommutative polynomial K-coalgebra KŒt ˘ D KŒt , P m r with comultiplication and counity " defined by .t / D rCsDm t ˝ t s , ".1/ D 1 and ".t s / D 0 for s 1. For any m 0, the subspace KŒt˘m D K ˚ Kt ˚ ˚ Kt m


593

S ˘ of KŒt˘ is a K-subcoalgebra of dimension m C 1 and KŒt ˘ D 1 mD0 KŒt m . Note ˘ ˘ ˘ that soc KŒt D KŒt0 D K and the Grothendieck group K0 .KŒt / of KŒt ˘ is infinite cyclic. It is easy to see that the K-algebra .KŒt ˘m / dual to KŒt ˘m is isomorphic to KŒŒt=.t m / Š KŒt =.t m / and the pseudocompact K-algebra (2.15) associated with C D KŒt˘ is .KŒt ˘ / Š lim .KŒt ˘m / Š lim KŒt =.t m / Š KŒŒt ; m m ˘ that is, .KŒt / Š KŒŒt is the power series K-algebra, with the Jacobson radical powers topology. It follows that, under the identification KŒt ˘ -Comod D Dis..KŒt ˘ / / in (2.23) the left KŒt˘ -comodules are just the t -torsion KŒt -modules. It follows from [80], Proposition 14.12, that • the category KŒt˘ -comod has almost split sequences, • every indecomposable comodule in KŒt˘ -comod of dimension m C 1 is isomorphic to the subcomodule KŒt˘m of KŒt˘ , and • the valued Auslander–Reiten quiver of KŒt ˘ -comod is a rank one homogeneous tube of the shape (see [98], Section X.1) 2 KŒt ˘ 1 D KŒt =.t /

2 KŒt ˘ 2 D KŒt =.t /

3 KŒt ˘ 3 D KŒt =.t /

4 KŒt ˘ 4 D KŒt =.t /

.. .

It is obvious that the coalgebra KŒt˘ is of tame comodule type and, for any d 1, there is precisely one isoclass of indecomposable KŒt ˘ -comodules of dimension d . 1 0 In other words, the growth function KŒt ˘ W Z ! N is zero, whereas KŒt˘ .d / D 1, for all d 1. This shows that the coalgebra KŒt ˘ is of discrete comodule type. Example 6.13. Consider the cocommutative K-coalgebra KŒt1 ; t2 ˘ D KŒt1 ; t2 =.t1 t2 / D K ˚

1 L nD1

Kt1n ˚

1 L mD1

Kt2m ;

˝ KŒt1 ; t2 ˘ and the counity where the comultiplication W KŒt1 ; t2 ˘ ! KŒt1 ; t2 ˘ P m ˘ " W KŒt1 ; t2 ! K are defined by the formulae .tj / D rCsDm tjr ˝tjs for j D 1; 2,

594

D. Simson

".1/ D 1 and ".tjs / D 0 for s 1 and j D 1; 2. For any m 0 the subspace KŒt1 ; t2 ˘m D K ˚ Kt1 ˚ Kt12 ˚ ˚ Kt1m ˚ Kt2 ˚ Kt22 ˚ ˚ Kt2m of KŒt1 ; t2 ˘ is a K-subcoalgebra of dimension 2m C 1, and we also have KŒt1 ; t2 ˘ D S1 ˘ ˘ ˘ mD0 KŒt1 ; t2 m . Note that soc KŒt1 ; t2 D KŒt1 ; t2 0 D K It is easy to see that the K-algebra .KŒt1 ; t2 ˘m / dual to KŒt1 ; t2 ˘m is isomorphic to the algebra ƒm D KŒt1 ; t2 =.t1 t2 ; t1m ; t2m / and the pseudocompact K-algebra of (2.15) associated with KŒt1 ; t2 ˘ is .KŒt1 ; t2 ˘ / Š lim .KŒt1 ; t2 ˘m / Š lim ƒm Š KŒŒt1 ; t2 =.t1 t2 /; m m that is, the algebra .KŒt1 ; t2 ˘ / Š KŒŒt1 ; t2 =.t1 t2 / is a quotient of the power series K-algebra KŒŒt1 ; t2 , with the Jacobson radical powers topology. It follows that the Grothendieck group K0 .KŒt1 ; t2 ˘ / of .KŒt1 ; t2 ˘ -comod is infinite cyclic, and under the identification KŒt1 ; t2 ˘ -Comod D Dis.KŒŒt1 ; t2 =.t1 t2 // in (2.23), the finite-dimensional KŒt1 ; t2 ˘ -comodules are just the finite-dimensional modules over KŒt1 ; t2 =.t1 t2 / that are annihilated by some powers of the cosets tN1 and tN2 of the indeterminates t1 and t2 in KŒŒt1 ; t2 . By applying a well-known description of indecomposable ƒm -modules (see [38] and [9]) we conclude that for any d 0 the indecomposable modules of dimension d in Dis.KŒŒt1 ; t2 =.t1 t2 // are ƒd -modules. Hence we get a category isomorphism KŒt1 ; t2 ˘ -comod Š nilmodfl .KŒt1 ; t2 =.t1 t2 //; where nilmodfl .KŒt1 ; t2 =.t1 t2 // consists of modules of finite length that are nilpotent, that is, annihilated by some powers of tN1 and tN2 . Hence we easily conclude that the indecomposable KŒt1 ; t2 ˘ -comodules of dimension d 1 lie in KŒt1 ; t2 ˘d -comod Š dis.ƒd / D mod.ƒd /. It follows from [84], Corollary 6.14 (see also Section 8), that the coalgebra KŒt1 ; t2 ˘ is of tame comodule type because according to [38] the algebra ƒd is of tame representation type for any d 1. Since, for d 3, the K-algebra ƒd is of non-polynomial growth (see [100] and [101]), the coalgebra KŒt1 ; t2 ˘ is of tame comodule type and of non-polynomial growth. One can show that KŒt1 ; t2 ˘ is a string coalgebra in the sense of Section 8. By applying [9], one can deduce that the category KŒt1 ; t2 ˘ -comod has no almost split sequences, see also [18], [21] and Section 7.6. Interesting classes of coalgebras of tame comodule type are presented later.


595

6.3 Coalgebras of fc-tame comodule type and of fc-wild comodule type. It was conjectured in [84] and [85] (see also 6.10) that the tame-wild dichotomy holds for any coalgebra C over an algebraically closed field K. Unfortunately, for infinite-dimensional coalgebras the dichotomy is still an open problem. One of our main aims in this section is to give an affirmative solution of the problem for a class of coalgebras, including semiperfect ones. Following [93], we do it by introducing a new concept of tameness and wildness for a coalgebra C that coincides with the previous one for semiperfect coalgebras. Namely, we define C to be fc-wild and fc-tame if the category C -Comodfc of finitely copresented left C -comodules is K-wild and K-tame, respectively (see Definition 6.15). One of the main results of this subsection is the following fc-tame-wild dichotomy result [93]. If C is a basic computable K-coalgebra over an algebraically closed field K, then C is either fc-tame or fc-wild, and these two types are mutually exclusive. Here C is said to be computable if dimK HomC .E 0 ; E 00 / is finite, for any pair of indecomposable direct summands E 0 ; E 00 of C C . We prove the result by a reduction to the tame-wild dichotomy theorem of Drozd [29] for finite-dimensional K-algebras, developed by Crawley-Boevey in [22] and [23] by applying representations of free triangular Roiter bocses and the bimodule problems technique. Here we follow the proof given in [93] and [95]. We recall that a comodule N in C -Comod is said to be (socle) finitely copresented if there is an exact sequence 0 ! N ! E 0 ! E 00 in C -Comod, where each of the comodules E 0 and E 00 is a finite direct sum of indecomposable injective comodules. If E 0 ; E 00 2 add.E/, for some socle-finite injective C -comodule E, the comodule N is called finitely E-copresented. We denote by C -Comodfc C -ComodE fc the full subcategories of C -Comod whose objects are the finitely copresented comodules and finitely E-copresented comodules, respectively. Assume that K is an arbitrary field and C is a basic K-coalgebra with a fixed decomposition (3.1). Following [29], [22], [80] and [93], given a finitely copresented C -comodule N in C -Comodfc , with a minimal injective copresentation 0 ! N ! E0N ! E1N , we define the coordinate vector of N to be the bipartite vector N .IC / Z.IC / ; cdn.N / D .cdnN 0 j cdn1 / 2 K0 .C / K0 .C / D Z

(6.14)

N N N where cdnN 0 D lgth.soc E0 / and cdn1 D lgth.soc E1 /. Note that an indecomposable comodule N in C -Comodfc is injective if and only if the vector cdn.N / has the form v D .ej j0/, for some j 2 IC , where ej is the j -th standard basis vector of the abelian group Z.IC / . Following [29] and [84], we have introduced in [93] the concept of fc-tameness and fc-wildness for K-coalgebras as follows.

Definition 6.15 ([93]). Let K be an arbitrary field. (a) A K-coalgebra C is defined to be of fc-wild comodule type (or briefly fcwild ) if the category C -Comodfc of finitely copresented C -comodules is of K-wild

596

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representation type ([80], [85], [90], [99]) in the sense that there exists an exact K-linear 3 representation embedding T W mod 3 .K/ ! C -Comodfc , where 3 .K/ D K0 KK . If, in addition, the functor T is fully faithful, we call C of fully fc-wild comodule type, or strictly fc-wild comodule type; see [29], [93]. (b) Given a K-algebra S, a C -S-bicomodule C LS is defined to be finitely copresented if there is a C -S -bicomodule exact sequence ! E 00 ˝ S; 0 ! C LS ! E 0 ˝ S such that E 0 , E 00 are socle-finite injective C -comodules in C -Comodfc . If E 0 , E 00 lie in add.E/, where E is socle-finite injective, we call CLS finitely E-copresented. (c) A K-coalgebra C is defined to be of fc-tame comodule type (or briefly fc-tame) if the category C -Comodfc is of tame representation type [80], Section 14.4, that is, for every bipartite vector v D .vjv 00 / 2 K0 .C / K0 .C / Š Z.IC / Z.IC / ; there exist C -KŒth -bicomodules L.1/ ; : : : ; L.rv / which are finitely copresented such that all but finitely many indecomposable left C -comodules N in C -Comodfc , with cdn.N / D v, are of the form N Š L.s/ ˝KŒth K1 , where s rv , K1 D KŒt =.t /; and 2 K. In this case, we say that L.1/ ; : : : ; L.rv / is a finitely copresented almost parametrising family for the category indv .C -Comodfc / of all indecomposable C comodules N with cdn.N / D v. (d) If there is a common bound for the numbers rv of such C -KŒt h -bicomodules L.1/ ; : : : ; L.rv / , for each bipartite vector v D .v 0 jv 00 /, the fc-tame coalgebra C is called fc-domestic, compare with [80], Section 14.4, [99], Section XIX.4. We prove later that the definitions of fc-tame coalgebra and of fc-wild coalgebra are left-right symmetric if the field K is algebraically closed and C is a computable K-coalgebra. If the category C -comod is not contained in C -Comodfc , there is no proper relation between the tame comodule type (resp. wild comodule type) of C and the fc-tame comodule type (resp. fc-wild comodule type) of C . However, if the simple C -comodules are finitely copresented, the category C -comod is contained in C -Comodfc and, by the following lemma, the fc-tameness of C implies the tameness of C . In particular, this is the case if C is a left locally artinian coalgebra. Lemma 6.16 ([93]). Let C be a basic coalgebra as in (3.1) over a field K. (a) If C is of fc-tame comodule type (resp. of fc-discrete comodule type) and C -comod C -Comodfc then C is of tame comodule type (resp. of discrete comodule type), too. (b) If C -comod C -Comodfc then C -comod is of tame comodule type if and only if, for every bipartite vector v D .v 0 jv 00 / 2 K0 .C / K0 .C /, there exist a non-zero h.t/ 2 KŒt and C -KŒth -bicomodules L.1/ ; : : : ; L.rv / which that are finitely generated free KŒth -modules and form an almost parametrising family for indv .C -comod/.


597

(c) If C is left locally artinian then C -Comodfc is the subcategory of C -Comod consisting of artinian left C -comodules. In particular, C -Comodfc is abelian, has enough injectives, and contains each of the categories C -inj and C -comod. Proof. (b) Assume that C -comod C -Comodfc . Given a vector w 2 N .IC / , we P set v.w/ D j 2IC .cdn S.j //wj . If N is indecomposable in C -comod, with w D lgth N , then an easy induction on dimK N shows that cdn N v.w/. Hence the sufficiency of (b) follows. To prove necessity, given a non-negative bipartite vector P 0 v D .v 0 jv 00 / 2 Z.IC / Z.IC / , we set w.v/ D j 2IC .lgth E.j //vj . It follows that for every indecomposable comodule M in C -comod, with v D cdn M , we have lgth M lgth E.M / D w.v/, and the necessity follows. (a) Assume that C is of fc-tame comodule type (resp. of fc-discrete comodule type) and C -comod C -Comodfc . By applying (b), we conclude that C is of tame comodule type (resp. of discrete comodule type), because finitely copresented C -KŒt h bicomodules are finitely generated KŒth -modules and one can localise them to free modules, see [82], Lemma 6.11. The proof of (c) is simple and we leave it to the reader. Following Skowroński [100], the polynomial growth and the fc-discrete comodule type, for fc-tame coalgebras, are defined as follows. Definition 6.17 ([93]). Assume that C is a K-coalgebra of fc-tame comodule type. (a) We define two growth functions O 0C W K0 .C / ! N O 1C ;

(6.18)

as follows. Given a bipartite vector v D .v 0 jv 00 / 2 K0 .C / K0 .C / Š Z.IC / O 1C .v/ D rv to be the minimal number rv 1 of non-zero C Z.IC / , we define KŒth -bicomodules L.1/ ; : : : ; L.rv / satisfying the conditions in the definition of fcO 1C .v/ D rv D 0 if there is no such a family, that is, tame comodule type. We set there is only a finite number of the isoclasses of indecomposable C -comodules N in C -Comodfc with cdn.N / D v. O 0C .v/ to be the minimal number of isoclasses of indecomposable disWe define crete C -comodules N in C -Comodfc , with cdn.N / D v, that is, C -comodules N O 1C .v/, in C -Comodfc which are not of the form N Š L.s/ ˝KŒth K1 , where s 1 K1 D KŒt=.t /, 2 K and L.1/ ; : : : ; L.O C .v// is a minimal family of C -KŒt h bicomodules satisfying the conditions in the definition of fc-tame comodule type. (b) We define an fc-tame coalgebra C to be of fc-discrete comodule type if the growth O 1C W K0 .C / ! N of C in (2.4) is zero, that is, for every bipartite vector function 0 00 v D .v jv / 2 K0 .C / K0 .C / Š Z.IC / Z.IC / , the number of the isomorphism classes of indecomposable C -comodules M in C -Comodfc , with cdn.M / D v, is finite.

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(c) A K-coalgebra C of fc-tame comodule type is said to be of finite fc-growth O 1C .v/ m for each v D .v 0 jv 00 / 2 if there exists an integer m 0 such that .IC / .IC / K0 .C / K0 .C / Š Z Z . (d) An fc-tame coalgebra C is said to be of polynomial fc-growth if there exist two formal power series F .z/ D G.z/ D

1 P

P

P

mD1 j1 ;:::;jm 2IC sj ;:::;sjm 2N 1 1 P

P

P

mD1 j1 ;:::;jm 2IC sj ;:::;sjm 2N 1


1 fj1 ;:::;j m


1 gj1 ;:::;j m

s

jm : : : zjm 2 KŒŒfzj gj 2I ;

s

jm : : : zjm 2 KŒŒfzj gj 2I

O 1C .v/ F .v 0 /G.v 00 /, for all vectors in the set fzj gj 2IC of indeterminates zj , such that P v D .vj0 jvj00 /j 2IC 2 K0 .C / K0 .C / Š Z.IC / Z.IC / with kvk ´ j 2IC vj0 C P .sj1 ;:::;sjm / 00 , j 2IC vj 2, where z D .zj /j 2IC , sj1 0; sj2 0; : : : ; sjm 0, fj1 ;:::;jm .sj ;:::;sj /

m 1 2 Z are non-negative coefficients, and j1 ; : : : ; jm are pairwise different. gj1 ;:::;j m (e) A K-coalgebra C of fc-tame comodule type is said to be of linear fc-growth if C is of polynomial fc-growth and the Ppower series F .z/; G.z/P2 KŒŒfzj gj 2I in part (d) are of the linear form F .z/ D j 2IC fj zj and G.z/ D j 2IC gj zj , where fj ; gj 2 N D f0; 1; 2; : : : g.

To formulate our main results of this subsection, we recall from [89], [90], [93], and [95] some notation. Given a socle-finite injective direct summand L E D EU D E.u/ u2U

L

of C C D j 2IC E.j /, with a finite subset U of IC , we define the category C -ComodE fc to be fc-tame if, for every bipartite vector v D .v 0 jv 00 / 2 K0 .C / K0 .C /, there is a finitely E-copresented almost parametrising family for indv .C -ComodE fc /. Consider the K-algebra L RE D EndC E D eu R E ; (6.19) u2U

where eu RE D HomC .E; E.u// is viewed as an indecomposable projective right ideal of RE and eu is the primitive idempotent ofPRE defined by the summand E.u/ of E. Since the set U IC is finite, the element u2U eu is the identity of RE . It is easy to see that the Jacobson radical J.RE / of RE has the form J.RE / D fh 2 EndC E j h.soc E/ D 0g. Let Map1 .E/ be the category whose objects are the triples .E0 ; E1 ; / with E0 , E1 comodules in add.E/ and W E0 ! E1 a homomorphism of C -comodules such that .soc E0 / D 0, and whose morphisms are the pairs .f0 ; f1 /, where f0 W E0 ! E00 , f1 W E1 ! E10 and 0 B f0 D f1 B .


599

Denote by Map2 .E/ the full subcategory of Map1 .E/ whose objects are the triples .E0 ; E1 ; / such that soc Im D soc E1 . We define the coordinate vector of .E0 ; E1 ; / to be the bipartite vector cdn.E0 ; E1 ; / D .lgth.soc E0 / j lgth.soc E1 // 2 K0 .C / K0 .C /: Following [29] and [22], Section 6, we denote by P1 .RE / the category whose objects are the triples .P1 ; P0 ; / with P0 , P1 finitely generated projective right RE modules and W P1 ! rad.P0 / D P0 J.RE / a homomorphism of RE -modules; and whose morphisms are the pairs .g1 ; g0 /, where g0 W P0 ! P0 , g1 W P1 ! P10 and

0 B g1 D g0 B . Denote by P2 .RE / the full subcategory of P1 .RE / whose objects are the triples .P1 ; P0 ; / with Ker rad.P1 /. We define the coordinate vector of .P1 ; P0 ; / to be the bipartite vector cdn.P1 ; P0 ; / D .lgth.top P1 / j lgth.top P0 // 2 ZU ZU D K0 .RE / K0 .RE /: Following [29] and [22], Section 6, we consider the diagram Map1 .E/ o kerE

C -ComodE fc

E ˝Rı ./ E

'

P1 .RE / cokE

(6.20)

mod.RE /,

ı ı ./ W P1 .RE / ! Map .E/ is the functor induced by D Rop , E ˝RE where RE 1 ı ./ W pr.RE / ! add.E/, ker E .E0 ; E1 ; the functor E ˝RE / D Ker , and finally cokE .P1 ; P0 ; / D Coker . Here we view the category pr.RE / of finitely generated op projective right RE -modules as the category of finitely generated projective left RE modules. We start with the following useful result.

Proposition 6.21. Under the notation made above, assume that E D EU is such that RE D EndC RE is a finite-dimensional K-algebra. ı ./ W pr.RE / ! add.E/ is an equivalence of categories (a) The functor E ˝RE ı ./ W P1 .RE / ! Map .E/ that preserves the and induces the equivalence E ˝RE 1 coordinate vectors and restricts to the equivalence ı ./ W P .RE / ! Map .E/; E ˝RE 2 2

where P2 .RE / and Map 2 .E/ are the full subcategories of P2 .RE / and Map2 .E/ with objets having no non-zero direct summand of the form .Z 0 ; 0; 0/ and .0; Z 00 ; 0/. (b) The functors kerE and cokE restrict to the representation equivalences kerE W Map2 .E/ ! C -ComodE fc and cokE W P2 .RE / ! mod.RE / preserving the coordinate vectors. Proof. Note that a triple .E0 ; E1 ; / lies in Map1 .E/ if and only if W E0 ! E1 has no non-zero direct summand of the form idZ W Z ! Z. Moreover, a triple .E0 ; E1 ; / of

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D. Simson

Map1 .E/ lies in Map2 .E/ if and only if

W E0 ! E1 has no non-zero direct summand

00

of the form 0 ! E , or equivalently, if and only if 0 ! Ker ! E0 ! E1 is a minimal E-injective presentation of Ker in C -ComodE . Similarly, a triple fc .P1 ; P0 ; / lies in P1 .RE / if and only if W P1 ! P0 has no non-zero direct summand of the form idZ W Z ! Z. Moreover, a triple .P1 ; P0 ; / of P1 .RE / is an object of P2 .RE / if and only if W P1 ! P0 has no non-zero direct summand of the form

P ! 0, or equivalently, if and only if P1 ! P0 ! Coker ! 0 is a minimal projective presentation of Coker in mod.RE /. Hence (a) and (b) easily follows. In the proof of the fc-tame-wild dichotomy theorem we use the following result. Theorem 6.22. Let K be an algebraically closed field, C a basic K-coalgebra with the decomposition (3.1), E a socle-finite injective left C -comodule as above, and assume that the K-algebra RE D EndC E in (6.19) is finite-dimensional. (a) The following three conditions are equivalent: (a1) The category C -ComodE fc is K-wild. (a2) The category C -ComodE fc is properly fc-wild (or smooth) [82], Section 6, that is, for every finitely generated K-algebra ƒ (equivalently, for ƒ D Kht1 ; t2 i, or ƒ D 3 .K// there exists a finitely E-copresented C -ƒ-bicomodule C Nƒ which is a finitely generated free ƒ-module and induces a K-linear representation embedding C N ˝ƒ ./ W fin.ƒop / ! C -ComodE fc . (a3) The finite-dimensional algebra RE is of K-wild representation type. (b) The following three conditions are equivalent: (b1) The category C -ComodE fc is fc-tame. (b2) The algebra RE is of K-tame representation type. (b3) The category Map1 .E/ is tame. Proof. By our assumption, the injective comodule E D EU is socle-finite and the K-algebra RE D EndC E is finite-dimensional. (a) To prove (a2) (H (a1) H) (a3), we consider the composite covariant functor

hE

op

D

C -ComodE ! mod.RE / ! mod.RE /; fc where D./ D HomK .; K/ is the standard duality and hE is the contravariant functor defined by setting hM E D Hom C .M; E/. The functor hE is a duality of categories, see [90], Proposition 2.13. Since E is injective then the composite functor DhE is exact and therefore the K-wildness of C -ComodE fc . implies the K-wildness of RE , because any K-linear exact representation embedding mod 3 .K/ ! C -Comodfc yields a representation embedding mod 3 .K ! mod RE . Since the implication (a2) H) (a1) is obvious, it remains to show that (a3) implies op ı (a2). Fix a finitely generated K-algebra S and set RE D RE . Throughout, we view ı op RE -S-bimodules as right modules over the algebra R ˝ S .

601


Following Drozd [29], we extend the diagram (6.20) to the following diagram Map1 .E ˝ S op / o

E ˝Rı ./ E

'

ker

˝S op .C ˝ S op /-ComodE fc

P1 .RE ˝ S op /

cok

(6.23)

mod.RE ˝ S op /.

We set Cy D C ˝ S op and view it as an S op -coalgebra with the comultiplication y D ˝ S op and the counit "O D " ˝ S op . Then Ey D E ˝ S op is an injective object in the category Cy -Comod of left Cy -comodules, which is projective, when viewed as y E ˝S op op a right S-module. We define Cy -ComodE to be the full fc D .C ˝ S /-Comod fc y y y subcategory C -Comod whose objects are the finitely E-copresented C -comodules, that is, finitely E ˝ S op -copresented Cy -bicomodules. The categories Map1 .E ˝ S op /, P1 .RE ˝ S op /, and the functors ker D kerE ˝S op , cok D cokRE ˝S op are defined in an obvious way. op ı ./ W P1 .RE ˝ S / ! Map1 .E ˝ S op / is an Now prove that the functor E ˝RE equivalence of categories. First we prove that the homomorphism E 0 ;E 00 W HomC .E 0 ; E 00 / ˝ S op ! HomCy .E 0 ˝ S op ; E 00 ˝ S op /;

(6.24)

given by g ˝ s 7! Œ.g ˝ id/ s W E 0 ˝ S op ! E 00 ˝ S op , is an isomorphism of S modules, for each pair E 0 ; E 00 of comodules in add.E/. Since E 0 ;E 00 is functorial with respect to homomorphisms E 0 ! E10 and E 00 ! E100 of C -comodules, it is sufficient to prove that E 0 ;E 00 is bijective, for E 0 D E 00 D E. We do it by showing that the homomorphism E;E W RE ˝S op ! EndCy .E ˝S op / of S -algebras is an isomorphism. We recall that there are isomorphisms EndC C Š C D HomK .C; K/ and

EndCy Cy Š HomS .C; S op /

given by the Yoneda maps ' 7! " B ' and g 7! "O B g, respectively. Following the notation in [90], Section 2, we denote by eE 2 EndC C Š C the idempotent defined by the direct summand E of C , and we set eOE D eE ˝ 1 2 EndCy Cy Š HomS .C; S op /. We view C and Cy as EndC C -EndC C -bimodule and EndCy Cy -EndCy Cy -bimodule in a natural way, see [59] and [90], Section 2. By [90], Proposition 2.7, CE D eE C eE is a coalgebra, there is an isomorphism CE Š EndC E of finite-dimensional K-algebras, and there is an isomorphism HomS .CE ˝ S op ; S op / Š HomK .CE ; K/ ˝ S op because we assume that dimK HomK .CE ; K/ D dimK EndC E is finite. Hence, fol-

602

D. Simson

lowing [90], Proposition 2.7, we have isomorphisms EndCy .E ˝ S op / Š EndCy .eOE Cy / Š eOE HomCy .Cy ; Cy / eOE Š eOE HomS .C ˝ S op ; S op / eOE Š HomS .eOE .C ˝ S op / eOE ; S op / Š HomS ..eE ˝ 1/ .C ˝ S op / .eE ˝ 1/; S op / Š HomS ..eE C eE / ˝ S op ; S op / D HomS .CE ˝ S op ; S op / Š HomK .CE ; K/ ˝ S op D CE ˝ S op Š .EndC E/ ˝ S op : It is easy to see that the inverse of the composite isomorphism is just the homomorphism E;E W RE ˝ S op ! EndCy .E ˝ S op /. It follows that E;E is an isomorphism and, consequently, E 0 ;E 00 is an isomorphism, for each pair E 0 ; E 00 of comodules in ˝S op add.E/. It follows that a left Cy -comodule Z lies in .C ˝ S op /-ComodE if and fc only if there is an exact sequence ! E 00 ˝ S op ; 0 ! Z ! E 0 ˝ S op with E 0 ; E 00 2 add.E/, or equivalently, if and only if Cy -comodule Z D kerE ˝S op . / ˝S op is contained in the category .C ˝ S op /-ComodE , where we write instead of fc E 0 ˝ S op ! E 00 ˝ S op . This shows that the functor ker in diagram (6.23) is well defined. We prove the implication (a3) H) (a2) by using the diagram (6.23) and by applying [29], Propositions 11 and 13, [30], Theorem 1.1, [22], Theorem B, [93], [95]. Assume that the algebra RE is K-wild. It follows from [29], Proposition 11 and 13, and [22], Theorem B (apply the arguments given in [22], pp. 478–479), that there op ı MS D .P1 ˝ S exists an object RE ; P0 ˝ S op ; / in P2 .RE ˝ S op /, with S D op KŒt1 ; t2 D S , such that MS is finitely generated free S -module and induces a ı M ˝S ./ W fin.S / ! P .RE /. Since the functor representation embedding RE 2 ı ./ W P1 .RE ˝ S/ ! Map .E ˝ S / is an equivalence of categories, the E ˝RE 1 ı P1 ˝ S; E ˝Rı P0 ˝ S; E ˝ / of Rı MS under the functor image C NS ´ .E ˝RE E E 0 00 ı ./ is an object of Map .E ˝ S/, that is, C NS Š .E ˝ S; E ˝ S; /, E ˝RE 1 0 00 with some E ; E 2 add.E/. Moreover, NS induces a representation embedding C N ˝S ./ W fin.S / ! Map2 .E/. Since, according to Proposition 6.21, the restriction kerE W Map2 .E/ ! C -ComodE fc of kerE to Map2 .E/ is a representation equivalence, the composite functor x ˝S ./ ´ kerE B .C N ˝S .// W fin.S / ! C -ComodE fc

