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

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

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

Zuhair Nashed University of Central Florida Orlando, Florida

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology S. Kobayashi University of California, Berkeley Marvin Marcus University of California, Santa Barbara W. S. Massey Yale University

Anil Nerode Cornell University Freddy van Oystaeyen University of Antwerp Donald Passman University of Wisconsin Fred S. Roberts Rutgers University David L. Russell Virginia Polytechnic Institute and State University Walter Schempp Universität Siegen

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS Recent Titles G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications W. A. Carnielli et al., Paraconsistency A. Benkirane and A. Touzani, Partial Differential Equations A. Illanes et al., Continuum Theory M. Fontana et al., Commutative Ring Theory and Applications D. Mond and M. J. Saia, Real and Complex Singularities V. Ancona and J. Vaillant, Hyperbolic Differential Operators and Related Problems G. R. Goldstein et al., Evolution Equations A. Giambruno et al., Polynomial Identities and Combinatorial Methods A. Facchini et al., Rings, Modules, Algebras, and Abelian Groups J. Bergen et al., Hopf Algebras A. C. Krinik and R. J. Swift, Stochastic Processes and Functional Analysis: A Volume of Recent Advances in Honor of M. M. Rao S. Caenepeel and F. van Oystaeyen, Hopf Algebras in Noncommutative Geometry and Physics J. Cagnol and J.-P. Zolésio, Control and Boundary Analysis S. T. Chapman, Arithmetical Properties of Commutative Rings and Monoids O. Imanuvilov, et al., Control Theory of Partial Differential Equations Corrado De Concini, et al., Noncommutative Algebra and Geometry A. Corso, et al., Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications – VII L. Sabinin, et al., Non-Associative Algebra and Its Application K. M. Furati, et al., Mathematical Models and Methods for Real World Systems A. Giambruno, et al., Groups, Rings and Group Rings P. Goeters and O. Jenda, Abelian Groups, Rings, Modules, and Homological Algebra J. Cannon and B. Shivamoggi, Mathematical and Physical Theory of Turbulence A. Favini and A. Lorenzi, Differential Equations: Inverse and Direct Problems R. Glowinski and J.-P. Zolesio, Free and Moving Boundries: Analysis, Simulation and Control S. Oda and K. Yoshida, Simple Extensions with the Minimum Degree Relations of Integral Domains I. Peeva, Syzygies and Hilbert Functions V. Burd, Method of Averaging for Differential Equations on an Infinite Interval: Theory and Applications F. Van Oystaeyen, Virtual Topology and Functor Geometry

Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-6056-0 (Softcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable eﬀorts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microﬁlming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-proﬁt organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identiﬁcation and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Oystaeyen, F. Van, 1947Virtual topology and functor geometry / Fred Van Oystaeyen. p. cm. -- (Lecture notes in pure and applied mathematics ; 256) Includes bibliographical references and index. ISBN 978-1-4200-6056-0 (pbk. : alk. paper) 1. Categories (Mathematics) 2. Grothendieck categories. 3. Representations of congruence lattices. 4. Sheaf theory. 5. Dynamics. 6. Noncommutative function spaces. I. Title. II. Series. QA169.O97 2007 512’.62--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1 A Taste of Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1

Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Examples and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Grothendieck Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Separable Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Noncommutative Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1

2.2

2.3

2.4 2.5

Small Categories, Posets, and Noncommutative Topologies . . . . . . . . . . . 11 2.1.1 Sheaves over Posets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 2.1.2 Directed Subsets and the Limit Poset . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.3 Poset Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 The Topology of Virtual Opens and Its Commutative Shadow . . . . . . . . . 19 2.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.2.1 More Noncommutative Topology . . . . . . . . . . . . . . . . . . . 28 2.2.2.2 Some Dimension Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Points and the Point Spectrum: Points in a Pointless World . . . . . . . . . . . 29 2.3.1 Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.1.1 The Relation between Quantum Points and Strong Idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.1.2 Functions on Sets of Quantum Points . . . . . . . . . . . . . . . 36 Presheaves and Sheaves over Noncommutative Topologies. . . . . . . . . . . .36 2.4.1 Project: Quantum Points and Sheaves . . . . . . . . . . . . . . . . . . . . . . . . 39 Noncommutative Grothendieck Topologies . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5.1 Warning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5.2 Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5.2.1 A Noncommutative Topos Theory . . . . . . . . . . . . . . . . . . 44 2.5.2.2 Noncommutative Probability (and Measure) Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 vii

viii

Contents 2.5.2.3 Covers and Cohomology Theories . . . . . . . . . . . . . . . . . . 45 2.5.2.4 The Derived Imperative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.6 The Fundamental Examples I: Torsion Theories . . . . . . . . . . . . . . . . . . . . . 45 2.6.1 Project: Microlocalization in a Grothendieck Category . . . . . . . . 63 2.7 The Fundamental Examples II: L(H ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.7.1 The Generalized Stone Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.7.2 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.7.3 Project: Noncommutative Gelfand Duality . . . . . . . . . . . . . . . . . . . 73 2.8 Ore Sets in Schematic Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3 Grothendieck Categorical Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.1 3.2

Spectral Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Affine Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.2.1 Observation and Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.3 Quotient Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.3.1 Project: Geometrically Graded Rings . . . . . . . . . . . . . . . . . . . . . . . 100 3.4 Noncommutative Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.4.1 Project: Extended Theory for Gabriel Dimension . . . . . . . . . . . . 107 3.4.2 Properties of Gabriel Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.4.3 Project: General Birationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4 Sheaves and Dynamical Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.1

Introducing Structure Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.1.1 Classical Example and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.1.2 Abstract Noncommutative Spaces and Schemes . . . . . . . . . . . . . 113 4.1.3 Project: Replacing Essential by Separable Functors . . . . . . . . . . 119 4.1.4 Example: Ore Sets in Schematic Algebras . . . . . . . . . . . . . . . . . . 119 4.2 Dynamical Presheaves and Temporal Points . . . . . . . . . . . . . . . . . . . . . . . . 121 4.2.1 Project: Monads in Bicategories . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.2.2 Project: Spectral Families on the Spectrum . . . . . . . . . . . . . . . . . . 133 ˇ 4.2.3 Project: Temporal Cech and Sheaf Cohomology . . . . . . . . . . . . . 134 4.2.3.1 Subproject 1: Temporal Grothendieck Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 ˇ 4.2.3.2 Subproject 2: Temporal Cech Cohomology and Sheaf Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.2.4 Project: Dynamical Grothendieck Topologies . . . . . . . . . . . . . . . 135 4.2.5 Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.3 The Spaced-Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.3.1 Noncommutative Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.3.1.1 Toward Real Noncommutative Manifolds . . . . . . . . . . 140 4.3.2 Food for Thought: From Physics to Philosophy . . . . . . . . . . . . . . 141 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Foreword

In order to arrive at a version of Serre’s global sections theorem in the noncommutative geometry of associative algebras, one is forced to introduce a noncommutative topology of Zariski type. Sheaves over such a noncommutative topology do not constitute a topos, but that is exactly the reason why sheaf theory in this generality can carry the essential noncommutative information generalizing to a satisfactory extent classical scheme theory. The noncommutativity forces, at places, a departure from set theory-based techniques resulting in a higher level of abstraction, because opens are not sets of points. Based on some intuition stemming mainly from noncommutative algebra and classical geometry, I strived for an axiomatic introduction of noncommutative topology allowing at least a minimalistic version of geometry involving actual “spaces” and not merely a mask for noncommutative algebra! Completely new problems appear already at the fundamental level, requiring new ideas that sometimes almost alienate a pure algebraist. Not all such ideas are completely developed here, often I restricted myself to bare necessities but left room for many projects ranging from the exercise level to possible research. The spirit of these notes is somewhat experimental reflecting the initial stage of the theory. This may occasionally result in a certain imbalance between novelty sections on new aspects of virtual topology and functor geometry on one hand versus well-established parts of noncommutative algebra on the other. In either case I tried to supply sufficient background material concerning localization theory or some facts on the classical lattice L(H ) of quantum mechanics for some Hilbert space H . On the other hand, I included a few topics that are, at this moment, only important for some of the research projects. In recent years “research training” for so-called young researchers became a trendy topic, and several of the included projects might be viewed in such a framework; however, some projects mentioned are probably hard and essential for better development of the theory and its applications. Intrinsic problems related to sheafification over a noncommutative space are the main topic in Section 4.2 and represent the introduction of a dynamic version of noncommutative topology and geometry. Since this construction is strictly related to the “absence” of points or of “enough points” in the noncommutative spaces, the dynamic theory as defined here is an exclusively noncommutative phenomenon; it is trivialized in the commutative case where space, and its topology, is described by sets of points. While reading Section 4.3 the reader should maintain a physics point of view because a noncommutative model for “reality” is hinted at; I included some observations related to this “spaced time,” resulting from recent interactions with several physicists, just as food for thought. I welcome all reactions and suggestions, for example, concerning the projects or the general philosophy of the topic. F. Van Oystaeyen

ix

Acknowledgments

Research in this work was financially supported by : r

An E.C. Marie Curie Network (RTN - 505078) LIEGRITS

r

A European Science Foundation Scientific Programme: NOG

The author fittingly supported these projects in return. I thank my students and some colleagues for keeping the fire burning somewhere, and the Department of Mathematics at the University of Antwerp for staying small, even after the fusion of the three former Antwerp universities. I especially appreciated moral support from E. Binz, B. Hiley, and C. Isham; they shared their vast knowledge in both physics and mathematics with me, and by showing their interest, motivated me to further the formal construction of noncommutative topology. Finally, thanks to my family for the life power line.

xi

Introduction

Noncommutativity of certain operations in nature as well as in mathematics has been observed since the early development of physics and mathematics. For example, compositions of rotations in space or multiplication of matrices are well-known examples often highlighted in elementary algebra courses. More recently even geometry became noncommutative, and nowadays motivation for the consideration of intrinsically noncommutative spaces stems from several branches of modern physics, for example, quantum gravity, some aspects of string theory, statistical physics, and so forth. From this point of view it seems to be necessary to have a concept of space and its geometry that is fundamentally noncommutative even to the extent that one would not expect that its mathematics is built on set theory or the theory of topoi. On the other hand, some branches of noncommutative geometry realize the noncommutative space solely via the consideration of noncommutative algebras as algebras of functions on an undefined fantasy object called the noncommutative variety or manifold. Nevertheless this technique is relatively successful, and it allows a perhaps surprising level of geometric intuition combined with algebraic formalism either in the algebraic or differential geometry setup. Further generalization may be obtained by conveniently replacing sheaf theory on the Zariski or real topologies by more abstract theoretical versions of it. In such a theory, the objects of interest on the algebraic level are either some types of quantized algebras or suitable C ∗ -algebras. The fact that noncommutativity may force a departure from set theoretic foundations creates a parallel development on the side of logic involving non-Boolean aspects as in quantum logic or quantales replacing Grothendieck’s locales. The different points of view fitting the abstract picture sketched above do not seem to fit together seamlessly; in particular, some desired applications in physics do not seem to follow from spaceless geometry, even if some applications do exist already. For example, the symbiosis between quantales and C ∗ algebras defies more general applicability for algebras of completely different type. We may now rephrase the ever-tantalizing dilemma as points or no points, that’s the question! On one hand, the introduction of a pointless geometry defined by posets with suitable operations extending the idea of a lattice to the noncommutative situation, with the partial order relation not necessarily related to set theoretic inclusion, seems to be very appropriate. After all, an abstract poset approach to quantum gravity seems to be at hand! On the other hand, there are points in a pointless geometry! In fact, there are different kinds of points, and in specific situations a certain type of point is more available than another. The problem then arises whether a noncommutative topology, defined on a

xiii

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Introduction

noncommutative space in terms of a noncommutative type of lattice that replaces the set of opens, is to some extent characterized by sets (!) of noncommutative points. Observe that when defining the Zariski topology on the prime (or maximal) ideal spectrum of a commutative (Noetherian) ring, one actually defines the opens by specifying their points and the spectrum is defined before the topology. The difference between presheaf and sheaf theory is completely encoded in the relations between sections on opens and stalks at points. The sheafification functor may be the ultimate example of this interplay, its construction depends on consecutive limit constructions from basic opens to points by direct limits, and from points to arbitrary opens by inverse limits. Even in classical commutative geometry there is a difference when prime ideals of the ring are viewed as points of the spectrum or only maximal ideals are considered as such. However, at the basic level there is absolutely nothing to worry about because the type of rings considered, for example, commutative affine algebras over a field, are Jacobson rings (and Hilbert rings); that is, every prime ideal is determined by the maximal ideals containing it, and in fact it is the intersection of them. So even the commutative case learns that once a topology is given there are still several consistent ways to decide what the points, but when a notion of points is fixed first, the topology has to be adjusted to this notion in order to obtain a useful sheaf theory. Another most important property in classical geometry is that varieties, schemes, or manifolds are locally affine in some sense; for example, every point has an affine neighborhood. In a pointless geometry the latter property is hard to understand and a serious modification seems to be necessary. It will turn out that for this reason, one has to introduce representational theoretic aspects in the abstract theory. Now, for the kind of noncommutative algebraic geometry in the sense of a generalization of scheme theory over noncommutative algebras, as promoted by the author (for example in [44]), the presence of module theory and a theory of quasicoherent sheaves make this possible. But what remains if we try to drop all unnecessary (?) restrictions concerning the presence of an algebra, modules, spectra, points, and so forth, and try to arrive at a barely abstract geometry based on a kind of topology equipped with some functors on a general but suitable category or family of categories? Well, perhaps virtual topology and functor geometry! In the following I try to indicate how such a general theory will have to deal with the issues raised above. First, noncommutative topology is introduced via the notion of a noncommutative lattice where the operations ∧ and ∨ are defined axiomatically and they are less strictly connected to the partial order than the meet and join in usual lattices. The noncommutative topologies may be considered as sets of opens, but an open can in no way be viewed as a set. Noncommutative topologies do fit in a theory of noncommutative Grothendieck topologies but not in topos theory; a noncommutative version of the latter remains to be developed. Then points and minimal points may be defined in a generalized Stone space associated to a noncommutative topology. There are not enough points to characterize an open to which they belong, but there is a well-behaved notion of commutative shadow of a noncommutative space, which is given by a real lattice in the usual sense, and where the commutative opens are characterized by sets of points. At this point

Introduction

xv

generalized function theory could be developed but we did not go into this; rather we introduced sheaf theory on noncommutative topologies and verified that they transfer nicely to the generalized Stone space. A complete symmetrization of Grothedieck’s definition of a Grothendieck topology leads to noncommutative (left, right, skew) versions of this, and the noncommutative topologies defined axiomatically fit into the latter framework by restricting to certain partial order relations, that is, the so-called generic relations. All of this is in Chapter 2 ending with two fundamental examples: the lattice of torsion theories or Serre quotient categories of a Grothendieck category, and the lattice of closed linear subspaces of a Hilbert space. The first one has enough points in the “prime” sense; the second has enough minimal points (maximal filters) in the Stone space. In Chapter 3, Grothendieck categorical representations are studied with the aim of arriving at an abstract notion of affine open. When applying this to the algebraic geometry of associative algebras, for example, schematic algebras, their modules and the localizations of module categories, the general notion of affine open describes exactly the opens corresponding to exact localization functors commuting with direct sums (functors with an adjoint of a specific type). The general notion of quotient representation may then be used to explain how noncommutative projective spaces arise from noncommutaive affine spaces. Some sheaf theory is developed; in a sense this is an extension of a theory of quantum sheaves considered earlier by changing from categories of opens in a topology to more general lattices, but now we even allow the suitable noncommutative version of lattices. The creation of a new theory sometimes opens many doors, maybe too many doors. For example, the further development of the noncommutative version of topology, for example, closed sets, compactness, convergence structure is possible in the generalized Stone space, but we have not even tried to go there. Even though it may well be that such theory is interesting in its own right, we have only mentioned this as a project for the zealous reader looking for an original way to test his/her skills. Even more haunting ideas about noncommutative probability or measure theory have suffered the same fate. Some projects, however, are more straightforward exercises leading to possible research projects. The final section starts out swinging — well at least we propose a dynamical version of topology and sheaf theory, providing at least one solution of the problem of sheafification independent of generalizations of topos theory. It required a rephrasing of continuities in a poset setup with a totally ordered set (time!) as a parameter set. The result is a spectrum with a classical topology existing at each moment but not varying in time the way the noncommutative topologies do. This may be seen as a mathematical uncertainty principle or better as mathematics of observation. One might hope that physical phenomena, in particular quantum theories, may suitably be phrased in terms of this observational mathematics — perhaps a dream. For the more algebraic, or more geometric, applications of ideas expounded in these notes, we may refer to earlier work in noncommutative geometry — in particular the theory of schematic algebras (see [48]). It is not surprising that the geometric structures stripped to their naked abstraction retain a somewhat esoteric character, highlighting mainly partial ordered sets with noncommutative operations but related

xvi

Introduction

to lattices (via the commutative shadow of spaces), categorical methods and functorial constructions, a further abstraction of sheaf theory and spectral constructions, and a categorical representation theory using Grothendieck categories. Some ideas in these notes have already inspired a few recent papers in physics, so without trying to claim more, I hope that the exercise of digging deep for the abstract skeleton of geometry may lead to a further unification of different kinds of noncommutative geometry and point to an actual space of an intrinsically noncommutative nature, perhaps allowing the expression of observations concerning natural phenomena.

Projects

2.2.2.1 More Noncommutative Topology 2.2.2.2 Some Dimension Theory 2.3.1.1 The Relation between Quantum Points and Strong Idempotents 2.3.1.2 Functions on Sets of Quantum Points 2.4.6 Quantum Points and Sheaves 2.5.2.1 A Noncommutative Topos Theory 2.5.2.2 Noncommutative Probability (and Measure) Theory 2.5.2.3 Covers and Cohomology Theories 2.5.2.4 The Derived Imperative 2.6.1 Microlocalization in a Grothendieck Category 2.7.3 Noncommutative Gelfand Duality 3.3.1 Geometrically Graded Rings 3.4.1 Extended Theory for Gabriel Dimension Exercise 3.3 Krull and Gabriel Dimension for a Skew Topology and Its Relation to Commutative Shadow Exercise 3.4 Develop a Theory of Representation Dimension in Connection with Grothendieck Quotient Representations Exercise 3.5 Gabriel Dimension for Sheaf Categories and Related Behavior with Respect to Separable Functors Exercise 3.6 Using the Gabriel Dimension for Noncommutative Valuation Rings of Arbitrary Rank 3.4.3 General Birationality 4.1.3 Replacing Essential by Separable Functors

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xviii

Projects

4.2.1 Monads in Bicategories 4.2.2 Spectral Families ˇ 4.2.3 Temporal Cech and Sheaf Cohomology 4.2.3.1 Temporal Grothendieck Representations ˇ 4.2.3.2 Temporal Cech Cohomology and Sheaf Cohomology 4.2.4 Dynamical Grothendieck Topologies

Chapter 1 A Taste of Category Theory

1.1

Basic Notions

We assume that the reader is familiar with the foundations of set theory, at least in its intuitive version. Acategory C consists of a class of objects together with sets HomC (X, Y ), for any pair of objects X, Y of C, satisfying suitable conditions listed hereafter. The elements of HomC are called C-morphisms or just morphisms, if there is no ambiguity concerning the category considered. For any object X of C there is a distinguished element I X ∈ HomC (X, X ), called the identity morphism of X . For any triple X, Y, Z of objects in C there is a composition map: HomC (X, Y ) × HomC (Y, Z ) → HomC (X, Z ), ( f, g) → g ◦ f such that the following properties hold: i. For f ∈ HomC (X, Y ), g ∈ HomC (Y, Z ), h ∈ HomC (Z , W ) we have: h◦(g◦ f ) = (h ◦ g) ◦ f . ii. For f ∈ HomC (X, Y ) we have f ◦ I X = f = IY ◦ f . iii. If (X, Y ) = (X , Y ), then HomC (X, Y ) and HomC (X , Y ) are disjoint sets.

1.1.1

Examples and Notation

i. The category Set is obtained by taking the class of all sets using maps for the morphisms. ii. The category Top is obtained by taking the class of all topological spaces using continuous functions for the morphisms. iii. The category Ab is obtained by taking the class of all abelian groups using groups homomorphisms for the morphisms. iv. The category Gr is obtained by taking the class of all groups using group homomorphisms for the morphisms.

1

2

A Taste of Category Theory v. The category Ring is obtained by taking the class of all rings using ring homomorphisms for the morphisms. vi. For any given ring R the class of left R-modules using left R-linear maps for the morphisms defines the category R-mod. The category defined in a similar way but using right R-modules and right R-linear maps is denoted by mod-R.

Definition 1.1 Consider a class D consisting of objects of C, then D is said to be a subcategory of C when the following properties hold: i. For objects X, Y in D we have: HomD (X, Y ) ⊂ HomC (X, Y ). ii. Composition of morphisms in D is as in C. iii. For X in D, I X is the same as in C. We say that D is a full subcategory of C when HomD (X, Y ) = HomC (X, Y ) for all X, Y of D. In the list of examples one easily checks that Ab is a subcategory of Gr but, for example, Gr is not a full subcategory of Set. Definition 1.2 For a set S and a family of categories (C)s∈S we define the direct product category C by taking the class of objects to consist of the families (X s )s∈S of objects X s of C s for s ∈ S. If X = (X s )s∈S , Y = (Ys )s∈S are such families, then HomC (X, Y ) = {( f s )s∈S , f s ∈ HomCs (X s .Ys ), s ∈ S}. Composition of morphismsis defined componentwise. This direct product category C will be denoted by s∈S C s ; in case C s = C for all s ∈ S, then we also write C S . In case S = {1, . . . , n} we also write C 1 × C 2 × · · · × Cn for the direct product. This paragraph deals with specific properties of morphisms. A C-morphism from an object X to an object Y will be denoted by X → Y , and if f ∈ HomC (X, Y ) we will write f : X → Y . A monomorphism (in C) is a morphism f : X → Y such that for any object Z and given morphisms h, g ∈ HomC (Z , X ) such that f ◦ h = f ◦ g we must have g = h. Dually, an epimorphism is a morphism f : X → Y such that for any object W of C, and given morphisms h, g ∈ HomC (Y, W ) such that h ◦ f = g ◦ f , we must have h = g. An isomorphism is a morphism f : X → Y for which there exists a morphism g : Y → X such that g ◦ f = I X , f ◦ g = IY . In case a g as above exists, then it is unique, as one easily checks; it is called the inverse of f and will often be denoted by f −1 . In a straightforward way, one verifies that an isomorphism is necessarily an epimorphism as well as a monomorphism. Observe that a morphism that is both a monomorphism and an epimorphism need not necessarily be an isomorphism; indeed in Ring the canonical inclusion Z → Q is both a monomorphism and an epimorphism! Composition of monomorphisms, respectively epimorphisms, respectively isomorphisms, yields again a monomorphism, respectively epimorphism,

1.1 Basic Notions

3

respectively isomorphism. The duality between monomorphisms and epimorphisms may best be phrased by passing to the dual category C o , having the same class of objects as C but with HomC o (X, Y ) = HomC (Y, X ) by definition. Now a morphism f : X → Y in C is a monomorphism, respectively an epimorphism, if and only if f is an epimorphism, respectively a monomorphism when seen as a morphism Y → X in the dual category C o . Definition 1.3: Subobjects in C In many examples and applications the objects considered need not be sets, hence a correct definition of the term subobjects requires some care. Fix an object X of C; for any W of C we have a set MonoC (W, X ) consisting of monomorphisms W → X . For objects U and W of C we have a product set MonoC (U, X ) × MonoC (W, X ); if (α, β) is in the latter, then we may define α ≤ β if there exists a morphism γ : U → W such that β ◦ γ = α. In case a γ as before exists, it is unique and also a monomorphism. We say that α and β are equivalent monomorphisms if α ≤ β and β ≤ α; indeed, the foregoing defines an equivalence relation! Since we are dealing with sets now, we may evoke the Zermelo axiom and choose a representative in every equivalence class of monomorphisms. The resulting monomorphism is called a subobject of X in C. Quotient objects may now also be defined in a formally dual way by passing from C to C o . It is not difficult to verify that a subobject of a subobject is again a subobject and a quotient object of a quotient object is a quotient object. Definition 1.4: Initial and Final Object An object I , respectively F, of C such that HomC (I, X ), respectively HOMC (X, F), is a singleton for every X of C is called an initial object of C, respectively a final object of C. Two initial, respectively final, objects of C are necessarily isomorphic. A zero object of C is an object that is initial and final. This allows us to distinguish zero morphisms as those f : X → Y that factorize through the (unique up to isomorphism) zero object. If a zero object exists, then we denote it by O; then each set HomC (X, Y ) has precisely one zero morphism denoted o X Y or just o when no confusion can arise. Definition 1.5: Product and Coproduct Objects To a family (X s )s∈S of objects in C we may associate the product s∈S X s = X if we can solve a universal construction problem in C. The object X we look for should come equipped with a family of morphisms (πs )s∈S , πs : X → X s for s ∈ S, such that for any object Y of C with given morphisms f s : Y → X s for s ∈ S, there exists a unique morphism f : Y → X such that πs ◦ f = f s for all s ∈ S. If an object X with these properties does exist in C then it is unique up to isomorphism and we use the notation X = s∈S X s . In case S = {1, . . . , n} then we also write X = X 1 ×· · ·× X n (suppressing C in the product notation). The notion of coproductis defined dually. If the coproduct of a family (X s )s∈S exists we will denote it by s∈S X s ; in case S = {1, . . . , n} it is customary to write it as X 1 ⊕ · · · ⊕ X n .

4

A Taste of Category Theory

Now that our categories need not be sets, we may not be able to talk about maps from one category to another, notwithstanding the fact that we may without any problem associate an object of a category to an object in another category. For categories D and C we let a covariant functor F from D to C be defined by associating it to an object X of D an object F(X ) of C and to a morphism f : X → Y in D a morphism F( f ) : F(X ) → F(Y ) in C such that the following properties hold: i. F(I X ) = I F(X ) for every X of D. ii. F(g ◦ f ) = F(g) ◦ F( f ) for f : X → Y, g : Y → Z in D. A contravariant functor from D to C is then just a covariant functor Do → C. Usually, when F is a covariant functor from D to C, set-theoretic-inspired notation is used to express this by F : D → C; the context makes it clear that we do not mean to imply by this that F is a map! Definition 1.6: Full and Faithful Functors A covariant functor F : D → C yields for any X, Y in C a map HomD (X, Y ) → HomC (F(X ), F(Y )), which we denote also by: f → F( f ). We say that F is faithful, respectively, full, respectively full and faithful, if the above map f → F( f ) is injective, respectively surjective, respectively bijective. Note that for any category C there exists an identity functor 1C : C → C defined by 1C (X ) = X for every object X of C, and 1C ( f ) = f for every morphism f ∈ HomC (X, Y ). Obviously, the functor 1C is always full and faithful. Can functors between categories make up a category, and then what should be the morphisms? We do not treat functor categories in depth here but restrict ourselves to recalling a few fundamental notions related to this idea. Definition 1.7: Functorial Morphisms Consider a pair of covariant functors F, G : D → C. A functorial morphism ϕ : F → G is given by morphisms ϕ(X ) : F(X ) → G(X ) for X an object of D, such that for f : X → Y in D we have: ϕ(Y ) ◦ F( f ) = G( f ) ◦ ϕ(X ); in other words we have a commutative diagram of morphisms in C: F( f )

F(X )−−−−→ F(Y ) ϕ(Y ) ϕ(X ) G(X )−−−−→ G(Y ) G( f )

In case ϕ(X ) is an isomorphism for all objects X of D, ϕ is said to be a functorial isomorphism and we denote this by F G. For functorial morphisms ϕ : F → G and ψ : G → H the composition ψ ◦ ϕ : F → H may be defined by (ψ ◦ ϕ)(X ) = ψ(X ) ◦ ϕ(X ) for all X , and this yields again a functorial morphism. Let Hom(F, G) stand for the class of functorial morphisms from F to G. There exists an identity functorial morphism 1 F : F → F defined by putting 1 F (X ) = 1 F(X ) for all X of D. Since Hom(F, G) need not be a set in

1.2 Grothendieck Categories

5

general, we meet a small problem here in fitting this again in the framework of a category where the morphisms between two objects should be a set. In case D is a small category, then Hom(F, G) is also a set. Definition 1.8: Equivalences and Dualities The covariant functor F : D → C is said to be an equivalence of categories when there exists a covariant functor G : C → D such that G ◦ F 1D and F ◦ G 1C . In case G ◦ F = 1D and F ◦ G = 1C , F is called an isomorphism of categories, and in that case D and C are said to be isomorphic categories. A contravariant functor F : D → C defining an equivalence between Do and C is said to be a duality of categories, and in that situation D and C are said to be dual. Theorem 1.1 A covariant functor F : D → C is an equivalence if and only if: i. F is full and faithful. ii. For any Y ∈ C there is an X ∈ D such that Y F(X ). To an object X of C we may associate a contravariant functor h X : C → Set by putting h X (Y ) = HomC (Y, X ) and for a morphism f : Y → Z in C we define h X ( f ) : h X (Z ) → h X (Y ) by h X ( f )(z) = z ◦ f for any z ∈ h X (Z ). A functor F : C → Set is said to be representable if there is an object X of C such that F is isomorphic to the functor HomC (X, −) = h X . Theorem 1.2: The Yoneda Lemma For objects A and B of C there exists a natural bijection of Hom(h A , h B ) to HomC (B, A). In particular Hom(h A , h B ) is a set. Corollary 1.1 The category C o is isomorphic to the category of representable functors C → Set with the functorial morphisms for the morphisms.

1.2

Grothendieck Categories

The categories appearing in algebraic geometry, be it commutative or not, have very special properties, for example, modules over a ring, graded modules over a graded ring and so forth. For several results the class of abelian categories is suitable, but the best behaved categories we shall use are the so-called Grothendieck categories. These are rather close to being categories of left modules over a ring; the extra generality allows us to include categories of graded modules as well as certain categories of presheaves or sheaves.

6

A Taste of Category Theory A category C is pre-additive if the following three properties hold: i. For X, Y in C, HomC (X, Y ) is an abelian group with zero element O X Y called the zero morphism. ii. For X, Y, Z in C and f, f 1 , f 2 in HomC (X, Y ), g, g1 , g2 in HomC (Y, Z ), we obtain: g ◦ ( f1 + f2) = g ◦ f1 + g ◦ f2 (g1 + g2 ) ◦ f = g1 ◦ f + g2 ◦ f iii. There is an object X of C such that 1 X = O X X . Clearly such X is a zero object, unique up to isomorphism, usually denoted by O.

It is obvious that the dual of a pre-additive category is again pre-additive. A functor between pre-additive categories may have some additivity properties, too; for example, we say that F : D → C, where D and C are pre-additive, is an additive functor if for f, g ∈ HomD (X, Y ), where X, Y are objects of D, we have: F( f + g) = F( f ) + F(g). If OD is the zero object if D, then F(OD ) is the zero object of C, say OC . An additive category is a pre-additive category C such that for any two objects of C a coproduct exists in C. Definition 1.9: Abelian Categories An additive category C is said to be an abelian category if it satisfies conditions AB.1 and AB.2: AB.1 For any morphism f : X → Y in C, both Ker( f ) and Coker( f ) exist in C; then f may be decomposed as indicated in the following diagram: i

f

π

Ker( f )−−−−→X Y −−−−→Coker( f ) −−−−→ µ λ Coim( f ) −→ Im( f ) f¯ where f = µ ◦ f ◦ λ and i and µ are monomorphisms and π, λ are epimorphisms. AB.2 For every f as in AB.1, f is an isomorphism. In any category verifying AB.2, a morphism is an isomorphism exactly when it is both a monomorphism and an epimorphism. Definition 1.10: Exact Sequences and Functors Suppose that C is pre-additive such that AB.1. and AB.2. hold. A sequence of morphism X −→ Y −→ Z in C is exact if Im( f ) = Ker(g) as subobjects of Y . An f

g

arbitrary (long) sequence is said to be exact if every subsequence of two consecutive morphisms is exact in the sense defined above. An additive functor F : D → C, where both categories are pre-additive and such that AB.1 and AB.2 hold, is said to

1.2 Grothendieck Categories

7

be left exact, and respectively right exact if for any exact sequence of morphisms in D: 0→X →Y →Z →0 the following sequence is exact: 0

→

respectively

F(X ) F(X )

→ →

F(Y ) F(Y )

→ →

F(Z ) F(Z )

→

0

When F is both left and right exact, then F is exact. Consider X, Y in an additive category C and let X ⊕ Y be their coproduct. By definition of the coproduct there exist natural morphisms: i X : X → X ⊕ Y , π X : X ⊕ Y → X , i Y : Y → X ⊕ Y , πY : X ⊕ Y → Y , such that π X ◦ i X = 1 X , πY ◦ i Y = 1Y , π X ◦ i Y = 0 = πY ◦ i X , 1 X ⊕Y = i X ◦ π X + i Y ◦ πY . This actually establishes that (X ⊕ Y, π X , πY ) is a product of X and Y in C. Consequently, if C is additive, respectively abelian, then C o is too. Lemma 1.1 A functor F between additive categories is an additive functor if and only if it commutes with finite coproducts. In [17] A. Grothendieck introduced several extra axioms on abelian categories, gradually strengthening the definition until the notion of the Grothendieck category, as we know it, appears. The axioms AB.3, AB.4, AB.5 and their duals (AB.3)∗ , (AB.4)∗ , (AB.5)∗ are not independent; in fact, AB.5 presupposes AB.3 and implies AB.4. We just recall definitions and basic facts. AB.3 Arbitrary coproducts exist in C. (AB.3)∗ Arbitrary products exist in C. In case AB.3 holds in C we may define for any nonempty set S a functor ⊕s∈S : C (S) → C, associating to a family of C-objects, indexed by S the coproduct (sometimes called the direct sum) of that family. The functor ⊕s∈S is always right exact. AB.4 For any nonempty set S, ⊕s∈S is an exact functor. (AB.4)∗ For a nonempty set S, s∈S is an exact functor. If C is abelian and satisfies AB.3., then for any family ofsubobjects (X s )s∈S of X we may define a “smallest” subobject of X , denoted by s∈S X s , such that all X s are subobjects of the latter. The quotation marks around smallest refer to the fact that some care is necessary with the interpretation in view of the definition of subobject; compare Definition 1.3. The object s∈S X s is called the sum of (X s )s∈S . Dually, if C is an abelian category satisfying (AB.3)∗ , then for every family (X s )s∈S of subobjects of X we may associate ∩s∈S X s , the largest subobject of X contained in each X s , s ∈ S. The subobject ∩s∈S Ss is called the intersection of (X s )s∈S . Observe that in any abelian

8

A Taste of Category Theory

category finite products do exist, hence the intersection of a finite family exists (it is enough to have an intersection of two objects). AB.5 Let C be an abelian category satisfying AB.3. Consider an object X of C and subobjects X , s ∈ S, and Y , such that the family (X ) is right filtered, then: ( s s s∈S s∈S X s ) ∩ Y = (X ∩ Y ). s s∈S (AB.5)∗ The dual of AB.5. Observation 1.1 An abelian category such that AB.3 and AB.5 hold also satisfies AB.4. Definition 1.11: Generators for an Abelian Category Consider the family (X s )s∈S in the abelian category C; we say that (X s )s∈S is a family of generators if for every object X and subobject Y = X in C there is an s ∈ S and a morphism f : X s → X such that Im( f ) is not a subobject of Y . An object U of C is said to be a generator if {U } is a family of generators. Definition 1.12: Grothendieck Category An additive category satisfying AB, 1, . . . AB.5, and having a generator is a Grothendieck category. Observe that an abelian category, such that both AB.5 and (AB.5)∗ hold, is necessarily the zero category (category consisting of the zero object with the zero morphism). Consequently the opposite of a Grothendieck category is never a Grothendieck category. To end this section we recall some facts about adjoint functors. The notion of adjointness is very fundamental, and it has applications in different areas of mathematics. Consider functors F : C → D, G : D → C. Definition 1.13: Adjoint Functors The functor F is a left adjoint of G, or G is a right adjoint of F, if there is a functorial isomorphism : HomD (F, −) → HomC (−, G) where HomD (F, −) : C o × D → Set associates to (X, Y ) the set HomD (F(X ), Y ); HomC (−, G) : C o × D → Set associates to (X, Y ) the set HomC (X, G(Y )). If case C and D are pre-additive and the functors F and G are assumed to be additive, then we assume that (X, Y ) is an isomorphism of abelian groups. The following sums up some basic facts concerning adjoint functors. Properties 1.1 When the functor F is a left adjoint for G, then the following hold: i. F commutes with coproducts, G commutes with products. ii. If C and D are abelian and F and G are additive functors, then F is right exact and G is left exact.

1.3 Separable Functors

9

iii. If for every object Y of D there exists an injective object Q of D and a monomorphism Y → Q, then F is exact if and only if G preserves injectivity. iv. If for every X of C there exists a projective object P of C together with an epimorphism P → X in C, then G is exact if and only if F preserves projectivity. Perhaps one of the most well-known pairs of adjoint functors appears in connection with module categories over associative rings R and T say. Consider the module categories R-mod and T -mod as well as R-mod-T , the category of left R-right-T -bimodules. For M in R-mod-T we may define the tensor-functor M ⊗T − : T -mod → R-mod by viewing M ⊗T N for a left T -module N as a left R-module in the obvious way. It is easy to verify that M ⊗T − is a left adjoint of Hom R (M, −) : R-mod → S-mod.

1.3

Separable Functors

The notion of separable functor has been introduced by M. Van den Bergh and the author; the concept has been applied to algebras and in particular graded algebras in [33]. Several other applications of ring theoretical nature stem from the paper by M. D. Rafael, an author among participants at a summer institute at Cortona. The separable functors are not absolutely necessary for the development of the theory in this work; nevertheless, we include a short presentation because they may be used in several applications and some research projects we cover. Consider a covariant functor F : D → C. The functor F is a separable funcF : HomC (F(M), F(N )) → tor if for all objects M, N in D there are maps ϕ M,N HomC (M, N ) satisfying the following properties: F SF.1 For f ∈ HomD (M, N ), ϕ M,N (F( f )) = f .

SF.2 For objects M , N in D and f ∈ HomD (M, M ), g ∈ HomD (N , N ), f ∈ HomC (F(M), F(N )), and g ∈ HomC (F(M ), F(N )) such that the following diagram is commutative in D: F(M)− → F(N ) −−f− F(g) F( f ) F(M )−−−− → F(N ) g

then the following is commutative in D: M−−−−→ N F ( f ) ϕ M,N g M −−−−→ N

f

F ϕM ,N (g )

Observe that SF.1 holds if case F is a full faithful functor, that is, whenever for M, N in D the map HomD (M, N ) → HomD (F(M), F(N )) is bijective.

10

A Taste of Category Theory

Lemma 1.2 1. An equivalence of categories is also a separable functor. 2. If F : D → C and G : C → B are separable functors, then GF is separable. Conversely, if GF is a separable functor, then F is a separable functor. Let us summarize some basic properties of separable functors in the following proposition. Proposition 1.1 Let F : D → C be a separable functor and consider objects M and N in D. i. If f ∈ HomD (M, N ) is such that F( f ) is a split map, then f itself is a split map. ii. If f ∈ HomD (M, N ) is such that F( f ) is co-split, that is to say that there exists a u ∈ HomC (F(N ), F(M)) such that F( f ) ◦ u = 1 F(N ) , then f is itself co-split. iii. Assume that both D and C are abelian categories. When F preserves epimorphisms, respectively monomorphisms, and F(M) is projective in C, respectively injective in C, then M is projective in D, respectively injective in D. iv. Assume that D and C are abelian categries. If F(M) is a quasi-simple object, that is, every subobject splits off, and F preserves monomorphisms, then M is itself a quasi-simple object in D. Proof Statement iii follows from ii and iv follows i. The proof of ii is very similar to the proof of i, so it suffices to establish i. The assumption in i implies that there exists a map u : F(N ) → F(M) such that u F( f ) = 1 F(M) . Put g = ϕ NF ,M (u). Condition SF.2. then implies that g f = 1 M because we do have a commutative diagram in C. F(M)− −→ F(M) −1− F(M) F(1 M ) F(N )−−−−→ F(M)

F({)

u

The claim follows. Corollary 1.2 Part i may be rephrased as a functorial version of Maschke’s theorem (used frequently in the representation theory of groups). We finish this section by pointing out that the terminology derives from the fact that the restriction of scalar functors associated to a ring morphism C → R, where C is commutative and central in R, is a separable functor when R is C-separable. When R is also commutative, this agrees with the classical notion of a separable extension.

Chapter 2 Noncommutative Spaces

2.1

Small Categories, Posets, and Noncommutative Topologies

Throughout this section, C stands for a fixed small category, that is, a category having a class of objects that is a set. A category with exactly one object is a monoid; this is because we may view this as an object with a given monoid of endomorphisms. A group is then a monoid where all endomorphisms are automorphisms. By O we denote the zero-object category with a unique object O and a unique morphism: the identity of O. For any category C there exists a unique functor C → O. A terminal object in C is an object, I say, such that for every object α of C there is a unique morphism α → 1 in C. If C does not have a terminal object, we can adjoin one to C and obtain a category C1 with an obvious functor C → C1 taking an object α in C to α in C1 . To C we may associate the opposite category C o having the same class of objects but with morphisms reversed. For an arbitrary category D a D-representation of C is just a functor R : C → D; a D-representation of C o is called a presheaf on C with values in D. Hence, a presheaf P : C o → D is given by a family of objects P(α) in D such that for each C-morphism α → β we have D-morphisms ραβ : P(β) → P(α) such that to the identity α → α corresponds the identity P(α) → P(α), and to γ α → β → γ in C we correspond ραγ = ραβ ρβ . If B is another small category and given an arbitrary functor F : B → C, we construct the left (and right) comma category as follows. For the objects of the right comma category (α, F) we take C-morphisms α → Fβ, α ∈ C, β ∈ B and a morphism (α → Fβ ) → (α → Fβ) in (α, F) as a B-morphism b : β → β making the following diagram commutative: Fβ' α

F (b)

Fβ

The left comma category (F, α) is defined likewise, using for the objects the C-morphisms Fβ → α, and so forth.

11

12

Noncommutative Spaces Any C-morphism a : α → α induces functors: (a, F) : (α, F) → (α , F), (α → Fβ) → (α → Fβ)a (F, a) : (F, α ) → (F, α)

A type of small category often considered is a poset. A poset, or partially ordered set, is just a set with a partial ordering: ≤. If is a poset, then we shall write for the category having as objects the elements λ ∈ , and hom (λ, µ) consists of the unique arrow λ → µ when λ ≤ µ, or it is empty. The categories are examples of delta categories, that is, small categories in which endomorphisms are necessarily the identity morphisms and hom(σ, τ ) = ∅ for σ = τ implies hom(τ, σ ) = ∅ (maps are one-way and no loops). We define D-representations of , presheaves on with values in D, and comma categories for λ ∈ , . . . by taking the corresponding definitions for . The mother of all posets is the set of natural numbers with its usual ordering. For n ∈ N we let [n] denote the linearly ordered set 0 < · · · < n viewed as a category (as for posets). A C-representation of [n] is called an n-simplex; if σ : [n] → C is a (covariant) functor, then we say that σ is an n-simplex or dimσ = n. We denote the C-morphism σ (r → s) by σ r s , for r ≤ s in [n]. Zero simplices are functors [O] → C; these may be viewed as the elements of C, up to a harmless “abuse of language.” For n > 0, an n-simplex σ is completely determined by the n-tuple (σ 01 , σ 12 , . . ., σ p−1, p ); therefore, it is unambiguous to write σ = (σ 01 , σ 12 , . . . ). When C is for some poset , then an n-simplex σ is completely determined by the ordered list of elements, called vertices, σ (0), . . ., σ (n), because any σ r s is then necessarily the unique -morphism σ (r ) → σ (s) corresponding to r ≤ s. If τ is an n-simplex and σ is an m-simplex such that τ (n) = σ (0), then we can form the cup-product τ ∨ σ , which is the (n + m)-simplex given by: (τ ∨ σ )r,r +1 = τ r,r +1 (τ ∨ σ )r,r +1 = σ r −n,r −n+1

when r < n when n ≤ r.

For n > 0 and o ≤ r ≤ n we define a functor ∂r : [n − 1] → [n], ∂r (s) = s ∂r (s) = s + 1

if s < r if s ≥ r.

A given n-simplex σ has an r -face defined by the composition of the functors σ and ∂r . For r ≥ s one easily computes ∂r ∂s = ∂s ∂r +1 , where composition is here written in the arrow-order (i.e., not the usual way of writing a composition of maps). Hence: 12 if r = 0 (σ , . . ., σ n−1,n ) σr = (σ 01 , . . ., σ r −1,r +1 , . . ., σ n−1,n ) if 0 < r < n 01 (σ , . . ., σ n−1,n−1 ) if r = n In case C = , then the faces are distinct but that need not be true in general for arbitrary C.

2.1 Small Categories, Posets, and Noncommutative Topologies

13

The ∂r are called face operators. The collection of n-simplices and the face operators connecting them are called the simplicial complex, (C). It will be useful for introducing homology theories in a formal way. A zero element, denoted by 0, of the poset is one for which 0 ≤ λ for all λ ∈ ; clearly, if a zero element exists it is unique. A unit element, denoted by 1, of the poset is one for which λ ≤ 1 for all λ ∈ ; if a unit element exists then it is unique. A poset with 0 and 1 is said to be a lattice if for any two elements λ and µ in there exists a maximum λ ∨ µ ∈ and a minimum λ ∧ µ ∈ . A lattice is said to be ∨-complete if for any family {λα , α ∈ A} in the maximum ∨α∈A λα exists in . The lattice is complete if it is both ∨-complete and ∧-complete. For the general theory of lattices we refer to [6][7]. Now let us introduce the notion of cover in an arbitrary poset . Definition 2.1 We say that λ ∈ is covered by {λα , α ∈ A} with λα ∈ for all α ∈ A, if λα ≤ λ for all α ∈ A and if λα ≤ µ for all α ∈ A for some µ ∈ then λ ≤ µ; in this case we also say that {λα , α ∈ A} is a cover for λ. If A is finite, then {λα , α ∈ A} is said to be a finite cover for λ ∈ . Example 2.1 i. If is a lattice, then {λ1 , . . ., λn } is a finite cover for λ ∈ exactly when λ = λ1 ∨ . . . ∨ λn ; in a complete lattice this holds for arbitrary covers. ii. If is a distributive lattice, then a given finite cover λ = λ1 ∨ . . . ∨ λn induces for every τ ≤ λ a cover: τ = (τ ∧ λ1 ) ∨ . . . ∨ (τ ∧ λn ); the latter is called the induced cover for τ ∈ λ. iii. If is a poset with 1, then a global cover is a set {λα , α ∈ A} such that µ ≥ λα for all α ∈ A entails µ = 1. In particular, if is a distributive lattice with 0 and 1, then a finite global cover is a set {λ1 , . . ., λ} such that 1 = λ1 ∨ . . . ∨ λn and every τ ∈ then allows a cover {τ ∧ λ1 , . . ., τ ∧ λn } induced by a global cover. iv. A cover {λ1 , . . ., λ} of λ is reduced when it is ∨-independent in the lattice . In a semi-atomic lattice with 0 and 1 that is upper continuous and has the property that 1 can be written as a finite join of atoms of , there always exists a reduced global cover.

2.1.1

Sheaves over Posets

In this section we restrict attention to a category D, the objects of which are sets; hence, morphisms in D are in particular set maps. A presheaf P : ()o → D is separated if for every cover {λα , α ∈ A} of λ in and every x, y ∈ P(λ) such that for all α ∈ A, ρλλα (x) = ρλλα (y), we must have x = y. In case no covers exist, then every presheaf is separated. A separated presheaf is a sheaf on (with values in D), if for every finite cover {λi , i} of λ and given

14

Noncommutative Spaces λ

xi ∈ P(λi ) such ρµλi (xi ) = ρµ j (x j ) for every µ ≤ λi , µ ≤ λ j , there exists an x ∈ P(λ) such that for all i, ρλλi (x) = xi . In order to introduce stalks of presheaves or sheaves, we first introduce a so-called limit poset C() associated to any given poset .

2.1.2

Directed Subsets and the Limit Poset

A subset X ⊂ is said to be directed if for every x, y in X there exists a z ∈ X such that z ≤ x and z ≤ y. Let D() be the set of directed subsets of . For A and B in D() we say that A is equivalent to B, written A ∼ B, if and only if the following conditions are satisfied: i. For a ∈ there exists a ∈ A, a ≤ a and b, b ∈ B such that: b ≤ a ≤ b . ii. For b ∈ B, there exists b ∈ B, b ∈ b and a, a ∈ A such that: a ≤ b ≤ a . By [A] we denote the ∼-equivalence class of A in D(). We let C() be the set of classes of directed subsets of . A directed set X ⊂ is said to be a filter in if x ≤ y with x ∈ X entails y ∈ X . To an arbitrary directed set Y in we associate a filter Y as follows: Y = {λ ∈ , there exists an x ∈ Y such that x ≤ λ}. For A, B in D() we say that A ≤ B if and only if: i. For a ∈ A there is an a ∈ A, a ≤ a, such that a ≤ b for some b ∈ B. ii. For b ∈ B, there is an a ∈ A such that a ≤ b. Lemma 2.1 With conventions and notation as above: 1. For A, B in D(), A ≤ B if and only if B ⊂ A. 2. For A, B in D(), A ∼ B if and only if A ≤ B and B ≤ A, if and only if A = B. 3. The set C() with ≤ induced from D() is a poset such that the canonical map → C(), λ → [{λ}], is a poset monomorphism. 4. If is a lattice with respect to ∧, ∨ then these operations induce a lattice structure on C() such that the canonical poset map → C() is a lattice monomorphism. 5. If is a lattice, then the following properties of transfer to the lattice C(): i. has 0 and 1. ii. is modular. iii. is complete.

