Noncommutative Deformation Theory [1 ed.] 149879601X, 978-1-4987-9601-9

Table of contents :
Content: Classical Deformation Theory. Noncommutative Algebras and Simple Modules. Noncommutative Deformation Theory. The Noncommutative Phase Space. A Cosmological Toy Model. Moduli of Endomorphisms of Rank Three.

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MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS

Noncommutative Deformation Theory

Eivind Eriksen Olav Arnfinn Laudal Arvid Siqveland

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20161208 International Standard Book Number-13: 978-1-4987-9601-9 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. 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 Names: Eriksen, Eivind. | Laudal, Olav Arnfinn. | Siqveland, Arvid, 1964Title: Noncommutative deformation theory / Eivind Eriksen, Olav Arnfinn Laudal, Arvid Siqveland. Description: Boca Raton : CRC Press, [2017] | Series: Chapman & Hall/CRC monographs and research notes in mathematics | Includes bibliographical references and index. Identifiers: LCCN 2016054372| ISBN 9781498796019 (hardback : alk. paper) | ISBN 9781498796026 (ebook) Subjects: LCSH: Geometry, Algebraic. | Mathematical physics. | Perturbation (Mathematics) Classification: LCC QA564 .E75 2017 | DDC 516.3/5--dc23 LC record available at https://lccn.loc.gov/2016054372 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Contents

Introduction

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How to Read This Book

xv

1

Classical Deformation Theory 1.1 1.2 1.3 1.4 1.5

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Noncommutative Algebras and Simple Modules 2.1 2.2 2.3 2.4 2.5

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General principles . . . . . . . . . . . . . . . . . . Formal deformations and infinitesimal deformations Functors of Artin rings . . . . . . . . . . . . . . . . 1.3.1 Tangent spaces . . . . . . . . . . . . . . . . 1.3.2 Obstruction calculus . . . . . . . . . . . . . Deformations of associative algebras . . . . . . . . 1.4.1 Tangent space and obstruction calculus . . . 1.4.2 Examples . . . . . . . . . . . . . . . . . . . Deformations of modules . . . . . . . . . . . . . . 1.5.1 Tangent space and obstruction calculus . . . 1.5.2 Examples . . . . . . . . . . . . . . . . . . .

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Noncommutative algebras . . . . . . . . . . . . . . . . . . Artin-Wedderburn theory . . . . . . . . . . . . . . . . . . . Simple modules and the Jacobson radical . . . . . . . . . . The classical theorems of Burnside, Wedderburn, and Malcev Finite dimensional simple modules . . . . . . . . . . . . .

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Noncommutative Deformation Theory 3.1

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Noncommutative deformation functors . . . . . . . . . 3.1.1 Flatness in Abelian categories . . . . . . . . . . 3.1.2 Commutative deformation functors . . . . . . . 3.1.3 Noncommutative deformation functors . . . . . Structure of noncommutative deformation functors . . . 3.2.1 Functors of noncommutative Artin rings . . . . . 3.2.2 Algebraizations . . . . . . . . . . . . . . . . . . 3.2.3 Tangent spaces . . . . . . . . . . . . . . . . . . 3.2.4 Obstruction calculus . . . . . . . . . . . . . . . 3.2.5 Swarms . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Relations with commutative deformation functors Examples of noncommutative deformation functors . . . 3.3.1 Modules . . . . . . . . . . . . . . . . . . . . . 3.3.2 Modules with group action . . . . . . . . . . . .

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Contents 3.4

3.5

3.6

3.7

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Noncommutative deformations of sheaves and presheaves . . 3.4.1 Deformations of presheaves of modules . . . . . . . . 3.4.2 Deformations of quasi-coherent sheaves of modules . 3.4.3 Quasi-coherent ringed schemes . . . . . . . . . . . . 3.4.4 Calculations for D-modules on elliptic curves . . . . . Matric Massey products and A-infinity structures . . . . . . . 3.5.1 Matric Massey products on differential graded algebras 3.5.2 Matric Massey products and obstruction calculus . . . 3.5.3 Matric A-infinity algebras . . . . . . . . . . . . . . . The Generalised Burnside Theorem . . . . . . . . . . . . . . 3.6.1 The algebra of observables . . . . . . . . . . . . . . . 3.6.2 The kernel of the miniversal morphism . . . . . . . . 3.6.3 Iterated extensions and matric Massey products . . . . 3.6.4 The Generalised Burnside Theorem . . . . . . . . . . 3.6.5 Properties of the algebra of observables . . . . . . . . Iterated extension . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Moduli of iterated extensions . . . . . . . . . . . . . . 3.7.2 The category of iterated extensions . . . . . . . . . .

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The Noncommutative Phase Space 4.1 4.2 4.3 4.4 4.5 4.6

4.7 4.8

Introduction to noncommutative phase spaces . . . . . . . . . . . . . . . . . . . 4.1.1 The noncommutative Kodaira-Spencer map . . . . . . . . . . . . . . . . 4.1.2 Generalised momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . The iterated phase space functor and the Dirac derivation . . . . . . . . . . . . . 4.2.1 The Dirac derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 The generalised de Rham complex . . . . . . . . . . . . . . . . . . . . . Differentiable structures on the moduli of representations . . . . . . . . . . . . . 4.3.1 Dynamical structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Representations of Ph∞ (A) . . . . . . . . . . . . . . . . . . . . . . . . . Gauge groups and invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . The generic dynamical structures associated to a metric . . . . . . . . . . . . . . 4.5.1 The commutative case and general relativity . . . . . . . . . . . . . . . . 4.5.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical gauge invariance and metric classification of representations . . . . . . 4.6.1 The classical gauge invariance . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Chern characters and Chern-Simons classes . . . . . . . . . . . . . . . . 4.6.3 A generalised Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . 4.6.4 The classical Yang-Mills equation . . . . . . . . . . . . . . . . . . . . . 4.6.5 Reuniting general relativity, Yang-Mills, and general quantum field theory Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 The classical commutative case . . . . . . . . . . . . . . . . . . . . . . 4.7.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Interaction and noncommutative deformations . . . . . . . . . . . . . . . 4.8.2 Ensembles, bialgebras, and quantum groups in our model . . . . . . . . .

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A Cosmological Toy Model 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

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Background and some remarks on philosophy of science . . . . Deformations of associative algebras . . . . . . . . . . . . . . Spin, isospin, and supersymmetry . . . . . . . . . . . . . . . . Newton’s and Kepler’s laws . . . . . . . . . . . . . . . . . . . The universe as a versal base space . . . . . . . . . . . . . . . Worked out formulas . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Action of the gauge group g ⊕ su(2) on the tangent space 5.6.2 Adjoint actions of g . . . . . . . . . . . . . . . . . . . . Summing up the model . . . . . . . . . . . . . . . . . . . . . . 5.7.1 The toy model . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Further results . . . . . . . . . . . . . . . . . . . . . . Elementary particles, bosons, and fermions . . . . . . . . . . . 5.8.1 Spin, charge, and chirality . . . . . . . . . . . . . . . . 5.8.2 The weak force . . . . . . . . . . . . . . . . . . . . . .

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Moduli of Endomorphisms of Rank 3 6.1 6.2 6.3 6.4

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Endomorphisms of vector spaces . . . . . . . . . . . . Moduli of endomorphisms . . . . . . . . . . . . . . . . Noncommutative moduli of endomorphisms of rank three The computations . . . . . . . . . . . . . . . . . . . . . 6.4.1 The orbits in Case II and Case III . . . . . . . . 6.4.2 The tangent space dimensions in Case I . . . . . 6.4.3 The tangent space dimensions in Case II and III . 6.4.4 Bases and Yoneda forms in Case I . . . . . . . . 6.4.5 Second-order Massey products in Case I . . . . . 6.4.6 Computation of the first lifting . . . . . . . . . . 6.4.7 Computation of the second-order defining system 6.4.8 The results of the computations . . . . . . . . . 6.4.9 The geometric picture . . . . . . . . . . . . . . 6.4.10 The isotropy groups . . . . . . . . . . . . . . . The noncommutative affine ring . . . . . . . . . . . . .

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Bibliography

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Index

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Introduction

Philosophy If we want to study a natural phenomenon, call it P, we must in the present scientific situation describe P in some mathematical terms, say as a mathematical object X depending upon some parameters, in such a way that the changing aspects of P would correspond to altered parameter values for X. The object X would be a model for P if X, with any choice of parameter values, corresponds to some possibly occurring aspect of P. Clearly, this definition of a model is basically circular, as all general definitions of this kind will have to be. The aspect of P would have to be defined in terms of something (here simply in terms of the parameters of the model, the mathematical object X). Nevertheless, the wording above turns out to be helpful for comparing our point of view with other mathematical models in use in science today. Two mathematical objects X1 , and X2 corresponding to the same aspect of P would be called equivalent, and the set P of equivalence classes of the objects P would correspond to the moduli space M of the models X, or possibly to a quotient of this moduli space. Here, the moduli space is assumed to be an algebraic scheme of some sort. The study of the natural phenomena P, and their changing aspects, would then be equivalent to the study of the structure of P, and therefore to the study of the geometry of the moduli space M. In particular, the notion of time would correspond to some metric defined on this space, in agreement with Aristotle and Saint Augustine; see Saint Augustine [2] and also Laudal [28], [29]. This is the backdrop of our interest in the mathematical tools to be developed in this book.

Noncommutative algebraic geometry Mathematics is, since the time of Galilei, the language of physics. And since Descartes, Newton, and Leibniz, differential geometry and algebra have been our best tools for making the universe understandable. The last centuries have seen an amazing development in science and technology, due to the parallel achievements in mathematics and physics. The theory of general relativity and the modern theory of quantum physics have transformed our worldview and our daily life in a way almost unimaginable just 50 years ago. And the pace of change is, seemingly, accelerating. And so is the pace of change of the mathematical bases for these two grand theories. The differential geometry and the operator algebra have served these two fundamental sciences well for centuries, but the human curiosity does not rest. The feeling that they should, somehow, be united has been there for a long time, and has produced a lot of new mathematics. Algebraic geometry is as old as geometry, but has seen a formidable development in the last 50 years, starting with the Grothendieck era, including a new fundament for the age-old theory of deformations. Operator algebra has in the

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same period, due to work of Von Neumann, Gelfand and Connes, been transformed into a fascinating noncommutative geometry. The physicists have, of course, taken advantage of these developments, and used the new mathematics to construct new models. At the moment, the situation is nevertheless that there are still two theories, the general relativity, treating gravitation and to some extent electroweak forces, and the quantum field theory, taking care of relativistic quantum theory. The result of the latter is the Standard Model, a marvel of an effective theory, for most of the forces of nature, but not including gravitation. The hope has therefore been that by creating some sort of fusion of classical algebraic geometry and the new noncommutative geometry, one would be able to create a mathematical model fusing the theory of gravitation and the standard model. This is, in our view, what a mature noncommutative algebraic geometry should be about. There are many attempts to create a noncommutative geometry, based on the classical algebraic geometry, modified by Serre, Chevalley, and Grothendieck, but where the algebra part is extended from commutative to associative (not necessarily commutative) algebras. In this book, we will give reasons for why we think this effort must include the study of noncommutative deformations of algebraic structures. The idea is to look at the common goal of quantum theory and Grothendieck’s scheme theory, which is the study of the local and global properties of the set of representations of algebras, together with their dynamical structure.

Moduli of representations In scheme theory, a point is a representation of a (commutative) ring A, i.e., a ring homomorphism ρ : A → R of A into another ring R. The scheme is, in a general sense, the moduli space Rep(A) of such representations. The object of scheme theory is then to study the properties of these moduli spaces and their categorical relations, and eventually to classify them. In quantum theory, the objects of interest are also representations ρ : A → R of a ring A of observables, but here R = Endk (V ) where k is a field we may use for measuring the eigenvalues of ρ (a) as operator on the k-vector space V for any observable a ∈ A. The aim is to study the structure of the moduli space Rep(A) of such representations, and in particular to understand the dynamical properties of this space. The local structure of the moduli space is defined via deformation theory in both cases. But here is where noncommutative deformation theory enters, not only because the rings we must work with are noncommutative, but also because the local structure we are interested in is no longer given by commutative algebras. In fact, the local structure of Rep(A), in a finite family V ⊂ Rep(A), is not the superposition of the local structure of each one of the representations in V . Noncommutative deformations of the family V = {Vi : i ∈ I} produces a localisation homomorphism

η : A → O(V ) which we shall refer to as the O-construction. This is not the product of the individual localisations ηi : A → O(Vi ) of the members Vi ∈ V of the family V .

Introduction

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Dynamical structure To be able to work with the geometry Rep(A), in physics we obviously need the possibility of introducing an action of time, and more generally, to introduce a differentiable structure. This is taken care of by introducing the noncommutative phase space functor Ph : Algk → Algk For any finitely generated associative k-algebra A in Algk , Ph(A) is an associative k-algebra with an injective algebra homomorphism A → Ph(A) and a universal derivation d : A → Ph(A), and the phase space functor Ph : Algk → Algk has the universal property that for any other representation ρ : A → R with a derivation ξ : A → R, there is a unique morphism ρξ : Ph(A) → R such that ρξ ◦ d = ξ . The iteration of the functor Ph on any finitely generated k-algebra A produces an algebra Ph∞ (A) with a universal Dirac derivation δ : Ph∞ (A) → Ph∞ (A) with properties like the time parameter we want to have in physics. A dynamical structure is now any quotient A(σ ) = Ph∞ (A)/σ , where σ ⊆ Ph∞ (A) is a δ -stable ideal. Depending on the choice of σ , this is the differentiable structure we need.

Invariant theory The algebra A may be outfitted with an action of a Lie group G, or of a Lie algebra g. The subspaces V ⊂ Rep(A) of representations that we should be interested in might consist of those that are invariant under the action of G, respectively of g. The corresponding localisation homomorphism η : A → O(V ) generalises the classical quotient Spec(A)/G, respectively Spec(A)/g. This immediately lends itself to a reformulation of the notion of moduli space in algebraic geometry. The fact that there are easy examples of (fine) moduli problems that cannot be tackled in classical algebraic geometry may now be reviewed in the new setting of invariant theory in noncommutative algebraic geometry.

Application to physics These ideas can now be put to use in creating new, and hopefully interesting, mathematical models in physics. The applications include models for quantum theory and for general relativity, together with a seemingly reasonable model for a Big Bang event, extending work in Laudal [29], [31]. Our toy model, the Hilbert scheme Hilb2 (A3 ) of subschemes of length two in the affine three-dimensional scheme over k, seems to include a mathematically sound cosmological model, including the gauge groups of the Standard Model, and a lot of other structures. The family of representations of this scheme contains models for most of the present day accepted elementary particles, and the forces concerning them. And these forces seem to fuse with the classical model of gravitation.

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Noncommutative deformation theory This sketch of our goals has hopefully shown that noncommutative deformation theory, together with the noncommutative phase space functors, are the essential tools in the construction of the proposed noncommutative algebraic geometry. The exposition of these topics is therefore the main subject of this book. Noncommutative deformation theory is a rather natural extension of the classical deformation theory, developed by Kodaira and Spencer in complex analytic geometry, and by Grothendieck, Schlessinger, Andre, Quillen, and their followers in the algebraic setting. The new element in noncommutative deformation theory is the realization that deforming a family of objects, not just one by one but as a family, necessarily introduces noncommutative parameters. The main example is the situation referred to above, where V ⊂ Rep(A) is a finite family of representations with some finiteness conditions imposed, making it a swarm. In this case, we find that the localization homomorphism η : A → O(V ) is an isomorphism when k is algebraically closed, A is Artinian and V consists of all the simple representations of A. This Generalised Burnside Theorem is the noncommutative analogue of the first local result in most texts about classical scheme theory: When A is a commutative Artinian algebra, we have that A ≃ ∏ Am m

where m is running through the maximal ideals of A. Most of the results in this book are based upon these simple ideas, where the O-construction produces the noncommutative scheme structure, and Ph-construction provides the dynamics of the resulting geometry.

How to Read This Book

Preliminaries. In Chapter 1, we give an introduction to classical deformation theory in algebraic geometry. By classical deformations, we mean deformations over base rings that are commutative local rings. We describe the deformation functor Def X in concrete terms when X is an associative algebra or a module over an associative algebra, and give several examples. This is meant as an introduction to noncommutative deformations in Chapter 3, where we generalise the deformation theory to the case of noncommutative base rings. In Chapter 2, we present some standard results for noncommutative algebras (that is, associative but not necessarily commutative algebras) over a field. These are essentially the results from noncommutative algebra that we use in this book. The exposition is meant to make the reading of the book easier for readers with a background in commutative algebra. Noncommutative deformation theory. In Chapter 3, we develop the noncommutative deformation theory. We first give general definitions and results for the noncommutative deformation functor Def X of a finite family X of objects in an Abelian k-category. Then we describe noncommutative deformations in concrete terms for many important types of algebraic objects, including modules over a noncommutative algebra (i.e., representations of algebras), modules over an algebra with a group action, and presheaves and sheaves of modules over a sheaf of algebras. The main result is that under weak assumptions, which are often satisfied in the cases mentioned above, the noncommutative deformation functor Def X has a pro-representing hull and a versal family. Moreover, we give an algorithm for computing the pro-representing hull and its versal family, and show the computations in concrete terms in many examples. In the main case considered, a finite family M of modules over an algebra A, we describe the O-construction that is central in this book. It is an algebra homomorphism η : A → O(M ) given in terms of the prorepresenting hull and its versal family. We prove the Generalised Burnside Theorem, which states that η is an isomorphism when k is algebraically closed, A is Artinan and M is the family of all simple modules over A. Noncommutative phase spaces. In Chapter 4, we introduce noncommutative phase spaces and the Ph-construction, consisting of the phase space functor Ph defined for associative k-algebras, and the infinitely iterated phase space functor Ph∗ with its cosimplicial structure. The inductive limit Ph∞ (A) of Ph∗ (A) has an induced universal derivation δ : Ph∞ (A) → Ph∞ (A), the Dirac derivation, and we study the quotients A(σ ) = Ph∞ (A)/σ by δ -invariant ideals. Applications to physics. In Chapter 5, which is more demanding and assumes a curiosity of mathematical applications in physics, we show that the structure of the miniversal base space of the four-dimensional associative algebra U = k[x1 , x2 , x3 ]/(x1 , x2 , x3 )2 , geometrically a fat point in threedimensional space, turns out to contain the cosmological toy model of Laudal [29], [31] providing new and unsuspected structures. The main result is the existence of a canonical gauge Lie algebra bundle, containing the gauge groups of the Standard Model, and a strange supersymmetry relating the spaces of our versions of bosonic and fermionic fields. Moduli of endomorphisms of rank three. In Chapter 6, we give a new noncommutative treatment of a basic example in classical algebraic geometry, see Mumford and Suominen [38], showing the xv

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strength of the invariant theory in noncommutative algebraic geometry. The computations in this theory are not simple, and this is the reason for treating this in a section of its own. The explicit computations can either be skipped, or they can be used as an example for those who would do similar computations.

Chapter 1 Classical Deformation Theory

In this chapter, we study classical deformations of algebraic objects defined over a fixed field k. By classical deformations, we understand deformations parametrized by a commutative k-algebra, and the objects we deform are algebraic objects, such as algebras or modules. The classical deformation theory described in this chapter is also called commutative deformation theory. It is the starting point for later generalization to noncommutative deformation theory in Chapter 3.

1.1 General principles Any algebraic object can be deformed in a naïve sense of the word by varying the coefficients that define it. For instance, the k-algebra A = k[x, y]/(xy) can be deformed to At = k[x, y]/(xy − t) for any value t 6= 0 of the parameter t, and the family {At : t ∈ k} of commutative k-algebras can be considered as a family of deformations of the commutative k-algebra A = A0 . Classical deformation theory is concerned with deformations in a more restricted sense of the word. A local deformation of an associative k-algebra A, parameterized by a local commutative kalgebra (R, m) with residue field k, is a pair (AR , η ), where AR is a flat R-algebra and η : k ⊗R AR → A is an isomorphism of k-algebras. Local deformations of modules and other algebraic objects can be defined in a similar manner. We illustrate the notion of local deformations with a concrete example: Let A be the commutative k-algebra A = k[x, y]/(xy), and let R = k[[t]] be the formal power series ring in one variable. Then AR = k[x, y][[t]]/(xy − t) is a local deformation of A parameterized by R since AR is R-flat, with isomorphism η : k ⊗R AR → A induced by the map AR → A given by t 7→ 0. Classical deformation theory is based on work of Kodaira, Spencer, Nirenberg and Kuranishi on small deformations of complex manifolds, and was formalized by Grothendieck in the language of algebraic schemes in a series of Bourbaki seminars often refered to as Fondements de la géométrie algébrique [14]; see in particular [15], [16]. Deformation theory in the algebraic setting is referred to as algebraic deformation theory. The general method of algebraic deformation theory consists of the following steps: First, we consider formal deformations, which are certain local deformations parameterized by complete local commutative rings. Our goal is to find a formal deformation that is universal, or at least semiuniversal. When the best possible formal deformation has been obtained, we attempt to find an algebraization of it. In an analytical setting, the corresponding steps would be to first look for formal solutions and then consider convergence. In the example above, AR = k[x, y][[t]]/(xy − t) is a formal deformation of A = k[x, y]/(xy) that is parameterized by R = k[[t]], and it turns out to be semi-universal. An algebraization of R and the deformation AR is given by R = k[t] and the deformation AR = k[x, y,t]/(xy − t). Note that R is an algebra of finite type over k, and that there is family {AR (τ ) : τ ∈ k} of deformations of the algebra A parameterized by the closed points of R = Spec R with residue field k (the k-rational points). This 1

2

Noncommutative Deformation Theory

family is given by

AR (τ ) = k(τ ) ⊗k[t] AR ∼ = k[x, y]/(xy − τ )

where k(τ ) = k[t]/(t − τ ) ∼ = k is the residue field of (t − τ ) for all τ ∈ k. In particular, we have that AR (0) ∼ = k[x, y]/(xy) = A. In the rest of this chapter, we will describe general results on the existence, uniqueness and the explicit construction of semi-universal formal deformations in classical deformation theory, with emphasis on classical deformations of algebras and modules. We will also give some examples, and study algebraizations of the semi-universal deformation in these examples.