CN

xS D preserves the indecomposability and reflects the isomorphism classes, where C N xS is a finitely E-copresented C -S -bicomodule kerE ˝S .C NS / D Ker. /. Hence C N


603

and there is an exact sequence xS ! E 0 ˝ S ! E 00 ˝ S 0 ! CN

. /

of C -S-bicomodules. It follows that C -ComodE fc is properly fc-wild. To see this, we recall that there are representation embeddings ./˝ƒ L0

./˝Kht1 ;t2 i L00

fin.ƒ/ ! fin.Kht1 ; t2 i/ ! fin.S /; 0 for every finitely generated K-algebra ƒ, where ƒ LKht and Kht1 ;t2 i L00S are bi1 ;t2 i modules that are finitely generated free over ƒ and over Kht1 ; t2 i, respectively; see [22], p. 479, [29], and [80], Proposition 14.10. It follows that the bimodules C Ny ˝S 00 0 00 y LKht op and C N ˝S L ˝Kht1 ;t2 iop Lƒop are finitely E-copresented, because the left 1 ;t2 i C -comodules 00 00 00 0 00 0 0 00 0 E 0 ˝LKht op ; E ˝LKht ;t iop ; E ˝L ˝Kht1 ;t2 iop Lƒop ; E ˝L ˝Kht1 ;t2 iop Lƒop 1 ;t2 i 1 2

lie in add.E/ and the sequence . / is exact. This finishes the proof of (a). (b) If A is any of the categories C -Comodfc , C -ComodE fc , Map1 .E/ and P1 .RE /, we denote by indv .A/ the family of indecomposable objects X of A, with cdn.X / D v. First, we observe that [22], Theorem B (and its proof), yields: (A) The finite-dimensional K-algebra RE is K-tame if and only if the family indv .P1 .RE // has an almost parametrising family 1 ; : : : ; r in P1 .RE ˝ S /, for some rational K-algebra S D KŒth . Next we prove the following statement: (B) Let v D .v 0 jv 00 / be a bipartite vector, S D KŒt h a rational K-algebra and

1 ; : : : ; r in P1 .RE ˝ S/. Then 1 ; : : : ; r is an almost parametrising family ı 1 ; : : : ; ı r is an for indv .P1 .RE // if and only if 1 D E ˝RE r D E ˝RE almost parametrising family for indv .Map1 .E//. Suppose that 1 ; : : : ; r is an almost parametrising family for indv .P1 .RE //. Take ı , for some

Š E ˝RE 2 indv .Map1 .E//, 2 indv .Map1 .E//. Then 1 see Proposition 6.21. By the assumption, Š j. / ˝S K. / , for almost all ı

2 indv .Map1 .E//, where K1 D KŒt =.t /. Hence, Š E ˝ RE Š 1 1 ı j. / ˝S K Š ˝ K . E ˝RE S j. / . / . / Conversely, suppose that 1 ; : : : ; r is an almost parametrising family for the ı in category indv .Map1 .E//. For 2 indv .P1 .RE ˝ S //, we take D E ˝RE indv .Map1 .E ˝ S//, see (6.20). Thus, for almost all 2 indv .P1 .RE ˝ S //, we 1 1 ı Š ı j./ ˝S K get E ˝RE , and therefore Š j./ ˝S K./ Š E ˝RE ./ 1

Š j./ ˝S K./ , by Proposition 6.21. This finishes the proof of (B). To prove the necessity of (a), assume that C -ComodE fc is fc-tame. Fix a bipartite vector v D .v 0 jv 00 / and let 1 ; : : : ; r 2 Map1 .E˝S / be such that the C -S -bimodules

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D. Simson

L.1/ D ker. 1 /; : : : ; L.r/ D ker. r /, with a rational K-algebra S D KŒt h , form an almost parametrising family for indv .C -ComodE fc /. It follows that 1 ; : : : ; r is an almost parametrising family for indv .Map1 .E//. Indeed, by Proposition 6.21, given 2 indv .Map1 .E// the kernel ker. / is an indecomposable comodule in C -ComodE 2 indv .Map1 .E ˝ S //, fc , with cdn ker. / D v. Then, for almost all we get ker. / Š Lj.

/

1 ˝S K.

/

Š Œker.

j. / /

1 ˝S K.

/

Š kerŒ

j. /

1 ˝S K. / ;

1 and Proposition 6.21 yields Š j. / ˝S K. . / op ı ./ W P1 .RE ˝S Since the functor E ˝RE / ! Map1 .E ˝S op / is an equivalence ı of categories then there exist 1 ; : : : ; r 2 P1 .RE ˝ S / such that 1 D E ˝RE ı r . By (B), 1 ; : : : ; r is an almost parametrising family for

1 ; : : : ; r D E ˝RE indv .P1 .RE //. Hence, in view of (A), the algebra RE is tame. Conversely, assume that the algebra RE is tame. Then, according to (A), given a bipartite vector v, indv .P1 .RE // has an almost parametrising family 1 ; : : : ; r . ı 1 ; : : : ; ı r is an almost parametrising family Then, by (B), 1 D E ˝RE r D E ˝RE for indv .Map1 .E//. Hence easily follows that ker. 1 /; : : : ; ker. r / is a finitely Ecopresented almost parametrising family for indv .C -ComodE fc /, that is, the category is fc-tame. This completes the proof of the theorem. C -ComodE fc

6.4 fc-tame-wild dichotomy for computable coalgebras. The main aim of this section is to prove the following fc-tame-wild dichotomy theorem for a computable coalgebra. Hence we deduce that the definitions of fc-tameness and fc-wildness are left-right symmetric for computable coalgebras. Theorem 6.25 ([93]). Assume that C is a basic computable coalgebra over an algebraically closed field K. Then C is either of fc-tame comodule type or of fc-wild comodule type, and these two types are mutually exclusive. Proof. Since C is basic, C C has the decomposition (3.1). We fix it throughout the proof. Assume that C is not of fc-wild comodule type. To show that C is of fc-tame comodule type, fix a non-negative bipartite vector v D .v 0 jv 00 / 2 Z.IC / Z.IC / Š K0 .C / K0 .C /. Since U D supp.v/ D fj 2 IC j vj0 ¤ 0 L or vj00 ¤ 0g of v is a finite subset of IC then the injective C -comodule E D EU D j 2U E.j / is soclefinite and, according to our assumption the algebra RE D EndC E is finite-dimensional. Moreover, every left C -comodule N , with cdn.N / D v lies in the subcategory E C -ComodE fc of C -Comod. It follows that ind v .C -Comodfc / D indv .C -Comod/. E Moreover, by our assumption, the category C -Comodfc is not of fc-wild comodule type. Then, by Theorem 6.22 (a), the algebra RE is not wild; hence RE is tame, by the tame-wild dichotomy for finite-dimensional algebras proved in [29] and [22], Theorem B. It follows from Theorem 6.22 (b) that the category C -ComodE fc is fctame, that is, there exists a finitely E-copresented almost parametrising family for


605

indv .C -ComodE fc / D ind v .C -Comod/. Consequently, the coalgebra C is of fc-tame comodule type. It remains to prove that the coalgebra C cannot be both of fc-tame and of fc-wild comodule type. Assume, to the contrary, that C is of fc-tame and of fc-wild comodule type. Then there exists an exact K-linear representation embedding T W mod 3 .K/ ! C -Comodfc . Let S1 be the unique simple injective right 3 .K/-module, and let S2 be the unique simple projective right 3 .K/-module, up to isomorphism. Since T .S1 / and T .S2 / lie in C -Comodfc then there are exact sequences 0 ! T .S1 / ! E0.1/ ! E1.1/ and 0 ! T .S2 / ! E0.2/ ! E1.2/ , where E0.1/ , E1.1/ , E0.2/ , E1.2/ are socle-finite injective C -modules. Let E be a socle-finite direct summand of C such that the comodules E0.1/ , E1.1/ , E0.2/ , E1.2/ lie in add.E/. We show that Im T C -ComodE fc . Indeed, if N D T .X / lies in Im T , where X is a module in mod 3 .K/, then there is an exact sequence 0 ! S2n ! X ! S1m ! 0, with n; m 0. Since T is exact, we get the exact sequence 0 ! T .S2 /n ! N ! T .S1 /m ! 0 in C -Comod. The comodules T .S1 /m E and T .S2 /n lie in C -ComodE fc and, hence, also N lies in C -Comod fc . This shows that E E Im T C -Comodfc . It follows that the category C -Comodfc is fc-wild and, according to Theorem 6.22 (a), the finite-dimensional algebra RE is wild. On the other hand, the assumption that C is of fc-tame comodule type implies that C -ComodE fc is fc-tame. Indeed, by the fc-parametrisation correction lemma [95], Lemma 3.1, given a proper bipartite vector v D .v 0 jv 00 / 2 K0 .C / K0 .C / D Z.IC / Z.IC / and a finitely copresented almost parametrising family of C -S-bicomodules L.1/ ; : : : ; L.rv / for the category indv .C -ComodE fc /, with S D KŒt h , there exists z .1/ ; : : : ; L z .rv / (in the sense of [95], an fc-localising v-corrected C -S-bicomodules L Construction (2.2)) forming a finitely E-copresented almost parametrising family for indv .C -ComodE fc /. Hence, by Theorem 6.22 (b), the finite-dimensional algebra RE is tame and we get a contradiction with the tame-wild dichotomy [29] for finite-dimensional K-algebras.

Now we show that the fc-wildness is left-right symmetric, for computable coalgebras, and that fc-wild computable coalgebras are properly fc-wild. Corollary 6.26. Let C be a basic computable coalgebra over an algebraically closed field K. The following conditions are equivalent. (a) The coalgebra C is of fc-wild comodule type. (a0 ) The coalgebra C op opposite to C is of fc-wild comodule type. (b) The category C -Comodfc is properly fc-wild, that is, for any finite-dimensional K-algebra ƒ (or equivalently, for ƒ D 3 .K//, there is a representation embedding T W mod ƒ ! C -Comodfc such that the C -ƒ-bicomodule C T .ƒ/ƒ is finitely copresented.

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(c) There exists a socle-finite direct summand E of C C such that the category C -ComodE fc is fc-wild. (d) There exists a socle-finite direct summand E of C C such that the finite-dimensional K-algebra RE D EndC E is wild. L Proof. Assume that C is basic and fix a decomposition C C D j 2IC E.j /. To prove that (a) implies (c), assume that the category C -Comodfc is wild, that is, there is a representation embedding T W mod 3 .K/ ! C -Comodfc : It was shown in the proof of Theorem 6.25 that there exists a socle-finite direct summand E of C C such that Im T C -ComodE fc , and (c) follows. (c) () (d): Apply Theorem 6.22 (a). (c) H) (b): Assume that C -ComodE fc is fc-wild, for some socle-finite direct summand E of C C . By Theorem 6.22 (a), the category C -ComodE fc is properly fc-wild. Then, for any finite-dimensional K-algebra ƒ (or equivalently, for ƒ D 3 .K/), there is a representation embedding T W mod.ƒ/ ! C -ComodE fc such that the C -ƒbicomodule C T .ƒ/ƒ is finitely E-copresented. It follows that category C -Comodfc is properly fc-wild since the embedding C -ComodE fc C -Comod fc is fully faithful and exact. Since (b) H) (a) is obvious, the conditions (a)–(d) are equivalent. L (a0 ) H) (d): Assume that the coalgebra C op is fc-wild. Let E D EU D u2U E.u/ be a socle-finite direct summand of C C . Since the coalgebra C is basic and there is a duality DC W C -inj ! C op -inj between the categories ofL socle-finite injective left C -comodules and right C -comodules [18], such that CC Š j 2IC DC .E.j //. Then DC .E/ is a socle-finite L injective right C -comodule isomorphic to theopdirect summand E 0 D EU0 D is fc-wild, u2U DC .E.u// of CC . Since the coalgebra C 0 op op 0 then the finite-dimensional K-algebra RE D EndC .E / Š EndC .DC .E// Š op .EndC E/op Š RE is wild, by the implication (a) H) (d) with C and C op interchanged. This implies that the algebra RE is wild and (d) follows. Since the inverse implication (a0 ) ( (d) follows in a similar way, the proof is complete. We call a coalgebra C properly fc-wild if the category C -Comodfc is properly fcwild in the sense of Corollary 6.26 (b), or equivalently, the category C op -Comodfc is properly fc-wild, by Corollary 6.26. Now we show that the fc-tameness for computable coalgebras is left-right symmetric. Corollary 6.27. Let C be a basic computable coalgebra over an algebraically closed field K. The following conditions are equivalent. (a) The coalgebra C is of fc-tame comodule type. (a0 ) The coalgebra C op opposite to C is of fc-tame comodule type. (b) For any socle-finite direct summand E of C C , the category C -ComodE fc is fctame.


607

(c) For any socle-finite direct summand E of C C , the finite-dimensional K-algebra RE D EndC E is tame. (d) For any socle-finite direct summand E of C C , the category C -ComodE fc is not fc-wild, or equivalently, the algebra RE is not wild. (e) The coalgebra C is not of fc-wild comodule type. Proof. The equivalences (a) () (a0 ) () (e) are a consequence of the fc-tame-wild dichotomy and the fact that fc-wildness of C is left-right symmetric, see Theorem 6.25 and Corollary 6.26. (a) H) (c): Assume that C is fc-tame. Let E be a socle-finite direct summand of C C . By Theorem 6.25, C is not fc-wild and, by Corollary 6.26, the algebra RE D EndC E is not K-wild. (b) () (c): Apply Theorem 6.22 (b). (c) () (d): Apply Theorem 6.22 (b) and the tame-wild dichotomy of Drozd [29]. (c) H) (a): Assume, to the contrary, that C is not fc-tame. Then, by Theorem 6.25, C is fc-wild and, according to Corollary 6.26, there exists a socle-finite direct summand E of C C such that the algebra RE D EndC E is K-wild. By the tame-wild dichotomy of Drozd [29], RE is not tame. This finishes the proof of (c) H) (a) and completes the proof of the corollary. Remark 6.28. We hope that the fc-tame-wild dichotomy holds for any basic coalgebra C over an algebraically closed field K. We finish this subsection by recalling from [93] that the fc-tameness and tameness coincide, for hereditary K-coalgebras, with acyclic Gabriel’s quiver. Proposition 6.29. Let K be an algebraically closed field and C a basic hereditary K-coalgebra such that the left Gabriel quiver CQ of C is acyclic, interval finite, and left locally finite. Then C is fc-tame if and only if C is tame. Proof. See [93], Proposition 3.4. 6.5 Tame-wild dichotomy for semiperfect coalgebras. By applying the fc-tamewild dichotomy given in Theorem 6.25, we show in this section that the K-tame-wild dichotomy (conjectured in [85]) holds for semiperfect coalgebras C . Throughout we write K-tame and K-wild, to distinguish them from the fc-tameness and fc-wildness. Theorem 6.30. Let C be a basic coalgebra over an algebraically closed field K. (a) If C is right semiperfect then C -comod D C -Comodfc , and C is of fc-tame comodule type if and only if C is of K-tame comodule type (that is, the category C -comod is K-tame). (b) Assume that C is right semiperfect or left semiperfect. Then C -comod is K-tame or K-wild, and not both.

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Proof. (a) Assume that C is a basic right semiperfect coalgebra. Then every indecomposable direct summand E.i / of C C is finite-dimensional and, hence, C -comod D C -Comodfc . By Lemma 6.16 (b), C -comod is K-tame if and only if, for every bipartite vector v D .v 0 jv 00 / 2 K0 .C / K0 .C /, there exist a non-zero h.t / 2 KŒt and C -KŒt h bicomodules L.1/ ; : : : ; L.rv / which are finitely generated free KŒt h -modules and form an almost parametrising family for indv .C -comod/. Moreover, in view of C -comod D C -Comodfc , we conclude from Lemma 6.16 (a) that the fc-tameness of C implies the K-tameness of C -comod. To prove the converse, assume to the contrary that C -comod is K-tame and C is not fc-tame. It follows from the fc-tame-wild dichotomy Theorem 6.25 that the category C -Comodfc D C -comod is K-wild and we get a contradiction to the weak version of the tame-wild dichotomy for C -comod; see Corollary 6.8. (b) If C is right semiperfect then (a), together with Theorem 6.25, gives the dichotomy. Assume that C is left semiperfect. Then every comodule N in C -comod has

a finite-dimensional minimal projective presentation P1 ! P0 ! N ! 0. If P is a finite-dimensional projective comodule in C -comod such that P0 ; P1 2 add.P /, we call N finitely P -presented. Given a projective comodule P in C -comod, we denote by C -comodP fp the category of finitely P -presented left C -comodules. The standard duality (2.25) D W C -comod ! C op -comod carries P to the socle-finite injective right C -comodule E D D.P / and E op induces the duality D W C -comodP fp ! C -comod fc . Then RP D EndC .P / is a finite-dimensional K-algebra and, as in Proposition 6.21, there is a commutative diagram Map1 .P / o cokP

P ˝Rı ./ P

'

P1 .RP / cokP

(6.31)

P ˝Rı ./ P o C -comodP mod.R P /, fp '

where RPı D RP , P is viewed as a C -RPı -bicomodule, Map1 .P / is the category of triples .P1 ; P0 ; /, with P0 ; P1 2 add.P /, W P1 ! P0 such that Im rad.P0 /, and the functor cokP is defined by passing to cokernels. The functors P ˝RPı ./ in (3.2) are K-linear equivalences of categories and their quasi-inverses are defined by the exact functor HomC .P; /. It follows that the K-wildness of C -comodP fp implies the K-wildness of the algebra RP . The inverse implication follows, by applying the arguments used in the proof of Theorem 6.22 (a). Given a C -comodule N in C -comod, we define the co-coordinate vector of N to be the bipartite vector ccdn N D cdn D.N /. We say that C -comod is fp-tame if for every co-coordinate vector v the category indv .C -comod/ of the indecomposable comodules N with ccdn N D v admits a finitely presented almost parametrising op


609

family. By applying the arguments used in the proof of Theorem 6.22 (b), one can prove that the category C -comodP fp is fp-tame if and only if the finite-dimensional algebra RP D EndC .P / is K-tame for every projective comodule P in C -comod. To prove the K-tame-wild dichotomy, first note that C cannot be both K-tame and K-wild by the weak version of the tame-wild dichotomy for C -comod, see Corollary 6.8. It remains to prove that C -comod is K-tame if it is not K-wild. Assume that C -comod is not K-wild. To show that C -comod is K-tame, it is enough to prove that, for every co-coordinate vector v, the category indv .C -comod/ of the indecomposable comodules N with ccdn N D v admits an almost parametrising family, as in the proof of (a). Let v be a co-coordinate vector. Since C is left semiperfect then every comodule in C -comod is finitely presented. Then, as in the proof of Theorem 6.25, there exists a finitely generated projective comodule P in C -comod such that indv .C -comod/ D P indv .C -comodP fp /. Since C -comod is not K-wild, the category C -comod fp is not Kwild. Hence, by the observation made earlier, the finite-dimensional algebra RP D EndC .P / is not K-wild. By the tame-wild dichotomy for algebras [29], RP is K-tame, and therefore the category C -comodP fp is fp-tame by the observation made above. It v P follows that ind .C -comodfp / D indv .C -comod/ admits an almost parametrising family. Hence C -comod is K-tame and the proof of the K-tame-wild dichotomy for left semiperfect coalgebras is complete. Corollary 6.32. The K-tameness is left-right symmetric for any left (or right) semiperfect coalgebra C . Proof. By Theorem 6.7 (c), the K-wildness of C is left-right symmetric. Then the corollary follows from the K-tame-wild dichotomy for left (or right) semiperfect coalgebras given in Theorem 6.30 (b). We finish this section by showing that left pure semisimple coalgebras are tame of discrete comodule type. We recall from [66] that a coalgebra C is left pure semisimple if every left C -comodule in C -Comod is a direct sum of finite-dimensional comodules. The definition is not left-right symmetric for infinite-dimensional coalgebras C . It is symmetric for coalgebras of finite K-dimension, and every such coalgebra C with dimK C < 1 is representation-finite, that is, the number of indecomposable comodules in C -comod (and in C -Comod) is finite, up to isomorphism (see [66] and [84]). Corollary 6.33. Let K be a field and C a left pure semisimple coalgebra. (a) C is computable, right semiperfect and C -Comodfc D C -comod. (b) C is K-tame of discrete comodule type. Proof. Assume that C is a left pure semisimple.

610

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(a) Since C is a direct sum of finite-dimensional comodules, all socle-finite comodules are finite-dimensional, that is, C -Comodfc D C -comod. It follows that C is computable and right semiperfect. (b) Since C -Comodfc D C -comod then, in view of Lemma 6.16, it is enough to show that C is fc-tame of fc-discrete comodule type. By Theorem 2.32, without loss of generality, we can assume that C is basic of the form (3.1). Given a bipartite vector v D .v 0 jv 00 / 2 K0 .C / K0 .C / Š Z.IC / Z.IC / , the 00 set U D supp.v/ ´ fj 2 IC j vj0 ¤ L 0 or vj ¤ 0g is a finite subset of IC , the injective direct summand E D EU D u2U E.u/ of C C is finite-dimensional, and the K-algebra RE D EndC E is finite-dimensional. By [90], Proposition 2.13 (a), the covariant composite functor

hE

op

D

! .mod.RE //op ! mod.R/; DhE W C -ComodE fc '

'

with hE ´ HomC .; E/ and D the standard duality, is an equivalence of categories. Since C -ComodE fc C -comod and C is left pure semisimple, by [84], Theorem 7.2, [75] and [76] every infinite sequence f1

fm

N1 ! N2 ! ! Nm ! NmC1 ! of non-zero non-isomorphisms f1 ; f2 ; : : : between indecomposable C -comodules N1 , N2 ; : : : in C -ComodE fc terminates, that is, there exists m0 1 such that fj is bijective, for all j m0 . Since this property is preserved by the equivalence DhE , the algebra RE is right pure semisimple and, hence, RE is representation-finite [4]. It follows that the category C -ComodE fc has only a finite number of indecomposable comodules, up to isomorphism. Since indv .C -comod/ D indv .C -Comodfc / C -ComodE fc , by the choice of E D EU , the number of the isomorphism classes of comodule indv .C -comod/ is finite. This finishes the proof of (b), see [95], Corollary 3.3. 6.6 On fc-tameness for arbitrary coalgebras. The fc-tame-wild dichotomy for an arbitrary basic coalgebra C over an algebraically closed field K remains an open problem. Some suggestions for the proof in case C is not computable is given in the following proposition that collects important consequences of the technique described in previous sections. In particular, it shows that the coalgebra C is fc-tame if and only if ı every socle-finite colocalisation CE Š RE of C (in the sense of [45], [89]) is fc-tame. Now we state without proof the following result proved in [95], Section 5. Theorem 6.34 ([95]). Assume that K is an algebraically L closed field and C is an arbitrary basic coalgebra with a decomposition C C D j 2IC E.j / (3.1). L (a) Given a socle-finite injective direct summand E D EU D u2U E.u/ of C , with a finite subset U of I , the K-algebra R D End E is semiperfect and C C E C

611


pseudocompact. There is a commutative diagram Map1 .E/

HE '

/ P1 .Rop / o E

kerE

CE -Comodfc Š C -ComodE fc

hE '

G0 '

yE - modpr R pr Š repK BE

cokE

/ modfp .Rop /, E

(6.35) op ı where CE Š RE is the colocalisation of C at E (see [45], [89]), modfp .RE / is the yE - modpr category of finitely presented left RE -modules, R of finitely pr is the category h i C R E J.RE / yE D generated propartite left modules over the bipartite K-algebra R 0

RE

in the sense of [82], with J.RE /C D HomRE .J.RE /; RE /, BE D .A; A VA / is the yE , HE and h D HomC .; E/ are K-linear additive Roiter bocs associated to R E contravariant equivalences of categories, G 0 is a covariant K-linear equivalence of categories, hE is an exact functor, kerE .E0 ; E1 ; / D Ker , cokE .P1 ; P0 ; / D Coker . (b) For any socle-finite comodule E D EU as in (a), the fc-tameness of the coalU gebra C implies that the category C -ComodE fc is fc-tame, that is, the coalgebra CEU is fc-tame. U (c) Conversely, if the category CEU -Comodfc Š C -ComodE fc is fc-tame, for all socle-finite injective direct summands E D EU , then the coalgebra C is fc-tame. For the proof and details we refer to [95], Section 5. Remark 6.36. Theorem 6.34 shows that the fc-tameness and fc-wildness of a basic K-coalgebra C (that is not necessarily computable) can be studied by means of the op yE - modpr tameness and wildness of the categories R pr and mod fp .RE / over the semiperyE , RE , and the bocs BE which are generally not finite-dimensional. fect algebras R 6.7 Comodule varieties of a computable coalgebra. Throughout we assume that K is an algebraically closed L field and C a basic computable K-coalgebra with a fixed decomposition C C D j 2IC E.j / (3.1). Following [95], Section 4 (see also [82]), we present in this section a geometry context for a coalgebra C ; compare with [54]. We use it in the study of comodules over a K-coalgebra C by applying the geometry of orbits. In particular, we give a geometric characterisation of computable coalgebras of fc-tame comodule type. Definition 6.37 ([95]). Given a basic computable K-coalgebra C as in (3.1) and a bipartite non-negative vector v D .v 0 jv 00 / 2 Z.IC / Z.IC / ; we define an action C W GvC MapC v ! Mapv

of an algebraic (parabolic) group GvC on an affine K-variety MapC v as follows.

(6.38)

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(a) GvC D AutC E .v 0 / AutC E .v 00 / is viewed as an algebraic group with respect L L 00 0 to Zariski topology, where E .v 0 / ´ i2IC E.j /vi and E .v 00 / ´ j 2IC E.j /vj are injective C -comodules, with lgth E .v 0 / D .v 0 j0/ and lgth E .v 00 / D .v 00 j0/. (b) The subset MapC 2 HomC .E .v 0 /; E .v 00 // j .soc E .v 0 // D 0g of v D f 0 00 HomC .E .v /; E .v // is viewed as an affine K-variety (Zariski closed subset of the affine space HomC .E .v 0 /; E .v 00 / of finite K-dimension). (c) The algebraic group (left) action (6.38) of GvC on MapC v is defined by the 0 0 conjugation .f 0 ; f 00 / D f 00 B B .f 0 /1 , where 2 MapC v , f 2 Aut C E .v / 00 00 and f 2 AutC E .v /. (d) The open subset ComodC v Df

00 2 MapC v j soc E .v / Im g

(6.39)

of the variety MapC v is called a variety of C -comodules N with cdn.N / D v. We start with the following useful facts. Lemma 6.40. Let C be a computable K-coalgebra and let v D .v 0 jv 00 / be a nonnegative bipartite vector in Z.IC / Z.IC / D K0 .C / K0 .C /. C (a) The set ComodC v is a Gv -invariant and Zariski open subset of the affine variety C Mapv . (b) The map 7! Ker defines a bijection between the GvC -orbits of ComodC v and the isomorphism classes of comodules N in C -Comodfc such that cdn.N / D v. C Proof. (a) To see that ComodC v is a Zariski open subset of Mapv , note that, given 00 00 a 2 supp.v / D fj 2 IC j vj ¤ 0g IC , the subset Da of MapC v consisting of all C 0 00 2 Mapv such that W E .v / ! E .v / has a factorisation through the subcomodule L 00 E .v 00 /a D j ¤a E.j /vj of E.v 00 / is Zariski closed. Since the set supp.v 00 / is finite S C then D D a2supp.v00 / Da is closed and therefore ComodC v D Mapv n D is open. C C The fact that ComodC v is a Gv -invariant subset of Mapv follows by applying the definitions. (b) Note that a C -comodule homomorphism W E .v 0 / ! E .v 00 / is an element of ComodC !E .v 0 / !E .v 00 / is a minimal injective v if and only if 0 ! Ker copresentation of Ker in C -Comodfc . Hence every comodule N in C -Comodfc , with cdn.N / D v, is isomorphic to Ker , for some W E .v 0 / ! E .v 00 / in ComodC v . Obviously, two elements W E .v 0 / ! E .v 00 / and 0 W E .v 0 / ! E .v 00 / of ComodC v lie in the same GvC -orbits if and only if the comodules Ker and Ker 0 are isomorphic. Hence (b) follows.