2.1 Small Categories, Posets, and Noncommutative Topologies

15

iv. is distributive. v. is Brouwerian, that is, for λ, µ ∈ the set {x ∈ , λ ∧ x ≤ µ} has a largest element then denoted by µ : λ. Proof 1, 2, and 3 are easy enough, and we leave these as exercises. 4. For A and B in D() define: ˙ = {a ∧ b, a ∈ A, b ∈ B}, A∨B ˙ = {a ∨ b, a ∈ A, b ∈ B} A∧B Since ∧ and ∨ are bicontinuous with respect to ≤, it is immediately clear that ˙ as well as A∨B ˙ is in D(). If A ∼ A and B ∼ B in D(), then A∧B ˙ ∼ A ∨B ˙ ˙ ∼ A ∧B ˙ ˙ . So if we define [A] ∧ [B] = [A∧B], and A∨B A∧B ˙ [A] ∨ [B] = [A∨B] in C(), then we obtain a well-defined lattice structure on C() making the canonical poset map → C(), λ → [{λ}] into a lattice monomorphism, as is easily checked. 5. Straightforward verification of the lattice properties considered. Exercise 2.1 1. For any set S we let L(S) be the poset of all subsets of S partially ordered by inclusion. A lattice presentation of a poset is just a poset map π : → L(S) for some set S; observe that L(S) is a lattice. Verify that it is always possible to present a poset by π : → L(), λ → {µ ∈ , µ ≤ λ}. Provide examples to show that π need not be injective or surjective in general. 2. For any poset a subset of the form {µ ∈ , µ ≤ λ} is called an interval of . For a lattice we have a lattice isomorphism I () described by λ ↔ {µ ∈ , µ ≤ λ}, where I () is the lattice of intervals in . 3. If π : → L(S) is a lattice presentation of the poset , then it induces a poset map C(π) : C() → C L(S) fitting in the following commutative diagram of poset maps: Λ C (Λ)

π

C (π)

L (S) CL (S)

Verify also that C(π ) actually defines a lattice presentation of C(), C() → L(C L(S)). Note that we may identify C L(S) and I (C L(S)) in view of Lemma 2.1(4). 4. Can you describe C() in case is the lattice of open sets a. for the real topology on Rn b. for the Zariski topology on Rn

16

Noncommutative Spaces 5. If is the lattice Open(X ) consisting of open sets of a topological space X, τ , can you express the Hausdorff, respectively T1 -property, in terms of the lattice morphism → C()? 6. If 1 , 2 are posets, respectively lattices, is it possible to define a product poset, respectively product lattice, 1 × 2 ? Can you relate C(1 ) × C(2 ) and C(1 × 2 )? In case 1 = Open(X ), 2 = Open(Y ), is 1 × 2 equal to Open(X × Y ) where Y × Y is equipped with the product topology? Relate C(Open(X × Y )) and C(1 ), C(2 ). In case X and Y are varieties with their respective Zariski topologies and X × Y is viewed with the Zariski topology (i.e., this need not be the product topology of the product variety X × Y ), is it true that C(Open(X × Y )) = C(OpenZar (X × Y ))? Is the relation between the lattices C(OpenZar (X )) × C(OpenZar (Y )) and C(OpenZar (X × Y )) essentially different from the relation between the topologies themselves?

2.1.3

Poset Dynamics

When deformations of algebras are introduced, even when using rather general abstract methods such as diagram algebras in the sense of M. Gerstenhaber, there is always some algebra or ring of formal power series in the picture. This is somewhat unsatisfactory because it is really not asking too much to hope that the use of some sheaf-like theory over a suitably general underlying “space” (topology, lattice, poset, diagram, etc.) should allow a visualization of the deformation action and an interpretation in terms of infinitesimal-like phenomena worded in terms of sheaves in suitable categories. Motivated by a question concerning the possibility of developing a dynamic version of the use of causets (posets with respect to a causality order) in the theory of quantum gravity, we propose a notion of poset dynamics. This structure may be viewed as an interesting toy; that is, a generic exercise arises when trying to apply any derived structural result in the usual, that is to say static, theory to the dynamic situation. Let T be a totally ordered poset. A poset T -dynamics consists of a class of posets {Pt , t ∈ T } together with poset maps: ϕtt : Pt → Pt for every t ≤ t in T , satisfying the following conditions: DP.1 For each t ∈ T , ϕtt = I Pt , the identity of Pt . DP.2 For each triple t ≤ t ≤ t in T we have: ϕtt ϕt t = ϕtt (where composition is written in the arrow-order). DP.3 For any t ∈ T and subset F ⊂ T such that for every f ∈ F we have t ≤ f , for every nontrivial x < y in Pt (i.e., x = 0, y = 1) such that for all f ∈ F, ϕt f (x) < z f < ϕt f (y) for some z f ∈ P f , there exists a z ∈ Pt with x < z < y such that ϕt f (x) < ϕt f (z) ≤ z f < ϕt f (y).

2.1 Small Categories, Posets, and Noncommutative Topologies

17

DP.4 For every t ∈ T and nontrivial x < z < y in Pt (i.e., x and y not 0 and 1), there exists a t1 ∈ T , t < t1 , such that for all t ∈ [t, t1 [ we have ϕtt (x) < ϕtt (z) < ϕtt (y) (in other words strict < has a future). DP.5 For every t ∈ T and x < z < y in Pt there exists a t1 ∈ T , t1 < t, such that for all t ∈]t1 , t] such that x , y ∈ Pt with ϕt t (x ) = x, ϕt t (y ) = y exist, we have x < z < y in Pt with ϕt t (z ) = z (in other words strict < has a past for elements with a past). Note the discrepancy between DP.3 and DP.5; in DP.3 we reach Pt but the condition ϕt f (z) ≤ z f is weaker than ϕt t (z ) = z in DP.5, which is only reached on Pt . Under conditions ensuring that the set of z having the property in DP.3 has maximal elements (e.g., ∨-complete, Noetherian posets, etc.), then we may by iteration in DP.3 arrive at a z such that ϕt f (z) = z f and then DP.3 implies DP.5. In that case Imϕt f is convex in f , (ϕt f ([x, y]) = [ϕt f (x), ϕt f (y)]). The remainder of this section is devoted to the introduction of the notion of dimension of a poset. This in turn may be used to define the Krull dimension of an abelian category. Both concepts will have a different meaning in applications, comparable to the difference between the notion of dimension of a topological space and the dimension of a variety or of a linear space. The Krull dimension of a poset was introduced by P. Gabriel and R. Rentschler [11] for ordinal numbers; a generalization for “higher” numbers is obtained later by G. Krause. Let , ≤ be a poset. If λ ≤ µ in , then we write [λ, µ] for the closed interval {α ∈ , λ ≤ α ≤ µ}. The set of intervals is denoted by I(). We may define on I() the following filtration by using transfinite recurrence I−1 () = {[λ, µ], λ = µ} I0 () = {[λ, µ] = I(), [λ, µ] is an Artinian poset}. (Artinian means that the descending chain condition holds with respect to subposets, or equivalently every nonempty family in it has a minimal element). Assuming that Iβ () has already been defined for all β < α, then we define Iα () as follows: Iα () = {[λ, µ], for every sequence λ ≥ λ1 ≥ . . . λn ≥ . . . µ there is an n ∈ N such that [λi+1 , λi ] ∈ ∪β m in N, then there exists an n α ∈ N such that all [Bα ]d with d ≤ n α are different in C() but [Bα ]n α = [Bα ]n α +k for any k ∈ N. Therefore we are either in the situation where all [Bα ]d for d ∈ N are different constituting the interval [[0], [Bα ]] and none of the [Bα ]d is ∧-idempotent, or else [[0], [Bα ]] is finite and [Bα ]n α is ∧-idempotent (and a minimal point) for some n α ∈ N. To every λ ∈ as in Proposition 2.8, there corresponds a set of quantum points {[Bα ], α ∈ Aλ } obtained from the decomposition [λ] = ∨α∈Aλ {[Bα ]n α , α ∈ Aλ } some Aλ ⊂ A; we denote that set by S(λ). The set S(λ) does not define λ; it defines uniquely the quantum closure λcl = ∨ {[Bα ], α ∈ Aλ }, but λ ≤ λcl cannot be reconstructed unless we know the “multiplicities” n α , which measure in some sense the noncommutativity of ; also these numbers measure how quantum points fail to be ∧-idempotent. A [Bα ] not dominating any nonzero ∧-idempotent, that is, if [A] = 0 is such that [A] ≤ [Bα ], then [A] is not idempotent in C(), which is called a radical point (short for radical quantum point). More generally in any an element λ = 0 such that λ does not dominate a nonzero idempotent is called a radical element. If [A] is a radical element of C(), then [A] = ∨ {[Bα ]n α , α ∈ A A } for some n α ∈ N, where all [Bα ], α ∈ A A are radical points.

2.3.1 2.3.1.1

Projects The Relation between Quantum Points and Strong Idempotents

The relation between quantum points and strong idempotents in I∧ (C()) is enigmatic to say the least; therefore mixing conditions expressing generation properties in terms of strong idempotents (using bracketed expressions in ∧ and ∨) and in terms of quantum points (using only expressions in ∨) must lead to complex situations. The final definition in this philosophy should be that of a quantum topology: a topology of virtual opens such that τ , the pattern topology, has a quantum basis (the elements of this should then be logically termed quantum patterns; what’s in a name?). The intrinsic technical problem is that we do not have, in the generality we consider here, any prescribed relation between the intervals [[0], [A]] and [[B], [A] ∨ [B]] in C(). One could put conditions in terms of Kdim, composition series involving critical elements, composition series involving ∧-powers of quantum points, etc. The topologies in classical examples are defined on sets with a lot of extra geometric or algebraic structure (e.g., groups, vector spaces); without fixing such extra structure one should try to find suitable abstract “lattice theory–type” properties that can replace the missing structure.

36

Noncommutative Spaces

2.3.1.2

Functions on Sets of Quantum Points

Generalized points suggest a beginning of spectrum theory, implications in sheaf theory (we shall come back to this later), and the use of set theoretic functions. If A is the set of quantum points (as before [Bα ] with α ∈ A), then C() may be obtained from functions n : A → N, α → n α , corresponding to n the element ∨ {[Bα ]n α , α ∈ A such that n(α) = 0}. Putting [Bα ]o = [0] we obtain a map F(A, N) → C(), which is surjective. We may also study maps between noncommutative topologies and as before, defined by functions on the sets of quantum points. Such functions do not necessarily lead to a well-defined map from C() to C( ) because the expression of [A] ∈ C() as a ∨ of ∧-powers of quantum points may not be unique; some care is necessary here. Determining formal links between sets of quantum points and functions on these and noncommutative topological features is an interesting project. Obvious links to sheaf theory and spectral theory can be exploited. Using functions to sets with extra structures may lead to interesting examples carrying algebra-like structures; for example, look at the appearance of P(H ), the projective Hilbert space, in the case of L(H ) studied in Section 2.7, as an effect of the linearity of the theory.

2.4

Presheaves and Sheaves over Noncommutative Topologies

In this section we consider a noncommutative topology . In view of Proposition 2.2 and Lemma 2.12, T (), C(), τ are noncommutative topologies too. We are given a category C and we are interested in sheaves over , C(), τ, . . . and relations among these; the category C is rather arbitrary. C must have sums, products, limits, and its objects are assumed to be sets. In Section 2.1.3, separated presheaves and sheaves over general posets were introduced. We apply these to , C(), T (), τ, . . . lim Given a presheaf P : o −→ C. Consider [A] ∈ C() and define: P[A] = −→ P(a) a∈A

lim

lim

in C. Observe that for B ∼ A we do obtain that −→ P(a) = −→ P(b), so that the a∈A

definition of P[A] is well defined. So we may look at P[ P[ ] ([A]) = P[A] for all [A] ∈ C(). Lemma 2.19 If P is a presheaf, then P[

]

b∈B

]

: C()o → C defined by

is a presheaf on C().

Proof If [A] ≤ [B], then for all b ∈ B there is an a ∈ A such that a ≤ b; hence there is lim lim a canonical map −→ Pb → −→ Pa , induced by the restriction maps ρab : Pb → Pa b∈B

a∈A

[B] defined by the presheaf P. We shall write ρ[A] for the map defined before. In case we [B] have [B] = [A] we do obtain that ρ[A] is the identity map 1 P[A] . For [C] ≤ [B] ≤ [A] [B] [A] [A] it is easy enough to check that the usual composition formula holds: ρ[C] ρ[B] = ρ[C] .

2.4 Presheaves and Sheaves over Noncommutative Topologies Corollary 2.6 Let P[ ] | be the restriction of P[ C(), λ → [λ], then P[ ] | = P.

]

37

to via the canonical inclusion →

Theorem 2.1 With notation and conventions as before. i. If P is a separated presheaf, then P[ ii. If P is a sheaf over , then P[

]

]

is a separated presheaf.

is a sheaf over C().

Proof i. Let P be a separated presheaf over and look at a finite cover {[Aα ], α ∈ A} of [A] in C(). Suppose that y, x in P[A] are such that ρ[α] (x) = ρ[α] (y), for [A] . Let x, respectively y in lim−→ P(a) all α ∈ A, where we write ρ[α] for ρ[A α] α∈A be represented by {xa ; a ∈ A, xa ∈ P(a)} respectively {ya ; a ∈ A, ya ∈ P(a)}. Since for all α ∈ A we have [A] ≥ [Aα ], there are aα ∈ Aα , aα ≤ a for a given a ∈ A. Fixing a ∈ A and xa , ya in P(a), ρaaα (xa ) = ρaaα (ya ) for some ˙ α aα ∈ Aα because we have that ρ[α] (x) = ρ[α] (y). Now ∨{aα , α ∈ A} ∈ ∨A ˙ α , so we may replace the aα by bα ∈ Aα such that bα ≤ aα and A ∼ ∨A and there are a and a in A such that a ≤ ∨{bα , α ∈ A} ≤ a . Then pick ˙ α ∼ A it a1 ∈ A such that a1 ≤ a and a1 ≤ a (A is directed). Again from ∨A follows that we may find bα , α ∈ A, bα ∈ A, bα ∈ Aα such that bα ≤ γα and ˙ α is such that ∨{bα , α ∈ A} ≤ a1 ≤ a. Similarly we may ∨{bα , α ∈ A} ∈ ∨A find bα ∈ Aα , bα < bα for all α ∈ A, such that there exist a2 and a2 in A such that a2 ≤ ∨bα ≤ a2 , but ∨bα ≤ ∨bα ≤ a1 ≤ a. For convenience, rewrite bα for bα and put b = ∨{bα , α ∈ A}. Since P is separated and ρbbα ρba (xa ) = ρbbα ρba (ya ) for all α ∈ A, it follows that ρba (xa ) = ρba (ya ). Since we may choose a2 ∈ A such that a2 ≤ b, we get ρaa (xa ) = ρaa (ya ); thus the classes of xa and ya 2 2 coincide in lim−→ P(a), i.e. x = y follows as desired. a∈A

ii. Start from a cover [A] = [A1 ] ∨ · · · ∨ [An ] in C() and suppose there is given a set x α ∈ P[Aα ] , α = 1, . . ., n, such that, writing [C] for either [Aβ ] ∧ [Aα ] or [Aα ] ∧ [Aβ ] we have the gluing condition: (∗)

[Aα ] α (x ) = ρ[C]β (x β ), ρ[C] [A ]

for α, β ∈ {1, . . ., n}

Let x α , respectively x β , be represented by xaαα , respectively xaββ for aα ∈ Aα , ˙ β ], condition (*) yields: respectively aβ ∈ Aβ . Since [Aα ] ∧ [Aβ ] = [Aα ∧A (•)

ργaα (xaαα ) = ργaβ (xaββ ) for some γ = aα ∧ aβ

with aα ∈ Aα , aβ ∈ Aβ . In a similar way: (••)

a

ργ α (xaαα ) = ργ β (xaββ ) for some γ = aβ ∧ aα a

38

Noncommutative Spaces with aα ∈ Aα , aβ ∈ Aβ . Take bα ≤ aα , aα and bβ ≤ aβ , aβ , put γαβ = ˙ β , respectively Aβ ∧A ˙ α . Since γαβ ≤ γ the bα ∧ bβ , γβα = bβ ∧ bα in Aα ∧A equality (•) also holds with γ replaced by γαβ and (••) holds with γ replaced by γβα . Since we considered a finite cover we may repeat the foregoing argument for all pairs Aα , Aβ until we obtain a set {a1 , . . ., an } and γαβ = aα ∧ aβ in ˙ β , γβα = aβ ∧ aα in Aβ ∧A ˙ α such that the equalities (•) and (••) hold Aα ∧A for all α and β in {1, . . ., n} with respect to γαβ , respectively γβα . Since P is a sheaf on it follows that there exists a z ∈ P(τ ), τ = a1 ∨ . . . ∨ an such that ρaτα (z) = xaαα for all α ∈ {1, . . ., n}. ˙ . . . ∨A ˙ n defines an element of the class [A] = [A1 ]∨. . .∨[An ] Now τ ∈ A1 ∨ [A] (x) and x α in and z defines an element, say x, in P[A] . Now both elements ρ[A α] τ τ P[Aα ] are represented by ρaα (xτ ), respectively ρaα (z) for all α ∈ A = {1, . . ., n}. Since the latter are the same and P[ ] is separated in view of i, it follows that xτ = z and x is the unique element of P[A] with the desired restrictions in each P[Aα ] , α = 1, . . ., n.

Observe that a separated presheaf P on does not necessarily induce a separated presheaf on S L() with respect to the operation ∧. in id∧ ()! We shall return to the sheaf theory in subsequent sections. The notion of sheaf as we have used it so far presents a drawback; indeed if we start from a cover of λ ∈ , say λ = λ1 ∨ . . . ∨ λn , and µ ≤ λ in , then µ = (µ ∧ λ1 ) ∨ . . . ∨ (µ ∧ λn ) in general. Even if is a lattice but not distributive we meet this problem. Consequently, the gluing condition in the definition of a sheaf is too strong. Similar to the trick used in defining Grothendieck topologies we may restrict the covers used in the definition of a sheaf, and for example, only consider covers of λ induced by finite global covers in . Let us write the more general definition in its functorial form; we do not demand that objects of C are sets from now on, but we do assume that C is a Grothendieck category. Consider a presheaf P : o → C. Obviously presheaves on with values in C together with presheaf morphisms make up a category Q(, C). For λ ∈ we let Cov (λ) consist of those covers for λ induced by a finite global cover of . We make Cov (λ) into a category by defining U → V if U = (λi )i∈I , V = (µ j ) j∈J and there is a map ε : I → J such that λi ≤ µε(i) for all i ∈ I . Given P and λ ∈ we define a contravariant functor [P, λ] : Cov (λ) → C as follows. First we have for U = (λi )i∈I in Cov (λ) projection morphisms: pi :

P(λi ) → P(λi ),

i∈I

then we have a morphism j : P(λ) → morphisms: pl , ql , pr , qr :

i∈P

P(λi )

i∈I

P(λi ) such that pi j = ρλλi . We also have ( j,k)∈I ×I

P(λi ∧ λ j )

2.4 Presheaves and Sheaves over Noncommutative Topologies λ

39

λ

where the ( j, k)-component of pl is ρλ jj ∧λk , of ql is ρλλkj ∧λk , of pr is ρλkj∧λ j and of qr is ρλλkk∧λ j , corresponding to a diagram in C:

P(λ j ) −→ P(λ j ∧ λk )

P(λk ) −→ P(λk ∧ λ j ) Thus, in C, we obtain a diagram, for (λi )i∈I in Cov (λ): P(λ) → j

(∗)

P(λi )

P(λ j ∧ λk )

( j,k)∈I ×I

i∈I

Definition 2.4 The presheaf P is a sheaf if and only if the diagrams (*) are equalizer diagrams. Now we put [P, λ](U) equal to the kernel of

P(λi )

P(λ j ∧ λk ).

( j,k)∈I ×I

i∈I

We can define L P ∈ Q(, C) by putting L P : o → C, λ →

lim

−→ U∈CoJ (λ)

([P, λ](U))

Lemma 2.20 The notions of separated presheaf and sheaf are now defined with respect to the fixed type of covers. If P is a separated presheaf, then the canonical morphism P → L P in Q(, C) is a monomorphism and L P is a sheaf; hence the functor L L may be viewed as a sheafification functor. Proof Similar to the classical case. The foregoing lemma (in particular its proof) depends heavily on axiom A.10; therefore sheafification does not necessarily work well even for lattices. This can be seen in Section 2.7 for the lattice L(H ) of a Hilbert space H . However, it turns out that for sheaf theoretic applications the absence of A.10 may be compensated for by the existence of enough quantum points (cf. Section 2.3).

2.4.1

Project: Quantum Points and Sheaves

In the situation of Definition 2.3, suppose has a weak quantum basis {[Bα ], α ∈ A} and consider a separated presheaf on , say P. Look at the separated presheaf P[]

40

Noncommutative Spaces

on C(). We have ∨{[Bα ], α ∈ A} = 1 and for all α, β in A, [Bα ] ∩ [Bβ ] = 0; up to self-intersections the [Bα ] allows us to reconstruct [λ] ∈ C() with λ ∈ . Starting from the product α∈A P[Bα ] , of all stalks at the [Bα ], imitate the construction of the e´ tale space of a separated presheaf P[ ] in two ways; first by putting restrictions on P suchthat P[Bα ] = P[Bα ] n], and then more generally by replacing the product above by α∈A P[Bα ] n α where [Bα ]n α is the idempotent dominated by [Bα ]. In case radical elements exist, it is quite harmless to assume that the stalk representing a radical element is the global P[0] . Study the (partial) sheafification results that follow. Connect this to Projects 2.3.1.1 and 2.3.1.2 and ii. Observe that for the example L(H ), introduced in Section 2.7, this leads to the classical sheafification on the Stone space (see also later for generalizations of the Stone space). Observe that x, y ∈ P[Bα ] m such that x = y but ρααm (x) = ρααm (y) in the P[Bα ] may exist. So the e´ tale space in the second approach will be a quotient of α∈A P[Bα ] identifying elements that are restricted to the same element after n self-intersections for some n ∈ N.

2.5

Noncommutative Grothendieck Topologies

In the definition of noncommutative topology we have included some conditions about covers, such as A.10. When trying to fit the noncommutative topologies into the framework of Grothendieck topologies this fact will pay off. First recall that a Grothedieck topology is a category C such that for every object x of C a set cov(x), consisting of subsets of morphisms with target x, is given such that the following conditions are satisfied: G.1. {x → x} ∈ cov(x). G.2. If {xi → x, i ∈ J } ∈ cov(x) and {xi j → xi , j ∈ J } ∈ cov(xi ) for all i ∈ I, then {xi j → x, i ∈ J , j ∈ J } ∈ cov(x), where xi j → x is obtained from the xi j → x j → x. G.3. If {xi → x, i ∈ J } ∈ cov(x) and x → x in C, then there exists a pull-back x ×x xi in C such that {x ×x xi → x , i ∈ J } ∈ cov(x ), which is called the fibre product over x. Now look at a noncommutative topology, say. For the objects of C there is little choice but to take λ ∈ . Since we have to be able to induce covers, we cannot just let any relation λ ≤ µ be a morhphism λ → µ in C. A first idea could be to allow only focused relations λ ≤ µ (as defined after Section 2.2.1.); however, this leads to problems such as if λ1 ∨ . . . ∨ λn is a global cover and x ≤ x is focused then we do get a diagram: x ^ λi

x

x' ^ λi

x'

2.5 Noncommutative Grothendieck Topologies

41

in the category C but x ∧ λ ≤ x ∧ λi is not necessarily focused; that is, it may happen that (x ∧ λi ) ∧ (x ∧ λi ) = x ∧ λi . We may solve the problems by looking at generic relations. A relation λ ≤ µ in is said to be generic if it is a consequence of the axioms of a noncommutative topology; that is a ∧ b ≤ a is a generic relation. When a and b are idempotent in λ, any relation that is just given as a ≤ b is not viewed as generic; however, if b = a ∨ c, then a ≤ b is viewed as generic. Now if 1 = λ1 ∨. . .∨λn is a global cover and x ≤ x is generic, then x ∧λi ≤ x ∧λi is generic too, so if we define C g to be the category with objects λ ∈ and generic relations for the morphisms (with x = x representing 1x ), then the diagram above is a diagram of morphisms in C g . A new problem appears, for example, for x ≤ x and 1 = λ1 ∨ . . . ∨ λn as above; for given morphisms t → x ∧ λi , t → x we do not find a generic relation t ≤ x ∧ λi . But from the philosophy of patterns we may learn that it is not natural to ask for t → x ∧ λi ; we should ask for some morphism between elements having a similar pattern. This is at the basis of the following definition. Definition 2.5 A category C with cov(x) defined for every object x of C as before is said to be a noncommutative Grothendieck topology if the following conditions hold: G.1. and G.2. as before, and the new condition: G.3. nc for given x → x and {xi → x, i ∈ J } ∈ cov(x) there is a cover {x ×x x → x , i ∈ J } ∈ cov(x ) satisfying the following pull-back property: for s → xi , s → x and t → xi , t → x there exist s ∧ t → x ×x xi and t ∧ s → x ×x xi fitting in a commutative diagram: x xi

x' x' ×

t

x

s

xi

t^s

s^t

The particular case when s = t yields:

t

xi

x

x' × x xi

x'

t^t

Clearly when t is idempotent, then the diagram reduces to the usual pull-back diagram in G.3. Observe that the noncommutative version of a Grothendieck topology is obtained by a complete symmetrization of the classical definition.

42

Noncommutative Spaces

Theorem 2.2 Let X be either , C(), τ , so certainly a noncommutative topology and let C g be constructed on X with respect to generic relations as before Definition 2.5. For the covers of x in X we take covl (x) = {x ∧ λi → x, {λ1 , . . ., λn } a global cover}. Then X becomes a noncommutative Grothendieck topology. Proof Look at a cover {xi = x ∧ λi → x, i = 1, . . ., n} ∈ covl (x) and x → x. Put x ×x xi = x ∧ λi , for i = 1, . . ., n. Since x → x, x ≤ x is generic; thus x ∧ λi ≤ x ∧ λi is generic too. We obtain the following diagrams in C g : x ^ λi

x

x' × x xi = x' ^ λi

x'

Suppose s → x ∧ λi , s → x and t → x ∧ λi , t → x are given. Since x ∧ λi ≤ λi is generic by definition and s → x ∧ λi , t → x ∧ λi are morphisms in C g , it follows that s ≤ x ∧ λi ≤ λi and t ≤ x ∧ λi ≤ λi yield generic relations s ≤ λi , t ≤ λi . Since s ≤ λi and s ≤ x are generic and similarly t ≤ λi , t ≤ x are generic, it follows that s ∧ t ≤ x ∧ λi and t ∧ s ≤ x ∧ λi are generic. Hence we arrive at the morphism in C g , s ∧ t → x ∧ λi = x ×x xi and t ∧ s −→ x ∧ λi = x ×x xi , fitting nicely in nc

the diagram defining the condition G.3 (G.1 and G.2 are obvious), so X (as C g ) is a noncommutative Grothendieck topology. Remark 2.1 1. Using covl we refer to X with this structure as the left topology of (or T , or C(), τ ); in a similar way the right topology can be defined. 2. The advantage of using generic relations is obvious: the generic relations are recognizable on sight; they transfer well from to C() and τ . The fact that not every generic relation needs to be focused does not interfere with that. 3. When viewing , C(), τ with the noncommutative Grothendieck topology structure, we shall write g , C()g , τ g , respectively. Observation 2.3 A cover λ = λ1 ∨ . . . ∨ λn in X (as before) is automatically a generic cover because λi ≤ λ is generic for i = 1, . . ., n. Now it is clear how presheaves, separated presheaves, or sheaves are defined over X with respect to generic covers. Indeed the operations appearing in the separateness and the sheaf property are compatible with the restriction to generic relations and generic covers. When passing from to C() or τ we may look at a stronger restriction on the covers (much like the strong idempotents replacing the idempotents). For [A], [B] in C() the relation [A] ≤ [B] is generically defined if for all b ∈ B there exists an a ∈ A such that a ≤ b is generic. A cover [A] = ∪α [Aα ] is generically defined if [Aα ] ≤ [A] is generically defined for all α.

2.5 Noncommutative Grothendieck Topologies

43

Lemma 2.21 For C() consider the category C()gd consisting of the objects [A] ∈ C() with generically defined relations for the morphisms; then C()gd is a noncommutative Grothendieck topology. Proof The covers used are generically defined covers; G.1 and G.2 are obvious. For G nc 3 the proof in Theorem 2.2 remains valid if we verify that a generically defined relation a ≤ b yields a generically defined x ∧ a ≤ x ∧ b for any x. Put a = [A], b = [B] and x = [C] in C(). Since [A] ≤ [B] is generically defined, we find for every b ∈ B some a ∈ A with a ≤ b generic. For any x ∈ C we then have a generic ˙ there is an a ∧ c ∈ A∧C ˙ such relation a ∧ c ≤ b ∧ c; hence for every b ∧ c ∈ B ∧C that a ∧ c ≤ b ∧ c is generic. Consequently, [a] ∧ [C] ≤ [B] ∧ [C] is generically defined. Proposition 2.9 If P is a presheaf on g , then P[ ] is a presheaf on C()gd . Moreover: i. If P is separated, then P[ ] is separated. ii. If P is a sheaf, then P[ ] is a sheaf on C()gd Proof That P[ ] is now a presheaf on C()gd follows from the (proof of) Lemma 2.19. i. The same proof as in Theorem 2.1 i holds still if one verifies at each step that the relations may be chosen to be generic; then the only operations occuring are ∧ or ∨ and these do not change the generic character. ii. Same as in i, checking along the lines of Theorem 2.1 ii. Corollary 2.7 The canonical inclusion → C(), λ → [λ] defines a faithful g → C()gd . Indeed a generic relation λ ≤ µ does define a generically defined [λ] ≤ [µ]. The restriction of P[ ] to g is P. Corollary 2.8 The pattern topology τ ⊂ C() is also a noncommutative Grothendieck topology, written τ g . Defining τ gd in the same way as C()gd , this also defines another noncommutative Grothendieck topology and a faithful functor τ gd → C()gd . Presheaves on C()gd restrict to τ gd in such a way that separateness and the sheaf property are respected.

2.5.1

Warning

If a and b are idempotents in a topology of virtual opens , then the fact that a ≤ b is generic in need not imply that a ≤ b is generic in S L()! For example if

44

Noncommutative Spaces

a = (b∨c)∧(d ∨c), then c∧c ≤ a is generic in but c and d are not even idempotent, so we must relate a and c via something like (b ∨ c)∧. (d ∨ c). This bad behavior of genericity with respect to the commutative shadow prompts us to work with sheaves on the lattice-type noncommutative topology, avoiding the Grothendieck topologies here for the moment, at least when one aims to relate and S L(). A good theory of sheaves over noncommutative Grothendieck topologies probably has to be developed in connection with noncommutative topos theory. If one studies noncommutative Grothendieck topologies without reference to another noncommutative topology from which it stems, the sheaf theory is of independent interest.

2.5.2 2.5.2.1

Projects A Noncommutative Topos Theory

What structure fits the philosophical equation: locales topoi = quantales ? The answer should lead to some version of noncommutative topos theory; however, let me point out that it is not clear to this author whether the above question is the correct one to put forward; quantales are too tightly related to C ∗ -algebras to obtain the right level of generality perhaps; nevertheless, the search for a noncommutative topos is worthwhile in its own right. One easily finds that the first main problem is to circumvent the notion of subobject classifier. A first generalized theory may be constructed by maintaining the “up to self-intersections” philosophy; a second approach may be to allow a family of “subobject classifiers” defined in a suitable way. It is clear that noncommutative topology reflects a kind of ordered logic; the ordering reflects the fact that “x ∈ A” and “x ∈ B” cannot be realized at the same time. We have not yet tried to write down a foundation for such ordered logic, but certainly a notion of noncommutative topos would fit perfectly in this. We leave the development of noncommutative topos theory as a project here; nevertheless we do know the main examples to be covered by such a theory: sheaves over a noncommutative Grothendieck topology, in particular those constructed on a generalized Stone space or even more specifically on a noncommutative Grothendieck topology constructed from a quantum topology (because of the presence of suitable sheafification techniques). Detail on topos theory may be found in R. Goldblatt’s [15]. 2.5.2.2

Noncommutative Probability (and Measure) Theory

The mathematician may be completely satisfied with the foundation of probability theory knowing that it means exactly what it means and no more! There is more than one question to be raised concerning certain applications in the real world or what passes for that frequently. You can only throw the dice until they crack! The certainty that some event in a given selection must happen (and to associate to this a number related to a total number of possible events in the selection) is rather ill founded. The notion of the “time necessary for some events to actually happen” is neglected, and at the level of theoretical foundation probability is based on the membership relations

2.6 The Fundamental Examples I: Torsion Theories

45

of set theory to be time or ordering independent. The idea of a noncommutative space is not consistent with such a probability theory, so it must be constructed ab initio in a noncommutative way too. An intersection A ∧ B in a noncommutative space should be related to a conditional type of probability in the sense that the probability for λ ≤ A ∧ B is expressed by p(λ, A) p(A, B) p(λ, B) where p(A, B) expresses the probability correction for A before B. Since a σ -algebra for some noncommutative topology may be defined in a straightforward way, noncommutative versions of Borelstems, etc. are not hard to develop. Probabilities could be taken in the free semigroup (generalizing N) over the set , making multiplication of probabilities formal and noncommutative but perhaps also unnatural. The approach suggested above, that is using the conditional approach retains a classical flavor, again the self-intersection introduces new phenomena, for example, p(A, A) may be nontrivial, that need to be fully integrated in the theory. 2.5.2.3

Covers and Cohomology Theories

Traditionally the idea of Grothendieck topology and the extra abstraction in the notion of cover allows us to introduce new interesting (co)homology theories; recall the use of e´ tale covers and e´ tale cohomology. In [46] and [50] we did a similar experiment ˇ with respect to Cech cohomology, which led to some interesting results in the algebraic theory of noncommutative geometry. For example, a result of L. Le Bruyn concerning the moduli space of left ideals in Weyl algebras has been reduced to a ˇ fairly straightforward calculation of Cech cohomology on the noncommutative site created from the noncommutative topology phrased in terms of Ore sets in the algebra considered. Similarly, V. Ginzburg and A. Berest have used the same technique in another situation. Of course one should be tempted to develop a noncommutative e´ tale cohomology or more cohomology theories with respect to other types of covers. Since examples of noncommutative topologies may be constructed in a completely functorial way (see also Chapter 4) one may start a theory from the consideration of covers by separable functors in the sense of Section 1.3. 2.5.2.4

The Derived Imperative

For compactness sake derived categories and derived functors have not been introduced in these notes; consequently, our sheaves are not perverse. Clearly, the latter are popular topics nowadays and they also provide strong methods of analysis, for example, in connection with rings of differential operators, Riemann-Hilbert correspondence, and so forth. It is a promising idea to combine the techniques of derived categories, perverse sheaves, and so forth with the noncommutative topology point of view. We say no more about this here.

2.6

The Fundamental Examples I: Torsion Theories

We need to recall the basic facts about torsion theory. What we say will be valid for an abelian category that is assumed to be complete, co-complete, and locally small, but we shall restrict attention to Grothendieck categories for convenience.

46

Noncommutative Spaces

Let C be a Grothendieck category. A preradical ρ of C is just a subfunctor of the identity functor. The class of preradicals of C, Q say, is partially ordered by ρ1 ≤ ρ2 if and only if ρ1 (C) ⊂ ρ2 (C) for all objects C of C. Any family of preradicals {ρα , α ∈ A} has at least an upperbound ∨ρα and a greatest lower bound ∧ρα defined in the obvious way. Consequently, Q(C) is a complete lattice with respect to ∧ and ∨. For preradicals ρ1 and ρ2 on C we may also define ρ1 ρ2 by putting ρ1 ρ2 (C) = ρ1 (ρ2 (C)) for all c ∈ C; we define ρ1 : ρ2 by taking (ρ1 : ρ2 )(C) for C ∈ C, to be the subobject of C for which we have (ρ1 : ρ2 )(C)/ρ1 (C) = ρ2 (C/ρ1 (C)). Definition 2.6 A preradical ρ such that ρρ = ρ is said to be idempotent. We say that a preradical is a radical if (ρ : ρ) = ρ; in other words ρ is radical if ρ((C)/ρ(C)) = 0 for C ∈ C. To a preradical ρ of C we may associate a preradical ρ −1 of C o by defining ρ −1 (X ) = X/ρ(X ); we call ρ −1 the dual preradical of ρ. It is easy to establish that if ρ is idempotent, respectively radical, then ρ −1 is respectively radical, idempotent. A preradical ρ of C gives rise to two classes of objects in C: Fρ = {C ∈ C, ρ(C) = 0} (pretorsion free class) Tρ = {C ∈ C, ρ(C) = C} (pretorsion class) Observe that Fρ = Tρ −1 objectwise. The relation between ρ and these classes is well summarized in the following. Theorem 2.3 With notation and conventions as above: i. Tρ is closed under quotient objects and coproducts; Fρ is closed under subobjects and products. ii. If T ∈ Tρ and F ∈ Fρ , then HomC (T, F) = 0. iii. Idempotent preradicals of C correspond bijectively to pretorsion classes of objects of C, that is, classes that are closed under quotient and coproducts. iv. Radicals of C correspond bijectively to pretorsion-free classes of objects of C, that is, classes that are closed under subobjects and products. v. For every ρ ∈ Q(C) there exists a largest idempotent preradical ρ o ≤ ρ and a smallest radical ρ c ≥ ρ. Proof i. Let us establish the first claim; the second follows by duality. That Tρ is closed under quotient objects is easily seen. Look at a family {Cα , α ∈ A} of objects in C and in Tρ . Because ρ(Cα ) = Cα for all α ∈ A, the Cα map under the canonical monomorphism Cα → ⊕α Cα into ρ(⊕α Cα ). The universal property of ⊕ then leads to the conclusion that ρ(⊕C2 ) = ⊕Cα ; that is, ⊕Cα is in Tρ too.

2.6 The Fundamental Examples I: Torsion Theories

47

ii. If f : T → F is a nonzero morphism, then Imf is in Tρ because of i, but since it is also a subobject of F, Imf = 0 follows, so no nonzero f can exist. iii. Consider a pretorsion class T . To an arbitrary object C of C we associate t(C) ∈ C by considering t(C) to be the sum of all subobjects of C that are objects of T ; then t(C) ∈ T because T is closed under coproducts and quotient objects. It is easy to see that t is an idempotent preradical of C. It is also clear that Tt = T . Now if we start with an idempotent preradical ρ, Tρ is a pretorsion class in C because of i; the idempotent preradical tρ associates to C in C the largest subobject C of C such that ρ(C ) = C , but that is exactly ρ(C); hence ρ = tρ . iv. Dual to iii. v. Starting from a preradical ρ of C we define a pretorsion class Tρ and an idempotent preradical tρ defined (see proof of iii.) by taking tρ (C) to be the largest subobject of C, say C , such that ρ(C ) = C . Therefore tρ (C) ⊂ ρ(C) for all C ∈ C, that is, tρ ≤ ρ in Q(C), and tρ is clearly the largest idempotent preradical of C with this property, so we may put ρ o = tρ . The second claim, concerning ρ o , follows by duality. A closure operator on a complete lattice L is a map (−)c : L → L , λ → λc , satisfying the following: c.1 If λ ≤ µ in L then λc ≤ µc c.2 For λ ∈ L we have that λ ≤ λc c.3 For λ ∈ L we have that (λc )c = λc . The set of closed elements of L, that is, those λ for which we have λ = λc , forms a complete lattice L c with respect to ≤ and as in L but with ∨ defined by ∨λα = (∨λα )c . Observation 2.4 If L is a complete modular lattice with closure operator (−)c satisfying (λ ∧ µ)c = λc ∧ µc , then L c , ≤, ∧, ∨ is a complete modular lattice too. Proof If λ, µ, γ are closed elements with λ ≤ µ then: µ ∧ (γ ∨λ) = µc ∧ (γ ∨ λ)c

= = =

(µ ∧ (γ ∨ λ))c (by the assumption) (λ ∨ (γ ∧ µ))c (L is modular) λ∨(γ ∧ µ)

Now considering Q(C), it is clear that (−)c : Q(C) → Q(C) is a closure operator. Therefore the foregoing observation implies that the idempotent radicals, respectively the radicals, form a complete lattice. Indeed Q(R) satisfies the condition (λ ∧ µ)c = λc ∧ µc as is easily verified and the second statement follows by duality.

48

Noncommutative Spaces

Proposition 2.10 i. If ρ is idempotent, then so is ρ c . ii. If ρ is a radical, then so is ρ c . Proof It suffices to establish i; then ii follows by duality. From ρρ = ρ we have to establish that ρ c ρ c = ρ c . Now for C ∈ C ρ c (C) is the smallest subobject of C such that ρ(C/ρ c (C)) = 0, and ρ c ρ c (C) is the smallest subobject of ρ c (C) such that ρ(ρ c (C)/ρ c ρ c (C)) = 0. Look at C ⊃ ρ c (C) ⊃ ρ c ρ c (C). If T ⊂ C is such that T /ρ c ρ c (C) is in Tρ , then also T + ρ c (C)/ρ c (C) ∈ Tρ and therefore T ⊂ ρ c (C). The latter yields that T /ρ c ρ c (C) is in Tρ and thus T = ρ c ρ c (C). Therefore we arrive at C/ρ c ρ c (C) ∈ Fρ . On the other hand, if D ⊂ ρ c ρ c (C) is such that C/D is in Fρρ = F, then (ρρ)c (C) ⊂ D and then ρ c (C) ⊂ D, but this entails that ρ c (C) = ρ c ρ c (C). Proposition 2.11 For ρ ∈ Q(C), the following are equivalent: i. ρ is left exact. ii. For every subset D of C in C, ρ(D) = ρ(C) ∩ D. iii. ρ is idempotent and Tρ is closed under subobjects. Proof Easy enough to be left as an exercise. A pretorsion class closed under subobjects is said to be hereditary; hence, by the proposition, hereditary pretorsion classes correspond bijectively to left exact preradicals. Note that the operation ρ1 ρ2 in Q(C) is noncommutative. Note also that (ρτ )−1 and τ −1 ρ −1 (in C o ) are different preradicals. Whereas duality works perfectly when ∨ and ∧ are being considered, it breaks down for the noncommutative operations. One may interpret this as if ρτ tries to be a noncommutative intersection while τ −1 ρ −1 tries to be a noncommutative union. Phrasing this in (C) = Q(C)o we reobtain the possibility of using a commutative union stemming from ∧ in Q(C) and a noncommutative “intersection” (when viewed in Q(C)o ) stemming from the preradical product τ −1 ρ −1 for τ, ρ ∈ Q(C). We shall make this more precise in Section 3.1. First we look now at those preradicals that are both idempotent and radical. Definition 2.7 A torsion theory for C is a pair (T , F) of classes of objects from C such that: tt1 For T ∈ T , F ∈ F, HomC (T, F) = 0. tt2 If HomC (C, F) = 0 for all F ∈ F then C ∈ T . tt3 If HomC (T, C) = 0 for T ∈ T then C ∈ F.

2.6 The Fundamental Examples I: Torsion Theories

49

We say that T is a torsion-class of C and its objects are (T , F)-torsion objects of C, while F is the (T , F) torsion-free class. A given class M in C cogenerates a torsion theory (T , F), which is the smallest torsion-free class containing M, FM = {F in C, HomC (C, F) = 0 for all C ∈ M} TM = {T in C, HomC (T, F) = 0 for all F ∈ FM } Proposition 2.12: Characterization of Torsion Classes a. For a class T of objects of C the following are equivalent: i. T is the torsion class of some torsion theory. ii. T is closed under quotient objects, coproducts, and extensions; that is, for every exact sequence 0 → C → C → C → 0 in C with C and C in T , then C ∈ T . b. For a class F of objects of C the following are equivalent: i. F is a torsion-free class for some torsion theory of C. ii. F is closed under subobjects, products, and extensions. Proof Old hat, see among others B. Stenstr¨om, Rings of Quotients, Springer Verlag, Heidelberg, 1975 [42]. A torsion theory (T , F) is in particular pretorsion, so it defines an idempotent preradical τ , which in view of the fact that T is closed under extensions, is easily seen to be a radical. Conversely, given an idempotent radical τ in Q(C), it determines a torsion theory of C by Fτ = {C in C, τ (C) = 0}, Tτ = {C in C, τ (C) = C}. Proposition 2.13 Torsion theories correspond bijectively to idempotent radicals; if ρ is an idempotent preradical, then ρ c is the idempotent radical corresponding to the torsion theory generated by Tρ . A torsion theory (T , F) is said to be hereditary if T is closed under submodules. In view of Proposition 2.11 and the foregoing proposition, it follows that there is a bijective correspondence between hereditary torsion theories and left exact radicals. If C is a Grothendieck category with enough injectives, then we can characterize hereditary torsion theories by the fact that (T , F) is hereditary if and only if F is closed under injective envelopes. Proposition 2.14 If M is a class closed under subobjects and quotient objects (and C is as mentioned above), then the torsion theory generated by M is hereditary.

50

Noncommutative Spaces

Proof Suppose F is torsion free and assume there is a nonzero f : C → E(F) for some C ∈ M where E(F) is the injective envelope of F. Then Imf ∈ M and F ∩ Imf is a nonzero subobject of F belonging to M as the latter is closed under subobjects—a contradiction. Corollary 2.9 i. If ρ is a left exact preradical, then ρ c is also left exact. ii. If ρ is a left exact preradical, then ρ(C) is an essential subobject of ρ c (C); that is, for every subobject D, nonzero, in ρ c (C) we have that D ∩ ρ(C) is nonzero. Proof i. By assumption Tρ is a hereditary pretorsion class. The foregoing proposition (and the proof of iii in Theorem 2.3) yields that ρ c is left exact. ii. Suppose D ∩ ρ(C) = 0. Then ρ(D) = 0; hence ρ c (D) = 0, but ρ c (D) = ρ c (C) ∩ D, and the latter is just D, thus D = ρ(D) = 0 follows. Note that a Grothendieck category with a generator has enough injectives. Definition 2.8 A left exact idempotent radical is called a kernel functor. It is clear from the foregoing that kernel functors correspond bijectively to the hereditary torsion theories. If κ denotes a kernel functor, then (Tκ , Fκ ) stands for the corresponding hereditary torsion theory. An object E of C is said to be κ-injective if every exact diagram in C with C ∈ Tκ , may be completed by a morphism g : C → E, such that gi = f . C'

0

i

f

C

C''

0

g

E

If g as above is unique as such, then E is said to be faithfully κ-injective. Proposition 2.15 The following statements are equivalent: 1. E is κ-injective and κ-torsion free. 2. E is faithfully κ-injective. Proof Consider the following exact diagram in C: 0

C' f

g

E

with c ∈ Tκ .