1.2 Formal deformations and infinitesimal deformations Consider a complete local Noetherian commutative k-algebra (R, m) with residue field k. We may view R as the projective limit R = lim R/mn ←− n

R/mn

since R is complete, and is a local Artinian k-algebra for all n ≥ 2. Conversely, any complete local commutative k-algebra (R, m) with residue field k with the property that the quotients R/mn are Artinian for all n ≥ 2 is Noetherian. This follows from the following result: Lemma 1.1. Let (R, m) be a complete local commutative k-algebra with residue field k, and assume that d = dimk (m/m2 ) is finite. Then R ∼ = k[[t1 ,t2 , . . . ,td ]]/I for an ideal I with I ⊆ (t1 ,t2 , . . . ,td )2 . In particular, R is Noetherian. Proof. We can find elements x1 , x2 , . . . , xd ∈ m such that {x1 , x2 , . . . , xd } is a k-linear base for m/m2 , since dimk (m/m2 ) = d is finite. Consider the natural algebra homomorphism f : k[[t1 ,t2 , . . . ,td ]] → R given by ti 7→ xi . It is surjective since its associated graded homomorphism gr( f ) is, and this implies that R ∼ = k[[t1 ,t2 , . . . ,td ]]/I with I = ker( f ) ⊆ (t1 ,t2 , . . . ,td )2 . We define l to be the category of local Artinian commutative k-algebras with residue field k, with local algebra homomorphisms as morphisms. Its pro-category ˆl is the category of local commutative k-algebras (R, m) with residue field k that satisfy the following conditions: 1. R is m-adic complete 2. Rn = R/mn is in l for all n ≥ 2 The morphisms in ˆl are the local algebra homomorphisms. In view of the comments above, the procategory ˆl coincides with the category of complete local Noetherian commutative k-algebras with residue field k. Let X be an algebraic object defined over k. An infinitesimal deformation of X is a local deformation of X parameterized by an algebra R in l, and we denote by Def X (R) the set of equivalence classes of local deformations of X parameterized by R. We shall give a concrete description of Def X (R) when X is an algebra or a module; see Section 1.4 and Section 1.5. In general, the assignment R 7→ Def X (R) defines a functor Def X : l → Sets

Classical Deformation Theory

3

called the classical deformation functor of X. ˆ : ˆl → Sets on the proGiven a functor D : l → Sets, there is a natural extension to a functor D category ˆl, given by ˆ D(R) = lim D(Rn ) ←− n

ˆ for any algebra R in ˆl. An element ξ ∈ D(R) is therefore a collection ξ = (ξn )n≥2 of elements ξn ∈ D(Rn ) that are compatible in the obvious sense. Let X be an algebraic object defined over k, with classical deformation functor Def X : l → Sets. A formal deformation of X is a deformation ξ ∈ Def X (R) parameterized by an algebra R in ˆl, where we by abuse of notation write Def X : ˆl → Sets for the extension of the classical deformation functor of X to ˆl. In other words, a formal deformation of X is a collection ξ = (ξn )n≥2 of infinitesimal deformations ξn ∈ Def X (Rn ) that are compatible. As indicated in Section 1.1, our goal is to find a universal or semi-universal formal deformation of the algebraic object X. In view of the definition of formal deformations, this leads us to the study of infinitesimal deformations of X, and therefore to the study of the classical deformation functor Def X : l → Sets. A systematic study of such functors can be undertaken in the framework of functors of Artin rings developed in Schlessinger [42].

1.3 Functors of Artin rings A functor of Artin rings is a functor D : l → Sets such that D(k) = {∗}, a singleton set. When X is an algebraic object defined over k, the classical deformation functor Def X : l → Sets is a functor of Artin rings, and the element ∗ ∈ Def X (k) corresponds to the trivial deformation of X to k. Lemma 1.2 (Yoneda’s Lemma). Let D : l → Sets be a functor of Artin rings and let H be an algebra in ˆl. Then there is a natural bijection b Hom(Mor(H, −), D) ≃ D(H)

b from the set of natural transformations Mor(H, −) → D of functors of Artin rings to the set D(H).

Proof. Let φ : Mor(H, −) → D be a natural transformation of functors of Artin rings. For each n ≥ 2, we write Hn = H/mn , where m is the maximal ideal of H. By definition, Hn is an algebra in l, and the natural quotient map ιn : H → Hn gives rise to an element ξn = φHn (ιn ) ∈ D(Hn ). It is clear that ξ = (ξn ) ∈ D(H) is a well-defined element, and the map φ 7→ ξ sets up the bijection b Hom(Mor(H, −), D) ≃ D(H).

b We define a pro-couple for D to be a pair (H, ξ ), where H is an algebra in ˆl and ξ ∈ D(H), ′ ′ and a morphism φ : (H, ξ ) → (H , ξ ) of pro-couples for D to be a morphism φ : H → H ′ in ˆl b φ )(ξ ) = ξ ′ . By Yoneda’s Lemma, ξ ∈ D(H) b such that D( corresponds to a natural transformation φξ : Mor(H, −) → D of functors of Artin rings. We say that a functor D : l → Sets of Artin rings is pro-representable if there is a pro-couple (H, ξ ) such that the natural transformation φξ : Mor(H, −) → D is an isomorphism. In this case, the pro-couple (H, ξ ) is unique up to unique isomorphism, and is called the universal pro-couple that pro-represents D.

4

Noncommutative Deformation Theory

We recall that a natural transformation φ : D → D′ of functors on Artin rings is smooth if the natural map D(R) → D(S) × D′ (R) D′ (S)

is surjective for any surjective morphism R → S in l. If φ is smooth, then it follows that the map b b′ (H) is surjective for all algebras H in ˆl. φH : D(H) →D We say that a pro-couple (H, ξ ) for a functor D : l → Sets of Artin rings is versal if the natural transformation φξ : Mor(H, −) → D is smooth, and semi-universal or miniversal if φξ is smooth and φξ (R) is an isomorphism for any algebra (R, m) in l with m2 = 0. When the pro-couple (H, ξ ) is versal, the element ξ is often called a versal family, and when (H, ξ ) is semi-universal, the algebra H is often called a pro-representing hull or a formal moduli for D. Let l(n) be the full subcategory of l consisting of algebras (R, m) with mn = 0. A pro-couple (H, ξ ) for D is the same as a projective system H2 ← H3 ← H4 ← · · · ← Hn ← . . . of algebras Hn in l(n) such that their projective limit is H, and a family ξ = (ξn ) of compatible elements ξn ∈ D(Hn ). Moreover, it is clear that H is a pro-representing hull for D with versal family ξ if and only if (H2 , ξ2 ) represents D restricted to l(2) and ξn corresponds to a smooth morphism φξn : Mor(Hn , −) → D between functors on l(n) for n ≥ 3. Proposition 1.3. If a functor D : l → Sets of Artin rings has a pro-representing hull, then it is unique up to a noncanonical isomorphism. Proof. Assume that H, H ′ are pro-representing hulls for D with versal families ξ , ξ ′ . Since the natural transformations φξ : Hom(H, −) → D and φξ ′ : Mor(H ′ , −) → D induced by the versal families are smooth, there are morphisms u : (H, ξ ) → (H ′ , ξ ′ ) and u′ : (H ′ , ξ ′ ) → (H, ξ ) of procouples for D. Moreover, the induced maps u2 : H2 → H2′ and u′2 : H2′ → H2 defined on H2 = H/m2 and H2′ = H ′ /(m′ )2 are isomorphisms. This means that u′ u and uu′ are surjective endomorphisms of Noetherian rings H and H ′ , hence u and u′ are isomorphisms. In Schlessinger [42], necessary and sufficient conditions for existence of pro-representing hulls were given. It turns out that the conditions are rather weak, and that many deformation functors (such as deformation functors of algebras or modules) have pro-representing hulls. In contrast, the additional condition that is necessary and sufficient for pro-representability is quite strong. Our goal is not only to show that a pro-representing hull of D exists when D is a deformation functor, but also to find the pro-representing hull and the versal family of D explicitly. We will therefore not work with Schlessinger’s conditions, but rather with conditions related to the tangent space and obstruction calculus of D that are better suited for explicit computations. We will give an explicit construction of the pro-representable hull of D when these conditions are satisfied, following the general construction in Laudal [27].

1.3.1 Tangent spaces Let D : l → Sets be a functor of Artin rings. We define t(D) = D(k[ε ]) to be the tangent space of D, where k[ε ] = k[x]/(x2 ) is the algebra of dual numbers. This definition is motivated by the following fact: If D is pro-represented by a complete local ring (H, m) in ˆl, then t(D) = D(k[ε ]) ∼ = Mor(H, k[ε ]) ∼ = Homk (m/m2 , k ε ) = (m/m2 )∗ = t(H) where t(H) is the Zariski tangent space of (H, m). Under the weaker assumption that H is a prorepresenting hull of D, the same argument shows that t(D) ∼ = t(H).

5

Classical Deformation Theory

We shall consider the following condition, which is necessary for the existence of a prorepresenting hull of D by the comments above: t(D) is a vector space over k with dimk t(D) < ∞

(TS)

When D = Def X is the classical deformation functor of an algebra or a module, we give a concrete description of the tangent space t(D) in Section 1.4 and Section 1.5. In particular, we will show that the tangent space has a natural vector space structure in these cases, and that dimk t(D) is finite for many interesting deformation functors. The condition (TS) is not sufficient for the existence of a pro-representing hull of D, but it gives us a starting point for the construction. Let us choose a k-linear base {t1∗ ,t2∗ , . . . ,td∗ } of t(D) with d = dimk t(D), so that {t1 ,t2 , . . . ,td } is a k-linear base of V = t(D)∗ . We define the algebra T1 in ˆl to be the completion of the symmetric algebra Symk (V ). Explicitly, T1 is the formal power series algebra T1 = k[[t1 ,t2 , . . . ,td ]] with maximal ideal (t1 ,t2 , . . . ,td ). We also define T12 = T1 /(t1 ,t2 , . . . ,td )2 . The algebra T12 is the best candidate for an algebra H2 in l(2) that represents D restricted to l(2). In fact, if D has a pro-representing hull H, then t(D) ∼ = t(H), and it follows from the proof of Lemma 1.1 that H ∼ = T1 /a = k[[t1 ,t2 , . . . ,td ]]/a, where a is an ideal such that a ⊆ (t1 ,t2 , . . . ,td )2 . Hence, the second order approximation H2 = H/m2H of H is given by H2 ∼ = T12 . 1 We therefore define H2 = T2 . As mentioned above, the condition (TS) is not sufficient for the existence of a pro-representing hull H for D, and several obstacles remain in order to construct H: First, we must find a versal family ξ2 ∈ D(H2 ) such that (H2 , ξ2 ) represents D restricted to l(2). Secondly, we must find a minimal ideal a ⊆ T1 of obstructions for lifting ξ2 to T1 , such that ξ2 ∈ D(H2 ) can be lifted to H = T1 /a. This means that in order to construct ξ2 ∈ D(H2 ) and the ideal a ⊆ T1 , we need an obstruction calculus for D. We will explain this concept in detail in the next section, and describe how to use an obstruction calculus to explicitly construct a pro-representing hull (H, ξ ) for D.

1.3.2 Obstruction calculus Let u : R → S be a morphism in l, and let ξS ∈ D(S) be an element. An obstruction calculus for D is a systematic method for computing the (possibly empty) set {ξR ∈ D(R) : D(u)(ξR ) = ξS } of liftings of ξS to R when (u, ξS ) is given. We call (u, ξS ) a lifting situation for D, and define a morphism (α , β ) : (u, ξ ) → (u′ , ξ ′ ) of lifting situations for D to be a commutative diagram R

α

u

/ R′ u′

 S

β

 / S′

in l such that D(β )(ξ ) = ξ ′ . It is clear that liftings are functorial in the following sense: If ξR is a lifting of ξ to R, then D(α )(ξR ) is a lifting of ξ ′ to R′ . It is therefore natural to require that an obstruction calculus should have a similar functoriality. We say that u : R → S is a small surjection in l if u is surjective and K · mR = 0, where mR is

6

Noncommutative Deformation Theory

the maximal ideal of R and K = ker(u) is the kernel of u. It is clear that any surjection in l is a composition of small surjections. It follows that it is sufficient to specify an obstruction calculus for small lifting situations; that is, lifting situations (u, ξS ) for D such that u is a small surjection. The notion of an obstruction calculus that we have sketched so far is not very precise. We shall therefore define the notion of an obstruction theory that is precise and sufficiently general for our purposes: Definition 1.1. We say that D has an obstruction theory with cohomology {H p } if there are vector spaces H1 and H2 over k such that the following conditions hold: 1. For any small lifting situation (u, ξ ) for D, with K = ker(u), there is a canonical obstruction o(u, ξ ) ∈ Homk ((H2 )∗ , K) such that o(u, ξ ) = 0 if and only if there exists a lifting of ξ to R. Moreover, if o(u, ξ ) = 0, then there is a transitive and free action of Homk ((H1 )∗ , K) on the set of liftings of ξ to R. 2. For any morphism (α , β ) : (u, ξ ) → (u′ , ξ ′ ) of small lifting situations for D, we have that α ∗ (o(u, ξ )) = o(u′ , ξ ′ ), where α ∗ : Homk ((H2 )∗ , K) → Homk ((H2 )∗ , K ′ ) is the natural map induced by α , with K = ker(u) and K ′ = ker(u′ ). We say that D has an obstruction theory with finite dimensional cohomology if in addition H1 and H2 are finite dimensional vector spaces. In this case, we have Homk ((H p )∗ , K) ∼ = K ⊗k H p . Remark 1.1. We recall that a group action of a group G on a set X is transitive if the map G → X defined by g 7→ g · x0 is surjective for any base point x0 ∈ X, and free if the map is injective for any x0 ∈ X. When the group action of G on X is transitive and free, X is also called a torsor over G. Remark 1.2. If D has an obstruction theory with finite dimensional cohomology {H p }, then we have that t(D) ∼ = H1 . In fact, for any algebra (R, m) in l(2), the natural map R → k is a small surjection with kernel m. Moreover, the element ∗ ∈ D(k) has a trivial lifting ∗R ∈ D(R) to R, given by ∗R = D(k → R)(∗) ∈ D(R). By the definition of an obstruction theory, the base point ∗R ∈ D(R) sets up an isomorphism D(R) ∼ = Homk ((H1 )∗ , m). In particular, with R = k[ε ], we find that the ∼ tangent space t(D) = D(R) = Homk ((H1 )∗ , k) ∼ = H1 since m = k · ε ∼ = k in this case. When D = Def X is the classical deformation functor of an algebra or a module, we describe an obstruction theory for D in Section 1.4 and Section 1.5. In each case, the obstruction theory and the corresponding cohomology theory is described in concrete terms. In particular, we will show that H1 and H2 are finite dimensional for many interesting deformation functors. Lemma 1.4. A functor D of Artin rings that is pro-representable has an obstruction theory with finite dimensional cohomology. ∼ Mor(H, −) is pro-representable, then there is an ideal a such that H is a quotient Proof. If D = H = T1 /a with a = ( f1 , f2 , . . . , fr ) ⊆ T1 = k[[t1 , . . . ,td ]]. Let T2 = k[[s1 , s2 , . . . , sr ]] and let o : T2 → T1 be the morphism in ˆl defined by si 7→ fi . We define H1 = t(T1 ) and H2 = t(T2 ), and consider a small lifting situation (u, ξS ) for D, where ξS corresponds to a morphism φS ∈ Mor(H, S). To study possible liftings of ξS to R, or equivalently morphisms φR ∈ Mor(H, R) with u ◦ φR = φS , we consider the following commutative diagram: T2

o

/ T1

ψR

/R u

 H

φS

 /S

Choose ri ∈ R such that u(ri ) = φS (t i ) for 1 ≤ i ≤ d, and define ψR : T1 → R by ψ (ti ) = ri .

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Then the morphism ψR factors through H if and only if ψR ◦ o is trivial, or equivalently if ψR (o(s j )) = ψR ( f j ) = f j (r1 , . . . , rd ) = 0 for 1 ≤ j ≤ r. We see that ψR ◦ o : T2 → R is a morphism such that ψR (o(s j )) ∈ K. Since u is small, we have that K 2 = K · mR = 0. This implies that ψR (o(s j )) = f j (r1 , . . . , rd ) is independent of the choice of ψR , and determines a canonical element in Homk ((H2 )∗ , K), the obstruction for lifting φS to Mor(H, R), and therefore for lifting ξS to D(R). Assume that a lifting φR ∈ Mor(H, R) of φS to R is given. If φR′ is another lifting, then the assignment ti 7→ φR′ (t i ) − φR (t i ) defines a morphism in Mor(T1 , K) ∼ = Homk ((H1 )∗ , K), and conversely, any ele1 ∗ ′ ment τ ∈ Homk ((H ) , K) defines a morphism φR : H → R, given by φR′ (t i ) = φR (t i ) + τ (ti ). This is the transitive and free action on the set of liftings of ξS to R. It is clear that the obstruction defined above has the required functoriality. We shall consider the following condition, which implies condition (TS) and is necessary for the pro-representability of D by Lemma 1.4: D has an obstruction theory with finite dimensional cohomology

(OT)

We claim that condition (OT) is sufficient for the existence of a pro-representable hull for D. Let us therefore assume that condition (OT) holds. Using this assumption, we shall give an explicit construction of the pro-representing hull H of D and its versal family ξ ∈ D(H). Let (R, m) be an algebra in l(2). By the argument in Remark 1.2, it follows that there is a canonical isomorphism D(R) ∼ = Homk ((H1 )∗ , m) ∼ = Mor(T1 , R) ∼ = Mor(H2 , R) induced by ∗R ∈ D(R). The corresponding natural transformation Mor(H2 , −) → D of functors on l(2) corresponds to an element ξ2 ∈ D(H2 ) by Yoneda’s Lemma, and it is clear that (H2 , ξ2 ) represents D restricted to l(2). Theorem 1.5. Let D : l → Sets be a functor of Artin rings. If D has an obstruction theory with finite dimensional cohomology {H p }, then D has a pro-repesenting hull. Moreover, if this is the case, then b T2 k is a pro-representing hull there is an obstruction morphism o : T2 → T1 in ˆl such that H = T1 ⊗ of D.