Now we characterise computable K-colagebras of fc-discrete comodule type in terms of the GvC -orbits of ComodC v as follows. Theorem 6.41 ([95]). Let K be an algebraically closed field and C a computable K-coalgebra. The following four conditions are equivalent.


613

(a) The coalgebra C is fc-tame of fc-discrete comodule type. (b) For any vector v D .v 0 jv 00 / 2 K0 .C / K0 .C /, there is only a finite number of indecomposable objects .E0 ; E1 ; / in Map1 .EUv / with cdn.E0 ; E1 ; / D v, up to isomorphism, where Uv D supp.v/ ´ fj 2 IC j vj0 ¤ 0 or vj00 ¤ 0g. (c) The number of GvC -orbits in ComodC v is finite, for every bipartite vector v D .v 0 jv 00 / 2 K0 .C / K0 .C /. (d) The number of GvC -orbits in MapC v is finite, for every bipartite vector v D .v 0 jv 00 / 2 K0 .C / K0 .C /. Proof. We follow the proof [95], Proposition 4.1, and the notation used there. (a) H) (b): Assume that C is fc-tame of fc-discrete comodule type. Let v D .v 0 jv 00 / be a bipartite vector in K0 .C /K0 .C / and let .E0 ; E1 ; / be an indecomposable object of Map1 .EU // such that cdn.E0 ; E1 ; / D .v 0 jv 00 /, where we set U D Uv D supp.v/. If v 0 D 0 then E0 D 0, E1 Š E.a/, with a 2 U , and therefore the number of the indecomposable objects .E0 ; E1 ; / of Map1 .EU // with cdn.E0 ; E1 ; / D .0jv 00 / equals the cardinality of the finite subset U D supp.v/ of IC . Assume that v 0 ¤ 0, that is, the vector v is proper. Since .E0 ; E1 ; / is indecomposable, it lies in Map2 .EU / because it has no non-zero direct summand of the form .0; Z; 0/. By Theorem 6.22 (a), with E and EU interchanged, the functor kerEU in the diagram (6.35) restricts to the representation equivalence kerEU W Map2 .EU / ! U C -ComodE D kerEU .E0 ; E1 ; / is an indecomposable comodule in fc . Then Ker EU C -Comodfc such that cdn.Ker / D cdn.E0 ; E1 ; / D v, see Theorem 6.22 (b). Since C is fc-tame of fc-discrete comodule type then the number of the isomorphism classes of such comodules is finite and, hence, the number of the isomorphism classes of indecomposable objects .E0 ; E1 ; / in Map1 .EU / with cdn.E0 ; E1 ; / D v is also finite. (b) H) (d): Let v D .v 0 jv 00 / 2 K0 .C / K0 .C / be a vector with non-negative coordinates and let .E0 ; E1 ; / be an object in Map1 .EU /. Since the coalgebra C is assumed to be computable then the endomorphism ring End. / of .E0 ; E1 ; / is a finite-dimensional K-algebra, and End. / is a local algebra if .E0 ; E1 ; / is indecomposable. It follows that Map1 .EU /, with U D supp.v/ IC , is a Krull– Schmidt category such that each of its objects is a finite direct sum of indecomposable objects, and every such a decomposition is unique, up to isomorphism and a permutation of the indecomposables. By our assumption, there is only a finite number of indecomposable objects E D .E00 ; E10 ; 0 / in Map1 .EUv / with cdn.E00 ; E10 ; 0 / v, up to isomorphism. Let E1 ; : : : ; Esv be a complete set of such indecomposable objects. Then, up to isomorphism, any object E D .E0 ; E1 ; / in Map1 .EUv /, with cdn.E0 ; E1 ; / D v, has the form ` .E .v 0 /; E .v 00 /; / Š E`11 ˚ ˚ Esvsv where `.E .v 0 /; E .v 00 /; / D .`1 ; : : : ; `sv / 2 N sv is a vector with non-negative coor-

614

D. Simson

dinates such that `1 cdn.E1 / C C `sv cdn.Esv / D v: Obviously, the number of such vectors .`1 ; : : : ; `sv / is finite. The unique decomposition property in Map1 .EUv / yields `.E .v 0 /; E .v 00 /; / D `.E .v 0 /; E .v 00 /; 0 / () .E .v 0 /; E .v 00 /; / Š .E .v 0 /; E .v 00 /;

0

/;

C or equivalently, if and only if the elements and 0 of MapC v lie in the same Gv -orbit. C C Hence the number of Gv -orbits in Mapv is finite and (d) follows. Since the implication (d) H) (c) is obvious and the implication (c) H) (a) follows from Lemma 6.40 (b), the proof is complete.

Now we present a characterisation of computable fc-tame colagebras in terms of geometry of the GvC -orbits of ComodC v . Theorem 6.42 ([95]). Let K be an algebraically closed field and C a computable K-coalgebra. The following four statements are equivalent. (a) The K-coalgebra C is of fc-tame comodule type. (b) For every bipartite vector v D .v 0 jv 00 / 2 K0 .C / K0 .C /, the category Map1 .EUv /, with Uv D supp.v/ ´ fj 2 IC j vj0 ¤ 0 or vj00 ¤ 0g, is tame. (c) For every bipartite vector v D .v 0 jv 00 / 2 K0 .C /K0 .C /, the subset indComodC v of ComodC v defined by the indecomposable C -comodules is constructible and there exists a constructible subset C.v/ of indComodC v such that GvC C .v/ D indComodC v

and

dim C .v/ 1:

(d) For every bipartite vector v D .v 0 jv 00 / 2 K0 .C / K0 .C /, the subset indMapC v of MapC v defined by the indecomposable C -comodules is constructible and there exists a constructible subset Cy .v/ of indMapC v such that GvC Cy .v/ D indMapC v

and

dim Cy .v/ 1:

Proof. (a) ( ) (b): Apply Corollary 6.27 and Theorem 6.22 (b) to the comodule L E D EU D j 2U E.j /, where U D supp.v/ IC . The equivalence of (b), (c) and (d) is proved in [95], Theorem 4.1, by applying the arguments used by Drozd [29], [80], Section 15.2, and [82], Theorem 6.5. The reader is referred to [95], Section 4.1, for details.

7 Quivers, profinite bound quivers, path coalgebras and locally nilpotent representations The aim of this section is to show that path coalgebras of quivers and locally nilpotent representations provide us with a powerful tool for the study of coalgebras and their comodule categories. Here we mainly follow [35], [36], [83], [84], and [111],


615

7.1 Graphs with non-negative quadratic form. By a graph (not necessarily finite) we mean a triple D .0 ; 1 /, where 0 is the set of vertices and 1 is the set Q1 of wedges, satisfying the usual conditions. In a characterisation of hereditary coalgebras C of tame comodule type we use the Euler quadratic form q W Z.0 / ! Z of any (not necessarily finite) graph D .0 ; 1 / defined by the formula P 2 P q .x/ D xj dij xi xj ; (7.1) j 20

i;j 20

where dij D j1 .i; j /j is the number of edges connecting the vertices i and j in , see [2] and [5]. Here we assume that dij is finite, for all vertices i; j 2 0 of . A special role in applications is played by the Euler quadratic form of each of the simply-laced Dynkin diagrams An W

:::

Dn W

2

:::

E6 W

4

;

E7 W

4

;

4

1

1

1

1

E8 W

1

2

3

2

2

2

3

4

3

3

3

5

5

5

(n vertices, n 1);

(n vertices, n 4);

n1

n1

n

n

6

6

6

7

7

; 8

and by the Euler quadratic form of each of the following simply-laced extended Dynkin diagrams (Euclidean graphs) hhh VVVVVVV VVVV hhhh h h h VVVV hh h h VVVV h hhh V (n C 1 vertices, n 1); h h z : : :

An W

zn W

:::

(n C 1 vertices, n 4); D

z

; E6 W

z7 W

; E

z8 W

. E z 1 is the Kronecker graph

Note that A

z 0 is the loop

, and A

.

616

D. Simson

We start with the following well-known characterisation of graphs with non-negative Euler form. Proposition 7.2. Assume that D .0 ; 1 / is a connected graph such that the set 1 .i; j / is finite, for each pair i; j 2 0 of vertices of . Let q W Z.0 / ! Z be the Euler quadratic form (7.1) of . (a) The form q W Z.0 / ! Z is positive definite if and only if either is finite and is isomorphic to one of the simply-laced Dynkin diagrams An , Dn , E6 , E7 , E8 , or is one of the following infinite locally Dynkin graphs A1 W

0

1 A1 W

D1 W

0

:::

2

2

1

1

:::

1

1

s1

0

1

:::

2

s :::

2

s1

s

::: ;

sC1

s1

s

sC1

::: ;

::: ;

sC1

(b) The form q W Z.0 / ! Z is positive semi-definite and not positive definite if z n , with n 0, D z n, and only if is any of the extended Dynkin (Euclidean) graphs A z z z with n 4, E6 , E7 , and E8 . Outline of the proof. The proof is purely combinatorial, see [2], Chapter VII. We need to know that given a connected finite graph , q is positive definite if and only if is any of the Dynkin graphs An , with n 1, Dn , with n 4, E6 , E7 , and E8 , see [2], and q is positive semi-definite and not positive definite if and only if is any of the z n , with n 0, D z n , with n 4, E z 6, E z 7 , and E z 8. extended Dynkin (Euclidean) graphs A To show the second statement, we need to know that if is an extension of any of the extended Dynkin graphs, then q .v/ < 0 for some vector v 2 Z.0 / . Note that if is any of the following two graphs L0 W

;

L00 W

;

then q is indefinite, because qL0 .x1 / D x12 and qL00 .x1 ; x2 / D x12 x1 x2 . Note also that the set RqL00 D fv 2 Z2 j qL00 .v/ D 1g of roots of the form qL00 is finite. 7.2 Quivers and path coalgebras. Following Gabriel [33], [35], and [36], by a quiver Q D .Q0 ; Q1 / we mean an oriented graph (not necessarily finite), with the set Q0 of vertices and the set Q1 of arrows. We associate to Q the quiver Qı D .Q0ı ; Q1ı / opposite to Q, where Q0ı D Q0 and Q1ı consists of arrows of the form ˇ ı W j ! i corresponding to the arrows ˇ W i ! j in Q1 . By the Euler quadratic form qQ W Z.Q0 / ! Z of the quiver Q we mean the form x obtained from Q by forgetting q W Z.0 / ! Z in (7.1) of its underlined graph D Q the orientation.


617

Let K be a field. We recall that a K-linear representation of the quiver Q D .Q0 ; Q1 / is a system X D .Xi ; 'ˇ /i2Q0 ;ˇ 2Q1 ; where Xi is a K-vector space and 'ˇ W Xi ! Xj is a K-linear map for any ˇ W i ! j . The integral vector dim X D .dimK Xj /j 2Q0 2 N .Q0 / is called the dimension vector of X . A representation X of Q is of finite length if Xj is a finite dimensional K-vector space for any j , and Xi D 0 for almost all indices i . A morphism f W X ! X 0 of representations of Q is a system f D .fi /i2Q0 of K-linear maps fi W Xi ! Xi0 , i 2 Q0 , such that 'ˇ0 fi D fj 'ˇ for all ˇ W i ! j in Q1 (see [2], [5], [80], Chapter 14). We denote by HomQ .X; Y / the K-linear space of all morphisms from X to Y , by RepK .Q/ the Grothendieck K-category of K-linear representations of Q, and by lf .Q/ the full Grothendieck K-subcategory of RepK .Q/ formed by locally finite RepK dimensional representations, that is, directed unions of representations of finite length. lf Finally, we denote by repK .Q/ repK .Q/ the full subcategories of RepK .Q/ formed by finitely generated objects and by locally finite dimensional representations, or equivalently, by representations of finite length. lf lf It is easy to see that the category RepK .Q/ is locally finite [68] and repK .Q/ lf consists of all objects of RepK .Q/ of finite length. By an oriented path in the quiver Q D .Q0 ; Q1 / of length m 1, with the source vertex i D i0 and the terminus j D im , we mean a formal composition ˇ1

ˇ2

ˇm

! D ˇ1 ˇ2 : : : ˇm .i0 ! i1 ! ! im /

(7.3)

of arrows ˇ1 ; : : : ; ˇm . The path !, with i D j , is called an oriented cycle in Q. The quiver Q is defined to be • acyclic if Q has no oriented cycles, • interval finite if, for all a; b 2 Q0 , the set Q.a; b/ of all oriented paths from a to b is finite, • left locally bounded if, for each b 2 Q0 , there are only finitely many arrows j ! b in Q that terminate in b, • right locally bounded if, for each a 2 Q0 , there are only finitely many arrows a ! j in Q that start from a. To any vertex i 2 Q0 we attach a stationary path i starting and ending at i . The stationary path at j in the quiver Qı opposite to Q is also denoted by j . If ! D ˇ1 ˇ2 : : : ˇm is the path (7.3) in Q we set ! D ˇm ˇm1 : : : ˇ1 and we view it as a path in Qı . We denote by Qm the set of all oriented paths in Q of length m 0. By the path K-algebra of a quiver Q we mean the graded K-vector space KQ D KQ0 ˚ KQ1 ˚ ˚ KQm ˚

(7.4)

618

D. Simson

L equipped with the obvious addition and multiplication, where L KQm D !2Qm K!. Obviously, for each m 1, the K-vector space KQm D j m KQj is the two-sided ideal of KQ generated by all paths of length m. Given two vertices a; b 2 Q0 , we denote by KQ.a; b/ the subspace of KQ generated by all oriented paths from a to b, and by KQm .a; b/ the subspace of KQ.a; b/ generated by paths of length m. It is clear that KQ in (7.4) is a graded K-algebra, the stationary paths ei , i 2 Q0 , form a complete set of primitive there is a right L orthogonal idempotents of KQ, andP ideal decomposition KQ D i2Q0 ei KQ: If Q0 is finite, the element i2Q0 ei is the identity of KQ. If Q0 is infinite, the algebra KQ has no identity element. It is clear that the dimension of KQ is finite if and only if Q is finite and acyclic. It is well known that there are equivalences of categories RepK .Q/ Š Mod.KQ/ and

repK .Q/ Š mod.KQ/I

(7.5)

see [2], Theorem III.1.6, [5], and [80], Section 14.1. The path K-algebra KQ endowed with the direct sum decomposition (7.4) can be viewed as a graded K-coalgebra K Q, with the comultiplication W KQ ! KQ ˝ KQ and the counity " W KQ ! K defined as follows (see [15], [20], [83], [111]). Given the stationary path ei at i , we set .ei / D ei ˝ ei and ".ei / D 1. Given any path ! D ˇ1 ˇ2 : : : ˇm as in (7.3) of length m 1, with the source i D i0 and the terminus j D im , we set .!/ D i ˝ ! C ! ˝ j C

m1 P sD1

.ˇ1 ˇ2 : : : ˇs / ˝ .ˇsC1 : : : ˇm / and

".!/ D 0;

where ˝ D ˝K . It is easy to see that and " define a K-coalgebra structure on KQ. We call the triple K Q ´ .KQ; ; "/ the path K-coalgebra of the quiver Q with coefficients in K. L It is clear that KQ0 D soc K Q D a2Q0 Kea is the socle of K Q, S.a/ D Kea is the simple left (and right) coideal of K Q and, for each m 0, the space .K Q/m ´ K .Qm / D KQ0 ˚ KQ1 ˚ ˚ KQm

(7.6)

is a subcoalgebra of K Q and .K Q/m is the m-th term of the coradical filtration of K Q. The notation K Q is inspired by the fact that K Q is isomorphic to the cotensor coalgebra TC0 .C1 / D C0 ˚ C1 ˚ C1 C1 ˚ C1 C1 C1 ˚ : : : ; where L sum of the one-dimensional C0 ´ .K Q/0 D a2Q0 Kea is viewed as a directL simple coalgebras S.a/ D Kea , and C1 ´ .K Q/1 D a;b2Q0 KQ1 .a; b/ is viewed as a C0 -C0 -bicomodule in a natural way; see [20], Remark 4.2, [70], and [111]. Now we collect the main properties of the path coalgebra K Q.


619

Proposition 7.7. Assume that Q is a connected quiver, K a field and K Q the path coalgebra of Q. (a) The coalgebra K Q is pointed (hence basic), G.K Q/LD fea j a 2 Q/ g is the set (4.13) of group-like elements of K Q, C0 ´ .K Q/0 D a2Q0 Kea is the socle of K Q, S.a/ D Kea is a simple subcoalgebra of K Q for each a 2 Q0 , and the subcoalgebra chain .K Q/0 .K Q/1 .K Q/m : : : ; with .K Q/m ´ K .Qm / in (7.6) is the coradical filtration of K Q. (b) Q is isomorphic to the left Gabriel quiver of K Q. (c) If Q has at least one arrow, the coalgebra K Q is hereditary and, for any j 2 Q0 , the simple comodule S.j / has a minimal injective copresentation L E.a/daj ! 0 . / 0 ! S.j / ! E.j / ! a2Q0

L in C -Comod, where E.j / D E.S.j // D .KQ/ej D s2Q0 KQ.s; j / (the space generated by all paths terminating at j ) is the injective envelope of S.j / and daj D dimK Ext1C .S.a/; S.j // D dimK KQ.a; j /. Moreover, there is a K-vector space isoŠ .ej .KQ/ea / Š .KQ.j; a// and the left K Qmorphism HomK Q .E.a/; E.j // L comodule decomposition K Q D a2Q0 E.a/ as in (3.1). (d) K Q is computable if and only if Q is interval finite. (e) K Q-comod K Q-Comodfc if and only if Q is interval finite and left locally bounded. (f) K Q is right semiperfect if and only if, for every b 2 Q0 , the set Q.! b/ of all oriented paths terminating at b is finite. Proof. For the proof of (a) and (b), we refer to [14], [20], [84], Section 8, [85], [111]. (c) Apply [14], [84], Proposition 8.13, [85], Lemma 2.12, and [111]. To prove the final statement of (c), we set C D K Q. By applying the duality (2.18), Remark 3.3, and Lemma 3.4 (a), we get the isomorphisms E.a/ Š C ea , E.j / Š C ej , and HomC .E.a/; E.j // Š homC .E.j / ; E.a/ / Š homC .C ej ; C ea / Š ej C ea Š .ej .KQ/ea / Š .KQ.j; a// . (d) Apply the final statement in (c). (e) The inclusion K Q-comod K Q-Comodfc holds if and only if every simple comodule S.j / lies in K Q-Comodfc , or equivalently, the final term in the minimal injective copresentation . / of S.j / is socle-finite. But this holds if and only if P a2Q0 daj is finite, for all j 2 Q0 , where the sum is taken over all a 2 Q0 being a source of an arrow a ! j in Q. Hence (e) easily follows. Since (f) is a consequence of [52], Theorem 4.7, the proof is complete. 7.3 Comodules over a path coalgebra and locally nilpotent representations of a quiver. We show in this section that, in an analogy to the equivalences (7.5) due to

620

D. Simson

Gabriel [36], the category K Q-Comod is equivalent to the category of locally nilpotent representations of Q. A correspondence between K Q-comodules and locally nilpotent representations of Q described explicitly in (7.15) is one of our main tools in the study of comodule categories over arbitrary pointed K-coalgebras. The first step for the correspondence (7.15) is the following lemma; compare with [59], [84], [103], [111]. Lemma 7.8 ([88], Lemma 2.3). Assume that Q is a quiver, K a field and let K Q be the path K-coalgebra of Q. (a) If Q is finite and acyclic, dimK K Q is finite, and there is a K-algebra isomorphism KQ Š .K Q/ defined by ! 7! ! for any path ! in Q. Moreover, there are equivalences of categories K Q-comod Š repK .Q/ Š mod KQ. (b) If Q is arbitrary and m 2, the subcoalgebra .K Q/m1 D KQ0 ˚ KQ1 ˚ ˚ KQm1 of K Q in (7.6) is finite-dimensional and there are a K-algebra isomorphism .K Q/m1 Š KQ=KQm and an equivalence of categories .K Q/m1 -comod Š mod.KQ=KQm /. Proof. (a) Since Q is finite and acyclic then the set Qm is finite for each m 0, and Qm D 0 for m sufficiently large. Consequently, dimK K Q is finite. It follows from the definition of the convolution product in .K Q/ D HomK .K Q; K/ that the K-linear map KQ ! .K Q/ , defined on the paths ! (the elements of the standard K-basis of K Q) by setting ! 7! ! , is a K-algebra isomorphism. To prove the second statement, apply the equality dis.K Q/ D mod .K Q/ , the equivalences K Q-comod Š rat.K Q/ D dis.K Q/ D dis.KQ/ in (2.23), we get K Q-comod Š mod.KQ/ Š repK .Q/, see (7.5). Statement (b) follows in a similar way. Given a quiver Q D .Q0 ; Q1 /, we denote by RepK .Q/ the category of K-linear representations of Q, and by repK .Q/ the full subcategory of RepK .Q/ whose objects are the finitely generated representations. For a representation X D .Xa ; 'ˇX /a2Q0 ;ˇ 2Q1 in RepK .Q/, we define its support to be the subquiver QX D .Q0X ; Q1X / of Q, with Q0X D fa 2 Q0 j Xa ¤ 0g and Q1X D fˇ 2 Q1 j 'ˇX ¤ 0g. We say that X is of finite length if the support QX of X is a finite subquiver of Q and dimK Xa is finite for any a 2 Q0X . fl We denote by repK .Q/ the full subcategory of repK .Q/ whose objects are the fl .Q/ the category of locally finite length finite length representations, and by RepK fl representations, that is, directed unions of the finite length representations in repK .Q/. Following [83] and [84] (see also [11], [12], [15], [111]), we define a K-linear representation X of Q to be nilpotent (or a small representation of Q, in the sense of


621

Gabriel [36], Section 7.4) if there exists an m 2 such that the composed K-linear map 'ˇ1

'ˇ2

'ˇm

Xi0 ! Xi1 ! ! Xim fl is zero for any path ˇ1 ˇ2 : : : ˇm in Q of length m. We denote by nilrepK .Q/ the full fl subcategory of repK .Q/ whose objects are all nilpotent representations of finite length, lnfl lf .Q/ the full subcategory of RepK and by RepK .Q/, whose objects are locally nilpotent representations of finite length, that is, directed unions of finite length representations that are nilpotent. In other words, a representation X of Q is locally nilpotent if, for each a 2 Q0 and x 2 Xa , there is only finitely many paths ! in Q starting from a such that '! .m/ D 0. lnfl In our description in (7.15) of a functor K Q-Comod ! RepK .Q/, we use an alternative description of the pseudocompact K-algebra .K Q/ associated to the path coalgebra K Q. In this description we need the following simple observation.

Lemma 7.9. Assume that Q is a quiver, K a field, KQ the path K-algebra of Q, and K Q be the path K-coalgebra of Q. (a) There is a canonical algebra embedding KQ ,! .K Q/ defined by ! 7! ! for any path ! in Q. (b) The non-degenerate bilinear form h; i W .K Q/ K Q ! K defined by the formula h'; ci D '.c/ for ' 2 .K Q/ and c 2 K Q, restricts to the non-degenerate bilinear form h; iı W KQ K Q ! K defined by h!; ! 0 iı D ı!;! 0 (the Kronecker delta) for any pair of paths !, ! 0 in KQ. Now we define a K-linear finite subquiver topology on the path K-algebra KQ and c of Q. a complete path K-algebra KQ Definition 7.10 ([88]). Assume that Q is an arbitrary quiver, K a field, KQ the path K-algebra of Q, and K Q the path K-coalgebra of Q. (a) Given m 2 and a finite subquiver Q.x/ of Q, we consider the finite-dimen.x/ .x/ sional subcoalgebra .K Q.x/ /m1 D KQ.x/ 0 ˚ KQ1 ˚ ˚ KQm1 , see (7.6), of .x/ K Q (2.2) and its dual K-algebra .K Q.x/ /m1 Š KQ.x/ =KQm ; see Lemma 7.8. The finite subquiver topology on KQ is the K-linear topology defined by the two.x/ .x/ sided ideals Um.x/ D Ker m , where m is the composite K-algebra surjection .x/ m

.x/

.um /

.x/ D ŒKQ ,! .K Q/ ! .K Qm1 / Š KQ.x/ =KQ.x/ m ;

.x/ u.x/ /m1 ,! K Q is the K-coalgebra embedding and KQ ,! .K Q/ is m W .K Q the canonical K-algebra embedding. (b) The complete path K-algebra of Q is the completion

c D KQ Q

lim .x/

KQ=Um.x/

;m2

of KQ, where Q.x/ runs through all finite subquivers of Q.

(7.11)

622

D. Simson

Proposition 7.12 ([88]). Let Q be an arbitrary quiver, K a field, KQ the path Kalgebra of Q, and K Q the path K-coalgebra of Q. (a) The coalgebra K Q is a directed union of the finite-dimensional subcoalgebras .K Q.x/ /m in (7.6), where m 2 and Q.x/ runs through all finite subquivers of Q. .x/ (b) The ideals Um.x/ D Ker m define a Hausdorff K-linear topology on the algebra KQ. c in (7.11) of KQ is a pseudocompact K-algebra and there (c) The completion KQ is an isomorphism of pseudocompact algebras

b

c Š Ty .KQ1 /; .K Q/ Š KQ c KQ 0

where

b

b Q KQ b Q KQ b b ˚ Q KQ b D KQ

TyKQ c .KQ1 / D KQ0 0

0

1

1

1

1

mD1

y2 ˝

Q

Q

(7.13)

b

m

y KQ1 ˝

Q

:::

ym ˝

is the complete tensor K-algebra [36], p. 96, of the topological vector K-space Q KQ1 D a;b2Q0 KQ1 .a; b/ viewed as a KQ0 -KQ0 -bimodule over the topological Q product KQ0 D a2Q0 Kea of Q0 copies of the field K. (d) The ideal J..K Q/ / D .K Q0 /? , see (2.14), is closed in .K Q/ and the c restricts to the isomorphisms isomorphism of pseudocompact algebras .K Q/ Š KQ

b

b

4

bb

2 1

c D J .KQ/ D KQ1 , and (d1) J..K Q/ / Š J.KQ/

b

2

1

(d2) .K Q0 ˚ K Q1 /? D J..K Q/ /2 Š J.KQ/2 D .J .KQ//2 D KQ2 . Proof. The proof is rather long, but a routine one. The reader is referred to [36], [84], Proposition 8.1, and [88], Proposition 2.5; for details, see also [42], Section 10, [55], [56], and [111]. Remarks 7.14. (a) In [42], Section 10, a description of the pseudocompact K-algebra c in (7.6) in terms of Cauchy nets is given by defining on the path algebra KQ a KQ K-precompact topology (which is equivalent to our finite subquiver topology on KQ); see also [55] and [56]. (b) It is shown by Keller and Yang in [48] (see also [69] and [105]) that the algebra c is very useful in the study of the Jacobian algebras of quivers with potentials in the KQ sense of [27], the Jacobi-finite quivers with potentials studied by Amiot [1], and their c extension to complete path algebras KQ. To describe a correspondence between left K Q-comodules and the K-linear representations of the quiver Q, we recall from Section 2 that, given a K-coalgebra C , any left C -comodule M D .M; ıM /, with ıM W M ! C ˝ M , can be viewed as a right rational (D discrete) C -module via the action P m ' D '.c.1/ /m.2/ ; (7.15) .m/


where ' 2 C , m 2 M , and ıM .m/ D

P

623

c.1/ ˝m.2/ ; see [25], [59], [84], [103], [110].