C

i

p

C''

0

2.6 The Fundamental Examples I: Torsion Theories

51

Since E is κ-injective at least one morphism g : C → E, such that gi = f , must exist. Suppose g1 , g2 both have that property, then (g1 − g2 )i = 0; hence g1 − g2 factorizes through C ; that is, there is a morphism h : C → E such that g1 −g2 = hp. Now C ∈ Tκ , R ∈ Fκ yields h = 0 or g1 = g2 . This establishes the implication 1. ⇒ 2. Conversely, consider the diagram in C: κ(E)

0

κ(E)

0

E

Since κ(E) ∈ Tκ , there is a unique extension of the zero map 0 → E to κ(E), which therefore has to be the zero map too! However, since κ(E) → E is a monomorphism it follows that κ(E) = 0. Proposition 2.16 Look at the exact sequence in C: E'

0

i

p

E

E''

0

where E is κ-injective and E is κ-torsion free, and E is κ-torsionfree. Then E is κ-injective too. Proof Consider the following diagram for a given morphism f : C → E , where the rows in the diagram are exact: E'

0

i

f'

0

p

E f

C'

j

C

E''

0

f ''

C''

0

where C ∈ Tκ . Note that f is obtained from the κ-injectivity of E and f is just the induced quotient map. Since E ∈ Fκ and C ∈ Tκ it follows that f = 0 of f factorizes through E and f = i f 1 for some f 1 : C → E . One easily checks that f 1 j = f and it follows that E is κ-injective. Corollary 2.10 p i Let 0 −→ E −→ E −→ E −→ 0 be exact in C and assume that E is κ-injective, E ∈ Tκ and E ∈ Fκ ; then E is isomorphic to E . Proof The conditions imply that i is an essential morphism; that is, if X is a subobject of E that is nonzero then X ∩ E is nonzero, as is easily seen (exercise). The assumption that

52

Noncommutative Spaces

E ∈ Tκ allows us to complete the following diagram in C by a morphism j : E → E such that ji = 1 E . E'

0 1E'

E

i

p

E''

0

j

E'

From the foregoing it follows that j is a monomorphism and then ji = 1 E entails that E ∼ = E . The class of all faithfully κ-injective objects in C is a full subcategory of C, which will be denoted by C(κ) and called the quotient category of C with respect to κ. The canonical inclusion is denoted by i κ : C(κ) → C. For C in Fκ the κ-injective hull of C is defined to be an essential extension C → E such that E is κ-injective and E/C ∈ Tκ . It is clear that any κ-injective hull is in C(κ). Proposition 2.17 Every C ∈ Fκ has an essentially unique κ-injective hull. Proof The object C of C has an injective hull E in C; since C ∈ Fκ it is clear that E ∈ Fκ too. Consider the exact sequence 0 −→ C −→ E −→ E/C −→ 0 in C, and define E = E × E/C κ(E/C), which may be viewed as a subobject of E by the classical pull-back properties in Grothendieck categories. Hence E ∈ Fκ and E/E ∼ = (E/C)/κ(E/C); hence κ(E/E ) = 0. Apply Proposition 2.16 to conclude E is κ-injective. On the other hand, E /C ∼ = κ(E/C) or E /C is κ-torsion. Then let us assume that E 1 , E 2 are κ-injective hulls of C. It follows that E 2 is isomorphic to a subobject E 2 of E 1 containing C as a subobject. Because E 1 is in Fκ and also an essential extension of E 2 that itself is faithfully κ-injective, we apply Corollary 2.10 and arrive at E 1 ∼ = E 2 ∼ = E 2 . The κ-injective hull of C ∈ C is denoted by E κ (C). Recall that in a Grothendieck category C the following are equivalent for any endofunctor F: a. F has a right adjoint. b. F is right exact and commutes with coproducts. Recall also that right adjoints preserve projective (inverse) limits while left adjoints preserve inductive (direct) limits. Theorem 2.4 With notations as before: i κ : C(κ) → C has a left adjoint. Proof For C ∈ C, define aκ (C) = E κ (C/κ(C)). This yields a functor aκ : C → C(κ). If f : C → i κ (D) is an arbitrary morphism, with C ∈ C, D ∈ C(κ), then f extends to a

2.6 The Fundamental Examples I: Torsion Theories

53

morphism f 1 : C/κ(C) → i κ (D), since i κ (D) ∈ Fκ . Now aκ (i κ (D)) is faithfully κinjective and aκ (C)/(C/κ(C)) ∈ Tκ , hence f 1 extends to f : aκ (C) → aκ i κ (D) = D. Finally it is easily verified that we obtain the following isomorphism: HomC (C, iκ (D)) ∼ = HomC(κ) (aκ (C), D). We shall write Q κ = i κ aκ . For c ∈ C, the object Q κ (c) together with the canonical morphism jκ : C → Q κ (C) is called the C- object of quotients of C with respect to κ. Proposition 2.18 Q κ is a left exact endofunctor in C. Proof j If 0 → C −→ C is exact in C, then so is the sequence 0 → C /κ(C ) → C/κ(C). Since Q κ (C ), Q κ (C) are essential extensions of C /κ(C ), respectively C/κ(C), it follows that Q κ (i) is a monomorphism. First let C ∈ Fκ and consider the following commutative diagram with exact top row: 0

C'

0

Q κ (C' )

f

Qκ ( f )

C Q κ (C )

g

Q κ ( g)

C'' Q κ (C'' )

Here Q κ ( f ) is a monomorphism and Q κ (g)Q κ ( f ) = Q κ (g f ) = 0; hence Q κ (C ) is a subobject of KerQ κ (g). Then consider the exact sequence: 0 −→ KerQ κ (g) −→ Q κ (C) −→ ImQ κ (g) −→ 0 Since Q κ (C) is κ-injective and ImQ κ (g) ∈ Fκ , we conclude that KerQ κ (g) is κinjective; hence faithfully κ-injective. Moreover KerQ κ (g)/C ∼ = Q κ (C)/C is in Tκ ; hence Corollary 2.10 yields Q κ (C ) = KerQ κ (g). In general, that is, when C is not necessarily in Fκ we consider f

0 −→ C −→ C −→ C −→ 0 and define D = C ×C κ(C ), the pre-image of κ(C ) in C. Then κ(C) is clearly a subobject of D in C. Also Im f is a subobject of D and D/Imf ∼ = κ(C ). Therefore D/κ(C) contains (Im f + κ(C))/κ(C) such that modulo the latter it is κ-torsion. We obtain an exact sequence: 0 −→ D/κ(C) −→ C/κ(C) −→ C /κ(C ) −→ 0 where κ(C/κ(C)) = 0. Now we have reduced the problem to the torsion-free case because we obtain an exact sequence 0 −→ Q κ (D/C) −→ Q κ (C) −→ Q κ (C ) where Q κ (D/C) = Q κ (Im f + κ(C)/κ(C)) = ImQ κ ( f ).

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Noncommutative Spaces

Note that aκ has right adjoint i κ but Q κ need not have a right adjoint, in fact Q κ need not even be a right exact functor. For further application of these techniques to categories of sheaves or presheaves it is worthwhile to present some basic facts about Giraud subcategories of (complete) Grothendieck categories. This allows us to apply the “reflector” approach to localization theory; the presentation here is close to Section 2.3 in F. Van Oystaeyen, A. Verschoren, Reflectors and Localization. Application to Sheaf Theory, Lect. Notes in Pure and Applied Mathematics Vol. 41, M. Dekker, New York, 1978 [48]. This approach also allows a general treatment of compatibility of kernel functors and commuting properties of localization functors, that is, exactly the topic recognized in noncommutative topology with respect to the relations between noncommutative space and the commutative shadow. Compatibility of localization goes back to F. Van Oystaeyen, “Compatibility of Kernel Functors and Localization Functors” [45]. The consideration of Giraud subcategories of (complete) Grothendieck categories prepares for the study of sheaves as a subcategory of presheaves. So we look at a complete Grothendieck category P; a full subcategory S of P is called reflective if the inclusion functor i : S → P has a left adjoint a, called the reflector of S in P; that is, for P ∈ P, S ∈ S there is a natural isomorphism HomP (P, iS) ∼ = HomS (aP, S) with canonical natural transforms p : ai → 1S and q : 1P → ia. The couple (a P, q P : P → ia P) has the following universal property: every P-morphism f : P → i S with S ∈ S factorizes in a unique way as follows: P

iaP

qP f˜

f iS

The morphism q P is called the reflection of P. Proposition 2.19 Let S be a reflective subcategory of a (complete) Grothendieck category P; then S is complete and co-complete. If the reflector of S in P is left exact, then S has exact direct limits and a generator. A subcategory of P with a left exact reflector is called a Giraud subcategory of P. From the definition it follows that a Giraud subcategory of a (complete) Grothendieck category is itself a (complete) Grothendieck category and the reflector is exact, whereas the inclusion functor S → P is in general only left exact. Let T be the class of objects C in P for which a(C) = 0 and let F consist of subobjects in P of objects of S. Proposition 2.20 P ∈ P is in F exactly when q P : P → iaP is a monomorphism.

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55

Proof If q P is a monomorphism, then P is in F since a(P) ∈ S. Conversely, consider P ∈ F and 0 → P → i(S) with S ∈ S. Commutativity of the following diagram in P: 0

i(S)

P ia(P)

leads to the conclusion that q p is a monomorphism. F may be viewed as a full complete subcategory of P easily verified to be a reflective subcategory of P with epimorphism reflector a j where a is the reflector of S and j : F → P the canonical inclusion function. The objects of F are said to be separated. Proposition 2.21 If P ∈ P is separated, then ia(P) is an essential extension of P in P. Proof Let Q be a subobject of ia(P) in P and assume that Q ×ia(P) P = 0. Then 0 = a(Q ×ia(P) P) = a(Q) ×a(P) a(P) ∼ = a(Q). Since Q is a subobject of a separated object, it is itself separated; that is, the canonical Q → ia(Q) is a monomorphism and thus Q = 0 follows from 0 = a(Q). Proposition 2.22 Consider an exact sequence in P: 0 −→ P −→ P −→ P −→ 0 1. If P ∈ S, P ∈ F, then P ∈ F. 2. If P ∈ F, P ∈ S, then P ∈ S. Proof Instead of providing a direct proof we can derive it directly from the next proposition, which reduces it to the torsion theory situation and Proposition 2.16 as well as Corollary 2.10 (with a slight rephrasing). Proposition 2.23 The couple (T , F) determines a torsion theory in P such that its quotient category is equivalent to S.

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Proof Since a is exact, T is closed under subobjects, quotient objects, and extensions. Since a has a right adjoint, it preserves coproducts, hence T is a torsion class. Obviously for T ∈ T , S ∈ S we have HomP (T, i(S)) = 0 and also HomP (T, F) = 0 for every F in F. Conversely, if HomP (T, P) = 0 for all T ∈ T , then from Kerq P ∈ T we obtain that Kerq p = 0, hence P ∈ F. Consequently F may be considered as the torsion-free class corresponding to T . The kernel functor associated to (T , F) will be denoted by α. Clearly if P ∈ P then:

α(P) = {P , 0 −→ P −→ P and a(P ) = P } A Giraud subcategory of P is said to be strict if it is closed under P-isomorphisms. Observation 2.5 If κ is a kernel functor for a (complete) Grothendieck category P, then P(κ) is a strict Giraud subcategory of P. Proof From Proposition 2.17. From what we have already learned it follows easily that there is a bijective correspondence between torsion theories for P and strict Giraud subcategories of P; this is otherwise known as Gabriel’s Theorem. At times we have used the term (complete) Grothendieck category; in fact this indicates that the original statement of the result used the completeness as an extra assumption. It was only after the proof of the Gabriel-Popescu, theorem (first by Popescu, but with a gap solved by Gabriel) that it followed that a Grothendieck category with a generator has enough injective objects and also that every Grothendieck category is complete. Theorem 2.5: Gabriel-Popescu Let C be any Grothendieck category with generator G; put R = HomC (G, G) and let M : C → R-mod be the functor C → HomC (G, C) = M(C). 1. M is full and faithful. 2. M induces an equivalence between C and (R − mod)(κ), where κ is the largest kernel functor in R-mod for which all modules M(C), C ∈ C, are faithfully κ-injective. Corollary 2.10 Every object C in C has an extension that is an injective object of C. Every C is complete! The categorical approach to localization theory has an undeniable elegance, but now we have a less obvious notion of the noncommutative composition at hand, unless we start to compare localization of a strict Giraud category S to localization of P.

2.6 The Fundamental Examples I: Torsion Theories

57

Proposition 2.24 An object E of S is injective in S if and only if i(E) is injective in P. Proof If i(E) is injective in P, then E is injective in S because i is left exact. Conversely, suppose that E is injective in S and consider a diagram in S: C'

0

j

C

f

i(E)

yielding a commutative diagram in P: j

C'

0

C

f

i(E)

qC'

0

qC g

ia( f )

ia(C' )

ia( j )

ia(C )

where existence of g follows from the injectivity of E in S : ia( f ) = g ◦ ia( j). Put g = gqC ; then we find: g j = gqC j = gia( j)qC = ia( f )qC = f , finishing the proof. For C ∈ S, respectively C ∈ P, the injective hull of C in S, respectively in P, will be denoted by E s (C), respectively E p (C). Lemma 2.22 1. For S ∈ S, i(E s (S)) is an essential extension of i(S) in P. 2. Let S ∈ S; then E p (i(S)) = i E s (S); that is, the hull in P of an object in S is in S too. Proof 1. Let P be a subobject of i(E s (S)) in P such that P ×i E s (S) i(S) = 0. Exactness of a yields 0 = a(0) = a(P ×i E s (S) i(S)) = a(P) ×ai(E s (S)) ai(S) = a(P) × E s (S) S, contradicting the fact that E s (S) is an essential extension of S in S since a(P) ∈ S. 2. The foregoing implies that both i(E s (S)) and E p (i(S)) are essential extensions of i(S) in P; thus we arrive at a commutative diagram in P: i(S) i(E s(S))

E p(i(S))

g f

58

Noncommutative Spaces where f exists by definition of E p , and it is a monomorphism, moreover g is a monomorphism too. Since E p (i(S)) is a maximal essential extension of i(S) in P, it follows from 0 −→ E p (i(S)) −→ i E s (S) f

that E (i(S)) ∼ = i(E (S)). p

s

Corollary 2.11 If P ∈ P is separated, then E p (P) is separated. Indeed we have a commutative diagram of monomorphisms in P: P

ia(P)

E p(P)

E p(ia(P))

Since E p (ia(P)) ∼ = i E s (a(P)), it follows that E p (P), being a subobject of E s (a(P)) in P, is separated. Proposition 2.25 If P ∈ P is separated, then we have: ia(E p (P)) = E p (ia(P)) = i(E s (a(P))) = E p (P) (1)

(2)

(3)

Proof 1. We obviously have the following monomorphisms: P −→ ia(P),

E p (P) −→ E p (ia(P)),

ia E p (P) −→ E p (ia(P))

Now E p (ia(P)) is essential over ia(P) and this in turn is essential over P in P; that is, E p (ia(P)) is essential over P. The equality (1) follows if we establish that ia(E P (P)) is injective in P; that is a E(P) is injective in S (see 2.24.). Consider an exact sequence 0 −→ S −→ S and a given f : S −→ a E p (P)in s S. The pull-back properties yield: S'1 = iS'

×

iaEp (P)

Ep(P)

s'

f1

iS' if

Ep(P)

is

iS ig

iaEp(P)

where s is a monomorphism, thus (is)s is monomorphic. Obviously: a(S1 ) = S × a E p (P) = S a E p (P)

2.6 The Fundamental Examples I: Torsion Theories

59

By the injectivity of E p (P) in P, there is a P-morphism g1 : i S → E p (P), such that g(is)s = f 1 . Put g = a(g1 ) : S → a E p (P). Let j be the isomorphism aS1 → S . Then we have a f 1 = f j = q(g1 )sa(s ) = gs j, with f j = gs j, hence f = gs since j is an isomorphism and thus in particular an epimorphism in S. 2. The equality (2) is a direct consequence of Lemma 2.2.2(2). 3. E p (P) is separated because P is separated (Corollary 2.11). Then ia(E p (P)) is essential over E p (P), hence over P in P. Therefore E p (P) = ia E p (P) because E p (P) is a maximal essential extension of P in P. Consider kernel functors κ, κ in P. Then κ ≥ κ if and only if Tκ ⊃ Tκ or equivalently κ(P) ⊃ κ (P) for every object P of P. For kernel functors κ and κ in P we say that κ is Q κ -compatible if κ Q κ = Q κ κ . Lemma 2.23 Suppose κ is Q κ -compatible. 1. If P is in Fκ , then Q κ (P) is in Fκ ; the converse holds when P is κ-torsion free. 2. If P is in Tκ , then Q κ (P) is in Tκ ; the converse holds in case P ∈ Fκ . Proof Exercise. To any strict Giraud subcategory S of P we have associated a kernel functor α (see remark Proposition 2.23). The α- compatible kernel functors (Q α -compatible) are sometimes called Scompatible kernel functors. Hence, κ in P is S-compatible exactly when iaκ = κia. If iaκia = κia, that is, κ takes objects of S to objects of S, then κ is said to be inner in S. If κ is inner in S, then the functor κ is denoted by κ S ; in general κ S need not be a kernel functor in S. Proposition 2.26 Let κ be S-compatible in P; then κ s is a kernel functor in S. Proof For S in S, κ(i S) = κ(iai(S)); hence κ S is inner in S. Therefore aκ(i(S)) = κ s (S) and it is clear that κ s is a left exact subfunctor of the identity in S. Furthermore we easily calculate: κ s (S/κ s (S)) = aκ(ia(i S/I κ s (S))) = aκ(i S/iκ s (S)) = aκ(i S/κ(i S)) = a(0) = 0 (Note: we simplified notation by dropping some brackets in the notation).

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Noncommutative Spaces

Earlier we mentioned the gen-topology induced on a lattice by taking the intervals [0, λ], λ ∈ ; now comparing the lattice of kernel functors on P to what the Zariski topology would be if P were C-mod for some commutative ring, we know we have to look at the opposite lattice, and therefore the sets [κ, 1] get our attention. These are well behaved in terms of compatibility condition because of the following result. Proposition 2.27 Look at kernel functors κ and κ , for P such that κ ≥ κ ; then κ is Q(κ )-compatible. Proof Let qκ(P) : κ(P) → Q κ (κ(P)) be the reflection of P ∈ P in P(κ ). The following sequence is exact: 0 → Imqκ(P) → Q κ (κ(P)) → Coimqκ(P) → 0 Clearly, Imqκ(P) is κ-torsion; moreover κ ≥ κ implies that Coimqκ(P) is κ-torsion. Hence, Q κ (κ(P)) is a subobject κ Q κ (P). Conversely, since Fκ ⊂ Fκ we have that P/κ(P) ∈ Fκ so we have a monomorphism: 0 → P/κ(P) → Q κ (P/κ(P)). Then P/κ(P) ∩ κ Q κ (P/κ(P)) = 0 implies that Q κ (P/κ(P)) = 0. Finally, by the exactness of 0 → Q κ (κ(P)) → Q κ (P) → Q κ (P/κ(P)) we obtain that κ Q κ (P) ⊂ Q κ (κ(P)). Theorem 2.6 For a strict Giraud subcategory S of P and an S-compatible kernel functor κ for P we have S ∈ S is (faithfully) κ s -injective if and only if i(S) is (faithfully) κ-injective. Proof Assume that i(S) is κ-injective. Consider a diagram in S: 0

S1

S2

S2/S1

0

f

S

where κ s (S2 /S1 ) = S2 /S1 . In P we obtain a diagram, applying i to the above: 0

i(S1)

i(S2)

i(S2)/i(S1)

if

i(S)

where i(S2 )/i(S1 ) is subobject of i(S2 /S1 ). Since S2 /S1 is κ s -torsion and S2 /S1 = a(i S2 /i S1 ), it follows Lemma 2.23(2) that i S2 /i S1 is κ-torsion, so there exists a g : i S2 → i S completing the above diagram. Then it is clear that a(g ) completes the diagram in S from which we started.

2.6 The Fundamental Examples I: Torsion Theories

61

Conversely, let S be κ s -injective and consider an exact sequence in P: 0

P1

P2

P2/P1

f

i(S)

where P2 /P1 ∈ Tκ . Since a is exact we obtain the following diagram in S: 0 −→ a P1 −→ a P2 −→ a(P2 /P1 ) a f S Again, by Lemma 2.23, it follows that a(P2 /P1 ) is κ s -torsion; therefore, there exists an S-morphism g : a P2 → S completing the diagram. Let g be the map (ig )q P2 : P2 → i(S), and it is easily verified that g extends f as desired. Since i(S) is κ-torsion free if and only if S is κ s -torsion free (because i(S) is separated and Lemma 2.23) we may apply Proposition 2.15. Proposition 2.28 With notation as before: Let κ be an S-compatible kerel functor for P and consider an object S of S that is κ s -torsion free, i(E κ s (S)) ∼ = E κ (i(S)). Proof Lemma 2.23 yields: i(S) ∈ Fκ . The foregoing and the fact that E κ s (S) is faithfully κ s -injective imply that i E κ s (S) is faithfully κ-injective. Furthermore, i E κ s (S)/i S is κ-torsion in P because E κ s (S)/S is κ s -torsion in S. But E κ (i S) is unique up to isomorphism in P with the properties mentioned above; therefore, we arrive at E κ (i S) ∼ = i E κ s (S). Corollary 2.12 If κ is an S-compatible kernel functor for P and S ∈ S, then i Q κ s (S) = Q κ (i S). Proof If S is κ s -torsion free, then the statement follows from the foregoing proposition. In general: i Q κ (S) = i E κ s (S/κ s (S)) = E κ (ia(i S/κ(i S))) Of course, i S/κ(i S) is separated; thus ia(i S/κ(i S)) is an essential extension in P of i S/κ(i S). Consequently i(S/κ s (S)) is κ-torsion free and Q κ (i S) = E κ (i S/κ(i S)) ∼ = E κ (ia(i S/κ(i S))). Proposition 2.29 Let S be a strict Giraud subcategory of P and let κ be a kernel functor in P such that κ ≥ α where α corresponds to S. Then we have: Q κ (P) ∈ i S for every P ∈ P.

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Proof Put P = P/κ(P). Since κ ≥ α, α(P) = 0; hence P is separated with respect to S. From a(ia(P)/P) = 0 it follows that ia P/P is α-torsion, hence κ-torsion. Faithful κ-injectivity of Q κ (P) gives rise to the following commutative diagram in P: 0

– P

– iaP qP–

Qκ(P)

– – iaP P

0

jP

Since q P is a monomorphism and since ia P is an essential extension of P in P, j P is a monomorphism and thus Q κ (ia P) = Q κ (P). Because of Proposition 2.27, κ is S-compatible; then Corollary 2.12 applies and we obtain i Q κ s (a P) = Q κ (ia P) = Q κ (P), and finally this yields that Q κ (P) is in i S. Let us now construct some kernel functors that are S-compatible. Consider a nonzero object P in P and let K (P) be the class of kernel functors κ for P such that P ∈ Fκ . If P ∈ P is essential over P in P, then obviously K (P) = K (P ). Define κP for P by putting: κ P (Q) = ∩{kerg, g ∈ HomP (Q, Ep (P))} Proposition 2.30 With notation as before: i. κ P is a kernel functor for P and κ P ∈ K (P). ii. If κ is a kernel functor for P, then κ ∈ K (P) if and only if κ ≤ κ P . Proof i. Straightforward (note that there is a monomorphism 0 → E P (P) → E P (P); hence κ P (E P (P)) = 0 and κP ∈ K (P)). ii. If κ ≤ κ P , then κ ∈ K (P) obviously. Conversely, if κ ∈ K (P) for some kernel functor κ for P, look at an arbitrary morphism g : Q → E P (P), in P. Clearly κ(Q) ⊂ Kerg; hence κ(Q) ⊂ κ P (Q), for every Q in P; thus κ ≤ κ P . Theorem 2.7 If S ∈ S, then κi(S) is an S-compatible kernel functor for P. Conversely, if P in P is such that κ P is S-compatible, then there is an S in S such that κ P = κi(S) ; moreover S = a(P). Proof We first check that iaκi(S) (P) = κi(S) (ia P), and we may assume that we have replaced S by E s (S) and i(S) by E P (i(S)). Then the problem reduces to proving: ∩Kerg, g ∈ HomP (iaP, iS) = ia(∩{Kerg, g ∈ HomP (P, iS)})

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63

Since S is a full subcategory of P and since a is a right adjoint of i, we do obtain the following isomorphisms in Ab: HomP (iaP, iS) ∼ = HomS (aP, S) = HomP (P, iS). Since S is a Giraud subcategory of P, we have that ∩{Kerg, g ∈ HomS (aP, S)} is an object of S. Conversely, if P ∈ P is such that κ P is S-compatible, then κ P (ia P) = iaκ P (P) = ia(0) = 0, hence κ P ≤ κia(P) . On the other hand: κia(P) (P) = ∩{Kerg, g ∈ HomP (P, EP (iaP))}. In view of Lemma 2.22(2) we arrive at: κia(P) (P)

= =

∩{Kerg, g ∈ HomP (P, i E s (a P))} ∩{Kerg, g ∈ HomP (a P, E s (a P))} = 0

Finally we find 0 = κia(P) (P); thus κia(P) ≤ κ P . The foregoing techniques may be applied to some interesting special cases. Of course P = R-mod for some noncommutative ring R is of interest, but so is the case where P is the category of presheaves over a small category X with values in a Grothendieck category C. We return to this later.

2.6.1

Project: Microlocalization in a Grothendieck Category

In the algebraic geometry of associative algebras (see [46]), a particularly interesting case is presented by filtered algebras that are “almost commutative” in the sense that the associated graded ring is a commutative ring. Their noncommutative site may be viewed as being quantum-commutative in the sense that the topology defined in terms of microlocalzation functors is in fact a commutative one. Roughly speaking (see [44] for full detail) the microlocalization is obtained from a completion with respect to a localized filtration. This project is to develop such a technique for arbitrary Grothendieck categories; this can then be continued along the lines of Chapter 3, leading to canonical microtopologies. There may be a benefit of this to sheaf theory, but at this moment there are no obvious applications of this technique outside the algebraic theory already covered in [46]; however, the consideration of categories of topologized objects is natural in the context we have developed, so it is not unlikely that new applications of the microlocalizations may be discovered. Let κ be a kernel functor on the Grothendieck category C and let Tκ denote its torsion class (see Definition 2.8). To an arbitrary object M of C we may associate a filter L(κ, M) consisting of all subobjects N of M in C such that M/N ∈ Tκ . It is clear that L(κ, M) is closed under the lattice operations ∧(= ∩) and ∨(= ) defined in L(M), the (big) lattice of subobjects of M in C (see also the remarks following Lemma 2.3). The (big) lattice L(κ, M) need not have a 0 but we may formally add such if we wish. In any case we may view

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Noncommutative Spaces

lim κ in C as ←− L(κ, M) as a topologization of M and we may define M (M/N ). The N ∈L(κ,M)

κ and (Q κ (M))∧ is easily investigated. This provides us with a relation between M general notion of microlocalization, denoted Q µκ for a torsion theory (Tκ , Fκ ). When Q κ is an exact functor the properties of Q µκ have to be investigated.

2.7

The Fundamental Examples II: L(H)

Let H be a complex Hilbert space and consider the set L(H ) of closed subspaces of H . The set L(H ) becomes a complete lattice if we define: for U, V ∈ L(H ), U ∧ V = U ∩ V , U ∨ V = (U + V ), where (−) denotes the closure in H . In L(H ) we also have a complement, associating the orthogonal complement U ⊥ to U in H . It is clear that L(H ) satisfies axioms A.1, . . . , A.9, but not A.10 (look at a space of finite codimension in H and a basis for its orthogonal complement; try to use this global cover to induce a cover on a line in the orthogonal complement disjoint from the chosen basis). Consequently it is impossible here to use Lemma 2.20 to obtain sheaves over L(H )! However, we shall have other techniques available that will allow the construction of sheaves (and sheafification) over the generalized Stone space, which will be introduced later in this section. Consider the algebra L(H ) of bounded linear operators on H . Associating to U in L(H ) the orthogonal projection PU onto U viewed as an element in L(H ), then we see that the lattice L(H ) is isomorphic to the lattice P(L(H )) of orthogonal projections in L(H ) which are exactly the idempotent elements of L(H ). It is well known (and easily verified) that L(H ) is not distributive due to the fact that PU and PV need not commute. From the theory on noncommutative topologies in Chapter 2, we expect that idempotency of PU PV and PV PU would lead to the commutativity of PU and PV . In fact this is the case, but an even stronger result holds because only one such product has to be considered (I thank my colleague Jan van Casteren for some discussions about the analytical aspects). Proposition 2.31 With notation as introduced above: if (PU PV )2 = PU PV then PU and PV commute and PU PV = PU ∩V . Proof Observe that PU = PU∗ , PV = PV∗ . Put T = PU PV − PV PU , then T ∗ = −T and we easily calculate: PU T PV = PV T PU = PU T PU = PV T PV = 0 For f ∈ U ⊥ ∩ V ⊥ we have: T f = PU PV f − PV PU f = PU .0 − PV .0 = 0

2.7 The Fundamental Examples II: L(H)

65

Clearly, if f ∈ U , then T f ∈ U + V but also T f ∈ U ⊥ , because PU T f = PU T PV f = 0, and similarly T f ∈ V ⊥ . Consequently, f ∈ U yields T f ∈ (U ⊥ ∩ V ⊥ )∩(U +V ) = (U +V )⊥ ∩(U +V ) = 0. A similar argument establishes that T f = 0 for f ∈ V . Since (U + V ) + (U + V )⊥ is dense in H , T must be the zero operator; hence PU PV = PV PU . Next consider S = PU PV − PU ∩V . A direct calculation yields: S2

= = =

(PU PV − PU ∩V )(PU PV − PU ∩V ) PU PV PU PV − PU ∩V PU PV − PU PV PU V + PU2 ∩V PU PV − PU ∩V pU PV − PU ∩V + PU ∩V = (I − PU ∩V )PU PV

Since we obviously have S = S ∗ , the foregoing yields: S2

= =

S ∗ S = (S ∗ S)∗ = (S 2 )∗ = ((I − PU ∩V )PU PV )∗ PV PU (I − PU ∩V ) = PU PV − PU ∩V = S

Therefore S is an orthogonal projection. If f ∈ U ∩ V + (U ⊥ + V ⊥ ), then S f = 0 (using that U ⊥ + V ⊥ ⊂ (U ∩ V )⊥ ). Since the closure of U ⊥ + V ⊥ is (U ∩ V )⊥ and since S is continuous, it follows that S f = 0 for any f in (U ∩ V ) + (U ∩ V )⊥ ; hence S = 0. Observe that for any linear subspace of H , U for example, the closure of U in H is given by U ⊥⊥ . The advantage of the analytic proof given above is that we do not have to verify the axioms of a noncommutative topology for the set of finite products of idempotent elements of L(H ). In that way we would arrive at the following result too; we again provide an analytic proof. Theorem 2.8 Let P be a family of normal operators acting on the Hilbert space H ; that is, P ∈ P implies that P P ∗ = P ∗ P. Suppose that for all finite {P1 , . . ., Pn } ⊂ P we have (P1 . . . Pn )2 = P1 . . . Pn ; then for any finite {P1 , . . ., Pn } ⊂ P we have P1 . . . Pn = (P1 . . . Pn )∗ PR(P1 ...Pn ) where for any operator T, R(T ) stands for the range of T ; in particular all elements of P commute with one another. Proof First we establish that every P ∈ P is idempotent. For P ∈ P, (P P ∗ − P)P ∗ = P(P ∗ )2 − P P ∗ = P P ∗ − P P ∗ = 0; if P f = 0, then (P P ∗ − P) f = P ∗ P f − P f = 0; thus P P ∗ − P is zero on R(P ∗ ) + ker(P). The latter space is dense in H ; hence by continuity of P, P = P ∗ = P P ∗ = P ∗ P = P 2 = PR(P) . We proceed by induction, supposing the result holds for any finite {P1 , . . ., Pn−1 } ⊂ P. Take P ∈ P, P ∈ {P1 , . . ., Pn−1 }, which is thus necessarily an orthogonal projection. The induction hypothesis implies P1 . . . Pn−1 = (P1 . . . Pn−1 )∗ = PR(P1 ...Pn−1 ) and P1 . . . Pn = PR(P1 ...Pn−1 ) Pn , with Pn = P, leads to P1 . . . Pn = PR(P1 ...Pn−1 ) Pn = (PR(P1 ...Pn−1 ) Pn )∗ = PR(P1 ...Pn ) . Also if {P1 , . . ., Pi−1 , P, Pi+1 , . . ., Pn−1 } is considered, then again the same argument implies, with an interchanging of P and Pi , and the claim follows. We point out that in general it is not obvious that products of comparable operators may be comparable. We have P ≤ Q whenever P = P Q(= Q P = P Q P) for

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orthogonal projectors. If P j ≤ Q j for j ∈ {1, . . ., n}, then P1 . . . Pn ≤ Q 1 . . . Q n , but equality does not entail P j = Q j for j = 1, . . ., n. For suitable operators such a result may be proved; we include an example. Example 2.3: (J. van Casteren) If 0 ≤ P j ≤ Q j , j = 1, . . ., n for orthogonal projections P j , Q j such that the following hold: i. (Q j . . . Q k − P j . . . Pk )Pk (Q k . . . Q j − Pk . . . P j ) ≥ (Q j . . . Q k − P j . . . Pk )(Q k . . . Q j − Pk . . . P j ), for 1 ≤ j ≤ k ≤ n − 1 ii. (Q k . . . Q j − Pk . . . P j )P j−1 (Q j . . . Q k − P j . . . Pk ) ≥ (Q k . . . Q j − Pk . . . P j )(Q j . . . Q k − P j . . . Pk ) for 2 ≤ j ≤ k ≤ n. Then Q 1 . . . Q n = P1 . . . Pn if and only if P j = Q j , j = 1, . . ., n. We defined points and quasipoints in Section 2.3. Let us point out some facts in the particular case of L(H ). First, one easily verifies that L(H ) has no points; indeed if [ A] is a point of L(H ) given by its filter A, then V {Cu α , α ∈ A} = H for some selected basis {u α , α ∈ A}; hence Cu α ∈ A for some suitable α ∈ A. If U ∈ A, then U ∩ Cu α ∈ A; hence Cu α ⊂ U because 0 ∈ A by assumption; choose V ∈ L(H ) such that V nor V ⊥ contains Cu α ; then V ∨ V ⊥ = H yields that either V or V ⊥ is in A, but that contradicts Cu α ∈ V, Cu α ∈ V ⊥ . Of course L(H ) has minimal points (quasipoints) since maximal filters always exist. We have observed that maximal filters define idempotent elements of C(L(H )) (see Lemma 2.4) and if λ ∈ A, β ∈ A, then λ ∧ β ∈ A follows. Let us also recall that a directed set A in a poset (with 0 and 1) is said to be pointed if for all λ ∈ A there exists a µ ∈ A such that γ ≤ λ, γ ≤ µ implies γ = 0. Proposition 2.32 The minimal points of L(H ) are exactly given by the pointed filters. There are two types of pointed filters: i. A = {U ∈ L(H ), u α ∈ U for some u α = 0 in H }. ii. A contains all V of finite codimension in H . Proof If some cofinite dimensional U is not in A, then there is a V ∈ A such that U ∩ V = 0; therefore, V is finite dimensional. Thus there must exist a W ∈ A with minimal dimension as such. If dimW > 1, then pick a subspace W ⊂ W with dimW = 1; by assumption W ∈ A; hence there is a U ∈ A such that W ∩ U = 0, but that contradicts U ⊃ W . Consequently dimW = 1 and A is as claimed in i. The remaining case ii is exactly the case where all cofinite dimensional V are in A. Note that in general a pointed filter is maximal; indeed if A is pointed and A B L(H ), then there is a V ∈ B such that V ∈ A, and thus there exists a W ∈ A such that W ∩ V = 0, contradicting W, V ∈ B and B = L(H ). Conversely, if B is a maximal

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67

filter in L(H ) such that U ∈ B, then A = {U ∩ V, V ∈ B} is a directed set because for V, W ∈ B, U ∩ (V ∩ W ) ⊂ U ∩ V , U ∩ W ; hence A ⊃ B is a strict inclusion of filters because U ∈ A, U ∈ B. The maximality assumption on B then implies 0 ∈ A; therefore U ∩ V = 0 for some V ∈ B and consequently B is pointed. The foregoing property of L(H ) is shared by more classical types of lattices, for example, topologies. Proposition 2.33 Let X be a topological space satisfying T1 ; write L(X ) for the lattice of open subsets of X (sometimes denoted Open(X )). If A is a pointed directed set for L(X ), then it is one of three possible types: i. ∩{U ∈ A} = {x} for some x ∈ X and every open neighborhood of x in A. ii. ∩{U ∈ A} = ∅ and ∩{U , U ∈ A} = {x} for some x ∈ X (where U is the closure of U in X ). iii. ∩{U ∈ A} = ∅ and X − K ∈ A for every closed compact set K in X (compact here means having the finite intersection property). Proof Suppose x, y ∈ I = ∩{U ∈ L(X )}. If y = x then, in view of the T1 -property, we may select an open neighborhood Vy of y such that x ∈ Vy ; thus Vy ∈ A. Note that we may replace A by its filter A because the pointedness assumption is preserved. Thus, there is a U ∈ A such that U ∩ Vy = ∅; hence y ∈ U and then y ∈ I . It follows that I = {x} and the claims in i follow. In the remaining cases we have I = ∅. Suppose there is a closed compact K such that X − K ∈ A. Since A is pointed there is a V ∈ A such that V ∩ (X − K ) = ∅; that is, V ⊂ K or V ⊂ K . Look at I = ∩{U , U ∈ A}. Since for V as above, V is compact, it follows that I = ∅ unless ∅ ∈ A, a case that may be excluded because ∅ ∈ A. If x = y are both in I and Vy is an open neighborhood of y such that x ∈ Vy , then U ∩ Vy = ∅ for some U in A, while on the other hand y ∈ V ∩ U . Because U is dense in U , this leads to a contradiction unless y = x; thus I = {x} and the claims of ii are proved. The remaining case is the one where X − K ∈ A for all closed compact subsets K of X , as stated in iii. The Stone space, originally constructed for Boolean algebras, has been defined also for arbitrary lattices (I do not recall where this first appeared in the literature), but for us the Stone space, as a set, is nothing but the part of C() corresponding to the pointed directed sets, so this definition extends to noncommutative topologies. It is also clear how to define a generalized Stone topology on the above defined set, SC() for example.

2.7.1

The Generalized Stone Topology

Consider a noncommutative topology and C(). For λ ∈ , let Oλ ⊂ C() be given by Oλ = {[A], λ ∈ A}. It is trivial to verify Oλ∧µ ⊂ Oλ ∩ Oµ , Oλ∨µ ⊃ Oλ ∪ Oµ , and therefore the Oλ define a basis for a topology on C(), called the generalized Stone topology. We may restrict attention to the point-spectrum Sp(), or the

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quasipoint spectrum Q S p (); the topology defined on these subsets of C() will again be called the generalized Stone topology. Observe that on the quasispectrum, writing Q Oλ = {[A], [A] ∈ Q S P (), λ ∈ A} we actually obtain Q Oλ∧µ = Q Oλ ∩ Q Oµ but still only Q Oλ∨µ ⊃ Q Oλ ∪ Q Oµ . On the other hand, writing P Oλ = {[A], [A] ∈ S p (), λ ∈ A} we have P Oλ∧µ ⊂ P Oλ ∩ P Oµ , P Oλ∨µ = P Oλ ∪ P Oµ . This follows from the fact that for [A] in Sp() we do have that λ, µ ∈ A entails λ ∧ µ ∈ A; indeed (cf. Definition 2.3.3.). On Sp() the generalized Stone topology is nothing but the point-topology. On S P(), writing S P Oλ = {[A], [A] ∈ S P(), λ ∈ A}, we have both equalities: S P Oλ∧µ = S P Oλ ∩ S P Oµ and S P Oλ∨µ = S P Oλ ∪ S P Oµ . In the foregoing one may replace by the pattern topology T (or by T () and similar restrictions SpT or SPT as defined earlier; in all cases we shall use the same label—generalized topology or generalized Stone space—and it will be clear from the context which one it is. Finally, the generalized Stone topology may also be defined on the commutative shadow S L() (see Proposition 2.1), which is a modular lattice, then of course its induced topology on Q S P (S L()) is exactly the Stone topology of the Stone space of S L(). In the special case = L(H ), the generalized Stone space defined on Q S P (L(H )) = Q S P(L(H )) is exactly the classical Stone space that can be used in Gelfand duality theory for L(H ) and L(H ). A word of warning perhaps; since L(H ) is not satisfying the weak F D I property, one may not expect a result like Corollary 2.4. In fact, whereas Q S P(L(H ) is rather big, S p (L(H )) = S P(L(H )) is empty (see remarks preceeding Proposition 2.3.2). This fact will have a deeper meaning when we aim to develop some sheaf theory over general . In Section 2.4. the basic properties were introduced and we stressed the transfer from (pre-)sheaves over to (pre-)sheaves over C(). This will turn out to be of essence in the case = L(H ) because there are no sheaves (there is not enough “cohesion” between the element of L(H ) if one tries to view them as open sets in some generalized topology) over L(H ). There will be many sheaves over C(L(H )) allowing sheafification of presheaves; in fact, this will already be possible over Q S P (L(H )) (see Chapter 4). An important notion in Gelfand duality theory for L(H ) is the notion of spectral family and of observable function. In the following we will see, to our surprise, that a spectral family is essentially just a separated filtration. We shall consider a totally ordered group in the sequel, however, it would be enough to consider a totally ordered poset with meet and join defined for every subfamily. In applications: ⊂ Rn+ . Definition 2.9: -Spectral Family Let be a noncommutative topology; then a -filtration of is a family {λα , α ∈ } such that for α ≤ β in , λα ≤ λβ in and ∨{λα , α ∈ } = 1 in (i.e., we consider exhaustive filtrations). A -filtration is separated whenever γ = inf{γα , α ∈ A} in entails that λγ = ∧{λγα , α ∈ A} in , and 0 = ∧{λγ , γ ∈ }. A -spectral family is just a separated -filtration; it may be seen as F : → , γ → λγ where F is a poset map with F(γ ) = λγ satisfying the separatedness condition. Note that, by definition, the order in ∧{λγα , α ∈ A} does not matter while on the other hand the λγα need not be idempotent.

2.7 The Fundamental Examples II: L(H)

69

The foregoing definition applied with = R, + and = L(H ) yields the usual notion of spectral family. A well-known example (connected to the Hamilton operator of the harmonic oscillator) is obtained as follows: let xn , n ∈ N be an orthogonal basis of a separable Hilbert space H and define for γ ∈ R, L(H )γ = ∨{Cxn , n ≤ γ }. This is in fact also an example of a -filtration with discrete support as introduced in [1]. First let us continue with some general facts. A -spectral family on is said to be idempotent if λγ ∈ id∧ () for every γ ∈ . We say that is indiscrete if for all γ ∈ , γ = inf{τ, γ < τ }, for example, = Rn , +. Proposition 2.34 If is indiscrete, then every -spectral family is idempotent. Proof Since obviously γ = inf{τ, γ ≤ τ } in , we have λγ = ∧{λτ , γ ≤ τ }. Since in the latter expression the order of the λτ is irrelevant, we may rephrase this as λγ = λγ ∧ ( λτ ) = λγ ∧ λγ ; consequently λγ ∈ id∧ (). γ , then for every homogeneous Ore set S ∈ O(A) such that S ∩ A+ = ∅, it follows that Q S (A) is strongly graded. Proof Put B = Q S (A) with b ∈ Bd if and only if sb ∈ An+d for some s ∈ S ∩ An . It is harmless to replace A by A/κ S (A); that is, we may assume that A → S (A); g also note that Q S (A) = Q S (A). If Z ∈ B0 , then sn z ∈ An for some sn ∈ S ∩ (i) An ; that is, sn z = i a1 . . . an(i) with a (i) j ∈ A1 , j = 1 . . . n. Rewrite this as z = −1 (i) (i) (i) (i) −1 (i) (s a . . . a )a with s a . . . a 1 n−1 n 1 n−1 ∈ B−1 . Thus z ∈ B−1 A1 ⊂ B−1 B1 ∈ B, n i n follows, or B0 = B−1 B1 . Since B−1 . . . B−1 .B1 . . . B1 = B0 we also obtain B−n Bn = B0 for positive n. In a similar way we derive that Bn = B0 An1 and then Bn = B1n follows too. Finally if z ∈ B0 and sm ∈ S ∩ Am is such that sm z ∈ Am , then z = (zsm )sm−1 ⊂ Bm B−m = B1m B−m ⊂ B1 (B1m−1 B−m ) ⊂ B1 B−1 , thus B0 = B1 B−1 as well. An extended version of the foregoing result can be found in Proposition 3.16. Example 2.4 The coordinate ring of quantum 2 × 2-matrices, Oq (M2 (C)), with q ∈ C, is schematic and a Noetherian domain. This algebra is generated over C by elements A, B, C, D subjected to the following relations: • B A = q −2 AB • C A = q −2 AC • BC = C B

• • •

D B =−2 B D DC = q −2 C D AD − D A = (q 2 − q −2 )BC

In fact one take S A , S B , SC , S D respectively generated by the powers of A, B, C, D; the schematic condition can be checked stepwise for the consecutive extensions C[B, C][A][D], which at each step is given by an Ore extension; that is, Oq (M2 (C)) is an iterated Ore extension. Example 2.5 Quantum Weyl algebras are schematic. Look at an n × n-matrix (αi j ) = A with αi j ∈ K ∗ = K − {0} and let q = (q1 , . . ., qn ) be a row with qi = 0 in K . Define An (q, A) as

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the K -algebra generated by x1 , . . ., xn, y1 , . . ., yn subjected to the following relations: putting µi j = λi j qi , xi x j = µi j x j xi xi y j = α ji yi xi y j yi = α ji yi y j x j yi = µi j yi x j x j y j = q j y j x j + 1 + i< j (qi − 1)yi xi Again this algebra may be obtained as an iterated Ore extension creating stepwise extensions by consecutively adding to K , x1 , y1 , x2 , y2 , . . ., xn , yn . Example 2.6 The Sklyanin algebra is schematic. Let SK (A, B, C) be the K -algebra generated by three homogeneous elements X, Y, Z of degree 1, with homogeneous defining relations: A × Y + BY X + C Z 2 = 0 AY Z + B Z Y + C X 2 = 0 AZ X + B X Z + CY 2 = 0 A proof for this, using a valuation reduction idea, is given in [44]. Example 2.7: E. Witten’s Gauge Algebras for SU (2) Consider the G-algebra W generated by X, Y, Z , subjected to the following relations: X Y + αY X + βY = 0 Y Z + γ ZY + δX2 + εX = 0 Z X + ξ X Z + ηZ = 0 for any α, β, γ , δ, ε, ξ, η ∈ C. This Witten-algebra (and its associated graded rings as well as the Rees ring with respect to the obvious filtration given in terms of the total degree in X, Y and Z ) is schematic. Example 2.8: Woronowicz’s Quantum sl2 Let Wq (sl2 ) be the G-algebra generated by X, Y, Z subjected to the following relations: √ √ −1 q + q −1 Z qXZ − q ZX = √ √ −1 q X Y − qY X = − q + q −1 Y √ √ Y Z − Z Y = ( q − q −1 )X 2 − q − q −1 X 2πi is the Chern coupling constant. where classically q = exp k+2 The algebra, as well as its Rees ring for the the standard filtration, is schematic.

Chapter 3 Grothendieck Categorical Representations

3.1

Spectral Representations

We start from a category R allowing products and coproducts. Typical examples we have in mind are amongst others: the category Ring of associative rings with unit, the category R-grG of G-graded associative rings with unit for some group G, the category Algk of k-algebras, the category R-filt of Z-filtered rings (an interesting non-abelian case), and so forth. To each object R of R we associate a Grothendieck category Rep(R). For every f ∈ HomR (S, R), f : S → R, we are given an exact functor f o = F : Rep(R) → Rep(S), which commutes with products and coproducts and satisfies the following conditions: i. (1 R )o = IRep(R) for every R ∈ R ii. For g : T → S, f : S → R in R, ( f ◦ g)o = g o ◦ f o . We did not demand that for R = S in R necessarily Rep(R) = Rep(S) If G is the class consisting of objects Rep(R), R ∈ R, we let HomG (Rep(R), Rep(S)) consist of functors of type h o provided these go from Rep(R) to Rep(S). Note that if Rep is separating objects of R, then we may write HomG (Rep(R), Rep(S)) = HomR (S, R)o . In any case G as defined above becomes a category. Definition 3.1 A Grothendieck categorical representation (a GC representation) is a contravariant functor Rep : R → G commuting with arbitrary products, associating to f : S → R an exact functor f o = F : Rep(R) → Rep(S) commuting with products and coproducts. Several obvious examples come to mind, for example, representing noncommutative rings by their category of left modules, groups by their G-modules, and graded algebras by their categories of graded modules. Such examples exhibit stronger properties than those used in the general definition. This is mainly due to the fact that the objects of R appear in some form also in the representing Grothendieck category, such as a ring as a left module over itself, and so forth. This may be formalized in the following definition.

79

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Definition 3.2 A GC representation R → G is said to measure R if for every R ∈ R we have an object R R in Rep(R) such that to any f : S → R in R there corresponds a morphism f ∗ : S S → Rep( f )( R R) = S R, satisfying: µ.1. If f = 1 S then f ∗ is the identity of S S in Rep(S). µ.2. Consider g : T → S, f : S → R in R; then we have : ( f ◦ g)∗ = Rep(g)( f ∗ ) ◦ g ∗ :T T →T S →T R = Rep(g)Rep( f )( R R) A GC representation Rep : R → G is said to be full if to an epimorphism π : S → R in R there corresponds a full functor Rep(π ) : Rep(R) → Rep(S). A GC representation is faithful if for f : S → R in R, Rep( f ) is faithful. An R ∈ R is Rep-Noetherian when Rep(R) is a Grothendieck category having a Noetherian generator. Similarly, R ∈ R is locally Rep-Noetherian whenever Rep(R) has a family of Noetherian generators. The relation between a GC representation and suitable topologies will be obtained from the hereditary torsion theories existing on the Grothendieck categories. In Section 2.6 we introduced general torsion theory in Grothendieck categories, but here we modify the notation somewhat in order to fit the notation fixed in the introduction of Grothendieck representations. For an arbitrary Grothendieck category M we let Tors(M) be the set of hereditary torsion theories on M; we know that Tors(M) is a modular lattice with respect to inf and sup of torsion theories (cf. the note following Proposition 2.11), but the operation “product” in the lattice of preradicals Q(M) is noncommutative. Torsion theories of M will be denoted by σ, τ, κ, . . . ; then Tσ , Fσ will denote the torsion, respectively the torsion-free class of σ . We write Tσ : M → M for the corresponding torsion functor (kernel functor) and (M, σ ) for the quotient category together with the canonical functors i σ : (M, σ ) → M, aσ : M → (M, σ ) (see Theorem 2.17). Then i σ aσ = Q σ is the localization functor M → M associated to σ . For R ∈ R we abbreviate TorsRep(R) to Top(R). In case R has an initial object, k say, then we call Top(k) the initial space for Rep(R). To a morphism f : S → R in R we have associated a functor F = Rep(R) → Rep(S). Since F is exact and commutes with coproducts, it defines a map F o : Top(S) → Top(R); indeed, if γ ∈ Top(S) we may define F o (γ ) by taking for T F o (γ ) the class of objects X in Rep(R) such that F(X ) ∈ Tγ ; when F derives from f we shall write f for F o . Definition 3.3 A faithful Grothendieck representation that measures R is said to be spectral if for all A ∈ R, γ ∈ Top(A) and τ ∈ gen(γ ) we are given the following: i. An object A(γ ) in R together with a morphism f γ : A → A(γ ) such that the morphism f γ∗ in Rep(A), f γ∗ : A A → Rep( f γ )( A(γ ) A(γ )) is exactly the localization morphism A A → Q γ (A).