Proof. For simplicity, we use the notation m = (t1 , . . . ,td ) for the maximal ideal in T1 , T1n = T1 /mn for n ≥ 2, and tn : T1n+1 → T1n for the natural morphisms. Let a2 = m2 and H2 = T1 /a2 = T12 . Then the restriction of D to l(2) is represented by (H2 , ξ2 ) and H2 ∼ = T12 ⊗T2 k, where the tensor product is 2 1 taken over the trivial morphism o2 : T → T2 . Using o2 and ξ2 as a starting point, we shall construct on+1 and ξn+1 for n ≥ 2 inductively. So let n ≥ 2, and assume that the morphism on : T2 → T1n and the lifting ξn ∈ D(Hn ) are given, with Hn = T1n ⊗T2 k. We may assume that tn−1 ◦ on = on−1 and that ξn is a lifting of ξn−1 . Let us first construct the morphism on+1 : T2 → T1n+1 . We define a′n to be the ideal in T1n generated by {on(s1 ), . . . , on (sr )}. Then a′n = an /mn for an ideal an ⊆ T1 with mn ⊆ an , and Hn ∼ = T1 /an . Let bn = m · an, then we obtain the following commutative diagram: T2 ❆ T1n+1 ❆❆ ❆❆ ❆ on ❆❆ ❆  T1n

/ T1 /bn  / T1 /an

There is an obstruction o′n+1 = o(T1 /bn → Hn , ξn ) for lifting ξn to T1 /bn since T1 /bn → T1 /an is

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Noncommutative Deformation Theory

a small surjection, and this obstruction corresponds to a morphism o′n+1 : T2 → T1 /bn . We define a′′n+1 to be the ideal in T1 /bn generated by {o′n+1(s1 ), . . . , o′n+1 (sr )}. Then a′′n+1 = an+1 /bn for an ideal an+1 ⊆ T1 with bn ⊆ an+1 ⊆ an . Let Hn+1 = T1 /an+1. We obtain the following commutative diagram: o′n+1

* / T1 /bn

T2 ❆ T1n+1 ❆❆ ❆❆ ❆ on ❆❆ ❆ 

/ Hn+1 = T1 /an+1 qq qqq q q qq  q qqq x / Hn

T1n

By the choice of an+1, the obstruction for lifting ξn to Hn+1 vanishes, and we may choose a lifting ξn+1 ∈ D(Hn+1 ) of ξn to Hn+1 . We claim that there is a morphism on+1 : T2 → T1n+1 that commutes with o′n+1 and on . Note that an−1 = mn−1 + an since tn−1 ◦ on = on−1. For simplicity, we write O(V ) = Homk (t(T2 )∗ ,V ) for any k-linear vector space V . The following diagram of k-linear spaces is commutative with exact columns: 0 0  O(bn /mn+1 )

jn

 / O(bn−1 /mn )

 O(an /mn+1 )

kn

 / O(an−1 /mn )

ln



rn+1



O(an /bn )  0

rn

/ O(an−1 /bn−1 )  0

We may consider on as an element in O(an−1 /mn ), and o′n+1 ∈ O(an /bn ). Since o′n commutes with o′n+1 and on , we get ln (o′n+1 ) = rn (on ). To prove the claim, it is enough to find an element on+1 ∈ O(an /mn+1 ) such that rn+1 (on+1 ) = o′n+1 and kn (on+1 ) = on . Since on (I(T2 )) ⊆ an , there is an element on+1 ∈ O(an /mn+1 ) such that kn (on+1 ) = on . But an−1 = an + mn−1 implies that jn is surjective, so the claim follows from the snake lemma. In particular, T1n+1 ⊗T2 k ∼ = Hn+1 when the tensor product is taken over on+1 . By induction, we find a morphism on : T2 → T1n and an element ξn ∈ D(Hn ) for all integers n ≥ 2, with Hn = T1n ⊗T2 k. Using the universal property of the projective limit, we obtain a morphism b T2 k. We claim o : T2 → T1 in ˆl and an element ξ ∈ D(H), where H is the projective limit H = T1 ⊗ that (H, ξ ) is a pro-representing hull for D. It is enough to prove that (Hn , ξn ) is a pro-representing hull for the restriction of D to l(n) for all n ≥ 3. So let φn : Mor(Hn , −) → D be the morphism of functors on l(n) corresponding to ξn for some n ≥ 3. We shall prove that φn is a smooth morphism. Let u : R → S be a small surjection in l(n) with kernel K, let ξR ∈ D(R) and v ∈ Mor(Hn , S) be elements such that D(u)(ξR ) = D(v)(ξn ) = ξS ,

Classical Deformation Theory

9

and consider the following commutative diagram: / Hn+1 T1 ❈ ❈❈ ❈❈ ❈❈ ❈❈ !  Hn

R u v

 /S

We can find a morphism v′ : T1 → R that makes the diagram commutative, hence v′ (an ) ⊆ K. Since u is small, it follows that v′ (bn ) = 0. But the induced map T1 /bn → R maps the obstruction o′n+1 to o(u, ξS ) = 0. It follows that v′ (an+1 ) = 0, hence v′ induces a morphism v′ : Hn+1 → R making the diagram commutative. Since v′ (mn /mn+1) = 0, we may consider v′ as a map from Hn . This proves that there is a morphism v′ : Hn → R such that u ◦ v′ = v. Let ξR′ = D(v′ )(ξn ), then ξR′ is a lifting of ξS to R, and the difference between ξR and ξR′ is given by an element τ ∈ Hom((H1 )∗ , K). Let v′′ : T1 → R be the morphism defined by v′′ (ti ) = v′ (ti )+ τ (ti ) for i = 1, 2, . . . , d. Since an+1 ⊆ m2 and u is small, we have that v′′ (an+1 ) ⊆ v′ (an+1 ) + m · K + K · m + K 2 = v′ (an+1 ) = 0 This implies that v′′ induces a morphism v′′ : Hn → R. By construction, u ◦ v′′ = u ◦ v′ = v and D(v′′ )(ξn ) = ξR , and this proves that φn is smooth. Remark 1.3. The obstruction morphism o : T2 → T1 in Theorem 1.5 is not uniquely defined by the obstruction theory for D. However, we see from the construction of o that on : T2 → T1n is unique up to the ideal m · im(on−1 ) ⊆ T1n for all n ≥ 2, where on : T2 → T1n is the composition of o with the canonical quotient map. In particular, the leading term of o is unique. When D has an obstruction theory with finite dimensional cohomology {H p }, there are certain cohomology operations on H1 with values in H2 , called generalised Massey products, that can be used to compute the pro-representing hull H of D. These generalised Massey products are in a sense dual to the obstruction morphism. In the case of classical deformation functors, these generalised Massey products were described in Section 4.3 of Laudal [27]; see also Laudal [23]. Let o : T2 → T1 be the obstruction morphism of Theorem 1.5. We may consider (H2 )∗ ⊆ T2 as a k-linear subspace, and obtain a k-linear map M ∗ : (H2 )∗ → ∏ Symi ((H1 )∗ ) i≥2

by restriction of o to (H2 )∗ . We write M n : (H2 )∗ → ∏ni=2 Symi ((H1 )∗ ) for the projection of M ∗ on the first n − 1 factors for n ≥ 2. The dual map of M n in the category of k-linear spaces is the k-linear map n a Mn : Symi (H1 ) → H2 i=2

for n ≥ 2. Remark 1.3 shows that M ∗ is not uniquely defined by the obstruction theory for D. However, M ∗ is unique modulo the ideal ∆ = (H1 )∗ · im(M ∗ ) in T1 . Therefore, the quotient map n

Mon : (H2 )∗ → ∏ Symi ((H1 )∗ )/∆n i=2

is uniquely defined, where ∆n = (H1 )∗ · im(M n−1 ). We define Dn to be the dual of ∏ni=2 Symi ((H1 )∗ )/∆n in the category of vector space over k, n

Dn = (∏ Symi ((H1 )∗ )/∆n )∗ i=2

10

Noncommutative Deformation Theory `n

Then Dn ⊆ i=2 Symi (H1 ) is a k-linear subspace for n ≥ 2. We consider the dual map of Mon for n ≥ 2, and obtain a sequence of k-linear maps Mno : Dn → H2 for n ≥ 2. These maps are called the generalised Massey products induced by the obstruction theory for D. It follows from the definition that D2 = Sym2 (H1 ). Corollary 1.6. Let D : l → Sets be a functor of Artin rings. If D has an obstruction theory with finite dimensional cohomology {H p}, then the pro-representing hull H of D is completely determined by the generalised Massey products induced by the obstruction morphism. Proof. We have that H = T1 /a = T1 /( f1 , f2 , . . . , fr ) where fi = o(si ). In terms of the generalised matric Massey products, the power series fi are given by fi = lim fin , ←−

with fin = Mon (si )

n≥2

where M0n is the dual of Mno . Remark 1.4. We say that D is unobstructed if o : T2 → T1 is trivial. In this case, we have that H = T1 is a pro-representing hull for D, and there is a lifting of ξ2 ∈ D(H2 ) to T1 . This lifting is the versal family ξ ∈ D(T1 ). Otherwise, D is obstructed, and it is a nontrivial task to compute H using the generalised Massey products. When H1 = 0, we say that D is rigid. This is a special case of an unobstructed functor D, with pro-representing hull H = k. When D = Def X is a rigid deformation functor, the object X has no deformations.

1.4 Deformations of associative algebras Let A be an associative k-algebra. In this section, we study the classical deformation functor Def A : l → Sets of the associative algebra A, and show how to construct its pro-representing hull in concrete terms. Definition 1.2. We define a lifting of A to an algebra R in l to be a pair (AR , η ), where AR is an associative R-algebra that is R-flat, and where η : k ⊗R AR → A is an isomorphism of associative k-algebras. We say that two liftings (AR , η ) and (A′R , η ′ ) are equivalent if there is an isomorphism τ : AR → A′R of R-algebras such that η ′ ◦ (id ⊗R τ ) = η . Let Def A (R) be the set of equivalence classes of liftings of A to R. Then Def A : l → Sets is the classical deformation functor of the associative k-algebra A. Remark 1.5. The associative R-algebra AR has an underlying structure as an R-R bimodule, where the left and right R-module structures coincide. It is wellknown that AR is R-flat if and only if TorR1 (k, AR ) = 0 (see Theorem 22.3 in Matsumura [35]). Furthermore, the associative algebra AR is R-flat if and only if the underlying R-R bimodule structure of AR is isomorphic to the R-R bimodule R ⊗k A with trivial left and right action of R (see Bourbaki [6], II.3.2, Corollary 2). In order to compute deformations of A in concrete terms, we shall use the following explicit description of the deformations: For any algebra R in l, a deformation in Def A (R) is given by the R-R bimodule AR = R ⊗k A endowed with an R-bilinear multiplication µR : AR ⊗R AR → AR such

Classical Deformation Theory

11

that the diagram AR ⊗ R AR  A ⊗k A

µR

µ

/ AR  /A

commutes, where the vertical maps are induced by the natural map R → k. The map µ : A ⊗k A → A is the multiplication in A. We say that two multiplications µR and µR′ are equivalent if there is an automorphism φR ∈ AutR (AR ) of the R-R bimodule AR inducing the identity on A such that the diagram µR / AR AR ⊗ R AR φR

φR

 AR ⊗ R AR

µR′

 / AR

commutes. Lemma 1.7. There is a bijective correspondence between Def A (R) and the set of equivalence classes of multiplications µR on the R-R bimodule AR = R ⊗k A lifting the multiplication µ to R. Let us fix a k-linear base {ri : 0 ≤ i ≤ l} of R such that r0 = 1. To simplify notation, we shall often write (r ⊗ a)·(s⊗ b) for the multiplication µR (r ⊗ a, s⊗ b) when r, s ∈ R, a, b ∈ A. The multiplication µR is determined by its value on elements of the form 1 ⊗ a with a ∈ A, since we have

µR (r ⊗ a, s ⊗ b) = (r ⊗ a) · (s ⊗ b) = r · (1 ⊗ a)(1 ⊗ b) · s for all r, s ∈ R, a, b ∈ A. It follows that the multiplication µR can be considered as an element in Homk (A ⊗k A, R ⊗k A) ∼ = R ⊗k Homk (A ⊗k A, A), described in concrete terms as l

µR = 1 ⊗ µ + ∑ ri ⊗ δ (ri ) i=1

where δ = {δ (ri ) : 0 ≤ i ≤ l} is a family of k-linear maps δ (ri ) : A ⊗k A → A with δ (1) = µ such that δ (ri )(1 ⊗ b) = δ (ri )(a ⊗ 1) = 0 for 1 ≤ i ≤ l, a, b ∈ A. The last condition comes from the fact that 1 ⊗ 1 is the unit of the ring AR . Conversely, a family δ of k-linear maps δ (ri ) : A ⊗k A → A with the properties that δ (1) = µ and that δ (ri )(1 ⊗ b) = δ (ri )(a ⊗ 1) = 0 for 1 ≤ i ≤ l defines a lifting µR of the multiplication µ to R if and only if the associativity condition [(1 ⊗ a)(1 ⊗ b)](1 ⊗ c) = (1 ⊗ a) [(1 ⊗ b)(1 ⊗ c)] holds for all a, b, c ∈ A. Explicitly, the associativity condition can be expressed in terms of δ as

∑

ri ⊗ [aδ (ri )(b, c) − δ (ri )(ab, c) + δ (ri )(a, bc) − δ (ri )(a, b)c]

1≤i≤l

+

∑

ri r j ⊗ [δ (r j )(δ (ri )(a, b), c) − δ (r j )(a, δ (ri )(b, c))] = 0

1≤i, j≤l

Notice that ri r j = c1i j r1 + c2i j r2 + · · · + cli j rl for all 1 ≤ i, j ≤ l, where c1i j , . . . , cli j ∈ k are constants given by the ring structure on R.

12

Noncommutative Deformation Theory

1.4.1 Tangent space and obstruction calculus The tangent space of Def A is t(Def A ) = Def A (k[ε ]). Since we have a natural k-base {1, ε } of k[ε ], it follows that an element of the tangent space is represented by a multiplication given by

µk[ε ] = 1 ⊗ µ + ε ⊗ δ (ε ) where δ (ε ) : A ⊗k A → A is a k-linear map. In order to describe the tangent space t(Def A ) explicitly in terms of δ (ε ), we recall the definition of the Hochschild complex of A with values in an A-A bimodule Q: Definition 1.3. The Hochschild cohomology HH• (A, Q) of the associative algebra A with values in an A-A bimodule Q is the cohomology of the Hochschild complex HC• (A, Q), with HCn (A, Q) = Homk (⊗nk A, Q) = Homk (A ⊗k A ⊗k · · · ⊗k A, Q) for all n ≥ 0, and with differential d n : HCn (A, Q) → HCn+1 (A, Q) given by d n (φ )(a0 , a1 , . . . , an ) = a0 φ (a1 , . . . , an ) +

∑

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

1≤i≤n

+ (−1)n+1φ (a0 , . . . , an−1 )an for all φ ∈ HCn (A, Q). We may consider A-A bimodules as left modules over the enveloping algebra Ae = A ⊗k Aop , and it is wellknown that HHn (A, Q) ∼ = ExtnAe (A, Q) for all n ≥ 0. In particular, we n n ∼ have HH (A, A) = ExtAe (A, A) for all n ≥ 0. Definition 1.4. Let HC•∗ (A, Q) be the subcomplex of the Hochschild complex HC• (A, Q) defined by the condition that φ (a1 , a2 , . . . , an ) = 0 if ai = 1 for some index i. We define HH•∗ (A, Q) to be the cohomology of this subcomplex. We notice that δ (ε ) is an element of HC2∗ (A, A), and that the associativity condition holds if and only if δ (ε ) is a 2-cocycle. Moreover, we see that if δ (ε ) and δ ′ (ε ) are two 2-cocycles, then the corresponding multiplications are equivalent if and only if δ (ε ) − δ ′ (ε ) ∈ HH2∗ (A, A) is a coboundary. In fact, if δ (ε ) − δ ′ (ε ) = d(ψ ) for some ψ ∈ Endk (A) with ψ (1) = 0, then a 7→ a + εψ (a) defines an isomorphism from (A + ε A, δ (ε )) to (A + ε A, δ ′ (ε )). This proves the following result: Proposition 1.8. The tangent space of the classical deformation functor Def A of an associative k-algebra A is given by t(Def A ) ∼ = HH2∗ (A, A) In particular, t(Def A ) has a natural structure as a vector space over k. In fact, the next result shows that the classical deformation functor Def A has an obstruction theory with cohomology {H p}, where H p = HH∗p+1 (A, A). This is consistent with the computation of the tangent space given above. Proposition 1.9. Let u : R → S be a small surjection in l with kernel K, and let AS ∈ Def A (S) be a deformation of A to S. Then there exists a canonical obstruction o(u, AS ) ∈ K ⊗k HH3∗ (A, A) such that o(u, AS ) = 0 if and only if there is a lifting AR ∈ Def A (R) of AS to R. Moreover, if this is the case, then there is a transitive and free action of K ⊗k HH2∗ (A, A) on the set of liftings of AS to R. Proof. Let the multiplication on AS be given by µS ∈ S ⊗k Homk (A ⊗k A, A). We may choose a klinear section σ : S → R such that σ (1) = 1, and define a multiplication µR on the R-R bimodule AR = R ⊗k A by µR = (σ ⊗ 1)(µS )

Classical Deformation Theory

13

Then it follows that µR lifts µ to R, and that µR (a, b) = µ (a, b) = ab if a = 1 or b = 1. Therefore, (AR , µR ) is a deformation of A to R lifting AS if and only if µR satisfy the associativity condition. We consider the assignment (a, b, c) 7→ µR (µR (a, b), c) − µR (a, µR (b, c)) Notice that this assignment defines an element in K ⊗k Homk (A ⊗k A ⊗k A, A) since µS is associative, and that it is independent of the choice of k-linear section σ since u is small. One may check that it is a 3-cocyle in K ⊗k HC•∗ (A, A). Its class is therefore a canonical element o(u, AS ) ∈ K ⊗k HH3∗ (A, A). Let us define µR′ = µR + τ , where τ ∈ K ⊗k Homk (A ⊗k A, A) such that τ (a, b) = 0 if a = 1 or b = 1. Since K 2 = 0, we have that

µR′ (µR′ (a, b), c) − µR′ (a, µR′ (b, c)) = µR (µR (a, b), c) − µR(a, µR (b, c)) − d(τ )(a, b, c) Therefore, µR′ is associative if and only if the obstruction o(u, AS ) = 0 in K ⊗k HH3∗ (A, A). For the last part, assume that (AR , µR ) is a deformation of A to R lifting AS . By the computation above, µR′ = µR + τ defines another deformation of A to R lifting AS if and only if τ is a 2-cocyle in K ⊗k HC•∗ (A, A). Moreover, we see that if τ , τ ′ are two 2-cocycles, then the deformations (AR , µR + τ ) and (AR , µR + τ ′ ) are equivalent if and only if τ − τ ′ is a coboundary. In fact, if τ − τ ′ = d(ψ ) for ψ ∈ K ⊗k HC1∗ (A, A) with ψ (1) = 0, then a 7→ a + ψ (a) defines an isomorphism from (AR , µR + τ ) to (AR , µR + τ ′ ). It is clear from the construction of the obstruction o(u, AS ) that it is functorial, and Homk ((H p )∗ , K) ∼ = K ⊗k (H p )∗∗ ∼ = K ⊗k H p when H p = HH∗p+1 (A, A). The classical deformation functor Def A of an associative k-algebra A therefore has an obstruction theory with cohomology {HH∗p+1 (A, A)}. By Theorem 1.5, this implies that Def A has a pro-representing hull H whenever dimk HH∗p+1 (A, A) is finite for p = 1 and p = 2. Moreover, there is a constructive method for finding H and its versal family AH in this case. In fact, there is an obstruction morphism in ˆl, o : T2 = k[[s1 , s2 , . . . , sr ]] → T1 = k[[t1 ,t2 , . . . ,td ]] such that H = T1 /( f1 , f2 , . . . , fr ), where d = dimk HH2∗ (A, A), r = dimk HH3∗ (A, A), and fi = o(si ) for 1 ≤ i ≤ r.

1.4.2 Examples We consider the classical deformation functor Def A of the associative k-algebra A in the three examples A = k[x], A = k[x]/(x2 ) and A = khx, yi/(x, y)2 . We compute the pro-representing hull of Def A and its versal family in the first two cases, and give a partial result in the third case. Example 1.1. Let A = k[x]. Then Ae = A ⊗k Aop = k[x] ⊗k k[y]. The left Ae -module A is cyclic with generator 1 ∈ A, and has a free resolution x−y

0 ← A ← Ae ←−− Ae ← 0 since x · 1 = 1 · x = y · 1. It follows that HHn (A, A) ∼ = ExtnAe (A, A) = 0 for n > 1. We therefore see that t(Def A ) = 0, which means that A = k[x] is a rigid associative algebra, with pro-representing hull H = k and versal family AH = A = k[x]. Example 1.2. Let A = k[x]/(x2 ). In this case, we may use the fact that A has a finite k-linear base {1, x} to compute the cohomology groups HH•∗ (A, A) directly from the Hochschild complex. We

14

Noncommutative Deformation Theory

have that f ∈ HCn∗ (A, A) is completely determined by f (x, x, . . . , x) since f (a1 , a2 , . . . , an ) = 0 if ai = 1 for some index i. Therefore, we have HC1∗ (A, A)

A

/ HC2∗ (A, A)

d1

/A

/ HC3∗ (A, A)

d2

/ HC4∗ (A, A)

/A

d3

/A

/ ...

d4

/ ...