.m/

By Theorem 2.22, there is a categorical isomorphism -Comod Š Rat.C / D Dis.C /, defined by associating to any left C -comodule M the underlying vector space M endowed with the rational right C -module structure. Here Rat.C / is the category of the rational right C -modules and Dis.C / is the category of discrete right C modules. c on left By applying (7.15) to C D K Q, we get the right action .K Q/ Š KQ K Q-comodules M and we define the K-linear functor F W K Q-Comod ! RepK .Q/

(7.16)

as follows, see [84], [88], (3.1), [111]. Given a comodule M in K Q-Comod (viewed c we define as a rational right module over the pseudocompact algebra .K Q/ Š KQ), F .X / in RepK .Q/ by setting F .M / D .Ma ; 'ˇM /a2Q0 ;ˇ 2Q1 ;

(7.17)

c is the stationary path at a; consult where Ma D M ea D M ea and ea 2 KQ KQ c is defined by ! 7! ! , Example 7.20 below. Here the algebra embedding KQ KQ for any path ! in Q. For any arrow ˇ W a ! b, we define the K-linear map 'ˇM W Ma ! Mb by the formula 'ˇM .x ea / D .x ea /ˇ D mea ˇeb , where m 2 M and mea 2 Ma D M ea . Since ˇ 2 Q1 is viewed as an element of KQ, we have ˇ D ea ˇ D ˇeb in KQ. Given a K Q-comodule homomorphism f W M ! N , we set F .f / D .fa /a2Q0 , where fa W Ma ! Na is the restriction of f to Ma (consult the proof of Theorem III.1.6 in [2]). It is clear that F .f / W F .M / ! F .N / is a morphism in RepK .Q/ and that we have defined a covariant K-linear exact functor F W K Q-Comod ! RepK .Q/, which restricts to a functor F W K Q-comod ! repK .Q/. Alternative descriptions of F .M / are given in the following proposition, see [11], [12], p. 870, [94], (4.7). Proposition 7.18. Let Q be an arbitrary quiver, K a field, and K Q the path Kcoalgebra of Q. (a) Given a left C -comodule M and a 2 Q0 , there are K-linear isomorphisms Ma D M ea D fm 2 M j M .m/ D ea ˝ mg Š ŒHomC .M; EI .a//ı ; M

0

where M is the composed K-linear map M ! .K I / ˝ M ! .K I /0 ˝ M and 0 W K I ! .K I /0 D soc K I is the canonical K-linear projection. Here we view HomC .M; EI .a// as a pseudocompact right C -module and ŒHomC .M; EI .a//ı D homC .HomC .M; EI .a//; K/ is its topological K-dual in the sense of (2.18).

624

D. Simson

(b) The functor F in (7.16) commutes with arbitrary direct sums and directed unions, and F restricts to two exact equivalences of categories K Q-Comod O ? K Q-comod

F

/ Replnfl .Q/ KO ? / nilrepfl .Q/ K

F

(7.19)

making the diagram commutative. (c) dim F .M / D lgth M D dim M for any M in K Q-Comod. (d) The functor F 1 inverse to F associates to any K-linearL representation X D lnfl .Xa ; 'ˇX /a2Q0 ;ˇ 2Q1 in RepK .Q/ the vector space F 1 .X / D a2Q0 Xa equipped L with the left K Q-comodule structure on a2Q0 Xa induced by the natural discrete c right module structure over the profinite K-algebra .K Q/ Š KQ. fl (e) For any representation X in nilrepK .Q/ there exists a finite subquiver Q.x/ of fl fl Q and an integer m 2 such that X lies in repK .Q.x/ ; KQ.x/ m / nilrepK .Q/. For the proof we refer to [88], Proposition 3.3, [94], Proposition 4.3, and [111]. The equality Ma D M ea D fm 2 M j M .m/ D ea ˝ mg in (a) is also proved in [11] and [12], p. 870. Now we show how the functor (7.16) acts on the injective comodule E.a/. Example 7.20. Assume that Q D .Q0 ; Q1 / is an arbitrary quiver, b 2 Q0 a fixed vertex in Q and E.b/ D .KQ/eb is the left indecomposable direct summand of the path K-coalgebra K Q. Note that .KQ/eb is the left ideal of the path algebra KQ generated by the stationary path eb at the vertex b; equivalently, .KQ/eb viewed as a K-vector space is generated by all path in Q that terminates at b. We view .KQ/eb as a right rational module over the pseudocompact algebra c K-dual to K Q, and we consider the canonical K-algebra embedding .K Q/ Š KQ c ! 7! ! , for any path ! in Q. Hence the left K Q-comodule KQ .K Q/ Š KQ, M ´ E.b/ D .KQ/eb has a right module structure over the path algebra KQ. It is easy to see that, given ˇ1

ˇ2

ˇm

! D ˇ1 ˇ2 : : : ˇm .a D i0 ! i1 ! ! im D b/ of length m 0 in Q D .Q0 ; Q1 /, we have ! ec D ! ec D ec !, where ec ! is the multiplication in KQ and ! ec D ! ec means the rational action of ec (identified with ec W K Q ! K) on the path ! 2 M D E.b/ D .KQ/eb . Moreover, given an arrow ˇ W c ! c 0 in Q1 and a path ! D ˇ1 ˇ2 : : : ˇm 2 M D E.b/ D .KQ/eb of length m 1, we have ´ ˇ2 : : : ˇm if ˇ D ˇ1 ; !ˇ D!ˇ D 0 if ˇ ¤ ˇ1 :


625

lnfl It follows that the K-linear representation F .M / D F .E.b// 2 RepK .Q/ in (7.17) of Q corresponding to the injective left K Q-comodule M D E.b/ has the form F .M / D F .E.b// D .Ma ; 'ˇM /a2Q0 ;ˇ 2Q1 , where Ma D M ea D E.b/ ea D ea .KQ/eb D KQ.a; b/ is the K-vector space spanned by all oriented paths from a to b. Given an arrow ˇ W a ! c, the K-linear map 'ˇX W Xa D KQ.a; b/ ! Xc D KQ.c; b/ is defined by the formula

´ 'ˇX .!/ D

ˇ2 : : : ˇm 0

if c D j1 and ˇ D ˇ1 ; if ˇ ¤ ˇ1 ; ˇ1

ˇ2

ˇm

for any path ! D ˇ1 ˇ2 : : : ˇm .a D i0 ! i1 ! ! im D b/ in Q.a; b/. The following result due to Gabriel [36], [37], and Woodcock [111] shows that any basic coalgebra C over an algebraically closed field admits a suitable coalgebra embedding C ,! K Q into the path coalgebra K Q of the left Gabriel quiver Q D CQ of C , see also [42]. Theorem 7.21. Assume that K is an algebraically closed field. Let C be a basic K-colagebra, let C0 and C1 be the first two terms of the coradical filtration of C , let Q D CQ be the left Gabriel quiver of C and K Q the path K-coalgebra of Q. (a) There exist a coideal J of C and a vector space decomposition C D C0 ˚ J . (b) There exists an injective K-coalgebra homomorphism f W C ! K Q such that f .C1 / D .K Q/1 D KQ0 ˚ KQ1 , and f induces a full, faithful, K-linear and exact fl embedding f W C -comod ! nilrepK .Q/. (c) The coalgebra C is hereditary if and only if f is an isomorphism. Proof. (a) Apply [111], (4.5). (b) Without loss of generality, we may assume that C is of the form (3.1), because C is basic. Hence we have a left comodule decomposition L S.j /; . / socC C D j 2IC

where IC is a set and S.j / are pairwise non-isomorphic simple comodules, for j 2 IC . Since K is algebraically closed, by Corollary 2.29, we have: dimK S.j / D 1 for all j 2 IC , each S.j / is a subcoalgebra of C , C0 D socC C and . / is a coalgebra direct sum decomposition of C0 . We have shown in Section 4 that the left Gabriel quiver Q D CQ has a description in terms of the quotient vector space W D C1 =C0 viewed as a C0 -C0 -bicomodule in a natural way, see [36], 7.5, and [111], Section 4. Since . / is a coalgebra direct sum decomposition of C0 , for eachL pair i; j 2 IC , there exists an S.i /-S.j /-sub-bicomodule i Wj of W such that W D i;j 2IC i Wj is a direct sum decomposition of C0 -C0 -bicomodules, where the C0 -C0 -bicomodule structure on i Wj is induced by its S.i /-S.j /-bicomodule structure. It follows from Theorem 4.6 that the arrows from the point i to the point j of the Gabriel quiver Q D CQ can be identified

626

D. Simson

with elements of a fixed basis of i Wj . It follows that the space KQ1 is isomorphic to W . Let 1 L W m TC0 W D mD0

be the (co)tensor coalgebra of the C0 -C0 -bicomodule W , with W 0 D C0 and W 1 D W (see [20], Remark 4.2, [65], [111], Section 4). Then there is a coalgebra isomorphism K Q Š TC0 W; that restricts to the isomorphism .K Q/0 / ˚ .K Q/1 / Š C0 ˚ W TC0 W: In view of the decomposition C D C0 ˚ J: given in (a), there is a natural projection f1 W C ! W such that C0 Ker f1 . It follows from the universal properties of TC0 W that f1 extends to a unique coalgebra homomorphism f 0 W C ! TC0 W Š K Q; see [36], [20], Remark 4.2, and [111], Proposition 4.6. We take for f W C ! K Q; the composition of f 0 with the coalgebra isomorphism TC0 W Š K Q. It follows from the construction of f that the restriction of f to C1 is injective, and therefore f is injective (see [59], Theorem 5.3.1). Since obviously .K Q/1 D KQ0 ˚ KQ1 f .C1 / and dimK .KQ0 ˚ KQ1 / D dimK f .C1 / D dimK C1 , then .K Q/1 D KQ0 ˚ KQ1 D f .C1 /. The second part of (b) is a consequence of Proposition 7.18. (c) If f is an isomorphism then C Š KQ is hereditary, by Proposition 7.7. Conversely, assume that C is hereditary. By applying (b) and [14], Lemma 3 (or the arguments of Gabriel [36], Sections 7.5, 8.4, 8.5), we conclude that f is surjective; see also [15] and [43] for alternative proofs. 7.4 Hereditary coalgebras of tame comodule type. In this subsection we present a diagrammatic characterisation of hereditary K-coalgebras of tame comodule type over an algebraically closed field K given in [84] and [92]. Theorem 7.22. Assume that K is an algebraically closed field and C is a basic connected hereditary K-coalgebra. Let Q D CQ be the left Gabriel quiver of C , D .0 ; 1 / be the graph obtained from Q D .Q0 ; Q1 / by forgetting the orientation, and let qQ D q W Z.0 / ! Z be the quadratic form (7.1) of . (a) If dimK C is finite then the following three conditions are equivalent. (a1) The coalgebra C is of tame comodule type. (a2) The quadratic form qQ W Z.IC / ! Z of C is positive semi-definite, that is, qQ .v/ 0, for all vectors v 2 Z.IC / . (a3) The left Gabriel quiver CQ of C is acyclic and the graph is of one of the simply-laced Dynkin diagrams An , Dn , E6 , E7 , E8 or one of the simply-laced z n, D z n, E z 6, E z 7, E z 8 presented in Section 7.1. extended Dynkin diagrams A (b) If dimK C is infinite then the following four conditions are equivalent.


627

(b1) The coalgebra C is of tame comodule type. (b2) The coalgebra C is tame of discrete comodule type, that is, the number of the isomorphism classes of comodules M in C -comod, with lgth M D v, is finite for every vector v 2 N .IC / . (b3) The quadratic form qQ W Z.0 / ! Z of C is positive semi-definite, that is, q .v/ 0 for all vectors v 2 Z.0 / . (b4) The left Gabriel quiver CQ of C is of one of the types: infinite of any of the locally Dynkin forms A1 , 1 A1 , D1 presented in Proposition 7.2, where means ! or , (ii) CQ is the finite cycle of length n 1 (i)

CQ

0

1

◦

◦n −1 ◦ n −2

2

z n1 W A

(with n 1)

◦

◦

Proof. Without loss of generality, we may assume that C is of the form (3.1) because C is basic. Since Q D CQ D .Q0 ; Q1 / then Q0 D IC is the index set in the decompositions (3.1). (a) Assume that dimK C is finite. Then there is an equivalence C -comod Š dis.C / D mod C and C is of tame comodule type if and only if the finite-dimensional K-algebra is tame, by Theorem 6.7 (b). Then (a) is a consequence of the well-known result of Gabriel [35] and Nazarova [64], see [2] and [98]. (b) Assume that dimK C is infinite. Since K is algebraically closed and C is hereditary, basic and connected. Then, by Theorem 7.21 (a), there is a coalgebra lf isomorphism C Š K Q and an equivalence of categories C -comod Š nilrepK .Q/, where K Q is the path coalgebra of the quiver Q D CQ. By Corollary 4.11, the quiver Q D CQ is connected because C is connected. (b1) H) (b4): Assume also that C is of tame comodule type. If L is a convex connected finite subquiver of Q, the path coalgebra K L is a subcoalgebra of K Q and by, Theorem 6.7 (e), K L is of tame comodule type. If Q is infinite, then, by (a), any convex connected finite subquiver L of Q is a Dynkin quiver or an extended Dynkin quiver. Hence easily follows that Q is a locally Dynkin quiver of one of the types A1 , z z 1 A1 , D1 . If Q is finite, it is the finite cycle An1 of length n 1, where A0 is the one loop quiver, because dimK K Q is infinite. Hence the implications (b1) H) (b4) follows. The equivalence (b3) () (b4) is a consequence of Proposition 7.2.

628

D. Simson

(b4) H) (b2): Assume first that Q D CQ infinite. Then Q is of any of the locally Dynkin form A1 , 1 A1 , D1 . It follows that any convex finite subquiver of Q is a Dynkin quiver 2 fAn ; Dn ; E6 ; E7 ; E8 g. In view of the equivalences lf C -comod Š K Q-comod Š nilrepK .Q/ any indecomposable C -comodule M of a given length vector v (viewed as a representation of Q) is an indecomposable representation of a fixed Dynkin subquiver v containing the finite set supp.v/ D fj 2 IC D Q0 j .lgth M /j ¤ 0g. By a theorem of Gabriel, the number of such indecomposable representations is finite; hence C is tame of discrete comodule type. z n1 of length Next we assume that Q D CQ finite, that is, Q is the oriented cycle A z n 1. Note that the quiver universal cover of Q D An1 , n 1, is the infinite quiver zW ::: Q

/

2

/

1

/ 0

/ 1

/

2

/ :::

/

s1

/

s

/

sC1

/ :::

of type 1 A1 , with the action of the infinite cyclic group Z given by the shift j 7! j n. z n1 is isomorphic to the residue quiver Q=Z z Obviously, the cycle Q D A and there is a well-known push-down functor lf z lf f W nilrepK .Q/ ! nilrepK .Q/: lf z z because the quiver Q z is acyclic. Since any finiteNote that nilrepK .Q/ D repK .Q/, z of Q z is a representation dimensional indecomposable representation M 2 repK .Q/ z of finite subquiver of a Dynkin type Ap of Q then M is isomorphic to a finite interval representation of the form mIt

id

id

id

id

W 0 ! Km ! KmC1 ! ! K t1 ! Kt ! 0 ! ;

where 1 < m t < 1 and Kj D K, for all m j t , see [31]. Hence easily lf z follows that the Auslander–Reiten translation quiver .nilrepK .Q// of the category lf z nilrepK .Q/ is connected and has the form (compare with [52], [66], and [83]) :> : : S.4/ S.3/ S.2/ S.1/ S.0/ GGG EEE BBB BBB > > > < EEE }} y | | | } y | | | B || B || E" E" G# yy }} || ::: I I I I I 3 4 2 3 1 2 0 1 1 0 EEE CCC CCC CCC AA v; y< y< |= |= AA C! C! C! E" ||| ||| yyy yyy vvv A :> : : I I I I 3 I5H 2 4C 1 3C 0 2E 1 1E } = = = < CCC CCC EEE EEE HHH y | | | }} y | | | } y | | | } ! ! " " | | | y # ::: I I I I I 2 5 1 4 0 3 1 2 2 1 >> BB < ?? |> |> ? ??? ? ??? BB yy >> ?? ?? ?? |||| BB |||| yy > ? y | | ! y :: :: :: :: :: :: : : : : : :

:::

::: ,

where the upper row consists of the simple representations S.a/ D a Ia with a 2 Z. z is uniquely deIt follows that any indecomposable representation M in repK .Q/ z Š termined by its composition length vector lgth M , and so the category K Q-comod


629

z is tame of discrete representation type. By applying the push-down funcrepK .Q/ lf tor f , we show that the category C -comod Š K Q-comod Š nilrepK .Q/ is z equivalent to the Z-orbit category repK .Q/=Z and the Auslander–Reiten translation lf lf quiver .C -comod/ Š .nilrepK .Q// of the category C -comod Š nilrepK .Q/ is isomorphic to the tube T .n/ of rank n 1 (see [98], Chapter X) obtained from lf z the infinite quiver .nilrepK .Q// presented earlier by passing to the Z-orbit quiver lf z .nilrepK .Q//=Z; see [2], Chapter IX. For n D 3, the tube T .3/ of rank 3 has the shape C2 M1

C M1 M1

T .3/ W

.. .

C M2

C2 M2

M2

C2 M3

C M3

M3

C2 M4

M4

C M 4

.. .

.. .

.. .

where M1 D S.0/ Š f .S.0// Š f .S.3//, C M1 Š S.1/ Š f .S.1// Š f .S.4//, C2 M1 Š S.2/ Š f .S.2// Š f .S.5// are the simple comodules. Analogously, for arbitrary n 1, the mouth of the tube T .n/ of rank n consists of the simple comodules M1 D S.0/ Š f .S.0//; C M1 Š S.1/ Š f .S.1//; : : : ; Cn1 M1 Š S.n 1/ Š f .S.n 1//, where the simple C -comodule S.a/ Š f .a Ia / is identified with the one-dimensional simple representation of the cycle Q with the space K over the vertex a and the zero space over remaining vertices, for a D 0; 1; 2; 3; : : : ; n 1. It is easy to check that C S.n 1/ Š S.0/. It follows that the coalgebra C Š K Q is tame of discrete comodule type and completes the proof of (b4) H) (b2). Since the implication (b2) H) (b1) is obvious, the proof is complete. We finish this subsection by showing that the fc-tameness and K-tameness coincide, for acyclic hereditary K-coalgebras.

630

D. Simson

Proposition 7.23. Let K be an algebraically closed field and C a basic hereditary K-coalgebra such that the left Gabriel quiver CQ of C is acyclic and interval finite. Then C is of fc-tame comodule type and CQ is left locally finite if and only if C is of K-tame comodule type. Proof. To prove the necessity, assume that C is of fc-tame comodule type and CQ is left locally finite. By Proposition 7.7, all simple left C -comodules are finitely copresented and we have C -comod C -Comodfc . Hence, by Lemma 6.16 (b), fc-tameness of C implies the K-tameness of C . Conversely, assume that C -comod is tame. If dimK C is finite then C -comod D C -Comodfc Š mod.RP /, for some finitely generated projective comodule P , and the K-tame-wild dichotomy of Drozd [29] applies. Then, without loss of generality, we may suppose that the quiver CQ is connected and infinite. It follows from Theorem 7.22 that C is tame of discrete comodule type and CQ is one of the quivers 1 A1 , A1 , D1 . Moreover, C is isomorphic to the path K-coalgebra K Q of the quiver Q D CQ. It follows that C is computable and CQ is left locally finite; hence Corollary 6.27 applies to C . P Let U be any finite convex subquiver of Q and consider the idempotent eU D u2U eu in the path algebra L KQ of Q. We view KQ as the subalgebra of C Š .K Q/ . Let E D EU D u2U E.u/. It follows form [90], Section 2, that CE D eU C eU is a coalgebra and there are coalgebra isomorphisms CE Š eU .K Q/eU Š K U , see Example in [90], p. 460. Since U is a finite convex subquiver of Q then U is a Dynkin quiver and therefore the coalgebra CE Š K U is finite-dimensional of finite comodule type. Then, by [90], Proposition 2.7, the algebra RE D EndC E Š CE is finite-dimensional and, according to [90], Proposition 3.3, there is an equivalence of E categories C -ComodE fc Š CU -comod Š mod.RE /. Then the category C -Comod fc is of finite representation type and, hence, is fc-tame. It follows that, for every socle-finite direct summand E of C C , the algebra RE D EndC E is representation-finite. Consequently, C is fc-tame of discrete comodule type, by Corollary 6.27. This completes the proof. 7.5 Bound quivers, profinite bound quivers, and their path coalgebras. Following Gabriel [37], 4.2, a bound quiver (or quiver with relations) is a pair .Q; /, where Q is a quiver (not necessarily finite) and is a two-sided ideal of the path K-algebra KQ such that KQ2 . Every such an ideal is called an ideal of relations, or a relation ideal fl of KQ. If .Q; / is a quiver with relations, we define repK .Q; / repK .Q; / fl fl nilrepK .Q; / to be the corresponding full subcategories of repK .Q/ repK .Q/ fl nilrepK .Q/ formed by the K-linear representations of Q satisfying all relations in , see [37], 4.2, and [2]. Definition 7.24 ([84]). Let K be a field and .Q; / a bound quiver. A path K-coalgebra of a .Q; / is the subcoalgebra K .Q; / ´ fa 2 K Q j ha; i D 0g

(7.25)


631

of the path K-coalgebra K Q, where h; i W KQ KQ ! K is the standard nondegenerate symmetric K-bilinear form defined by the formula hu; wi D ıu;w for all paths u; w in Q. Here ıu;w is the Kronecker delta. Definition 7.26 ([85]). A relational subcoalgebra (or admissible subcoalgebra [20]) of a path K-coalgebra K Q is any subcoalgebra H of K Q satisfying the following two conditions. (a) The subcoalgebra .K Q/1 D KQ0 ˚ KQ1 of K Q is contained in H . L (b) H D a;b2Q0 H.a; b/, where H.a; b/ D H \ KQ.a; b/, compare with [37], 4.2. Given a quiver Q and a field K, a connection between relation ideals of the path algebra KQ and relational subcoalgebras of the path coalgebra C D K Q is described as follows. Theorem 7.27. Let K be a field, .Q; / a bound quiver. (a) The subspace K .Q; / D ? in (7.25) of K Q is a basic relational subcoalgebra of K Q and the Gabriel quiver H Q of H D K .Q; / is isomorphic to Q. (b) The category equivalences (7.19) restrict to the category equivalences K .Q; /-Comod O ? K .Q; /-comod

'

/ Replnfl .Q; / K O

F

' F

?

(7.28)

/ nilrepfl .Q; /. K

(c) If H is a relational subcoalgebra of the path coalgebra K Q, then H ? D fa 2 KQ j hH; ai D 0g is a two-sided relation ideal of the path algebra KQ. (d) If Q is interval finite and acyclic, the map 7! K .Q; / defines a bijection between the set of relation ideals of the path K-algebra K Q and the set of relational subcoalgebras H of the path coalgebra K Q. The inverse is given by H 7! H ? . (e) If K is algebraically closed, C a basic K-coalgebra such that the Gabriel quiver Q D CQ of C is locally finite and acyclic and f W C ! K Q is the coalgebra injection of Theorem 7.21, then f .C / is a relational subcoalgebra of the path coalgebra K Q, the space D f .C /? is a relation ideal of the path K-algebra KQ, and the homomorphism f defines a coalgebra isomorphism C Š K .Q; /. Proof. We refer to [85], Theorems 3.14 and 4.9. Remarks 7.29. (a) In [44] an example is given of a basic wild coalgebra C such that the Gabriel quiver Q D CQ of C is acyclic, not locally finite, and C is not of the form K .Q; /. Moreover, a criterion is given allowing us to decide whether or not a relational subcoalgebra of K Q is of the form K .Q; /.

632

D. Simson

(b) It is shown in [45] that any tame relational subcoalgebra of K Q is of the form K .Q; / if Q is acyclic. (c) The problem whether or not any tame relational subcoalgebra of K Q is of the form K .Q; / remains open.

The answer to (c) is affirmative if we extend the definition of bound quiver as follows, see [88]. Definition 7.30 ([88]). (a) A profinite bound quiver (or a quiver with profinite relations) is a pair .Q; B/, where Q is a quiver and B is a closed two-sided ideal (called a profinite c Š .K Q/ in (7.11) such that relation ideal) of the pseudocompact K-algebra KQ B KQ2 . (b) A path K-coalgebra of a profinite bound quiver .Q; B/ is the subcoalgebra

1

K .Q; B/ D fc 2 K Q j hB0 ; ci D 0g K Q

(7.31)

of the path K-coalgebra K Q, where h; i W .K Q/ K Q ! K is the nondegenerate symmetric K-bilinear form defined by the formula h'; ci D '.c/, and c under the isomorphism KQ c Š .K Q/ in (7.11) of B0 is the image of B KQ pseudocompact K-algebras. (c) A K-coalgebra C is defined to be a bound quiver coalgebra if there are a bound quiver .Q; / and a coalgebra isomorphism C Š K .Q; /; and C is defined to be a profinite bound quiver coalgebra if there is a profinite bound quiver .Q; B/ and a coalgebra isomorphism C Š K .Q; B/. The following result extends Theorem 7.27. Theorem 7.32. Let K be a field, Q a quiver and .Q; B/ a profinite bound quiver. (a) The subspace K .Q; B/ in (7.31) of the path K-coalgebra K Q is a basic relational subcoalgebra of K Q such that the Gabriel quiver H Q of H D K .Q; B/ is isomorphic to Q. (b) The category equivalences (7.19) restrict to the category equivalences '

K .Q; B/-Comod O ? K .Q; B/-comod

F

' F

/ Replnfl .Q; B/ K O ?

(7.33)

/ nilrepfl .Q; B/. K

(c)The map B 7! K .Q; B/ defines a bijection between the set of profinite relation c and the set of relational subcoalgebras H of ideals B of the profinite K-algebra KQ the path coalgebra K .Q; B/. The inverse map is given by H 7! H ? . (d) If K is an algebraically closed field, C a basic K-coalgebra, and Q D CQ is the c Š .K Q/ left Gabriel quiver of C , then there exist a profinite relation ideal B of KQ and a coalgebra isomorphism C Š K .Q; B/. For the proof we refer to [88], Section 4.

633


7.6 String coalgebras. Following [9], we introduce a special class of path K-coalgebras K .Q; /, called string coalgebras (see also [16]), and we show that string coalgebras are of tame comodule type. Definition 7.34 ([84], [85]). A string K-coalgebra is the path K-coalgebra K .Q; / of a bound quiver .Q; / satisfying the following conditions. (a) Any vertex of Q is a source of at most two arrows and is a terminus of at most two arrows. (b) The ideal of the path K-algebra KQ is generated by a set 0 of zero relations, that is, the elements of 0 are paths of length 2. (c) Given an arrow ˇ W i ! j in Q, there is at most one arrow ˛ W s ! i and at most one arrow W j ! r in Q such that ˛ˇ 62 0 and ˇ 62 0 . The following lemma shows that string coalgebras have “biserial properties”. Lemma 7.35 ([16]). Assume that C D K .Q; / is a string coalgebra. Then C is pointed and, given a simple comodule S.a/, the injective envelope E.a/ of S.a/ is a uniserial comodule or is the sum of two uniserial comodules. Proof. See [16], Section 1.1. We recall from [84], Section 6, that a vector v 2 K0 .C / Š Z.IC / has a finite-dimensional support subcoalgebra Hv of C if every indecomposable C -comodule M in C -comod, with dim M D v, lies in the category Hv -comod C -comod, that is, indv .Hv -comod/ D indv .C -comod/. The following theorem collects the main properties of string coalgebras. Theorem 7.36 ([84], [85]). Let K be an algebraically closed field and C D K .Q; / a string K-coalgebra. (a) C is a graded subcoalgebra of K Q in (7.25) and has the form

C D KQ0 ˚ KQ1 ˚ KQ

2 ˚ KQ3 ˚ ˚ KQm ˚ ;

(7.37)

is the subset of Qm consisting of the oriented paths ! of length m that do where Qm not contain any subpath from . fl (b) There is a K-linear category isomorphism K .Q; /-comod Š nilrepK .Q; /. (c) Every vector v 2 K0 .C / admits a finite dimensional support string subcoalcv is the y v / of C , where Qv is a finite convex subquiver of Q and gebra C.Qv ; v v v v v ideal of the path algebra KQ generated by C KQ0 C KQ1 C C KQkvk1 and P kvk D j jvj j. (d) Every indecomposable K .Q; /-comodule L of finite K-dimension, viewed as fl .Q; /, is a string representation M.u/ or a band representation an object of nilrepK M.u; '/ (in the terminology of [9]), and L is annihilated by all paths ! of in Q of length kvk D dimK L. (d) The coalgebra K .Q; / is of tame comodule type.