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81

ii. A morphism f τγ : A(γ ) → A(τ ) in R fitting into a commutative diagram: A(γ) fγ

A

fτγ fτ

A(τ)

iii. If ξo (A) stands for the trivial element of Top(A), that is, the zero element of the lattice Tors(Rep(A)), then A = A(ξo (A)). Proposition 3.1 Consider an exact functor F, F : Rep(S) → Rep(R), commuting with direct sums; then F o : Top(R) → Top(S) has the following properties: i. For γ ≤ σ in Top(R), F o (γ ) ≤ F o (σ ) in Top(S). ii. If U ⊂ Top(R), then F o (∧U ) = ∧F o (U ). iii. For τ in Top(S), let ξτ be the minimal torsion theory having all F(Tτ ), Tτ ∈ Tτ ; in its torsion class, in other words Tξτ is the torsion class generated by the F(Tτ ), Tτ ∈ Tτ . We have (F o )−1 (gen(τ )) = gen(ξτ ). Proof An easy exercise. In general, for U ⊂ Top(A) : gen(∧∪) ⊃ U {gen(τ ), τ ∈ U }, gen(∨U ) = ∩{gen, τ ∈ U }; moreover there is a trivial torsion theory ξ defined by Tξ = 0, and a maximal torsion theory χ defined by Tχ = Rep(A). Consequently, the sets gen(τ ), τ ∈ Top(A) define a topology on Top(A). Corollary 3.1 The gen-topology. F o as in Proposition 3.1 is continuous in the gen-topology. For γ in Top(A) we have the reflector aγ : Rep(A) → (Rep(A), γ ) and the associated map aγo : Tors(Rep(A), γ ) → Top(A). Note that the forgetful functor : (Rep(A), γ ) → Rep(A) is not exact in general, so it does not yield an associated map Top(A) → Tors(Rep(A), γ ). In the part of this section we only consider GC representations that are faithful. The categories considered are assumed to have a zero object. Proposition 3.2 Suppose that Rep is spectral and γ ∈ Top(A) for A in R; take τ ∈ gen(γ ) and write τ for f γ (τ ) ∈ Top(A(γ ) and fτ for the corresponding morphism in R, fτ : A(γ ) → A(γ )( τ ), which exists because of the spectral property of Rep applied to A(γ ) in R.

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Then we have: i. Rep( f τγ )( A(τ ) A(τ )) = Qτ ( A(γ ) A(γ )). ii. Rep( f γ )(Qτ ( A(γ ) A(γ ))) = Q τ ( A A). Proof i. By the spectral property of Rep, there exists an A(γ )( τ ) in R together with a morphism fτ : A(γ ) −→ A(γ )( τ ) with corresponding morphism in Rep(A(γ )), fτ∗ say, fτ∗ : A(γ ) A(γ ) −→ Rep( fτ )( A(γ )(τ ) A(γ )( τ )), which is exactly the localization homomorphism in Rep(A(γ )), A(γ ) A(γ ) −→ Qτ ( A(γ ) A(γ )). On the other hand we may consider: Rep( f γ )( fτ∗ ) : Q γ ( A A) → Rep( f γ )(Qτ ( A(γ ) A(γ ))), the latter being equal to Rep( f γ )Rep( fτ )( A(γ )(τ ) A(γ )( τ )) = Rep( fτ f γ )( A(γ )(τ ) A(γ )( τ )). From the composition A −→ A(γ ) −→ A(γ )( τ ), we obtain: fγ

fτ

Rep( f γ )( fτ∗ ) f γ∗ : A A −→ ∗

Rep( fτ f γ )( A(γ )(τ ) A(γ )( τ )) ∼ = Rep( f γ )(Qτ ( A(γ ) A(γ )))

Now A(τ ) A(τ ) in Rep(A(τ )) is such that the localization morphism A A → is exactly given by f τ∗ :A A −→ Rep(τ ) Q τ (A) ( A(τ ) A(τ )). Consider Mτ (γ ) in rep(A(γ )) defined as follows: Mτ (γ ) = Rep( f τγ )( A(τ ) A(τ )) and let f τ∗γ : A(γ ) A(γ ) −→ Mτ (γ ) be the morphism in Rep(A(γ )) obtained from the measuring property of Rep. Obviously, τ (Mτ (γ )) ⊂ Mτ (γ ) in Rep(A(γ )). The definition of τ yields: Rep( f γ )( τ Mτ (γ )) is τ -torsion in Rep(A), and moreover, the exactness of Rep( f γ ) yields Rep( f γ )( τ Mτ (γ ) ⊂ Rep( f γ )(Mτ (γ )) = Rep( f τ )( A(τ ) A(τ )), where the τ Mτ (γ )) latter is τ -torsion free because it equals Q τ ( A A). Thus Rep( f γ )( = 0 and the faithfulness of Rep then implies that τ Mτ (γ ) = 0. It follows that τ ( A(γ ) A(γ )). we may factorize f τ∗γ | A(γ ) A(γ ) → Mτ (γ ), via Bτ (γ ) = A(γ ) A(γ )/ So we have the following sequence in Rep(A): A A −→ f γ∗

Q γ ( A A)

−→

Rep( f γ )( f τ∗γ )

Q τ ( A A)

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It is therefore clear that Rep( f γ )(Mτ (γ )/Bτ (γ )) is τ -torsion in Rep(A) because the co-kernel of Rep( f γ )( f τ∗γ ) is τ -torsion, then Mτ (γ )/Bτ (γ ) is τ -torsion in Rep(A(γ )). It follows that Mτ (γ ) ⊂ Qτ ( A(γ ) A(γ )) in Rep(A(γ )). The definition of Qτ ( A(γ ) A(γ )) makes it τ -torsion over Bτ (γ ) in Rep(A(γ )); consequently Rep( f γ )(Qτ ( A(γ ) (A(γ ))) is contained in Q τ ( A A) as it is τ -torsion over A A/τ ( A A). Since Mτ (γ ) ⊂ Q τ ( A(γ ) A(γ )), the functor Rep( f γ ) takes the value Q τ (A) for both objects, so the faithfulness and exactness of Rep( f γ ) entails that Mτ (γ ) = Qτ ( A(γ ) A(γ )). This establishes i above. Observe also that (*) entails that the morphisms Rep( f γ )( fτ∗ ) f γ∗ and Rep( f γ )( f τ∗γ ) f γ∗ are the same. ii. This follows from i by applying Rep( f γ ) to both members and then again applying the exactness and faithfulness of Rep( f γ ).

Corollary 3.2 i. Consider δ ≤ τ, γ ≤ τ in Top(A) and τ1 ∈ Top(A(δ)), τ2 ∈ Top(A(γ )) constructed as before (we prefer to write τ1 , τ2 rather than τδ , τγ ). We obtain the following commutative diagram of morphisms in R: A(δ)(τ1)

A(γ)(τ~2)

A(τ) A(δ )

A(γ) A

ii. Consider the following objects: M1 = A(δ)(τ1 ) A(δ)( τ1 ) in Rep (A(δ)( τ1 )) M2 = A(γ )(τ2 ) A(γ )( τ2 ) in Rep (A(γ )( τ2 )) M = A(τ ) A(τ )in Rep (A(τ )) Then the following relations hold: a. Rep( fτ1 )(M1 ) = Qτ ( A(δ) A(δ)) = Rep( f τδ )(M) b. Rep( fτ2 )(M2 ) = Qτ ( A(γ ) A(γ )) = Rep( f τγ )(M) c. Rep( f δ )Rep( f τ1 )(M1 ) = Q τ ( A A) = Rep( f γ )Rep( fτ2 )(M2 ) = Rep( f τ )( A(τ ) A(τ ) = Rep( f τ )(M) Proof Apply the foregoing result (twice).

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The foregoing may be compared to the classical fact that for a ring R and τ ≥ γ in Tors(R−mod) we have Q τ (R) = Qτ Q γ (R); in our abstract setting Rep(A(γ )) replaces Q γ (R)-mod. This shows that we have traced exactly the property of a GC representation, that is, spectrality, necessary to extend the foregoing classical fact to the general categorical situation. Note that A(δ)( τ1 ) and A(γ )( τ2 ) in the foregoing corollary need not be isomorphic in R (the relations in the corollary sum up what we do know). Write Top(A)o for the opposite lattice of Top(A). We would like to consider the functor A P : Top(A) → Rep(A), τ → Q τ ( A A) as a structural presheaf (or in fact a sheaf) for A A with values in Rep(A). Exactly the spectral property of Rep would then allow us to “realize” this structure sheaf in R by considering P : Top(A) → R, τ → A(τ ) with structure morphism f τ : A → A(τ ). It is clear that P(ξ0 (A)) = A with I A : A → A as the structure morphism. Moreover, for γ ≤ τ in Top(A) we take f τγ : A(γ ) → A(τ ) for the restriction morphism from γ to τ (in Top(A)o the partial order is reversed when viewing γ and τ in Top(A) as “opens”). For P to be a presheaf we do need an extra property for Rep! Definition 3.4 A spectral Grothendieck representation Rep is said to be schematic if for every triple γ ≤ τ ≤ δ in Top(A), for every A in R, we have a commutative diagram in R: A(γ) fτγ

fγ fδγ

A

A(τ) fδτ

fδ

A(δ)

Proposition 3.3 If Rep is schematic, then with notation as before, P : Top(A) → R is a presheaf with values in R over the lattice Top(A)o , for every A in R. Proof The composition property of sections follows from f δγ = f δτ f τγ , and the claim follows easily. Note that for a schematic GC representation Rep, the structure presheaf obtained in Proposition 3.3 is constructed via localizations in the representing Grothendieck categories described as in Proposition 3.2. In the foregoing we have restricted attention to Tors(Rep( A)); that is, we considered a lattice in the usual sense; hence this should be viewed as the commutative shadow of a suitable noncommutative theory. For A in R we shall write Q(A) for the set of preradicals (see remark before Definition 2.6; note Observation 2.4 too). Warning, in earlier work we (and several other authors) have approached hereditary torsion theory

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via radicals; that is, via the opposite Q(A)op , the notion Q(A) expresses the topology aspects of the theory more directly! Applying definitions (e.g., 2.6) and properties of preradicals derived in Section 2.6 to the Grothendieck category C = Rep(A), we obtain the complete lattice Q(A) and a duality expressed by an order-reversing bijection: (−)−1 : Q(A) → Q((Rep(A))o ). First let us point out that (Rep(A))o is not a Grothendieck category! It is additive and has a projective generator; moreover, it is known to be a varietal category (also called triplable) in the sense that it has a projective regular generator P, it is co-complete and has kernel pairs with respect to the functor Hom(P, −), and moreover every equivalence relation in the category is a kernel pair. A concrete description of the opposite of a Grothendieck category is given by U. Oberst (“Duality Theory for Grothendieck Categories and Linearly Compact Rings,” J. Algebra 15, 1970, 473–542) but sacrificing the varietal aspect for a topological approach. General localization techniques can be developed via the Eilenberg-Moore category of a triple (S. MacLane, Categories for the Working Mathematician, Springer-Verlag, Berlin, New York, 1971 [28]). The latter depends on the so-called comparison functor constructed via Hom(P, −) as a functor to the category of sets. It works well for the category of set-valued sheaves over a Grothendieck topology. As yet we have not investigated whether the approach via the Eilenberg-Moore category remains valid in the case of a noncommutative Grothendieck topology; this may be an interesting project of abstract value. Now (−)−1 defined as an order-reversing bijection between idempotent radicals on Rep(A) and (Rep(A))o , we write (Top(A))−1 for the image of Top( A) in Q((Rep(A))o ). This is encoded in the exact sequence in Rep(A): 0 −→ ρ(M) −→ M −→ ρ −1 (M) −→ 0 (reversed in (Rep(A))o ). By restricting attention to hereditary torsion theories (kernel functors) when defining Tors(−), we introduce an asymmetry that breaks the duality because Top(A)−1 is not in Tors((Rep(A))op ). Write T T (G) for the complete lattice of torsion theories (not necessarily hereditary) of the category G; then (T T (G))−1 ∼ = T T (G op ). Hence we may view Tors(G)−1 as a complete sublattice of T T (G op ). For preradicals ρ1 and ρ2 in Q(G) we have defined the lattice operations ρ1 ∧ preceding ρ2 , ρ1 ∨ρ2 , as well as the product ρ1 ρ2 . Inspired by the duality (see remarks Definition 2.6 and those following Proposition 2.11) we define ρ ρ = ρ1 : ρ2 . 1 2 Hence an object M of G is ρ1 ρ2 -torsion if and only if there is a subobject N ⊂ M such that N is ρ1 -torsion and M/N is ρ2 -torsion. The notation suggests that it is topologically an intersection, but as a preradical ρ1 ρ2 islarger than ρ1 and ρ2 . Let o op us denote the similar operation but defined we let −1in Q((G) )o by−1 ; for σ, τ ∈ Q(G) −1 σ . The notation suggests σ τ be the preradical such that (σ τ ) = τ that it is topologically a (noncommutative) union. Proposition 3.4 With notation as above: a. For σ, τ ∈ Q(G), Tσ τ = Tσ ∩ Tτ . Clearly σ τ ≤ σ ∧ τ . If σ ∧ τ is idempotent, then σ τ = τ σ = σ ∧ τ = τ ∧ σ . In particular, when σ and τ are left exact preradicals, then σ τ = τ σ = σ ∧ τ , and this is a left exact preradical. Also if

86

Grothendieck Categorical Representations τ is left exact and σ is idempotent, then σ τ is idempotent and σ τ = (σ ∧ τ )o (notation of Theorem 2.3(v)). b. If σ and τ are idempotent preradicals, then σ τ and τ σare idempotent o o preradicals. When only σ is idempotent, then (σ τ ) = σ τ . If σ and τ are left exact preradicals, then σ τ andτ σ are left exact preradicals too. Note also that Fσ τ = Fσ ∩ Fτ (but σ τ is not determined by Fσ τ ).

Proof a. The statements are obvious; let us establish the last one. Let τ be left exact, σ idempotent. For any M in G we have σ τ (M) = σ (τ (M)), hence a subobject of τ (M); therefore the left exactness of τ implies that σ τ (M) is in the τ -pretorsion class. Then σ τ (σ τ (M)) = σ (σ τ (M)) = σ τ (M); hence σ τ is an idempotent preradical. Therefore (σ ∧ τ )o ≤ σ τ . As observed before, σ τ (M) is in Tσ as well as in Tτ (τ is left exact); thus σ τ ≤ σ ∧ τ is clear; the definition of (σ ∧ τ )o then implies that σ τ = (σ ∧ τ )o . b. Let σ and τ be idempotent preradicals. By definition (σ τ )(M) = N , the largest subobject of M such that M ⊃ N ⊃ σ (M) with N /σ (M) being τ pretorsion. Since σ is idempotent σ (N ) = σ (M) hence (σ τ )(N ) = N1 , the largest subobject of N , N ⊃ N1 ⊃ σ (N ) = σ (M) such that N1 /σ (M) is τ -pretorsion. However, the latter implies N1 = N in view of the foregoing. In case σ is idempotentbut not necessarily τ , then by the foregoing σ τ o is idempotent, hence σ τ o ≤ (σ τ )o . Since both idempotent preradicals correspond to the same pretorsion class, it follows that σ τ o = (σ τ )o . In case both σ and τ are left exact, hence certainly idempotent, we have that σ τ (and τ σ ) is idempotent. is a Hence it suffices to check that Tσ τ hereditary class (the result for τ σ follows by symmetry). If M ∈ Tσ τ , then M/σ (M) is τ -pretorsion; consider a subobject M of M in G. Since σ is left exact, M ∩ σ (M) = σ (M ). Thus M /σ (M ) is isomorphic to a subobject of M/σ (M) and therefore the left exactness of τ entails that M /σ (M ) ∈ Tτ or M ∈ Tσ τ . Consequently σ τ is a left exact preradical. Proposition 3.5 With notation as before: a. If σ, τ ∈ Q(G) are radicals, then σ τ is a radical; when τ is a radical, then for any σ ∈ Q(G), (σ τ )c = σ c τ . b. If σ, τ are radicals, then σ τ and τ σ are radicals. If moreover σ and τ are kernel functors, then σ τ = τ σ = σ ∧ τ (= σ τ = τ σ in view of Proposition 3.4). Proof a. The statements in a follow by duality from the statements in Proposition 3.4a. b. The first statement follows by duality from the first statement in Proposition 3.4b. Before dealing with the second statement, let us describe Fσ τ ;

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now that we are considering radicals σ, τ and σ τ is a radical too, they are determined by their pre-torsion-free classes. By dualization of the definition of o in Q(G op ), we observe that an object of G, M say, is in Fσ τ if it fits in a G-exact sequence: 0 −→ N −→ M −→ M/N −→ 0 with N ∈ Fσ and M/N ∈ Fτ . First let us check that σ τ is a torsion theory, that is, that Fσ τ is closed under extensions. So, suppose we are given the following exact sequences in G: 0 −→ M1 −→ M −→ M2 −→ 0 π

0 −→ N1 −→ M1 −→ M1 /N1 −→ 0 π1

0 −→ N2 −→ M2 −→ M2 /N2 −→ 0 π2

where N1 and N2 are in Fσ , M1 /N1 and M2 /N2 are in Fτ . View N1 as a subobject of M and let X ⊂ M be the subobject such that X/N1 = τ (M/N1 ). Observe that X ∩ M1 = N1 because X ∩ M1 /N1 is isomorphic to a subobject of X/N1 , and hence a τ -torsion object because τ is a hereditary torsion theory radical, while on the other hand it is isomorphic to a subobject of M1 /N1 , hence τ -torsion free. Since π (X ) is a quotient of X/N1 it must be τ -torsion. But then π2 (π (X )) = 0; hence π (X ) ⊂ N2 , and we obtain an exact sequence in G: 0 −→ N1 = X ∩ M1 −→ X −→ π (X ) −→ 0 resπ

with N1 and π (X ) in Fσ . Consequently, X ∈ Fσ because it is closed under extensions. We also obtain the exact sequence: 0 −→ M1 /N1 −→ M/ X −→ M2 /N2 −→ 0 γ

where γ is a factorization of π2 π obtained from π (X ) ⊂ N2 . Since M1 /N1 and M2 /N2 are in Fτ , so must be M/ X . In the exact sequence in G : 0 −→ X −→ M −→ M/ X −→ 0 we now have X ∈ Fσ and M/ X ∈ Fτ , that σ τ is a torsion thus M ∈ Fσ τ . This establishes theory. We can go on and establish that σ τ is left exact and that σ τ is left exact and that σ τ = τ σ , but all of this will follow if we establish directly that σ τ = σ ∧ τ (∧ denoting the lattice operation in the lattice of idempotent preradicals, being the same as the one inthe lattice of idempotent radicals, that is, torsion theories). Since σ ∧ τ and σ τ are radicals, it suffices to establish that Fσ ∧τ = Fσ τ . Start with M ∈ Fσ τ and assume (σ ∧ τ )(M) = 0; put X = (σ ∧ τ )(M) ⊂ τ (M). We know there exists an exact sequence in G : 0 −→ N −→ M −→ M/N −→ 0, with N ∈ Fσ and M/N ∈ Fτ . Since γ

τ is left exact, X ⊂ τ (M) entails that X ∈ Tτ , hence γ (X ) ∈ Tτ , but as γ (X ) ⊂ M/N this entails γ (X ) = 0 or X ⊂ N . Then X ⊂ σ (M) leads to X being σ -torsion; hence X = 0 as N ∈ Fσ , that is, (σ ∧ τ )(M) = 0.

88

Grothendieck Categorical Representations Conversely, start from M ∈ Fσ ∧τ . If σ (M) = 0, then τ (σ (M)) = 0 because otherwise τ σ (M) = 0 would be σ ∧ τ -torsion (Proposition 3.4a.), σ τ and τ σ are ≤ σ ∧ τ ) in M, a contradiction. Of course M/σ (M) is in Fσ because σ is a radical and so, from the exact sequence in G: 0 −→ σ (M) −→ M −→ M/σ (M) −→ 0 with σ (M) ∈ Fτ and M/σ (M) ∈ Fσ , M ∈ Fσ τ follows. Combining both parts yields Fσ τ = Fσ ∧τ .

In the last part of the proof, symmetry has obviously been broken; indeed, for nonhereditary torsion theories we cannot use the proof above to arrive at commutativity of on torsion theories. The problem is that for (nonhereditary) torsion theories we do not know whether τ −1 ◦ σ −1 is radical. Since Fτ −1 o σ −1 = Fτ −1 ∩ Fσ −1 duality implies that Tσ τ = Tσ ∩ Tτ ; hence if σ τ isidempotent, or if σ τ or τ σ is idempotent (note that Tσ τ = Tσ ∩ Tτ too!), then σ τ = σ τ = τ σ = σ ∧ τ . In particular, when we are only interested in left exact preradicals we might define σ τ = σ τ from the beginning, neglecting the duality with the (noncommutative) topological intersection in Q(G op ). In the philosophy of the pattern as in Chapter 2 we are interested in brack topology eted expressions involving , , and the -idempotent elements (hence radicals). So we look atQh (G),the setof hereditary preradicals of G; this is a subset of Q(G) closed under and (and agrees with the preradical product and is a commutative operation in o = Q h (G)). Clearly i (Qh (G)) consists of theleft exact radicals, that is, the kernel functors. Conversely, bracketed expressions p( , , σi ) (notation of Section 2.2. see after Lemma 2.12) with σi ∈ id (), always yield left exact preradicals, hence it is in complete agreement with our interest in noncommutative topologies generated by their intersection-idempotent elements, to look at Qh (G). We write T for the set of left exact preradicals obtained as finite bracketed expressions as defined above. Are and T now noncommutative topologies, in fact topologies of virtual opens? This follows from the following easy lemma. Lemma 3.1 With notation and conventions as before: 1. and T , have 1 and 0 (the zero preradical, respectively the identity functor) 2. If σ, τ, γ are left exact preradicals, then (σ τ )(γ ) = σ (τ γ ). 3. If σ ≤ τ and γ are left exact preradicals, then σ γ ≤ τ γ , γ σ ≤ γ τ . 4. If σ is a left exact preradical such that σ n = 0, then σ = 0 (observe that Tσ n = Tσ ). 5. If σ σ . . . σ = 1, then σ = 1. 6. If σ ≤ τ and γ are left exact preradicals, then we have: γ σ ≤ γ τ, π γ ≤ τ γ. 7. For left exact preradicals σ, τ, γ , (σ τ ) γ = σ (τ γ ).

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8. For σ, τ, γ as before: (σ τ )γ = (σ γ τ )γ . 9. For σ, τ γ as before: (σ τ )γ = σ γ τ γ whenever γ is a radical, that is, in id (). 10. σ τ σ = σ = σ τ σ , whenever σ is radical. 11. σ (σ τ ) = σ = (τ σ )σ . 12. For left exact preradicals σ τ = τ σ (see earlier). 13. satisfies the F D I property as defined after Definition 1.9, (hence satisfies axiom A.10 as defined before Definition 1.9, as well as axiom VOT.3 as defined at the beginning of Section 1.2). Proof All properties follow easily from the preradical calculus; only 13 may need a little explanation. Recall that σ ≤ τ is a focused relation if σ τ = τ σ = σ (note that we are using ≤ in , which is poset opposite to Qh (G)); the FDI property holds if for a focused relation σ ≤ λ with λ = λ1 λ2 we have σ = (σ λ1 ) (σ λ 2 ). Now since σ ≤ λ is focused, we have Tσ = Tσ λ = Tλ σ and because λ = λ1 λ2 we obtain Tλ = Tλ1 ∩ Tλ2 . Therefore, we obtain Tσ = Tσ λ ∩ Tσ λ ⊂ Tσ λ1 ∩ Tσ λ2 . For the converse, look at an object M in the latter. That is, we have: M ⊃ M1 ⊃ 0 with M1 being λ1 -torsion and M/M1 being σ -torsion M ⊃ M2 ⊃ 0 with M2 being λ2 -torsion and M/M2 being σ -torsion Look at M ⊃ M1 ∩ M2 ⊃ 0; since we are considering left exact preradicals, M1 ∩ M2 is in Tλ1 ∩ Tλ2 = Tλ , while on the other hand M/M1 ∩ M2 is σ -torsion (it embeds in M/M1 ⊕ M/M2 ). We arrive at M ⊂ Tσ λ = Tσ . Corollary 3.3 is a noncommutative topology; T is a topology of virtual opens. The commutative shadow of (and then also of T ) is Tors(G) with its usual lattice operations (∧ = . , ∨ = , in the notation of Proposition 2.1). Returning to the setting of Grothendieck representations, we have for every object A of R, the Grothendieck category Rep(A) and the set of preradicals Q(A) of Rep(A) containing Top(A) = Tors(Rep(A)). The foregoing construction of (A) in Qh (A) leads to a noncommutative topology (A) having Top(A) = id ((A)) with its canonical lattice structure for the commutative shadow. The behavior of (A) with respect to morphisms B → A in R would be satisfactory if we can generalize Proposition 3.1 because then a GC representation leads to canonical topologization of the objects of R by noncommutative topologies having the classical lattice of kernel functors for its commutative shadows. A noncommutative space (spectrum) could then be understood as a localization functor on the level of Rep(A)-valued sheaves on (A), using structure sheaves, with natural transforms between these functors corresponding to morphisms in the base category R.

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To a morphism f : S → R in R we have associated a functor F = Rep( f ) : Rep(R) → Rep(S), which is exact and commutates with coproducts. Define a map : Qid (S) → Qid (R), where Qid denotes the lattice of idempotent preradicals by F ) defined by the pretorsion associating to γ ∈ Qid (S) the idempotent preradical F(γ class of objects X in Rep(R) such that F(X ) ∈ Tγ . When F derives from f , we shall in order to highlight the connection. also write f for F Lemma 3.2 maps Qh (S) to Qh (R). With notation as introduced above: F Proof Take γ ∈ Qh (S) and look at M ∈ T F(γ ) and a subobject N of M. Thus F(M) ∈ Tγ and because F is exact we have F(N ) ⊂ F(M); the fact that γ is hereditary then ). yields that F(N ) ∈ Tγ and hence N ∈ T F(γ ) by definition of F(γ Proposition 3.6 Consider any functor F : Rep(R) → Rep(S) such that F is exact and commutes with has the following properties: coproducts; then F : Qh (S) → Qh (R) to Top(S), then we obtain F o , as defined 1. If we restrict F before Definition 3.3. is a poset morphism. 2. F {ϕ ∈ F}) = ∧{ F(ϕ), 3. If F ⊂ Qh (S), then F(∧ ϕ ∈ F}. ϕ

ϕ

Proof Easy enough. On Qh (A) for any A in R we define the gen-topology in formally the same way as it was introduced on Top(A), that is, for ρ ∈ Qh (A) we let gen(ρ) = {τ ∈ Qh (A), ρ ≤ τ }. For any F ⊂ Qh (A) we have: gen(∧{ϕ, ϕ ∈ F}) ⊃ ∪{gen(ϕ), ϕ ∈ F}, gen(∨{ϕ, ϕ ∈ F}) = ∩{gen(ϕ), ϕ ∈ F} Together with the existence of a minimal left exact preradical ξ and a maximal one χ , the foregoing relations do establish that the sets gen(ϕ), ϕ ∈ Qh (A) generate (the open sets of) a topology on Qh (A). This topology induces on Top( A) the gen-topology of Top(A) (see also Corollary 3.1). Proposition 3.7 In the situation of Proposition 3.6 −1 (gen(ρ)) = gen(ξρ ), where ξρ is the preradical corre1. For ρ ∈ Qh (R), ( F) sponding to the hereditary pretorsion class generated by the F(Tρ ), Tρ ∈ Tρ . 2. F ∧ : Qh (S) → Qh (R) is continuous in the gen-topology. Proof Straightforward (like Proposition 3.1).

3.1 Spectral Representations With respect to the operations behavior of F.

and

91

, we have the following rules for the

Proposition 3.8 Consider an exact functor, commuting with coproducts: F : Rep(R) → Rep(S), and : Qh (S) → Qh (R) be the corresponding poset map. For σ, τ in Qh (S) we have: let F ) )=F σ F(σ F(τ τ , ) )≤F σ F(σ F(τ τ Proof Theproperty with respect to follows from Proposition 3.6(3) (in fact it extends coincides with ∧ on left exact to of a family F ⊂ Qh (S)) and the fact that ) F(τ ), then we have an preradicals. If M in Rep(R) is a pretorsion object for F(σ exact sequence in Rep(R): )(M) −→ M −→ M/ F(σ )(M) −→ 0 0 −→ F(σ )(M) is F(τ )-pretorsion. By exactness of F we then obtain an exact where M/ F(σ sequence in Rep(S): )(M)) −→ F(M) −→ F(M)/F( F(σ )(M)) −→ 0 0 −→ F( F(σ )(M)) is σ -pretorsion by definition of F(σ ), and F(M)/F( F(σ )(M)) where F( F(σ is τ -pretorsion by definition of F(τ ) (and exactness of F). Consequently, F(M) ∈ Tσ τ or M is F(σ τ )-pretorsion. ); hence for an arbitrary morphism ) F(τ In Qh (S)op we have F(σ τ ) ≤ F(σ f : S → R we need not obtain a map F taking (S) to (R). This is related to the fundamental problem concerning functoriality, also appearing in noncommutative geometry (scheme theory for associative algebras). Certain morphisms yield better e.g. when f in an epimorphism in R with Rep( f ) = F being a full behavior of F, functor. Definition 3.5 Given a morphism f : S → R in R, then σ ∈ Qh (S) is said to center f if σ F = ) (composition of functors written in antiorder of application): Rep(R) → F F(σ )(M)). The set of all σ in Qh (S) that center f will be Rep(S), M → σ F(M) = F( F(σ denoted by Z h ( f ). For Z n ( f ) ∩ Top(S) we write Z top ( f ) and Z h ( f ) ∩ (S) = Z ( f ). Corollary 3.4 With notation as above: ) 1. If σ ∈ Z h ( f ), then for any τ in Qh (S) we have: F(σ ) ∨ F(τ ) = F(σ ∨ τ ). F(σ

) = F(σ F(τ

τ ),

92

Grothendieck Categorical Representations ) 2. If σ, τ ∈ Z h ( f ), then we obtain the following equalities: F(σ F(σ τ ), F(τ ) F(σ ) = F(τ σ ), F(σ ) ∨ F(τ ) = F(σ ∨ τ ).

) = F(τ

Proof Clearly Proposition 3.8 that 1 so let us prove 1. We know from 2 follows from ) ≤ F(σ ) F(τ τ ). Consider M in Rep(R) that is F(σ τ )-pretorsion, that F(σ is, F(M) ∈ Tσ τ . We arrive at the existence of an exact sequence in Rep(S): 0 −→ σ (F(M)) −→ F(M) −→ F(M)/σ (F(M)) −→ 0 where F(M)/σ (F(M)) is τ -pretorsion. )(M)). Therefore we have that F(M)/σ (F(M)) = Since σ ∈ Z h ( f ), σ (F(M)) = F( F(σ )(M)) and consequently M/ F(σ )(M) is F(τ )-pretorsion. From the exact F(M/ F(σ sequence in Rep(R): (M)) −→ M −→ M/ F(σ )(M) −→ 0 0 −→ F(σ ) F(τ ). The statement we may conclude that M is in the pretorsion class for F(σ concerning the commutative operation ∨ follows in a similar way. Proposition 3.9 For any morphism, f : S → R, Z h ( f ) is a lattice with respect to ∧(= ) and ∨. Moreover, is inner in Z h ( f ). Proof Using Corollary 3.4 we obtain: for σ, τ ∈ Z h ( f ), (σ ∧ τ )(F) = σ τ F = σ (τ F) = )) = σ F F(τ ) = F(σ ) F(τ ) = F(σ ) ∧ F(τ ). Now consider M in Rep(R); σ (F F(τ then we have an exact sequence in Rep(R): )(M) −→ F(σ ) ) (M) −→ F(τ )(M/ F(σ )(M)) −→ 0. 0 −→ F(σ F(τ By applying the exact functor F we obtain an exact sequence in Rep(S), where we ) F(τ ))(M) in F(M), put X = F( F(σ )(M)) −→ 0, or 0 −→ σ F(M) −→ X −→ τ F(M/ F(σ 0 −→ σ F(M) −→ X −→ τ (F(M)/σ F(M)) −→ 0 By definition of , (σ τ )(F(M)) is defined by the exact sequence: 0 −→ σ F(M) −→ σ τ )(F(M) −→ τ (F(M)/σ F(M)) −→ 0 It follows that X = (σ τ )(F(M). This leads to the equalities: ) F(τ )) = F F(σ F( F(σ τ ) = (σ τ )F. The statements with respect to ∨ follow in a similar way.

3.1 Spectral Representations

93

Corollary 3.5 Z h ( f ), as well as Z top ( f ), Z ( f ), have the structure of a noncommutative topology induced by the corresponding structure on Qh (S). We have before avoided working with filters of left ideals in a ring in order to try to describe the noncommutative intersection in case of a ring A and torsion theories on A-mod. Since the preradicals of interest appear as compositions of torsion theories, it may be interesting anyway to mention a few technical facts. First, let us stay in the generality of a given Grothendieck category G. For torsion theories τ and κ on G, we define a class Tκτ as the class of objects M in G such that there is a subobject N of M; N is κ-torsion and M/N is τ -torsion, or equivalently M/κ M is τ -torsion. The class Tκτ is closed for taking subobjects, direct sums, and images but not necessarily closed under extensions. Of course Tτ κ ⊃ Tκ , Tτ and similar for Tκτ . Let us rephrase some of our statements about the closure operator (avoiding the terminology of Proposition 3.4 and following) and let Tκτid be the closure of Tκτ under extensions; then Tκτid is a torsion class for a hereditary torsion theory on G. The following lemma is along the lines of earlier observations. Lemma 3.3 With notations as above, Tτidκ = Tκτid . If τ κ and κτ are idempotent, that is, Tτidκ = Tτ κ , respectively Tκτid = Tκτ , then τ and κ are compatible, that is, τ κ = κτ and Q τ Q κ = Qκ Qτ . Proof We know this already; however, observe that we can obtain Tτidκ as the union of T(τ κ)...(τ κ) for compositions of finitely many factors τ κ, and similar for Tκτid . Hence first Tτidκ ⊃ Tκτ follows and then also Tτidκ ⊃ Tκτid . Symmetry in τ and κ then yields Tτidκ = Tκτid . For M 0, m < 0; thus δ = 0 if and only if R is either positively or negatively graded depending upon whether R1 = 0 or R−1 = 0. We deal with the positively graded case; the other case is similar. Put R = ⊕n≥0 Rn , I = ⊕n>0 Rn = R+ . The assumption I∈ Lg (τ ), τ being perfect, yields S = S I . Hence S0 = n>0 S−n Tn = S R . Look at s r with s ∈ S , R ∈ R . For some L ∈ Lg (τ ) −n n −n n −n −n n n n>0 we have that L p s−n rn ⊃ R p−n Rn ⊂ R p

(∗) thus for any p > 0:

p−1

S− p L p s−n rn ⊂ S− p T p = S− p R1

R1 ⊂ S−1 R1 ⊂ S−1 S1

3.3 Quotient Representations

103

Observe that L ⊂ Lg (τ )); consequently, p>0 S− p L p = (S L)0 = S0 (as S0 s−n rn ⊂ S−1 S1 with n arbitrary. From S0 = n>0 S−n Rn we obtain S0 = S−1 S1 . Note that in (*) L 0 = 0 because if L ∈ Lg (τ ), then I ∩ L ∈ Lg (τ ) with (I ∩ L)0 = 0, so we may replace L by L ∩ I without loss of generality. Observe that the foregoing does not imply that also S1 S−1 = S0 ! However, when τ is associated with a homogeneous nontrivial (left) Ore set T , then for y ∈ S0 look at ytm with tm ∈ T ∩ Rm for m > 0. Since tm is invertible in S with tm−1 ∈ S−m we may look at (ytm )tm−1 = y ∈ Sm S−m . Now from S−1 S1 = S0 we derive that Sm = S1m (because Sm = Sm S0 = Sm S−1 S1 , thus Sm = Sm−1 S1 and by repetition of this argumentation we obtain Sm = S1m ). Finally we obtain y ∈ S1m S−m = S1 (S1m−1 S−m ) ⊂ S1 S−1 and consequently S0 = S1 S−1 as desired. To the graded ring R we associate a rigid torsion theory κ R defined by its graded filter Lg (κ R ), which is the graded filter (Gabriel topology) generated by Rδ and I . Observe that I = δ ⊕ (⊕n=0 Rn ) is automatically in Lg (κ R ) because it contains Rδ when δ = 0. The situation of geometrically graded rings R with rigid torsion theories κ R is also interesting because it provides us with a new example of a topological nerve. Lemma 3.5 Let R and S be Z-graded rings such that δ R and δ S are both either nonzero or both zero. If f : R → S is a morphism of graded rings, then κ S ≤ f (κ R ). Proof Since f (Rn ) ⊂ Sn for every n ∈ Z, it is clear that f (δ R ) ⊂ δ S . By definition L ∈ L( f (κ R )) means that S/L is κ R -torsion as an R-module; that is, L contains some L , L ∈ L(κ R ). Consider the category B of Z-graded rings R with δ R = 0, taking just graded ring morphisms for the morphisms. Associating R-gr to R defines a Grothendieck representation such that {κ R , R ∈ B} is ε nerve. Therefore we may consider the quotient Grothendieck representation with respect to the nerve {κ R , R ∈ B}. Exercise 3.2 Develop the noncommutative geometry of “ProjR”, which is defined by the noncommutative topology of the quotient category (R-gr, κ R ), together with the corresponding sheaf theory. Define schematically graded rings as the class of graded (Noetherian) rings R such that there exists a finite set of homogeneous Ore sets T1 , . . . , Tn such that κ R = κT1 ∧. . .∧κTm ; suitable generalizations may be defined by replacing the κTi by perfect rigid torsion theories not necessarily stemming from Ore sets. In this case ProjR defined on (R-gr, κ R ) satisfies all properties valid in the positively graded case. In particular, the schematic condition entails the existence of an affine cover (invoking Proposition 3.16). It is possible to establish a proof of Serre’s global section theorem for ProjR. A new ingredient in this project consists in the study of the relation between the affine geometry of R0 , in terms of SpecR0 say, and the projective geometry in

104

Grothendieck Categorical Representations

terms of ProjR, that is, (R-gr, κ R ). A subproject of this consists of a concrete algebraic approach when R is a ring satisfying polynomial identities (in particular when R is a finite module over its center) where a relation with the theory of maximal (R0 -)orders has to be investigated. More concretely, study the geometry when R is an R0 -order, R0 is a Noetherian integrally closed domain of dimension n and δ defines a closed subvariety of SpecR0 of dimension n 1 < n. Even more concrete, n = 1 and n 0 = 0, or n = 2 and n 1 = 1. There are new phenomena here when compared to the theory started in [47] or in L. Le Bruyn, M. Van den Bergh, and F. Van Oystaeyen, Graded Orders, Birkhauser Monographs (xxxx).

3.4

Noncommutative Projective Space

As an example of the quotient representations introduced in the foregoing section we point out how the construction of projective spaces fits in that theory. For the category R we now restrict attention to the category of positively graded k-algebras with graded k-algebra morphisms of degree zero for the morphisms; recall that a graded k-algebra is said to be connected if its part of degree zero is k; that is, A is a graded connected kalgebra if A = k⊕ A1 ⊕ A2 ⊕. . . . For geometry-oriented purposes we restrict attention to finite gradations in the sense that each An is a finite dimensional k-space and A is generated as a k-algebra by A1 . In that case, the positive part A+ = A1 ⊕ A2 ⊕ . . . is finitely generated as a left (or right) ideal of A and moreover A A1 = A+ and the powers Am + form the basis of a Gabriel topology of a torsion theory that we denote by κ+ and we write L(κ+ ) for the Gabriel topology. Any τ ∈ A-tors is said to be rigid if the graded torsion class Tτg is shift invariant; that is, if a graded A-module M is τ -torsion, then for every n ∈ Z : T (n)M is also τ -torsion, and conversely. The set of graded left ideals in L(τ ) is denoted by Lg (τ ); it is called the graded filter or Gabriel topology. Now if we start with τ , even one such that τ (M) is a graded submodule of M whenever M is graded, then τ need not be characterized by Lg (τ ). This is due to the fact that A need not be a generator for A-gr. On the positive side, if τ is rigid, then it is characterized by Lg (τ ). In any case κ+ is a rigid torsion theory, so it is completely determined by the graded Gabriel topology Lg (κ+ ). For an arbitrary graded ring R we denote by R-rig the sublattices of R-tors of rigid graded torsion theories. g Let us write Rk instead of R in this section, in order to reflect the graded character g and to fix the field k. If g : R → S is a morphism in Rk , then g(R+ ) ⊂ S+ . Let us write κ+ (R) respectively κ+ (S) for the rigid graded torsion theory in R-tors, respectively S-tors, associated to R+ , respectively S+ . Obviously, S/S+ is κ+ (R)-torsion, so it follows easily that κ+ (S) ≤ g (κ+ (R)) where g : R-tors→ S-tors corresponds to g. Lemma 3.6 The restriction of g to R-rig defines g : R-rig→ S-rig.

3.4 Noncommutative Projective Space

105

Proof Take τ ∈ R-rig and look at g (τ ). If N is a g (τ )-torsion graded S-module, then R N is τ -torsion and every T (m) R N is then τ -torsion because τ is rigid. Now it is clear that R (T (m)N ) = T (m) R N ; thus T (m)N is g (τ )-torsion for every m ∈ Z or g (τ ) is rigid. For a k-algebra A graded as before, we let Proj(A) be the Grothendieck category obtained as the Serre quotient category of finitely generated graded A-modules modulo graded A-modules of finite length, that is, if A-gr f denotes the category of finitely generated graded A-modules, then the localizing functor A-gr f → Proj(A) corresponds to the torsion class of the κ+ -torsion objects, which in this case are finite dimensional over k. The functor A-gr f → Proj(A) defines a lattice morphism: Torsg (Proj(A)) → Torsg (A−gr f ) where Torsg (−) stands for the lattice of graded torsion theories on the category specified. The latter morphism restricts to Rig(Proj(A)) → Rig(A−gr f ), where Rig(−) stands for the lattice of rigid graded torsion theories. For full detail on graded localization theory we refer the reader to C. Nˇastˇasescu and F. Van Oystaeyen, Graded Rings and Modules, LNM 758, Springer Verlag [31], or Graded Ring Theory, North Holland. In this section we restrict attention to the commutative shadow; that is, we deal with the torsion theories and leave the extension to graded radicals and noncommutative topology to the reader (this is a fairly straightforward graded version of the arguments of part of Section 3.1, after Proposition 3.3). Lemma 3.7 With notation as before we have: 1. Rig(A−gr f ) = A-rig. 2. Rig(Proj(A)) = genrig (κ+ ), the latter denoting the set of rigid graded torsion theories τ in A-tors such that τ ≥ κ+ , that is, genrig (κ+ ) = gen(κ+ ) ∩ A-rig. Proof 1. For any torsion theory on modules it is true that a module is torsion if and only if every finitely generated submodule is torsion. Hence the restriction of a torsion class to A-gr f does determine the torsion class in A-gr f . Rigidity of the (graded) torsion class in A-gr is obviously equivalent to the rigidity of the corresponding torsion class in A-gr f . From foregoing observations it follows easily that Rig(A-gr f ) = A-rig. 2. The map Rig(Proj(A)) → A-rig associates to a rigid (graded) torsion theory on the quotient category Proj(A) of A-gr f , the rigid torsion theory it induces on Agr f , and thus on A-gr, that is, an element of gen(κ+ )∩ A-rig. Conversely, that any τ ∈ genrig (κ+ ) induces a rigid torsion theory on Proj(A) follows by checking

106

Grothendieck Categorical Representations the transfer of rigidity; the bijective correspondence Tors(Proj(A)) = gen(κ+ ) follows from earlier observations (this also follows from Proposition 2.26) The lattice genrig (κ+ ), or in fact the category corresponding to it in the usual way, may be viewed as the projective version of Top(A) introduced in the ungraded situation as Tors(Rep(A)). So it makes sense to write Topproj (A) = g : Rgenrig (κ+ ). If g : R → S is in Rκg , then we have already introduced rig→ S-rig in Lemma 3.6, which may be viewed as a functor when the lattices are considered as categories in the usual way. However, there is a problem in constructing an associated map: Topproj (R) −→ Topproj (S). Indeed g : R → S does not necessarily define a “restriction of scalars” functor Proj(S) −→ Proj(R)! On the positive side we have g (κ+ (R)) in S-rig such that κ+ (S) ≤ g (κ+ (R)); therefore, the quotient category (S-gr f , g (κ+ (R))) is also a quotient category of Proj(S) such that Rig(S-gr f , g (κ+ (R))) may be identified to genrig ( g (κ+ (R))) in S-rig via the map associated to the localizing g (κ+ (R))). So we may conclude that we obtain a functor S-gr f −→(S-gr f , functor Proj(S), g (κ+ (R)) −→ Proj(R), induced by the restriction of scalars with respect to g. By transitivity of the localization functors associated to κ+ (S) ≤ g (κ+ (R)) we actually find that: (Proj(S), g (κ+ (R))) = (S−gr f , g (κ+ (R))) (or by the compatibility property deriving from κ+ (S) ≤ g (κ+ (R))). In any case, the functor (Proj(S), g (κ+ (R))) −→ Proj(R) induces a lattice morphism: Rig(Proj(R))) −→ Rig(S−gr f , g (κ+ (R))). Observing that Rig(Proj(R)) = genrig (κ+ (R)) in R-rig, Rig(S-gr f , g (κ+ (R))) = genrig ( g (κ+ (R))) in S-rig, we may conclude that the morphism g gives rise to a lattice morphism: genrig (κ+ (R)) −→ genrig ( g (κ+ (R))) → gen(κ+ (S)) defining a lattice morphism Topproj (R) −→ gen( g (κ+ (R))) → Topproj (S). In other words, the lattice morphism that we obtain here is obtained from g but not from a functor Proj(S) → Proj(R). The above phenomenon is also present in the commutative scheme theory. It expresses the fact that, even when suitable localizations do carry over from R to S via g , the scheme theory has to take into account that the underlying topological morphism can only be defined on an open subset of Proj(S) in fact given by gen( g (κ+ (R))) (viewed in the opposite lattice).

The foregoing establishes the following proposition.

3.4 Noncommutative Projective Space

107

Proposition 3.17 g To a morphism g : R → S in Rk there corresponds a functor (deriving from a lattice morphism) Topproj (R) −→ Topproj (S). An arbitrary g : R → S does not allow us to relate finitely generated (graded) S-modules to finitely generated (graded) R-modules when S itself is not even finitely generated as an R-module. This makes it more natural to consider (A-gr, κ+ (A)) g for any A in Rk , that is, without restricting to A-g f . Let us write PROJ(A) for the latter quotient category. In sheaf theoretical language this would mean that we focus on quasi-coherent sheaves rather than on coherent sheaves. The torsion objects with respect to κ+ (A) in A-gr need not have finite length, but this does not affect any of the statements and results derived earlier. g In the language of Section 3.3, where we put R = Rk , now we consider a g Grothendieck representation associating A-gr (or A-gr f ) to A in Rk . A topological nerve κ+ can now be obtained by letting n A (as in Section 3.3) be κ+ (A) as g defined earlier in this section. For a morphism g : A → B in Rk we do have g (κ+ (A)) and so we arrive at a generalized Grothendieck representathat κ+ (B) ≤ g tion; (A-gr, κ+ (A)) is then associated to A in Rk , which is the quotient generalized Grothendieck representation of Section 3.3 and in particular, from Theorem 3.1 it follows that it is measuring and weakly spectral; moreover, it satisfies the statements of Proposition 3.13. With notation as introduced in this section, the GC representag tion gr, respectively gr f , associating A-gr, respectively A-gr f , to A in Rk , allows the quotient representation PROJ, respectively Proj, associating PROJ(A), respectively g Proj(A) to A in Rk .