Moreover, the fact that A is commutative and that x2 = 0 gives that d n = 0 when n is even, and that d n (a) = 2x · a when n is odd. This implies that ( k·1 ∼ = k, n is even n HH∗ (A, A) = ∼ k · x = k, n is odd ∼ k. It follows that the classical deformation functor Def A has a proIn particular, t(Def A ) = representing hull H = k[[t]]/( f ), where the power series f = o(s) is determined by the obstruction morphism o : k[[s]] → k[[t]]. At the tangent level, H2 = k[[t]]/(t 2 ), and the versal family ξ2 ∈ Def A (H2 ) is given by a multiplication µ2 : A ⊗k A → H2 ⊗k A lifting µ to H2 . In concrete terms, µ2 is given by

µ2 (a, b) = 1 ⊗ µ (a, b) + t ⊗ δ (t)(a, b) for a, b ∈ A. We let t ∗ = 1 such that {t ∗ } is a k-linear base for t(Def A ), and let δ (t) be a 2-cocycle in the Hochschild complex HC∗∗ (A, A) that represents t ∗ . We choose ( 0, a = 1 or b = 1 δ (t)(a, b) = 1, a = b = x Then the multiplication µ2 is explicitly given by

µ2 (a, b) = 1 ⊗ µ (a, b) + t ⊗ δ (t)(a, b) =

(

1 ⊗ ab, a = 1 or b = 1 t ⊗ 1, a=b=x

In other words, µ2 (1, 1) = 1, µ2 (1, x) = µ2 (x, 1) = x and µ2 (x, x) = t. By construction, this multiplication defined on H2 is associative, and we may check that this is the case:

µ2 (µ2 (x, x), x) − µ2 (x, µ2 (x, x)) = (x · x)x − x(x · x) = tx − xt = 0 In fact, we do not use that t 2 = 0 in H2 to show that the obstruction above vanishes. This means that we may lift ξ2 ∈ Def A (H2 ) from H2 = k[[t]]/(t 2 ) to T1 = k[[t]]. In particular, H = k[[t]] is a prorepresenting hull for Def A (with f = 0) and Def A is unobstructed. The versal family ξ ∈ Def A (H) is given by a lifting µH of the multiplication µ2 to H. In concrete terms, the multiplication µH is given by ( ab, a = 1 or b = 1 µH (a, b) = t, a=b=x This means that the versal family ξ ∈ Def A (H) is the H-algebra given by AH = k[x][[t]]/(x2 − t) We see that there is an algebraization H = k[t] of H, and an algebraization AH = k[x,t]/(x2 − t) of the versal family AH . The family {AH (τ ) : τ ∈ k}, with AH (τ ) ∼ = k[x]/(x2 − τ ), is the corresponding family of deformations of A parameterized by the k-rational points of H = Spec H.

15

Classical Deformation Theory

Example 1.3. Let A = khx, yi/(x, y)2 . We may use the fact that A has a finite k-linear base {1, x, y} to compute the cohomology groups HH•∗ (A, A) directly from the Hochschild complex. We have that f ∈ HCn∗ (A, A) is completely determined by f (a1 , a2 , . . . , an ) with ai ∈ {x, y} for 1 ≤ i ≤ n since f (a1 , a2 , . . . , an ) = 0 if ai = 1 for some index i. Therefore, we have / HC2∗ (A, A)

HC1∗ (A, A)

A2

d1

/ A4

/ HC3∗ (A, A)

d2

/ A8

/ HC4∗ (A, A)

d3

/ A16

/ ...

d4

/ ...

The differentials are a bit more complicated in this case. Since (x, y)2 = 0 in A, we have that d n ( f )(a0 , . . . , an ) = a0 f (a1 , . . . , an ) + (−1)n+1 f (a0 , . . . , an−1 )an when ai ∈ {x, y} for 0 ≤ i ≤ n. This implies that n ker(d n ) = HCn∗ (A, (x, y)) = (x, y)2 Hence ker(d 2 ) = (x, y)4 , and the image of d 1 is the two-dimensional vector space d 1 (k2 ) generated by the two elements (2x, y, y, 0) and (0, x, x, 2y). In particular, t(Def A ) ∼ = HH2∗ (A, A) ∼ = (x, y)4 /h(2x, y, y, 0), (0, x, x, 2y)i ∼ = k6 Similarly, ker(d 3 ) = (x, y)8 , and the image of d 2 is the four-dimensional vector space d 2 (k4 ). This implies that HH3∗ (A, A) = (x, y)8 /d 2 (k4 ) ∼ = k12 Since t(Def A ) ∼ = k6 and HH3∗ (A, A) ∼ = k12 , it follows that the classical deformation functor Def A has a pro-representing hull H = k[[t1 , . . . ,t6 ]]/( f1 , . . . , f12 ), where the power series fi = o(si ) are determined by the obstruction morphism o : k[[s1 , . . . , s12 ]] → k[[t1 , . . . ,t6 ]]. At the tangent level, H2 = k[[t1 ,t2 ,t3 ,t4 ,t5 ,t6 ]]/(t1 , . . . ,t6 )2 , and the versal family ξ2 ∈ Def A (H2 ) is given by a multiplication µ2 : A ⊗k A → H2 ⊗k A lifting µ to H2 . In concrete terms, µ2 is given by

µ2 (a, b) = 1 ⊗ µ (a, b) +

∑

ti ⊗ δ (ti )(a, b)

1≤i≤6

for a, b ∈ A. We choose a k-linear base {t1∗ , . . . ,t6∗ } of t(Def A ) defined by the k-linear maps c is given by Fxxx , Fxxy , Fxyx , Fxyy , Fyyx , Fyyy ∈ Homk (A ⊗k A, A), where Fab ( c, (a′ , b′ ) = (a, b) c ′ ′ Fab (a , b ) = 0, (a′ , b′ ) 6= (a, b) for all a, b, c ∈ {x, y}. Then the relations in t(Def A ) are Fyxy = −2Fxxx − Fxyy and Fyxx = −2Fyyy − Fxyx . c that represents t ∗ . Explicitly, we have We let δ (ti ) be the 2-cocycle of the form Fab i

δ (t1 ) = Fxxx , δ (t2 ) = Fxxy , δ (t3 ) = Fxyx , δ (t4 ) = Fxyy , δ (t5 ) = Fyyx , δ (t6 ) = Fyyy Then the multiplication µ2 is explicitly given by µ2 (a, b) = ab when a = 1 or b = 1, and

µ2 (x, x) = t1 x + t2y µ2 (y, x) = 0

µ2 (x, y) = t3 x + t4 y µ2 (y, y) = t5 x + t6 y

since x2 = xy = yx = y2 = 0. By construction, this multiplication defined on H2 is associative, and we may check that this is the case. For instance, when (a1 , a2 , a3 ) = (x, y, x), the value of the obstruction µ2 (µ2 (a1 , a2 ), a3 ) − µ2(a1 , µ2 (a2 , a3 )) is

µ2 (µ2 (x, y), x) − µ2 (x, µ2 (y, x)) = (x · y)x − x(y · x) = (t3 x + t4 y)x − x · 0 = t3 (t1 x + t2y) + t4 · 0 = t1t3 x + t2t3 y

16

Noncommutative Deformation Theory

Since (t1 , . . . ,t6 )2 = 0 in H2 , this value of the obstruction vanishes. A similar computation for the value of the obstruction when (a1 , a2 , a3 ) 6= (x, y, x) shows that the obstruction vanishes in H2 . We claim that the obstruction for lifting µ2 to an associative multiplication µ3 on T13 does not vanish. By the computation of HH3∗ (A, A) above, it follows that the elements in (x, y)8 that are zero in HH3∗ (A, A) are given by d 2 (k4 ). Let us consider a k-linear map f ∈ HC2∗ (A, A) that corresponds to an element (α1 , . . . , α4 ) ∈ k4 . Then f is explicitly given by f (x, x) = α1 ,

f (x, y) = α2 ,

f (y, x) = α3 ,

f (y, y) = α4

and we have that the value of d 2 ( f ) on (x, y, x) is d 2 ( f )(x, y, x) = x f (y, x) − f (x, y)x = (α3 − α2 )x This show that the obstruction for lifting µ2 to T13 is non-zero since y 6∈ d 2 (k4 )(x, y, x). In particular, the deformation functor Def A is obstructed. In fact, one of the power series fi that define its pro-representing hull H = k[[t1 , . . . ,t6 ]]/( f1 , . . . , f12 ) has leading term t2t3 . We shall not complete the computation of the pro-representing hull H in this case.

1.5 Deformations of modules Let A be an associative k-algebra, and let M be a right A-module. In this section, we study the classical deformation functor Def M : l → Sets of the A-module M, and show how to construct its pro-representing hull in concrete terms. Definition 1.5. We define a lifting of M to an algebra R in l to be a pair (MR , η ), where MR is an R-A bimodule, on which k acts centrally, such that MR is R-flat, and where η : k ⊗R MR → M is an isomorphism of right A-modules. We say that two liftings (MR , η ) and (MR′ , η ′ ) are equivalent if there is an isomorphism τ : MR → MR′ of R-A bimodules such that η ′ ◦ (id ⊗R τ ) = η . Let Def M (R) be the set of equivalence classes of liftings of M to R. Then Def M : l → Sets is the classical deformation functor of the right A-module M. In order to compute with deformations of M in concrete terms, we shall use free resolutions. Let us therefore fix a free resolution of the right A-module M of the form d

d

0 1 0←M← − L0 ←− L1 ←− L2 ← · · ·

We often use the shorter notation (L• , d• ) to refer to this free resolution. Let R be an algebra in l. For any free right A-module L, we write LR = R ⊗k L for the trivial lifting of the right A-module L to R. Notice that LR is projective in the category of R-A bimodules. We define a lifting of complexes of (L• , d• ) to R to be a complex (LR• , d•R ) of R-A bimodules, where LRm = R ⊗k Lm is a projective module of the type defined above for all m ≥ 0, such that the natural diagram LR0 o  L0 o

d0R

d0

LR1 o  L1 o

d1R

d1

LR2 o  L2 o

d2R

d2

···

···

commutes. The vertical maps in this diagram are induced by the natural map R → k. We say that two liftings of complexes (LR• , d•R ) and (LR• , ′ d•R ) are equivalent if there is an isomorphism of complexes of R-A bimodules (LR• , d•R ) ∼ = (LR• , ′ d•R ) inducing the identity on (L• , d• ).

17

Classical Deformation Theory

Proposition 1.10. There is a bijective correspondence between Def M (R) and the set of equivalence classes of complexes (LR• , d•R ) lifting the complex (L• , d• ) to R. Proof. When (LR• , d•R ) is a lifting of complexes of (L• , d• ) to R, one may show that it is a projective resolution of the R-A bimodule MR = coker(d0R ), hence MR ∈ Def M (R) is a deformation. Conversely, any deformation MR ∈ Def M (R) has a projective resolution of R-A bimodules (LR• , d•R ) that is a lifting of complexes of (L• , d• ) to R. We shall not prove these claims here. In Chapter 3, we give a detailed proof in the more general noncommutative setting; see Lemma 3.9 and Lemma 3.10. Let us fix a k-linear base {ri : 0 ≤ i ≤ l} for R such that r0 = 1. The differential dmR is determined by its value on elements of the form 1 ⊗ f with f ∈ Lm+1 since we have dmR (r ⊗ f ) = r · dmR (1 ⊗ f ) for all r ∈ R, f ∈ Lm+1 . It follows that the differential dmR can be considered as an element in HomA (Lm+1 , R ⊗k Lm ) ∼ = R ⊗k HomA (Lm+1 , Lm ), described in concrete terms as l

dmR = 1 ⊗ dm + ∑ ri ⊗ α (ri )m i=1

where α = {α (ri )m : m ≥ 0, 0 ≤ i ≤ l} is a family of A-linear homomorphisms α (ri )m : Lm+1 → Lm with α (1)m = dm . Conversely, such a family α of A-linear homomorphisms represents a lifting of R complexes of (L• , d• ) to R if and only if dmR ◦ dm+1 = 0 for all m ≥ 0. Explicitly, this condition can be expressed in terms of α as

∑

ri ⊗ (α (ri )m dm+1 + dm α (ri )m+1 ) +

1≤i≤l

∑

ri r j ⊗ α (ri )m α (r j )m+1 = 0

1≤i, j≤l

Notice that ri r j = c1i j r1 + c2i j r2 + · · · + cli j rl for all 1 ≤ i, j ≤ l, where c1i j , . . . , cli j ∈ k are constants given by the ring structure on R.

1.5.1 Tangent space and obstruction calculus The tangent space of Def M is t(Def M ) = Def M (k[ε ]). Since we have a natural k-base {1, ε } of k[ε ], it follows from Proposition 1.10 that an element of the tangent space is represented by a lifting of complexes given by k[ε ] dm = 1 ⊗ dm + ε ⊗ α (ε )m where α (ε )m : Lm+1 → Lm is an A-linear map for m ≥ 0. In order to describe the tangent space t(Def M ) explicitly in terms of α (ε ), we recall the definition of the Yoneda complex: Definition 1.6. The Yoneda cohomology YH• (M, M) of M is the cohomology of the Yoneda complex YC• (L• , L• ), defined in terms of the free resolution (L• , d• ) of M, with YCn (L• , L• ) =

∏ HomA (Lm+n , Lm )

m≥0

for all n ≥ 0, and with differential d n : YCn (L• , L• ) → YCn+1 (L• , L• ) given by d n (φ )m = φm dn+m + (−1)n+1dm φm+1

for m ≥ 0

for all φ = (φm )m≥0 ∈ YCn (L• , L• ). We have that YHn (M, M) ∼ = ExtnA (M, M) for all n ≥ 0, hence n YH (M, M) is independent of the chosen free resolution.

18

Noncommutative Deformation Theory

We note that α (ε ) = (α (ε )m )m≥0 is an element of YC1 (L• , L• ), and observe that the condition = 0 holds if and only if α (ε ) is a 1-cocycle. Moreover, the lifting of complexes corresponding to the 1-cocycles α (ε ) and α ′ (ε ) are equivalent liftings of complexes of (L• , d• ) to k[ε ] if and only if α (ε ) − α ′ (ε ) ∈ YC1 (L• , L• ) is a coboundary. In fact, if α (ε ) − α ′ (ε ) = d(ψ ) for some ψ = (ψm ) ∈ YC0 (L• , L• ) with ψm ∈ EndA (Lm ) for m ≥ 0, then ψ defines an isomorphism (LR• , d•R ) → (LR• , ′ d•R ) of complexes of R-A bimodules. This gives the following characterization of t(Def M ): R dmR ◦ dm+1

Proposition 1.11. The tangent space of the classical deformation functor Def M of a right A-module M is given by t(Def M ) ∼ = YH1 (M, M) ∼ = Ext1A (M, M) In particular, t(Def M ) has a natural structure as a vector space over k. In fact, the next result shows that the classical deformation functor Def M has an obstruction p theory with cohomology {H p }, where H p = YH p (M, M) ∼ = ExtA (M, M). This is consistent with the computation of the tangent space given above. Proposition 1.12. Let u : R → S be a small surjection in l with kernel K, and let MS ∈ Def M (S) be a deformation of M to S. Then there exists a canonical obstruction o(u, MS ) ∈ K ⊗k YH2 (M, M) such that o(u, MS ) = 0 if and only if there is a lifting MR ∈ Def M (R) of MS to R. Moreover, if this is the case, then there is a transitive and free action of K ⊗k YH1 (M, M) on the set of liftings of MS to R. Proof. The deformation MS ∈ Def M (S) corresponds to a lifting (LS• , d•S ) of the complex (L• , d• ) to S. We choose a k-linear section σ : S → R of u such that σ (1) = 1, and define an R-A linear map dmR ∈ R ⊗k HomA (Lm+1 , Lm ) by dmR = (σ ⊗ 1) ◦ dmS for all m ≥ 0. Then it follows that dmR lifts dmS to R. Therefore, (LR• , d•R ) corresponds to a deformation R S MR ∈ Def M (R) lifting MS to R if and only if dmR dm+1 = 0 for all m ≥ 0. Since dmS dm+1 = 0 for all R R m ≥ 0, we may consider dm dm+1 as an element in K ⊗k HomA (Lm+2 , Lm ), and we see that it is a 2-cocycle in K ⊗k YC• (L• , L• ). It is independent of the choice of section σ since u is small. Its class is therefore a canonical element o(u, MS ) ∈ K ⊗k YH2 (M, M). If we define ′ dmR = dmR + τm , where τm ∈ K ⊗k HomA (Lm+1 , Lm ), then we have that ′ R′ R dm dm+1

R = dmR dm+1 + d(τ )m

Therefore d•R defines a lifting of complexes to R if and only if o(u, MS ) = 0. For the last part, assume that d•R defines a lifting of complexes to R. By the computation above, ′ dmR = dmR + τm defines another lifting of complexes to R is and only if τ is a 1-cocycle. Moreover, we see that if τ , τ ′ are two 1cocycles, then they define equivalent lifting of complexes if and only if τ − τ ′ is a coboundary. In fact, if τ − τ ′ = d(ψ ) for some ψ = (ψm ) with ψm ∈ K ⊗k EndA (Lm ), then ψ defines an isomorphism (LR• , d•R ) → (LR• , ′ d•R ) of complexes of R-A bimodules. It is clear from the construction of the obstruction o(u, MS ) that it is functorial, and Homk ((H p )∗ , K) ∼ = K ⊗k (H p )∗∗ ∼ = K ⊗k H p when H p = YH p (M, M). The classical deformation functor Def M of the right A-module M therefore has an obstruction theory with cohomology {ExtAp (M, M)}. By Theorem 1.5, this implies that Def M has a pro-representing hull (H, MH ) whenever dimk ExtAp (M, M) is finite for p = 1 and p = 2. Moreover, there is a constructive method for finding H and its versal family MH in this case. In fact, there is an obstruction morphism in ˆl, o : T2 = k[[s1 , s2 , . . . , sr ]] → T1 = k[[t1 ,t2 , . . . ,td ]] such that H = T1 /( f1 , f2 , . . . , fr ), where d = dimk Ext1A (M, M), r = dimk Ext2A (M, M), and fi = o(si ) for 1 ≤ i ≤ r.

19

Classical Deformation Theory

1.5.2 Examples We consider the classical deformation functor of the following right modules, and compute the pro-representing hull and its versal family in each case: 1. M = k[x, y]/(x, y) considered as a right module over A = k[x, y] 2. M = (x, y) considered as a right module over A = k[x, y]/(x2 − y3 ) 3. M = (x, y2 ) considered as a right module over A = k[x, y]/(x3 − y4) Example 1.4. Let A = k[x, y], and let M be the right A-module M = A/(x, y) with free resolution 

( x y )·

 y −x ·

0 ← M ← A ←−−− A2 ←−−−− A ← 0 p To compute YH p (M, M) ∼ = ExtA (M, M) for p = 1 and p = 2, we consider the complex HomA (L• , M):   y · −x

·( x y )

M −−−→ M 2 −−−−→ M → 0 Note that the differentials in this complex are zero. Since M = k[x, y]/(x, y) ≃ k has dimension one, we see that ( k2 , p = 1 p YH p (M, M) ∼ = ExtA (M, M) = k, p = 2 It follows that the classical deformation functor Def M has a pro-representing hull H = k[[t1 ,t2 ]]/( f ), where the power series f = o(s) is determined by the obstruction morphism o : k[[s]] → k[[t1 ,t2 ]]. At the tangent level, H2 = k[[t1 ,t2 ]]/(t1 ,t2 )2 , and the versal family ξ2 ∈ Def M (H2 ) is given by H2 H2 H 2 a lifting of complexes (LH • , d• ) of (L• , d• ) to H2 . In concrete terms, the differential d 2 = (dm ), with dmH2 ∈ H2 ⊗k HomA (Lm+1 , Lm ), is given by dmH2 = 1 ⊗ dm +

∑

ti ⊗ α (ti )m

1≤i≤2

for all m ≥ 0. We let t1∗ = (1, 0) and t2∗ = (0, 1) such that {t1∗ ,t2∗ } is a k-linear base for t(Def M ), and let α (ti ) be a 1-cocycle in the Yoneda complex YC• (L• , L• ) that represents ti∗ ∈ YH1 (M, M). Note that a 1-cocyle φ ∈ YC1 (L• , L• ) is a pair (φ0 , φ1 ) of A-linear maps φi : Li+1 → Li such that d0 φ1 + φ0 d1 = 0 since Li = 0 for i > 2. We may therefore choose        

0 1 α (t1 ) = 1 0 · , · α (t2 ) = 0 1 · , · −1 0 Then, the differential d H2 = (d0H2 , d1H2 ) is explicitly given by d0H2 = d0 + d1H2 = d1 +

∑

ti α (ti )0 = x + t1

∑

ti α (ti )1 =

1≤i≤2

1≤i≤2




y + t2 ·

 y + t2 · −x − t1

By construction, d0H2 ◦ d1H2 = 0 in H2 ⊗k HomA (L2 , L0 ) and we may check that this is the case:  
y + t2 d0H2 ◦ d1H2 = x + t1 y + t2 · = (x + t1 )(y + t2) + (y + t2)(−x − t1) = 0 −x − t1

20

Noncommutative Deformation Theory

In fact, we do not use that (t1 ,t2 )2 = 0 in H2 to show that the obstruction above vanishes. This means that we may lift ξ2 ∈ Def M (H2 ) from H2 = k[[t1 ,t2 ]]/(t1 ,t2 )2 to T1 = k[[t1 ,t2 ]]. In particular, H = k[[t1 ,t2 ]] is a pro-representing hull of Def M (with f = 0) and Def M is unobstructed. The versal H2 H2 H family ξ ∈ Def M (H) is given by a lifting of complexes (LH • , d• ) from (L• , d• ) to H. In concrete terms, the differential d H = (d0H , d1H ) is given by  
y + t2 H H d0 = x + t1 y + t2 · and d1 = · −x − t1 This means that the versal family ξ ∈ Def M (H) is the H-A bimodule MH given by MH = coker(d0H ) = k[[t1 ,t2 ]][x, y]/ (x + t1, y + t2 ) We see that there is an algebraization H = k[t1 ,t2 ] of H and an algebraization MH of the versal family MH , given by MH = k[t1 ,t2 , x, y]/ (x + t1, y + t2 ). The family {MH (τ ) : τ = (τ1 , τ2 ) ∈ k2 }, with MH (τ ) ∼ = k[x, y]/ (x + τ1 , y + τ2 ), is the corresponding family of deformations of M parameterized by the k-rational points of H = SpecH. We recognize this family as the family of simple modules S over A = k[x, y] with dimk S = 1. Example 1.5. Let A = k[x, y]/(x2 − y3 ), and let M be the ideal M = (x, y) in A considered as a right A-module, with free resolution (obtained using the generators y and −x) 

 x y2 · y x



 x −y2 · −y x



 x y2 · y x

0 ← M ← A2 ←−−−− A2 ←−−−−−−− A2 ←−−−− A2 ← . . . We notice that this free resolution has period two, with di = d0 when i is even and di = d1 when p i is odd. To compute YH p (M, M) ∼ = ExtA (M, M) for p = 1 and p = 2, we consider the complex HomA (L• , M):    2 · xy y x 2

·

x −y2 −y x

 2 · xy y x 2

M −−−−→ M 2 −−−−−−−→ M −−−−→ M 2 → . . .