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Proof. See [85], Theorem 6.2, and [84], Proposition 6.14. An alternative proof is given in [16]. Remarks 7.38. (a) In [16], a relation between string coalgebras, special biserial coalgebras, their indecomposable comodules and Auslander–Reiten quivers is discussed. In particular, it is shown there that any special biserial coalgebra is a relational coalgebra. (b) The classification of indecomposable comodules over string coalgebras and the tameness is applied [12] and [16] to get a classification of indecomposable modules of quantum SLq .2/ and comodules of Uq .sl3 /. The problem leads to the classification of indecomposable comodules over a string coalgebra K .Q; /, where Q is the multiple //

//

//

// : : : Kronecker quiver : : :

8 Euler coalgebras, Cartan matrices and Euler characteristic In this section we study a class of pointed computable Ext-finite coalgebras C (with an additional property), called left Euler coalgebras, by means of the Cartan matrix CC 2 MIC .Z/ of C , the Euler Z-bilinear form bC W Z.IC / Z.IC / ! Z in (8.8), the Euler characteristic C .M; N / D

1 P

.1/j dimK Ext jC .M; N /;

(8.1)

j D0

and the defect @C .M; N / 2 Z, defined in [90] and [97], for any computable Euler pair .M; N / of left C -comodules. Following [90] and [97], we show that bC .dim M; dim N / D C .M; N / C @C .M; N / and the defect @C .M; N / vanishes for all M , N of finite K-dimension. 8.1 The Cartan matrix of a pointed computable coalgebra. Throughout this section we assume that K is a field and C is a pointed (hence, basic) computable Kcolagebra of the form (3.1). We recall that a left C -comodule is said to be computable if its dimension vector dim M D ŒdimK M ej j 2IC ; with dimK M ej , has values in N. Since C is pointed, we have dimK M ej D dimK HomC .M; C ej / D dimK HomC .M; E.j // and dim M coincides with the composition length vector lgth M D Œ`j .M /j 2IC of M . A pointed coalgebra C is computable if the injective comodule E.i / D C ej is computable, or equivalently, if the dimension vector e.i/ D dim E.i / D ŒdimK ei C ej j 2IC D ŒdimK HomC .E.i /; E.j //j 2IC has finite coordinates for every i 2 IC .

(8.2)


635

Definition 8.3 ([90]). (a) Given a pointed computable K-coalgebra C , with soc C C D L j 2IC S.j /, we define the left Cartan matrix of C to be the integral IC IC matrix 3 :: : 7 6 7 2 MI .Z/; e.i / D6 C 5 4 :: : 2

CC D Œcij i;j 2IC

(8.4)

whose (i j )-entry is the composition length multiplicity cij D e.i /j D dimK ei C ej of S.j / in E.i /. In other words, the i -th row of CC is the dimension vector e.i / D dim E.i / of E.i /. (b) We say that a row (or a column) of a matrix is finite if the number of its non-zero coordinates is finite. A matrix is called row-finite (or column-finite) if each of its rows (columns) is finite. Lemma 8.5. Let C be a pointed computable K-coalgebra and let CC 2 MIC .Z/ be the left Cartan matrix (8.4) of C . (a) CC op D CtrC , that is, CC op 2 MIC .Z/ is the transpose of the matrix CC . (b) The i -th row of the matrix CC is finite if and only if the indecomposable injective left C -comodule E.i / is finite-dimensional. (c) The j -th column of the matrix CC is finite if and only if the indecomposable y / D rC .E.j // D ej C in (2.26) is finite-dimensional. injective right C -comodule E.j (d) The matrix CC of C is row-finite if and only if C is right semiperfect. (e) The matrix CC of C is column-finite if and only if C is left semiperfect. Proof. See [21] and [90]. 8.2 Euler bilinear form of an Euler coalgebra and Euler characteristic Definition 8.6 ([90]). A pointed K-coalgebra C as in (3.1) is defined to be a left Euler coalgebra if C has the following three properties: (a) C is computable, (b) Extm C .S.i /; S.j // is zero, for m sufficiently large, and (c) every simple left C -comodule S.j / admits an injective resolution .j / 0 ! S.j / ! E0.j / ! E1.j / ! ! En.j / ! EnC1 ! .j / such that Em is socle-finite for m 0 and for each i 2 IC there exists mj i 0 with HomC .Er.j / ; E.i // D 0 for all r mj i .

Note that the multiplication in the matrix algebra MIC .Z/ is not associative. The matrices in MIC .Z/ may have unequal left and right inverses and that one-sided inverse of a matrix may not be unique. Moreover, the left inverse may exist, without a right

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inverse existing. Also being invertible as a Z-linear map is not equivalent to being invertible as a matrix, see [109]. The existence of the inverses the Cartan matrix CC of a left Euler coalgebra C is established in the following theorem proved in [90] and [97]. Theorem 8.7 ([90], [97]). Let C be a pointed 2 3 left Euler K-coalgebra. :: :7 6 (a) The matrix FC D Œ`jp i;p2IC D 4 `j 5 2 MIC .Z/, with `jp D C .S.p/; S.j // :: : and `j D Œ: : : ; `jp ; : : : p2IC , is a left inverse of the Cartan matrix CC in MIC .Z/. (b) If j 2 IC is such that inj.dim S.j / is finite then the j -th row `j of the matrix FC has at most finitely many non-zero entries. 0 (c) If C is both left and right Euler coalgebra and the equality dimK Ext m C .S; S / D m 0 dimK ExtC op .DS ; DS/ hold, for all m 0 and all simple left C -comodules S and S 0 , where D W C -comod ! C op -comod is the duality (2.25), then FC is also a right inverse of CC , which is both row-finite and column-finite, and we set C1 C ´ FC . Following [89], Section 4, and [97], (3.3), we define the Euler bilinear form bOC .v; w/ ´ v .FCtr w tr / D v .w FC /tr ;

(8.8)

for any pair of vectors v; w 2 ZIC such that the multiplication v .FCtr w tr / D v .w FC /tr is a well-defined integer. We set bC .v; w/ D bOC .v; w/ for v; w 2 Z.IC / . Definition 8.9 ([97]). (a) The pair M , N in C -Comod is called a computable Euler pair if: (i) M and N are computable, (ii) the minimal injective resolution of N is socle-finite and computable, and (iii) there exists a minimal integer m0 0 (denoted by N deg.M; N /) such that Ext m C .M; N / D 0 and Hom C .M; Em / D 0 for any m 1Cm0 , N where Em is the m-th term of a minimal injective resolution of N . (b) Assume that M , N is a computable Euler pair such that the multiplication .dim M / ŒFCtr .dim N /tr is defined. The Euler defect of M , N is the integer @C .M; N / D .1/m0 C1 bOC .dim M; dim m0 C1 N /;

(8.10)

where m0 D deg.M; N / 0 and m0 C1 N is the .m0 C 1/-cosyzygy of N . The main result of this subsection is the following theorem proved in [90], [97]. Theorem 8.11 ([90], [97]). Let C be a pointed left Euler K-coalgebra, CC 2 MIC .Z/ the left Cartan matrix of C , and FC the left inverse of CC given in Theorem 8.7 (a). Assume that CC FC D E and M , N is a computable Euler pair of comodules in C -Comod such that .dim M / ŒFCtr .dim N /tr is defined. (a) If C .M; N / is the Euler characteristic (8.1) of the pair .M; N / then bOC .dim M; dim N / D C .M; N / C @C .M; N /: (b) If any of the conditions

(8.12)

637


(b1) both comodules M and N are finite-dimensional, (b2) CC FC D E, dimK M is finite, and dimK N is infinite, (b3) CC FC D E, dimK M is infinite, and inj.dim N is finite, is satisfied, we have @C .M; N / D 0 and the equality bOC .dim M; dim N / D C .M; N / holds. Remark 8.13. We do not know if there exists a computable Euler pair M , N of indecomposable socle-finite comodules over a left Euler pointed K-coalgebra C , with CC FC D E, such that the Euler defect @C .M; N / is non-zero. It follows from Theorem 8.11 (b) that if M , N is such a pair, then dimK M D 1 and inj.dim N D 1. 8.3 Incidence coalgebras of interval finite posets. In this section we study comodules of the incidence coalgebra K I of interval finite posets I of left locally finite width by means of the Cartan matrix CK I and the Euler Z-bilinear form bK I W Z.I / Z.I / ! Z of K I . We show that any such a coalgebra K I is pointed, left artinian, left Euler coalgebra, and the defect @K I of K I vanishes. Moreover, the tame-wild dichotomy 6.10 holds for K I if K is algebraically closed. Infinite posets I such that K I is tame of discrete comodule type are listed. Throughout this section we assume that I .I; / be a poset, that is, I is a partially ordered set with respect to the partial order relation ; see [80]. We write i j if i j and i ¤ j . We say that I is connected if I is not a disjoint union I 0 [ I 00 of two subposets I 0 and I 00 , with all pairs i 0 2 I 0 and i 00 2 I 00 incomparable in I . By the width w.I / of I we mean the maximal number of pairwise incomparable elements of I if it is finite; otherwise we set w.I / D 1. We recall that the Hasse quiver of I is the quiver QI D .Q0I ; Q1I /, where Q0I D I is the set of points of QI ; and there is a unique arrow p ! q from p 2 I to q 2 I in Q1I if and only if p q and there is no t 2 I such that p t q. We say that I is of left locally finite width (resp. of right locally finite width) if given b 2 I , the subposet D b D fj 2 I j j bg of I , called the left cone at b (resp. b E D fj 2 I j b j g; called the right cone at b), is of finite width, that is, has no infinitely many pairwise incomparable elements. A subposet I 0 of I is defined to be convex, or interval closed, if given a b in I 0 , the interval Œa; b D fs 2 I j a s bg D aE \ D b is contained in I 0 . Finally, we say that I is interval finite if the interval Œa; b is finite for all a b in I . We visualise the cone E b and the interval Œa; b D aE \ D b, with a b, as follows: ... D

b: ...

◦ ◦ .. . ◦

◦ ◦b

[a, b] :

a◦

◦

◦ ◦ .. . ◦

◦b

638

D. Simson

The relation of the poset I is uniquely determined by the incidence matrix CI of I , that is, the integral square I I matrix (see [80]) ´ 1 for i j; CI D Œcij i;j 2I 2 MI .Z/; with cij D (8.14) 0 for i 6 j; where Z is the ring of integers and the abelian group MI .Z/ D fc D Œcpq p;q2I 2 MI .Z/ j cpq D 0 if p 6 qg

(8.15)

is viewed as a partial subalgebra of MI .Z/. If I is interval finite, then, given two matrices 0 D Œ0ij and 00 D Œ00ij in MI .Z/, 0 00 D Œab a;b2I , with ab D P P 0 00 0 00 j 2I aj jb D aj b aj jb , is a well-defined matrix lying in MI .Z/. Hence, MI .Z/ is an associative K-algebra and the matrix E, with the identity 1 on the main diagonal and zeros elsewhere, is the identity of MI .Z/. Given a non-empty set I , we denote by MI .K/ the set of all square I by I matrices D Œpq p;q2I , with coefficients pq 2 K. The set MI .K/ is equipped with the usual the K-vector space structure and the usual (partial!) matrix multiplication (which is not associative and 0 is not defined for all ; 0 2 MI .K/ if I is infinite). We denote by MI .K/ MI .K/ the associative matrix K-algebra consisting of all matrices D Œpq 2 MI .K/ such that pq D 0 for all but a finite number of indices p; q 2 I . Definition 8.16 ([94]). Let I .I; / be an interval finite poset (not necessarily finite) and K a field. (a) The incidence K-algebra of I is the associative algebra

KI D f D Œpq 2 MI .K/ j pq D 0 if p 6 qg MI .K/:

(8.17)

(b) The complete incidence K-algebra is the associative unitary K-algebra ´ f D Œpq 2 MI .K/ j pq D 0 if p 6 qg; KI

(8.18)

with the matrix multiplication defined as in MI .Z/, see (8.15). It is easy to see that KI is an associative K-subalgebra of MI .K/, and the matrix units epq , with p q, having the identity in the .p; q/ entry and zeros elsewhere, form a K-basis of KI. Given j 2 I , the matrix unit ej D ejj 2 KI is a primitive idempotent of the K-algebra KI and fej gj 2I is a complete set of pairwise orthogonal primitive idempotents of KI. Obviously, the algebra KI has an identity element if and only if I is a finite poset. In this case we have KI D KI. Example. Let I D Z be the set of integers, equipped with the linear order of Z. Then the Hasse quiver QI of I has the form QI W ! 2 ! 1 ! 0 ! 1 ! 2 ! ! r C 1 ! r C 2 ! r C 3 ! :


639

The incidence K-algebra KI of the poset I D Z consists of the upper triangular matrices consists of the upper triangular 2 MZ .K/, the complete incidence K-algebra KI matrices 2 MZ .K/, and the incidence matrix CI D Œcpq 2 M Z .Z/ of I has the identities on the main diagonal and over it, and has cpq D 0 for p > q. Definition 8.19. Let K be a field and let I be an interval finite poset. The incidence K-coalgebra of I is the triple K I D .KI; I ; "I /;

(8.20)

where KI is the incidence K-algebra of I , and the counit "I W KI ! K and the comultiplication I W KI ! KI ˝ KI are defined by the formulae ´ P rl0 for p ¤ q; ept ˝ e tq ; "I .epq / D I .epq / D 1 for p D q: ptq Since I is interval finite, the K-linear map I is well-defined. We note that dimK K I @0 if the poset I is connected. Remark 8.21. Incidence coalgebras were defined by Sweedler [103]. In [46] and [47], Joni and Rota explain how incidence coalgebras provide a suitable framework for an interpretation of some combinatorial problems in terms of coalgebras. To get a description of the coalgebra K I as a path coalgebra of a bound quiver, we consider the Hasse quiver QI of I and note that the K-algebra homomorphism KQI ! KI associating to any arrow p ! q of QI the matrix unit epq 2 KI induces a K-algebra isomorphism KQI = I Š KI, where I is the two-sided ideal of the path K-algebra KQI of QI generated by all commutativity relations, that is, by all differences w 0 w 00 2 KQI of paths w 0 ; w 00 of length m 2 with a common source and a common terminus, see [2], Chapter II, and [80], Chapter 14. Recall that K .QI ; I / D I? D fu 2 KQI j hu; I i D 0g K QI

(8.22)

is the path K-coalgebra of the bound quiver .QI ; I /; see (7.25). We call .QI ; I / the Hasse bound quiver of the poset I , and view K .QI ; I / as a relational subcoalgebra of K QI . We show that there is a coalgebra isomorphism ' W K I ! K .QI ; I /

(8.23)

fl and an equivalence K I -Comod Š RepK .I /. To define , we consider the space ?

I .p; q/ D fu 2 KQI .p; q/ j hu; I i P D 0g for any pair p; q 2 I . It is easy to check ! is the sum of all oriented paths ! in QI that I? .p; q/ D K eOpq , where eOpq D starting in p and ending with q.PIt then follows that the K-linear map defined by attaching to epq the sum eOpq D ! is a K-coalgebra isomorphism.

640

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Remark. Usually we study the comodule category K I -Comod by means of K-linear representations of I (equivalently, representations of .QI ; I /), which are the systems X D .Xp ; q 'p /pq , where Xp is a K-vector space for each p 2 I , q 'p W Xp ! Xq is a K-linear map for all p q, and s 'q B q 'p D s 'p for all p q s. A morphism f W X ! X 0 is a system f D .fp /p2I of K-linear maps fp W Xp ! Xp0 such that 0 q 'p B fp D fq B q 'p for p q. We denote by RepK .I / Š RepK .QI ; I / the Grothendieck K-category of Klf .I / the abelian full linear representations of the poset I , and by repK .I / repK subcategories of RepK .I / formed by finitely generated representations and by finitely generated representations of finite length, respectively. lf Finally, we denote by RepK .I / the full Grothendieck subcategory of RepK .I / lf formed by locally finite representations, that is, directed unions of objects from repK .I /; fl fl and by nilrepK .I / the full subcategory of repK .I / formed by all nilpotent representalnfl lf tions of finite length, and by RepK .I / the full subcategory of RepK .I / formed by all fl fl .I / D repK .I / locally nilpotent representations. Since I is a poset, we have nilrepK lnfl lf and, hence, RepK .I / D RepK .I /; see [94] for details. Then the category equivalences (7.28) restrict to the category equivalences K I -Comod O ? K I -comod

' F

' F

/ Replnfl .I / D Repfl .I / Š Repfl .QI ; I / K K O K ? / nilrepfl .I / D repfl .I / Š repfl .QI ; I /. K K K

(8.24)

Basic properties of coalgebras K I are collected in the following propositions. Proposition 8.25 ([91], [94]). Let K be a field and let K I be the incidence Kcoalgebra (8.17) of a connected and interval finite poset I . (a) K I is pointed, connected, computable, and dimK K I @0 . Moreover, given a pair of indecomposable injective left (or right) K I -comodules E, E 0 , every non-zero homomorphism E ! E 0 is surjective. (8.18) of I is the completion of KI in the (b) The complete incidence algebra KI Š .K I / of pseudocompact Kcofinite topology, and there is an isomorphism KI algebras. (c) G.K I / D fej j j 2 I g and, given j 2 I , SI .j / D ej .KI/ ej Š Kej is a simple left coideal (and a subcoalgebra) of K I , the left ideal EI .j / D KI ej of the path K-algebra KI is a left coideal (and a summand) of K I , with soc EI .j / D SI .j / and EndK I SI .j / Š EndK I EI .j / Š K. (d) The dimension vector dim EI .j / of the injective left ´ K I -comodule EI .j / 1 if p j; has the form e.j / ´ dim EI .j / 2 ZI , where e.j /p D and we 0 if p 6 j; ´ qp Kepq if p q; and the left K I -comodule have HomK I .EI .q/; EI .p// ! ' ' 0 if p 6 q;


641

decompositions soc K I D

L j 2I

SI .j / and K I D

L j 2I

EI .j /;

(8.26)

(e) The incidence matrix CI 2 MI .Z/ of I has a unique two-sided inverse CI1 in MI .Z/ such that, given a pair a b in I , the restriction CI1 jŒa;b 2 M .Z/ of Œa;b 1 the matrix CI to the interval Œa; b is the inverse of the restriction CŒa;b 2 MŒa;b .Z/ of CI to the finite subposet Œa; b of I . (f) CK I D CItr , that is, the Cartan matrix CK I of K I is the transpose of the incidence matrix CI . Moreover, CItr is a unique two-sided inverse of CK I . Examples. (a) Let I D Z be the set of integers, equipped with L the linear L order of Z, as in the previous example. Then SI .j / D Kej , EI .j / D sj sij Kesi , dim SI .j / D ej , e.j / ´ dim EI .j / D .: : : ; 1; 1; 1j ; 0; 0; 0; : : : / 2 ZZ , for any j 2 I where 1j D 1 is the j -th coordinate of dim EI .j /, and 2

6 : 6 6 6: : : 6 6: : : CI D 6 6: : : 6 6: : : 6 6: : : 4 2

CI1

:: : 1 1 0 0 0 0 :: :

::

::

6 : 6 6 6: : : 6 6: : : D6 6: : : 6 6: : : 6 6: : : 4

1 0 0 0 0 0 :: : ::

:: : 1 1 1 0 0 0 :: :

: 1 1 0 1 0 0 0 0 0 0 0 0 :: :: : :

:: : 1 1 1 1 0 0 :: :

:: : 1 1 1 1 1 0 :: :

:: : 1 1 1 1 1 1 :: :

:: : 1 1 1 1 1 1 :: :

3 7 : : :7 7 : : :7 7 : : :7 7; : : :7 7 : : :7 7 : : :7 5

:: :: :: :: :: : : : : : 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 :: :: :: :: : : : : : : :

3 7 : : :7 7 : : :7 7 : : :7 7: : : :7 7 : : :7 7 7 5 :: :

One can show that the Grothendieck group K0 .K I -Comodfc / of the category K I -Comodfc is free, with Z-basis fŒEI .j /gj 2Z , that the Grothendieck group K0 .K I -comod/ is free, with Z-basis fŒSI .j /gj 2Z , and that there is a monomorphism of abelian groups K0 .K I -comod/ ! K0 .K I -Comodfc / defined by ŒSI .j / 7! ŒEI .j / ŒEI .j 1/.

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(b) Let C D K I be the incidence K-coalgebra of the poset /0 /2 /4 / :> : :H / 2r / :: : : I W : : :CC / = 4 D / = 2 @ II uu DD zz CC {{ @@ ~~> === @ === @ @@@ ~~~ HHHHvvv: I u Iu ~@~@ {{CC zzDD == == uuII vvHH ~~@@ uu II$ {{ C! zz D! ~~ @ = = ~~ @@ vvv H$ { / 3 / 1 /1 /3 /5 / ::: / 2r C 1 / :::. :::

We set e.b/ ´ dim EI .b/. By [94], Theorem 5.7, there is a group isomorphism ' dim W K0 .K I -Comodfc / !

L a22 ZC1

Z ea ˚

L

Z e.b/ ZI ;

b22 Z

that is, the Grothendieck group K0 .K I -Comodfc / is free and the elements ŒSI .a/, with a 2 Z odd, and ŒEI .b/, with b 2 Z even, form a Z-basis of K0 .K I -Comodfc /. Proposition 8.27 ([94]). Let K be a field and let K I be the incidence K-coalgebra (8.17) of a connected and interval finite poset I . (a) The coalgebra K I is hereditary if and only if the Hasse quiver of the left cone D b is an oriented tree, for every b 2 I . (b) The coalgebra K I is left semiperfect if and only if the right cone b E is finite, for every b 2 I . (c) The coalgebra K I is left perfect [24] if and only if the right cone b E is finite and there is no infinite path ! b4 ! b3 ! b2 ! b1 ! b in the Hasse quiver of I for every b 2 I . Proof. The statement (a) follows from the description of minimal injective resolutions of simple comodules in K I -Comod given in [94], Section 5. The statements (b) and (c) are consequence of [24], Theorem 5.6, and [94], Proposition 5.2. Theorem 8.28 ([94]). Let K be a field, I a connected interval finite poset I that is of left locally finite width, and let K I be the incidence K-coalgebra (8.17). (a) The coalgebra K I is locally left artinian and left cocoherent. (b) K I is a left Euler coalgebra and the left Cartan matrix CK I of K I has a 1 tr unique two-sided inverse CK given in Proposition 8.25 (e). I D CI (c) The category K I -Comodfc is abelian and coincides with the category of artinian left K I -comodules. It is closed under taking extensions, contains the categories K I -comod and K I -inj, and every comodule N in K I -Comodfc has an injective resolution in K I -Comodfc . (d) The defect @K I .M; N / 2 Z vanishes and equality (8.12) reduces to bK I .dim M; dim N / D K I .M; N / for all M , N in K I -comod, where K I .M; N / is the Euler characteristic (8.1).

643


(e) The tame-wild dichotomy (6.10) holds for the incidence K-coalgebras K I if the field is algebraically closed. Proof. For the proof of (a)–(d) we refer to [94], Section 5. Now we outline the proof of (e). Since K I is computable and K I -Comodfc contains K I -comod, the fc-tameness of K I implies the tameness of K I , by Lemma 6.16. Conversely, by Theorem 6.7 (e), the tameness of K I implies the tameness of the finite-dimensional coalgebra K U , for any convex finite subposet U of I , because K U is a subcoalgebra of K I . Hence we conclude, that the finite-dimensional K-algebra REU D EndK I EU is tame for every L such U where EU D j 2U EI .j /. Then, in view of Corollary 6.27, the coalgebra K I is of fc-tame comodule type. This shows that K I is fc-tame comodule type if and only if K I is tame comodule type; hence (e) is a consequence of the fc-tame-wild dichotomy Theorem 6.25. Now we give a description of infinite posets I such that the coalgebra K I is tame of discrete comodule type. The reader is referred to [91] for a proof. Theorem 8.29 ([91]). Assume that I is a connected interval finite poset, K an algebraically closed field, and K I is the incidence coalgebra of I . Let qNI W Z.I / ! Z be the Euler quadratic form of the poset I defined by the formula qNI .x/ D x CI1 x tr , see [94], [96]. If the poset I is infinite, the following conditions are equivalent. (a) The coalgebra K I is tame of discrete comodule type. (b) The Euler quadratic form qNI W Z.I / ! Z of I is weakly positive. (c) The coalgebra K I is left representation-directed, see [91]. (d) Given a finite and convex subposet U of I , the incidence K-algebra KU of U is representation-finite and U is a subposet of one of the representation-finite Loupias–Zavadskij–Shkabara posets presented in [31]. (e) The poset I has one of the following two properties: (e1) gl.dim K I D 1 and I is one of the locally Dynkin posets A1 , presented in Proposition (7.2), and (e2) gl.dim K I D 2, I contains a subposet isomorphic to

% &

& %

,

1 A1 , 1 D1

and I , or the

poset opposite to I , is a subposet of any of the following three posets: 1 DAn W

5 4 3 2 1 1 2 x

! x

? ? ? ?

! ! ! !

0 3 n1 n

644

D. Simson

1 DD5 W

6 5 4 3 2 1 1 2 x

!

? ? ? ? y 0 ! ! 6 5 & %

!

3 4 1 DD1 W

6 5 4 3 2 1 ?

! ! ?

? ? y y

! ! : 0 1 2 s sC1 where n 3 and means ! or ! . A characterisation of incidence coalgebras of tame comodule type is given in the following theorem by means of the reduced Euler form P 2 P P qI .x/ D (8.30) xi xi xj C cij xi xj i2I

i!j

iJj

of I [96], (3.4), where cij D cij is the .i; j / entry of the matrix CI1 D Œcij 2 MI .Z/ inverse to CI 2 MI .Z/, and the relation a J b holds in I if a b and there are two 0

00

00

0

pairs .a ; b / and .a ; b / of incomparable elements in I such that

a

% &

a0 b 0

a00 b 00

& %

b.

Theorem 8.31. Assume that I is a connected interval finite poset, K an algebraically closed field, K I the incidence coalgebra in (8.17), and qI W Z.I / ! Z the reduced Euler form (8.30) of I . If I does not contain a poset of the form : : : _?? ?O _?? ?O _?? ?O _?? ?? ?O _??? ?O ?? ?? ?? ?? bbb?b?bb0 ?? ?? ?? ?? b?b?bbbbbbbbbb?b? b ? ? b b b b b b b b b b b z m W bbb

:::

with m 2; A 1

2

3

m1

m

then the following three conditions are equivalent. (a) The coalgebra K I is of tame comodule type. (b) The coalgebra K U is tame, for any finite convex subposet U of I . (c) The quadratic form qI W Z.I / ! Z of I is weakly non-negative. Proof. Apply [96], Theorem 1.5, [96], Proposition 4.2, and its proof, and follows the proof of Theorem 8.28 (e).


645

The following example shows that Theorem 8.31 is not valid for posets I that z m , with m 2. contain a subposet A Example 8.32. The reduced Euler form qJ W Z7 ! Z of the poset

TTTTT j/4

TTTTjjjjjjj 3 jjjTTTTTT TT*/ jjjj / j j

77

C 1 7 6 7 77 7 /

2

4

5

can be viewed as follows:

qJ .x/ D x12 C x22 C x32 C x42 C x52 C x62 C x72 .x1 C x2 /x3 .x1 C x5 /x4 x1 x6 C .x1 x2 x5 x6 /x7 D .x1 12 x4 12 x5 12 x6 C 12 x7 /2 C .x2 21 x3 12 x7 /2 C C

5 .x 25 x5 45 x6 C 15 x7 /2 12 3 3 .x 12 x6 12 x7 /2 C 14 .x6 5 5

C 34 . 13 x3 C x4 23 x5 31 x6 C 13 x7 /2 x7 /2 :

Then qJ is non-negative and Ker qJ D Z h, where h D .1; 1; 1; 1; 1; 1; 1/. This shows that qJ is critical in the sense of Ovsijenko [67]; see also [58]. Now we show that the finite-dimensional incidence K-coalgebra K J of J is of wild comodule type; hence, in view of Corollary 6.8, K J is not of tame comodule type if K is algebraically closed. For let .QI ; I / be the Hasse bound quiver of I and zI ; z I / ! .QI ; I / be a universal covering of bound quivers. It derives a let f W .Q zI ; z I /-Comod ! K .QI ; I /-Comod, see [13], [17], push-down functor f W K .Q [53], (4.17). z I / contains a wild subquiver Q of type zI ; One can show that .Q z z7W D

:

Hence, the finite-dimensional K-coalgebra K Q of the quiver Q is a subcoalgebra of zI ; z I / and f restricts to the functor K .Q f_ W K Q-comod ! K .QI ; I /-comod Š K I -comod preserving wildness. We recall that, by (8.23), there is a coalgebra isomorphism K J Š K .QI ; I /. It follows that the coalgebra K J is of wild comodule type since the K-coalgebra K Q is of wild comodule type by Theorem 7.22 (a) and Corollary 6.8; see also [64]. Consequently, the coalgebra K J is not of tame comodule type by Corollary 6.8.