3.4.1

Project: Extended Theory for Gabriel Dimension

In Section 2.6 we established how localization functors or torsion theories appear as a major example of noncommutative topology. In fact in view of the constructed scheme theory for schematic algebras (cf. [49]), the latter example has been the main motivation for the introduction of noncommutative topology in a more axiomatic way. Now unlike the Krull dimension, the Gabriel dimension is defined exactly in terms of torsion theories, so it is a possible instrument for calculating certain dimensions of noncommutative algebras, topologies, or other categorical structures. Let us recall some basic facts along the way to describing some possible projects. The name Gabriel dimension is attributed by Gordon, Robson to a notion introduced by P. Gabriel in his thesis, [10], but here termed the Krull dimension. Since several notions of generalized Krull dimension became available later, the different names were used; in the book Dimensions of Ring Theory [33], the notion of Gabriel dimension is given for an arbitrary modular lattice. A first project could be to generalize this to noncommutative topologies or virtual topologies. We do not go into this, but turn to Grothendieck categories instead. Let G be a Grothendieck category. An object M of G is said to be semi-Artinian if for every subobject M of M such that M = M there exists a simple subobject in M/M . The full subcategory of G consisting of all semi-Artinian objects is easily seen to be a localizing subcategory, in other words to determine a torsion theory of G. Indeed it is the smallest localizing subcategory of G containing all the simple objects.

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Grothendieck Categorical Representations

By transfinite recursion we now define an ascending sequence of localizing subcategories of G: G0 ⊂ G ⊂ · · · ⊂ Gα ⊂ · · · ⊂ G such that G 0 = {0}, and G 1 is the localizing subcategory of all semi-Artinian objects of G as defined above. If α is an ordinal such that for every β < α we have already defined G β , then: 1. If α is not a limit ordinal, that is, we may view α = β + 1, we write G/Gβ for the quotient category of G with respect to Gβ and Qβ : G → G/G β for the canonical functor, which is known to be an exact functor; 2. if α is a limit ordinal, then we let G α be the smallest subcategory containing ∪β

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

Zuhair Nashed University of Central Florida Orlando, Florida

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology S. Kobayashi University of California, Berkeley Marvin Marcus University of California, Santa Barbara W. S. Massey Yale University

Anil Nerode Cornell University Freddy van Oystaeyen University of Antwerp Donald Passman University of Wisconsin Fred S. Roberts Rutgers University David L. Russell Virginia Polytechnic Institute and State University Walter Schempp Universität Siegen

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2007022732

Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1 A Taste of Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1

Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Examples and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Grothendieck Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Separable Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Noncommutative Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1

2.2

2.3

2.4 2.5

Small Categories, Posets, and Noncommutative Topologies . . . . . . . . . . . 11 2.1.1 Sheaves over Posets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 2.1.2 Directed Subsets and the Limit Poset . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.3 Poset Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 The Topology of Virtual Opens and Its Commutative Shadow . . . . . . . . . 19 2.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.2.1 More Noncommutative Topology . . . . . . . . . . . . . . . . . . . 28 2.2.2.2 Some Dimension Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Points and the Point Spectrum: Points in a Pointless World . . . . . . . . . . . 29 2.3.1 Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.1.1 The Relation between Quantum Points and Strong Idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.1.2 Functions on Sets of Quantum Points . . . . . . . . . . . . . . . 36 Presheaves and Sheaves over Noncommutative Topologies. . . . . . . . . . . .36 2.4.1 Project: Quantum Points and Sheaves . . . . . . . . . . . . . . . . . . . . . . . . 39 Noncommutative Grothendieck Topologies . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5.1 Warning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5.2 Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5.2.1 A Noncommutative Topos Theory . . . . . . . . . . . . . . . . . . 44 2.5.2.2 Noncommutative Probability (and Measure) Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 vii

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Contents 2.5.2.3 Covers and Cohomology Theories . . . . . . . . . . . . . . . . . . 45 2.5.2.4 The Derived Imperative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.6 The Fundamental Examples I: Torsion Theories . . . . . . . . . . . . . . . . . . . . . 45 2.6.1 Project: Microlocalization in a Grothendieck Category . . . . . . . . 63 2.7 The Fundamental Examples II: L(H ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.7.1 The Generalized Stone Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.7.2 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.7.3 Project: Noncommutative Gelfand Duality . . . . . . . . . . . . . . . . . . . 73 2.8 Ore Sets in Schematic Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3 Grothendieck Categorical Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.1 3.2

Spectral Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Affine Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.2.1 Observation and Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.3 Quotient Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.3.1 Project: Geometrically Graded Rings . . . . . . . . . . . . . . . . . . . . . . . 100 3.4 Noncommutative Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.4.1 Project: Extended Theory for Gabriel Dimension . . . . . . . . . . . . 107 3.4.2 Properties of Gabriel Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.4.3 Project: General Birationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4 Sheaves and Dynamical Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.1

Introducing Structure Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.1.1 Classical Example and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.1.2 Abstract Noncommutative Spaces and Schemes . . . . . . . . . . . . . 113 4.1.3 Project: Replacing Essential by Separable Functors . . . . . . . . . . 119 4.1.4 Example: Ore Sets in Schematic Algebras . . . . . . . . . . . . . . . . . . 119 4.2 Dynamical Presheaves and Temporal Points . . . . . . . . . . . . . . . . . . . . . . . . 121 4.2.1 Project: Monads in Bicategories . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.2.2 Project: Spectral Families on the Spectrum . . . . . . . . . . . . . . . . . . 133 ˇ 4.2.3 Project: Temporal Cech and Sheaf Cohomology . . . . . . . . . . . . . 134 4.2.3.1 Subproject 1: Temporal Grothendieck Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 ˇ 4.2.3.2 Subproject 2: Temporal Cech Cohomology and Sheaf Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.2.4 Project: Dynamical Grothendieck Topologies . . . . . . . . . . . . . . . 135 4.2.5 Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.3 The Spaced-Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.3.1 Noncommutative Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.3.1.1 Toward Real Noncommutative Manifolds . . . . . . . . . . 140 4.3.2 Food for Thought: From Physics to Philosophy . . . . . . . . . . . . . . 141 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Foreword

In order to arrive at a version of Serre’s global sections theorem in the noncommutative geometry of associative algebras, one is forced to introduce a noncommutative topology of Zariski type. Sheaves over such a noncommutative topology do not constitute a topos, but that is exactly the reason why sheaf theory in this generality can carry the essential noncommutative information generalizing to a satisfactory extent classical scheme theory. The noncommutativity forces, at places, a departure from set theory-based techniques resulting in a higher level of abstraction, because opens are not sets of points. Based on some intuition stemming mainly from noncommutative algebra and classical geometry, I strived for an axiomatic introduction of noncommutative topology allowing at least a minimalistic version of geometry involving actual “spaces” and not merely a mask for noncommutative algebra! Completely new problems appear already at the fundamental level, requiring new ideas that sometimes almost alienate a pure algebraist. Not all such ideas are completely developed here, often I restricted myself to bare necessities but left room for many projects ranging from the exercise level to possible research. The spirit of these notes is somewhat experimental reflecting the initial stage of the theory. This may occasionally result in a certain imbalance between novelty sections on new aspects of virtual topology and functor geometry on one hand versus well-established parts of noncommutative algebra on the other. In either case I tried to supply sufficient background material concerning localization theory or some facts on the classical lattice L(H ) of quantum mechanics for some Hilbert space H . On the other hand, I included a few topics that are, at this moment, only important for some of the research projects. In recent years “research training” for so-called young researchers became a trendy topic, and several of the included projects might be viewed in such a framework; however, some projects mentioned are probably hard and essential for better development of the theory and its applications. Intrinsic problems related to sheafification over a noncommutative space are the main topic in Section 4.2 and represent the introduction of a dynamic version of noncommutative topology and geometry. Since this construction is strictly related to the “absence” of points or of “enough points” in the noncommutative spaces, the dynamic theory as defined here is an exclusively noncommutative phenomenon; it is trivialized in the commutative case where space, and its topology, is described by sets of points. While reading Section 4.3 the reader should maintain a physics point of view because a noncommutative model for “reality” is hinted at; I included some observations related to this “spaced time,” resulting from recent interactions with several physicists, just as food for thought. I welcome all reactions and suggestions, for example, concerning the projects or the general philosophy of the topic. F. Van Oystaeyen

ix

Acknowledgments

Research in this work was financially supported by : r

An E.C. Marie Curie Network (RTN - 505078) LIEGRITS

r

A European Science Foundation Scientific Programme: NOG

The author fittingly supported these projects in return. I thank my students and some colleagues for keeping the fire burning somewhere, and the Department of Mathematics at the University of Antwerp for staying small, even after the fusion of the three former Antwerp universities. I especially appreciated moral support from E. Binz, B. Hiley, and C. Isham; they shared their vast knowledge in both physics and mathematics with me, and by showing their interest, motivated me to further the formal construction of noncommutative topology. Finally, thanks to my family for the life power line.

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Introduction

Noncommutativity of certain operations in nature as well as in mathematics has been observed since the early development of physics and mathematics. For example, compositions of rotations in space or multiplication of matrices are well-known examples often highlighted in elementary algebra courses. More recently even geometry became noncommutative, and nowadays motivation for the consideration of intrinsically noncommutative spaces stems from several branches of modern physics, for example, quantum gravity, some aspects of string theory, statistical physics, and so forth. From this point of view it seems to be necessary to have a concept of space and its geometry that is fundamentally noncommutative even to the extent that one would not expect that its mathematics is built on set theory or the theory of topoi. On the other hand, some branches of noncommutative geometry realize the noncommutative space solely via the consideration of noncommutative algebras as algebras of functions on an undefined fantasy object called the noncommutative variety or manifold. Nevertheless this technique is relatively successful, and it allows a perhaps surprising level of geometric intuition combined with algebraic formalism either in the algebraic or differential geometry setup. Further generalization may be obtained by conveniently replacing sheaf theory on the Zariski or real topologies by more abstract theoretical versions of it. In such a theory, the objects of interest on the algebraic level are either some types of quantized algebras or suitable C ∗ -algebras. The fact that noncommutativity may force a departure from set theoretic foundations creates a parallel development on the side of logic involving non-Boolean aspects as in quantum logic or quantales replacing Grothendieck’s locales. The different points of view fitting the abstract picture sketched above do not seem to fit together seamlessly; in particular, some desired applications in physics do not seem to follow from spaceless geometry, even if some applications do exist already. For example, the symbiosis between quantales and C ∗ algebras defies more general applicability for algebras of completely different type. We may now rephrase the ever-tantalizing dilemma as points or no points, that’s the question! On one hand, the introduction of a pointless geometry defined by posets with suitable operations extending the idea of a lattice to the noncommutative situation, with the partial order relation not necessarily related to set theoretic inclusion, seems to be very appropriate. After all, an abstract poset approach to quantum gravity seems to be at hand! On the other hand, there are points in a pointless geometry! In fact, there are different kinds of points, and in specific situations a certain type of point is more available than another. The problem then arises whether a noncommutative topology, defined on a

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noncommutative space in terms of a noncommutative type of lattice that replaces the set of opens, is to some extent characterized by sets (!) of noncommutative points. Observe that when defining the Zariski topology on the prime (or maximal) ideal spectrum of a commutative (Noetherian) ring, one actually defines the opens by specifying their points and the spectrum is defined before the topology. The difference between presheaf and sheaf theory is completely encoded in the relations between sections on opens and stalks at points. The sheafification functor may be the ultimate example of this interplay, its construction depends on consecutive limit constructions from basic opens to points by direct limits, and from points to arbitrary opens by inverse limits. Even in classical commutative geometry there is a difference when prime ideals of the ring are viewed as points of the spectrum or only maximal ideals are considered as such. However, at the basic level there is absolutely nothing to worry about because the type of rings considered, for example, commutative affine algebras over a field, are Jacobson rings (and Hilbert rings); that is, every prime ideal is determined by the maximal ideals containing it, and in fact it is the intersection of them. So even the commutative case learns that once a topology is given there are still several consistent ways to decide what the points, but when a notion of points is fixed first, the topology has to be adjusted to this notion in order to obtain a useful sheaf theory. Another most important property in classical geometry is that varieties, schemes, or manifolds are locally affine in some sense; for example, every point has an affine neighborhood. In a pointless geometry the latter property is hard to understand and a serious modification seems to be necessary. It will turn out that for this reason, one has to introduce representational theoretic aspects in the abstract theory. Now, for the kind of noncommutative algebraic geometry in the sense of a generalization of scheme theory over noncommutative algebras, as promoted by the author (for example in [44]), the presence of module theory and a theory of quasicoherent sheaves make this possible. But what remains if we try to drop all unnecessary (?) restrictions concerning the presence of an algebra, modules, spectra, points, and so forth, and try to arrive at a barely abstract geometry based on a kind of topology equipped with some functors on a general but suitable category or family of categories? Well, perhaps virtual topology and functor geometry! In the following I try to indicate how such a general theory will have to deal with the issues raised above. First, noncommutative topology is introduced via the notion of a noncommutative lattice where the operations ∧ and ∨ are defined axiomatically and they are less strictly connected to the partial order than the meet and join in usual lattices. The noncommutative topologies may be considered as sets of opens, but an open can in no way be viewed as a set. Noncommutative topologies do fit in a theory of noncommutative Grothendieck topologies but not in topos theory; a noncommutative version of the latter remains to be developed. Then points and minimal points may be defined in a generalized Stone space associated to a noncommutative topology. There are not enough points to characterize an open to which they belong, but there is a well-behaved notion of commutative shadow of a noncommutative space, which is given by a real lattice in the usual sense, and where the commutative opens are characterized by sets of points. At this point

Introduction

xv

generalized function theory could be developed but we did not go into this; rather we introduced sheaf theory on noncommutative topologies and verified that they transfer nicely to the generalized Stone space. A complete symmetrization of Grothedieck’s definition of a Grothendieck topology leads to noncommutative (left, right, skew) versions of this, and the noncommutative topologies defined axiomatically fit into the latter framework by restricting to certain partial order relations, that is, the so-called generic relations. All of this is in Chapter 2 ending with two fundamental examples: the lattice of torsion theories or Serre quotient categories of a Grothendieck category, and the lattice of closed linear subspaces of a Hilbert space. The first one has enough points in the “prime” sense; the second has enough minimal points (maximal filters) in the Stone space. In Chapter 3, Grothendieck categorical representations are studied with the aim of arriving at an abstract notion of affine open. When applying this to the algebraic geometry of associative algebras, for example, schematic algebras, their modules and the localizations of module categories, the general notion of affine open describes exactly the opens corresponding to exact localization functors commuting with direct sums (functors with an adjoint of a specific type). The general notion of quotient representation may then be used to explain how noncommutative projective spaces arise from noncommutaive affine spaces. Some sheaf theory is developed; in a sense this is an extension of a theory of quantum sheaves considered earlier by changing from categories of opens in a topology to more general lattices, but now we even allow the suitable noncommutative version of lattices. The creation of a new theory sometimes opens many doors, maybe too many doors. For example, the further development of the noncommutative version of topology, for example, closed sets, compactness, convergence structure is possible in the generalized Stone space, but we have not even tried to go there. Even though it may well be that such theory is interesting in its own right, we have only mentioned this as a project for the zealous reader looking for an original way to test his/her skills. Even more haunting ideas about noncommutative probability or measure theory have suffered the same fate. Some projects, however, are more straightforward exercises leading to possible research projects. The final section starts out swinging — well at least we propose a dynamical version of topology and sheaf theory, providing at least one solution of the problem of sheafification independent of generalizations of topos theory. It required a rephrasing of continuities in a poset setup with a totally ordered set (time!) as a parameter set. The result is a spectrum with a classical topology existing at each moment but not varying in time the way the noncommutative topologies do. This may be seen as a mathematical uncertainty principle or better as mathematics of observation. One might hope that physical phenomena, in particular quantum theories, may suitably be phrased in terms of this observational mathematics — perhaps a dream. For the more algebraic, or more geometric, applications of ideas expounded in these notes, we may refer to earlier work in noncommutative geometry — in particular the theory of schematic algebras (see [48]). It is not surprising that the geometric structures stripped to their naked abstraction retain a somewhat esoteric character, highlighting mainly partial ordered sets with noncommutative operations but related

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Introduction

to lattices (via the commutative shadow of spaces), categorical methods and functorial constructions, a further abstraction of sheaf theory and spectral constructions, and a categorical representation theory using Grothendieck categories. Some ideas in these notes have already inspired a few recent papers in physics, so without trying to claim more, I hope that the exercise of digging deep for the abstract skeleton of geometry may lead to a further unification of different kinds of noncommutative geometry and point to an actual space of an intrinsically noncommutative nature, perhaps allowing the expression of observations concerning natural phenomena.

Projects

2.2.2.1 More Noncommutative Topology 2.2.2.2 Some Dimension Theory 2.3.1.1 The Relation between Quantum Points and Strong Idempotents 2.3.1.2 Functions on Sets of Quantum Points 2.4.6 Quantum Points and Sheaves 2.5.2.1 A Noncommutative Topos Theory 2.5.2.2 Noncommutative Probability (and Measure) Theory 2.5.2.3 Covers and Cohomology Theories 2.5.2.4 The Derived Imperative 2.6.1 Microlocalization in a Grothendieck Category 2.7.3 Noncommutative Gelfand Duality 3.3.1 Geometrically Graded Rings 3.4.1 Extended Theory for Gabriel Dimension Exercise 3.3 Krull and Gabriel Dimension for a Skew Topology and Its Relation to Commutative Shadow Exercise 3.4 Develop a Theory of Representation Dimension in Connection with Grothendieck Quotient Representations Exercise 3.5 Gabriel Dimension for Sheaf Categories and Related Behavior with Respect to Separable Functors Exercise 3.6 Using the Gabriel Dimension for Noncommutative Valuation Rings of Arbitrary Rank 3.4.3 General Birationality 4.1.3 Replacing Essential by Separable Functors

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Projects

4.2.1 Monads in Bicategories 4.2.2 Spectral Families ˇ 4.2.3 Temporal Cech and Sheaf Cohomology 4.2.3.1 Temporal Grothendieck Representations ˇ 4.2.3.2 Temporal Cech Cohomology and Sheaf Cohomology 4.2.4 Dynamical Grothendieck Topologies

Chapter 1 A Taste of Category Theory

1.1

Basic Notions

We assume that the reader is familiar with the foundations of set theory, at least in its intuitive version. Acategory C consists of a class of objects together with sets HomC (X, Y ), for any pair of objects X, Y of C, satisfying suitable conditions listed hereafter. The elements of HomC are called C-morphisms or just morphisms, if there is no ambiguity concerning the category considered. For any object X of C there is a distinguished element I X ∈ HomC (X, X ), called the identity morphism of X . For any triple X, Y, Z of objects in C there is a composition map: HomC (X, Y ) × HomC (Y, Z ) → HomC (X, Z ), ( f, g) → g ◦ f such that the following properties hold: i. For f ∈ HomC (X, Y ), g ∈ HomC (Y, Z ), h ∈ HomC (Z , W ) we have: h◦(g◦ f ) = (h ◦ g) ◦ f . ii. For f ∈ HomC (X, Y ) we have f ◦ I X = f = IY ◦ f . iii. If (X, Y ) = (X , Y ), then HomC (X, Y ) and HomC (X , Y ) are disjoint sets.

1.1.1

Examples and Notation

i. The category Set is obtained by taking the class of all sets using maps for the morphisms. ii. The category Top is obtained by taking the class of all topological spaces using continuous functions for the morphisms. iii. The category Ab is obtained by taking the class of all abelian groups using groups homomorphisms for the morphisms. iv. The category Gr is obtained by taking the class of all groups using group homomorphisms for the morphisms.

1

2

A Taste of Category Theory v. The category Ring is obtained by taking the class of all rings using ring homomorphisms for the morphisms. vi. For any given ring R the class of left R-modules using left R-linear maps for the morphisms defines the category R-mod. The category defined in a similar way but using right R-modules and right R-linear maps is denoted by mod-R.

Definition 1.1 Consider a class D consisting of objects of C, then D is said to be a subcategory of C when the following properties hold: i. For objects X, Y in D we have: HomD (X, Y ) ⊂ HomC (X, Y ). ii. Composition of morphisms in D is as in C. iii. For X in D, I X is the same as in C. We say that D is a full subcategory of C when HomD (X, Y ) = HomC (X, Y ) for all X, Y of D. In the list of examples one easily checks that Ab is a subcategory of Gr but, for example, Gr is not a full subcategory of Set. Definition 1.2 For a set S and a family of categories (C)s∈S we define the direct product category C by taking the class of objects to consist of the families (X s )s∈S of objects X s of C s for s ∈ S. If X = (X s )s∈S , Y = (Ys )s∈S are such families, then HomC (X, Y ) = {( f s )s∈S , f s ∈ HomCs (X s .Ys ), s ∈ S}. Composition of morphismsis defined componentwise. This direct product category C will be denoted by s∈S C s ; in case C s = C for all s ∈ S, then we also write C S . In case S = {1, . . . , n} we also write C 1 × C 2 × · · · × Cn for the direct product. This paragraph deals with specific properties of morphisms. A C-morphism from an object X to an object Y will be denoted by X → Y , and if f ∈ HomC (X, Y ) we will write f : X → Y . A monomorphism (in C) is a morphism f : X → Y such that for any object Z and given morphisms h, g ∈ HomC (Z , X ) such that f ◦ h = f ◦ g we must have g = h. Dually, an epimorphism is a morphism f : X → Y such that for any object W of C, and given morphisms h, g ∈ HomC (Y, W ) such that h ◦ f = g ◦ f , we must have h = g. An isomorphism is a morphism f : X → Y for which there exists a morphism g : Y → X such that g ◦ f = I X , f ◦ g = IY . In case a g as above exists, then it is unique, as one easily checks; it is called the inverse of f and will often be denoted by f −1 . In a straightforward way, one verifies that an isomorphism is necessarily an epimorphism as well as a monomorphism. Observe that a morphism that is both a monomorphism and an epimorphism need not necessarily be an isomorphism; indeed in Ring the canonical inclusion Z → Q is both a monomorphism and an epimorphism! Composition of monomorphisms, respectively epimorphisms, respectively isomorphisms, yields again a monomorphism, respectively epimorphism,

1.1 Basic Notions

3

respectively isomorphism. The duality between monomorphisms and epimorphisms may best be phrased by passing to the dual category C o , having the same class of objects as C but with HomC o (X, Y ) = HomC (Y, X ) by definition. Now a morphism f : X → Y in C is a monomorphism, respectively an epimorphism, if and only if f is an epimorphism, respectively a monomorphism when seen as a morphism Y → X in the dual category C o . Definition 1.3: Subobjects in C In many examples and applications the objects considered need not be sets, hence a correct definition of the term subobjects requires some care. Fix an object X of C; for any W of C we have a set MonoC (W, X ) consisting of monomorphisms W → X . For objects U and W of C we have a product set MonoC (U, X ) × MonoC (W, X ); if (α, β) is in the latter, then we may define α ≤ β if there exists a morphism γ : U → W such that β ◦ γ = α. In case a γ as before exists, it is unique and also a monomorphism. We say that α and β are equivalent monomorphisms if α ≤ β and β ≤ α; indeed, the foregoing defines an equivalence relation! Since we are dealing with sets now, we may evoke the Zermelo axiom and choose a representative in every equivalence class of monomorphisms. The resulting monomorphism is called a subobject of X in C. Quotient objects may now also be defined in a formally dual way by passing from C to C o . It is not difficult to verify that a subobject of a subobject is again a subobject and a quotient object of a quotient object is a quotient object. Definition 1.4: Initial and Final Object An object I , respectively F, of C such that HomC (I, X ), respectively HOMC (X, F), is a singleton for every X of C is called an initial object of C, respectively a final object of C. Two initial, respectively final, objects of C are necessarily isomorphic. A zero object of C is an object that is initial and final. This allows us to distinguish zero morphisms as those f : X → Y that factorize through the (unique up to isomorphism) zero object. If a zero object exists, then we denote it by O; then each set HomC (X, Y ) has precisely one zero morphism denoted o X Y or just o when no confusion can arise. Definition 1.5: Product and Coproduct Objects To a family (X s )s∈S of objects in C we may associate the product s∈S X s = X if we can solve a universal construction problem in C. The object X we look for should come equipped with a family of morphisms (πs )s∈S , πs : X → X s for s ∈ S, such that for any object Y of C with given morphisms f s : Y → X s for s ∈ S, there exists a unique morphism f : Y → X such that πs ◦ f = f s for all s ∈ S. If an object X with these properties does exist in C then it is unique up to isomorphism and we use the notation X = s∈S X s . In case S = {1, . . . , n} then we also write X = X 1 ×· · ·× X n (suppressing C in the product notation). The notion of coproductis defined dually. If the coproduct of a family (X s )s∈S exists we will denote it by s∈S X s ; in case S = {1, . . . , n} it is customary to write it as X 1 ⊕ · · · ⊕ X n .

4

A Taste of Category Theory

Now that our categories need not be sets, we may not be able to talk about maps from one category to another, notwithstanding the fact that we may without any problem associate an object of a category to an object in another category. For categories D and C we let a covariant functor F from D to C be defined by associating it to an object X of D an object F(X ) of C and to a morphism f : X → Y in D a morphism F( f ) : F(X ) → F(Y ) in C such that the following properties hold: i. F(I X ) = I F(X ) for every X of D. ii. F(g ◦ f ) = F(g) ◦ F( f ) for f : X → Y, g : Y → Z in D. A contravariant functor from D to C is then just a covariant functor Do → C. Usually, when F is a covariant functor from D to C, set-theoretic-inspired notation is used to express this by F : D → C; the context makes it clear that we do not mean to imply by this that F is a map! Definition 1.6: Full and Faithful Functors A covariant functor F : D → C yields for any X, Y in C a map HomD (X, Y ) → HomC (F(X ), F(Y )), which we denote also by: f → F( f ). We say that F is faithful, respectively, full, respectively full and faithful, if the above map f → F( f ) is injective, respectively surjective, respectively bijective. Note that for any category C there exists an identity functor 1C : C → C defined by 1C (X ) = X for every object X of C, and 1C ( f ) = f for every morphism f ∈ HomC (X, Y ). Obviously, the functor 1C is always full and faithful. Can functors between categories make up a category, and then what should be the morphisms? We do not treat functor categories in depth here but restrict ourselves to recalling a few fundamental notions related to this idea. Definition 1.7: Functorial Morphisms Consider a pair of covariant functors F, G : D → C. A functorial morphism ϕ : F → G is given by morphisms ϕ(X ) : F(X ) → G(X ) for X an object of D, such that for f : X → Y in D we have: ϕ(Y ) ◦ F( f ) = G( f ) ◦ ϕ(X ); in other words we have a commutative diagram of morphisms in C: F( f )

F(X )−−−−→ F(Y ) ϕ(Y ) ϕ(X ) G(X )−−−−→ G(Y ) G( f )

In case ϕ(X ) is an isomorphism for all objects X of D, ϕ is said to be a functorial isomorphism and we denote this by F G. For functorial morphisms ϕ : F → G and ψ : G → H the composition ψ ◦ ϕ : F → H may be defined by (ψ ◦ ϕ)(X ) = ψ(X ) ◦ ϕ(X ) for all X , and this yields again a functorial morphism. Let Hom(F, G) stand for the class of functorial morphisms from F to G. There exists an identity functorial morphism 1 F : F → F defined by putting 1 F (X ) = 1 F(X ) for all X of D. Since Hom(F, G) need not be a set in

1.2 Grothendieck Categories

5

general, we meet a small problem here in fitting this again in the framework of a category where the morphisms between two objects should be a set. In case D is a small category, then Hom(F, G) is also a set. Definition 1.8: Equivalences and Dualities The covariant functor F : D → C is said to be an equivalence of categories when there exists a covariant functor G : C → D such that G ◦ F 1D and F ◦ G 1C . In case G ◦ F = 1D and F ◦ G = 1C , F is called an isomorphism of categories, and in that case D and C are said to be isomorphic categories. A contravariant functor F : D → C defining an equivalence between Do and C is said to be a duality of categories, and in that situation D and C are said to be dual. Theorem 1.1 A covariant functor F : D → C is an equivalence if and only if: i. F is full and faithful. ii. For any Y ∈ C there is an X ∈ D such that Y F(X ). To an object X of C we may associate a contravariant functor h X : C → Set by putting h X (Y ) = HomC (Y, X ) and for a morphism f : Y → Z in C we define h X ( f ) : h X (Z ) → h X (Y ) by h X ( f )(z) = z ◦ f for any z ∈ h X (Z ). A functor F : C → Set is said to be representable if there is an object X of C such that F is isomorphic to the functor HomC (X, −) = h X . Theorem 1.2: The Yoneda Lemma For objects A and B of C there exists a natural bijection of Hom(h A , h B ) to HomC (B, A). In particular Hom(h A , h B ) is a set. Corollary 1.1 The category C o is isomorphic to the category of representable functors C → Set with the functorial morphisms for the morphisms.

1.2

Grothendieck Categories

The categories appearing in algebraic geometry, be it commutative or not, have very special properties, for example, modules over a ring, graded modules over a graded ring and so forth. For several results the class of abelian categories is suitable, but the best behaved categories we shall use are the so-called Grothendieck categories. These are rather close to being categories of left modules over a ring; the extra generality allows us to include categories of graded modules as well as certain categories of presheaves or sheaves.

6

A Taste of Category Theory A category C is pre-additive if the following three properties hold: i. For X, Y in C, HomC (X, Y ) is an abelian group with zero element O X Y called the zero morphism. ii. For X, Y, Z in C and f, f 1 , f 2 in HomC (X, Y ), g, g1 , g2 in HomC (Y, Z ), we obtain: g ◦ ( f1 + f2) = g ◦ f1 + g ◦ f2 (g1 + g2 ) ◦ f = g1 ◦ f + g2 ◦ f iii. There is an object X of C such that 1 X = O X X . Clearly such X is a zero object, unique up to isomorphism, usually denoted by O.

It is obvious that the dual of a pre-additive category is again pre-additive. A functor between pre-additive categories may have some additivity properties, too; for example, we say that F : D → C, where D and C are pre-additive, is an additive functor if for f, g ∈ HomD (X, Y ), where X, Y are objects of D, we have: F( f + g) = F( f ) + F(g). If OD is the zero object if D, then F(OD ) is the zero object of C, say OC . An additive category is a pre-additive category C such that for any two objects of C a coproduct exists in C. Definition 1.9: Abelian Categories An additive category C is said to be an abelian category if it satisfies conditions AB.1 and AB.2: AB.1 For any morphism f : X → Y in C, both Ker( f ) and Coker( f ) exist in C; then f may be decomposed as indicated in the following diagram: i

f

π

Ker( f )−−−−→X Y −−−−→Coker( f ) −−−−→ µ λ Coim( f ) −→ Im( f ) f¯ where f = µ ◦ f ◦ λ and i and µ are monomorphisms and π, λ are epimorphisms. AB.2 For every f as in AB.1, f is an isomorphism. In any category verifying AB.2, a morphism is an isomorphism exactly when it is both a monomorphism and an epimorphism. Definition 1.10: Exact Sequences and Functors Suppose that C is pre-additive such that AB.1. and AB.2. hold. A sequence of morphism X −→ Y −→ Z in C is exact if Im( f ) = Ker(g) as subobjects of Y . An f

g

arbitrary (long) sequence is said to be exact if every subsequence of two consecutive morphisms is exact in the sense defined above. An additive functor F : D → C, where both categories are pre-additive and such that AB.1 and AB.2 hold, is said to

1.2 Grothendieck Categories

7

be left exact, and respectively right exact if for any exact sequence of morphisms in D: 0→X →Y →Z →0 the following sequence is exact: 0

→

respectively

F(X ) F(X )

→ →

F(Y ) F(Y )

→ →

F(Z ) F(Z )

→

0

When F is both left and right exact, then F is exact. Consider X, Y in an additive category C and let X ⊕ Y be their coproduct. By definition of the coproduct there exist natural morphisms: i X : X → X ⊕ Y , π X : X ⊕ Y → X , i Y : Y → X ⊕ Y , πY : X ⊕ Y → Y , such that π X ◦ i X = 1 X , πY ◦ i Y = 1Y , π X ◦ i Y = 0 = πY ◦ i X , 1 X ⊕Y = i X ◦ π X + i Y ◦ πY . This actually establishes that (X ⊕ Y, π X , πY ) is a product of X and Y in C. Consequently, if C is additive, respectively abelian, then C o is too. Lemma 1.1 A functor F between additive categories is an additive functor if and only if it commutes with finite coproducts. In [17] A. Grothendieck introduced several extra axioms on abelian categories, gradually strengthening the definition until the notion of the Grothendieck category, as we know it, appears. The axioms AB.3, AB.4, AB.5 and their duals (AB.3)∗ , (AB.4)∗ , (AB.5)∗ are not independent; in fact, AB.5 presupposes AB.3 and implies AB.4. We just recall definitions and basic facts. AB.3 Arbitrary coproducts exist in C. (AB.3)∗ Arbitrary products exist in C. In case AB.3 holds in C we may define for any nonempty set S a functor ⊕s∈S : C (S) → C, associating to a family of C-objects, indexed by S the coproduct (sometimes called the direct sum) of that family. The functor ⊕s∈S is always right exact. AB.4 For any nonempty set S, ⊕s∈S is an exact functor. (AB.4)∗ For a nonempty set S, s∈S is an exact functor. If C is abelian and satisfies AB.3., then for any family ofsubobjects (X s )s∈S of X we may define a “smallest” subobject of X , denoted by s∈S X s , such that all X s are subobjects of the latter. The quotation marks around smallest refer to the fact that some care is necessary with the interpretation in view of the definition of subobject; compare Definition 1.3. The object s∈S X s is called the sum of (X s )s∈S . Dually, if C is an abelian category satisfying (AB.3)∗ , then for every family (X s )s∈S of subobjects of X we may associate ∩s∈S X s , the largest subobject of X contained in each X s , s ∈ S. The subobject ∩s∈S Ss is called the intersection of (X s )s∈S . Observe that in any abelian

8

A Taste of Category Theory

category finite products do exist, hence the intersection of a finite family exists (it is enough to have an intersection of two objects). AB.5 Let C be an abelian category satisfying AB.3. Consider an object X of C and subobjects X , s ∈ S, and Y , such that the family (X ) is right filtered, then: ( s s s∈S s∈S X s ) ∩ Y = (X ∩ Y ). s s∈S (AB.5)∗ The dual of AB.5. Observation 1.1 An abelian category such that AB.3 and AB.5 hold also satisfies AB.4. Definition 1.11: Generators for an Abelian Category Consider the family (X s )s∈S in the abelian category C; we say that (X s )s∈S is a family of generators if for every object X and subobject Y = X in C there is an s ∈ S and a morphism f : X s → X such that Im( f ) is not a subobject of Y . An object U of C is said to be a generator if {U } is a family of generators. Definition 1.12: Grothendieck Category An additive category satisfying AB, 1, . . . AB.5, and having a generator is a Grothendieck category. Observe that an abelian category, such that both AB.5 and (AB.5)∗ hold, is necessarily the zero category (category consisting of the zero object with the zero morphism). Consequently the opposite of a Grothendieck category is never a Grothendieck category. To end this section we recall some facts about adjoint functors. The notion of adjointness is very fundamental, and it has applications in different areas of mathematics. Consider functors F : C → D, G : D → C. Definition 1.13: Adjoint Functors The functor F is a left adjoint of G, or G is a right adjoint of F, if there is a functorial isomorphism : HomD (F, −) → HomC (−, G) where HomD (F, −) : C o × D → Set associates to (X, Y ) the set HomD (F(X ), Y ); HomC (−, G) : C o × D → Set associates to (X, Y ) the set HomC (X, G(Y )). If case C and D are pre-additive and the functors F and G are assumed to be additive, then we assume that (X, Y ) is an isomorphism of abelian groups. The following sums up some basic facts concerning adjoint functors. Properties 1.1 When the functor F is a left adjoint for G, then the following hold: i. F commutes with coproducts, G commutes with products. ii. If C and D are abelian and F and G are additive functors, then F is right exact and G is left exact.

1.3 Separable Functors

9

iii. If for every object Y of D there exists an injective object Q of D and a monomorphism Y → Q, then F is exact if and only if G preserves injectivity. iv. If for every X of C there exists a projective object P of C together with an epimorphism P → X in C, then G is exact if and only if F preserves projectivity. Perhaps one of the most well-known pairs of adjoint functors appears in connection with module categories over associative rings R and T say. Consider the module categories R-mod and T -mod as well as R-mod-T , the category of left R-right-T -bimodules. For M in R-mod-T we may define the tensor-functor M ⊗T − : T -mod → R-mod by viewing M ⊗T N for a left T -module N as a left R-module in the obvious way. It is easy to verify that M ⊗T − is a left adjoint of Hom R (M, −) : R-mod → S-mod.

1.3

Separable Functors

The notion of separable functor has been introduced by M. Van den Bergh and the author; the concept has been applied to algebras and in particular graded algebras in [33]. Several other applications of ring theoretical nature stem from the paper by M. D. Rafael, an author among participants at a summer institute at Cortona. The separable functors are not absolutely necessary for the development of the theory in this work; nevertheless, we include a short presentation because they may be used in several applications and some research projects we cover. Consider a covariant functor F : D → C. The functor F is a separable funcF : HomC (F(M), F(N )) → tor if for all objects M, N in D there are maps ϕ M,N HomC (M, N ) satisfying the following properties: F SF.1 For f ∈ HomD (M, N ), ϕ M,N (F( f )) = f .

SF.2 For objects M , N in D and f ∈ HomD (M, M ), g ∈ HomD (N , N ), f ∈ HomC (F(M), F(N )), and g ∈ HomC (F(M ), F(N )) such that the following diagram is commutative in D: F(M)− → F(N ) −−f− F(g) F( f ) F(M )−−−− → F(N ) g

then the following is commutative in D: M−−−−→ N F ( f ) ϕ M,N g M −−−−→ N

f

F ϕM ,N (g )

Observe that SF.1 holds if case F is a full faithful functor, that is, whenever for M, N in D the map HomD (M, N ) → HomD (F(M), F(N )) is bijective.

10

A Taste of Category Theory

Lemma 1.2 1. An equivalence of categories is also a separable functor. 2. If F : D → C and G : C → B are separable functors, then GF is separable. Conversely, if GF is a separable functor, then F is a separable functor. Let us summarize some basic properties of separable functors in the following proposition. Proposition 1.1 Let F : D → C be a separable functor and consider objects M and N in D. i. If f ∈ HomD (M, N ) is such that F( f ) is a split map, then f itself is a split map. ii. If f ∈ HomD (M, N ) is such that F( f ) is co-split, that is to say that there exists a u ∈ HomC (F(N ), F(M)) such that F( f ) ◦ u = 1 F(N ) , then f is itself co-split. iii. Assume that both D and C are abelian categories. When F preserves epimorphisms, respectively monomorphisms, and F(M) is projective in C, respectively injective in C, then M is projective in D, respectively injective in D. iv. Assume that D and C are abelian categries. If F(M) is a quasi-simple object, that is, every subobject splits off, and F preserves monomorphisms, then M is itself a quasi-simple object in D. Proof Statement iii follows from ii and iv follows i. The proof of ii is very similar to the proof of i, so it suffices to establish i. The assumption in i implies that there exists a map u : F(N ) → F(M) such that u F( f ) = 1 F(M) . Put g = ϕ NF ,M (u). Condition SF.2. then implies that g f = 1 M because we do have a commutative diagram in C. F(M)− −→ F(M) −1− F(M) F(1 M ) F(N )−−−−→ F(M)

F({)

u

The claim follows. Corollary 1.2 Part i may be rephrased as a functorial version of Maschke’s theorem (used frequently in the representation theory of groups). We finish this section by pointing out that the terminology derives from the fact that the restriction of scalar functors associated to a ring morphism C → R, where C is commutative and central in R, is a separable functor when R is C-separable. When R is also commutative, this agrees with the classical notion of a separable extension.

Chapter 2 Noncommutative Spaces

2.1

Small Categories, Posets, and Noncommutative Topologies

Throughout this section, C stands for a fixed small category, that is, a category having a class of objects that is a set. A category with exactly one object is a monoid; this is because we may view this as an object with a given monoid of endomorphisms. A group is then a monoid where all endomorphisms are automorphisms. By O we denote the zero-object category with a unique object O and a unique morphism: the identity of O. For any category C there exists a unique functor C → O. A terminal object in C is an object, I say, such that for every object α of C there is a unique morphism α → 1 in C. If C does not have a terminal object, we can adjoin one to C and obtain a category C1 with an obvious functor C → C1 taking an object α in C to α in C1 . To C we may associate the opposite category C o having the same class of objects but with morphisms reversed. For an arbitrary category D a D-representation of C is just a functor R : C → D; a D-representation of C o is called a presheaf on C with values in D. Hence, a presheaf P : C o → D is given by a family of objects P(α) in D such that for each C-morphism α → β we have D-morphisms ραβ : P(β) → P(α) such that to the identity α → α corresponds the identity P(α) → P(α), and to γ α → β → γ in C we correspond ραγ = ραβ ρβ . If B is another small category and given an arbitrary functor F : B → C, we construct the left (and right) comma category as follows. For the objects of the right comma category (α, F) we take C-morphisms α → Fβ, α ∈ C, β ∈ B and a morphism (α → Fβ ) → (α → Fβ) in (α, F) as a B-morphism b : β → β making the following diagram commutative: Fβ' α

F (b)

Fβ

The left comma category (F, α) is defined likewise, using for the objects the C-morphisms Fβ → α, and so forth.

11

12

Noncommutative Spaces Any C-morphism a : α → α induces functors: (a, F) : (α, F) → (α , F), (α → Fβ) → (α → Fβ)a (F, a) : (F, α ) → (F, α)

A type of small category often considered is a poset. A poset, or partially ordered set, is just a set with a partial ordering: ≤. If is a poset, then we shall write for the category having as objects the elements λ ∈ , and hom (λ, µ) consists of the unique arrow λ → µ when λ ≤ µ, or it is empty. The categories are examples of delta categories, that is, small categories in which endomorphisms are necessarily the identity morphisms and hom(σ, τ ) = ∅ for σ = τ implies hom(τ, σ ) = ∅ (maps are one-way and no loops). We define D-representations of , presheaves on with values in D, and comma categories for λ ∈ , . . . by taking the corresponding definitions for . The mother of all posets is the set of natural numbers with its usual ordering. For n ∈ N we let [n] denote the linearly ordered set 0 < · · · < n viewed as a category (as for posets). A C-representation of [n] is called an n-simplex; if σ : [n] → C is a (covariant) functor, then we say that σ is an n-simplex or dimσ = n. We denote the C-morphism σ (r → s) by σ r s , for r ≤ s in [n]. Zero simplices are functors [O] → C; these may be viewed as the elements of C, up to a harmless “abuse of language.” For n > 0, an n-simplex σ is completely determined by the n-tuple (σ 01 , σ 12 , . . ., σ p−1, p ); therefore, it is unambiguous to write σ = (σ 01 , σ 12 , . . . ). When C is for some poset , then an n-simplex σ is completely determined by the ordered list of elements, called vertices, σ (0), . . ., σ (n), because any σ r s is then necessarily the unique -morphism σ (r ) → σ (s) corresponding to r ≤ s. If τ is an n-simplex and σ is an m-simplex such that τ (n) = σ (0), then we can form the cup-product τ ∨ σ , which is the (n + m)-simplex given by: (τ ∨ σ )r,r +1 = τ r,r +1 (τ ∨ σ )r,r +1 = σ r −n,r −n+1

when r < n when n ≤ r.

For n > 0 and o ≤ r ≤ n we define a functor ∂r : [n − 1] → [n], ∂r (s) = s ∂r (s) = s + 1

if s < r if s ≥ r.

A given n-simplex σ has an r -face defined by the composition of the functors σ and ∂r . For r ≥ s one easily computes ∂r ∂s = ∂s ∂r +1 , where composition is here written in the arrow-order (i.e., not the usual way of writing a composition of maps). Hence: 12 if r = 0 (σ , . . ., σ n−1,n ) σr = (σ 01 , . . ., σ r −1,r +1 , . . ., σ n−1,n ) if 0 < r < n 01 (σ , . . ., σ n−1,n−1 ) if r = n In case C = , then the faces are distinct but that need not be true in general for arbitrary C.

2.1 Small Categories, Posets, and Noncommutative Topologies

13

The ∂r are called face operators. The collection of n-simplices and the face operators connecting them are called the simplicial complex, (C). It will be useful for introducing homology theories in a formal way. A zero element, denoted by 0, of the poset is one for which 0 ≤ λ for all λ ∈ ; clearly, if a zero element exists it is unique. A unit element, denoted by 1, of the poset is one for which λ ≤ 1 for all λ ∈ ; if a unit element exists then it is unique. A poset with 0 and 1 is said to be a lattice if for any two elements λ and µ in there exists a maximum λ ∨ µ ∈ and a minimum λ ∧ µ ∈ . A lattice is said to be ∨-complete if for any family {λα , α ∈ A} in the maximum ∨α∈A λα exists in . The lattice is complete if it is both ∨-complete and ∧-complete. For the general theory of lattices we refer to [6][7]. Now let us introduce the notion of cover in an arbitrary poset . Definition 2.1 We say that λ ∈ is covered by {λα , α ∈ A} with λα ∈ for all α ∈ A, if λα ≤ λ for all α ∈ A and if λα ≤ µ for all α ∈ A for some µ ∈ then λ ≤ µ; in this case we also say that {λα , α ∈ A} is a cover for λ. If A is finite, then {λα , α ∈ A} is said to be a finite cover for λ ∈ . Example 2.1 i. If is a lattice, then {λ1 , . . ., λn } is a finite cover for λ ∈ exactly when λ = λ1 ∨ . . . ∨ λn ; in a complete lattice this holds for arbitrary covers. ii. If is a distributive lattice, then a given finite cover λ = λ1 ∨ . . . ∨ λn induces for every τ ≤ λ a cover: τ = (τ ∧ λ1 ) ∨ . . . ∨ (τ ∧ λn ); the latter is called the induced cover for τ ∈ λ. iii. If is a poset with 1, then a global cover is a set {λα , α ∈ A} such that µ ≥ λα for all α ∈ A entails µ = 1. In particular, if is a distributive lattice with 0 and 1, then a finite global cover is a set {λ1 , . . ., λ} such that 1 = λ1 ∨ . . . ∨ λn and every τ ∈ then allows a cover {τ ∧ λ1 , . . ., τ ∧ λn } induced by a global cover. iv. A cover {λ1 , . . ., λ} of λ is reduced when it is ∨-independent in the lattice . In a semi-atomic lattice with 0 and 1 that is upper continuous and has the property that 1 can be written as a finite join of atoms of , there always exists a reduced global cover.

2.1.1

Sheaves over Posets

In this section we restrict attention to a category D, the objects of which are sets; hence, morphisms in D are in particular set maps. A presheaf P : ()o → D is separated if for every cover {λα , α ∈ A} of λ in and every x, y ∈ P(λ) such that for all α ∈ A, ρλλα (x) = ρλλα (y), we must have x = y. In case no covers exist, then every presheaf is separated. A separated presheaf is a sheaf on (with values in D), if for every finite cover {λi , i} of λ and given

14

Noncommutative Spaces λ

xi ∈ P(λi ) such ρµλi (xi ) = ρµ j (x j ) for every µ ≤ λi , µ ≤ λ j , there exists an x ∈ P(λ) such that for all i, ρλλi (x) = xi . In order to introduce stalks of presheaves or sheaves, we first introduce a so-called limit poset C() associated to any given poset .

2.1.2

Directed Subsets and the Limit Poset

A subset X ⊂ is said to be directed if for every x, y in X there exists a z ∈ X such that z ≤ x and z ≤ y. Let D() be the set of directed subsets of . For A and B in D() we say that A is equivalent to B, written A ∼ B, if and only if the following conditions are satisfied: i. For a ∈ there exists a ∈ A, a ≤ a and b, b ∈ B such that: b ≤ a ≤ b . ii. For b ∈ B, there exists b ∈ B, b ∈ b and a, a ∈ A such that: a ≤ b ≤ a . By [A] we denote the ∼-equivalence class of A in D(). We let C() be the set of classes of directed subsets of . A directed set X ⊂ is said to be a filter in if x ≤ y with x ∈ X entails y ∈ X . To an arbitrary directed set Y in we associate a filter Y as follows: Y = {λ ∈ , there exists an x ∈ Y such that x ≤ λ}. For A, B in D() we say that A ≤ B if and only if: i. For a ∈ A there is an a ∈ A, a ≤ a, such that a ≤ b for some b ∈ B. ii. For b ∈ B, there is an a ∈ A such that a ≤ b. Lemma 2.1 With conventions and notation as above: 1. For A, B in D(), A ≤ B if and only if B ⊂ A. 2. For A, B in D(), A ∼ B if and only if A ≤ B and B ≤ A, if and only if A = B. 3. The set C() with ≤ induced from D() is a poset such that the canonical map → C(), λ → [{λ}], is a poset monomorphism. 4. If is a lattice with respect to ∧, ∨ then these operations induce a lattice structure on C() such that the canonical poset map → C() is a lattice monomorphism. 5. If is a lattice, then the following properties of transfer to the lattice C(): i. has 0 and 1. ii. is modular. iii. is complete.