We first compute Ext1A (M, M) = ker(d 1 )/ im(d 0 ). Let (m1 , m2 ) ∈ M 2 , where mi is represented 1 i1 by ( m mi2 ) with mi j ∈ k[x, y]. Then (m1 , m2 ) ∈ ker(d ) if and only if there are polynomials f1 , f2 , f3 , f4 , h1 , h2 , h3 , h4 ∈ k[x, y] such that xm11 − ym21 = f1 x + f2 y2 + h1(x2 − y3 ) xm12 − ym22 = f1 y + f2 x + h2(x2 − y3 ) −y2 m11 + xm21 = f3 x + f4 y2 + h3(x2 − y3 ) −y2 m12 + xm22 = f3 y + f4 x + h4(x2 − y3 ) We solve the first two equations of this system using unique factorization in k[x, y], and find that      
m11 m21 1 0 0 −y m1 m2 = = f1 + f2 + im(d 0 ) m12 m22 0 −1 1 0

We check that the elements in M 2 of this form are also solutions of the last two equations. Let us define     1 0 0 −y ∗ ∗ t1 = · and t2 = · 0 −1 1 0

Since we have that xti∗ , yti∗ ∈ im(d 0 ) for 1 ≤ i ≤ 2, we see that Ext1A (M, M) ∼ = k2 with k-linear 2 ∗ ∗ 2 ∼ base {t1 ,t2 }. In particular, t(Def M ) = k . We may compute ExtA (M, M) in a similar way. Since

21

Classical Deformation Theory

Ext2A (M, M) = ker(d 2 )/ im(d 1 ), and (m1 , m2 ) ∈ ker(d 2 ) if and only if (m1 , −m2 ) ∈ ker(d 1 ), it follows that Ext2A (M, M) ∼ = k2 with k-linear base {s∗1 , s∗2 }, where     1 0 0 y s∗1 = · and s∗2 = · 0 1 1 0 The classical deformation functor Def M therefore has a pro-representing hull H = k[[t1 ,t2 ]]/( f1 , f2 ), where the power series fi = o(si ) are given by the obstruction morphism o : k[[s1 , s2 ]] → k[[t1 ,t2 ]]. At the tangent level, H2 = k[[t1 ,t2 ]]/(t1 ,t2 )2 and the versal family ξ2 ∈ Def M (H2 ) is given by H2 H2 H 2 a lifting of complexes (LH • , d• ) of (L• , d• ) to H2 . In concrete terms, the differential d 2 = (dm ), with dmH2 ∈ H2 ⊗k HomA (Lm+1 , Lm ), is given by dmH2 = 1 ⊗ dm +

∑

ti ⊗ α (ti )m

1≤i≤2

for all m ≥ 0, where α (ti ) is a 1-cocycle in the Yoneda complex YC• (L• , L• ) that represents the cohomology class ti∗ ∈ YH1 (M, M). Since the free resolution (L• , d• ) has period two, we see that a pair (φ0 , φ1 ) of A-linear maps φi : Li+1 → Li such that d0 φ1 + φ0 d1 = 0 and d1 φ0 + φ1 d0 = 0 determines a 1-cocycle φ = (φi ) ∈ YC1 (L• , L• ) with φi = φ0 when i is even and φi = φ1 when i is odd. We write {φ0 , φ1 } for such a 1-cocycle, and choose           1 0 −1 0 0 −y 0 y α (t1 ) = ·, · α (t2 ) = ·, · 0 −1 0 1 1 0 −1 0 The differential d H2 = (dmH2 ) is explicitly given by dmH2 = d0H2 when m is even and dmH2 = d1H2 when m is odd, where   x + t1 y2 − t2 y d0H2 = d0 + ∑ ti α (ti )0 = · y + t2 x − t1 1≤i≤2   x − t1 −y2 + t2 y H2 d1 = d1 + ∑ ti α (ti )1 = · −y − t2 x + t1 1≤i≤2 By construction, d0H2 ◦ d1H2 = 0 and d1H2 ◦ d0H2 = 0 in H2 ⊗k HomA (L2 , L0 ). We may check that this is the case:     x + t1 y2 − t2 y x − t1 −y2 + t2 y · d0H2 ◦ d1H2 = y + t2 x − t1 −y − t2 x + t1  2  x − y3 − t12 + t22y 0 = =0 0 x2 − y3 − t12 + t22 y d1H2 ◦ d0H2 =



x − t1 −y − t2

  −y2 + t2 y x + t1 · x + t1 y + t2

 y2 − t2 y x − t1  2 x − y3 − t12 + t22y = 0

In this case, we have used that (t1 ,t2 )2 = 0 in H2 fact, the obstruction is given by  2 2   −t1 + t2 y 0 2 1 = −t1 0 0 −t12 + t22y

 0 =0 x2 − y3 − t12 + t22 y

to show that the obstruction above vanishes. In    1 0 0 2 + t2 y = −t12 s∗1 + t22ys∗1 0 1 1

22

Noncommutative Deformation Theory

We know that s∗1 6∈ im(d 1 ) while y s∗1 ∈ im(d 1 ). This implies that H3 = k[[t1 ,t2 ]]/a3 , where a3 is the ideal a3 = ( f12 ) + (t1 ,t2 )3 and f12 = −t12 is the second-order approximation of f1 . To lift ξ2 to H3 , we choose α (t22 ) such that d 1 α (t22 ) = −ys∗1 , and find that          0 1 0 −1 0 −1 0 1 ·, · d1 ·, · = ys∗1 ⇒ α (t22 ) = 0 0 0 0 0 0 0 0 Explicitly, the lifting ξ3 is represented by the differential d H3 , given by   x + t1 y2 − t2 y + t22 H d0 3 = · y + t2 x − t1   x − t1 −y2 + t2 y − t22 H d1 3 = · −y − t2 x + t1 Again, we compute the obstruction given by d0H3 d1H3 and d1H3 d0H3 , and find that d0H3

◦ d1H3

 x + t1 = y + t2

   y2 − t2 y + t22 x − t1 −y2 + t2 y − t22 · x − t1 −y − t2 x + t1  2  x − y3 − t12 − t23 0 H H = = d1 3 ◦ d0 3 0 x2 − y3 − t12 − t23

This obstruction is given by (−t12 − t23 )s∗1 . Hence H4 = k[[t1 ,t2 ]]/a4 , where a4 = ( f13 ) + (t1 ,t2 )4 and f13 = −t12 − t23 is the third order approximation of f1 . We have not used that (t1 ,t2 )4 = 0 in H4 to show that the obstruction above vanishes. Therefore, we may lift ξ3 to H = k[[t1 ,t2 ]]/(−t12 − t23 ), and this implies that H is the pro-representing hull of Def M (with f1 = −t12 − t23 and f2 = 0), and H that Def M is obstructed. The versal family ξ ∈ Def M (H) is given by a lifting of complexes (LH • , d• ) H2 H2 from (L• , d• ) to H. In concrete terms, the differential d H = (dmH ) is given by     x + t1 y2 − t2 y + t22 x − t1 −y2 + t2 y − t22 H H d0 = · and d1 = · y + t2 x − t1 −y − t2 x + t1 This means that the versal family ξ ∈ Def M (H) is the H-A bimodule MH given by   x + t1 y2 − t2 y + t22 H 2 MH = coker(d0 ) = k[[t1 ,t2 ]][x, y] / y + t2 x − t1 We see that there is an algebraization of H, the cusp H = k[t1 ,t2 ]/(−t12 − t23 ), and an algebraization MH of the versal family MH , given by   x + t1 y2 − t2 y + t22 2 MH = k[t1 ,t2 , x, y] / y + t2 x − t1 The family {MH (τ ) : τ = (τ1 , τ2 ) ∈ k2 with τ12 + τ23 = 0}, where   x + τ1 y2 − τ2 y + τ22 ∼ MH (τ ) ∼ = k[x, y]2 / = (x + τ1 , y + τ2 ) y + τ2 x − τ1 is the corresponding family of deformations of M, and it is parameterized by the k-rational points of H = Spec H. We recognize this as the family of maximal ideals m in A = k[x, y]/(x2 − y3 ) such that dimk (A/m) = 1.

23

Classical Deformation Theory

Example 1.6. Let A = k[x, y]/(x3 − y4), and let M be the ideal M = (x, y2 ) in A considered as a right A-module, with free resolution (obtained using the generators y2 and −x) 

 x y2 · 2 2 y x

 2  x −y2 · 2 −y x 2



 x y2 · 2 2 y x

0 ← M ← A2 ←−−−−− A ←−−−−−−− A2 ←−−−−− A2 ← . . . We notice that this free resolution has period two, with di = d0 when i is even and di = d1 when p i is odd. To compute YH p (M, M) ∼ = ExtA (M, M) for p = 1 and p = 2, we consider the complex HomA (L• , M):       ·

x y2

y2 x2

·

x2 −y2 −y2 x

·

x y2

y2 x2

M 2 −−−−−→ M 2 −−−−−−−→ M 2 −−−−−→ M 2 → . . .

We first compute Ext1A (M, M) = ker(d 1 )/ im(d 0 ). Let (m1 , m2 ) ∈ M 2 , where mi is represented 1 i1 by ( m mi2 ) with mi j ∈ k[x, y]. Then (m1 , m2 ) ∈ ker(d ) if and only if there are polynomials f1 , f2 , f3 , f4 , h1 , h2 , h3 , h4 ∈ k[x, y] such that x2 m11 − y2 m21 = f1 x + f2 y2 + h1 (x3 − y4) x2 m12 − y2 m22 = f1 y2 + f2 x2 + h2 (x3 − y4 ) −y2 m11 + xm21 = f3 x + f4 y2 + h3 (x3 − y4) −y2 m12 + xm22 = f3 y2 + f4 x2 + h4 (x3 − y4 ) We solve the last two equations of this system using unique factorization in k[x, y], and find that      
m11 m21 −1 0 0 1 m1 m2 = = f4 + f3 + im(d 0 ) m12 m22 0 x −1 0

We check that the elements in M 2 of this form are also solutions of the first two equations. Let us define     −1 0 0 1 ∗ ∗ t1 = · and t2 = · 0 x −1 0 Since we have that xti∗ , y2ti∗ ∈ im(d 0 ) but that ti∗ , yti∗ 6∈ im(d 0 ) for i = 1 and i = 2, we also define     −y 0 0 y ∗ ∗ ∗ ∗ t3 = yt1 = · and t4 = yt2 = · 0 xy −y 0

∼ Ext1 (M, M) ∼ We see that t(Def M ) = = k4 with k-linear base {t1∗ ,t2∗ ,t3∗ ,t4∗ }. We compute Ext2A (M, M) A 2 in a similar way since ExtA (M, M) = ker(d 2 )/ im(d 1 ), and note that (m1 , m2 ) ∈ ker(d 2 ) if and only if (m2 , −m1 ) ∈ ker(d 1 ). Hence Ext2A (M, M) ∼ = k4 with k-linear base {s∗1 , s∗2 , s∗3 , s∗4 }, where         0 1 1 0 0 y y 0 s∗1 = · , s∗2 = · , s∗3 = · , s∗4 = · x 0 0 1 xy 0 0 y The classical deformation functor Def M therefore has a pro-representing hull H = k[[t1 ,t2 ,t3 ,t4 ]]/a, where a = ( f1 , f2 , f3 , f4 ) and the power series fi = o(si ) are given by the obstruction morphism o : k[[s1 , s2 , s3 , s4 ]] → k[[t1 ,t2 ,t3 ,t4 ]]. At the tangent level, H2 = k[[t1 ,t2 ,t3 ,t4 ]]/(t1 ,t2 ,t3 ,t4 )2 and the versal family ξ2 in Def M (H2 ) H2 2 is given by a lifting of complexes (LH • , d• ) of (L• , d• ) to H2 . In concrete terms, the differential d H2 = (dmH2 ), with dmH2 ∈ H2 ⊗k HomA (Lm+1 , Lm ), is given by dmH2 = 1 ⊗ dm +

∑

1≤i≤4

ti ⊗ α (ti )m

24

Noncommutative Deformation Theory

for all m ≥ 0, where α (ti ) is a 1-cocycle in the Yoneda complex YC• (L• , L• ) that represents the cohomology class ti∗ ∈ YH1 (M, M). Since the free resolution (L• , d• ) has period two, we see that a pair (φ0 , φ1 ) of A-linear maps φi : Li+1 → Li such that d0 φ1 + φ0 d1 = 0 and d1 φ0 + φ1 d0 = 0 determines a 1-cocycle φ = (φi ) ∈ YC1 (L• , L• ) with φi = φ0 when i is even and φi = φ1 when i is odd. We write {φ0 , φ1 } for such a 1-cocycle, and choose           −1 0 x 0 0 1 0 −1 α (t1 ) = ·, · α (t2 ) = ·, · 0 x 0 −1 −1 0 1 0           −y 0 xy 0 0 y 0 −y α (t3 ) = ·, · α (t4 ) = ·, · 0 xy 0 −y −y 0 y 0

The differential d H2 = (dmH2 ) is explicitly given by dmH2 = d0H2 when m is even and dmH2 = d1H2 when m is odd, where   x − t1 − t3 y y2 + t2 + t4 y H2 d0 = d0 + ∑ ti α (ti )0 = 2 · y − t2 − t4 y x2 + t1 x + t3xy 1≤i≤4   x + t1 x + t3xy −y2 − t2 − t4 y H2 d1 = d1 + ∑ ti α (ti )1 = · −y2 + t2 + t4 y x − t1 − t3 y 1≤i≤4 By construction, d0H2 ◦ d1H2 = 0 and d1H2 ◦ d0H2 = 0 in H2 ⊗k HomA (L2 , L0 ). We may check that this is the case:     x−t −t y y2 + t2 + t4 y x + t1 x + t3xy −y2 − t2 − t4 y d0H2 ◦ d1H2 = 2 1 3 · y − t2 − t4y x2 + t1 x + t3xy −y2 + t2 + t4 y x − t1 − t3 y  
1 0 = x3 − y4 − (t1 + t3 y)2 x + (t2 + t4 y)2 · = 0 = d1H2 ◦ d0H2 0 1

In this case, we have used that (t1 ,t2 ,t3 ,t4 )2 = 0 in H2 to show that the obstruction above vanishes. In fact, the obstruction is given by −t12 · xs∗2 − 2t1t3 · xys∗2 − t32 · xy2 s∗2 + t22 · s∗2 + 2t2t4 · ys∗2 + t42 · y2 s∗2 Since s∗2 , ys∗2 = s∗4 6∈ im(d 1 ) while xs∗2 , xys∗2 , xy2 s∗2 , y2 s∗2 ∈ im(d 1 ), we have that H3 = k[[t1 ,t2 ,t3 ,t4 ]]/a3 with a3 = ( f22 , f42 )+(t1 ,t2 ,t3 ,t4 )3 , where f22 = t22 and f42 = 2t2t4 are the second-order approximations of f2 and f4 . To lift ξ2 to H3 , we choose α (t12 ), α (t1 t3 ), α (t32 ), α (t42 ) such that d 1 α (t12 ) = xs∗2 , and find that  0 d1 0

d 1 α (t1t3 ) = 2xys∗2 ,

    0 1 0 ·, · = xs∗2 1 0 0

d 1 α (t32 ) = xy2 s∗2 ,

⇒ ⇒ ⇒

 0 1 d 0

  −1 0 ·, 0 0

  1 · = y2 s∗2 0

⇒

d 1 α (t42 ) = −y2 s∗2

 0 0  0 α (t1t3 ) = 0  0 α (t32 ) = 0  0 α (t42 ) = 0

α (t12 ) =

   0 1 0 ·, · 1 0 0    0 2y 0 ·, · 2y 0 0   2  0 y 0 ·, · y2 0 0    1 0 −1 ·, · 0 0 0

Explicitly, the lifting ξ3 is represented by the differential d H3 , given by   x − t1 − t3 y y2 + t2 + t4 y + t42 H3 d0 = 2 · y − t2 − t4 y x2 + t1 x + t3 xy + t12 + 2t1t3 y + t32y2   x + t1 x + t3 xy + t12 + 2t1t3 y + t32y2 −y2 − t2 − t4 y − t42 H3 d1 = · −y2 + t2 + t4 y x − t1 − t3 y

25

Classical Deformation Theory Again, we compute the obstruction given by d0H3 d1H3 and d1H3 d0H3 , and find that H

H

H

H

d0 3 ◦ d1 3 = d1 3 ◦ d0 3 = (x3 − y4 − t13 − 3t12t3 y − 3t1t32 y2 − t33y3 + t22 + 2t2t4 y + t2t42 + t43 y)I This obstruction is given by (t22 − t13 + t2t42 )s∗2 + (2t2t4 − 3t12t3 + t43)s∗4 − 3t1t32 y2 s∗2 − t33y3 s∗2 We know that s∗2 , s∗4 6∈ im(d 1 ) while y2 s∗2 , y3 s∗2 ∈ im(d 1 ). It follows that H4 = k[[t1 ,t2 ,t3 ,t4 ]]/a4 with a4 = ( f23 , f43 ) + (t1 ,t2 ,t3 ,t4 )4 , where f23 = t22 − t13 + t2t42 and f43 = 2t2t4 − 3t12t3 + t43 are the third-order approximations of f2 and f4 . To lift ξ3 to H4 , we choose α (t1t32 ), α (t33 ) such that d 1 α (t1t32 ) = 3y2 s∗2 , and find, based on the choice of α (t42 ) above, that     0 −3 0 3 α (t1t32 ) = ·, · , 0 0 0 0

d 1 α (t33 ) = y3 s∗2

α (t33 ) =

 0 0

  −y 0 ·, 0 0

 y · 0

Explicitly, the lifting ξ4 is represented by the differential d H4 , given by   x − t1 − t3 y y2 + t2 + t4 y + t42 − 3t1t32 − t33 y H4 d0 = 2 · y − t2 − t4 y x2 + t1 x + t3 xy + t12 + 2t1t3 y + t32y2   x + t1 x + t3xy + t12 + 2t1t3 y + t32y2 −y2 − t2 − t4 y − t42 + 3t1t32 + t33 y H4 d1 = · −y2 + t2 + t4 y x − t1 − t3 y Yet again, we compute the obstruction given by d0H4 d1H4 and d1H4 d0H4 , and find that d0H4 ◦ d1H4 = d1H4 ◦ d0H4 = (x3 − y4 − t13 − 3t12t3 y + t22 + 2t2t4 y + t2t42 + t43 y − 3t1t2t32 − t2t33 y − 3t1t32t4 y − t33t4 y2 )I This obstruction is given by (t22 − t13 + t2t42 − 3t1t2t32 )s∗2 + (2t2t4 − 3t12t3 + t43 − t2t33 − 3t1t32t4 )s∗4 − t33t4 y2 s∗2 Hence H5 = k[[t1 ,t2 ,t3 ,t4 ]]/a5 with a5 = ( f24 , f44 ) + (t1 ,t2 ,t3 ,t4 )5 , where f24 = t22 − t13 + t2t42 − 3t1t2t32 and f44 = 2t2t4 − 3t12t3 + t43 − t2t33 − 3t1t32t4 are the fourth-order approximations of f2 and f4 . To lift ξ4 to H5 , we choose α (t33t4 ) such that d 1 α (t33t4 ) = y2 s∗2 and find, based on the choice of α (t42 ) above, that     0 −1 0 1 α (t33t4 ) = ·, · 0 0 0 0 Explicitly, the lifting ξ5 is represented by the differential d H5 , given by d0H5 = d1H5 =





x − t1 − t3 y y2 − t2 − t4 y

 y2 + t2 + t4 y + t42 − 3t1 t32 − t33 y − t33 t4 · x2 + t1 x + t3 xy + t12 + 2t1 t3 y + t32 y2

x + t1 x + t3 xy + t12 + 2t1 t3 y + t32 y2 −y2 + t2 + t4 y

 −y2 − t2 − t4 y − t42 + 3t1t32 + t33 y + t33 t4 · x − t1 − t3 y

26

Noncommutative Deformation Theory H

H

H

H

Yet again, we compute the obstruction given by d0 5 d1 5 and d1 5 d0 5 , and find that H