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9 Almost split sequences in comodule categories The existence of almost split sequences in comodule categories is successfully discussed in [18], [19], and recently in [21], by applying the well-known ideas of Auslander–Reiten theory for finite-dimensional algebras, see [2] and [5]. In this section we discuss the existence of almost split sequences in the category C -Comodfc of finitely copresented left C -comodules, that is, the comodules M that admit a socle-finite injective copresentation g 0 ! M ! E0 ! E1 (9.1) because the category C -Comodfc plays a crucial role in our discussion of the tame comodule type presented in Section 6. Here we follow mainly the presentation given in [21], Section 1. Throughout, K is a field and C a pointed K-colagebra. We set C -comodfc D C -comod \ C -Comodfc , and we denote by C -Comodfc D Comodfc = the quotient category of C -Comodfc modulo the two-sided ideal D ŒC -inj consisting of all f 2 HomC .N; N 0 /, with N and N 0 in C -Comodfc , that have a factorisation through a socle-finite injective comodule. We recall that a comodule M is quasi-finite if dimK HomC .X; M / is finite, for any X in C -comod; equivalently, if the simple summands of soc M have finite (but perhaps unbounded) multiplicities [104]. It follows that every socle-finite comodule is quasi-finite and, hence, all comodules in C -Comodfc are quasi-finite. Given a left quasi-finite C -comodule M , the covariant cohom functor defined in [104], hC .M; / W C -Comod ! Mod.K/;

(9.2)

associates to any N in C -Comod the vector space hC .M; N / D lim D HomC .N ; M /, ! where fN g is the family of all finite-dimensional subcomodules of N . Denote by C op -Comodfp the full subcategory of C op -Comod whose objects are the (injectively) finitely presented C op -comodules, that is, the C op -comodules L that admit g0

a short exact sequence E10 ! E00 ! L ! 0 in C op -Comod, with socle-finite injective comodules E10 and E00 , called a socle-finite injective presentation of L. Following [21], Section 1, we define a pair of contravariant left exact functors rC op ! C -Comodfc C -Comodfp r0

(9.3)

C

to be the composite functors making the following diagrams commutative: C -Comodfc

z D '

rC

C op -Comodfp o

/ C -PCfp ./C

./ı '

C op -PCfc ,

o C -Comod fc O

./ı '

0 rC

C op -Comodfp

C -PCfp O ./C

z D '

/ C op -PCfc ,

(9.4)


647

where C -PCfp (resp. C op -PCfc ) is the category of pseudocompact (top-) finitely z D HomK .; K/, presented (resp. (top-) finitely copresented) modules, D ./C D homC .; C / W C -PCfp ! C -PCfc op

is a contravariant functor which associates to any X in C -PCfp (see Section 2.2), with f1

the top-finite pseudocompact projective presentation P1 ! P0 ! X ! 0 where P1 , P0 are finite direct sums of indecomposable projective C -modules, the right C module X C D homC .X; C / of all continuous C -homomorphisms X ! C , with the top-finite pseudocompact projective copresentation f1C

0 ! X C ! P0C ! P1C : Finally, the functor ./ı D homK .; K/ associates to X C the right C -comodule .X C /ı in C op -Comodfp , with the socle-finite injective presentation .f1C /ı

.P1C /ı ! .P0C /ı ! .X C /ı ! 0: The functors in the right hand diagram of (9.4) are defined analogously. Sometimes, for simplicity of the notation, we write rC instead of rC0 . Following Auslander [3], we define the transpose operator Tr D Tr C W C -Comodfc ! C op -Comodfc

(9.5)

(on objects only!) which associates to any comodule M in C -Comodfc , with a minimal socle-finite injective copresentation (9.1), the comodule rC .g/

Tr C M D KerŒrC E1 ! rC E0 in C op -Comodfc . The existence of almost split sequences in C -Comodfc depends on Theorem 9.6. Let C be a K-coalgebra and rC the functor in (9.3). (a) There are functorial isomorphisms rC M Š HomC .C; M /ı Š hC .M; C / for any comodule M in C -Comodfc . (b) The functors rC , rC0 are left exact and restrict to the dualities (2.26): rC

! C op -inj: C -inj 0 rC

(9.7)

(c) For any comodule M in C -Comodfc with a minimal socle-finite injective copresentation (9.1), the comodules Tr C M , rC E1 , rC E0 lie in C op -Comodfc , rC M lies in C op -Comodfp , and the following sequence is exact in C op -Comod: rC .g/

0 ! Tr C M ! rC E1 ! rC E0 ! rC M ! 0:

(9.8)

(d) The transpose operator Tr C , together with the functor rC , induces the equiva' ! C op -Comodfc . lence of quotient categories Tr C W C -Comodfc

648

D. Simson

Proof. For the convenience of the reader, we outline the proof. (a) Let fC g is the family of all finite-dimensional subcoalgebras of C and let M be a comodule in C -Comodfc . Then M is quasi-finite, C Š lim C , and we get ! z /C /ı Š homC .DM; z C /ı rC M D ..DM Š HomC .C; M /ı Š Œlim HomC .C ; M /ı Š lim HomC .C ; M /ı ! Š lim D HomC .C ; M / ! D hC .M; C /: One can easily see that the composite isomorphism is functorial at M . Statement (b) follows from the definition of rC . To prove (c) and (d), we note that the exact functors, see (9.4), z W C -Comodfp ! C -PCfp D

and

./ı W C -PCfc ! C op -Comodfc ; op

defining the functor rC , are equivalences of categories carrying injectives to projectives and projectives to injectives, respectively. Recall that C is a topological semiperfect algebra. Now, given an indecomposable comodule M in C -Comodfc , with a minimal socle-finite injective copresentation (9.1), we get a pseudocompact z

Dg z 1 z 0 ! DM z minimal top-finite projective presentation DE ! DE ! 0 in C -PC, z 1 D E , DE z 0 D E finite direct sums of indecomposable projective C with DE 1 0 z . Hence, by applying the left modules, of the right pseudocompact C -module DM z / exact functor homC .; C / and the definition of the Auslander transpose Tr C .DM z of the pseudocompact left C -module DM , we get the exact sequence z C .Dg/

z /C ! .DE z 0 /C ! .DE z 1 /C ! Tr C .DM z /!0 0 ! .DM

(9.9)

z C .Dg/

z / ! .DE z 0 / ! .DE z 1 /C in C -PC and the projective copresentation 0 ! .DM z /C , where .DE z 0 /C and .DE z 1 /C are of the right pseudocompact C -module .DM finitely generated projective top-finite right C -modules. The sequence (9.9) induces the sequence (9.8) and (c) follows. Statement (d) follows from the properties of the transpose operator Tr C W C -PCfp ! C op -PCfp on the pseudocompact finitely presented top-finite modules over C ; consult the proof of [2], Proposition IV.2.2. or [5], Section IV.1. op

C

C

Denote by C -Comodfc and by C -Comod fc the full subcategory of C -Comodfc z /C is consisting of the comodules M such that dimK Tr C .M / is finite and dimK .DM finite, respectively. Following the representation theory of finite-dimensional algebras, we define the Nakayama functor (covariant) C W C -Comod fc ! C -comod

(9.10)


649

by the formula C ./ D DrC ./. For a left semiperfect coalgebra C , the functor C restricts to the equivalence of categories ' ! C -proj; C W C -inj

(9.11)

where C -proj is the category of top-finite projective comodules in C -comod. We denote by C -comodfP the full subcategory of C -comod consisting of the left comodules N which, viewed as rational right C -modules, have a minimal top-finite projective presentation P1 ! P0 ! N ! 0 in C op -PC D PC-C , that is, P0 and P1 are top-finite projective modules in PC-C . Here we make the identification C -comod rat-C D dis-C PC-C in the notation of Section 2.3. Finally, C -comodfP D C -comodfP =P is the quotient category of C -comodfP modulo the two-sided ideal P of C -comodfP consisting of all f 2 HomC .N; N 0 /, with N and N 0 in C -comodfP , that have a factorisation through a projective right C -module, when f is viewed as a C -homomorphism between the rational right C -modules N and N 0 . If C is left semiperfect, in view of the exact sequence (9.8) in C op -Comod, we have C -comodfP D C -comod, C -Comodfc D C -Comodfc , C -Comod fc D C -Comodfc and, by applying C to (9.8), we get the exact sequence C .g/

0 ! C .M / ! C .E0 / ! C .E1 / ! D Tr C .M / ! 0

(9.12)

in C -comod Lemma 9.13 ([21]). Let C be a pointed K-coalgebra and let CQ be the left Gabriel quiver of C . (a) The duality D W C -comod ! C op -comod in (2.25) restricts to the duality D W C -comodfP ! C op -comodfc D C op -comod \ C op -Comodfc ; In particular, a left C -comodule N lies in C -comodfP if and only if the right C comodule DN is finitely copresented. (b) The following four conditions are equivalent: (b1) C -comodfP D C -comod, (b2) C op -comod C op -Comodfc , (b3) every simple comodule in C op -Comod is finitely copresented, the quiver CQ is right locally bounded, that is, for every vertex a of CQ there is only a finite number of arrows a ! j in CQ. (c) If C is right locally artinian, we have C -comodfP D C -comod.

650

D. Simson

Proof. (a) Since we make the identification C -comod rat-C D dis-C PC-C (in the notation of Section 2.3), there is a commutative diagram C -comod

id

./ı Š

D Š

C op -comod

/ dis-C PC-C

id

./ı Š

/ C -dis C op -Comod.

Then a left C -comodule N lies in C -comodfP if and only if there is an exact sequence P1 ! P0 ! N ! 0 in PC-C , where P0 and P1 are top-finite projective modules in PC-C , or equivalently, N lies in C op -PCfp D PCfp -C . By applying the duality ./ı W C op -PC ! C op -Comod (9.4), we get an exact sequence 0 ! N ı ! P0ı ! P1ı in C op -Comod. Since dimK N is finite, we have N ı D DN . This shows that DN lies in C op -comodfc because P0ı and P1ı are socle-finite injective right C -comodules. It follows that the duality (2.25) restricts to the duality D W C -comodfP ! C op -comodfc . (b) By (a), the equality C -comodfP D C -comod holds if and only if the equality C op -comodfc D C op -comod holds, that is, the conditions (b1) and (b2) are equivalent. The implication (b2) H) (b3) is obvious. To prove the inverse implication (b3) H) (b2), we assume that the simple right C -comodules lie in C op -Comodfc and let X be a comodule in C op -comod. By standard arguments and the induction on the K-dimension of X , we show that X Llies in C -Comodfc . y / be a direct sum decomposition of the (b3) H) (b4): Let soc CC D j 2IC S.j right socle soc CC of C , where IC is an index set and fSy.j /gj 2IC is a set of pairwise y / D E.Sy.j // the injective non-isomorphic simple right C -coideals. Denote by E.j envelope of Sy.j /. By Corollary 4.10, the left Gabriel quiver CQ of C is dual to the right Gabriel quiver QC of C . Hence, by the assumption (b2), for every vertex a of the quiver QC , there is only a finite number of arrows j ! a in QC . In other words, y dimK Ext1C .Sy.j /; S.a// is finite for all j 2 IC , and dimK Ext 1C .Sy.j /; Sy.a// D 0 for all but a finite number of indices j 2 IC . y y Fix a 2 IC and let 0 ! S.a/ ! E.a/ ! Ey1 ! be a minimal injective y resolution of Sy.a/ in C op -Comod, with Ey1 Š E.soc.E.a/= Sy.a///. y /; Sy.a// the number of times the comodGiven j 2 IC , we denote by 1 .S.j y / appears as a direct summand in Ey1 . Since C is assumed to be pointed, ule E.j y /; S.a// y D dimK Ext 1C .Sy.j /; Sy.a//; by [89], dimK EndC Sy.j / D 1 and 1 .S.j op (4.23). Thus the injective C -comodule Ey1 is socle finite, by the observation made earlier, it follows that the simple right C -comodule Sy.a/ is finitely copresented. This shows that (b3) implies (b4). Since the inverse implication follows in a similar way, the proof of (b) is complete. (c) Apply (b) and the easily seen fact that simple right comodules over any right locally artinian coalgebra are finitely copresented. Proposition 9.14 ([21]). Let C be a pointed coalgebra and D the duality (2.25).

651


(a) The transpose equivalence of Theorem 9.6 (d) defines the equivalence

' Tr C W C -Comodfc ! C op -comodfc ; ' and together with the duality D W C op -comodfc ! C -comodfP defined by (2.25) induces the translate operator

C D D Tr C W C -Comodfc ! C -comodfP ;

(9.15)

' and an equivalence of quotient categories NC D D Tr C W C -Comodfc !C -comodfP . Moreover, for any M in Comodfc , with a presentation (9.1), the following sequence z C .Dg/

z 1 /C ! .DE z 0 /C ! C M ! 0 z /C ! .DE 0 ! .DM

(9.16)

is exact in C op -PC and the comodule C M lies in the category C -comodfP C op -rat fp C op -PC. ' ! C op -comodfc (b) The duality (2.25) restricts to the duality D W C -comodfP op and together with the transpose operator Tr C op W C -comodfc ! C -Comodfc defines the translate operator

C D Tr C op D W C -comodfP ! C -Comodfc

(9.17)

' and induces the equivalence NC D Tr C op D W C -comodfP ! C -Comodfc which is ' ! C -comodfP in (a). quasi-inverse to the equivalence NC D D Tr C W C -Comodfc (c) Let M be an indecomposable comodule in C -Comodfc . Then C M D 0 if and only if M is injective. If C M ¤ 0 then C M is indecomposable, non-projective, of finite K-dimension, and there is an isomorphism M Š C C M . (d) Let N be an indecomposable comodule in C -comodfP . Then C N D 0 if and only if N is projective. If C N ¤ 0 then C N is indecomposable, non-injective, finitely copresented, and there is an isomorphism N Š C C N . ' ! C -comod (2.25) restricts Proof. By Lemma 9.13 (a), the duality D W C op -comod op ' to the duality D W C -comodfc ! C -comodfP . One also shows, by applying foregoing definitions, that a homomorphism f W X ! X 0 in C op -comodfc has a factorisation through a socle-finite injective comodule if and only if the homomorphism Df W D.X / ! DX 0 in C -comodfP belongs to P .DX; DX 0 /. This shows that the du' ! C -comodfP induces an equivalence of quotient categories ality D W C op -comodfc op ' D W C -comodfc ! C -comodfP . It follows from the definition of the category C -Comodfc that the transpose equivalence of Theorem 9.6 (d), defines the equivalence ' Tr C W C -Comodfc ! C op -comodfc . This together with an earlier observation implies (a) and (b). The statements (c) and (d) are obtained by a straightforward calculation and by using the definition of translates C and C , consult [2] and [5].

652

D. Simson

We call the operators C D Tr C op D in (9.17) and C D D Tr C in (9.15) the Auslander–Reiten translations of C . It follows from Proposition 9.14 that the image of C is the subcategory C -comodfP of the category C -comod. Under some assumption on the Gabriel quiver CQ of C , we get the following result on the existence of almost split sequences in the category C -Comodfc of (socle) finitely copresented left C -comodules. Theorem 9.18 ([18], [21]). Let C be a K-coalgebra such that its left Gabriel quiver CQ is left locally bounded, that is, for every vertex a of CQ there is only a finite number of arrows j ! a in CQ. (a) C -comod C -Comodfc and, for any indecomposable non-injective comodule M in C -Comodfc , there exists a unique almost split sequence 0 ! M ! M 0 ! C M ! 0

(9.19)

in C -Comodfc , with an indecomposable comodule C M lying in C -comodfP . The sequence (9.19) is almost split in the whole comodule category C -Comod. (b) For an arbitray indecomposable non-projective comodule N in the category C -comodfP C -Comodfc , there exists a unique almost split sequence 0 ! C N ! N 0 ! N ! 0

(9.20)

in C -Comodfc , with an indecomposable comodule C N lying in C -Comodfc . The sequence (9.20) is almost split in the whole comodule category C -Comod. (c) If, in addition, C is left semiperfect then C -Comodfc D C -Comodfc and C -comodfP D C -comod, the Auslander–Reiten translate operators act as follows

C ! C -comod C -Comodfc C

and the almost split sequences (9.19) and (9.20) do exist in the category C -Comodfc , for any indecomposable non-injective comodule M in C -Comodfc and for any indecomposable non-projective comodule N in C -comod. Moreover, if the comodule M lies in C -comod then the almost split sequence (9.19) lies in C -comod. Proof. (a) As in the proof of Lemma 9.13 (b), we conclude that every simple left C -comodule admits a minimal socle-finite injective copresentation (9.1). Hence (a) follows as in Lemma 9.13 (b). The statement (b) follow from Theorem 4.2 and Corollary 4.3 in [18]. Since any comodule M lying in C -Comodfc is quasi-finite, Proposition 9.14 (a) yields that C M lies in C -comodfP for any indecomposable comodule M in C -Comodfc , and the inclusions C -comodfP C -comod C -Comodfc hold by (a). (c) Assume that C is left semiperfect and let M be an indecomposable comodule in C -Comodfc with a minimal socle-finite injective copresentation (9.1). By Theorem 9.6, the induced sequence (9.8) is exact and the comodules rC E0 and rC E1 lie in C op -inj. Since C is left semiperfect, the comodules rC E0 and rC E1 are finite-dimensional


653

and, hence, dimK Tr C .M / is finite for any comodule M in C -Comodfc . It follows that C -Comodfc D C -Comodfc . Since C is left semiperfect, any comodule N in C -comod has a projective presentation P1 ! P0 ! N ! 0, with P1 , P0 finite-dimensional projective C -comodules. It follows that N lies in C -comodfP and, hence, we get C -comodfP D C -comod. Corollary 9.21. Let C be a pointed K-coalgebra such that the left Gabriel quiver CQ of C is both left and right locally bounded. (a) C -comodfP D C -comod C -Comod fc and the Auslander–Reiten translate C ! operators act as follows: C -Comodfc C -comod. C

(b) For any indecomposable non-injective comodule M in C -Comodfc , there exists a unique almost split sequence 0 ! M ! M 0 ! C M ! 0 in C -Comodfc , with an indecomposable comodule C M lying in C -comod. (c) For any indecomposable non-projective comodule N in C -comod, there exists a unique almost split sequence 0 ! C N ! N 0 ! N ! 0 in C -Comodfc , with an indecomposable comodule C N lying in C -Comodfc . (d) The exact sequences in (b) and (c) are almost split in the whole comodule category C -Comod. Proof. Apply Lemma 9.13 and Theorem 9.18. Remark 9.22. Under the assumption that the left Gabriel quiver CQ of C is both left and right locally bounded the almost split sequences (9.19) and (9.20) lie in C -Comodfc . If we drop the assumption then the term C M lies in C -comodfP C -comod, but not necessarily lies in C -Comodfc . Let C be a pointed connected K-algebra. Following [21], C is called a sharp Euler coalgebra if C is computable, every simple left (resp. right) C -comodule S admits a fim 0 y0 y nite and socle-finite injective resolution, and dimK Extm C .S; S / D dimK Ext C op .S ; S / 0 for all m 0 and all simple left C -comodules S and S , where Sy D DS and Sy0 D DS 0 are the dual simple right C -comodules. It is shown in [21], Section 2, that (i) the path coalgebra K Q of a quiver is a sharp Euler coalgebra if and only if every vertex of Q has only a finite number or neighbours, (ii) every pointed left (or right) semi-perfect coalgebra of finite global dimension is a sharp Euler coalgebra. It follows from [94], Sections 5–6, that a path coalgebra K I of an interval finite poset I having no infinitely many of pairwise incomparable elements is a sharp Euler coalgebra if every simple left comodule and every simple right comodule has finite injective dimension. By Theorem 8.7 (c), the Cartan matrix CC of a sharp Euler coalgebra C has twosided inverse C1 C , which is both row-finite and column-finite. In this case, we define

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(i) the Coxeter matrix of C to be the IC IC square matrix CoxC D Ctr C CC , 1 tr tr 1 D . C / D . C / , and where we set Ctr C C C (ii) the Coxeter transformations of C to be the group homomorphisms ˆ

IC C ! ZIIC ZJ ˆC

(9.23)

IC given by the formulae ˆC .x/ D .x Ctr C / CC for x 2 ZI , and ˆC .y/ D IC IC tr IC is the subgroup of ZIC generated .y C1 C / CC for y 2 ZJ , where ZI Z IC IC y by the subset fy e.a/ D dim E.a/g is the subgroup of ZIC a2IC and ZJ Z generated by the subset fe.a/ D dim E.a/ga2IC .

One shows that the transformations (9.23) are well defined and mutually inverse. Moreover, we have the following useful result. Theorem 9.24 ([21], [90]). Suppose L that C is a pointed sharp Euler K-coalgebra with fixed decomposition soc C C D j 2IC S.j /. Let ˆC and ˆ C be the Coxeter transformations of C . (a) Assume that M is an indecomposable left C -comodule in C -Comodfc such that inj.dim M D 1 and HomC .C; M / D 0. If 0 ! M ! M 0 ! C M ! 0 is the unique almost split sequence (9.19) in C -Comodfc , with an indecomposable comodule C M lying in C -comodfP , then dim C M D ˆ C .dim M /. (b) Assume that N is an indecomposable non-projective left C -comodule in C -comodfP C -Comodfc such that inj.dim DN D 1 and HomC .C; DN / D 0. If 0 ! C N ! N 0 ! N ! 0 is the unique almost split sequence (9.20) in C -Comodfc , with an indecomposable comodule C N lying in C -Comodfc , then dim C N D ˆC .dim N /. Remark 9.25. (a) If C is a hereditary sharp Euler coalgebra and M (resp. N ) is an indecomposable non-injective comodule (resp. non-projective comodule), then Theorem 9.24 applies to M (resp. to N ). (b) Examples can be found in [16], [17], [21], [83], [90], [92]. Note also that the cotilting procedure for coalgebras discussed in [92] (see also [41], [106], and [107]) produces a class of examples of almost split sequences for coalgebras of global dimension two from module categories of hereditary coalgebras. (c) Unfortunately, the cotilting procedure for coalgebras is not well developed. In view of recent results by Keller and Yang [48], Section 7.9, a better framework for cotilting of coalgebras are derived categories of comodule categories or pseudocompact module categories; see also [69] and [105].


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Singularities of orbit closures in module varieties Grzegorz Zwara

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . 1 Varieties of representations and modules . . . . . . 2 Types of singularities . . . . . . . . . . . . . . . . 3 Orbit closures of small dimensions . . . . . . . . . 4 Orbit closures which are hypersurfaces . . . . . . . 5 Characterizations of degenerations . . . . . . . . . 6 Hom-order . . . . . . . . . . . . . . . . . . . . . . 7 Transversal slices . . . . . . . . . . . . . . . . . . 8 Desingularizations and unibranch varieties . . . . . 9 Hom-controlled exact functors . . . . . . . . . . . 10 Schubert varieties . . . . . . . . . . . . . . . . . . 11 Equations of orbit closures . . . . . . . . . . . . . 12 Tangent spaces . . . . . . . . . . . . . . . . . . . 13 Singularities in codimension one . . . . . . . . . . 14 Singularities for degenerations of codimension two 15 Generic singularities . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction Throughout the paper, k denotes an algebraically closed field of arbitrary characteristic. One of the main aims of the representation theory of algebras is to understand the category mod A of finite dimensional (left) modules over an associative k-algebra A, and the category rep.Q/ (or rep.Q; I /) of finite dimensional representations of a finite quiver Q (or a bound quiver .Q; I /) over k. There are algebraic varieties equipped with regular actions of algebraic groups, which occur in a natural way in this context. Given a natural number d , we denote by modA .d / the set of A-module structures on the vector space k d . Hence we may write modA .d / D fM W A ! M.d / j M is a homomorphism of k-algebrasg; Supported by the Research Grant No. N N201 269135 of the Polish Ministry of Science and Higher Education.

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where M.d / stands for the algebra of d d -matrices with coefficients in k. The set modA .d /, called a module variety, has a natural structure of an affine variety, provided the k-algebra A is finitely generated (for more details see Section 1). The general linear group GL.d / acts regularly on modA .d / via Œg M .a/ D g M.a/ g 1 ; for any a 2 A. The orbits correspond bijectively to the isomorphism classes of d dimensional A-modules. Let Q D .Q0 ; Q1 ; s; t / be a finite quiver, i.e. Q0 is a finite set of vertices and Q1 is a finite set of arrows ˛ W s.˛/ ! t .˛/. For a dimension vector d D .di /i2Q0 2 N Q0 , the representations of Q with the underlying vector spaces k di at the vertex i , for any i 2 Q0 , form the vector space Y repQ .d/ D M.d t.˛/ ds.˛/ /; ˛2Q1

where M.d 0 d 00 / denotes the vector space of d 0 d 00 -matrices with coefficients in k, for any d 0 ; d 00 1. The product Y GL.d/ D GL.di / i2Q0

of general linear groups acts on repQ .d/ via 1 .gi /i2Q0 .M˛ /˛2Q1 D .g t.˛/ M˛ gs.˛/ /˛2Q1 :

As in the case of modules over algebras, the orbits correspond bijectively to the isomorphism classes of representations of Q with dimension vector d. Let M be either a finite dimensional A-module or a finite dimensional representation of Q. We denote the corresponding orbit by OM , it is a nonsingular quasi-affine variety. xM (in the Zariski topology) of the orbit The main object of our interest is the closure O OM . There are two main reasons for studying such orbit closures. By inspecting them as affine varieties by methods of algebraic geometry we can achieve deeper understanding of the categories of modules and representations. On the other hand, our orbit closures provide many interesting examples of affine varieties, whose geometric properties are derived from known properties of the categories of modules or representations. For more general program of applying geometric methods to the representation theory of algebras we refer the reader to [31] and [14]. Let us illustrate the situation with two fundamental examples that students meet in linear algebra courses, namely the classification of linear maps and linear endomorphisms up to base change. In our language, we speak about representations of connected quivers with one arrow. ˛ First, let Q D 1 2 and d D .d1 ; d2 / 2 N 2 . The group GL.d/ D GL.d1 / GL.d2 / acts on repQ .d/ D M.d1 d2 / via .g1 ; g2 / .M˛ / D .g1 M˛ g21 /:

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The vector space repQ .d/ D M.d1 d2 / decomposes into a union of .m C 1/ orbits xr Or , 0 r m D minfd1 ; d2 g, consisting of matrices of rank r. The orbit closure O consists of the matrices of rank at most r: [ xr D Oj : O j r

Thus we obtain well-known fundamental determinantal varieties, and as general references on them the reader can take the book [19]. Let us remark that the minors of xr in the coordinate ring of M.d1 d2 /. The affine size .r C 1/ generate the ideal of O x variety Or is normal, Cohen–Macaulay, with rational singularities, and Frobenius split in positive characteristics (see Section 8 for a discussion on Frobenius split varieties). xr It is Gorenstein if and only if r D 0 or r D m or d1 D d2 . If r D 0 or r D m then O is a regular variety; else the singular locus consists of the matrices of rank less than r. ˛ with one The second example leads to representations of the quiver Q: b vertex and a loop. Let d D .d / 2 N. Any representation M in repQ .d/ is a square matrix in M.d /. The group GL.d/ D GL.d / acts by conjugation g M D g M g 1 ; and the orbits are just the conjugacy classes of matrices in M.d /. Any orbit in M.d / contains a matrix in the canonical Jordan form X M ri D d; J.i ; ri /; i 2 k; ri 1; which is unique, up to an order of the blocks J.i ; ri /. In particular, there are infinitely many orbits. However, the characteristic polynomial det.t 1d X / of a matrix X 2 M.d / leads to a GL.d /-invariant regular morphism, and therefore any orbit xM closure contains only finitely many orbits. It is known, that the orbit closure O contains N if and only if rk. Id M j / rk. Id N j /;

for all 2 k and j 1:

The orbit closures are normal and Cohen–Macaulay varieties with rational singularities. xM n OM . Assume that M is nilpotent (M d D 0). xM equals O The singular locus of O xM is a Frobenius split variety, otherwise O xM is If k is of positive characteristic, then O Gorenstein and Weyman described in [55] (see also [56], Section 8) the defining ideal xM ; the generators are linear combinations of minors of various sizes. of O The above two examples give a flavour of geometry of orbit closures in varieties of representations. However, they do not show some pathology that can occur for more complicated quivers. For instance, there are orbit closures which are neither normal nor Cohen–Macaulay (see Example 8.14). In the case of wild quivers there is no hope of classifying orbits for large dimension vectors. We describe briefly the content of the sections in this article. In Section 1 we introduce the varieties of quiver representations annihilated by a two-sided ideal, and the

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nilpotent representations. Then the invariants are applied to get information on closed orbits and orbit closures. Section 2 contains definitions and discussions on smooth morphisms, types of singularities and associated fibre bundles. The section is concluded by the geometric version of Morita equivalence due to Bongartz. In Section 3 we classify the varieties of dimension at most four, which appear as orbit closures of nilpotent representations. Section 4 contains classifications of modules whose orbit closures are regular varieties or local hypersurfaces. In Section 5 we characterize the degeneration order (the partial order of inclusions between the orbit closures) in terms of short exact sequences. In Section 6 we define the hom-order, which is a partial order defined in terms of dimensions of homomorphisms spaces. Then the equivalence of the degeneration order and the hom-order for algebras of finite representation type is proved. Applications of the equivalence are presented, including a construction of short exact sequences of special form, and algorithmic computations of degenerations. In Section 7 we introduce transversal slices to orbit closures. This concept gives a method to compute interesting examples of types of singularities occurring in orbit closures. Section 8 contains two constructions of desingularizations of orbit closures followed by discussions on unibranch varieties. In Section 9 we introduce the concept of hom-controlled exact functors, which is applied to show that the types of singularities for some modules over different algebras coincide. In Section 10 we discuss relations between Schubert varieties and orbit closures for quivers. The problem of equations describing orbit closures is discussed in Section 11. In Section 12 we relate tangent spaces to orbit closures, to the groups of selfextensions, in the spirit of Voigt’s isomorphism. Sections 13 and 14 concerns the types of singularities of orbit closures in codimension one and two, respectively. In the final Section 15 we discuss generic singularities of orbit closures. For background on the representation theory of algebras and quivers we refer to [4], [6], [49] and [51]; and on algebraic geometry we refer to [23], [24], [27] and [43].