2.1 Small Categories, Posets, and Noncommutative Topologies

15

iv. is distributive. v. is Brouwerian, that is, for λ, µ ∈ the set {x ∈ , λ ∧ x ≤ µ} has a largest element then denoted by µ : λ. Proof 1, 2, and 3 are easy enough, and we leave these as exercises. 4. For A and B in D() define: ˙ = {a ∧ b, a ∈ A, b ∈ B}, A∨B ˙ = {a ∨ b, a ∈ A, b ∈ B} A∧B Since ∧ and ∨ are bicontinuous with respect to ≤, it is immediately clear that ˙ as well as A∨B ˙ is in D(). If A ∼ A and B ∼ B in D(), then A∧B ˙ ∼ A ∨B ˙ ˙ ∼ A ∧B ˙ ˙ . So if we define [A] ∧ [B] = [A∧B], and A∨B A∧B ˙ [A] ∨ [B] = [A∨B] in C(), then we obtain a well-defined lattice structure on C() making the canonical poset map → C(), λ → [{λ}] into a lattice monomorphism, as is easily checked. 5. Straightforward verification of the lattice properties considered. Exercise 2.1 1. For any set S we let L(S) be the poset of all subsets of S partially ordered by inclusion. A lattice presentation of a poset is just a poset map π : → L(S) for some set S; observe that L(S) is a lattice. Verify that it is always possible to present a poset by π : → L(), λ → {µ ∈ , µ ≤ λ}. Provide examples to show that π need not be injective or surjective in general. 2. For any poset a subset of the form {µ ∈ , µ ≤ λ} is called an interval of . For a lattice we have a lattice isomorphism I () described by λ ↔ {µ ∈ , µ ≤ λ}, where I () is the lattice of intervals in . 3. If π : → L(S) is a lattice presentation of the poset , then it induces a poset map C(π) : C() → C L(S) fitting in the following commutative diagram of poset maps: Λ C (Λ)

π

C (π)

L (S) CL (S)

Verify also that C(π ) actually defines a lattice presentation of C(), C() → L(C L(S)). Note that we may identify C L(S) and I (C L(S)) in view of Lemma 2.1(4). 4. Can you describe C() in case is the lattice of open sets a. for the real topology on Rn b. for the Zariski topology on Rn

16

Noncommutative Spaces 5. If is the lattice Open(X ) consisting of open sets of a topological space X, τ , can you express the Hausdorff, respectively T1 -property, in terms of the lattice morphism → C()? 6. If 1 , 2 are posets, respectively lattices, is it possible to define a product poset, respectively product lattice, 1 × 2 ? Can you relate C(1 ) × C(2 ) and C(1 × 2 )? In case 1 = Open(X ), 2 = Open(Y ), is 1 × 2 equal to Open(X × Y ) where Y × Y is equipped with the product topology? Relate C(Open(X × Y )) and C(1 ), C(2 ). In case X and Y are varieties with their respective Zariski topologies and X × Y is viewed with the Zariski topology (i.e., this need not be the product topology of the product variety X × Y ), is it true that C(Open(X × Y )) = C(OpenZar (X × Y ))? Is the relation between the lattices C(OpenZar (X )) × C(OpenZar (Y )) and C(OpenZar (X × Y )) essentially different from the relation between the topologies themselves?

2.1.3

Poset Dynamics

When deformations of algebras are introduced, even when using rather general abstract methods such as diagram algebras in the sense of M. Gerstenhaber, there is always some algebra or ring of formal power series in the picture. This is somewhat unsatisfactory because it is really not asking too much to hope that the use of some sheaf-like theory over a suitably general underlying “space” (topology, lattice, poset, diagram, etc.) should allow a visualization of the deformation action and an interpretation in terms of infinitesimal-like phenomena worded in terms of sheaves in suitable categories. Motivated by a question concerning the possibility of developing a dynamic version of the use of causets (posets with respect to a causality order) in the theory of quantum gravity, we propose a notion of poset dynamics. This structure may be viewed as an interesting toy; that is, a generic exercise arises when trying to apply any derived structural result in the usual, that is to say static, theory to the dynamic situation. Let T be a totally ordered poset. A poset T -dynamics consists of a class of posets {Pt , t ∈ T } together with poset maps: ϕtt : Pt → Pt for every t ≤ t in T , satisfying the following conditions: DP.1 For each t ∈ T , ϕtt = I Pt , the identity of Pt . DP.2 For each triple t ≤ t ≤ t in T we have: ϕtt ϕt t = ϕtt (where composition is written in the arrow-order). DP.3 For any t ∈ T and subset F ⊂ T such that for every f ∈ F we have t ≤ f , for every nontrivial x < y in Pt (i.e., x = 0, y = 1) such that for all f ∈ F, ϕt f (x) < z f < ϕt f (y) for some z f ∈ P f , there exists a z ∈ Pt with x < z < y such that ϕt f (x) < ϕt f (z) ≤ z f < ϕt f (y).

2.1 Small Categories, Posets, and Noncommutative Topologies

17

DP.4 For every t ∈ T and nontrivial x < z < y in Pt (i.e., x and y not 0 and 1), there exists a t1 ∈ T , t < t1 , such that for all t ∈ [t, t1 [ we have ϕtt (x) < ϕtt (z) < ϕtt (y) (in other words strict < has a future). DP.5 For every t ∈ T and x < z < y in Pt there exists a t1 ∈ T , t1 < t, such that for all t ∈]t1 , t] such that x , y ∈ Pt with ϕt t (x ) = x, ϕt t (y ) = y exist, we have x < z < y in Pt with ϕt t (z ) = z (in other words strict < has a past for elements with a past). Note the discrepancy between DP.3 and DP.5; in DP.3 we reach Pt but the condition ϕt f (z) ≤ z f is weaker than ϕt t (z ) = z in DP.5, which is only reached on Pt . Under conditions ensuring that the set of z having the property in DP.3 has maximal elements (e.g., ∨-complete, Noetherian posets, etc.), then we may by iteration in DP.3 arrive at a z such that ϕt f (z) = z f and then DP.3 implies DP.5. In that case Imϕt f is convex in f , (ϕt f ([x, y]) = [ϕt f (x), ϕt f (y)]). The remainder of this section is devoted to the introduction of the notion of dimension of a poset. This in turn may be used to define the Krull dimension of an abelian category. Both concepts will have a different meaning in applications, comparable to the difference between the notion of dimension of a topological space and the dimension of a variety or of a linear space. The Krull dimension of a poset was introduced by P. Gabriel and R. Rentschler [11] for ordinal numbers; a generalization for “higher” numbers is obtained later by G. Krause. Let , ≤ be a poset. If λ ≤ µ in , then we write [λ, µ] for the closed interval {α ∈ , λ ≤ α ≤ µ}. The set of intervals is denoted by I(). We may define on I() the following filtration by using transfinite recurrence I−1 () = {[λ, µ], λ = µ} I0 () = {[λ, µ] = I(), [λ, µ] is an Artinian poset}. (Artinian means that the descending chain condition holds with respect to subposets, or equivalently every nonempty family in it has a minimal element). Assuming that Iβ () has already been defined for all β < α, then we define Iα () as follows: Iα () = {[λ, µ], for every sequence λ ≥ λ1 ≥ . . . λn ≥ . . . µ there is an n ∈ N such that [λi+1 , λi ] ∈ ∪β m in N, then there exists an n α ∈ N such that all [Bα ]d with d ≤ n α are different in C() but [Bα ]n α = [Bα ]n α +k for any k ∈ N. Therefore we are either in the situation where all [Bα ]d for d ∈ N are different constituting the interval [[0], [Bα ]] and none of the [Bα ]d is ∧-idempotent, or else [[0], [Bα ]] is finite and [Bα ]n α is ∧-idempotent (and a minimal point) for some n α ∈ N. To every λ ∈ as in Proposition 2.8, there corresponds a set of quantum points {[Bα ], α ∈ Aλ } obtained from the decomposition [λ] = ∨α∈Aλ {[Bα ]n α , α ∈ Aλ } some Aλ ⊂ A; we denote that set by S(λ). The set S(λ) does not define λ; it defines uniquely the quantum closure λcl = ∨ {[Bα ], α ∈ Aλ }, but λ ≤ λcl cannot be reconstructed unless we know the “multiplicities” n α , which measure in some sense the noncommutativity of ; also these numbers measure how quantum points fail to be ∧-idempotent. A [Bα ] not dominating any nonzero ∧-idempotent, that is, if [A] = 0 is such that [A] ≤ [Bα ], then [A] is not idempotent in C(), which is called a radical point (short for radical quantum point). More generally in any an element λ = 0 such that λ does not dominate a nonzero idempotent is called a radical element. If [A] is a radical element of C(), then [A] = ∨ {[Bα ]n α , α ∈ A A } for some n α ∈ N, where all [Bα ], α ∈ A A are radical points.

2.3.1 2.3.1.1

Projects The Relation between Quantum Points and Strong Idempotents

The relation between quantum points and strong idempotents in I∧ (C()) is enigmatic to say the least; therefore mixing conditions expressing generation properties in terms of strong idempotents (using bracketed expressions in ∧ and ∨) and in terms of quantum points (using only expressions in ∨) must lead to complex situations. The final definition in this philosophy should be that of a quantum topology: a topology of virtual opens such that τ , the pattern topology, has a quantum basis (the elements of this should then be logically termed quantum patterns; what’s in a name?). The intrinsic technical problem is that we do not have, in the generality we consider here, any prescribed relation between the intervals [[0], [A]] and [[B], [A] ∨ [B]] in C(). One could put conditions in terms of Kdim, composition series involving critical elements, composition series involving ∧-powers of quantum points, etc. The topologies in classical examples are defined on sets with a lot of extra geometric or algebraic structure (e.g., groups, vector spaces); without fixing such extra structure one should try to find suitable abstract “lattice theory–type” properties that can replace the missing structure.

36

Noncommutative Spaces

2.3.1.2

Functions on Sets of Quantum Points

Generalized points suggest a beginning of spectrum theory, implications in sheaf theory (we shall come back to this later), and the use of set theoretic functions. If A is the set of quantum points (as before [Bα ] with α ∈ A), then C() may be obtained from functions n : A → N, α → n α , corresponding to n the element ∨ {[Bα ]n α , α ∈ A such that n(α) = 0}. Putting [Bα ]o = [0] we obtain a map F(A, N) → C(), which is surjective. We may also study maps between noncommutative topologies and as before, defined by functions on the sets of quantum points. Such functions do not necessarily lead to a well-defined map from C() to C( ) because the expression of [A] ∈ C() as a ∨ of ∧-powers of quantum points may not be unique; some care is necessary here. Determining formal links between sets of quantum points and functions on these and noncommutative topological features is an interesting project. Obvious links to sheaf theory and spectral theory can be exploited. Using functions to sets with extra structures may lead to interesting examples carrying algebra-like structures; for example, look at the appearance of P(H ), the projective Hilbert space, in the case of L(H ) studied in Section 2.7, as an effect of the linearity of the theory.

2.4

Presheaves and Sheaves over Noncommutative Topologies

In this section we consider a noncommutative topology . In view of Proposition 2.2 and Lemma 2.12, T (), C(), τ are noncommutative topologies too. We are given a category C and we are interested in sheaves over , C(), τ, . . . and relations among these; the category C is rather arbitrary. C must have sums, products, limits, and its objects are assumed to be sets. In Section 2.1.3, separated presheaves and sheaves over general posets were introduced. We apply these to , C(), T (), τ, . . . lim Given a presheaf P : o −→ C. Consider [A] ∈ C() and define: P[A] = −→ P(a) a∈A

lim

lim

in C. Observe that for B ∼ A we do obtain that −→ P(a) = −→ P(b), so that the a∈A

definition of P[A] is well defined. So we may look at P[ P[ ] ([A]) = P[A] for all [A] ∈ C(). Lemma 2.19 If P is a presheaf, then P[

]

b∈B

]

: C()o → C defined by

is a presheaf on C().

Proof If [A] ≤ [B], then for all b ∈ B there is an a ∈ A such that a ≤ b; hence there is lim lim a canonical map −→ Pb → −→ Pa , induced by the restriction maps ρab : Pb → Pa b∈B

a∈A

[B] defined by the presheaf P. We shall write ρ[A] for the map defined before. In case we [B] have [B] = [A] we do obtain that ρ[A] is the identity map 1 P[A] . For [C] ≤ [B] ≤ [A] [B] [A] [A] it is easy enough to check that the usual composition formula holds: ρ[C] ρ[B] = ρ[C] .

2.4 Presheaves and Sheaves over Noncommutative Topologies Corollary 2.6 Let P[ ] | be the restriction of P[ C(), λ → [λ], then P[ ] | = P.

]

37

to via the canonical inclusion →

Theorem 2.1 With notation and conventions as before. i. If P is a separated presheaf, then P[ ii. If P is a sheaf over , then P[

]

]

is a separated presheaf.

is a sheaf over C().

Proof i. Let P be a separated presheaf over and look at a finite cover {[Aα ], α ∈ A} of [A] in C(). Suppose that y, x in P[A] are such that ρ[α] (x) = ρ[α] (y), for [A] . Let x, respectively y in lim−→ P(a) all α ∈ A, where we write ρ[α] for ρ[A α] α∈A be represented by {xa ; a ∈ A, xa ∈ P(a)} respectively {ya ; a ∈ A, ya ∈ P(a)}. Since for all α ∈ A we have [A] ≥ [Aα ], there are aα ∈ Aα , aα ≤ a for a given a ∈ A. Fixing a ∈ A and xa , ya in P(a), ρaaα (xa ) = ρaaα (ya ) for some ˙ α aα ∈ Aα because we have that ρ[α] (x) = ρ[α] (y). Now ∨{aα , α ∈ A} ∈ ∨A ˙ α , so we may replace the aα by bα ∈ Aα such that bα ≤ aα and A ∼ ∨A and there are a and a in A such that a ≤ ∨{bα , α ∈ A} ≤ a . Then pick ˙ α ∼ A it a1 ∈ A such that a1 ≤ a and a1 ≤ a (A is directed). Again from ∨A follows that we may find bα , α ∈ A, bα ∈ A, bα ∈ Aα such that bα ≤ γα and ˙ α is such that ∨{bα , α ∈ A} ≤ a1 ≤ a. Similarly we may ∨{bα , α ∈ A} ∈ ∨A find bα ∈ Aα , bα < bα for all α ∈ A, such that there exist a2 and a2 in A such that a2 ≤ ∨bα ≤ a2 , but ∨bα ≤ ∨bα ≤ a1 ≤ a. For convenience, rewrite bα for bα and put b = ∨{bα , α ∈ A}. Since P is separated and ρbbα ρba (xa ) = ρbbα ρba (ya ) for all α ∈ A, it follows that ρba (xa ) = ρba (ya ). Since we may choose a2 ∈ A such that a2 ≤ b, we get ρaa (xa ) = ρaa (ya ); thus the classes of xa and ya 2 2 coincide in lim−→ P(a), i.e. x = y follows as desired. a∈A

ii. Start from a cover [A] = [A1 ] ∨ · · · ∨ [An ] in C() and suppose there is given a set x α ∈ P[Aα ] , α = 1, . . ., n, such that, writing [C] for either [Aβ ] ∧ [Aα ] or [Aα ] ∧ [Aβ ] we have the gluing condition: (∗)

[Aα ] α (x ) = ρ[C]β (x β ), ρ[C] [A ]

for α, β ∈ {1, . . ., n}

Let x α , respectively x β , be represented by xaαα , respectively xaββ for aα ∈ Aα , ˙ β ], condition (*) yields: respectively aβ ∈ Aβ . Since [Aα ] ∧ [Aβ ] = [Aα ∧A (•)

ργaα (xaαα ) = ργaβ (xaββ ) for some γ = aα ∧ aβ

with aα ∈ Aα , aβ ∈ Aβ . In a similar way: (••)

a

ργ α (xaαα ) = ργ β (xaββ ) for some γ = aβ ∧ aα a

38

Noncommutative Spaces with aα ∈ Aα , aβ ∈ Aβ . Take bα ≤ aα , aα and bβ ≤ aβ , aβ , put γαβ = ˙ β , respectively Aβ ∧A ˙ α . Since γαβ ≤ γ the bα ∧ bβ , γβα = bβ ∧ bα in Aα ∧A equality (•) also holds with γ replaced by γαβ and (••) holds with γ replaced by γβα . Since we considered a finite cover we may repeat the foregoing argument for all pairs Aα , Aβ until we obtain a set {a1 , . . ., an } and γαβ = aα ∧ aβ in ˙ β , γβα = aβ ∧ aα in Aβ ∧A ˙ α such that the equalities (•) and (••) hold Aα ∧A for all α and β in {1, . . ., n} with respect to γαβ , respectively γβα . Since P is a sheaf on it follows that there exists a z ∈ P(τ ), τ = a1 ∨ . . . ∨ an such that ρaτα (z) = xaαα for all α ∈ {1, . . ., n}. ˙ . . . ∨A ˙ n defines an element of the class [A] = [A1 ]∨. . .∨[An ] Now τ ∈ A1 ∨ [A] (x) and x α in and z defines an element, say x, in P[A] . Now both elements ρ[A α] τ τ P[Aα ] are represented by ρaα (xτ ), respectively ρaα (z) for all α ∈ A = {1, . . ., n}. Since the latter are the same and P[ ] is separated in view of i, it follows that xτ = z and x is the unique element of P[A] with the desired restrictions in each P[Aα ] , α = 1, . . ., n.

Observe that a separated presheaf P on does not necessarily induce a separated presheaf on S L() with respect to the operation ∧. in id∧ ()! We shall return to the sheaf theory in subsequent sections. The notion of sheaf as we have used it so far presents a drawback; indeed if we start from a cover of λ ∈ , say λ = λ1 ∨ . . . ∨ λn , and µ ≤ λ in , then µ = (µ ∧ λ1 ) ∨ . . . ∨ (µ ∧ λn ) in general. Even if is a lattice but not distributive we meet this problem. Consequently, the gluing condition in the definition of a sheaf is too strong. Similar to the trick used in defining Grothendieck topologies we may restrict the covers used in the definition of a sheaf, and for example, only consider covers of λ induced by finite global covers in . Let us write the more general definition in its functorial form; we do not demand that objects of C are sets from now on, but we do assume that C is a Grothendieck category. Consider a presheaf P : o → C. Obviously presheaves on with values in C together with presheaf morphisms make up a category Q(, C). For λ ∈ we let Cov (λ) consist of those covers for λ induced by a finite global cover of . We make Cov (λ) into a category by defining U → V if U = (λi )i∈I , V = (µ j ) j∈J and there is a map ε : I → J such that λi ≤ µε(i) for all i ∈ I . Given P and λ ∈ we define a contravariant functor [P, λ] : Cov (λ) → C as follows. First we have for U = (λi )i∈I in Cov (λ) projection morphisms: pi :

P(λi ) → P(λi ),

i∈I

then we have a morphism j : P(λ) → morphisms: pl , ql , pr , qr :

i∈P

P(λi )

i∈I

P(λi ) such that pi j = ρλλi . We also have ( j,k)∈I ×I

P(λi ∧ λ j )

2.4 Presheaves and Sheaves over Noncommutative Topologies λ

39

λ

where the ( j, k)-component of pl is ρλ jj ∧λk , of ql is ρλλkj ∧λk , of pr is ρλkj∧λ j and of qr is ρλλkk∧λ j , corresponding to a diagram in C:

P(λ j ) −→ P(λ j ∧ λk )

P(λk ) −→ P(λk ∧ λ j ) Thus, in C, we obtain a diagram, for (λi )i∈I in Cov (λ): P(λ) → j

(∗)

P(λi )

P(λ j ∧ λk )

( j,k)∈I ×I

i∈I

Definition 2.4 The presheaf P is a sheaf if and only if the diagrams (*) are equalizer diagrams. Now we put [P, λ](U) equal to the kernel of

P(λi )

P(λ j ∧ λk ).

( j,k)∈I ×I

i∈I

We can define L P ∈ Q(, C) by putting L P : o → C, λ →

lim

−→ U∈CoJ (λ)

([P, λ](U))

Lemma 2.20 The notions of separated presheaf and sheaf are now defined with respect to the fixed type of covers. If P is a separated presheaf, then the canonical morphism P → L P in Q(, C) is a monomorphism and L P is a sheaf; hence the functor L L may be viewed as a sheafification functor. Proof Similar to the classical case. The foregoing lemma (in particular its proof) depends heavily on axiom A.10; therefore sheafification does not necessarily work well even for lattices. This can be seen in Section 2.7 for the lattice L(H ) of a Hilbert space H . However, it turns out that for sheaf theoretic applications the absence of A.10 may be compensated for by the existence of enough quantum points (cf. Section 2.3).

2.4.1

Project: Quantum Points and Sheaves

In the situation of Definition 2.3, suppose has a weak quantum basis {[Bα ], α ∈ A} and consider a separated presheaf on , say P. Look at the separated presheaf P[]

40

Noncommutative Spaces

on C(). We have ∨{[Bα ], α ∈ A} = 1 and for all α, β in A, [Bα ] ∩ [Bβ ] = 0; up to self-intersections the [Bα ] allows us to reconstruct [λ] ∈ C() with λ ∈ . Starting from the product α∈A P[Bα ] , of all stalks at the [Bα ], imitate the construction of the e´ tale space of a separated presheaf P[ ] in two ways; first by putting restrictions on P suchthat P[Bα ] = P[Bα ] n], and then more generally by replacing the product above by α∈A P[Bα ] n α where [Bα ]n α is the idempotent dominated by [Bα ]. In case radical elements exist, it is quite harmless to assume that the stalk representing a radical element is the global P[0] . Study the (partial) sheafification results that follow. Connect this to Projects 2.3.1.1 and 2.3.1.2 and ii. Observe that for the example L(H ), introduced in Section 2.7, this leads to the classical sheafification on the Stone space (see also later for generalizations of the Stone space). Observe that x, y ∈ P[Bα ] m such that x = y but ρααm (x) = ρααm (y) in the P[Bα ] may exist. So the e´ tale space in the second approach will be a quotient of α∈A P[Bα ] identifying elements that are restricted to the same element after n self-intersections for some n ∈ N.

2.5

Noncommutative Grothendieck Topologies

In the definition of noncommutative topology we have included some conditions about covers, such as A.10. When trying to fit the noncommutative topologies into the framework of Grothendieck topologies this fact will pay off. First recall that a Grothedieck topology is a category C such that for every object x of C a set cov(x), consisting of subsets of morphisms with target x, is given such that the following conditions are satisfied: G.1. {x → x} ∈ cov(x). G.2. If {xi → x, i ∈ J } ∈ cov(x) and {xi j → xi , j ∈ J } ∈ cov(xi ) for all i ∈ I, then {xi j → x, i ∈ J , j ∈ J } ∈ cov(x), where xi j → x is obtained from the xi j → x j → x. G.3. If {xi → x, i ∈ J } ∈ cov(x) and x → x in C, then there exists a pull-back x ×x xi in C such that {x ×x xi → x , i ∈ J } ∈ cov(x ), which is called the fibre product over x. Now look at a noncommutative topology, say. For the objects of C there is little choice but to take λ ∈ . Since we have to be able to induce covers, we cannot just let any relation λ ≤ µ be a morhphism λ → µ in C. A first idea could be to allow only focused relations λ ≤ µ (as defined after Section 2.2.1.); however, this leads to problems such as if λ1 ∨ . . . ∨ λn is a global cover and x ≤ x is focused then we do get a diagram: x ^ λi

x

x' ^ λi

x'

2.5 Noncommutative Grothendieck Topologies

41

in the category C but x ∧ λ ≤ x ∧ λi is not necessarily focused; that is, it may happen that (x ∧ λi ) ∧ (x ∧ λi ) = x ∧ λi . We may solve the problems by looking at generic relations. A relation λ ≤ µ in is said to be generic if it is a consequence of the axioms of a noncommutative topology; that is a ∧ b ≤ a is a generic relation. When a and b are idempotent in λ, any relation that is just given as a ≤ b is not viewed as generic; however, if b = a ∨ c, then a ≤ b is viewed as generic. Now if 1 = λ1 ∨. . .∨λn is a global cover and x ≤ x is generic, then x ∧λi ≤ x ∧λi is generic too, so if we define C g to be the category with objects λ ∈ and generic relations for the morphisms (with x = x representing 1x ), then the diagram above is a diagram of morphisms in C g . A new problem appears, for example, for x ≤ x and 1 = λ1 ∨ . . . ∨ λn as above; for given morphisms t → x ∧ λi , t → x we do not find a generic relation t ≤ x ∧ λi . But from the philosophy of patterns we may learn that it is not natural to ask for t → x ∧ λi ; we should ask for some morphism between elements having a similar pattern. This is at the basis of the following definition. Definition 2.5 A category C with cov(x) defined for every object x of C as before is said to be a noncommutative Grothendieck topology if the following conditions hold: G.1. and G.2. as before, and the new condition: G.3. nc for given x → x and {xi → x, i ∈ J } ∈ cov(x) there is a cover {x ×x x → x , i ∈ J } ∈ cov(x ) satisfying the following pull-back property: for s → xi , s → x and t → xi , t → x there exist s ∧ t → x ×x xi and t ∧ s → x ×x xi fitting in a commutative diagram: x xi

x' x' ×

t

x

s

xi

t^s

s^t

The particular case when s = t yields:

t

xi

x

x' × x xi

x'

t^t

Clearly when t is idempotent, then the diagram reduces to the usual pull-back diagram in G.3. Observe that the noncommutative version of a Grothendieck topology is obtained by a complete symmetrization of the classical definition.

42

Noncommutative Spaces

Theorem 2.2 Let X be either , C(), τ , so certainly a noncommutative topology and let C g be constructed on X with respect to generic relations as before Definition 2.5. For the covers of x in X we take covl (x) = {x ∧ λi → x, {λ1 , . . ., λn } a global cover}. Then X becomes a noncommutative Grothendieck topology. Proof Look at a cover {xi = x ∧ λi → x, i = 1, . . ., n} ∈ covl (x) and x → x. Put x ×x xi = x ∧ λi , for i = 1, . . ., n. Since x → x, x ≤ x is generic; thus x ∧ λi ≤ x ∧ λi is generic too. We obtain the following diagrams in C g : x ^ λi

x

x' × x xi = x' ^ λi

x'

Suppose s → x ∧ λi , s → x and t → x ∧ λi , t → x are given. Since x ∧ λi ≤ λi is generic by definition and s → x ∧ λi , t → x ∧ λi are morphisms in C g , it follows that s ≤ x ∧ λi ≤ λi and t ≤ x ∧ λi ≤ λi yield generic relations s ≤ λi , t ≤ λi . Since s ≤ λi and s ≤ x are generic and similarly t ≤ λi , t ≤ x are generic, it follows that s ∧ t ≤ x ∧ λi and t ∧ s ≤ x ∧ λi are generic. Hence we arrive at the morphism in C g , s ∧ t → x ∧ λi = x ×x xi and t ∧ s −→ x ∧ λi = x ×x xi , fitting nicely in nc

the diagram defining the condition G.3 (G.1 and G.2 are obvious), so X (as C g ) is a noncommutative Grothendieck topology. Remark 2.1 1. Using covl we refer to X with this structure as the left topology of (or T , or C(), τ ); in a similar way the right topology can be defined. 2. The advantage of using generic relations is obvious: the generic relations are recognizable on sight; they transfer well from to C() and τ . The fact that not every generic relation needs to be focused does not interfere with that. 3. When viewing , C(), τ with the noncommutative Grothendieck topology structure, we shall write g , C()g , τ g , respectively. Observation 2.3 A cover λ = λ1 ∨ . . . ∨ λn in X (as before) is automatically a generic cover because λi ≤ λ is generic for i = 1, . . ., n. Now it is clear how presheaves, separated presheaves, or sheaves are defined over X with respect to generic covers. Indeed the operations appearing in the separateness and the sheaf property are compatible with the restriction to generic relations and generic covers. When passing from to C() or τ we may look at a stronger restriction on the covers (much like the strong idempotents replacing the idempotents). For [A], [B] in C() the relation [A] ≤ [B] is generically defined if for all b ∈ B there exists an a ∈ A such that a ≤ b is generic. A cover [A] = ∪α [Aα ] is generically defined if [Aα ] ≤ [A] is generically defined for all α.

2.5 Noncommutative Grothendieck Topologies

43

Lemma 2.21 For C() consider the category C()gd consisting of the objects [A] ∈ C() with generically defined relations for the morphisms; then C()gd is a noncommutative Grothendieck topology. Proof The covers used are generically defined covers; G.1 and G.2 are obvious. For G nc 3 the proof in Theorem 2.2 remains valid if we verify that a generically defined relation a ≤ b yields a generically defined x ∧ a ≤ x ∧ b for any x. Put a = [A], b = [B] and x = [C] in C(). Since [A] ≤ [B] is generically defined, we find for every b ∈ B some a ∈ A with a ≤ b generic. For any x ∈ C we then have a generic ˙ there is an a ∧ c ∈ A∧C ˙ such relation a ∧ c ≤ b ∧ c; hence for every b ∧ c ∈ B ∧C that a ∧ c ≤ b ∧ c is generic. Consequently, [a] ∧ [C] ≤ [B] ∧ [C] is generically defined. Proposition 2.9 If P is a presheaf on g , then P[ ] is a presheaf on C()gd . Moreover: i. If P is separated, then P[ ] is separated. ii. If P is a sheaf, then P[ ] is a sheaf on C()gd Proof That P[ ] is now a presheaf on C()gd follows from the (proof of) Lemma 2.19. i. The same proof as in Theorem 2.1 i holds still if one verifies at each step that the relations may be chosen to be generic; then the only operations occuring are ∧ or ∨ and these do not change the generic character. ii. Same as in i, checking along the lines of Theorem 2.1 ii. Corollary 2.7 The canonical inclusion → C(), λ → [λ] defines a faithful g → C()gd . Indeed a generic relation λ ≤ µ does define a generically defined [λ] ≤ [µ]. The restriction of P[ ] to g is P. Corollary 2.8 The pattern topology τ ⊂ C() is also a noncommutative Grothendieck topology, written τ g . Defining τ gd in the same way as C()gd , this also defines another noncommutative Grothendieck topology and a faithful functor τ gd → C()gd . Presheaves on C()gd restrict to τ gd in such a way that separateness and the sheaf property are respected.

2.5.1

Warning

If a and b are idempotents in a topology of virtual opens , then the fact that a ≤ b is generic in need not imply that a ≤ b is generic in S L()! For example if

44

Noncommutative Spaces

a = (b∨c)∧(d ∨c), then c∧c ≤ a is generic in but c and d are not even idempotent, so we must relate a and c via something like (b ∨ c)∧. (d ∨ c). This bad behavior of genericity with respect to the commutative shadow prompts us to work with sheaves on the lattice-type noncommutative topology, avoiding the Grothendieck topologies here for the moment, at least when one aims to relate and S L(). A good theory of sheaves over noncommutative Grothendieck topologies probably has to be developed in connection with noncommutative topos theory. If one studies noncommutative Grothendieck topologies without reference to another noncommutative topology from which it stems, the sheaf theory is of independent interest.

2.5.2 2.5.2.1

Projects A Noncommutative Topos Theory

What structure fits the philosophical equation: locales topoi = quantales ? The answer should lead to some version of noncommutative topos theory; however, let me point out that it is not clear to this author whether the above question is the correct one to put forward; quantales are too tightly related to C ∗ -algebras to obtain the right level of generality perhaps; nevertheless, the search for a noncommutative topos is worthwhile in its own right. One easily finds that the first main problem is to circumvent the notion of subobject classifier. A first generalized theory may be constructed by maintaining the “up to self-intersections” philosophy; a second approach may be to allow a family of “subobject classifiers” defined in a suitable way. It is clear that noncommutative topology reflects a kind of ordered logic; the ordering reflects the fact that “x ∈ A” and “x ∈ B” cannot be realized at the same time. We have not yet tried to write down a foundation for such ordered logic, but certainly a notion of noncommutative topos would fit perfectly in this. We leave the development of noncommutative topos theory as a project here; nevertheless we do know the main examples to be covered by such a theory: sheaves over a noncommutative Grothendieck topology, in particular those constructed on a generalized Stone space or even more specifically on a noncommutative Grothendieck topology constructed from a quantum topology (because of the presence of suitable sheafification techniques). Detail on topos theory may be found in R. Goldblatt’s [15]. 2.5.2.2

Noncommutative Probability (and Measure) Theory

The mathematician may be completely satisfied with the foundation of probability theory knowing that it means exactly what it means and no more! There is more than one question to be raised concerning certain applications in the real world or what passes for that frequently. You can only throw the dice until they crack! The certainty that some event in a given selection must happen (and to associate to this a number related to a total number of possible events in the selection) is rather ill founded. The notion of the “time necessary for some events to actually happen” is neglected, and at the level of theoretical foundation probability is based on the membership relations

2.6 The Fundamental Examples I: Torsion Theories

45

of set theory to be time or ordering independent. The idea of a noncommutative space is not consistent with such a probability theory, so it must be constructed ab initio in a noncommutative way too. An intersection A ∧ B in a noncommutative space should be related to a conditional type of probability in the sense that the probability for λ ≤ A ∧ B is expressed by p(λ, A) p(A, B) p(λ, B) where p(A, B) expresses the probability correction for A before B. Since a σ -algebra for some noncommutative topology may be defined in a straightforward way, noncommutative versions of Borelstems, etc. are not hard to develop. Probabilities could be taken in the free semigroup (generalizing N) over the set , making multiplication of probabilities formal and noncommutative but perhaps also unnatural. The approach suggested above, that is using the conditional approach retains a classical flavor, again the self-intersection introduces new phenomena, for example, p(A, A) may be nontrivial, that need to be fully integrated in the theory. 2.5.2.3

Covers and Cohomology Theories

Traditionally the idea of Grothendieck topology and the extra abstraction in the notion of cover allows us to introduce new interesting (co)homology theories; recall the use of e´ tale covers and e´ tale cohomology. In [46] and [50] we did a similar experiment ˇ with respect to Cech cohomology, which led to some interesting results in the algebraic theory of noncommutative geometry. For example, a result of L. Le Bruyn concerning the moduli space of left ideals in Weyl algebras has been reduced to a ˇ fairly straightforward calculation of Cech cohomology on the noncommutative site created from the noncommutative topology phrased in terms of Ore sets in the algebra considered. Similarly, V. Ginzburg and A. Berest have used the same technique in another situation. Of course one should be tempted to develop a noncommutative e´ tale cohomology or more cohomology theories with respect to other types of covers. Since examples of noncommutative topologies may be constructed in a completely functorial way (see also Chapter 4) one may start a theory from the consideration of covers by separable functors in the sense of Section 1.3. 2.5.2.4

The Derived Imperative

For compactness sake derived categories and derived functors have not been introduced in these notes; consequently, our sheaves are not perverse. Clearly, the latter are popular topics nowadays and they also provide strong methods of analysis, for example, in connection with rings of differential operators, Riemann-Hilbert correspondence, and so forth. It is a promising idea to combine the techniques of derived categories, perverse sheaves, and so forth with the noncommutative topology point of view. We say no more about this here.

2.6

The Fundamental Examples I: Torsion Theories

We need to recall the basic facts about torsion theory. What we say will be valid for an abelian category that is assumed to be complete, co-complete, and locally small, but we shall restrict attention to Grothendieck categories for convenience.

46

Noncommutative Spaces

Let C be a Grothendieck category. A preradical ρ of C is just a subfunctor of the identity functor. The class of preradicals of C, Q say, is partially ordered by ρ1 ≤ ρ2 if and only if ρ1 (C) ⊂ ρ2 (C) for all objects C of C. Any family of preradicals {ρα , α ∈ A} has at least an upperbound ∨ρα and a greatest lower bound ∧ρα defined in the obvious way. Consequently, Q(C) is a complete lattice with respect to ∧ and ∨. For preradicals ρ1 and ρ2 on C we may also define ρ1 ρ2 by putting ρ1 ρ2 (C) = ρ1 (ρ2 (C)) for all c ∈ C; we define ρ1 : ρ2 by taking (ρ1 : ρ2 )(C) for C ∈ C, to be the subobject of C for which we have (ρ1 : ρ2 )(C)/ρ1 (C) = ρ2 (C/ρ1 (C)). Definition 2.6 A preradical ρ such that ρρ = ρ is said to be idempotent. We say that a preradical is a radical if (ρ : ρ) = ρ; in other words ρ is radical if ρ((C)/ρ(C)) = 0 for C ∈ C. To a preradical ρ of C we may associate a preradical ρ −1 of C o by defining ρ −1 (X ) = X/ρ(X ); we call ρ −1 the dual preradical of ρ. It is easy to establish that if ρ is idempotent, respectively radical, then ρ −1 is respectively radical, idempotent. A preradical ρ of C gives rise to two classes of objects in C: Fρ = {C ∈ C, ρ(C) = 0} (pretorsion free class) Tρ = {C ∈ C, ρ(C) = C} (pretorsion class) Observe that Fρ = Tρ −1 objectwise. The relation between ρ and these classes is well summarized in the following. Theorem 2.3 With notation and conventions as above: i. Tρ is closed under quotient objects and coproducts; Fρ is closed under subobjects and products. ii. If T ∈ Tρ and F ∈ Fρ , then HomC (T, F) = 0. iii. Idempotent preradicals of C correspond bijectively to pretorsion classes of objects of C, that is, classes that are closed under quotient and coproducts. iv. Radicals of C correspond bijectively to pretorsion-free classes of objects of C, that is, classes that are closed under subobjects and products. v. For every ρ ∈ Q(C) there exists a largest idempotent preradical ρ o ≤ ρ and a smallest radical ρ c ≥ ρ. Proof i. Let us establish the first claim; the second follows by duality. That Tρ is closed under quotient objects is easily seen. Look at a family {Cα , α ∈ A} of objects in C and in Tρ . Because ρ(Cα ) = Cα for all α ∈ A, the Cα map under the canonical monomorphism Cα → ⊕α Cα into ρ(⊕α Cα ). The universal property of ⊕ then leads to the conclusion that ρ(⊕C2 ) = ⊕Cα ; that is, ⊕Cα is in Tρ too.

2.6 The Fundamental Examples I: Torsion Theories

47

ii. If f : T → F is a nonzero morphism, then Imf is in Tρ because of i, but since it is also a subobject of F, Imf = 0 follows, so no nonzero f can exist. iii. Consider a pretorsion class T . To an arbitrary object C of C we associate t(C) ∈ C by considering t(C) to be the sum of all subobjects of C that are objects of T ; then t(C) ∈ T because T is closed under coproducts and quotient objects. It is easy to see that t is an idempotent preradical of C. It is also clear that Tt = T . Now if we start with an idempotent preradical ρ, Tρ is a pretorsion class in C because of i; the idempotent preradical tρ associates to C in C the largest subobject C of C such that ρ(C ) = C , but that is exactly ρ(C); hence ρ = tρ . iv. Dual to iii. v. Starting from a preradical ρ of C we define a pretorsion class Tρ and an idempotent preradical tρ defined (see proof of iii.) by taking tρ (C) to be the largest subobject of C, say C , such that ρ(C ) = C . Therefore tρ (C) ⊂ ρ(C) for all C ∈ C, that is, tρ ≤ ρ in Q(C), and tρ is clearly the largest idempotent preradical of C with this property, so we may put ρ o = tρ . The second claim, concerning ρ o , follows by duality. A closure operator on a complete lattice L is a map (−)c : L → L , λ → λc , satisfying the following: c.1 If λ ≤ µ in L then λc ≤ µc c.2 For λ ∈ L we have that λ ≤ λc c.3 For λ ∈ L we have that (λc )c = λc . The set of closed elements of L, that is, those λ for which we have λ = λc , forms a complete lattice L c with respect to ≤ and as in L but with ∨ defined by ∨λα = (∨λα )c . Observation 2.4 If L is a complete modular lattice with closure operator (−)c satisfying (λ ∧ µ)c = λc ∧ µc , then L c , ≤, ∧, ∨ is a complete modular lattice too. Proof If λ, µ, γ are closed elements with λ ≤ µ then: µ ∧ (γ ∨λ) = µc ∧ (γ ∨ λ)c

= = =

(µ ∧ (γ ∨ λ))c (by the assumption) (λ ∨ (γ ∧ µ))c (L is modular) λ∨(γ ∧ µ)

Now considering Q(C), it is clear that (−)c : Q(C) → Q(C) is a closure operator. Therefore the foregoing observation implies that the idempotent radicals, respectively the radicals, form a complete lattice. Indeed Q(R) satisfies the condition (λ ∧ µ)c = λc ∧ µc as is easily verified and the second statement follows by duality.

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Proposition 2.10 i. If ρ is idempotent, then so is ρ c . ii. If ρ is a radical, then so is ρ c . Proof It suffices to establish i; then ii follows by duality. From ρρ = ρ we have to establish that ρ c ρ c = ρ c . Now for C ∈ C ρ c (C) is the smallest subobject of C such that ρ(C/ρ c (C)) = 0, and ρ c ρ c (C) is the smallest subobject of ρ c (C) such that ρ(ρ c (C)/ρ c ρ c (C)) = 0. Look at C ⊃ ρ c (C) ⊃ ρ c ρ c (C). If T ⊂ C is such that T /ρ c ρ c (C) is in Tρ , then also T + ρ c (C)/ρ c (C) ∈ Tρ and therefore T ⊂ ρ c (C). The latter yields that T /ρ c ρ c (C) is in Tρ and thus T = ρ c ρ c (C). Therefore we arrive at C/ρ c ρ c (C) ∈ Fρ . On the other hand, if D ⊂ ρ c ρ c (C) is such that C/D is in Fρρ = F, then (ρρ)c (C) ⊂ D and then ρ c (C) ⊂ D, but this entails that ρ c (C) = ρ c ρ c (C). Proposition 2.11 For ρ ∈ Q(C), the following are equivalent: i. ρ is left exact. ii. For every subset D of C in C, ρ(D) = ρ(C) ∩ D. iii. ρ is idempotent and Tρ is closed under subobjects. Proof Easy enough to be left as an exercise. A pretorsion class closed under subobjects is said to be hereditary; hence, by the proposition, hereditary pretorsion classes correspond bijectively to left exact preradicals. Note that the operation ρ1 ρ2 in Q(C) is noncommutative. Note also that (ρτ )−1 and τ −1 ρ −1 (in C o ) are different preradicals. Whereas duality works perfectly when ∨ and ∧ are being considered, it breaks down for the noncommutative operations. One may interpret this as if ρτ tries to be a noncommutative intersection while τ −1 ρ −1 tries to be a noncommutative union. Phrasing this in (C) = Q(C)o we reobtain the possibility of using a commutative union stemming from ∧ in Q(C) and a noncommutative “intersection” (when viewed in Q(C)o ) stemming from the preradical product τ −1 ρ −1 for τ, ρ ∈ Q(C). We shall make this more precise in Section 3.1. First we look now at those preradicals that are both idempotent and radical. Definition 2.7 A torsion theory for C is a pair (T , F) of classes of objects from C such that: tt1 For T ∈ T , F ∈ F, HomC (T, F) = 0. tt2 If HomC (C, F) = 0 for all F ∈ F then C ∈ T . tt3 If HomC (T, C) = 0 for T ∈ T then C ∈ F.

2.6 The Fundamental Examples I: Torsion Theories

49

We say that T is a torsion-class of C and its objects are (T , F)-torsion objects of C, while F is the (T , F) torsion-free class. A given class M in C cogenerates a torsion theory (T , F), which is the smallest torsion-free class containing M, FM = {F in C, HomC (C, F) = 0 for all C ∈ M} TM = {T in C, HomC (T, F) = 0 for all F ∈ FM } Proposition 2.12: Characterization of Torsion Classes a. For a class T of objects of C the following are equivalent: i. T is the torsion class of some torsion theory. ii. T is closed under quotient objects, coproducts, and extensions; that is, for every exact sequence 0 → C → C → C → 0 in C with C and C in T , then C ∈ T . b. For a class F of objects of C the following are equivalent: i. F is a torsion-free class for some torsion theory of C. ii. F is closed under subobjects, products, and extensions. Proof Old hat, see among others B. Stenstr¨om, Rings of Quotients, Springer Verlag, Heidelberg, 1975 [42]. A torsion theory (T , F) is in particular pretorsion, so it defines an idempotent preradical τ , which in view of the fact that T is closed under extensions, is easily seen to be a radical. Conversely, given an idempotent radical τ in Q(C), it determines a torsion theory of C by Fτ = {C in C, τ (C) = 0}, Tτ = {C in C, τ (C) = C}. Proposition 2.13 Torsion theories correspond bijectively to idempotent radicals; if ρ is an idempotent preradical, then ρ c is the idempotent radical corresponding to the torsion theory generated by Tρ . A torsion theory (T , F) is said to be hereditary if T is closed under submodules. In view of Proposition 2.11 and the foregoing proposition, it follows that there is a bijective correspondence between hereditary torsion theories and left exact radicals. If C is a Grothendieck category with enough injectives, then we can characterize hereditary torsion theories by the fact that (T , F) is hereditary if and only if F is closed under injective envelopes. Proposition 2.14 If M is a class closed under subobjects and quotient objects (and C is as mentioned above), then the torsion theory generated by M is hereditary.

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Noncommutative Spaces

Proof Suppose F is torsion free and assume there is a nonzero f : C → E(F) for some C ∈ M where E(F) is the injective envelope of F. Then Imf ∈ M and F ∩ Imf is a nonzero subobject of F belonging to M as the latter is closed under subobjects—a contradiction. Corollary 2.9 i. If ρ is a left exact preradical, then ρ c is also left exact. ii. If ρ is a left exact preradical, then ρ(C) is an essential subobject of ρ c (C); that is, for every subobject D, nonzero, in ρ c (C) we have that D ∩ ρ(C) is nonzero. Proof i. By assumption Tρ is a hereditary pretorsion class. The foregoing proposition (and the proof of iii in Theorem 2.3) yields that ρ c is left exact. ii. Suppose D ∩ ρ(C) = 0. Then ρ(D) = 0; hence ρ c (D) = 0, but ρ c (D) = ρ c (C) ∩ D, and the latter is just D, thus D = ρ(D) = 0 follows. Note that a Grothendieck category with a generator has enough injectives. Definition 2.8 A left exact idempotent radical is called a kernel functor. It is clear from the foregoing that kernel functors correspond bijectively to the hereditary torsion theories. If κ denotes a kernel functor, then (Tκ , Fκ ) stands for the corresponding hereditary torsion theory. An object E of C is said to be κ-injective if every exact diagram in C with C ∈ Tκ , may be completed by a morphism g : C → E, such that gi = f . C'

0

i

f

C

C''

0

g

E

If g as above is unique as such, then E is said to be faithfully κ-injective. Proposition 2.15 The following statements are equivalent: 1. E is κ-injective and κ-torsion free. 2. E is faithfully κ-injective. Proof Consider the following exact diagram in C: 0

C' f

g

E

with c ∈ Tκ .