H

H

H

d0 5 ◦ d1 5 = d1 5 ◦ d0 5 = (x3 − y4 − t13 − 3t12t3 y + t22 + 2t2t4 y + t2t42 + t43 y − 3t1t2t32 − t2t33 y − 3t1t32t4 y − t2t33t4 − t33t42 y)I This obstruction is given by (t22 − t13 + t2t42 − 3t1t2t32 − t2t33t4 )s∗2 + (2t2t4 − 3t12t3 + t43 − t2t33 − 3t1t32t4 − t33t42 )s∗4 Hence it follows that H6 = k[[t1 ,t2 ,t3 ,t4 ]]/a6 with a6 = ( f25 , f45 ) + (t1,t2 ,t3 ,t4 )6 , where f25 = t22 − t13 + t2t42 − 3t1t2t32 − t2t33t4 , and f45 = 2t2t4 − 3t12t3 + t43 − t2t33 − 3t1t32t4 − t33t42 are the fifth-order approximations of f2 and f4 . We have not used that (t1 ,t2 ,t3 ,t4 )6 = 0 in H6 to show that the obstruction above vanishes. Therefore, we may lift ξ5 ∈ Def M (H5 ) to the algebra H = k[[t1 ,t2 ,t3 ,t4 ]]/( f2 , f4 ) defined by f2 = t22 − t13 + t2t42 − 3t1t2t32 − t2t33t4 f4 = 2t2t4 − 3t12t3 + t43 − t2t33 − 3t1t32t4 − t33t42 This implies that H is the pro-representing hull of Def M (with f2 , f4 given above and f1 = f3 = 0), and that Def M is obstructed. The versal family ξ ∈ Def M (H) is given by a lifting of complexes H2 H2 H H H (LH • , d• ) from (L• , d• ) to H. In concrete terms, the differential d = (dm ) is given by d0H = d1H =





x − t1 − t3 y y2 − t2 − t4 y

 y2 + t2 + t4 y + t42 − 3t1 t32 − t33 y − t33 t4 · x2 + t1 x + t3 xy + t12 + 2t1t3 y + t32 y2

x + t1 x + t3 xy + t12 + 2t1 t3 y + t32 y2 −y2 + t2 + t4 y

 −y2 − t2 − t4 y − t42 + 3t1 t32 + t33 y + t33 t4 · x − t1 − t3 y

This means that the versal family ξ ∈ Def M (H) is the H-A bimodule MH given by MH = coker(d0H ) =



x−t −t y k[[t1 ,t2 ,t3 ,t4 ]][x, y] / 2 1 3 y − t2 − t4 y 2

y2 + t2 + t4 y + t42 − 3t1t32 − t33 y − t33t4 x2 + t1 x + t3xy + t12 + 2t1t3 y + t32y2



We see that there is an algebraization H = k[t1 ,t2 ,t3 ,t4 ]/( f2 , f4 ) of H and an algebraization MH of the versal family MH , given by MH

=



x−t −t y k[t1 ,t2 ,t3 ,t4 ][x, y] / 2 1 3 y − t2 − t4 y 2

y2 + t2 + t4 y + t42 − 3t1t32 − t33 y − t33t4 x2 + t1 x + t3xy + t12 + 2t1t3 y + t32y2



The family {MH (τ ) : τ = (τ1 , . . . , τ4 ) ∈ k4 with f2 (τ ) = f4 (τ ) = 0}, where   x − τ1 − τ3 y y2 + τ2 + τ4 y + τ42 − 3τ1 τ32 − τ33 y − τ33 τ4 2 ∼ MH (τ ) = k[x, y] / 2 y − τ2 − τ4 y x2 + τ1 x + τ3 xy + τ12 + 2τ1 τ3 y + τ32 y2 ∼ = (x − τ1 − τ3 y, y2 − τ2 − τ4 y) is the corresponding family of deformations of M, and it is parameterized by the k-rational points of H = Spec H. This is a family of ideals I in A = k[x, y]/(x3 − y4 ) such that dimk (A/I) = 2.

Chapter 2 Noncommutative Algebras and Simple Modules

In this chapter, we review some basic facts about noncommutative algebras and their modules. By the term noncommutative algebra, we understand an associative algebra that is not necessarily commutative. All algebras are defined over a fixed field k. The results included in this chapter are mostly those needed to develop the noncommutative deformation theory in subsequent chapters. We have included some proofs and omitted others, and refer the reader to standard texts such as Lam [22] for a more complete account and full proofs.

2.1 Noncommutative algebras Let k be a field, and let A be an associative k-algebra. We do not assume that A is commutative, and shall sometimes call it a noncommutative algebra. Notice that the coefficients in k commute with all elements in A, since there is an algebra homomorphism k → C(A) into the centre of A by the definition of a noncommutative k-algebra. An Abelian subgroup I ⊆ A is called a left ideal in A if aI ⊆ I for any a ∈ A, and a right ideal in A if Ia ⊆ I for any a ∈ A. It is called an ideal in A if it is both a left and a right ideal. Notice that the quotient A/I is an algebra only if I is an ideal. The free associative k-algebra khx1 , x2 , . . . , xn i on n symbols is an example of a noncommutative algebra. It is also called the algebra of noncommutative polynomials in x1 , x2 , . . . , xn with coefficients in k, and can be realized as the tensor algebra T(V ) = ⊕ Tn (V ) = k ⊕ V ⊕ (V ⊗k V ) ⊕ (V ⊗k V ⊗k V ) ⊕ . . . n≥0

of a vector space V over k of dimension n. Let us consider the case n = 2 as an example. The algebra khx, yi has a k-base consisting of the words 1, x, y, x2 , xy, yx, y2 , x3 , x2 y, xyx, yx2 , xy2 , yxy, y2 x, y3 , . . . and multiplication of words is defined using juxtapositioning. For example, we have x · (yx) = xyx. The matrix algebra A = Mr (R) of r × r-matrices with coefficients in an algebra R is another example of a noncommutative algebra. Multiplication in A = Mr (R) is defined using usual matrix
multiplication and the ring structure of R. For instance, the matrix algebra A = M2 (k) = kk kk is an algebra of dimension four with k-linear base         1 0 0 1 0 0 0 0 e11 = , e12 = , e21 = , e22 = 0 0 0 0 1 0 0 1 and multiplication given by ei j ers = 0 if j 6= r, and ei j e js = eis . We say that a noncommutative algebra A is a matric algebra over kr if there is an injective algebra homomorphism φ : kr → A, or equivalently, if there are non-zero idempotents {e1 , e2 , . . . , er } in A such that e1 + e2 + · · · + er = 1 and ei e j = 0 when i 6= j. We notice that the matric algebras over 27

28

Noncommutative Deformation Theory

kr includes any matrix ring A = Mr (R) of r × r-matrices with coefficients in a noncommutative algebra R. Of course, any noncommutative algebra has a unique structure as a matric algebra over k. This is the case r = 1. For a matric algebra over kr , we define Ai j = ei Ae j ⊆ A for all i, j. Then there is a direct sum decomposition A = ⊕ Ai j i, j

and we write A = (Ai j ). We notice that Aii is a subalgebra of A with unit ei , and that Ai j is an Aii -A j j bimodule. Moreover, the multiplication in A is the usual matrix multiplication obtained using the ring structure of Aii and the bimodule structure of Ai j . Let W be a kr -kr bimodule. Then it has a decomposition W = (Wi j ), where Wi j = eiWe j , and the bimodule W is completely determined by the k-linear vector spaces Wi j for 1 ≤ i, j ≤ r. We consider the tensor algebra T(W ) = kr ⊕W ⊕(W ⊗ W ) ⊕ · · · = ⊕ Ti (W ) i≥0

kr ,

Ti (W )

⊗i W

of W over where = = W ⊗ · · · ⊗ W and all tensor products are taken over kr . The tensor algebra T(W ) is a matric algebra over kr , and it is finitely generated as a k-algebra if and only if W is finitely generated as a bimodule over kr . It is easy to see that this condition is equivalent to the condition that Wi j is a finite-dimensional vector space over k for 1 ≤ i, j ≤ r. The tensor algebras T(W ) are called the free matric algebras. An example of a free matric algebra over k2 is given by the bimodule W defined by W12 = k t12 , and Wi j = 0 for (i, j) 6= (1, 2). In this case, the tensor algebra is the algebra of upper triangular matrices:     0 k k k 2 W= ⇒ T(W ) = k ⊕W = 0 0 0 k since t12 ⊗ t12 = 0. It has a k-linear base consisting of e1 , e2 and t12 . Another example over k2 is ′ = k t , W ′ = k t and W ′ = W ′ = 0. In this case, the given by the bimodule W ′ defined by W12 12 21 21 11 22 tensor algebra is not finite dimensional:     0 k k[t12t21 ] ht12 i W′ = ⇒ T(W ′ ) = k 0 ht21 i k[t21t12 ]

It has a finite generating set consisting of e1 , e2 ,t12 and t21 . An associative algebra is called a division algebra if any non-zero element is a unit. It follows that the commutative division algebras are the fields, and many of the results for vector spaces over fields remain true for modules over division algebras. For example, any finitely generated left module V over a division algebra D has a well-defined dimension. We say that A is a finitely generated k-algebra if there is a surjective algebra homomorphism φ : khx1 , x2 , . . . , xn i → A, or equivalently, if there is a finite set {a1 , a2 , . . . , an } of generators of A such that any element a ∈ A can be written as a = f (a1 , a2 , . . . , an ) for a noncommutative polynomial f ∈ khx1 , x2 , . . . , xn i with coefficients in k. A left A-module M is called Artinian if its submodules satisfy the descending chain condition, and Noetherian if its submodules satisfy the ascending chain condition. Artinian and Noetherian right modules are defined similarly. We say that A is an Artinian (Noetherian) ring if it is Artinian (Noetherian) as a left and as a right A-module. In other words, A is Artinian (Noetherian) if its left ideals and its right ideals satisfy the descending (ascending) chain condition. It is clear that if A has finite dimension as a vector space over k, then it is Artinian. If A is a finitely generated commutative k-algebra, then it is a Noetherian ring. This is not true in the noncommutative case. For example, the free associative k-algebra A = khx, yi is finitely generated, but not Noetherian. In fact, the ascending chain of left ideals Ax ⊆ A(x, xy) ⊆ A(x, xy, xy2 ) ⊆ . . .

Noncommutative Algebras and Simple Modules

29

is not stationary, since xyn+1 6∈ A(x, xy, xy2 , . . . , xyn ) for any integer n ≥ 0. However, we have the following result, which we will not prove in this book: Theorem 2.1 (Hopkins-Levitzki Theorem). Any Artinian ring is Noetherian.

2.2 Artin-Wedderburn theory Let M be a left A-module. We say that M is a simple A-module if M 6= 0 has no submodules other than M and 0, and that M is a semisimple A-module if any submodule of M is a direct summand. Simple and semisimple right modules are defined similarly. It follows from this definition that any simple module is semisimple, and that 0 is a semisimple but not a simple module. Lemma 2.2. Any submodule and quotient module of a semisimple module is semisimple. Moreover, a module is semisimple if and only if it is a direct sum of simple modules. We say that A is a simple ring if there are no ideals in A other than A and 0. If A is a commutative ring, it is simple if and only if it is a field. In the noncommutative case, there are many more simple rings. In fact, when D is a division algebra and n ≥ 1 is a positive integer, the matrix algebra Mn (D) is a simple ring. This follows from the fact that for any algebra R, the ideals in Mn (R) have the form Mn (I) for an ideal I ⊆ R. Proposition 2.3. Let A = Mn (D) for a division algebra D and a positive integer n ≥ 1. Then A is a simple Artinian ring with a unique simple left A-module V . Moreover, V has the properties that EndA (V ) = D and that A ∼ = V n as left A-modules. Proof. We have seen that A = Mn (D) is a simple ring, and it is Artinian since it is a left and right module over D of finite dimension. Let V be the column vector V = Dn , considered as a right Dmodule with the natural left action of A = Mn (D). Then EndD (V ) ∼ = Mn (D) = A, and this implies that V is a simple left A-module, since A · v = V for any non-zero vector v ∈ V . There is a direct sum decomposition A = Mn (D) = I1 ⊕ I2 ⊕ · · · ⊕ In where Ii ⊆ Mn (D) is the left ideal consisting of matrices where all entries except those in the i’th column are zero. As a left A-module, it is clear that Ii ∼ = V , and therefore A ∼ = V n . To prove the ′ ′ uniqueness of V , assume that V is another simple left A-module. Then V = Av′ for any non-zero element v′ ∈ V ′ . Hence V ′ is a quotient of A, and therefore a decomposition factor of A. Since A ∼ = V n, ′ it follows by the Jordan-Hölder Theorem that V ∼ = V . It remains to prove that EndA (V ) ∼ = D. There is a natural injective map D → EndA (V ), given by d 7→ {v 7→ vd}. To show that it is surjective, let φ ∈ EndA (V ), and let d ∈ D be the first component of φ (e1 ). For any vector v ∈ V , we may choose a matrix M ∈ Mn (D) with v as its first column vector, and all other column vectors equal to zero. Then Mei = v, and   d 
∗   φ (v) = φ (Mei ) = M φ (ei ) = v 0 . . . 0 ·  .  = vd  ..  ∗

This shows that D → EndA (V ) is surjective, and therefore it is an isomorphism.

We say that A is a left semisimple ring if A is semisimple as a left A-module, or equivalently, if

30

Noncommutative Deformation Theory

A considered as a left A-module is a direct sum of simple left A-modules. In that case, A is a direct sum of a finite number of simple left A-modules, and therefore left Artinian. A right semisimple ring is defined similarly. Lemma 2.4. All left A-modules are semisimple if and only if A is left semisimple. When D is a division ring and n ≥ 1 is a positive integer, it follows from Proposition 2.3 that A = Mn (D) is a left semisimple ring. Moreover, finite direct products of left semisimple rings are clearly left semisimple. Hence, it follows that any algebra of the form A = Mn1 (D1 ) × Mn2 (D2 ) × · · · × Mnr (Dr ) is left semisimple when D1 , D2 , . . . , Dr are division algebras and n1 , n2 , . . . , nr are positive integers. Lemma 2.5 (Schur’s Lemma). If V is a simple left A-module, then EndA (V ) is a division algebra. Proof. Let φ ∈ EndA (V ) be a non-zero endomorphism. Then ker(φ ) and im(φ ) are submodules of the simple module V , and it follows that ker(φ ) = 0 and im(φ ) = V . Hence φ is an isomorphism, and therefore a unit in EndA (V ). Theorem 2.6 (Wedderburn-Artin Theorem). Let A be a left semisimple k-algebra. Then there is an isomorphism A∼ = Mn1 (D1 ) × Mn2 (D2 ) × · · · × Mnr (Dr ) where D1 , D2 , . . . , Dr are division algebras defined over k and n1 , n2 , . . . , nr ≥ 1 are positive integers. The number r and the pairs (D1 , n1 ), . . . , (Dr , nr ) are uniquely determined, and there are exactly r simple left A-modules, up to isomorphism. n Proof. Since A is left semisimple, there is an isomorphism A ∼ = V1 1 ⊕ · · · ⊕ Vrnr of left A-modules, where V1 , . . . ,Vr are non-isomorphic simple A-modules and n1 , . . . , nr ≥ 1 are positive integers. Since any simple left A-module is a quotient of A, it is isomorphic to one of the simple modules {V1 ,V2 , . . . ,Vr } by the Jordan-Hölder Theorem, so {V1 ,V2 , . . . ,Vr } is the full set of non-isomorphic simple left A-modules. Let Di = EndA (Vi ) for 1 ≤ i ≤ r. Then Di is a division algebra by Schur’s Lemma, and we have that n n A∼ = EndA (A) ∼ = EndA (V1 1 ⊕ V2 2 ⊕ · · · ⊕ Vrnr ) ∼ = Mn1 (D1 ) × Mn2 (D2 ) × · · · × Mnr (Dr ) nj n since EndA (Vini ) ∼ = Mni (Di ) and HomA (Vi i ,V j ) = 0 for i 6= j. The uniqueness follows from the uniqueness of the decomposition factors of A.

Since A = Mn1 (D1 ) × Mn2 (D2 ) × · · · × Mnr (Dr ) is both left and right semisimple, we have that A is right semisimple if and only if it is left semisimple. We shall therefore refer to left semisimple rings as semisimple rings. It follows from the comments above that all semisimple rings are Artinian. Corollary 2.7. Let A be a simple Artinian k-algebra. Then A ∼ = Mn (D) for a division algebra D and an integer n ≥ 1. Moreover, D and n are uniquely determined by A.

2.3 Simple modules and the Jacobson radical Let M be a simple left A-module. Then M = A/m for a maximal left ideal m ⊆ A. We write Ann(M) = {a ∈ A : aM = 0} for the annihilator of M. It is clear that this is an ideal, and that Ann(M) ⊆ m. But notice that in general, Ann(M) 6= m.

Noncommutative Algebras and Simple Modules

31

We define the left Jacobson radical of A to be the intersection of all maximal left ideals of A, and write J(A) for this left ideal. The right Jacobson radical is defined similarly. Lemma 2.8. For any ring A, we have that J(A) =

\

Ann(M)

M simple left A-module

Proof. Assume that aM = 0 for any simple left A-module M. Since M = A/m for a maximal left ideal m ⊆ A, it follows that a · 1 = a = 0, and therefore a ∈ m for any maximal left ideal, and this means that a ∈ J(A). To prove the converse, assume that a ∈ J(A). We claim that A(1 − xa) = A for any x ∈ A. In fact, if A(1 − xa) 6= A, then there is a maximal left ideal m such that A(1 − xa) ⊆ m, and therefore 1 − xa ∈ m. Since a ∈ m by assumption, this implies that 1 ∈ m, and this is a contradiction. Therefore, A(1 − xa) = A for any x ∈ A. We will use this to show that aM = 0 for any simple left module M. Assume that am 6= 0 for some element m ∈ M. Then A(am) = M, so there is an element x ∈ A such that x(am) = m. This implies that (1 − xa)m = 0, and this is a contradiction since A(1 − xa) = A. Therefore, aM = 0 for any simple left A-module M. Since Ann(M) ⊆ A is an ideal for any simple left A-module, it follows that the left Jacobson ideal J(A) ⊆ A is in fact an ideal. Moreover, it follows from the left-right symmetric characterization of J(A) in Lemma 2.9 below that the right Jacobson ideal of A is equal to the left Jacobson ideal J(A). We shall therefore refer to J(A) as the Jacobson radical of A. Notice that the quotient algebra A = A/J(A) has the same simple left modules as A (and also the same simple right modules as A). Lemma 2.9. We have that a ∈ J(A) if and only if 1 − xay is a unit in A for all x, y ∈ A. Proof. If 1 − xay is a unit in A for all x, y ∈ A, then in particular A(1 − xa) = A for all x ∈ A, and this implies that a ∈ J(A) by the proof of Lemma 2.8. To prove the converse, assume that a ∈ J(A). Since J(A) is an ideal, it follows that ay ∈ J(A) for any y ∈ A, and again by the proof of Lemma 2.8, this implies that A(1 − xay) = A for any x, y ∈ A. In particular, there exists u ∈ A such that u(1 − xay) = 1, and therefore u = 1 + u(xay). Since J(A) is an ideal, xay ∈ J(A), and therefore Au = A(1 + u(xay)) = A. Since u is both left and right invertible, it is a unit with inverse 1 − xay. We say that an element x ∈ A is nilpotent if xn = 0 for a positive integer n ≥ 1, and that an ideal (or a left or right ideal) I ⊆ A is nilpotent if I n = 0 for a positive integer n ≥ 1. Proposition 2.10. If A is Artinian, then J(A) is nilpotent. Moreover, if I ⊇ J(A) is a nilpotent left or right ideal in A, then I = J(A). Proof. By the descending chain condition, we have that J(A)n = J(A)n+1 = · · · = I for a positive integer n ≥ 1. To show that J(A) is nilpotent, we show that I = 0: If I 6= 0, then there is a minimal left ideal m with Im 6= 0. Fix an a ∈ m with Ia 6= 0. Then I(Ia) = I 2 a = Ia, and m = Ia by minimality, so that a = ya for some y ∈ I ⊆ J(A). This implies that (1 − y)a = 0, and therefore that a = 0 since 1 − y is a unit. This is a contradiction, and I = J(A)n = 0. Finally, if I is a nilpotent left ideal and y ∈ I, then xy ∈ I is nilpotent for any x ∈ A. This means that 1 − xy is a unit, and that y ∈ J(A). A similar argument holds if I is a right ideal. It follows that if I ⊇ J(A), then I = J(A). Theorem 2.11. An associative algebra A is semisimple if and only if A is Artinian and J(A) = 0.