1 Varieties of representations and modules A path in Q of length n 1: ˛n1

˛n

t.˛n / s.˛n / D t .˛n1 /

˛2

˛1

s.˛2 / D t .˛1 / s.˛1 /

will be denoted by ! D ˛n ˛n1 : : : ˛1 . We put s.!/ D s.˛1 / and t .!/ D t .˛n /. We define a path "i of length zero with s."i / D t ."i / D i , for each vertex i 2 Q0 . The paths in Q form a k-linear basis of the path algebra kQ with the multiplication induced by the composition of paths. The paths "i , i 2 Q0 , are idempotents summing up to 1kQ , and therefore we have a k-vector space decomposition: kQ D

M i;j 2Q0

"j kQ "i :

665


Let V D .V˛ /˛2Q1 be an element of repQ .d/ for a dimension vector d D .di / 2 N Q0 . An element in "j kQ "i is a linear combination of paths starting at i and terminating at j , and therefore we can define, in an obvious way, the matrix V 2 M.dj di /: P We associate with V the kQ-module Vz in modkQ .d /, where d D i2Q0 di , such that the matrix Vz ./, 2 kQ is build of the blocks V"j "i for i; j 2 Q0 . This assignment extends to a well-known equivalence between the category rep.Q/ and mod.kQ/. Let I be a two-sided ideal of kQ. Let rep.Q; I / denote the full subcategory of rep.Q/ consisting of the representations of Q, which, as kQ-modules, are annihilated by I . Let repQ;I .d/ be the subset of repQ .d/ consisting of the elements V D .V˛ /˛2Q1 which, treated as representations of Q, belong to rep.Q; I /, or equivalently, such that V D 0, for any i; j 2 Q0 and 2 "j I "i . It is not difficult to prove that it suffices to consider elements generating the ideal I . Obviously repQ;I .d/ is an affine GL.d/-variety. ˛

ˇ

Example 1.1. Let Q be the quiver 1 2 3, I D h˛ˇi be the ideal generated by ˛ˇ, and d D .d1 ; d2 ; d3 / 2 N Q0 . Then repQ;I .d/ D f.V˛ ; Vˇ / 2 M.d1 d2 / M.d2 d3 / j V˛ Vˇ D 0g and the group GL.d/ acts via .g1 ; g2 ; g3 / .V˛ ; Vˇ / D .g1 V˛ g21 ; g2 Vˇ g31 /: In particular, repQ;I .1; 1; 1/ is a union of three GL.1; 1; 1/-orbits of the representations Œ1

Œ0

k k k;

Œ0

Œ1

k k k

and

Œ0

Œ0

k k k:

We introduce an affine variety structure on the module varieties. Let A be a finitely generated k-algebra. We choose generators a1 ; a2 ; : : : ; a t of A, or equivalently, a surjective homomorphism from a free k-algebra to A: W khx1 ; x2 ; : : : ; x t i ! A: Let J denote the kernel of and d be a positive integer. The module variety modA .d / is in GL.d /-equivariant bijection with the GL.d/-variety repQ;I .d/, where Q is a quiver with one vertex and t loops attached to the vertex (hence kQ can be identified with khx1 ; x2 ; : : : ; x t i), I D J and d D .d /. One can show that another choice of generators of A leads to the same GL.d /-variety structure on modA .d /. Example 1.2. Let A D kŒx; y be a polynomial ring in two variables and d 1. Then modA .d / can be identified with the variety fM D .M1 ; M2 / 2 M.d /2 j M1 M2 M2 M1 D 0g

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equipped with the following action of GL.d /: g .M1 ; M2 / D .g M1 g 1 ; g M2 g 1 /: Let d 2 N Q0 be a dimension vector. We apply the invariant theory in order to study repQ .d/. Let kŒrepQ .d/ denote the coordinate ring of the affine space repQ .d/. We consider the induced action of GL.d/ on kŒrepQ .d/ given by .gf /.X / D f .g 1 X /. If ! is an oriented cycle in Q and i D s.!/ D t .!/, then the coefficients of the characteristic polynomial det.t 1di X! / are GL.d/-invariants. By [22] and [34], these coefficients generate the k-algebra of invariants kŒrepQ .d/GL.d/ . Note that kŒrepQ .d/GL.d/ D k if and only if the quiver Q has no oriented cycles. Let repQ .d/== GL.d/ denote the affine variety with kŒrepQ .d/GL.d/ as a coordinate ring. The inclusion of the coordinate rings induces a quotient map W repQ .d/ ! repQ .d/== GL.d/;

(1.1)

which is regular and surjective. The fibres of are obviously unions of GL.d/-orbits. The map has nice properties in terms of representations: (1) Given two representations M and N in repQ .d/, .M / D .N / if and only if M and N have the same Jordan–Hölder factors including multiplicities. (2) Each fibre of contains exactly one closed orbit, the orbit of a semisimple representation. We see that the variety repQ .d/== GL.d/ parameterizes the semisimple representations in repQ .d/. As a direct consequence we get the following result (compare [25], Corollaries 1.3 and 1.4). Corollary 1.3. Let M be a representation in repQ .d/. Then: (1) The orbit OM is closed if and only if M is semisimple. xM contains a unique closed orbit ON , where N is the semisimple representation (2) O associated with M . Let M and N be two representations (or modules). If the orbit ON is contained xM then we say that M degenerates to N , or N is a degeneration of M , and write in O M deg N . Then deg is a partial order on the set of isomorphism classes of objects in rep.Q/ (or in mod.A/). The above corollary can be partially derived from the following simple fact: Lemma 1.4. Let 0 ! U ! M ! V ! 0 be a short exact sequence in rep.Q/ or mod.A/. Then M degenerates to U ˚ V . We return to the quotient map (1.1). Let rep0Q .d/ denote the fibre of containing 0. The representations in rep0Q .d/ are called nilpotent. Hence a representation M 2 repQ .d/ is nilpotent if and only if one of the following equivalent conditions is satisfied:


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(1) M! is a nilpotent matrix in M.ds.!/ / for any oriented cycle ! in Q. xM . (2) 0 belongs to the orbit closure O (3) Any Jordan–Hölder factor of M is isomorphic to Si for some i 2 Q0 . (4) .kQC /n AnnQ .M / for n large enough. Here, Si is the one-dimensional representation such that .Si /i D k and .Si /ˇ D 0 for any loop ˇ W i ! i ; kQC denotes the ideal of kQ generated by arrows; and z associated with M . Obviously any AnnQ .M / is the annihilator of the kQ-module M representation is nilpotent if Q has no oriented cycles. A two-sided ideal I of kQ is said to be admissible if .kQC /n I .kQC /2 for some n 2. Then the pair .Q; I / is called a bound quiver. Obviously repQ;I .d/ is contained in rep0Q .d/ for any bound quiver .Q; I /. We call a representation M 2 repQ .d/ admissible if the annihilator AnnQ .M / of M is admissible. We will explain in the following section that essentially one needs to focus only on nilpotent representations, or even only on admissible representations.

2 Types of singularities The orbit closures in varieties of modules or representations are not smooth varieties in general. A natural problem is to describe the singularities occurring in the orbit closures and their connections with representation theory. Obviously the singularities should be classified under some equivalence relation. We discuss smooth morphisms before the definition of types of singularities will be presented. The concept of smooth morphisms known in mathematical analysis, has a more complicated equivalent in algebraic geometry. Let us recall from Definition III.10.3 of [43] that (1) The particular morphisms X D Spec.RŒx1 ; : : : ; xnCr =.f1 ; : : : ; fn // Y D Spec.R/

are said to be smooth at a point x 2 X (of relative dimension r) if

rk

@fi .x/ D n: @xj

(2) An arbitrary morphism f W X ! Y of finite type is smooth (of relative dimension r), if for all x 2 X, there are open neighbourhoods U X of x and V Y of f .x/ such that f .U/ V and such that f restricted to U, looks like a morphism of the above type which is smooth at x, i.e. there is a commutative

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diagram U

/ Spec.RŒx1 ; : : : ; xnCr =.f1 ; : : : ; fn //

V

/ Spec.R/

where the horizontal arrows represent open immersions. Typical examples of smooth morphisms are open immersions and vector bundles. Moreover, the composition of smooth morphisms is smooth, and smoothness is closed under base change. A morphism f W X ! Y is smooth if and only if it is flat and, for any point y 2 Y, the schematic fibre f 1 .y/ is a regular variety ([26], Corollary 17.5.2). In case when X and Y are regular varieties, the smoothness at x means that the induced linear map between the tangent spaces Tx;X ! Tf .x/;Y is surjective ([26], Theorem 17.11.1). For a smooth morphism f W X ! Y, the variety X is smooth, normal, Cohen–Macaulay, Gorenstein or unibranch at its point x if and only if the same holds for the variety Y at the point f .x/ ([26], Propositions 17.5.7, 17.5.8). In characteristic 0 rational singularities are preserved by smooth morphisms (see for instance Section 3 of [53]). Thus the smooth morphisms preserve many geometric properties (except irreducibility) interesting for us in the context of orbit closures. We write Ox;X for the local ring of a variety X at a point x 2 X, and we denote its yx;X . A morphism f W X ! Y is smooth at x (of relative dimension r) completion by O if and only if the induced homomorphism of completions of the local rings has the form yf .x/;Y ! O yf .x/;Y ŒŒt1 ; : : : ; tr D O yx;X : O Following the definition given by Hesselink ([29], (1.7)) two pointed varieties .X; x/ and .Y; y/ are smoothly equivalent if there exists the third pointed variety .Z; z/ together with two smooth morphisms f W Z ! X, g W Z ! Y sending the point z to x and y, respectively. Smooth equivalence is an equivalence relation and its equivalence classes will be called the types of singularities and denoted by Sing.X; x/. Example 2.1. We assume that the characteristic of k is different than 2 and consider two plane curves C D f.x; y/ 2 k 2 j x 2 D y 2 C y 3 g

and

D D f.x; y/ 2 k 2 j x 2 D y 2 g:

Then Sing.C ; .0; 0// D Sing.D; .0; 0//. Indeed, we define the third plane curve E D f.x; t/ j x 2 D .t t 3 /2 g and two regular morphisms u E III g II uu u I$ zuu f

C

D,

.x; t / HHHH v6 v v H# zvv .x; t 2 1/ .x; t 3 t /.

Then f .0; 1/ D .0; 0/ and g.0; 1/ D .0; 0/. Moreover, f is étale (i.e. smooth of relative dimension 0) at the point .x; t / provided t ¤ 0, and g is étale at .x; t / provided


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3t 2 ¤ 1. Observe that the first curve is irreducible while 0 belongs to two irreducible components of D. The pointed varieties .X; x/ such that X is regular at x form one type of singularity, denoted by Reg. Lemma 2.2. Let .X; x/ and .Y; y/ be pointed varieties and assume that dimx X dimy Y D r 0. Then Sing.X; x/ D Sing.Y; y/ if and only if yy;Y ŒŒt 0 ; : : : ; t 0 yx;X ŒŒt1 ; : : : ; ts ' O O 1 sCr for some s 0. If k is of characteristic 0, then we may assume that s D 0. Proof. One implication follows directly from the characterization of smooth morphisms in terms of the induced homomorphisms of the completions of local rings. The converse implication can be derived from Artin’s approximation theorem (see [3], Corollary 2.6). The remaining part of the claim is related to the cancellation problem RŒŒt ' SŒŒt H) R ' S for special complete rings R and S , which has a positive answer at least in characteristic 0 (see [28], especially Lemma 2). For a degeneration M deg N , we denote by Sing.M; N / the type of singularity xM ; N 0 /, where N 0 is an arbitrary point of the orbit ON . The problem of clasSing.O sification of the types of singularities Sing.M; N / for all degenerations of modules and representations seems to be hopeless. Thus one should restrict to some particular classes of degenerations. Now we want to define associated fibre bundles (see [54], Section 3.7, and [14], Section 5.2). Let G be a connected algebraic group with a closed subgroup H . Then the quotient G=H exists and is called a homogeneous space (see §6 in [16]). Let X be an H -variety. Then the group H acts freely on the product G X via h .g; x/ D .g h1 ; h x/; and there exists the geometric quotient denoted by G H X. Furthermore, the map G H X ! G=H;

Œ.g; x/ 7! g H;

is a G-equivariant fibre bundle having X as a typical fibre. The G-varieties G H X and X are connected by the following two surjective smooth morphisms

G H

l lll vlll X

G XN NNN NNN '

X,

.g; 9 x/ y x888 y 8 |yy x, Œ.g; x/

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G. Zwara

the first morphism is G-equivariant and the second one is H -equivariant. Therefore the map U 7! G H U induces a bijection, between H -invariant subvarieties of X and G-invariant subvarieties of G H X, preserving closures, inclusions and types of singularities (see [14], Lemma 5.16). We describe a method to detect associated fibre bundles (see [54], Lemma 3.7.4). Lemma 2.3. Let f W Y ! G=H be a G-equivariant morphism of varieties. Assume that the schematic fibre X D f 1 .H / is reduced (hence a variety). Then Y is isomorphic to the associated fibre bundle G H X. Now we can explain the geometric version of Morita equivalence established by Bongartz in [10]. Assume that the algebra A is finite dimensional. Then there are only finitely many pairwise nonisomorphic simple A-modules, say S1 ; : : : ; Sn . The Grothendieck group K0 .A/ of mod A can be identified with Zn , such that the coordinates of the dimension vector dim M of a module M 2 mod A are multiplicities of Si ’s as Jordan–Hölder factors of M . Thus the underlying vector space of any A-module with dimension vector d has dimension X jdj D di dimk Si : in

Given d 1, we have a decomposition of the module variety a modA .d/ modA .d / D d2N n jdjDd

into connected components containing modules of fixed dimension vector (see [25], Corollary 1.4). Let B be a maximal semisimple subalgebra of A. We have a similar decomposition a modB .d / D modB .d/; d2N n jdjDd

but in addition, each connected component modB .d/ D GL.d /= GL.d/ is a homogeneous space. The inclusion B A induces a GL.d /-equivariant morphism modA .d/ ! modB .d/ with the schematic fibres being reduced. It turns out that the fibres are isomorphic to the GL.d/-variety repQ;I .d/ for a bound quiver .Q; I / such that the closed immersion of repQ;I .d/ into modA .d / is compatible with a categorical (Morita) equivalence F W rep.Q; I / ! mod A: By Lemma 2.3, modA .d/ ' GL.d / GL.d/ repQ;I .d/:


Therefore

671

xF M ' GL.d / GL.d/ O xM ; O

for any representation M in rep.Q; I /, and we obtain the following result. Theorem 2.4. Let F W rep.Q; I / ! mod A be a Morita equivalence and M and N be two representations in rep.Q; I /. Then M deg N if and only if F M deg F N . If this is the case then Sing.M; N / D Sing.F M; F N /: The above theorem shows that we can switch easily between “the world” of modules and “the world” of representations. It is often more convenient to formulate theoretical results in terms of modules, but concrete matrix calculations are easier for xM is usually strict. xF M dim O representations as the inequality dim O Let M be a representation in repQ .d/ for some quiver Q and a dimension vector d. xM is contained in repQ;I .d/ for I D AnnQ .M /. The quotient algebra Observe that O kQ=I is finite dimensional as the image of the last map in the exact sequence X z M 0 ! AnnQ .M / ! kQ ! M.d /; d D di : i2Q0

Consequently, the variety repQ;I .d/ decomposes into connected components, each having exactly one closed GL.d/-orbit. One can easily generalize the Bongartz construction to get the following fact. Theorem 2.5. Let M be a representation in repQ .d/. There is a quiver Q0 , a dimension vector d 0 2 N Q0 , a group homomorphism GL.d 0 / ! GL.d/ being a closed immersion, and an admissible (hence nilpotent) representation N in repQ0 .d 0 /, such that xM ' GL.d/ GL.d 0 / O xN : O This shows that we can restrict our attention to nilpotent (or even admissible) representations of quivers.

3 Orbit closures of small dimensions It is an interesting problem to determine which affine varieties appear as orbit closures in varieties of representations. We have no chance to solve the problem in general. One idea is to restrict the dimension of affine varieties. There is a simple formula for the dimension of orbits and their closures. Namely, dim OM , for a representation M in repQ .d/, is the difference of the dimension of the acting group and the dimension of the stabilizer of M . The stabilizer of M is just the automorphism group of M , which is an open subset in the vector space EndQ .M / of the endomorphisms of M . Therefore, X xM D dim O di2 dimk EndQ .M /: (3.1) i2Q0

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xM for nilpotent representations Up to fibre bundles, we may consider orbit closures O only. Rochman classified in [50] the affine varieties of dimension at most 4 which appear as orbit closures for nilpotent representations. xM is Theorem 3.1. Let M be a nilpotent representation such that its orbit closure O x of dimension less or equal 4. Then OM is isomorphic to a product of the following 11 varieties: dim 1 2 3 4

k H D Œ2 .2; 2/ D.2; 2/, H D Œ2 .2; 3/ D.2; 3/, H D Œ2 .2; 4/, H D Œ2 .3; 3/, H D Œ3 .3; 3/, D.2; 2; 2/,C .2; 3/, C .2; 2; 2/

Now we define the affine varieties used in the above theorem and show examples of their realizations as orbit closures. The affine line k is the orbit closure for the representation Œ1

k k: We take p; q 2 and denote by D.p; q/ the closed subset of M.p q/ consisting of the matrices of rank at most 1. This is the orbit closure for the representation

kp

"0 0 1

0 0 0

0# 0 0

k q

with the coordinate ring kŒD.p; q/ ' kŒxi;j j i p; j q=.xi 0 ;j 0 xi 00 ;j 00 xi 0 ;j 00 xi 00 ;j 0 j i 0 < i 00 ; j 0 < j 00 /: Let H be a linear hyperplane in M.p q/ given by XX

bi;j xi;j D 0;

ip j q

where the matrix B D .bi;j / 2 M.p q/ is nonzero. Since D.p; q/ is invariant under the action of GL.p/ GL.q/, the intersection H \ D.p; q/ depends, up to isomorphism, only on the rank of B. Thus we obtain the varieties H D Œr .p; q/ D fM D Œmi;j 2 D.p; q/ j m1;1 C m2;2 C C mr;r D 0g for r D 1; 2; : : : ; minfp; qg. The variety H D Œ1 .p; q/ has two irreducible components isomorphic to D.p 1; q/ and D.p; q 1/. If r 2 then H D Œr .p; q/ is the orbit

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closure for the representation "0 0 1 "0 0 1

0 0 0

0# 0 0

0 0 0

0# k qr 0 y 0 yyy

yy "0 |yy r 0 6 k EE 1 EE E " 0 0 0 # EE E" 0 0 0 k pr 1 0 0

0 0 0

0# 0 0

with the coordinate ring kŒH D Œr .p; q/ D kŒD.p; q/=.x1;1 C x2;2 C C xr;r /: If p D r or q D r, then we cancel appropriate vertices and arrows in the above picture. Let D.2; 2; 2/ denote the orbit closure for h

1 oh

00 10 00 10

i i

/

2:

The coordinate ring kŒD.2; 2; 2/ is isomorphic to the quotient of the polynomial algebra kŒxi;j ; yi;j j i; j 2 modulo the ideal generated by 10 polynomials x1;1 x2;2 x1;2 x2;1 ;

y1;1 y2;2 y1;2 y2;1 ;

xi;1 y1;j Cxi;2 y2;j ;

yi;1 x1;j Cyi;2 x2;j ;

where i; j 2 f1; 2g. The generators are obtained from the following conditions on 2 2-matrices X D Œxi;j and Y D Œyi;j : rk.X / 1;

rk.Y / 1;

X Y D 0;

Y X D 0:

To obtain a minimal set of generators one needs to cancel one of the following 4 generators: xi;1 y1;i C xi;2 y2;i ;

yi;1 x1;i C yi;2 x2;i ;

i 2 f1; 2g:

Furthermore, D.2; 2; 2/ is isomorphic to the affine cone over the product P 1 P 1 P 1 of projective lines embedded (by Segre map) in P 7 . Let C .2; 3/ and C .2; 2; 2/ be the orbit closure of the representations { k CCC Œ1 CC {{ { CC { }{{ ! k1 k 11

1

Œ1 1

Œ1 Œ1 o k k

k ~ @@ ~~ Œ1@@@Œ1 ~ @@ ~ ~~~ k, k @@ k @@ ~~ @@ Œ1~~~ Œ1 @ ~~~ Œ1 k Œ1

Œ1

and

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G. Zwara

with the coordinate rings kŒC .2; 3/ ' kŒx1 ; x2 ; x3 ; x4 ; x5 =.x1 x2 x3 x4 x5 / and kŒC .2; 2; 2/ ' kŒx1 ; x2 ; x3 ; x4 ; x5 ; x6 =.x1 x2 x3 x4 ; x3 x4 x5 x6 /; respectively. In order to obtain products of the above varieties we use a simple observation that xM 0 M 00 ' O xM 0 O xM 00 O for any representations M 0 2 repQ0 .d 0 / and M 00 2 repQ00` .d 00 /, where the product M 0 M 00 is considered as a representation of the quiver Q0 Q00 . Complexity of the problem of describing varieties occurring as orbit closures of nilpotent representations grows drastically with the dimension of varieties. All varieties occurring in Theorem 3.1 are normal and Cohen–Macaulay (see arguments in Section 2 of [50]) and hence we derive the following result. Corollary 3.2. Let M be a representation in rep.Q/ such that the orbit OM has xM is normal and Cohen–Macaulay. dimension at most 4. Then the variety O We will see later that there is an example of 12-dimensional orbit closure which is not normal (see Example 8.13) and an example of 14-dimensional orbit closure which is not Cohen–Macaulay (see Example 8.14). Problem 3.3. Let V .d / be a complete set of pairwise non-isomorphic varieties of dimension d which are isomorphic to an orbit closure for a quiver representation. Find minimal dimension d such that: (1) the set V .d / is infinite (we know that d 5); (2) there is a variety in V .d / which is not normal (5 d 12); (3) there is a variety in V .d / which is not Cohen–Macaulay (5 d 14). Note that V .3/ contains a variety which is not Gorenstein. Indeed, the variety D.p; q/ (and hence also H D Œr .p; q/, by [18], Proposition 3.1.19) is Gorenstein if and only if p D q.

4 Orbit closures which are hypersurfaces An interesting problem is to characterize representations and modules whose orbit closures are regular varieties. First we can reduce to the case of an admissible quiver representation, say M 2 repQ .d/. Next, since the set of regular points of a GL.d/xM is regular if and only if it is variety is open and GL.d/-invariant, the orbit closure O xM at 0 coincide regular at 0. The latter condition means that the tangent space T0;OxM of O


675

xM at 0 (see [43], III.4). Observe that the tangent space T x , with the tangent cone of O 0;OM naturally identified with a k-linear subspace of repQ .d/, is GL.d/-invariant, so it can be considered as a GL.d/-submodule of repQ .d/. xM is regular at 0. Since O xM is contained in the set rep0 .d/ of Assume that O Q xM at 0 nilpotent representations and the latter is an affine cone, the tangent cone of O is contained in rep0Q .d/ and T0;OxM rep0Q .d/: This is a key information in showing (see [35], Section 3) that the quiver Q has no oriented cycles. Then, using the assumption Ann.M / .kQC /2 , one can prove (see xM D T x D repQ .d/ and Ann.M / D 0. [36], Section 4) that O 0;OM xM is a regular variety. xM D repQ .d/, then obviously O On the other hand, if O Hence we obtain the following characterization. xM is regular Theorem 4.1. Let M be an admissible representation in repQ .d/. Then O xM D repQ .d/. Furthermore, if this is the case, then AnnQ .M / D 0. if and only if O By the Artin–Voigt formula (see [48]) xM D dimk Ext1 .M; M /; codimrepQ .d/ O Q xM D repQ .d/ is equivalent to vanishing of Ext 1 .M; M /. Applying the equality O Q geometric equivalences described in Section 2 we get the following result. xM is a regular variety if and only Corollary 4.2. Let M be a module in mod A. Then O 1 if the algebra B D A= Ann.M / is hereditary and ExtB .M; M / D 0. We call a variety X a local hypersurface at x 2 X if the local ring Ox;X is isomorphic to the quotient of a local regular ring by a principal ideal. A variety X is said to be a local hypersurface provided it is a local hypersurface at any point of X. This property is preserved by smooth morphisms. Of course, regular varieties are local hypersurfaces. Other simple examples of local hypersurfaces are hypersurfaces in affine spaces, i.e. the zero sets of a nonconstant polynomial. It is an interesting question which orbit closures are local hypersurfaces. Example 4.3. Among the 11 varieties listed in Theorem 3.1 there are exactly 4 local hypersurfaces: k; H Œ2 .2; 2/; D.2; 2/ and C .2; 3/: xM , where M 2 repQ .d/ is an adWe reduce the problem to the orbit closure O missible representation. Here the tangent space T0;OxM is not necessarily contained in rep0Q .d/ and the quiver Q can have oriented cycles. However, the GL.d/-submodule T0;OxM of repQ .d/ is contained in reptrace Q .d/ D f.W˛ /˛2Q1 2 repQ .d/ j trace.W˛ / D 0; for any loop ˛ 2 Q1 g:

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If k is of characteristic 0, the group GL.d/ is linearly reductive and we can use Reynolds operator kŒX ! kŒXGL.d/ for any affine GL.d/-variety X. This is applied to obtain the following result: Proposition 4.4. Assume that the field k is of characteristic 0. Let M be an admissible representation in repQ .d/. Then T0;OxM D reptrace Q .d/: We do not know if the proposition remains true without the assumption on the xM is a singular local characteristic of k. Assume now that k is of characteristic 0 and O xM is singular at 0 and therefore dim O xM D dimk T x 1. hypersurface. Then O 0;OM xM is contained in rep0 .d/ and the latter is contained in reptrace .d/, we get that Since O Q Q xM is a hypersurface in the affine space reptrace .d/. Then a detailed analysis leads to O Q the following characterization. Theorem 4.5. Assume that k is of characteristic 0. Let M be an admissible represenxM is a singular local tation in repQ .d/ for a dimension vector d D .di /i2Q0 . Then O hypersurface if and only if one of the following conditions holds: (1) Ann.M / D 0 and Ext1Q .M; M / ' k. (2) Ann.M / D h 2 i, where is a loop in Q1 at a vertex i with di D 2, and Ext1.Q;h 2 i/ .M; M / D 0. (3) Ann.M / D hi, where is a relation in Q starting at a vertex i and terminating at a vertex j with di D dj D 1 and Ext 1.Q;hi/ .M; M / D 0. Example 4.6. The following 3 representations: h

00 10

i

k 2 k 2 ;

k2 e

h

00 10

i

and

k

Œ0 1

k

h i 1 0 2

k

satisfy the conditions .1/, .2/ and .3/, respectively, of the above theorem. Their orbit closures are isomorphic to D.2; 2/, H Œ2 .2; 2/ and D.2; 2/, respectively. Local hypersurfaces are examples of Cohen–Macaulay varieties. By Serre criterion, they are normal if and only if they are regular in codimension 1. Since the orbit closures are irreducible varieties, this means that the singular locus has codimension at least 2. xM is a normal variety if M satisfies condition .2/ or It is relatively easy to show that O xM is normal also in the case .1/. The positive answer is .3/. It is an open question if O given in [36] provided repQ .d/ contains an open GL.d/-orbit (such dimension vector d is called prehomogeneous).