C

i

p

C''

0

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51

Since E is κ-injective at least one morphism g : C → E, such that gi = f , must exist. Suppose g1 , g2 both have that property, then (g1 − g2 )i = 0; hence g1 − g2 factorizes through C ; that is, there is a morphism h : C → E such that g1 −g2 = hp. Now C ∈ Tκ , R ∈ Fκ yields h = 0 or g1 = g2 . This establishes the implication 1. ⇒ 2. Conversely, consider the diagram in C: κ(E)

0

κ(E)

0

E

Since κ(E) ∈ Tκ , there is a unique extension of the zero map 0 → E to κ(E), which therefore has to be the zero map too! However, since κ(E) → E is a monomorphism it follows that κ(E) = 0. Proposition 2.16 Look at the exact sequence in C: E'

0

i

p

E

E''

0

where E is κ-injective and E is κ-torsion free, and E is κ-torsionfree. Then E is κ-injective too. Proof Consider the following diagram for a given morphism f : C → E , where the rows in the diagram are exact: E'

0

i

f'

0

p

E f

C'

j

C

E''

0

f ''

C''

0

where C ∈ Tκ . Note that f is obtained from the κ-injectivity of E and f is just the induced quotient map. Since E ∈ Fκ and C ∈ Tκ it follows that f = 0 of f factorizes through E and f = i f 1 for some f 1 : C → E . One easily checks that f 1 j = f and it follows that E is κ-injective. Corollary 2.10 p i Let 0 −→ E −→ E −→ E −→ 0 be exact in C and assume that E is κ-injective, E ∈ Tκ and E ∈ Fκ ; then E is isomorphic to E . Proof The conditions imply that i is an essential morphism; that is, if X is a subobject of E that is nonzero then X ∩ E is nonzero, as is easily seen (exercise). The assumption that

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E ∈ Tκ allows us to complete the following diagram in C by a morphism j : E → E such that ji = 1 E . E'

0 1E'

E

i

p

E''

0

j

E'

From the foregoing it follows that j is a monomorphism and then ji = 1 E entails that E ∼ = E . The class of all faithfully κ-injective objects in C is a full subcategory of C, which will be denoted by C(κ) and called the quotient category of C with respect to κ. The canonical inclusion is denoted by i κ : C(κ) → C. For C in Fκ the κ-injective hull of C is defined to be an essential extension C → E such that E is κ-injective and E/C ∈ Tκ . It is clear that any κ-injective hull is in C(κ). Proposition 2.17 Every C ∈ Fκ has an essentially unique κ-injective hull. Proof The object C of C has an injective hull E in C; since C ∈ Fκ it is clear that E ∈ Fκ too. Consider the exact sequence 0 −→ C −→ E −→ E/C −→ 0 in C, and define E = E × E/C κ(E/C), which may be viewed as a subobject of E by the classical pull-back properties in Grothendieck categories. Hence E ∈ Fκ and E/E ∼ = (E/C)/κ(E/C); hence κ(E/E ) = 0. Apply Proposition 2.16 to conclude E is κ-injective. On the other hand, E /C ∼ = κ(E/C) or E /C is κ-torsion. Then let us assume that E 1 , E 2 are κ-injective hulls of C. It follows that E 2 is isomorphic to a subobject E 2 of E 1 containing C as a subobject. Because E 1 is in Fκ and also an essential extension of E 2 that itself is faithfully κ-injective, we apply Corollary 2.10 and arrive at E 1 ∼ = E 2 ∼ = E 2 . The κ-injective hull of C ∈ C is denoted by E κ (C). Recall that in a Grothendieck category C the following are equivalent for any endofunctor F: a. F has a right adjoint. b. F is right exact and commutes with coproducts. Recall also that right adjoints preserve projective (inverse) limits while left adjoints preserve inductive (direct) limits. Theorem 2.4 With notations as before: i κ : C(κ) → C has a left adjoint. Proof For C ∈ C, define aκ (C) = E κ (C/κ(C)). This yields a functor aκ : C → C(κ). If f : C → i κ (D) is an arbitrary morphism, with C ∈ C, D ∈ C(κ), then f extends to a

2.6 The Fundamental Examples I: Torsion Theories

53

morphism f 1 : C/κ(C) → i κ (D), since i κ (D) ∈ Fκ . Now aκ (i κ (D)) is faithfully κinjective and aκ (C)/(C/κ(C)) ∈ Tκ , hence f 1 extends to f : aκ (C) → aκ i κ (D) = D. Finally it is easily verified that we obtain the following isomorphism: HomC (C, iκ (D)) ∼ = HomC(κ) (aκ (C), D). We shall write Q κ = i κ aκ . For c ∈ C, the object Q κ (c) together with the canonical morphism jκ : C → Q κ (C) is called the C- object of quotients of C with respect to κ. Proposition 2.18 Q κ is a left exact endofunctor in C. Proof j If 0 → C −→ C is exact in C, then so is the sequence 0 → C /κ(C ) → C/κ(C). Since Q κ (C ), Q κ (C) are essential extensions of C /κ(C ), respectively C/κ(C), it follows that Q κ (i) is a monomorphism. First let C ∈ Fκ and consider the following commutative diagram with exact top row: 0

C'

0

Q κ (C' )

f

Qκ ( f )

C Q κ (C )

g

Q κ ( g)

C'' Q κ (C'' )

Here Q κ ( f ) is a monomorphism and Q κ (g)Q κ ( f ) = Q κ (g f ) = 0; hence Q κ (C ) is a subobject of KerQ κ (g). Then consider the exact sequence: 0 −→ KerQ κ (g) −→ Q κ (C) −→ ImQ κ (g) −→ 0 Since Q κ (C) is κ-injective and ImQ κ (g) ∈ Fκ , we conclude that KerQ κ (g) is κinjective; hence faithfully κ-injective. Moreover KerQ κ (g)/C ∼ = Q κ (C)/C is in Tκ ; hence Corollary 2.10 yields Q κ (C ) = KerQ κ (g). In general, that is, when C is not necessarily in Fκ we consider f

0 −→ C −→ C −→ C −→ 0 and define D = C ×C κ(C ), the pre-image of κ(C ) in C. Then κ(C) is clearly a subobject of D in C. Also Im f is a subobject of D and D/Imf ∼ = κ(C ). Therefore D/κ(C) contains (Im f + κ(C))/κ(C) such that modulo the latter it is κ-torsion. We obtain an exact sequence: 0 −→ D/κ(C) −→ C/κ(C) −→ C /κ(C ) −→ 0 where κ(C/κ(C)) = 0. Now we have reduced the problem to the torsion-free case because we obtain an exact sequence 0 −→ Q κ (D/C) −→ Q κ (C) −→ Q κ (C ) where Q κ (D/C) = Q κ (Im f + κ(C)/κ(C)) = ImQ κ ( f ).

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Note that aκ has right adjoint i κ but Q κ need not have a right adjoint, in fact Q κ need not even be a right exact functor. For further application of these techniques to categories of sheaves or presheaves it is worthwhile to present some basic facts about Giraud subcategories of (complete) Grothendieck categories. This allows us to apply the “reflector” approach to localization theory; the presentation here is close to Section 2.3 in F. Van Oystaeyen, A. Verschoren, Reflectors and Localization. Application to Sheaf Theory, Lect. Notes in Pure and Applied Mathematics Vol. 41, M. Dekker, New York, 1978 [48]. This approach also allows a general treatment of compatibility of kernel functors and commuting properties of localization functors, that is, exactly the topic recognized in noncommutative topology with respect to the relations between noncommutative space and the commutative shadow. Compatibility of localization goes back to F. Van Oystaeyen, “Compatibility of Kernel Functors and Localization Functors” [45]. The consideration of Giraud subcategories of (complete) Grothendieck categories prepares for the study of sheaves as a subcategory of presheaves. So we look at a complete Grothendieck category P; a full subcategory S of P is called reflective if the inclusion functor i : S → P has a left adjoint a, called the reflector of S in P; that is, for P ∈ P, S ∈ S there is a natural isomorphism HomP (P, iS) ∼ = HomS (aP, S) with canonical natural transforms p : ai → 1S and q : 1P → ia. The couple (a P, q P : P → ia P) has the following universal property: every P-morphism f : P → i S with S ∈ S factorizes in a unique way as follows: P

iaP

qP f˜

f iS

The morphism q P is called the reflection of P. Proposition 2.19 Let S be a reflective subcategory of a (complete) Grothendieck category P; then S is complete and co-complete. If the reflector of S in P is left exact, then S has exact direct limits and a generator. A subcategory of P with a left exact reflector is called a Giraud subcategory of P. From the definition it follows that a Giraud subcategory of a (complete) Grothendieck category is itself a (complete) Grothendieck category and the reflector is exact, whereas the inclusion functor S → P is in general only left exact. Let T be the class of objects C in P for which a(C) = 0 and let F consist of subobjects in P of objects of S. Proposition 2.20 P ∈ P is in F exactly when q P : P → iaP is a monomorphism.

2.6 The Fundamental Examples I: Torsion Theories

55

Proof If q P is a monomorphism, then P is in F since a(P) ∈ S. Conversely, consider P ∈ F and 0 → P → i(S) with S ∈ S. Commutativity of the following diagram in P: 0

i(S)

P ia(P)

leads to the conclusion that q p is a monomorphism. F may be viewed as a full complete subcategory of P easily verified to be a reflective subcategory of P with epimorphism reflector a j where a is the reflector of S and j : F → P the canonical inclusion function. The objects of F are said to be separated. Proposition 2.21 If P ∈ P is separated, then ia(P) is an essential extension of P in P. Proof Let Q be a subobject of ia(P) in P and assume that Q ×ia(P) P = 0. Then 0 = a(Q ×ia(P) P) = a(Q) ×a(P) a(P) ∼ = a(Q). Since Q is a subobject of a separated object, it is itself separated; that is, the canonical Q → ia(Q) is a monomorphism and thus Q = 0 follows from 0 = a(Q). Proposition 2.22 Consider an exact sequence in P: 0 −→ P −→ P −→ P −→ 0 1. If P ∈ S, P ∈ F, then P ∈ F. 2. If P ∈ F, P ∈ S, then P ∈ S. Proof Instead of providing a direct proof we can derive it directly from the next proposition, which reduces it to the torsion theory situation and Proposition 2.16 as well as Corollary 2.10 (with a slight rephrasing). Proposition 2.23 The couple (T , F) determines a torsion theory in P such that its quotient category is equivalent to S.

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Proof Since a is exact, T is closed under subobjects, quotient objects, and extensions. Since a has a right adjoint, it preserves coproducts, hence T is a torsion class. Obviously for T ∈ T , S ∈ S we have HomP (T, i(S)) = 0 and also HomP (T, F) = 0 for every F in F. Conversely, if HomP (T, P) = 0 for all T ∈ T , then from Kerq P ∈ T we obtain that Kerq p = 0, hence P ∈ F. Consequently F may be considered as the torsion-free class corresponding to T . The kernel functor associated to (T , F) will be denoted by α. Clearly if P ∈ P then:

α(P) = {P , 0 −→ P −→ P and a(P ) = P } A Giraud subcategory of P is said to be strict if it is closed under P-isomorphisms. Observation 2.5 If κ is a kernel functor for a (complete) Grothendieck category P, then P(κ) is a strict Giraud subcategory of P. Proof From Proposition 2.17. From what we have already learned it follows easily that there is a bijective correspondence between torsion theories for P and strict Giraud subcategories of P; this is otherwise known as Gabriel’s Theorem. At times we have used the term (complete) Grothendieck category; in fact this indicates that the original statement of the result used the completeness as an extra assumption. It was only after the proof of the Gabriel-Popescu, theorem (first by Popescu, but with a gap solved by Gabriel) that it followed that a Grothendieck category with a generator has enough injective objects and also that every Grothendieck category is complete. Theorem 2.5: Gabriel-Popescu Let C be any Grothendieck category with generator G; put R = HomC (G, G) and let M : C → R-mod be the functor C → HomC (G, C) = M(C). 1. M is full and faithful. 2. M induces an equivalence between C and (R − mod)(κ), where κ is the largest kernel functor in R-mod for which all modules M(C), C ∈ C, are faithfully κ-injective. Corollary 2.10 Every object C in C has an extension that is an injective object of C. Every C is complete! The categorical approach to localization theory has an undeniable elegance, but now we have a less obvious notion of the noncommutative composition at hand, unless we start to compare localization of a strict Giraud category S to localization of P.

2.6 The Fundamental Examples I: Torsion Theories

57

Proposition 2.24 An object E of S is injective in S if and only if i(E) is injective in P. Proof If i(E) is injective in P, then E is injective in S because i is left exact. Conversely, suppose that E is injective in S and consider a diagram in S: C'

0

j

C

f

i(E)

yielding a commutative diagram in P: j

C'

0

C

f

i(E)

qC'

0

qC g

ia( f )

ia(C' )

ia( j )

ia(C )

where existence of g follows from the injectivity of E in S : ia( f ) = g ◦ ia( j). Put g = gqC ; then we find: g j = gqC j = gia( j)qC = ia( f )qC = f , finishing the proof. For C ∈ S, respectively C ∈ P, the injective hull of C in S, respectively in P, will be denoted by E s (C), respectively E p (C). Lemma 2.22 1. For S ∈ S, i(E s (S)) is an essential extension of i(S) in P. 2. Let S ∈ S; then E p (i(S)) = i E s (S); that is, the hull in P of an object in S is in S too. Proof 1. Let P be a subobject of i(E s (S)) in P such that P ×i E s (S) i(S) = 0. Exactness of a yields 0 = a(0) = a(P ×i E s (S) i(S)) = a(P) ×ai(E s (S)) ai(S) = a(P) × E s (S) S, contradicting the fact that E s (S) is an essential extension of S in S since a(P) ∈ S. 2. The foregoing implies that both i(E s (S)) and E p (i(S)) are essential extensions of i(S) in P; thus we arrive at a commutative diagram in P: i(S) i(E s(S))

E p(i(S))

g f

58

Noncommutative Spaces where f exists by definition of E p , and it is a monomorphism, moreover g is a monomorphism too. Since E p (i(S)) is a maximal essential extension of i(S) in P, it follows from 0 −→ E p (i(S)) −→ i E s (S) f

that E (i(S)) ∼ = i(E (S)). p

s

Corollary 2.11 If P ∈ P is separated, then E p (P) is separated. Indeed we have a commutative diagram of monomorphisms in P: P

ia(P)

E p(P)

E p(ia(P))

Since E p (ia(P)) ∼ = i E s (a(P)), it follows that E p (P), being a subobject of E s (a(P)) in P, is separated. Proposition 2.25 If P ∈ P is separated, then we have: ia(E p (P)) = E p (ia(P)) = i(E s (a(P))) = E p (P) (1)

(2)

(3)

Proof 1. We obviously have the following monomorphisms: P −→ ia(P),

E p (P) −→ E p (ia(P)),

ia E p (P) −→ E p (ia(P))

Now E p (ia(P)) is essential over ia(P) and this in turn is essential over P in P; that is, E p (ia(P)) is essential over P. The equality (1) follows if we establish that ia(E P (P)) is injective in P; that is a E(P) is injective in S (see 2.24.). Consider an exact sequence 0 −→ S −→ S and a given f : S −→ a E p (P)in s S. The pull-back properties yield: S'1 = iS'

×

iaEp (P)

Ep(P)

s'

f1

iS' if

Ep(P)

is

iS ig

iaEp(P)

where s is a monomorphism, thus (is)s is monomorphic. Obviously: a(S1 ) = S × a E p (P) = S a E p (P)

2.6 The Fundamental Examples I: Torsion Theories

59

By the injectivity of E p (P) in P, there is a P-morphism g1 : i S → E p (P), such that g(is)s = f 1 . Put g = a(g1 ) : S → a E p (P). Let j be the isomorphism aS1 → S . Then we have a f 1 = f j = q(g1 )sa(s ) = gs j, with f j = gs j, hence f = gs since j is an isomorphism and thus in particular an epimorphism in S. 2. The equality (2) is a direct consequence of Lemma 2.2.2(2). 3. E p (P) is separated because P is separated (Corollary 2.11). Then ia(E p (P)) is essential over E p (P), hence over P in P. Therefore E p (P) = ia E p (P) because E p (P) is a maximal essential extension of P in P. Consider kernel functors κ, κ in P. Then κ ≥ κ if and only if Tκ ⊃ Tκ or equivalently κ(P) ⊃ κ (P) for every object P of P. For kernel functors κ and κ in P we say that κ is Q κ -compatible if κ Q κ = Q κ κ . Lemma 2.23 Suppose κ is Q κ -compatible. 1. If P is in Fκ , then Q κ (P) is in Fκ ; the converse holds when P is κ-torsion free. 2. If P is in Tκ , then Q κ (P) is in Tκ ; the converse holds in case P ∈ Fκ . Proof Exercise. To any strict Giraud subcategory S of P we have associated a kernel functor α (see remark Proposition 2.23). The α- compatible kernel functors (Q α -compatible) are sometimes called Scompatible kernel functors. Hence, κ in P is S-compatible exactly when iaκ = κia. If iaκia = κia, that is, κ takes objects of S to objects of S, then κ is said to be inner in S. If κ is inner in S, then the functor κ is denoted by κ S ; in general κ S need not be a kernel functor in S. Proposition 2.26 Let κ be S-compatible in P; then κ s is a kernel functor in S. Proof For S in S, κ(i S) = κ(iai(S)); hence κ S is inner in S. Therefore aκ(i(S)) = κ s (S) and it is clear that κ s is a left exact subfunctor of the identity in S. Furthermore we easily calculate: κ s (S/κ s (S)) = aκ(ia(i S/I κ s (S))) = aκ(i S/iκ s (S)) = aκ(i S/κ(i S)) = a(0) = 0 (Note: we simplified notation by dropping some brackets in the notation).

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Earlier we mentioned the gen-topology induced on a lattice by taking the intervals [0, λ], λ ∈ ; now comparing the lattice of kernel functors on P to what the Zariski topology would be if P were C-mod for some commutative ring, we know we have to look at the opposite lattice, and therefore the sets [κ, 1] get our attention. These are well behaved in terms of compatibility condition because of the following result. Proposition 2.27 Look at kernel functors κ and κ , for P such that κ ≥ κ ; then κ is Q(κ )-compatible. Proof Let qκ(P) : κ(P) → Q κ (κ(P)) be the reflection of P ∈ P in P(κ ). The following sequence is exact: 0 → Imqκ(P) → Q κ (κ(P)) → Coimqκ(P) → 0 Clearly, Imqκ(P) is κ-torsion; moreover κ ≥ κ implies that Coimqκ(P) is κ-torsion. Hence, Q κ (κ(P)) is a subobject κ Q κ (P). Conversely, since Fκ ⊂ Fκ we have that P/κ(P) ∈ Fκ so we have a monomorphism: 0 → P/κ(P) → Q κ (P/κ(P)). Then P/κ(P) ∩ κ Q κ (P/κ(P)) = 0 implies that Q κ (P/κ(P)) = 0. Finally, by the exactness of 0 → Q κ (κ(P)) → Q κ (P) → Q κ (P/κ(P)) we obtain that κ Q κ (P) ⊂ Q κ (κ(P)). Theorem 2.6 For a strict Giraud subcategory S of P and an S-compatible kernel functor κ for P we have S ∈ S is (faithfully) κ s -injective if and only if i(S) is (faithfully) κ-injective. Proof Assume that i(S) is κ-injective. Consider a diagram in S: 0

S1

S2

S2/S1

0

f

S

where κ s (S2 /S1 ) = S2 /S1 . In P we obtain a diagram, applying i to the above: 0

i(S1)

i(S2)

i(S2)/i(S1)

if

i(S)

where i(S2 )/i(S1 ) is subobject of i(S2 /S1 ). Since S2 /S1 is κ s -torsion and S2 /S1 = a(i S2 /i S1 ), it follows Lemma 2.23(2) that i S2 /i S1 is κ-torsion, so there exists a g : i S2 → i S completing the above diagram. Then it is clear that a(g ) completes the diagram in S from which we started.

2.6 The Fundamental Examples I: Torsion Theories

61

Conversely, let S be κ s -injective and consider an exact sequence in P: 0

P1

P2

P2/P1

f

i(S)

where P2 /P1 ∈ Tκ . Since a is exact we obtain the following diagram in S: 0 −→ a P1 −→ a P2 −→ a(P2 /P1 ) a f S Again, by Lemma 2.23, it follows that a(P2 /P1 ) is κ s -torsion; therefore, there exists an S-morphism g : a P2 → S completing the diagram. Let g be the map (ig )q P2 : P2 → i(S), and it is easily verified that g extends f as desired. Since i(S) is κ-torsion free if and only if S is κ s -torsion free (because i(S) is separated and Lemma 2.23) we may apply Proposition 2.15. Proposition 2.28 With notation as before: Let κ be an S-compatible kerel functor for P and consider an object S of S that is κ s -torsion free, i(E κ s (S)) ∼ = E κ (i(S)). Proof Lemma 2.23 yields: i(S) ∈ Fκ . The foregoing and the fact that E κ s (S) is faithfully κ s -injective imply that i E κ s (S) is faithfully κ-injective. Furthermore, i E κ s (S)/i S is κ-torsion in P because E κ s (S)/S is κ s -torsion in S. But E κ (i S) is unique up to isomorphism in P with the properties mentioned above; therefore, we arrive at E κ (i S) ∼ = i E κ s (S). Corollary 2.12 If κ is an S-compatible kernel functor for P and S ∈ S, then i Q κ s (S) = Q κ (i S). Proof If S is κ s -torsion free, then the statement follows from the foregoing proposition. In general: i Q κ (S) = i E κ s (S/κ s (S)) = E κ (ia(i S/κ(i S))) Of course, i S/κ(i S) is separated; thus ia(i S/κ(i S)) is an essential extension in P of i S/κ(i S). Consequently i(S/κ s (S)) is κ-torsion free and Q κ (i S) = E κ (i S/κ(i S)) ∼ = E κ (ia(i S/κ(i S))). Proposition 2.29 Let S be a strict Giraud subcategory of P and let κ be a kernel functor in P such that κ ≥ α where α corresponds to S. Then we have: Q κ (P) ∈ i S for every P ∈ P.

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Proof Put P = P/κ(P). Since κ ≥ α, α(P) = 0; hence P is separated with respect to S. From a(ia(P)/P) = 0 it follows that ia P/P is α-torsion, hence κ-torsion. Faithful κ-injectivity of Q κ (P) gives rise to the following commutative diagram in P: 0

– P

– iaP qP–

Qκ(P)

– – iaP P

0

jP

Since q P is a monomorphism and since ia P is an essential extension of P in P, j P is a monomorphism and thus Q κ (ia P) = Q κ (P). Because of Proposition 2.27, κ is S-compatible; then Corollary 2.12 applies and we obtain i Q κ s (a P) = Q κ (ia P) = Q κ (P), and finally this yields that Q κ (P) is in i S. Let us now construct some kernel functors that are S-compatible. Consider a nonzero object P in P and let K (P) be the class of kernel functors κ for P such that P ∈ Fκ . If P ∈ P is essential over P in P, then obviously K (P) = K (P ). Define κP for P by putting: κ P (Q) = ∩{kerg, g ∈ HomP (Q, Ep (P))} Proposition 2.30 With notation as before: i. κ P is a kernel functor for P and κ P ∈ K (P). ii. If κ is a kernel functor for P, then κ ∈ K (P) if and only if κ ≤ κ P . Proof i. Straightforward (note that there is a monomorphism 0 → E P (P) → E P (P); hence κ P (E P (P)) = 0 and κP ∈ K (P)). ii. If κ ≤ κ P , then κ ∈ K (P) obviously. Conversely, if κ ∈ K (P) for some kernel functor κ for P, look at an arbitrary morphism g : Q → E P (P), in P. Clearly κ(Q) ⊂ Kerg; hence κ(Q) ⊂ κ P (Q), for every Q in P; thus κ ≤ κ P . Theorem 2.7 If S ∈ S, then κi(S) is an S-compatible kernel functor for P. Conversely, if P in P is such that κ P is S-compatible, then there is an S in S such that κ P = κi(S) ; moreover S = a(P). Proof We first check that iaκi(S) (P) = κi(S) (ia P), and we may assume that we have replaced S by E s (S) and i(S) by E P (i(S)). Then the problem reduces to proving: ∩Kerg, g ∈ HomP (iaP, iS) = ia(∩{Kerg, g ∈ HomP (P, iS)})

2.6 The Fundamental Examples I: Torsion Theories

63

Since S is a full subcategory of P and since a is a right adjoint of i, we do obtain the following isomorphisms in Ab: HomP (iaP, iS) ∼ = HomS (aP, S) = HomP (P, iS). Since S is a Giraud subcategory of P, we have that ∩{Kerg, g ∈ HomS (aP, S)} is an object of S. Conversely, if P ∈ P is such that κ P is S-compatible, then κ P (ia P) = iaκ P (P) = ia(0) = 0, hence κ P ≤ κia(P) . On the other hand: κia(P) (P) = ∩{Kerg, g ∈ HomP (P, EP (iaP))}. In view of Lemma 2.22(2) we arrive at: κia(P) (P)

= =

∩{Kerg, g ∈ HomP (P, i E s (a P))} ∩{Kerg, g ∈ HomP (a P, E s (a P))} = 0

Finally we find 0 = κia(P) (P); thus κia(P) ≤ κ P . The foregoing techniques may be applied to some interesting special cases. Of course P = R-mod for some noncommutative ring R is of interest, but so is the case where P is the category of presheaves over a small category X with values in a Grothendieck category C. We return to this later.

2.6.1

Project: Microlocalization in a Grothendieck Category

In the algebraic geometry of associative algebras (see [46]), a particularly interesting case is presented by filtered algebras that are “almost commutative” in the sense that the associated graded ring is a commutative ring. Their noncommutative site may be viewed as being quantum-commutative in the sense that the topology defined in terms of microlocalzation functors is in fact a commutative one. Roughly speaking (see [44] for full detail) the microlocalization is obtained from a completion with respect to a localized filtration. This project is to develop such a technique for arbitrary Grothendieck categories; this can then be continued along the lines of Chapter 3, leading to canonical microtopologies. There may be a benefit of this to sheaf theory, but at this moment there are no obvious applications of this technique outside the algebraic theory already covered in [46]; however, the consideration of categories of topologized objects is natural in the context we have developed, so it is not unlikely that new applications of the microlocalizations may be discovered. Let κ be a kernel functor on the Grothendieck category C and let Tκ denote its torsion class (see Definition 2.8). To an arbitrary object M of C we may associate a filter L(κ, M) consisting of all subobjects N of M in C such that M/N ∈ Tκ . It is clear that L(κ, M) is closed under the lattice operations ∧(= ∩) and ∨(= ) defined in L(M), the (big) lattice of subobjects of M in C (see also the remarks following Lemma 2.3). The (big) lattice L(κ, M) need not have a 0 but we may formally add such if we wish. In any case we may view

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lim κ in C as ←− L(κ, M) as a topologization of M and we may define M (M/N ). The N ∈L(κ,M)

κ and (Q κ (M))∧ is easily investigated. This provides us with a relation between M general notion of microlocalization, denoted Q µκ for a torsion theory (Tκ , Fκ ). When Q κ is an exact functor the properties of Q µκ have to be investigated.

2.7

The Fundamental Examples II: L(H)

Let H be a complex Hilbert space and consider the set L(H ) of closed subspaces of H . The set L(H ) becomes a complete lattice if we define: for U, V ∈ L(H ), U ∧ V = U ∩ V , U ∨ V = (U + V ), where (−) denotes the closure in H . In L(H ) we also have a complement, associating the orthogonal complement U ⊥ to U in H . It is clear that L(H ) satisfies axioms A.1, . . . , A.9, but not A.10 (look at a space of finite codimension in H and a basis for its orthogonal complement; try to use this global cover to induce a cover on a line in the orthogonal complement disjoint from the chosen basis). Consequently it is impossible here to use Lemma 2.20 to obtain sheaves over L(H )! However, we shall have other techniques available that will allow the construction of sheaves (and sheafification) over the generalized Stone space, which will be introduced later in this section. Consider the algebra L(H ) of bounded linear operators on H . Associating to U in L(H ) the orthogonal projection PU onto U viewed as an element in L(H ), then we see that the lattice L(H ) is isomorphic to the lattice P(L(H )) of orthogonal projections in L(H ) which are exactly the idempotent elements of L(H ). It is well known (and easily verified) that L(H ) is not distributive due to the fact that PU and PV need not commute. From the theory on noncommutative topologies in Chapter 2, we expect that idempotency of PU PV and PV PU would lead to the commutativity of PU and PV . In fact this is the case, but an even stronger result holds because only one such product has to be considered (I thank my colleague Jan van Casteren for some discussions about the analytical aspects). Proposition 2.31 With notation as introduced above: if (PU PV )2 = PU PV then PU and PV commute and PU PV = PU ∩V . Proof Observe that PU = PU∗ , PV = PV∗ . Put T = PU PV − PV PU , then T ∗ = −T and we easily calculate: PU T PV = PV T PU = PU T PU = PV T PV = 0 For f ∈ U ⊥ ∩ V ⊥ we have: T f = PU PV f − PV PU f = PU .0 − PV .0 = 0

2.7 The Fundamental Examples II: L(H)

65

Clearly, if f ∈ U , then T f ∈ U + V but also T f ∈ U ⊥ , because PU T f = PU T PV f = 0, and similarly T f ∈ V ⊥ . Consequently, f ∈ U yields T f ∈ (U ⊥ ∩ V ⊥ )∩(U +V ) = (U +V )⊥ ∩(U +V ) = 0. A similar argument establishes that T f = 0 for f ∈ V . Since (U + V ) + (U + V )⊥ is dense in H , T must be the zero operator; hence PU PV = PV PU . Next consider S = PU PV − PU ∩V . A direct calculation yields: S2

= = =

(PU PV − PU ∩V )(PU PV − PU ∩V ) PU PV PU PV − PU ∩V PU PV − PU PV PU V + PU2 ∩V PU PV − PU ∩V pU PV − PU ∩V + PU ∩V = (I − PU ∩V )PU PV

Since we obviously have S = S ∗ , the foregoing yields: S2

= =

S ∗ S = (S ∗ S)∗ = (S 2 )∗ = ((I − PU ∩V )PU PV )∗ PV PU (I − PU ∩V ) = PU PV − PU ∩V = S

Therefore S is an orthogonal projection. If f ∈ U ∩ V + (U ⊥ + V ⊥ ), then S f = 0 (using that U ⊥ + V ⊥ ⊂ (U ∩ V )⊥ ). Since the closure of U ⊥ + V ⊥ is (U ∩ V )⊥ and since S is continuous, it follows that S f = 0 for any f in (U ∩ V ) + (U ∩ V )⊥ ; hence S = 0. Observe that for any linear subspace of H , U for example, the closure of U in H is given by U ⊥⊥ . The advantage of the analytic proof given above is that we do not have to verify the axioms of a noncommutative topology for the set of finite products of idempotent elements of L(H ). In that way we would arrive at the following result too; we again provide an analytic proof. Theorem 2.8 Let P be a family of normal operators acting on the Hilbert space H ; that is, P ∈ P implies that P P ∗ = P ∗ P. Suppose that for all finite {P1 , . . ., Pn } ⊂ P we have (P1 . . . Pn )2 = P1 . . . Pn ; then for any finite {P1 , . . ., Pn } ⊂ P we have P1 . . . Pn = (P1 . . . Pn )∗ PR(P1 ...Pn ) where for any operator T, R(T ) stands for the range of T ; in particular all elements of P commute with one another. Proof First we establish that every P ∈ P is idempotent. For P ∈ P, (P P ∗ − P)P ∗ = P(P ∗ )2 − P P ∗ = P P ∗ − P P ∗ = 0; if P f = 0, then (P P ∗ − P) f = P ∗ P f − P f = 0; thus P P ∗ − P is zero on R(P ∗ ) + ker(P). The latter space is dense in H ; hence by continuity of P, P = P ∗ = P P ∗ = P ∗ P = P 2 = PR(P) . We proceed by induction, supposing the result holds for any finite {P1 , . . ., Pn−1 } ⊂ P. Take P ∈ P, P ∈ {P1 , . . ., Pn−1 }, which is thus necessarily an orthogonal projection. The induction hypothesis implies P1 . . . Pn−1 = (P1 . . . Pn−1 )∗ = PR(P1 ...Pn−1 ) and P1 . . . Pn = PR(P1 ...Pn−1 ) Pn , with Pn = P, leads to P1 . . . Pn = PR(P1 ...Pn−1 ) Pn = (PR(P1 ...Pn−1 ) Pn )∗ = PR(P1 ...Pn ) . Also if {P1 , . . ., Pi−1 , P, Pi+1 , . . ., Pn−1 } is considered, then again the same argument implies, with an interchanging of P and Pi , and the claim follows. We point out that in general it is not obvious that products of comparable operators may be comparable. We have P ≤ Q whenever P = P Q(= Q P = P Q P) for

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orthogonal projectors. If P j ≤ Q j for j ∈ {1, . . ., n}, then P1 . . . Pn ≤ Q 1 . . . Q n , but equality does not entail P j = Q j for j = 1, . . ., n. For suitable operators such a result may be proved; we include an example. Example 2.3: (J. van Casteren) If 0 ≤ P j ≤ Q j , j = 1, . . ., n for orthogonal projections P j , Q j such that the following hold: i. (Q j . . . Q k − P j . . . Pk )Pk (Q k . . . Q j − Pk . . . P j ) ≥ (Q j . . . Q k − P j . . . Pk )(Q k . . . Q j − Pk . . . P j ), for 1 ≤ j ≤ k ≤ n − 1 ii. (Q k . . . Q j − Pk . . . P j )P j−1 (Q j . . . Q k − P j . . . Pk ) ≥ (Q k . . . Q j − Pk . . . P j )(Q j . . . Q k − P j . . . Pk ) for 2 ≤ j ≤ k ≤ n. Then Q 1 . . . Q n = P1 . . . Pn if and only if P j = Q j , j = 1, . . ., n. We defined points and quasipoints in Section 2.3. Let us point out some facts in the particular case of L(H ). First, one easily verifies that L(H ) has no points; indeed if [ A] is a point of L(H ) given by its filter A, then V {Cu α , α ∈ A} = H for some selected basis {u α , α ∈ A}; hence Cu α ∈ A for some suitable α ∈ A. If U ∈ A, then U ∩ Cu α ∈ A; hence Cu α ⊂ U because 0 ∈ A by assumption; choose V ∈ L(H ) such that V nor V ⊥ contains Cu α ; then V ∨ V ⊥ = H yields that either V or V ⊥ is in A, but that contradicts Cu α ∈ V, Cu α ∈ V ⊥ . Of course L(H ) has minimal points (quasipoints) since maximal filters always exist. We have observed that maximal filters define idempotent elements of C(L(H )) (see Lemma 2.4) and if λ ∈ A, β ∈ A, then λ ∧ β ∈ A follows. Let us also recall that a directed set A in a poset (with 0 and 1) is said to be pointed if for all λ ∈ A there exists a µ ∈ A such that γ ≤ λ, γ ≤ µ implies γ = 0. Proposition 2.32 The minimal points of L(H ) are exactly given by the pointed filters. There are two types of pointed filters: i. A = {U ∈ L(H ), u α ∈ U for some u α = 0 in H }. ii. A contains all V of finite codimension in H . Proof If some cofinite dimensional U is not in A, then there is a V ∈ A such that U ∩ V = 0; therefore, V is finite dimensional. Thus there must exist a W ∈ A with minimal dimension as such. If dimW > 1, then pick a subspace W ⊂ W with dimW = 1; by assumption W ∈ A; hence there is a U ∈ A such that W ∩ U = 0, but that contradicts U ⊃ W . Consequently dimW = 1 and A is as claimed in i. The remaining case ii is exactly the case where all cofinite dimensional V are in A. Note that in general a pointed filter is maximal; indeed if A is pointed and A B L(H ), then there is a V ∈ B such that V ∈ A, and thus there exists a W ∈ A such that W ∩ V = 0, contradicting W, V ∈ B and B = L(H ). Conversely, if B is a maximal

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filter in L(H ) such that U ∈ B, then A = {U ∩ V, V ∈ B} is a directed set because for V, W ∈ B, U ∩ (V ∩ W ) ⊂ U ∩ V , U ∩ W ; hence A ⊃ B is a strict inclusion of filters because U ∈ A, U ∈ B. The maximality assumption on B then implies 0 ∈ A; therefore U ∩ V = 0 for some V ∈ B and consequently B is pointed. The foregoing property of L(H ) is shared by more classical types of lattices, for example, topologies. Proposition 2.33 Let X be a topological space satisfying T1 ; write L(X ) for the lattice of open subsets of X (sometimes denoted Open(X )). If A is a pointed directed set for L(X ), then it is one of three possible types: i. ∩{U ∈ A} = {x} for some x ∈ X and every open neighborhood of x in A. ii. ∩{U ∈ A} = ∅ and ∩{U , U ∈ A} = {x} for some x ∈ X (where U is the closure of U in X ). iii. ∩{U ∈ A} = ∅ and X − K ∈ A for every closed compact set K in X (compact here means having the finite intersection property). Proof Suppose x, y ∈ I = ∩{U ∈ L(X )}. If y = x then, in view of the T1 -property, we may select an open neighborhood Vy of y such that x ∈ Vy ; thus Vy ∈ A. Note that we may replace A by its filter A because the pointedness assumption is preserved. Thus, there is a U ∈ A such that U ∩ Vy = ∅; hence y ∈ U and then y ∈ I . It follows that I = {x} and the claims in i follow. In the remaining cases we have I = ∅. Suppose there is a closed compact K such that X − K ∈ A. Since A is pointed there is a V ∈ A such that V ∩ (X − K ) = ∅; that is, V ⊂ K or V ⊂ K . Look at I = ∩{U , U ∈ A}. Since for V as above, V is compact, it follows that I = ∅ unless ∅ ∈ A, a case that may be excluded because ∅ ∈ A. If x = y are both in I and Vy is an open neighborhood of y such that x ∈ Vy , then U ∩ Vy = ∅ for some U in A, while on the other hand y ∈ V ∩ U . Because U is dense in U , this leads to a contradiction unless y = x; thus I = {x} and the claims of ii are proved. The remaining case is the one where X − K ∈ A for all closed compact subsets K of X , as stated in iii. The Stone space, originally constructed for Boolean algebras, has been defined also for arbitrary lattices (I do not recall where this first appeared in the literature), but for us the Stone space, as a set, is nothing but the part of C() corresponding to the pointed directed sets, so this definition extends to noncommutative topologies. It is also clear how to define a generalized Stone topology on the above defined set, SC() for example.

2.7.1

The Generalized Stone Topology

Consider a noncommutative topology and C(). For λ ∈ , let Oλ ⊂ C() be given by Oλ = {[A], λ ∈ A}. It is trivial to verify Oλ∧µ ⊂ Oλ ∩ Oµ , Oλ∨µ ⊃ Oλ ∪ Oµ , and therefore the Oλ define a basis for a topology on C(), called the generalized Stone topology. We may restrict attention to the point-spectrum Sp(), or the

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quasipoint spectrum Q S p (); the topology defined on these subsets of C() will again be called the generalized Stone topology. Observe that on the quasispectrum, writing Q Oλ = {[A], [A] ∈ Q S P (), λ ∈ A} we actually obtain Q Oλ∧µ = Q Oλ ∩ Q Oµ but still only Q Oλ∨µ ⊃ Q Oλ ∪ Q Oµ . On the other hand, writing P Oλ = {[A], [A] ∈ S p (), λ ∈ A} we have P Oλ∧µ ⊂ P Oλ ∩ P Oµ , P Oλ∨µ = P Oλ ∪ P Oµ . This follows from the fact that for [A] in Sp() we do have that λ, µ ∈ A entails λ ∧ µ ∈ A; indeed (cf. Definition 2.3.3.). On Sp() the generalized Stone topology is nothing but the point-topology. On S P(), writing S P Oλ = {[A], [A] ∈ S P(), λ ∈ A}, we have both equalities: S P Oλ∧µ = S P Oλ ∩ S P Oµ and S P Oλ∨µ = S P Oλ ∪ S P Oµ . In the foregoing one may replace by the pattern topology T (or by T () and similar restrictions SpT or SPT as defined earlier; in all cases we shall use the same label—generalized topology or generalized Stone space—and it will be clear from the context which one it is. Finally, the generalized Stone topology may also be defined on the commutative shadow S L() (see Proposition 2.1), which is a modular lattice, then of course its induced topology on Q S P (S L()) is exactly the Stone topology of the Stone space of S L(). In the special case = L(H ), the generalized Stone space defined on Q S P (L(H )) = Q S P(L(H )) is exactly the classical Stone space that can be used in Gelfand duality theory for L(H ) and L(H ). A word of warning perhaps; since L(H ) is not satisfying the weak F D I property, one may not expect a result like Corollary 2.4. In fact, whereas Q S P(L(H ) is rather big, S p (L(H )) = S P(L(H )) is empty (see remarks preceeding Proposition 2.3.2). This fact will have a deeper meaning when we aim to develop some sheaf theory over general . In Section 2.4. the basic properties were introduced and we stressed the transfer from (pre-)sheaves over to (pre-)sheaves over C(). This will turn out to be of essence in the case = L(H ) because there are no sheaves (there is not enough “cohesion” between the element of L(H ) if one tries to view them as open sets in some generalized topology) over L(H ). There will be many sheaves over C(L(H )) allowing sheafification of presheaves; in fact, this will already be possible over Q S P (L(H )) (see Chapter 4). An important notion in Gelfand duality theory for L(H ) is the notion of spectral family and of observable function. In the following we will see, to our surprise, that a spectral family is essentially just a separated filtration. We shall consider a totally ordered group in the sequel, however, it would be enough to consider a totally ordered poset with meet and join defined for every subfamily. In applications: ⊂ Rn+ . Definition 2.9: -Spectral Family Let be a noncommutative topology; then a -filtration of is a family {λα , α ∈ } such that for α ≤ β in , λα ≤ λβ in and ∨{λα , α ∈ } = 1 in (i.e., we consider exhaustive filtrations). A -filtration is separated whenever γ = inf{γα , α ∈ A} in entails that λγ = ∧{λγα , α ∈ A} in , and 0 = ∧{λγ , γ ∈ }. A -spectral family is just a separated -filtration; it may be seen as F : → , γ → λγ where F is a poset map with F(γ ) = λγ satisfying the separatedness condition. Note that, by definition, the order in ∧{λγα , α ∈ A} does not matter while on the other hand the λγα need not be idempotent.

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69

The foregoing definition applied with = R, + and = L(H ) yields the usual notion of spectral family. A well-known example (connected to the Hamilton operator of the harmonic oscillator) is obtained as follows: let xn , n ∈ N be an orthogonal basis of a separable Hilbert space H and define for γ ∈ R, L(H )γ = ∨{Cxn , n ≤ γ }. This is in fact also an example of a -filtration with discrete support as introduced in [1]. First let us continue with some general facts. A -spectral family on is said to be idempotent if λγ ∈ id∧ () for every γ ∈ . We say that is indiscrete if for all γ ∈ , γ = inf{τ, γ < τ }, for example, = Rn , +. Proposition 2.34 If is indiscrete, then every -spectral family is idempotent. Proof Since obviously γ = inf{τ, γ ≤ τ } in , we have λγ = ∧{λτ , γ ≤ τ }. Since in the latter expression the order of the λτ is irrelevant, we may rephrase this as λγ = λγ ∧ ( λτ ) = λγ ∧ λγ ; consequently λγ ∈ id∧ (). γ , then for every homogeneous Ore set S ∈ O(A) such that S ∩ A+ = ∅, it follows that Q S (A) is strongly graded. Proof Put B = Q S (A) with b ∈ Bd if and only if sb ∈ An+d for some s ∈ S ∩ An . It is harmless to replace A by A/κ S (A); that is, we may assume that A → S (A); g also note that Q S (A) = Q S (A). If Z ∈ B0 , then sn z ∈ An for some sn ∈ S ∩ (i) An ; that is, sn z = i a1 . . . an(i) with a (i) j ∈ A1 , j = 1 . . . n. Rewrite this as z = −1 (i) (i) (i) (i) −1 (i) (s a . . . a )a with s a . . . a 1 n−1 n 1 n−1 ∈ B−1 . Thus z ∈ B−1 A1 ⊂ B−1 B1 ∈ B, n i n follows, or B0 = B−1 B1 . Since B−1 . . . B−1 .B1 . . . B1 = B0 we also obtain B−n Bn = B0 for positive n. In a similar way we derive that Bn = B0 An1 and then Bn = B1n follows too. Finally if z ∈ B0 and sm ∈ S ∩ Am is such that sm z ∈ Am , then z = (zsm )sm−1 ⊂ Bm B−m = B1m B−m ⊂ B1 (B1m−1 B−m ) ⊂ B1 B−1 , thus B0 = B1 B−1 as well. An extended version of the foregoing result can be found in Proposition 3.16. Example 2.4 The coordinate ring of quantum 2 × 2-matrices, Oq (M2 (C)), with q ∈ C, is schematic and a Noetherian domain. This algebra is generated over C by elements A, B, C, D subjected to the following relations: • B A = q −2 AB • C A = q −2 AC • BC = C B

• • •

D B =−2 B D DC = q −2 C D AD − D A = (q 2 − q −2 )BC

In fact one take S A , S B , SC , S D respectively generated by the powers of A, B, C, D; the schematic condition can be checked stepwise for the consecutive extensions C[B, C][A][D], which at each step is given by an Ore extension; that is, Oq (M2 (C)) is an iterated Ore extension. Example 2.5 Quantum Weyl algebras are schematic. Look at an n × n-matrix (αi j ) = A with αi j ∈ K ∗ = K − {0} and let q = (q1 , . . ., qn ) be a row with qi = 0 in K . Define An (q, A) as

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the K -algebra generated by x1 , . . ., xn, y1 , . . ., yn subjected to the following relations: putting µi j = λi j qi , xi x j = µi j x j xi xi y j = α ji yi xi y j yi = α ji yi y j x j yi = µi j yi x j x j y j = q j y j x j + 1 + i< j (qi − 1)yi xi Again this algebra may be obtained as an iterated Ore extension creating stepwise extensions by consecutively adding to K , x1 , y1 , x2 , y2 , . . ., xn , yn . Example 2.6 The Sklyanin algebra is schematic. Let SK (A, B, C) be the K -algebra generated by three homogeneous elements X, Y, Z of degree 1, with homogeneous defining relations: A × Y + BY X + C Z 2 = 0 AY Z + B Z Y + C X 2 = 0 AZ X + B X Z + CY 2 = 0 A proof for this, using a valuation reduction idea, is given in [44]. Example 2.7: E. Witten’s Gauge Algebras for SU (2) Consider the G-algebra W generated by X, Y, Z , subjected to the following relations: X Y + αY X + βY = 0 Y Z + γ ZY + δX2 + εX = 0 Z X + ξ X Z + ηZ = 0 for any α, β, γ , δ, ε, ξ, η ∈ C. This Witten-algebra (and its associated graded rings as well as the Rees ring with respect to the obvious filtration given in terms of the total degree in X, Y and Z ) is schematic. Example 2.8: Woronowicz’s Quantum sl2 Let Wq (sl2 ) be the G-algebra generated by X, Y, Z subjected to the following relations: √ √ −1 q + q −1 Z qXZ − q ZX = √ √ −1 q X Y − qY X = − q + q −1 Y √ √ Y Z − Z Y = ( q − q −1 )X 2 − q − q −1 X 2πi is the Chern coupling constant. where classically q = exp k+2 The algebra, as well as its Rees ring for the the standard filtration, is schematic.

Chapter 3 Grothendieck Categorical Representations

3.1

Spectral Representations

We start from a category R allowing products and coproducts. Typical examples we have in mind are amongst others: the category Ring of associative rings with unit, the category R-grG of G-graded associative rings with unit for some group G, the category Algk of k-algebras, the category R-filt of Z-filtered rings (an interesting non-abelian case), and so forth. To each object R of R we associate a Grothendieck category Rep(R). For every f ∈ HomR (S, R), f : S → R, we are given an exact functor f o = F : Rep(R) → Rep(S), which commutes with products and coproducts and satisfies the following conditions: i. (1 R )o = IRep(R) for every R ∈ R ii. For g : T → S, f : S → R in R, ( f ◦ g)o = g o ◦ f o . We did not demand that for R = S in R necessarily Rep(R) = Rep(S) If G is the class consisting of objects Rep(R), R ∈ R, we let HomG (Rep(R), Rep(S)) consist of functors of type h o provided these go from Rep(R) to Rep(S). Note that if Rep is separating objects of R, then we may write HomG (Rep(R), Rep(S)) = HomR (S, R)o . In any case G as defined above becomes a category. Definition 3.1 A Grothendieck categorical representation (a GC representation) is a contravariant functor Rep : R → G commuting with arbitrary products, associating to f : S → R an exact functor f o = F : Rep(R) → Rep(S) commuting with products and coproducts. Several obvious examples come to mind, for example, representing noncommutative rings by their category of left modules, groups by their G-modules, and graded algebras by their categories of graded modules. Such examples exhibit stronger properties than those used in the general definition. This is mainly due to the fact that the objects of R appear in some form also in the representing Grothendieck category, such as a ring as a left module over itself, and so forth. This may be formalized in the following definition.

79

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Definition 3.2 A GC representation R → G is said to measure R if for every R ∈ R we have an object R R in Rep(R) such that to any f : S → R in R there corresponds a morphism f ∗ : S S → Rep( f )( R R) = S R, satisfying: µ.1. If f = 1 S then f ∗ is the identity of S S in Rep(S). µ.2. Consider g : T → S, f : S → R in R; then we have : ( f ◦ g)∗ = Rep(g)( f ∗ ) ◦ g ∗ :T T →T S →T R = Rep(g)Rep( f )( R R) A GC representation Rep : R → G is said to be full if to an epimorphism π : S → R in R there corresponds a full functor Rep(π ) : Rep(R) → Rep(S). A GC representation is faithful if for f : S → R in R, Rep( f ) is faithful. An R ∈ R is Rep-Noetherian when Rep(R) is a Grothendieck category having a Noetherian generator. Similarly, R ∈ R is locally Rep-Noetherian whenever Rep(R) has a family of Noetherian generators. The relation between a GC representation and suitable topologies will be obtained from the hereditary torsion theories existing on the Grothendieck categories. In Section 2.6 we introduced general torsion theory in Grothendieck categories, but here we modify the notation somewhat in order to fit the notation fixed in the introduction of Grothendieck representations. For an arbitrary Grothendieck category M we let Tors(M) be the set of hereditary torsion theories on M; we know that Tors(M) is a modular lattice with respect to inf and sup of torsion theories (cf. the note following Proposition 2.11), but the operation “product” in the lattice of preradicals Q(M) is noncommutative. Torsion theories of M will be denoted by σ, τ, κ, . . . ; then Tσ , Fσ will denote the torsion, respectively the torsion-free class of σ . We write Tσ : M → M for the corresponding torsion functor (kernel functor) and (M, σ ) for the quotient category together with the canonical functors i σ : (M, σ ) → M, aσ : M → (M, σ ) (see Theorem 2.17). Then i σ aσ = Q σ is the localization functor M → M associated to σ . For R ∈ R we abbreviate TorsRep(R) to Top(R). In case R has an initial object, k say, then we call Top(k) the initial space for Rep(R). To a morphism f : S → R in R we have associated a functor F = Rep(R) → Rep(S). Since F is exact and commutes with coproducts, it defines a map F o : Top(S) → Top(R); indeed, if γ ∈ Top(S) we may define F o (γ ) by taking for T F o (γ ) the class of objects X in Rep(R) such that F(X ) ∈ Tγ ; when F derives from f we shall write f for F o . Definition 3.3 A faithful Grothendieck representation that measures R is said to be spectral if for all A ∈ R, γ ∈ Top(A) and τ ∈ gen(γ ) we are given the following: i. An object A(γ ) in R together with a morphism f γ : A → A(γ ) such that the morphism f γ∗ in Rep(A), f γ∗ : A A → Rep( f γ )( A(γ ) A(γ )) is exactly the localization morphism A A → Q γ (A).