32

Noncommutative Deformation Theory

2.4 The classical theorems of Burnside, Wedderburn, and Malcev Let A be an associative k-algebra that is finite dimensional. Then A has the same simple modules as the semisimple algebra A = A/J(A). In particular, A has a finite number of non-isomorphic simple modules. We write {M1 , M2 , . . . , Mr } for the family of simple right A-modules, and

ρ : A → ⊕ Endk (Mi ) 1≤i≤r

for the algebra homomorphism given by ρ (a)(mi ) = mi a for a ∈ A, mi ∈ Mi . Theorem 2.12 (Burnside’s Theorem). Let A be a finite dimensional associative k-algebra, and let {M1 , M2 , . . . , Mr } be the family of simple right A-modules. If EndA (Mi ) = k for 1 ≤ i ≤ r, then ρ is surjective. In particular, ρ is surjective if k is algebraically closed. Proof. There is an obvious factorization A → A → ⊕i Endk (Mi ) of ρ , where we write A = A/J(A). If EndA (Mi ) = k for 1 ≤ i ≤ r, then A → ⊕i Endk (Mi ) is an isomorphism by the Artin-Wedderburn Theorem, and therefore ρ is surjective. If k is algebraically closed, then EndA (Mi ) = k for all i, since EndA (Mi ) is a finite dimensional division algebra over k. We say that A is a separable k-algebra if A ⊗k K is semisimple for any field extension k ⊆ K. Notice that the matrix algebra A = Mn (k) is separable, since A ⊗k K ∼ = Mn (K) is semisimple. Theorem 2.13 (Wedderburn-Malcev Theorem). If A is a finite dimensional associative k-algebra such that A = A/J(A) is a separable k-algebra, then there is a semisimple subalgebra S ⊆ A such that A = J(A) ⊕ S. If S′ ⊆ A is another semisimple subalgebra such that A = J(A) ⊕ S′ , then there is an inner automorphism φ : A → A with φ (S) = S′ given by φ (a) = (1 − x)−1 a(1 − x) with x ∈ J(A). Proof. See Albert [1] and Tihomirov [47].

2.5 Finite dimensional simple modules Let A be finitely generated associative k-algebra. We denote the space of isomorphism classes of finite dimensional simple right A-modules by X = Simp(A), and write Xn ⊆ X for the subspace Xn = {M ∈ X : dimk (M) = n} of n-dimensional simple modules. We shall think of X = Simp(A) as a space associated with the algebra A, with the simple modules M ∈ X as its points. The space X is equipped with the Jacobson topology, with closed sets given by V (I) = {M ∈ X : MI = 0} for ideals I ⊆ A. For any element f ∈ A, we define the open set D( f ) to be the complement of V (I) with I = ( f ) = A f A. In concrete terms, we have that V ( f ) = {M ∈ X : M f = M} = {M ∈ X : ρ ( f ) ∈ Endk (M) is an automorphism} where we write ρ : A → Endk (M) for the algebra homomorphism given by ρ (a) = {m 7→ m · a}. Notice that D( f ) ∩ D(g) = D( f g) for all f , g ∈ A. In particular, the family {D( f ) : f ∈ A} of open sets is a base for the Jacobson topology. Let M be a simple module in X = Simp(A). By Schur’s Lemma, we know that EndA (M) is a division algebra, and it has finite dimension since M is finite dimensional. If k is algebraically closed, it follows that EndA (M) = k. We define the residue division algebra of the point M to be EndA (M), and denote the space of k-rational points in X by X(k) = {M ∈ X : EndA (M) = k}.

Noncommutative Algebras and Simple Modules

33

Any right A-module M corresponds to a representation (V, ρ ), where V is the underlying vector space of M and ρ : A → Endk (V ) is the algebra homomorphism defined by ρ (a) = {v 7→ va} for any a ∈ A, v ∈ V . Clearly, we have that Ann(M) = ker(ρ ), and this is a maximal ideal in A if and only if im(ρ ) ⊆ Endk (V ) is a simple ring. Proposition 2.14. Let M be finite dimensional right A-module. Then M ∈ X(k) if and only if ρ is surjective. If this is the case, then Ann(M) is a maximal ideal in A. Proof. If M ∈ X(k), then M is simple with EndA (M) = k. This implies that ρ is surjective by Burnside’s Theorem. Conversely, if ρ is surjective, then M is clearly simple, and EndA (M) = k since the central elements in Endk (V ) ∼ = Mn (k) are kIn ∼ = k. The last part follows from the fact that Mn (k) is a simple ring. It follows from the Amitsur-Levitzki Theorem that the representation ρ : A → Endk (V ) factors through the quotient algebra A(n) = A/I(n) for any simple module M ∈ Xn (k), where I(n) is the ideal ) ( I(n) =

∑

(−1)|σ | aσ (1) aσ (2) · · · aσ (2n) : a1 , a2 , . . . , a2n ∈ A

σ ∈S2n

In particular, for any one-dimensional simple module M ∈ X1 (k), the representation factors through A(1) = A/I(1) = A/({ab − ba : a, b ∈ A}), the commutativization of A. We remark that it may happen that A(1) = 0, for instance when A = khx, yi/(xy − yx − 1) is the first Weyl algebra. Proposition 2.15. If A(1) 6= 0, then there is a natural homeomorphism between X1 (k) and the set of closed k-rational points in Spec(A(1)). Let A be a matric algebra over kr that if finitely generated. Then A has a finite set of generators of the form {e1 , e2 , . . . , er } ∪ {ti j (l) : 1 ≤ i, j ≤ r, 1 ≤ l ≤ di j } where ti j (l) ∈ Ai j , since t ∈ A implies that ti j = eite j ∈ Ai j for 1 ≤ i, j ≤ r. Therefore, it follows that A can be realized as a quotient A = T(W )/I, where T(W ) is the tensor algebra of a bimodule W over kr with dimk (Wi j ) = di j , and I ⊆ T(W ) is an ideal. Proposition 2.16. Let A be algebra over kr and let X = Simp(A). If A is a finitely generated `a matric ` ` algebra, then X1 (k) = V1 V2 · · · Vr is a disjoint union of r classical affine algebraic varieties.

Proof. The matric algebra A ∼ = B/I, where B = T(W ) is the tensor algebra of a finitely generated bimodule W over kr and I ⊆ B is an ideal. If {ti j (l) : 1 ≤ l ≤ di j } is a k-base for Wi j for 1 ≤ i, j ≤ r, then we have that B(1)ii = k[tii (1), . . . ,tii (dii )]

⇒

A(1)ii = k[tii (1), . . . ,tii (dii )]/Iii′

where I ′ is the image of I in B(1). Since A(1) = ⊕i A(1)ii and the closed k-rational points of Spec(A(1)ii ) form a classical algebraic variety Vi = V (Iii′ ) ⊆ Adii , it follows that X1 (k) is a ` affine ` ` disjoint union X1 (k) = V1 V2 · · · Vr of these classical affine algebraic varieties.

Finally, we remark that there are matric algebras over kr such that X(k) 6= X1 (k). However, we have the following result: Proposition 2.17. Let A be a matric algebra over kr that is finitely generated, and let X = Simp(A). If Ai j = 0 for i > j and Aii = (A/J(A))ii is commutative, then X(k) = X1 (k).

Proof. If a ∈ Ai j with i < j, then a ∈ J(A). To see this, notice that 1 − xay is a unit in A for any x, y ∈ A, since any element t ∈ AaA satisfy t 2 = 0. This implies that A = A/J(A) = A11 ⊕ . . . ⊕ Arr is a commutative ring. Since A and A have the same simple modules, the result follows.

Chapter 3 Noncommutative Deformation Theory

In this chapter, we study noncommutative deformations of algebraic objects defined over a fixed field k. By noncommutative deformations, we understand deformations parametrized by noncommutative algebras. In contrast, the classical (or commutative) deformations in Chapter 1 are parametrized by commutative algebras. Noncommutative deformations of modules (representations) is an important case to consider. For any associative k-algebra A, we study the noncommutative deformation functor Def M of a finite family M = {M1 , M2 , . . . , Mr } of right A-modules. However, we want to include deformations of other algebraic objects, such as sheaves or presheaves over a sheaf of algebras, and modules with an action of a group or a Lie algebra. We therefore start this chapter in the more general context of an Abelian k-category A, and define the noncommutative deformation functor Def X of a finite family X = {X1 , X2 , . . . , Xr } in A. Recall that an Abelian k-category is a category A with a k-linear vector space structure on MorA (X,Y ) for any objects X,Y in A such that composition of morphism is k-bilinear. For a full account of Abelian categories, we refer to Freyd [11].

3.1 Noncommutative deformation functors In this section, we shall rephrase the definition of commutative (classical) deformation functors from Chapter 1 using the language of Abelian categories, and define noncommutative deformation functors by a natural generalization. To do this, we need a notion of flatness in Abelian categories.

3.1.1 Flatness in Abelian categories Let R be an associative k-algebra and let A be an Abelian k-category. We define the category R A of left R-objects in A in the following way: An object in R A is a pair (X, φ ), where X is an object of A and φ : R → MorA (X, X) is a k-algebra homomorphism. A morphism f : (X, φ ) → (X ′ , φ ′ ) in R A is a morphism f : X → X ′ in A such that f ◦ φ (r) = φ ′ (r) ◦ f for all r ∈ R. Hence R A is an Abelian k-category (but in general not an Abelian R-category, unless R is commutative). We shall often write X for (X, φ ) when the R-linear structure on X is understood from the context. Let R be an Artinian k-algebra, and let modR be the category of finitely generated right Rmodules. For any object Y in R A, there is a unique right exact functor − ⊗R Y : modR → A that maps R to Y . When f : Rm → Rn is a presentation of M in modR , with M = coker( f ), then M ⊗R Y = coker(F), where F : Y m → Y n is the morphism in A induced by f and the R-linear structure on Y . We say that an object Y in R A is R-flat if the functor − ⊗R Y is exact.

35

36

Noncommutative Deformation Theory

3.1.2 Commutative deformation functors Let X be an object of the Abelian k-category A. We recall the definition of the commutative deformation functor Def X : l → Sets of X in A. Commutative deformation functors are defined on the category l of commutative local Artinian k-algebras with residue field k. We recall that an object in l is a local commutative Artinian k-algebra (R, m) with a k-rational point R → k, and that a morphism in l is a commutative diagram of the form k

/R

/k

k

 / R′

φ

/k

That is, φ : R → R′ is a local k-algebra homomorphism. Definition 3.1. Let X be an object in A. For any algebra R in l, we define a lifting of X to R to be an object XR in R A which is R-flat, together with an isomorphism η : k ⊗R XR → X in A, and we say that two liftings (XR , η ) and (XR′ , η ′ ) are equivalent if there is an isomorphism τ : XR → XR′ in ′ R A such that η ◦ (id ⊗R τ ) = η . Let Def X (R) be the set of equivalence classes of liftings of X to R. Then Def X : l → Sets is the commutative deformation functor of X in A. When A = ModA is the category of right modules over an associative k-algebra A, the category is the category of R-A bimodules on which k acts centrally, and − ⊗R MR : modR → ModA is the usual tensor product for any bimodule MR in R A. Hence, the commutative deformation functor Def M : l → Sets of a right A-module M defined above coincides with the usual classical deformation functor Def M of M defined in Section 1.5. RA

3.1.3 Noncommutative deformation functors Let r ≥ 1 be a positive integer and let X = {X1 , X2 , . . . , Xr } be a family of objects in the Abelian k-category A. We shall define the noncommutative deformation functor Def X : ar → Sets of the family X in A. Let us introduce some notation. We write e1 , e2 , . . . , er for the indecomposable idempotents in the semisimple ring kr . Then kr = k1 ⊕ k2 ⊕ · · · ⊕ kr , where ki = k ·ei for 1 ≤ i ≤ r, and {k1 , k2 , . . . , kr } are the simple modules over kr . Let us define the category ar as a generalization of the category l. For expository purposes, we first define the category Ar of r-pointed algebras. The objects of Ar are the commutative diagrams id

kr

ι

/R

π

'/

kr

of associative algebras, and the morphisms of Ar are commutative diagrams of the form kr

/R

kr

 / R′

/ kr φ

/ kr

In particular, the objects R = (Ri j ) in Ar are matric algebras over kr in the sense of Chapter 2, and the morphisms φ = (φi j ) consist of k-linear maps φi j : Ri j → R′i j for 1 ≤ i, j ≤ r. ι

π

For any object kr − →R− → kr in Ar , we define the radical of R to be the ideal I(R) = ker(π ). We remark that the Jacobson radical J(R) ⊆ I(R), since R/ I(R) ∼ = kr is semisimple. We also notice that r {k1 , k2 , . . . , kr } are simple right R-modules via π : R → k .

Noncommutative Deformation Theory

37

Lemma 3.1. Let kr → R → kr be an object of Ar such that R is an Artinian ring. Then, the following conditions are equivalent: 1. I(R) is the Jacobson radical of R 2. I(R) is nilpotent 3. R is complete in the I(R)-adic topology 4. R has exactly r simple right modules Proof. We see that the first two statements are equivalent, since J(R) is the maximal nilpotent ideal in R when R is Artinian. Moreover, it is clear that R is complete if and only if I(R) is nilpotent, since I(R) ⊇ I(R)2 ⊇ . . . is stationary. Finally, we see that I(R) = J(R) if and only if {k1 , . . . , kr } is the full set of simple right R-modules, since R/ I(R) ∼ = kr . We define the category ar to be the full subcategory of Ar consisting of Artinian r-pointed algebras that satisfy the equivalent conditions of Lemma 3.1. It follows that any algebra R in ar has finite dimension over k. We write ar (n) for the full subcategory of ar consisting of r-pointed algebras R with I(R)n = 0. Furthermore, we define the pro-category aˆ r to be the full subcategory of Ar consisting of rpointed algebras R such that Rn = R/ I(R)n is Artinian for any integer n ≥ 1 and such that R is complete in the I(R)-adic topology, i.e., such that the natural homomorphism R → Rb = lim Rn ←− n≥1

is an isomorphism in Ar . We notice that if R is an algebra in the pro-category aˆ r , then its radical I(R) is the Jacobson radical of R. In fact, for any element r ∈ I(R), we have that (1 − r)−1 = 1 + r + r2 + . . . is an element of R, since R is complete in the I(R)-adic topology, and therefore I(R) = J(R). Consequently, R has exactly r simple right modules {k1 , k2 , . . . , kr }. Definition 3.2. Let X = {X1 , X2 , . . . , Xr } be a family of r objects in A. For any algebra R in ar , we define a lifting of X to R to be an object XR in R A which is R-flat, together with a family {ηi }1≤i≤r of isomorphisms ηi : ki ⊗R XR → Xi in A, and we say that two liftings (XR , {ηi }) and (XR′ , {ηi′ }) are equivalent if there is an isomorphism τ : XR → XR′ in R A such that ηi′ ◦ (id ⊗R τ ) = ηi for 1 ≤ i ≤ r. Let Def X (R) be the set of equivalence classes of liftings of the family X to R. Then, Def X : ar → Sets is the noncommutative deformation functor of X in A. When A = ModA is the category of right modules over an associative k-algebra A, the category the category of R-A bimodules on which k acts centrally, and − ⊗R MR : modR → ModA is the usual tensor product for any R-A bimodule MR . In this case, the definition of the noncommutative deformation functor can be rephrased in more familiar terms as follows: R A is

Definition 3.3. Let A be an associative k-algebra, and let M = {M1 , M2 , . . . , Mr } be a family of r right A-modules. For any algebra R in ar , we define a lifting of M to R to be an R-A bimodule MR on which k acts centrally such that MR is R-flat, together with a family {ηi }1≤i≤r of isomorphisms ηi : ki ⊗R MR → Mi of right A-modules, and we say that two liftings (MR , {ηi }) and (MR′ , {ηi′ }) are equivalent if there is an isomorphism τ : MR → MR′ of R-A bimodules such that ηi′ ◦ (id ⊗R τ ) = ηi for 1 ≤ i ≤ r. Let Def M (R) be the set of equivalence classes of liftings of the family M to R. Then Def M : ar → Sets is the noncommutative deformation functor of M in ModA .

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Noncommutative Deformation Theory

In Definition 3.3, the requirement that MR is R-flat simply means that − ⊗R MR is an exact functor. We shall describe what this means in concrete terms. We forget the right A-module structure of MR and of M1 , M2 , . . . , Mr , and consider the left R-module L = ⊕ (Ri j ⊗k M j ) i, j

with the trivial left R-module structure induced by the multiplication in R. Using matrix notation, we often write L = (Ri j ⊗k M j ) for this module. Notice that L is a flat R-module. In fact, for any right R-module P, we have that P ⊗R L = (Pi ⊗k Ri j ⊗k M j ) with Pi = Pei for 1 ≤ i ≤ r. If P → Q is an injection of right R-modules, then the induced map Pi → Qi is injective for 1 ≤ i ≤ r. Since k is a field, it follows that P ⊗R L → Q ⊗R L is injective. Lemma 3.2. Let R be an algebra in ar , and let MR be a left R module, with k-linear isomorphisms ηi : ki ⊗R MR → Mi of vector spaces for 1 ≤ i ≤ r. Then MR is a flat R-module if and only if there is an isomorphism MR ∼ = L of left R-modules. Proof. Since L is a flat R-module, one implication is clear. To prove the other implication, we assume that MR is a flat R-module. For 1 ≤ i ≤ r, we choose a k-linear base {mi (l) : l ∈ I(i)} for Mi . Since any element in Mi is of the form ηi (ei ⊗ x) for an element x ∈ MR , we may choose a family {xi (l) : l ∈ I(i)} of elements in MR such that ηi (ei ⊗ xi (l)) = mi (l). Consider the left Rmodule homomorphism ψ : L → MR given by ψ (ri j ⊗ m j (l)) = ri j x j (l). By construction, kr ⊗R ψ is surjective, and by Bourbaki [6], II.3.2.2, Corollary 2 of Proposition 4, it follows that ψ is surjective. It is therefore enough to prove that K = ker(ψ ) = 0. Since MR is a flat left R-module, it follows that the homomorphism I(R)⊗R MR → MR is injective. We consider the following commutative diagram: 0

0

I(R) ⊗R K

/ I(R) ⊗R L

 /K

 /L

 k r ⊗R K

 / k r ⊗R L

 / I(R) ⊗R MR

/0

 / MR

/0

ψ

 / kr ⊗R MR

Since kr ⊗R ψ is an isomorphism, it follows from the Snake Lemma that kr ⊗R K ∼ = K/ I(R)K = 0, and therefore that K = I(R)K. Since I(R) is the Jacobson radical of R, Nakayama’s Lemma implies that K = 0, and therefore that ψ : L → MR is an isomophism. It follows that any lifting MR of M to R is an R-A bimodule that is isomorphic to L = (Ri j ⊗k M j ) as a left R-module. However, there is no canonical right A-module structure on L that is preserved by this isomorphism. In particular, the trivial right A-module structure on L, given by (ri j ⊗ m j ) · a = ri j ⊗ (m j · a) for ri j ∈ Ri j , m j ∈ M j , a ∈ A, is not preserved by the isomorphism L ∼ = MR in general. If it is preserved, then MR is called a trivial deformation.

Noncommutative Deformation Theory

39

3.2 Structure of noncommutative deformation functors In this section, we study functors of noncommutative Artin rings. This is the natural framework for studying noncommutative deformation functors, and we obtain generalizations of Schlessinger’s structural results for functors of (commutative) Artin rings described in Chapter 1.