5 Characterizations of degenerations Let M and N be two representations in repQ .d/. It is a fundamental problem when N xM , and what does it mean in module-theoretical terms. belongs to the orbit closure O


677

We present here a characterization. It is worth to point out that this characterization is also useful in studying geometric properties of orbit closures, for instance to construct desingularizations in Section 8. The result below is a slight extension of Theorem 1 of [60]. Theorem 5.1. Let M and N be two d -dimensional modules over a finitely generated algebra A. Then the following conditions are equivalent: xM . (1) M degenerates to N , i.e. ON O (2) There exists a short exact sequence 0 ! Z ! Z ˚ M ! N ! 0 in mod A, for some module Z. (3) There exists a short exact sequence 0 ! N ! M ˚ Z ! Z ! 0 in mod A, for some module Z. xM such that .t / belongs to the orbit (4) There is a regular morphism W k ! O ON , if t D 0, and to OM , otherwise. We present here some ideas of the proof. The implication .4/ ) .1/ follows immediately from the continuity of regular morphisms. We can assume in .2/ that the injective homomorphism f W Z ! Z ˚ M is radical. Indeed, this follows from the following simple fact. Lemma 5.2. Any short exact sequence 0 ! U ! W ! V ! 0 of modules (or 1C

representations) is isomorphic to a direct sum of 0 ! C ! C ! 0 ! 0 and a short exact sequence f0

g0

0 ! U 0 ! W 0 ! V ! 0 with a radical homomorphism f 0 . In particular, U ' C ˚ U 0 and W ' C ˚ W 0 . Proof of .2/ ) .4/. Let f fD 1 f2

h

0 ! Z ! Z ˚ M ! N !0 be a short exact sequence in mod A, such that the homomorphism f is radical. In particular, the endomorphism f1 W Z ! Z is nilpotent. This implies that any non-zero k-linear combination of f and the section j W Z ! Z ˚ M is still a monomorphism. Let W D Z ˚ M . We claim that there is a d -dimensional vector subspace C in W which is complementary to all images of the monomorphisms f C t j W Z ! W , t 2 k. We consider the following representation of the Kronecker quiver RW Z

f j

/

/W .

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G. Zwara

Since all non-zero linear combinations of f and j are injective, the representation R is preprojective. Since dimk W dimk Z D d , R is a direct sum of d indecomposable preprojective representations, say Ri W Z i

fi ji

/

/ Wi .

It is an easy exercise to show that there is a unique 1-dimensional linear subspace Ci of Wi which is complementary to the image of the injective linear map fi Ct ji W Zi ! Wi for each t 2 k. The claim holds for C being the direct sum of Ci ’s. Without loss of generality we may assume that Z 2 modA .e/, where e D dimk Z. We define a regular map g W k ! M.d C e/ as follows. The first e-columns of g.t / are determined by the linear map f C t j , while the remaining d columns are chosen independently of t 2 k and we assume that they are a k-linear basis of C . By our assumptions, the matrix g.t / is invertible, for any t 2 k. Observe that the module g.t/1 W in modA .d C e/ has the block form Z Yt ; 0 Mt which means that g.t /1 W .a/ g.t / D

Z.a/ Y t .a/ ; 0 M t .a/

for any a 2 A:

Here, M t is a module in modA .d /, which is isomorphic to the cokernel of f C t j , thus M t is isomorphic to N if t D 0, and to M otherwise. We define as the regular morphism sending t to M t . Riedtmann showed in [45], in terms of quiver representations, that .2/ implies a slightly weaker version of .4/, where the affine line k is replaced by a cofinite subset of k. The proof of Proposition 3.4 in [45] was implemented above with two modifications: a reduction to the case f W Z ! Z ˚ M is radical and a careful choice of linear complement C . We get the implication .3/ ) .4/ by duality. Assume .1/ xM . It holds, i.e. there are modules M and N in modA .d / such that N belongs to O remains to show the existence of short exact sequences in mod A of special forms as in .2/ and .3/. We use the following general fact from algebraic geometry. Lemma 5.3. Let x be a point on an irreducible variety X and U be a nonempty open subset of X. Then there is an irreducible curve C in X containing x and intersecting U (hence generically contained in U). Proof. Taking an affine open neighbourhood of x if necessary, we may assume that the variety X is affine. By Noether’s normalization theorem there exists a surjective finite morphism ' W X ! k e , where e D dim X. Since U is open in X and the latter


679

is an irreducible variety, dim.X n U/ < e. Consequently, dim '.X n U/ < e. We choose an affine line L in k e containing '.x/ and a point y not contained in '.X n U/. This implies that ' 1 .y/ is contained in U. Applying the going-down theorem we get an irreducible curve C in X containing x and such that '.C/ D L. In particular, C contains a point from ' 1 .y/. xM containing N and intersecting OM (hence C is Let C be an affine curve in O generically contained in OM ). Let W C 0 ! C be the normalization morphism. Then C 0 is a smooth affine curve and is a finite (surjective) map. In particular, there is a point c 0 on C 0 such that .c 0 / D N , and .c/ in contained in OM , for all but a finite number of points c 2 C 0 . We denote by Md .R/ the k-algebra of d d -matrices with coefficients in a commutative k-algebra R. In fact, Md ./ can be considered as a functor from the category of commutative k-algebras to the category of sets. Considering as a map from C 0 ! modA .d / we get a k-algebra homomorphism from A to Md .kŒC 0 /, where kŒC 0 is the coordinate ring of C 0 . Using the composition of the canonical homomorphisms ' yc 0 ;C 0 kŒC 0 ! Oc 0 ;C 0 ! O ! kŒŒt ;

we get a k-algebra homomorphism Y W A ! Md .kŒŒt /: This homomorphism defines an A-module structure on kŒŒt d which is compatible with a canonical kŒŒt -module structure on kŒŒt d . Therefore we may consider Y as an A-kŒŒt-bimodule or, equivalently, as an A ˝k kŒŒt -module. One can derive from properties of the following two isomorphisms Y ˝kŒŒt k..t // ' M ˝k k..t //

and

Y =Y t ' N;

of A-k..t//-bimodules and A-modules, respectively, where k..t // denotes the field of fractions of kŒŒt . The key step is to show that there is an A-module isomorphism Y =Y t hC1 ' M ˚ Y =Y t h

(5.1)

for h large enough (see [60], Proposition 3.5). Applying the functor Y ˝kŒŒt ./ to the short exact sequence 0 ! kŒŒt =.t h / ! kŒŒt =.t hC1 / ! kŒŒt =.t / ! 0 and its dual, we get two short exact sequences in mod A of the form 0!Z !Z˚M !N !0

and

0 ! N ! M ˚ Z ! Z ! 0;

(5.2)

where Z D Y =Y t h . The smallest h 0 satisfying (5.1) is called the complexity of degeneration M to N (see [2] for results concerning the complexity of degenerations). We illustrate Theorem 5.1 with several examples.

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Example 5.4. Let 0 ! U ! M ! V ! 0 be a short exact sequence in mod A. By Lemma 1.4, M degenerates to N D U ˚ V . Indeed, taking direct sums with short exact sequences 0 ! 0 ! U ' ! U ! 0 and 0 ! V ' ! V ! 0 ! 0, respectively, we get exact sequences of the form (5.2). Example 5.5. Let Q W 1 2 3 be the equioriented Dynkin quiver of type A3 . We have 6 pairwise non-isomorphic indecomposable representations of Q, namely, the simple representations S1 , S2 , S3 and P2 W k

Œ1

MW k

Œ1

k k

I2 W 0

0; Œ1

k;

Œ1

k

k:

The representation M is projective and injective. Taking a direct sum of the following exact sequences in rep.Q/: 0 ! 0 ! S1 ! S1 ! 0; 0 ! S1 ! P2 ! S2 ! 0; 0 ! P2 ! M ! S3 ! 0; we get a short exact sequence 0 ! Z ! Z ˚ M ! N ! 0, where Z D S1 ˚ P2 and N D S1 ˚ S2 ˚ S3 . Hence M degenerates to NWk

Œ0

Œ0

k

k:

Example 5.6. We generalize the previous example as follows. Assume that the algebra i A is finite L dimensional and let M 2 mod A. Let Mi D .rad A/ M for i 0. Then N D i0 Mi =MiC1 is the semisimple A-module associated with M . The module M degenerates toL N , and we can obtain a short exact sequence 0 ! Z ! Z ˚ M ! N ! 0 for Z D i1 Mi by taking a direct sum of 0 ! MiC1 ! Mi ! Mi =MiC1 ! 0;

i 0:

The following example is taken from [45], Section 3.4. ˛

Example 5.7. Let Q W 1 representations of .Q; I / ZW 0

/ k2 e

h

00 10

i

;

/2b

MW k

ˇ

and I D hˇ 2 i. We consider the following 3

h i 1 0

/ k2 e

h

00 10

i

;

NW k

h i 0 1

/ k2 e

h

00 10

i

;

681


and 4 homomorphisms ZW fW ZW ZW hW NW

h

00 10

i

/ k2 h

0

h

h i 0 1

h

h

/ k2 h

0 k

h

/ k2 h

0

10 01

i

/ k2 h

h

00 10

i

ZW gW

00 10 00 10

00 10

i

i

i

MW MW lW NW

/ k2 h

0

h

h i 1 0

k

h i 1 0

k k

/ k2 h / k2 h h

Œ 1 h i 0 1

10 01

00 10

/ k2 h

h i h

h i h

00 10

00 10

00 10

00 10

i

i

i

i

The short sequence

f g

. h l /

0 ! Z ! Z ˚ M ! N ! 0

(5.3)

is exact, and so M degenerates to N . Since the endomorphism algebra End.Q;I / .N / is local (isomorphic to kŒx=.x 2 /), the representation N is indecomposable. The representations M and N are not isomorphic (ˇ˛ belongs to AnnQ .N / but not to AnnQ .M /), hence we found a proper degeneration to an indecomposable representation. Let ext be the partial order on the set of isomorphism classes of objects in mod A (or rep.Q; I /) generated by M ext U ˚ V; for any short exact sequence 0 ! U ! M ! V ! 0. By Lemma 1.4, M ext N H) M deg N: The converse implication does not hold in general as M <ext N implies that N is decomposable (see Example 5.7). In fact, the following result (see [60], Corollary 5) characterizes when the partial orders ext and deg coincide. Theorem 5.8. The partial orders ext and deg are equivalent for all modules in mod A if and only if there is no proper degeneration to an indecomposable A-module. Applying the above theorem, the proofs of [13], Proposition 3.2, and the main result of [58] can be shortened. Theorem 5.9. Let Q be a Dynkin or an extended Dynkin quiver (more generally, let A be a representation-directed or a tame quasi-tilted algebra). Then the partial orders ext and deg coincide.

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G. Zwara

ˇ and I D h˛ 2 ; ˇ 2 ; ˛ˇ ˇ˛i. Note that the Example 5.10. Let Q W ˛ S3 | AA | } || }} S2 B I2 A AAf6 BBf4 f5 }}> A B! } } P3 _ _ _ _ S1 f1

where fi ’s are canonical irreducible homomorphisms inducing the almost split sequences "S3 W

f2

f3

0 ! P1 ! I2 ! S3 ! 0;

f1 f4

. f2 f5 /

"I2 W

0 ! S2 ! P1 ˚ P3 ! I2 ! 0;

"S1 W

0 ! P3 ! I2 ! S1 ! 0:

f5

f6

The module M D I2 degenerates to N D S1 ˚ S2 ˚ S3 . By direct calculations, ŒN; P1 ŒM; P1 D ŒN; S2 ŒM; S2 D ŒN; P3 ŒM; P3 D 1; and hence we associate with each almost split sequence the value 1: _ _ _ _ S3 = P1 AA > | AA 1 }}} | || } S2 B 1 > I2 AA BB A }} B! } } 1 A P3 _ _ _ _ S1

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G. Zwara

We take the following splittable short exact sequence ending at N :

0 W

100 010 001

0 ! 0 ! S1 ˚ S2 ˚ S3 ! S1 ˚ S2 ˚ S3 ! 0:

We want to glue 0 with an almost split sequence whose end term is isomorphic to a direct summand of the middle term of 0 , and such that the attached integer is positive. There are 2 possibilities for an almost split sequence: "S3 or "S1 . We use "S3 and obtain

1 W

0 0 f2

!

10 0 01 0 0 0 f3

!

0 ! P1 ! S1 ˚ S2 ˚ I2 ! S1 ˚ S2 ˚ S3 ! 0:

We decrease by 1 the integer associated with "S3 : P1 A_ _ _ _ > S3 AA 0 }} |= | A }} || S2 B 1 > I2 AA BB A }} B! } } 1 A _ _ _ _ S1 P3 Next we can use either "I2 or "S1 . We add "S1 and get

2 W

0 f5 0 0 f2 0

!

f6 0 0 0 1 0 0 0 f3

!

0 ! P1 ˚ P3 ! I2 ˚ S2 ˚ I2 ! S1 ˚ S2 ˚ S3 ! 0:

After decreasing by 1:

P1 A_ _ _ _ > S3 AA 0 }} |= | A }} || S2 B 1 > I2 AA BB A }} B! } } 0 A P3 _ _ _ _ S1

We have to glue 2 with "S1 . But there are 2 direct summands of the middle term of

2 isomorphic to I2 , so we need to choose a section s W I2 ! I2 ˚ S2 ˚ I2 . Using the section s D .1; 0; 0/T we obtain the sequence 0

f1 B f4 @ 0 0

0 0 0 f2

1 0 1C A 0 0

f6 f2 0 0 0 0 01 0 0 0 0 f3

!

0 ! S2 ˚P1 ˚P3 ! P1 ˚P3 ˚S2 ˚I2 ! S1 ˚S2 ˚S3 ! 0: We split off an isomorphism from the injective homomorphism S2 ˚ P1 ˚ P3 ! P1 ˚ P3 ˚ S2 ˚ I2 (see Lemma 5.2) and get

3 W

0 f1 0 0 f2 0

!

f6 f2 0 0 0 1 0 0 0 f3

!

0 ! P1 ˚ S2 ! P1 ˚ S2 ˚ I2 ! S1 ˚ S2 ˚ S3 ! 0;

689


which has the form 0 ! Z ! Z ˚ M ! N ! 0 for Z D P1 ˚ S2 . As we have seen the above construction is not unique. For example, we could use "S1 instead of "S3 at the beginning. Furthermore, the sequence 3 being the result of gluing 2 and "I2 depends on choice of a section s W I2 ! I2 ˚ S2 ˚ I2 . The sections s D .0; 0; 1/T and s D .1; 0; 1/T lead to an exact sequence of the form 0 ! Z ! Z ˚ M ! N ! 0 with Z D P3 ˚ S2 and Z D S2 , respectively. Example 6.8. Let Q be the quiver /2b

˛

1

ˇ

and I D hˇ 2 i. There are, up to isomorphism, 7 indecomposable representations in rep.Q; I /: P1 W k

h i 1 0

h

/ k2 e

00 10

i

/ k2

P2 W 0 h

/k c

S2 W 0 NW k

h i 0 1

/ k2

I2 W k

Œ0

e

h

00 10

2

10 01

h

e

00 10

i

/0b

S1 W k

i

/ k2 e

h

00 10

i

TW k

Œ1

/k c

Œ0

i

The Auslander–Reiten quiver .Q;I / has the form T _ _ _ _ _ S2 B |= BBBB˛9 ˛10 ||= BB˛8 | B! || B || S2 B N T BB˛4 ˛5 ||= BBB˛6 ˛7 ||> B! | B | | | ! I P2 B 2 BB ˛3 BB˛1 ˛2 ||= BB B! || _ _ _ _ _ S1 P1 ˛8

with the canonical irreducible A-homomorphisms ˛i , 1 i 10, inducing the almost split sequences ˛2

˛3

"S1 W

0 ! P1 ! I2 ! S1 ! 0;

"I2 W

0 ! P2 ! P1 ˚ N ! I2 ! 0;

"N W

0 ! S2 ! P2 ˚ T ! N ! 0;

"T W

0 ! N ! I2 ˚ S2 ! T ! 0;

"S2 W

0 ! T ! N ! S2 ! 0:

.˛1 ;˛5 /T

.˛4 ;˛8 /T

.˛6 ;˛10 /T

˛9

˛10

.˛2 ;˛6 /

.˛5 ;˛9 /

.˛7 ;˛8 /

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The representation M D P1 degenerates to N and after some calculations we get T _ _ _ _ _ = S2 B |= BBBB BB | | B! 0 ||| B || S2 B 1 >T = N BB 1 BB BB || || B! || || ! P2 B 1 = I2 BB BB B || B! | | 0 B P1 _ _ _ _ _ S1 We begin with the splittable sequence ending at N :

0 W

.1/

0 ! 0 ! N ! N ! 0:

Now we glue 0 with "N : .˛4 ;˛8 /T

1 W

.˛5 ;˛9 /

0 ! S2 ! P2 ˚ T ! N ! 0I

then with "T :

2 W

.˛4 ˛10 ;˛6 /T

.˛5 ;˛9 ˛7 /

0 ! N ! P2 ˚ I2 ! N ! 0I

and finally we use "I2 :

3 W

.˛4 ˛10 ˛5 ;˛1 /T

.˛5 ;˛9 ˛7 ˛2 /

0 ! P2 ! P2 ˚ P1 ! N ! 0:

The last sequence is isomorphic to (5.3) in Example 5.7. We finish the section with an algorithmic applications of Theorem 6.6 for degenerations of modules over algebras of finite representation type. We identify ind A D fX1 ; : : : ; Xn g with the set of vertices of the Auslander–Reiten quiver A . The first question is: how to compute dimensions ŒXi ; Xj ? We do not need matrix forms of the modules Xi ’s, as we can read those dimensions from the structure of A . Let 8 ˆ Xi D Xj or Xi D Xj ; ŒM; N ŒM; M and

ŒN; N ŒM; M D 1

that ŒM; M D ŒM; N . Applying the characterization of degenerations we get a short exact sequence of the form f

0!Z !Z˚M !N !0 with f being a radical homomorphism. Using the assumption codim.M; N / D 1, one can show that Z is an indecomposable module. The key fact to be proved is that ŒZ; M D ŒZ; N . Then Sing.M; N / D Reg, by Theorem 12.7. Suppose the contrary holds, namely ŒZ; M < ŒZ; N . Then there exists a short exact sequence of the form

fQ gQ

. fQ hQ / 0 ! Z ! Z ˚ Y ! Z ! 0

(13.1)

for some module Y and a nilpotent endomorphism fQ (see the proof of Proposition 2.7 in [64]). Furthermore, ŒZ; Z ŒY; Y D 1, which means Y <deg Z is a codimension one degeneration. We shall prove that this leads to a contradiction, the key idea is to consider how the endomorphisms gQ hQ and gQ fQhQ act on the algebra EndA .Y /. Since Q 3 D .gQ fQh/ Q 2 . Let R denote the subalgebra kŒŒm2 ; m3 of the hQ gQ D fQ2 , we have .gQ h/

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G. Zwara

power series ring kŒŒm. Then EndA .Y / becomes an R-R-bimodule, where m2 and m3 Q respectively. In [64] the subalgebra kŒm2 ; m3 of act as composition with gQ hQ and gQ fQh, the polynomial ring kŒm was considered instead of kŒŒm2 ; m3 , but all considerations presented there can be repeated with respect to the subalgebra kŒŒm2 ; m3 . One says that a left R-module W has property [P1] if the sequence

W W

m3 m2 m4 m3

W ! W

m3 m2 m4 m3

W ! W

is exact. Dually, one says that a right R-module W has property [P10 ] if the sequence

W

m3 m4 m2 m3

W ! W

m3 m4 m2 m3

W ! W

W

0 is exact. In fact, the R-R-bimodule W W E D EndA .Y / satisfies [P1] and [P1 ]. Let W be an R-R-bimodule, N D W W and consider the maps

m3 m2 4 3 m m

W N ! N

and

m3 m4 m2 m3

W N ! N;

given by left and right, respectively, multiplications of N by 2 2-matrices. Observe that D D 0 and D . We say that W has property [P2] if the sequence

N ! N ! N ˚ N is exact. In a similar way we can introduce the concept of the property [P2] for a left S -module, where S is the subalgebra kŒŒm2 ; m3 ; n2 ; n3 of the power series ring kŒŒm; n. Here, the multiplications by n2 and n3 correspond to the right R-module structure. 2 0 Observe that N D M M is a kŒ=. /-module, for any R-module M and N D M0 M0 is a kŒ; =. 2 ; 2 /-module, for any S-module M 0 , where the residue classes M0 M0 3 m2 and the right multiplication by of and denote the left multiplication by m m4 m3 3 4 m m , respectively. Since the algebras kŒ=. 2 / and kŒ; =. 2 ; 2 / are local m2 m3 and Frobenius, the free modules over them coincide with the projective modules and with the injective ones. One can prove that M has property [P1] if and only if N is a free kŒ=. 2 /-module, and M 0 has property [P2] if and only if N 0 is a free kŒ; =. 2 ; 2 /module. Treating R as an S-module, we consider a short exact sequence of S -modules: 0 ! I ! R ˚ S i ! E ! 0;


715

where the S-homomorphism R ! E sends 1R to 1E . Applying the condition codim.Y; Z/ D 1, one can prove that I satisfies [P2] (see [64], Lemma 4.4). Finally, considering a free resolution of an S -module I leads to a contradiction (see [64], Section 4). Consequently, Sing.M; N / D Reg. The gaps in [64] are incorrect applications of Nakayama lemma in the proofs of Lemma 3.1 and Proposition 3.6. A proper formulation of these two facts should be: Lemma 13.2. Let M be a finitely generated submodule of a free kŒŒm2 ; m3 -module. If M has property [P1] then it is free. Lemma 13.3. Let M be a finitely generated kŒŒm2 ; m3 ; n2 ; n3 -module having property [P2]. If M is torsion free as a kŒŒn2 ; n3 -module, then there is a short exact sequence of kŒŒm2 ; m3 ; n2 ; n3 -modules 0!U !W !M !0 for some finitely generated free kŒŒm2 ; m3 ; n2 ; n3 -modules U and W . The statements of Corollaries 5.2 and 5.3 in [64] (which were not applied in that paper) must be corrected as well: Corollary 13.4. Let M be a finitely generated kŒŒm2 ; m3 -module. Then the following conditions are equivalent: (1) M has property [P1]; (2) there is a free resolution 0 ! F1 ! F0 ! M ! 0 of M ; (3) M has a finite projective dimension. Corollary 13.5. Let M be a finitely generated kŒŒm2 ; m3 ; n2 ; n3 -module. Then the following conditions are equivalent: (1) M has property [P2]; (2) there is a free resolution 0 ! F2 ! F1 ! F0 ! M ! 0 of M ; (3) M has a finite projective dimension. The proof of Theorem 13.1 is complicated, it would be nice to find a simpler one. Note that the assumption codim.Y; Z/ D 1 was crucial in the proof. If codim.Y; Z/ D 2 then there are exact sequences (Example 5.10) of the form (13.1). Problem 13.6. Let 0 ! Z ! Z ˚ Y ! Z ! 0 be a nonsplittable exact sequence in mod A with Z indecomposable. Is the algebra A necessarily of infinite representation type? There is a direct consequence of Theorem 13.1: xM contains only finitely many orbits Corollary 13.7. Let M be a module such that O (for instance, if the algebra A is of finite representation type or A is the path algebra xM is regular in codimension 1. of a cyclic quiver). Then the variety O

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Bender and Bongartz showed in [7], Section 4.2, that the orbit closures for the Kronecker quiver are regular in codimension 1. They gave also indications how to generalize that for other extended Dynkin quivers. We will follow their ideas. Theorem 13.8. Let Q be an extended Dynkin quiver and M 2 rep.Q/. Then the xM is regular in codimension 1. variety O xM n OM which is of codimension 1 in Proof. Let be an irreducible component of O x xM . If contains a dense OM . We have to show that contains a smooth point of O orbit, then the claim follows from Theorem 13.1. As pointed out in [7], Section 4.2, by taking an open subset of if necessary and canceling common direct summands, we may assume that contains only minimal degenerations N such that M D M 0 ˚ U p1 ˚ V q1 deg N D U p ˚ V q and there exists a short exact sequence W 0 ! U ! M 0 ! V ! 0, for some indecomposable representations U , V and p; q 1 depending on N . Up to duality, the only problem is when U is preprojective and V is regular. Using standard arguments, one can prove that ŒU; M D ŒU; N : (13.2) Replacing by its open subset if necessary, we get that q D 1, M is a preprojective representation, consists of orbits of representations N D U p1 ˚ V , where runs through a cofinite subset of k, and V are -periodic indecomposable representations. xM , then codim.M; N / D 2, for any (observe that Since has codimension 1 in O dimk EndQ .N / does not depend on ). The sequence does not split and therefore ı .U / 1 and ı .V / D ı0 . V / D ı0 .V / 1; by the Auslander–Reiten formula (6.1). Consequently, 2 D ŒN ; N ŒM; M D .ŒN ; N ŒM; N / C .ŒM; N ŒM; M / D p ı .U / C ı .V / C .ŒM; N ŒM; M / p C 1 C 0 2: This implies that p D 1 and ŒM; N D ŒM; M . Therefore M D M 0 , N D U ˚ V , and ŒU ˚ M; M D ŒU ˚ M; N , by (13.2). Finally, Sing.M; N / D Reg, by Corollary 12.8. It follows from Example 8.13 that there exist orbit closures which are not regular in codimension 1, even for algebras of tame domestic type.

14 Singularities for degenerations of codimension two Let M <deg N be a degeneration of codim.M; N / D 2. It is interesting to study the type Sing.M; N /. We have seen in Example 7.1 that the Kleinian singularities of type


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A can occur. We start with other examples of types of singularities for codimension two degenerations that can be calculated directly using the method of transversal slices described in Section 7. o

Example 14.1. Let Q be the Kronecker quiver 1 o

˛ ˇ

2 . Let Pn denote the prepro-

jective indecomposable representation with dim Pn D .n; n 1/ for n 1. Let d 1 and take M D P2 ˚ Pd C2 and N D P1 ˚ Pd C3 . Then (see [7]) Sing.M; N / D Sing.f.s d ; s d 1 t; : : : ; t d / 2 k d C1 j s; t 2 kg; 0/ D Sing.f.x0 ; : : : ; xd / 2 k d C1 j xi xj D xi 0 xj 0 for i C j D i 0 C j 0 g; 0/ is the type of the affine cone over a rational normal curve of degree d . Here Sing.M; N / is Reg for d D 1 and Kleinian of type A1 for d D 2. Example 14.2. We return to the algebra A D kŒx; y=.xy; x 2 y 2 / and the modules ´ 0 0 0 0 0 0 0 0 00 ; 0 0 ; 2 k; 1 0 0 0 0 0 0 0 M D AA D ; U D 10 00 0 00 0000 ; 1000 ; D 1: 0100 0010 00 ; 10 Recall that if 2 ¤ 1 and N D U ˚ U1= then Sing.M; N / D Sing.fxy D 0g; 0/ is the type of isolated singularity of two crossing lines. We assume that 2 D 1 and define 0 0 0 0 0 0 0 0 1000 ; 0 0 0 N D 0000 0 0 0 0 0010

1 00

(there is an almost split sequence 0 ! U ! N ! U ! 0). Then Sing.M; N / D Sing.fx 2 C xyz C y 2 z D 0g; 0/: Observe that the surface fx 2 C xyz C y 2 z D 0g has 1-dimensional singular locus fx D y D 0g and the singularity at .0; 0; z/, for z ¤ 0, is smoothly equivalent to two crossing lines. Moreover, Sing.fx 2 C xyz C y 2 z D 0g; 0/ D Sing.fx 2 C y 2 z D 0g; 0/ if k is not of characteristic 2. The latter singularity is denoted by D1 . Example 14.3. Let A D kŒx; y=.x 2 ; y 2 / and M D A A. This algebra is isomorphic to the one from the previous example provided k is not of characteristic 2. We assume here that k is of characteristic 2 and consider the family of indecomposable representations U , 2 k [ f1g, defined as in the previous example. Then there are exact sequences 0 ! U ! M ! U ! 0

and

0 ! U ! N ! U ! 0

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such that the latter is almost split and N is indecomposable for any . Then M degenerates to N , by [13], Lemma 1.1. Surprisingly, we get Sing.M; N / D Sing.fx 2 C y 2 z D 0g; 0/: Note that in characteristic 2 the surface fx 2 Cy 2 z D 0g has the same type of singularity at any singular point .0; 0; z/, z 2 k. Example 14.4. We consider the quiver QW

Representations of Algebras and Related Topics (EMS Series of Congress Reports)

Representations of Algebras and Related Topics

Representations of Algebras and Related Topics (ICRA X, Toronto 2002)

Representations of Algebras and Related Topics (ICRA V, Tsukuba 1990)

Surveys in Stochastic Processes (EMS Series of Congress Reports)

Lie Algebras and Related Topics