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ii. A morphism f τγ : A(γ ) → A(τ ) in R fitting into a commutative diagram: A(γ) fγ

A

fτγ fτ

A(τ)

iii. If ξo (A) stands for the trivial element of Top(A), that is, the zero element of the lattice Tors(Rep(A)), then A = A(ξo (A)). Proposition 3.1 Consider an exact functor F, F : Rep(S) → Rep(R), commuting with direct sums; then F o : Top(R) → Top(S) has the following properties: i. For γ ≤ σ in Top(R), F o (γ ) ≤ F o (σ ) in Top(S). ii. If U ⊂ Top(R), then F o (∧U ) = ∧F o (U ). iii. For τ in Top(S), let ξτ be the minimal torsion theory having all F(Tτ ), Tτ ∈ Tτ ; in its torsion class, in other words Tξτ is the torsion class generated by the F(Tτ ), Tτ ∈ Tτ . We have (F o )−1 (gen(τ )) = gen(ξτ ). Proof An easy exercise. In general, for U ⊂ Top(A) : gen(∧∪) ⊃ U {gen(τ ), τ ∈ U }, gen(∨U ) = ∩{gen, τ ∈ U }; moreover there is a trivial torsion theory ξ defined by Tξ = 0, and a maximal torsion theory χ defined by Tχ = Rep(A). Consequently, the sets gen(τ ), τ ∈ Top(A) define a topology on Top(A). Corollary 3.1 The gen-topology. F o as in Proposition 3.1 is continuous in the gen-topology. For γ in Top(A) we have the reflector aγ : Rep(A) → (Rep(A), γ ) and the associated map aγo : Tors(Rep(A), γ ) → Top(A). Note that the forgetful functor : (Rep(A), γ ) → Rep(A) is not exact in general, so it does not yield an associated map Top(A) → Tors(Rep(A), γ ). In the part of this section we only consider GC representations that are faithful. The categories considered are assumed to have a zero object. Proposition 3.2 Suppose that Rep is spectral and γ ∈ Top(A) for A in R; take τ ∈ gen(γ ) and write τ for f γ (τ ) ∈ Top(A(γ ) and fτ for the corresponding morphism in R, fτ : A(γ ) → A(γ )( τ ), which exists because of the spectral property of Rep applied to A(γ ) in R.

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Then we have: i. Rep( f τγ )( A(τ ) A(τ )) = Qτ ( A(γ ) A(γ )). ii. Rep( f γ )(Qτ ( A(γ ) A(γ ))) = Q τ ( A A). Proof i. By the spectral property of Rep, there exists an A(γ )( τ ) in R together with a morphism fτ : A(γ ) −→ A(γ )( τ ) with corresponding morphism in Rep(A(γ )), fτ∗ say, fτ∗ : A(γ ) A(γ ) −→ Rep( fτ )( A(γ )(τ ) A(γ )( τ )), which is exactly the localization homomorphism in Rep(A(γ )), A(γ ) A(γ ) −→ Qτ ( A(γ ) A(γ )). On the other hand we may consider: Rep( f γ )( fτ∗ ) : Q γ ( A A) → Rep( f γ )(Qτ ( A(γ ) A(γ ))), the latter being equal to Rep( f γ )Rep( fτ )( A(γ )(τ ) A(γ )( τ )) = Rep( fτ f γ )( A(γ )(τ ) A(γ )( τ )). From the composition A −→ A(γ ) −→ A(γ )( τ ), we obtain: fγ

fτ

Rep( f γ )( fτ∗ ) f γ∗ : A A −→ ∗

Rep( fτ f γ )( A(γ )(τ ) A(γ )( τ )) ∼ = Rep( f γ )(Qτ ( A(γ ) A(γ )))

Now A(τ ) A(τ ) in Rep(A(τ )) is such that the localization morphism A A → is exactly given by f τ∗ :A A −→ Rep(τ ) Q τ (A) ( A(τ ) A(τ )). Consider Mτ (γ ) in rep(A(γ )) defined as follows: Mτ (γ ) = Rep( f τγ )( A(τ ) A(τ )) and let f τ∗γ : A(γ ) A(γ ) −→ Mτ (γ ) be the morphism in Rep(A(γ )) obtained from the measuring property of Rep. Obviously, τ (Mτ (γ )) ⊂ Mτ (γ ) in Rep(A(γ )). The definition of τ yields: Rep( f γ )( τ Mτ (γ )) is τ -torsion in Rep(A), and moreover, the exactness of Rep( f γ ) yields Rep( f γ )( τ Mτ (γ ) ⊂ Rep( f γ )(Mτ (γ )) = Rep( f τ )( A(τ ) A(τ )), where the τ Mτ (γ )) latter is τ -torsion free because it equals Q τ ( A A). Thus Rep( f γ )( = 0 and the faithfulness of Rep then implies that τ Mτ (γ ) = 0. It follows that τ ( A(γ ) A(γ )). we may factorize f τ∗γ | A(γ ) A(γ ) → Mτ (γ ), via Bτ (γ ) = A(γ ) A(γ )/ So we have the following sequence in Rep(A): A A −→ f γ∗

Q γ ( A A)

−→

Rep( f γ )( f τ∗γ )

Q τ ( A A)

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83

It is therefore clear that Rep( f γ )(Mτ (γ )/Bτ (γ )) is τ -torsion in Rep(A) because the co-kernel of Rep( f γ )( f τ∗γ ) is τ -torsion, then Mτ (γ )/Bτ (γ ) is τ -torsion in Rep(A(γ )). It follows that Mτ (γ ) ⊂ Qτ ( A(γ ) A(γ )) in Rep(A(γ )). The definition of Qτ ( A(γ ) A(γ )) makes it τ -torsion over Bτ (γ ) in Rep(A(γ )); consequently Rep( f γ )(Qτ ( A(γ ) (A(γ ))) is contained in Q τ ( A A) as it is τ -torsion over A A/τ ( A A). Since Mτ (γ ) ⊂ Q τ ( A(γ ) A(γ )), the functor Rep( f γ ) takes the value Q τ (A) for both objects, so the faithfulness and exactness of Rep( f γ ) entails that Mτ (γ ) = Qτ ( A(γ ) A(γ )). This establishes i above. Observe also that (*) entails that the morphisms Rep( f γ )( fτ∗ ) f γ∗ and Rep( f γ )( f τ∗γ ) f γ∗ are the same. ii. This follows from i by applying Rep( f γ ) to both members and then again applying the exactness and faithfulness of Rep( f γ ).

Corollary 3.2 i. Consider δ ≤ τ, γ ≤ τ in Top(A) and τ1 ∈ Top(A(δ)), τ2 ∈ Top(A(γ )) constructed as before (we prefer to write τ1 , τ2 rather than τδ , τγ ). We obtain the following commutative diagram of morphisms in R: A(δ)(τ1)

A(γ)(τ~2)

A(τ) A(δ )

A(γ) A

ii. Consider the following objects: M1 = A(δ)(τ1 ) A(δ)( τ1 ) in Rep (A(δ)( τ1 )) M2 = A(γ )(τ2 ) A(γ )( τ2 ) in Rep (A(γ )( τ2 )) M = A(τ ) A(τ )in Rep (A(τ )) Then the following relations hold: a. Rep( fτ1 )(M1 ) = Qτ ( A(δ) A(δ)) = Rep( f τδ )(M) b. Rep( fτ2 )(M2 ) = Qτ ( A(γ ) A(γ )) = Rep( f τγ )(M) c. Rep( f δ )Rep( f τ1 )(M1 ) = Q τ ( A A) = Rep( f γ )Rep( fτ2 )(M2 ) = Rep( f τ )( A(τ ) A(τ ) = Rep( f τ )(M) Proof Apply the foregoing result (twice).

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The foregoing may be compared to the classical fact that for a ring R and τ ≥ γ in Tors(R−mod) we have Q τ (R) = Qτ Q γ (R); in our abstract setting Rep(A(γ )) replaces Q γ (R)-mod. This shows that we have traced exactly the property of a GC representation, that is, spectrality, necessary to extend the foregoing classical fact to the general categorical situation. Note that A(δ)( τ1 ) and A(γ )( τ2 ) in the foregoing corollary need not be isomorphic in R (the relations in the corollary sum up what we do know). Write Top(A)o for the opposite lattice of Top(A). We would like to consider the functor A P : Top(A) → Rep(A), τ → Q τ ( A A) as a structural presheaf (or in fact a sheaf) for A A with values in Rep(A). Exactly the spectral property of Rep would then allow us to “realize” this structure sheaf in R by considering P : Top(A) → R, τ → A(τ ) with structure morphism f τ : A → A(τ ). It is clear that P(ξ0 (A)) = A with I A : A → A as the structure morphism. Moreover, for γ ≤ τ in Top(A) we take f τγ : A(γ ) → A(τ ) for the restriction morphism from γ to τ (in Top(A)o the partial order is reversed when viewing γ and τ in Top(A) as “opens”). For P to be a presheaf we do need an extra property for Rep! Definition 3.4 A spectral Grothendieck representation Rep is said to be schematic if for every triple γ ≤ τ ≤ δ in Top(A), for every A in R, we have a commutative diagram in R: A(γ) fτγ

fγ fδγ

A

A(τ) fδτ

fδ

A(δ)

Proposition 3.3 If Rep is schematic, then with notation as before, P : Top(A) → R is a presheaf with values in R over the lattice Top(A)o , for every A in R. Proof The composition property of sections follows from f δγ = f δτ f τγ , and the claim follows easily. Note that for a schematic GC representation Rep, the structure presheaf obtained in Proposition 3.3 is constructed via localizations in the representing Grothendieck categories described as in Proposition 3.2. In the foregoing we have restricted attention to Tors(Rep( A)); that is, we considered a lattice in the usual sense; hence this should be viewed as the commutative shadow of a suitable noncommutative theory. For A in R we shall write Q(A) for the set of preradicals (see remark before Definition 2.6; note Observation 2.4 too). Warning, in earlier work we (and several other authors) have approached hereditary torsion theory

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via radicals; that is, via the opposite Q(A)op , the notion Q(A) expresses the topology aspects of the theory more directly! Applying definitions (e.g., 2.6) and properties of preradicals derived in Section 2.6 to the Grothendieck category C = Rep(A), we obtain the complete lattice Q(A) and a duality expressed by an order-reversing bijection: (−)−1 : Q(A) → Q((Rep(A))o ). First let us point out that (Rep(A))o is not a Grothendieck category! It is additive and has a projective generator; moreover, it is known to be a varietal category (also called triplable) in the sense that it has a projective regular generator P, it is co-complete and has kernel pairs with respect to the functor Hom(P, −), and moreover every equivalence relation in the category is a kernel pair. A concrete description of the opposite of a Grothendieck category is given by U. Oberst (“Duality Theory for Grothendieck Categories and Linearly Compact Rings,” J. Algebra 15, 1970, 473–542) but sacrificing the varietal aspect for a topological approach. General localization techniques can be developed via the Eilenberg-Moore category of a triple (S. MacLane, Categories for the Working Mathematician, Springer-Verlag, Berlin, New York, 1971 [28]). The latter depends on the so-called comparison functor constructed via Hom(P, −) as a functor to the category of sets. It works well for the category of set-valued sheaves over a Grothendieck topology. As yet we have not investigated whether the approach via the Eilenberg-Moore category remains valid in the case of a noncommutative Grothendieck topology; this may be an interesting project of abstract value. Now (−)−1 defined as an order-reversing bijection between idempotent radicals on Rep(A) and (Rep(A))o , we write (Top(A))−1 for the image of Top( A) in Q((Rep(A))o ). This is encoded in the exact sequence in Rep(A): 0 −→ ρ(M) −→ M −→ ρ −1 (M) −→ 0 (reversed in (Rep(A))o ). By restricting attention to hereditary torsion theories (kernel functors) when defining Tors(−), we introduce an asymmetry that breaks the duality because Top(A)−1 is not in Tors((Rep(A))op ). Write T T (G) for the complete lattice of torsion theories (not necessarily hereditary) of the category G; then (T T (G))−1 ∼ = T T (G op ). Hence we may view Tors(G)−1 as a complete sublattice of T T (G op ). For preradicals ρ1 and ρ2 in Q(G) we have defined the lattice operations ρ1 ∧ preceding ρ2 , ρ1 ∨ρ2 , as well as the product ρ1 ρ2 . Inspired by the duality (see remarks Definition 2.6 and those following Proposition 2.11) we define ρ ρ = ρ1 : ρ2 . 1 2 Hence an object M of G is ρ1 ρ2 -torsion if and only if there is a subobject N ⊂ M such that N is ρ1 -torsion and M/N is ρ2 -torsion. The notation suggests that it is topologically an intersection, but as a preradical ρ1 ρ2 islarger than ρ1 and ρ2 . Let o op us denote the similar operation but defined we let −1in Q((G) )o by−1 ; for σ, τ ∈ Q(G) −1 σ . The notation suggests σ τ be the preradical such that (σ τ ) = τ that it is topologically a (noncommutative) union. Proposition 3.4 With notation as above: a. For σ, τ ∈ Q(G), Tσ τ = Tσ ∩ Tτ . Clearly σ τ ≤ σ ∧ τ . If σ ∧ τ is idempotent, then σ τ = τ σ = σ ∧ τ = τ ∧ σ . In particular, when σ and τ are left exact preradicals, then σ τ = τ σ = σ ∧ τ , and this is a left exact preradical. Also if

86

Grothendieck Categorical Representations τ is left exact and σ is idempotent, then σ τ is idempotent and σ τ = (σ ∧ τ )o (notation of Theorem 2.3(v)). b. If σ and τ are idempotent preradicals, then σ τ and τ σare idempotent o o preradicals. When only σ is idempotent, then (σ τ ) = σ τ . If σ and τ are left exact preradicals, then σ τ andτ σ are left exact preradicals too. Note also that Fσ τ = Fσ ∩ Fτ (but σ τ is not determined by Fσ τ ).

Proof a. The statements are obvious; let us establish the last one. Let τ be left exact, σ idempotent. For any M in G we have σ τ (M) = σ (τ (M)), hence a subobject of τ (M); therefore the left exactness of τ implies that σ τ (M) is in the τ -pretorsion class. Then σ τ (σ τ (M)) = σ (σ τ (M)) = σ τ (M); hence σ τ is an idempotent preradical. Therefore (σ ∧ τ )o ≤ σ τ . As observed before, σ τ (M) is in Tσ as well as in Tτ (τ is left exact); thus σ τ ≤ σ ∧ τ is clear; the definition of (σ ∧ τ )o then implies that σ τ = (σ ∧ τ )o . b. Let σ and τ be idempotent preradicals. By definition (σ τ )(M) = N , the largest subobject of M such that M ⊃ N ⊃ σ (M) with N /σ (M) being τ pretorsion. Since σ is idempotent σ (N ) = σ (M) hence (σ τ )(N ) = N1 , the largest subobject of N , N ⊃ N1 ⊃ σ (N ) = σ (M) such that N1 /σ (M) is τ -pretorsion. However, the latter implies N1 = N in view of the foregoing. In case σ is idempotentbut not necessarily τ , then by the foregoing σ τ o is idempotent, hence σ τ o ≤ (σ τ )o . Since both idempotent preradicals correspond to the same pretorsion class, it follows that σ τ o = (σ τ )o . In case both σ and τ are left exact, hence certainly idempotent, we have that σ τ (and τ σ ) is idempotent. is a Hence it suffices to check that Tσ τ hereditary class (the result for τ σ follows by symmetry). If M ∈ Tσ τ , then M/σ (M) is τ -pretorsion; consider a subobject M of M in G. Since σ is left exact, M ∩ σ (M) = σ (M ). Thus M /σ (M ) is isomorphic to a subobject of M/σ (M) and therefore the left exactness of τ entails that M /σ (M ) ∈ Tτ or M ∈ Tσ τ . Consequently σ τ is a left exact preradical. Proposition 3.5 With notation as before: a. If σ, τ ∈ Q(G) are radicals, then σ τ is a radical; when τ is a radical, then for any σ ∈ Q(G), (σ τ )c = σ c τ . b. If σ, τ are radicals, then σ τ and τ σ are radicals. If moreover σ and τ are kernel functors, then σ τ = τ σ = σ ∧ τ (= σ τ = τ σ in view of Proposition 3.4). Proof a. The statements in a follow by duality from the statements in Proposition 3.4a. b. The first statement follows by duality from the first statement in Proposition 3.4b. Before dealing with the second statement, let us describe Fσ τ ;

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now that we are considering radicals σ, τ and σ τ is a radical too, they are determined by their pre-torsion-free classes. By dualization of the definition of o in Q(G op ), we observe that an object of G, M say, is in Fσ τ if it fits in a G-exact sequence: 0 −→ N −→ M −→ M/N −→ 0 with N ∈ Fσ and M/N ∈ Fτ . First let us check that σ τ is a torsion theory, that is, that Fσ τ is closed under extensions. So, suppose we are given the following exact sequences in G: 0 −→ M1 −→ M −→ M2 −→ 0 π

0 −→ N1 −→ M1 −→ M1 /N1 −→ 0 π1

0 −→ N2 −→ M2 −→ M2 /N2 −→ 0 π2

where N1 and N2 are in Fσ , M1 /N1 and M2 /N2 are in Fτ . View N1 as a subobject of M and let X ⊂ M be the subobject such that X/N1 = τ (M/N1 ). Observe that X ∩ M1 = N1 because X ∩ M1 /N1 is isomorphic to a subobject of X/N1 , and hence a τ -torsion object because τ is a hereditary torsion theory radical, while on the other hand it is isomorphic to a subobject of M1 /N1 , hence τ -torsion free. Since π (X ) is a quotient of X/N1 it must be τ -torsion. But then π2 (π (X )) = 0; hence π (X ) ⊂ N2 , and we obtain an exact sequence in G: 0 −→ N1 = X ∩ M1 −→ X −→ π (X ) −→ 0 resπ

with N1 and π (X ) in Fσ . Consequently, X ∈ Fσ because it is closed under extensions. We also obtain the exact sequence: 0 −→ M1 /N1 −→ M/ X −→ M2 /N2 −→ 0 γ

where γ is a factorization of π2 π obtained from π (X ) ⊂ N2 . Since M1 /N1 and M2 /N2 are in Fτ , so must be M/ X . In the exact sequence in G : 0 −→ X −→ M −→ M/ X −→ 0 we now have X ∈ Fσ and M/ X ∈ Fτ , that σ τ is a torsion thus M ∈ Fσ τ . This establishes theory. We can go on and establish that σ τ is left exact and that σ τ is left exact and that σ τ = τ σ , but all of this will follow if we establish directly that σ τ = σ ∧ τ (∧ denoting the lattice operation in the lattice of idempotent preradicals, being the same as the one inthe lattice of idempotent radicals, that is, torsion theories). Since σ ∧ τ and σ τ are radicals, it suffices to establish that Fσ ∧τ = Fσ τ . Start with M ∈ Fσ τ and assume (σ ∧ τ )(M) = 0; put X = (σ ∧ τ )(M) ⊂ τ (M). We know there exists an exact sequence in G : 0 −→ N −→ M −→ M/N −→ 0, with N ∈ Fσ and M/N ∈ Fτ . Since γ

τ is left exact, X ⊂ τ (M) entails that X ∈ Tτ , hence γ (X ) ∈ Tτ , but as γ (X ) ⊂ M/N this entails γ (X ) = 0 or X ⊂ N . Then X ⊂ σ (M) leads to X being σ -torsion; hence X = 0 as N ∈ Fσ , that is, (σ ∧ τ )(M) = 0.

88

Grothendieck Categorical Representations Conversely, start from M ∈ Fσ ∧τ . If σ (M) = 0, then τ (σ (M)) = 0 because otherwise τ σ (M) = 0 would be σ ∧ τ -torsion (Proposition 3.4a.), σ τ and τ σ are ≤ σ ∧ τ ) in M, a contradiction. Of course M/σ (M) is in Fσ because σ is a radical and so, from the exact sequence in G: 0 −→ σ (M) −→ M −→ M/σ (M) −→ 0 with σ (M) ∈ Fτ and M/σ (M) ∈ Fσ , M ∈ Fσ τ follows. Combining both parts yields Fσ τ = Fσ ∧τ .

In the last part of the proof, symmetry has obviously been broken; indeed, for nonhereditary torsion theories we cannot use the proof above to arrive at commutativity of on torsion theories. The problem is that for (nonhereditary) torsion theories we do not know whether τ −1 ◦ σ −1 is radical. Since Fτ −1 o σ −1 = Fτ −1 ∩ Fσ −1 duality implies that Tσ τ = Tσ ∩ Tτ ; hence if σ τ isidempotent, or if σ τ or τ σ is idempotent (note that Tσ τ = Tσ ∩ Tτ too!), then σ τ = σ τ = τ σ = σ ∧ τ . In particular, when we are only interested in left exact preradicals we might define σ τ = σ τ from the beginning, neglecting the duality with the (noncommutative) topological intersection in Q(G op ). In the philosophy of the pattern as in Chapter 2 we are interested in brack topology eted expressions involving , , and the -idempotent elements (hence radicals). So we look atQh (G),the setof hereditary preradicals of G; this is a subset of Q(G) closed under and (and agrees with the preradical product and is a commutative operation in o = Q h (G)). Clearly i (Qh (G)) consists of theleft exact radicals, that is, the kernel functors. Conversely, bracketed expressions p( , , σi ) (notation of Section 2.2. see after Lemma 2.12) with σi ∈ id (), always yield left exact preradicals, hence it is in complete agreement with our interest in noncommutative topologies generated by their intersection-idempotent elements, to look at Qh (G). We write T for the set of left exact preradicals obtained as finite bracketed expressions as defined above. Are and T now noncommutative topologies, in fact topologies of virtual opens? This follows from the following easy lemma. Lemma 3.1 With notation and conventions as before: 1. and T , have 1 and 0 (the zero preradical, respectively the identity functor) 2. If σ, τ, γ are left exact preradicals, then (σ τ )(γ ) = σ (τ γ ). 3. If σ ≤ τ and γ are left exact preradicals, then σ γ ≤ τ γ , γ σ ≤ γ τ . 4. If σ is a left exact preradical such that σ n = 0, then σ = 0 (observe that Tσ n = Tσ ). 5. If σ σ . . . σ = 1, then σ = 1. 6. If σ ≤ τ and γ are left exact preradicals, then we have: γ σ ≤ γ τ, π γ ≤ τ γ. 7. For left exact preradicals σ, τ, γ , (σ τ ) γ = σ (τ γ ).

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8. For σ, τ, γ as before: (σ τ )γ = (σ γ τ )γ . 9. For σ, τ γ as before: (σ τ )γ = σ γ τ γ whenever γ is a radical, that is, in id (). 10. σ τ σ = σ = σ τ σ , whenever σ is radical. 11. σ (σ τ ) = σ = (τ σ )σ . 12. For left exact preradicals σ τ = τ σ (see earlier). 13. satisfies the F D I property as defined after Definition 1.9, (hence satisfies axiom A.10 as defined before Definition 1.9, as well as axiom VOT.3 as defined at the beginning of Section 1.2). Proof All properties follow easily from the preradical calculus; only 13 may need a little explanation. Recall that σ ≤ τ is a focused relation if σ τ = τ σ = σ (note that we are using ≤ in , which is poset opposite to Qh (G)); the FDI property holds if for a focused relation σ ≤ λ with λ = λ1 λ2 we have σ = (σ λ1 ) (σ λ 2 ). Now since σ ≤ λ is focused, we have Tσ = Tσ λ = Tλ σ and because λ = λ1 λ2 we obtain Tλ = Tλ1 ∩ Tλ2 . Therefore, we obtain Tσ = Tσ λ ∩ Tσ λ ⊂ Tσ λ1 ∩ Tσ λ2 . For the converse, look at an object M in the latter. That is, we have: M ⊃ M1 ⊃ 0 with M1 being λ1 -torsion and M/M1 being σ -torsion M ⊃ M2 ⊃ 0 with M2 being λ2 -torsion and M/M2 being σ -torsion Look at M ⊃ M1 ∩ M2 ⊃ 0; since we are considering left exact preradicals, M1 ∩ M2 is in Tλ1 ∩ Tλ2 = Tλ , while on the other hand M/M1 ∩ M2 is σ -torsion (it embeds in M/M1 ⊕ M/M2 ). We arrive at M ⊂ Tσ λ = Tσ . Corollary 3.3 is a noncommutative topology; T is a topology of virtual opens. The commutative shadow of (and then also of T ) is Tors(G) with its usual lattice operations (∧ = . , ∨ = , in the notation of Proposition 2.1). Returning to the setting of Grothendieck representations, we have for every object A of R, the Grothendieck category Rep(A) and the set of preradicals Q(A) of Rep(A) containing Top(A) = Tors(Rep(A)). The foregoing construction of (A) in Qh (A) leads to a noncommutative topology (A) having Top(A) = id ((A)) with its canonical lattice structure for the commutative shadow. The behavior of (A) with respect to morphisms B → A in R would be satisfactory if we can generalize Proposition 3.1 because then a GC representation leads to canonical topologization of the objects of R by noncommutative topologies having the classical lattice of kernel functors for its commutative shadows. A noncommutative space (spectrum) could then be understood as a localization functor on the level of Rep(A)-valued sheaves on (A), using structure sheaves, with natural transforms between these functors corresponding to morphisms in the base category R.

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To a morphism f : S → R in R we have associated a functor F = Rep( f ) : Rep(R) → Rep(S), which is exact and commutates with coproducts. Define a map : Qid (S) → Qid (R), where Qid denotes the lattice of idempotent preradicals by F ) defined by the pretorsion associating to γ ∈ Qid (S) the idempotent preradical F(γ class of objects X in Rep(R) such that F(X ) ∈ Tγ . When F derives from f , we shall in order to highlight the connection. also write f for F Lemma 3.2 maps Qh (S) to Qh (R). With notation as introduced above: F Proof Take γ ∈ Qh (S) and look at M ∈ T F(γ ) and a subobject N of M. Thus F(M) ∈ Tγ and because F is exact we have F(N ) ⊂ F(M); the fact that γ is hereditary then ). yields that F(N ) ∈ Tγ and hence N ∈ T F(γ ) by definition of F(γ Proposition 3.6 Consider any functor F : Rep(R) → Rep(S) such that F is exact and commutes with has the following properties: coproducts; then F : Qh (S) → Qh (R) to Top(S), then we obtain F o , as defined 1. If we restrict F before Definition 3.3. is a poset morphism. 2. F {ϕ ∈ F}) = ∧{ F(ϕ), 3. If F ⊂ Qh (S), then F(∧ ϕ ∈ F}. ϕ

ϕ

Proof Easy enough. On Qh (A) for any A in R we define the gen-topology in formally the same way as it was introduced on Top(A), that is, for ρ ∈ Qh (A) we let gen(ρ) = {τ ∈ Qh (A), ρ ≤ τ }. For any F ⊂ Qh (A) we have: gen(∧{ϕ, ϕ ∈ F}) ⊃ ∪{gen(ϕ), ϕ ∈ F}, gen(∨{ϕ, ϕ ∈ F}) = ∩{gen(ϕ), ϕ ∈ F} Together with the existence of a minimal left exact preradical ξ and a maximal one χ , the foregoing relations do establish that the sets gen(ϕ), ϕ ∈ Qh (A) generate (the open sets of) a topology on Qh (A). This topology induces on Top( A) the gen-topology of Top(A) (see also Corollary 3.1). Proposition 3.7 In the situation of Proposition 3.6 −1 (gen(ρ)) = gen(ξρ ), where ξρ is the preradical corre1. For ρ ∈ Qh (R), ( F) sponding to the hereditary pretorsion class generated by the F(Tρ ), Tρ ∈ Tρ . 2. F ∧ : Qh (S) → Qh (R) is continuous in the gen-topology. Proof Straightforward (like Proposition 3.1).

3.1 Spectral Representations With respect to the operations behavior of F.

and

91

, we have the following rules for the

Proposition 3.8 Consider an exact functor, commuting with coproducts: F : Rep(R) → Rep(S), and : Qh (S) → Qh (R) be the corresponding poset map. For σ, τ in Qh (S) we have: let F ) )=F σ F(σ F(τ τ , ) )≤F σ F(σ F(τ τ Proof Theproperty with respect to follows from Proposition 3.6(3) (in fact it extends coincides with ∧ on left exact to of a family F ⊂ Qh (S)) and the fact that ) F(τ ), then we have an preradicals. If M in Rep(R) is a pretorsion object for F(σ exact sequence in Rep(R): )(M) −→ M −→ M/ F(σ )(M) −→ 0 0 −→ F(σ )(M) is F(τ )-pretorsion. By exactness of F we then obtain an exact where M/ F(σ sequence in Rep(S): )(M)) −→ F(M) −→ F(M)/F( F(σ )(M)) −→ 0 0 −→ F( F(σ )(M)) is σ -pretorsion by definition of F(σ ), and F(M)/F( F(σ )(M)) where F( F(σ is τ -pretorsion by definition of F(τ ) (and exactness of F). Consequently, F(M) ∈ Tσ τ or M is F(σ τ )-pretorsion. ); hence for an arbitrary morphism ) F(τ In Qh (S)op we have F(σ τ ) ≤ F(σ f : S → R we need not obtain a map F taking (S) to (R). This is related to the fundamental problem concerning functoriality, also appearing in noncommutative geometry (scheme theory for associative algebras). Certain morphisms yield better e.g. when f in an epimorphism in R with Rep( f ) = F being a full behavior of F, functor. Definition 3.5 Given a morphism f : S → R in R, then σ ∈ Qh (S) is said to center f if σ F = ) (composition of functors written in antiorder of application): Rep(R) → F F(σ )(M)). The set of all σ in Qh (S) that center f will be Rep(S), M → σ F(M) = F( F(σ denoted by Z h ( f ). For Z n ( f ) ∩ Top(S) we write Z top ( f ) and Z h ( f ) ∩ (S) = Z ( f ). Corollary 3.4 With notation as above: ) 1. If σ ∈ Z h ( f ), then for any τ in Qh (S) we have: F(σ ) ∨ F(τ ) = F(σ ∨ τ ). F(σ

) = F(σ F(τ

τ ),

92

Grothendieck Categorical Representations ) 2. If σ, τ ∈ Z h ( f ), then we obtain the following equalities: F(σ F(σ τ ), F(τ ) F(σ ) = F(τ σ ), F(σ ) ∨ F(τ ) = F(σ ∨ τ ).

) = F(τ

Proof Clearly Proposition 3.8 that 1 so let us prove 1. We know from 2 follows from ) ≤ F(σ ) F(τ τ ). Consider M in Rep(R) that is F(σ τ )-pretorsion, that F(σ is, F(M) ∈ Tσ τ . We arrive at the existence of an exact sequence in Rep(S): 0 −→ σ (F(M)) −→ F(M) −→ F(M)/σ (F(M)) −→ 0 where F(M)/σ (F(M)) is τ -pretorsion. )(M)). Therefore we have that F(M)/σ (F(M)) = Since σ ∈ Z h ( f ), σ (F(M)) = F( F(σ )(M)) and consequently M/ F(σ )(M) is F(τ )-pretorsion. From the exact F(M/ F(σ sequence in Rep(R): (M)) −→ M −→ M/ F(σ )(M) −→ 0 0 −→ F(σ ) F(τ ). The statement we may conclude that M is in the pretorsion class for F(σ concerning the commutative operation ∨ follows in a similar way. Proposition 3.9 For any morphism, f : S → R, Z h ( f ) is a lattice with respect to ∧(= ) and ∨. Moreover, is inner in Z h ( f ). Proof Using Corollary 3.4 we obtain: for σ, τ ∈ Z h ( f ), (σ ∧ τ )(F) = σ τ F = σ (τ F) = )) = σ F F(τ ) = F(σ ) F(τ ) = F(σ ) ∧ F(τ ). Now consider M in Rep(R); σ (F F(τ then we have an exact sequence in Rep(R): )(M) −→ F(σ ) ) (M) −→ F(τ )(M/ F(σ )(M)) −→ 0. 0 −→ F(σ F(τ By applying the exact functor F we obtain an exact sequence in Rep(S), where we ) F(τ ))(M) in F(M), put X = F( F(σ )(M)) −→ 0, or 0 −→ σ F(M) −→ X −→ τ F(M/ F(σ 0 −→ σ F(M) −→ X −→ τ (F(M)/σ F(M)) −→ 0 By definition of , (σ τ )(F(M)) is defined by the exact sequence: 0 −→ σ F(M) −→ σ τ )(F(M) −→ τ (F(M)/σ F(M)) −→ 0 It follows that X = (σ τ )(F(M). This leads to the equalities: ) F(τ )) = F F(σ F( F(σ τ ) = (σ τ )F. The statements with respect to ∨ follow in a similar way.

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Corollary 3.5 Z h ( f ), as well as Z top ( f ), Z ( f ), have the structure of a noncommutative topology induced by the corresponding structure on Qh (S). We have before avoided working with filters of left ideals in a ring in order to try to describe the noncommutative intersection in case of a ring A and torsion theories on A-mod. Since the preradicals of interest appear as compositions of torsion theories, it may be interesting anyway to mention a few technical facts. First, let us stay in the generality of a given Grothendieck category G. For torsion theories τ and κ on G, we define a class Tκτ as the class of objects M in G such that there is a subobject N of M; N is κ-torsion and M/N is τ -torsion, or equivalently M/κ M is τ -torsion. The class Tκτ is closed for taking subobjects, direct sums, and images but not necessarily closed under extensions. Of course Tτ κ ⊃ Tκ , Tτ and similar for Tκτ . Let us rephrase some of our statements about the closure operator (avoiding the terminology of Proposition 3.4 and following) and let Tκτid be the closure of Tκτ under extensions; then Tκτid is a torsion class for a hereditary torsion theory on G. The following lemma is along the lines of earlier observations. Lemma 3.3 With notations as above, Tτidκ = Tκτid . If τ κ and κτ are idempotent, that is, Tτidκ = Tτ κ , respectively Tκτid = Tκτ , then τ and κ are compatible, that is, τ κ = κτ and Q τ Q κ = Qκ Qτ . Proof We know this already; however, observe that we can obtain Tτidκ as the union of T(τ κ)...(τ κ) for compositions of finitely many factors τ κ, and similar for Tκτid . Hence first Tτidκ ⊃ Tκτ follows and then also Tτidκ ⊃ Tκτid . Symmetry in τ and κ then yields Tτidκ = Tκτid . For M 0, m < 0; thus δ = 0 if and only if R is either positively or negatively graded depending upon whether R1 = 0 or R−1 = 0. We deal with the positively graded case; the other case is similar. Put R = ⊕n≥0 Rn , I = ⊕n>0 Rn = R+ . The assumption I∈ Lg (τ ), τ being perfect, yields S = S I . Hence S0 = n>0 S−n Tn = S R . Look at s r with s ∈ S , R ∈ R . For some L ∈ Lg (τ ) −n n −n n −n −n n n n>0 we have that L p s−n rn ⊃ R p−n Rn ⊂ R p

(∗) thus for any p > 0:

p−1

S− p L p s−n rn ⊂ S− p T p = S− p R1

R1 ⊂ S−1 R1 ⊂ S−1 S1

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Observe that L ⊂ Lg (τ )); consequently, p>0 S− p L p = (S L)0 = S0 (as S0 s−n rn ⊂ S−1 S1 with n arbitrary. From S0 = n>0 S−n Rn we obtain S0 = S−1 S1 . Note that in (*) L 0 = 0 because if L ∈ Lg (τ ), then I ∩ L ∈ Lg (τ ) with (I ∩ L)0 = 0, so we may replace L by L ∩ I without loss of generality. Observe that the foregoing does not imply that also S1 S−1 = S0 ! However, when τ is associated with a homogeneous nontrivial (left) Ore set T , then for y ∈ S0 look at ytm with tm ∈ T ∩ Rm for m > 0. Since tm is invertible in S with tm−1 ∈ S−m we may look at (ytm )tm−1 = y ∈ Sm S−m . Now from S−1 S1 = S0 we derive that Sm = S1m (because Sm = Sm S0 = Sm S−1 S1 , thus Sm = Sm−1 S1 and by repetition of this argumentation we obtain Sm = S1m ). Finally we obtain y ∈ S1m S−m = S1 (S1m−1 S−m ) ⊂ S1 S−1 and consequently S0 = S1 S−1 as desired. To the graded ring R we associate a rigid torsion theory κ R defined by its graded filter Lg (κ R ), which is the graded filter (Gabriel topology) generated by Rδ and I . Observe that I = δ ⊕ (⊕n=0 Rn ) is automatically in Lg (κ R ) because it contains Rδ when δ = 0. The situation of geometrically graded rings R with rigid torsion theories κ R is also interesting because it provides us with a new example of a topological nerve. Lemma 3.5 Let R and S be Z-graded rings such that δ R and δ S are both either nonzero or both zero. If f : R → S is a morphism of graded rings, then κ S ≤ f (κ R ). Proof Since f (Rn ) ⊂ Sn for every n ∈ Z, it is clear that f (δ R ) ⊂ δ S . By definition L ∈ L( f (κ R )) means that S/L is κ R -torsion as an R-module; that is, L contains some L , L ∈ L(κ R ). Consider the category B of Z-graded rings R with δ R = 0, taking just graded ring morphisms for the morphisms. Associating R-gr to R defines a Grothendieck representation such that {κ R , R ∈ B} is ε nerve. Therefore we may consider the quotient Grothendieck representation with respect to the nerve {κ R , R ∈ B}. Exercise 3.2 Develop the noncommutative geometry of “ProjR”, which is defined by the noncommutative topology of the quotient category (R-gr, κ R ), together with the corresponding sheaf theory. Define schematically graded rings as the class of graded (Noetherian) rings R such that there exists a finite set of homogeneous Ore sets T1 , . . . , Tn such that κ R = κT1 ∧. . .∧κTm ; suitable generalizations may be defined by replacing the κTi by perfect rigid torsion theories not necessarily stemming from Ore sets. In this case ProjR defined on (R-gr, κ R ) satisfies all properties valid in the positively graded case. In particular, the schematic condition entails the existence of an affine cover (invoking Proposition 3.16). It is possible to establish a proof of Serre’s global section theorem for ProjR. A new ingredient in this project consists in the study of the relation between the affine geometry of R0 , in terms of SpecR0 say, and the projective geometry in

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terms of ProjR, that is, (R-gr, κ R ). A subproject of this consists of a concrete algebraic approach when R is a ring satisfying polynomial identities (in particular when R is a finite module over its center) where a relation with the theory of maximal (R0 -)orders has to be investigated. More concretely, study the geometry when R is an R0 -order, R0 is a Noetherian integrally closed domain of dimension n and δ defines a closed subvariety of SpecR0 of dimension n 1 < n. Even more concrete, n = 1 and n 0 = 0, or n = 2 and n 1 = 1. There are new phenomena here when compared to the theory started in [47] or in L. Le Bruyn, M. Van den Bergh, and F. Van Oystaeyen, Graded Orders, Birkhauser Monographs (xxxx).

3.4

Noncommutative Projective Space

As an example of the quotient representations introduced in the foregoing section we point out how the construction of projective spaces fits in that theory. For the category R we now restrict attention to the category of positively graded k-algebras with graded k-algebra morphisms of degree zero for the morphisms; recall that a graded k-algebra is said to be connected if its part of degree zero is k; that is, A is a graded connected kalgebra if A = k⊕ A1 ⊕ A2 ⊕. . . . For geometry-oriented purposes we restrict attention to finite gradations in the sense that each An is a finite dimensional k-space and A is generated as a k-algebra by A1 . In that case, the positive part A+ = A1 ⊕ A2 ⊕ . . . is finitely generated as a left (or right) ideal of A and moreover A A1 = A+ and the powers Am + form the basis of a Gabriel topology of a torsion theory that we denote by κ+ and we write L(κ+ ) for the Gabriel topology. Any τ ∈ A-tors is said to be rigid if the graded torsion class Tτg is shift invariant; that is, if a graded A-module M is τ -torsion, then for every n ∈ Z : T (n)M is also τ -torsion, and conversely. The set of graded left ideals in L(τ ) is denoted by Lg (τ ); it is called the graded filter or Gabriel topology. Now if we start with τ , even one such that τ (M) is a graded submodule of M whenever M is graded, then τ need not be characterized by Lg (τ ). This is due to the fact that A need not be a generator for A-gr. On the positive side, if τ is rigid, then it is characterized by Lg (τ ). In any case κ+ is a rigid torsion theory, so it is completely determined by the graded Gabriel topology Lg (κ+ ). For an arbitrary graded ring R we denote by R-rig the sublattices of R-tors of rigid graded torsion theories. g Let us write Rk instead of R in this section, in order to reflect the graded character g and to fix the field k. If g : R → S is a morphism in Rk , then g(R+ ) ⊂ S+ . Let us write κ+ (R) respectively κ+ (S) for the rigid graded torsion theory in R-tors, respectively S-tors, associated to R+ , respectively S+ . Obviously, S/S+ is κ+ (R)-torsion, so it follows easily that κ+ (S) ≤ g (κ+ (R)) where g : R-tors→ S-tors corresponds to g. Lemma 3.6 The restriction of g to R-rig defines g : R-rig→ S-rig.

3.4 Noncommutative Projective Space

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Proof Take τ ∈ R-rig and look at g (τ ). If N is a g (τ )-torsion graded S-module, then R N is τ -torsion and every T (m) R N is then τ -torsion because τ is rigid. Now it is clear that R (T (m)N ) = T (m) R N ; thus T (m)N is g (τ )-torsion for every m ∈ Z or g (τ ) is rigid. For a k-algebra A graded as before, we let Proj(A) be the Grothendieck category obtained as the Serre quotient category of finitely generated graded A-modules modulo graded A-modules of finite length, that is, if A-gr f denotes the category of finitely generated graded A-modules, then the localizing functor A-gr f → Proj(A) corresponds to the torsion class of the κ+ -torsion objects, which in this case are finite dimensional over k. The functor A-gr f → Proj(A) defines a lattice morphism: Torsg (Proj(A)) → Torsg (A−gr f ) where Torsg (−) stands for the lattice of graded torsion theories on the category specified. The latter morphism restricts to Rig(Proj(A)) → Rig(A−gr f ), where Rig(−) stands for the lattice of rigid graded torsion theories. For full detail on graded localization theory we refer the reader to C. Nˇastˇasescu and F. Van Oystaeyen, Graded Rings and Modules, LNM 758, Springer Verlag [31], or Graded Ring Theory, North Holland. In this section we restrict attention to the commutative shadow; that is, we deal with the torsion theories and leave the extension to graded radicals and noncommutative topology to the reader (this is a fairly straightforward graded version of the arguments of part of Section 3.1, after Proposition 3.3). Lemma 3.7 With notation as before we have: 1. Rig(A−gr f ) = A-rig. 2. Rig(Proj(A)) = genrig (κ+ ), the latter denoting the set of rigid graded torsion theories τ in A-tors such that τ ≥ κ+ , that is, genrig (κ+ ) = gen(κ+ ) ∩ A-rig. Proof 1. For any torsion theory on modules it is true that a module is torsion if and only if every finitely generated submodule is torsion. Hence the restriction of a torsion class to A-gr f does determine the torsion class in A-gr f . Rigidity of the (graded) torsion class in A-gr is obviously equivalent to the rigidity of the corresponding torsion class in A-gr f . From foregoing observations it follows easily that Rig(A-gr f ) = A-rig. 2. The map Rig(Proj(A)) → A-rig associates to a rigid (graded) torsion theory on the quotient category Proj(A) of A-gr f , the rigid torsion theory it induces on Agr f , and thus on A-gr, that is, an element of gen(κ+ )∩ A-rig. Conversely, that any τ ∈ genrig (κ+ ) induces a rigid torsion theory on Proj(A) follows by checking

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Grothendieck Categorical Representations the transfer of rigidity; the bijective correspondence Tors(Proj(A)) = gen(κ+ ) follows from earlier observations (this also follows from Proposition 2.26) The lattice genrig (κ+ ), or in fact the category corresponding to it in the usual way, may be viewed as the projective version of Top(A) introduced in the ungraded situation as Tors(Rep(A)). So it makes sense to write Topproj (A) = g : Rgenrig (κ+ ). If g : R → S is in Rκg , then we have already introduced rig→ S-rig in Lemma 3.6, which may be viewed as a functor when the lattices are considered as categories in the usual way. However, there is a problem in constructing an associated map: Topproj (R) −→ Topproj (S). Indeed g : R → S does not necessarily define a “restriction of scalars” functor Proj(S) −→ Proj(R)! On the positive side we have g (κ+ (R)) in S-rig such that κ+ (S) ≤ g (κ+ (R)); therefore, the quotient category (S-gr f , g (κ+ (R))) is also a quotient category of Proj(S) such that Rig(S-gr f , g (κ+ (R))) may be identified to genrig ( g (κ+ (R))) in S-rig via the map associated to the localizing g (κ+ (R))). So we may conclude that we obtain a functor S-gr f −→(S-gr f , functor Proj(S), g (κ+ (R)) −→ Proj(R), induced by the restriction of scalars with respect to g. By transitivity of the localization functors associated to κ+ (S) ≤ g (κ+ (R)) we actually find that: (Proj(S), g (κ+ (R))) = (S−gr f , g (κ+ (R))) (or by the compatibility property deriving from κ+ (S) ≤ g (κ+ (R))). In any case, the functor (Proj(S), g (κ+ (R))) −→ Proj(R) induces a lattice morphism: Rig(Proj(R))) −→ Rig(S−gr f , g (κ+ (R))). Observing that Rig(Proj(R)) = genrig (κ+ (R)) in R-rig, Rig(S-gr f , g (κ+ (R))) = genrig ( g (κ+ (R))) in S-rig, we may conclude that the morphism g gives rise to a lattice morphism: genrig (κ+ (R)) −→ genrig ( g (κ+ (R))) → gen(κ+ (S)) defining a lattice morphism Topproj (R) −→ gen( g (κ+ (R))) → Topproj (S). In other words, the lattice morphism that we obtain here is obtained from g but not from a functor Proj(S) → Proj(R). The above phenomenon is also present in the commutative scheme theory. It expresses the fact that, even when suitable localizations do carry over from R to S via g , the scheme theory has to take into account that the underlying topological morphism can only be defined on an open subset of Proj(S) in fact given by gen( g (κ+ (R))) (viewed in the opposite lattice).

The foregoing establishes the following proposition.

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Proposition 3.17 g To a morphism g : R → S in Rk there corresponds a functor (deriving from a lattice morphism) Topproj (R) −→ Topproj (S). An arbitrary g : R → S does not allow us to relate finitely generated (graded) S-modules to finitely generated (graded) R-modules when S itself is not even finitely generated as an R-module. This makes it more natural to consider (A-gr, κ+ (A)) g for any A in Rk , that is, without restricting to A-g f . Let us write PROJ(A) for the latter quotient category. In sheaf theoretical language this would mean that we focus on quasi-coherent sheaves rather than on coherent sheaves. The torsion objects with respect to κ+ (A) in A-gr need not have finite length, but this does not affect any of the statements and results derived earlier. g In the language of Section 3.3, where we put R = Rk , now we consider a g Grothendieck representation associating A-gr (or A-gr f ) to A in Rk . A topological nerve κ+ can now be obtained by letting n A (as in Section 3.3) be κ+ (A) as g defined earlier in this section. For a morphism g : A → B in Rk we do have g (κ+ (A)) and so we arrive at a generalized Grothendieck representathat κ+ (B) ≤ g tion; (A-gr, κ+ (A)) is then associated to A in Rk , which is the quotient generalized Grothendieck representation of Section 3.3 and in particular, from Theorem 3.1 it follows that it is measuring and weakly spectral; moreover, it satisfies the statements of Proposition 3.13. With notation as introduced in this section, the GC representag tion gr, respectively gr f , associating A-gr, respectively A-gr f , to A in Rk , allows the quotient representation PROJ, respectively Proj, associating PROJ(A), respectively g Proj(A) to A in Rk .

3.4.1

Project: Extended Theory for Gabriel Dimension

In Section 2.6 we established how localization functors or torsion theories appear as a major example of noncommutative topology. In fact in view of the constructed scheme theory for schematic algebras (cf. [49]), the latter example has been the main motivation for the introduction of noncommutative topology in a more axiomatic way. Now unlike the Krull dimension, the Gabriel dimension is defined exactly in terms of torsion theories, so it is a possible instrument for calculating certain dimensions of noncommutative algebras, topologies, or other categorical structures. Let us recall some basic facts along the way to describing some possible projects. The name Gabriel dimension is attributed by Gordon, Robson to a notion introduced by P. Gabriel in his thesis, [10], but here termed the Krull dimension. Since several notions of generalized Krull dimension became available later, the different names were used; in the book Dimensions of Ring Theory [33], the notion of Gabriel dimension is given for an arbitrary modular lattice. A first project could be to generalize this to noncommutative topologies or virtual topologies. We do not go into this, but turn to Grothendieck categories instead. Let G be a Grothendieck category. An object M of G is said to be semi-Artinian if for every subobject M of M such that M = M there exists a simple subobject in M/M . The full subcategory of G consisting of all semi-Artinian objects is easily seen to be a localizing subcategory, in other words to determine a torsion theory of G. Indeed it is the smallest localizing subcategory of G containing all the simple objects.

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By transfinite recursion we now define an ascending sequence of localizing subcategories of G: G0 ⊂ G ⊂ · · · ⊂ Gα ⊂ · · · ⊂ G such that G 0 = {0}, and G 1 is the localizing subcategory of all semi-Artinian objects of G as defined above. If α is an ordinal such that for every β < α we have already defined G β , then: 1. If α is not a limit ordinal, that is, we may view α = β + 1, we write G/Gβ for the quotient category of G with respect to Gβ and Qβ : G → G/G β for the canonical functor, which is known to be an exact functor; 2. if α is a limit ordinal, then we let G α be the smallest subcategory containing ∪β

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