3.2.1 Functors of noncommutative Artin rings Let r ≥ 1 be a positive integer. A functor of r-pointed noncommutative Artin rings is a covariant functor D : ar → Sets such that D(kr ) = {∗} is reduced to one element. For any Abelian k-category A and any family X = {X1 , X2 , . . . , Xr } of r objects in A, the noncommutative deformation functor Def X : ar → Sets is an example of a functor of r-pointed noncommutative Artin rings. Let D : ar → Sets be a functor of noncommutative Artin rings. For any object R in ar , there is a distinguished element ∗R ∈ D(R), given by ∗R = D(kr → R)(∗). Any element ξR ∈ D(R) is a lifting of ∗ ∈ D(kr ) to R, in the sense that D(R → kr )(ξR ) = ∗. In particular, ∗R ∈ D(R) is a lifting of ∗ to R, and we call it the trivial lifting. For any functor D : ar → Sets of noncommutative Artin rings, there is an extension of D to a b : aˆ r → Sets defined on the pro-category aˆ r , given by functor D b D(R) = lim D(Rn ) ←− n≥1

b for any algebra R in aˆ r . By abuse of notation, we often write D for D. A pro-couple for D : ar → Sets is a pair (H, ξ ), where H is an algebra in aˆ r and ξ ∈ D(H), and a morphism u : (H, ξ ) → (H ′ , ξ ′ ) of pro-couples for D is a morphism u : H → H ′ in aˆ r such that D(u)(ξ ) = ξ ′ . For any pro-couple (H, ξ ) of D, the element ξ ∈ D(H) corresponds to a morphism φξ : Mor(H, −) → D of functors on ar , since there is a version of Yoneda’s Lemma for functors of noncommutative Artin rings: Lemma 3.3 (Yoneda’s Lemma). Let D : ar → Sets be a functor of noncommutative Artin rings and let H be an algebra in aˆ r . Then there is a natural bijection Hom(Mor(H, −), D) ≃ D(H) from the set of natural transformations Mor(H, −) → D of functors on ar to the set D(H). Definition 3.4. We say that D : ar → Sets is pro-representable if there is a pro-couple (H, ξ ) of D such that the natural transformation φξ : Mor(H, −) → D is an isomorphism of functors on ar . In this case, we also say that (H, ξ ) is a universal pro-couple. If D is pro-representable, then the pro-couple (H, ξ ) is unique up to unique isomorphism. The notion of smooth morphisms also has a natural extension to functors of noncommutative Artin rings. We say that a morphism φ : D → D′ of functors on ar is smooth if the natural settheoretic map D(R) → D(S) × D′ (R) D′ (S)

is surjective for any surjection u : R → S in ar . Notice that if φ is smooth, then φR : D(R) → D′ (R) is surjective for any algebra R in aˆ r . Definition 3.5. We say that a pro-couple (H, ξ ) for D is versal if the natural transformation φξ : Mor(H, −) → D is smooth, and semi-universal or miniversal if φξ is smooth and φξ (R) is

40

Noncommutative Deformation Theory

an isomorphism for any algebra R in ar (2). When the pro-couple (H, ξ ) is versal, the element ξ is often called a versal family, and when (H, ξ ) is semi-universal, the algebra H is often called a pro-representing hull or a formal moduli of D. We remark that the definition of smoothness implies that a miniversal pro-couple (H, ξ ) of D has the following versal property: For any algebra R in aˆ r and any element ξR ∈ D(R), there exists a morphism u : H → R such that D(u)(ξ ) = ξR . However, the morphism u is not uniquely determined by ξR and (H, ξ ) in general. Lemma 3.4. Let D : ar → Sets be a functor of noncommutative Artin rings. If D has a prorepresenting hull, then it is unique up to a (noncanonical) isomorphism of pro-couples. Proof. Let H, H ′ be pro-representing hulls of D with versal families ξ , ξ ′ , and let φ , φ ′ be the morphisms of functors on ar corresponding to ξ , ξ ′ . By the smoothness of φ , φ ′ , it follows that φH ′ and φH′ are surjective. Hence, there are morphisms of pro-couples u : (H, ξ ) → (H ′ , ξ ′ ),

v : (H ′ , ξ ′ ) → (H, ξ )

with φH ′ (u) = ξ ′ and φH (v) = ξ . Restriction to ar (2) gives two morphisms u2 : (H2 , ξ2 ) → (H2′ , ξ2′ ) and v2 : (H2′ , ξ2′ ) → (H2 , ξ2 ). But both (H2 , ξ2 ) and (H2′ , ξ2′ ) represent the restriction of D to ar (2), so u2 and v2 are mutual inverses. We write grn (R) = I(R)n / I(R)n+1 for R ∈ aˆ p and n ≥ 1. By the above argument, it follows that gr1 (u) and gr1 (v) are mutual inverses, and in particular, it follows that gr1 (v ◦ u) = gr1 (v) ◦ gr1 (u) is surjective. This implies that grn (v ◦ u) is a surjective endomorphism of the finite dimensional vector space grn (H) for all n ≥ 1, and therefore it is an automorphism of grn (H) for all n ≥ 1. So v ◦ u is an automorphism, and by a symmetric argument, u ◦ v is an automorphism as well. It follows that u and v are isomorphisms of pro-couples.

3.2.2 Algebraizations Let D : ar → Sets be a functor of noncommutative Artin rings, and assume that it has a prorepresenting hull H with versal family ξH ∈ D(H). An algebraization of (H, ξH ) is pair (H, ξH ) b∼ where H is a finitely generated r-pointed algebra such that its completion H = H, and ξH ∈ D(H) is an element such that ξH = D(H → H)(ξH ). We recall that an r-pointed algebra H comes with an algebra homomorphism π : H → kr such that I(H) = ker(π ), and that the completion of H denotes its I(H)-adic completion. If the noncommutative deformation functor Def X : ar → Sets of a family X = {X1 , X2 , . . . , Xr } in A has a pro-representing hull H, with versal family XH ∈ Def X (H), and an algebraization (H, XH ) of (H, XH ), it follows that there is a family {MX (S) : S ∈ X(k)} of deformations of the family X parametrized by the k-rational points in X = Simp(H). In fact, if S ∈ X(k) is simple right H-module with EndH (S) = k, then we define MX (S) = S ⊗H XH in A. It is clear that MX (ki ) ∼ = Xi for 1 ≤ i ≤ r.

3.2.3 Tangent spaces We recall that the category of bimodules over kr has internal Hom(−, −) and − ⊗ − bifunctors. For any kr -bimodules V and W , we write r

Hom(V,W ) = (Homk (Vi j ,Wi j )) and V ⊗ W = ( ⊕ (Vil ⊗k Wl j )) l=1

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41

for these functors. Notice that we use matrix notation, and that we often write Hom(V,W ) for Homkr (V,W ) and V ⊗ W for V ⊗kr W . The kr -bimodule   k k ... k k k . . . k    Mr (k) =  . . . . . ...   .. ..  k k ... k

plays the same role for bimodules over kr as k = M1 (k) does for vector spaces over k. In particular, we write V ∗ = Hom(V, Mr (k)) = (Vi∗j ) for the dual of V in the category of bimodules over kr . Let R be any object in aˆ r . Notice that I(R)/ I(R)2 has a natural kr -bimodule structure since R/ I(R) ∼ = kr . We define the tangent space of R to be the kr -bimodule t(R) = (I(R)/ I(R)2 )∗

Any morphism φ : R → S in aˆ r induces a homomorphism I(R)/ I(R)2 → I(S)/ I(S)2 of bimodules over kr , and therefore a homomorphism t(φ ) : t(S) → t(R). Using the internal tensor product in the category of bimodules over kr , we can define the tensor algebra T(V ) for any kr -bimodule V : For any integer i ≥ 0, we consider the iterated tensor product Ti (V ) = ⊗i V = V ⊗ V ⊗ · · · ⊗ V , and define T(V ) = ⊕ Ti (V ) i≥0

with the natural product given by juxtaposition. If V is a finitely generated bimodule (that is, if Vi j is finite-dimensional for 1 ≤ i, j ≤ r), then the tensor algebra T(V ) is an object of Ar with radical I(T(V )) = ⊕i>0 Ti (V ), and the formal tensor algebra b ) = lim T(V )n = lim T(V )/ I(T(V ))n T(V ←− ←− n≥1

n≥1

is an object of aˆ r . Let D : ar → Sets be any functor of noncommutative Artin rings. We define the tangent space of D to be t(D) = D(R) with R = T(Mr (k))2 ∼ Mor(H, −) for an algebra H in aˆ r , then it follows If D : ar → Sets is pro-representable, with D = from this definition that t(D) ∼ = Mor(H, T(Mr (k))2 ) = Hom(t(H)∗ , Mr (k)) = t(H)∗∗ ∼ = t(H). Under the weaker assumption that D has a pro-representing hull H, the same argument shows that there is an isomorphism of kr -bimodules t(D) ∼ = t(H).

3.2.4 Obstruction calculus Let u : R → S be a morphism in ar with kernel K = ker(u). It is called a small surjection if it is surjective and satisfies K · I(R) = I(R) · K = 0. Notice that any surjection in ar is a composition of small surjections. It follows that a morphism φ : F → G of functors on ar is smooth if and only if F(R) → F(S) × G(R) G(S)

is surjective for any small surjection u : R → S in ar . Let D : ar → Sets be a functor of noncommutative Artin rings. A small lifting situation for D is a pair (u, ξ ), where u : R → S is a small surjection in ar and ξ ∈ D(S). When (u, ξ ) and (u′ , ξ ′ ) are

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two small lifting situations for D, we define a morphism (α , β ) : (u, ξ ) → (u′ , ξ ′ ) of small lifting situations to be a commutative diagram R

α

u

/ R′ u′

 S

β

 / S′

in ar such that D(β )(ξ ) = ξ ′ . We say that the functor D : ar → Sets of noncommutative Artin rings has an obstruction theory with cohomology {H p} if there are kr -bimodules H1 and H2 such that the following conditions hold: 1. For any small lifting situation (u, ξ ) for D, with K = ker(u), there is a canonical obstruction o(u, ξ ) ∈ Hom((H2 )∗ , K) such that o(u, ξ ) = 0 if and only if there exists a lifting of ξ to R. Moreover, if o(u, ξ ) = 0, then there is a transitive and free action of Hom((H1 )∗ , K) on the set of liftings of ξ to R. 2. For any morphism (α , β ) : (u, ξ ) → (u′ , ξ ′ ) of small lifting situations for D, with K = ker(u) and K ′ = ker(u′ ), and where α ∗ : Hom((H2 )∗ , K) → Hom((H2 )∗ , K ′ ) is the natural map induced by α , we have that α ∗ (o(u, ξ )) = o(u′ , ξ ′ ). We say that D has an obstruction theory with finite dimensional cohomology if in addition H1 and H2 p are finitely generated bimodules over kr (that is, Hi j are finite dimensional vector spaces). Notice that p p p in this case, we have that Hom((H p )∗ , K) ∼ = (Homk ((Hi j )∗ , Ki j )) ∼ = (Ki j ⊗k (Hi j )∗∗ ) ∼ = (Ki j ⊗k Hi j ) for p = 1, 2. Remark 3.1. If D has an obstruction theory with finite dimensional cohomology {H p }, then we have that t(D) ∼ = H1 . In fact, for any algebra R in ar (2), the natural map R → k is a small surjection with kernel I(R). Moreover, since the element ∗ ∈ D(kr ) has the trivial lifting ∗R ∈ D(R) to R, the base point ∗R ∈ D(R) sets up an isomorphism D(R) ∼ = Homk ((H1 )∗ , I(R)) by the definition of an obstruction theory. In particular, with R = T(Mr (k))2 , we find that the tangent space t(D) = D(R) ∼ = Homk ((H1 )∗ , Mr (k)) = (H1 )∗∗ ∼ = H1 since I(R) = Mr (k) in this case. Let us assume that D : ar → Sets has an obstruction theory with finite dimensional cohomology {H p }. Using this assumption, we shall give an explicit construction of the pro-representing hull H of D and its versal family ξ ∈ D(H). 1 )∗ ) and T2 = T((H 2 )∗ ) to be the formal tensor algebras in a b b ˆ r of the kr We define T1 = T((H 1 ∗ 2 ∗ bimodules (H ) and (H ) . For 1 ≤ i, j ≤ r, we may choose k-linear bases {ti j (l) : 1 ≤ l ≤ di j } for H1i j and {si j (l) : 1 ≤ l ≤ ri j } for H2i j , with di j = dimk H1i j and ri j = dimk H2i j . Then T1 is the formal kr -algebra generated by {ti j (l)}, and T2 is the formal kr -algebra generated by {si j (l)}. Let H2 = T12 = T1 / I(T1 )2 . For any algebra R in ar (2), it follows from Remark 3.1 that there is an isomorphism D(R) ∼ = Hom((H1 )∗ , I(R)) ∼ = Mor(H2 , R), and we therefore have an isomorphism 1 Mor(T2 , −) → D of functors on ar (2). By Yoneda’s Lemma, this isomorphism corresponds to an element ξ ∈ D(H2 ), and the restriction of D to ar (2) is represented by (H2 , ξ2 ). Theorem 3.5. Let D : ar → Sets be a functor of noncommutative Artin rings. If D has an obstruction theory with finite dimensional cohomology {H p }, then there is an obstruction morphism o : T2 → T1 b T2 kr is a pro-representing hull of D. in aˆ r such that H = T1 ⊗

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43

Proof. For simplicity, we use the notation I = I(T1 ), T1n = T1 /I n for n ≥ 2, and tn : T1n+1 → T1n for the natural morphisms. Let a2 = I 2 and H2 = T1 /a2 = T12 . Then the restriction of D to ar (2) is represented by (H2 , ξ2 ) and H2 ∼ = T12 ⊗T2 kr , where the tensor product is taken over the trivial 2 1 morphism o2 : T → T2 . Using o2 and ξ2 as a starting point, we shall construct on+1 and ξn+1 for n ≥ 2 inductively. So let n ≥ 2, and assume that the morphism on : T2 → T1n and the lifting ξn ∈ D(Hn ) are given, with Hn = T1n ⊗T2 kr . We may assume that tn−1 ◦ on = on−1 and that ξn is a lifting of ξn−1 . Let us first construct the morphism on+1 : T2 → T1n+1 . We define a′n to be the ideal in T1n generated by on (I(T2 )). Then a′n = an /I n for an ideal an ⊆ T1 with I n ⊆ an , and Hn ∼ = T1 /an . We define the ideal bn = I · an + an · I, and obtain the following commutative diagram: T2 ❆ T1n+1 ❆❆ ❆❆ ❆ on ❆❆ ❆ 

/ T1 /bn  / T1 /an

T1n

There is an obstruction o′n+1 = o(T1 /bn → Hn , ξn ) for lifting ξn to T1 /bn since T1 /bn → T1 /an is a small surjection, giving rise to a morphism o′n+1 : T2 → T1 /bn. Let a′′n+1 be the ideal in T1 /bn generated by o′n+1 (I(T2 )). Then a′′n+1 = an+1 /bn for an ideal an+1 ⊆ T1 with bn ⊆ an+1 ⊆ an . Let Hn+1 = T1 /an+1 . We obtain the following commutative diagram: o′n+1

* / T1 /bn

T2 ❆ T1n+1 ❆❆ ❆❆ ❆ on ❆❆❆ 

/ Hn+1 = T1 /an+1 qq qqq q q qq  q qqq x / Hn

T1n

By the choice of an+1, the obstruction for lifting ξn to Hn+1 vanishes, and we may choose a lifting ξn+1 ∈ D(Hn+1 ) of ξn to Hn+1 . We claim that there is a morphism on+1 : T2 → T1n+1 that commutes with o′n+1 and on . Note that an−1 = I n−1 + an since tn−1 ◦ on = on−1 . For simplicity, we write O(V ) = Hom(t(T2 )∗ ,V ) for any kr -bimodule V . The following diagram of kr -bimodules is commutative with exact columns: 0

0

 O(bn /I n+1 )

jn

 / O(bn−1 /I n )

 O(an /I n+1 )

kn

 / O(an−1 /I n )

ln



rn+1



O(an /bn )  0

rn

/ O(an−1 /bn−1 )  0

We may consider on as an element in O(an−1 /I n ), and o′n+1 ∈ O(an /bn ). Since o′n commutes

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Noncommutative Deformation Theory

with o′n+1 and on , we get ln (o′n+1 ) = rn (on ). To prove the claim, it is enough to find an element on+1 ∈ O(an /I n+1) such that rn+1 (on+1 ) = o′n+1 and kn (on+1 ) = on . Since on (I(T2 )) ⊆ an , there is an element on+1 ∈ O(an /I n+1 ) such that kn (on+1 ) = on . But an−1 = an + I n−1 implies that jn is surjective, so the claim follows from the snake lemma. In particular, T1n+1 ⊗T2 kr ∼ = Hn+1 when the tensor product is taken over on+1 . By induction, we find a morphism on : T2 → T1n and an element ξn ∈ D(Hn ) for all integers n ≥ 2, with Hn = T1n ⊗T2 kr . Using the universal property of the projective limit, we obtain a morphism b T2 kr . We claim that (H, ξ ) is a proo : T2 → T1 in aˆ r and an element ξ ∈ D(H), with H = T1 ⊗ representing hull for D. It is enough to prove that (Hn , ξn ) is a pro-representing hull for the restriction of D to ar (n) for all n ≥ 3. So let φn : Mor(Hn , −) → D be the morphism of functors on ar (n) corresponding to ξn for some n ≥ 3. We shall prove that φn is a smooth morphism. Let u : R → S be a small surjection in ar (n) with kernel K, let ξR ∈ D(R) and v ∈ Mor(Hn , S) be elements such that D(u)(ξR ) = D(v)(ξn ) = ξS , and consider the following commutative diagram: / Hn+1 T1 ❈ ❈❈ ❈❈ ❈❈ ❈❈ !  Hn

R u v

 /S

We can choose a morphism v′ : T1 → R that makes this diagram commutative, and this implies that v′ (an ) ⊆ K. Since u is small, this means that v′ (bn ) = 0. But the induced map T1 /bn → R maps the obstruction o′n+1 to o(u, ξS ) = 0. It follows that v′ (an+1 ) = 0, hence v′ induces a morphism v′ : Hn+1 → R making the diagram commutative. Since v′ (I(Hn+1 )n ) = 0, we may consider v′ as a map from Hn+1 / I(Hn+1 )n ∼ = Hn . This proves that there is a morphism v′ : Hn → R such that u ◦ v′ = v. ′ ′ We define ξR = D(v )(ξn ). Then ξR′ is a lifting of ES to R, and the difference between ξR and ′ ξR is given by an element d ∈ Hom((H1 )∗ , K). We consider t(v′ )∗ + d as a homomorphism of kr bimodules in Hom((H1 )∗ , t(R)∗ ). It defines a morphism v′′ : T1 → R, and since an+1 ⊆ I 2 and u is small, we have that v′′ (an+1 ) ⊆ v′ (an+1 ) + I(R) · K + K · I(R) + K 2 = v′ (an+1 ) = 0. This implies that v′′ induces a morphism v′′ : Hn → R. By construction, u ◦ v′′ = u ◦ v′ = v and D(v′′ )(ξn ) = ξR , and this proves that φn is smooth. Remark 3.2. If D has an obstruction theory with cohomology {H p } such that H1 is a countably generated kr -bimodule, a generalization of Theorem 3.5 can be proved using the topological methods of Laudal; see Section 4.3 in Laudal [27]. Remark 3.3. The obstruction morphism o : T2 → T1 in Theorem 3.5 is not uniquely defined by the obstruction theory for D. However, we see from the construction of o that on : T2 → T1n is unique up to the ideal I(T1 ) · im(on−1 ) + im(on−1 ) · I(T1 ) ⊆ T1n for all n ≥ 2, where on : T2 → T1n is the composition of o with the canonical quotient map. In particular, the leading term of o is unique. When D has an obstruction theory with finite dimensional cohomology {H p }, there are certain cohomology operations on H1 with values in H2 , called generalised matric Massey products, that can be used to compute the pro-representing hull H of D. These generalised matric Massey products are in a sense dual to the obstruction morphism. Let o : T2 → T1 be the obstruction morphism of Theorem 3.5. We may consider (H2 )∗ ⊆ T2 as a subspace of kr -bimodules, and obtain a homomorphism M ∗ : (H2 )∗ → ∏ Ti ((H1 )∗ ) i≥2

of kr -bimodules by restriction of o to (H2 )∗ . Let M n : (H2 )∗ → ∏ni=2 Ti ((H1 )∗ ) be the projection of

Noncommutative Deformation Theory

45

M ∗ on the first n − 1 factors for n ≥ 2. The dual map of M n in the category of kr -bimodules is the homomorphism n a Mn : Ti (H1 ) → H2 i=2

kr -bimodules

of for n ≥ 2. Remark 3.3 shows that M ∗ is not uniquely defined by the obstruction theory for D. However, M ∗ is unique modulo the ideal ∆ = (H1 )∗ · im(M ∗ ) + im(M ∗ ) · (H1 )∗ in T1 . Therefore, the quotient map n

Mon : (H2 )∗ → ∏ Ti ((H1 )∗ )/∆n i=2

1 ∗

is uniquely defined, where ∆n = (H ) · im(M ) + im(M n−1 ) · (H1 )∗ . We define Dn to be the dual of ∏ni=2 Ti ((H1 )∗ )/∆n in the category of kr -bimodules, !∗ n−1

n

Dn =

∏ Ti ((H1 )∗ ))/∆n i=2

`n

Then Dn ⊆ i=2 Ti (H1 ) is a subspace of kr -bimodules for n ≥ 2. We consider the dual map of Mon for n ≥ 2, and obtain a sequence of kr -bimodule homomorphisms Mno : Dn → H2 for n ≥ 2. These maps are called the generalised matric Massey products induced by the obstruction morphism o. It follows from the definition that D2 = T2 (H1 ), and M2o : H1 ⊗ H1 → H2 is called the cup product. Corollary 3.6. Let D : ar → Sets be a functor of noncommutative Artin rings. If D has an obstruction theory with finite dimensional cohomology {H p }, then the pro-representing hull H of D is completely determined by the generalised matric Massey products Mno : Dn → H2 induced by the obstruction morphism. Proof. We have that H = T1 /a, where a is the ideal in T1 generated by the noncommutative power series fi j (l) = o(si j (l)) ∈ T1i j for 1 ≤ i, j ≤ r, 1 ≤ l ≤ ri j . In terms of the generalised matric Massey products, the power series fi j (l) are given by fi j (l) = lim fi j (l)n , ←−

with fi j (l)n = Mon (si j (l))

n≥2

where M0n is the dual of Mno . Remark 3.4. We say that D is unobstructed if o : T2 → T1 is trivial. In this case, H = T1 is a pro-representing hull for D, and there is a lifting of ξ2 ∈ D(H2 ) to T1 . This lifting is the versal family ξ ∈ D(T1 ). Otherwise, D is obstructed, and it is a nontrivial task to compute H using the generalised matric Massey products. When H1 = 0 we say that D is rigid. This is a special case of an unobstructed functor D, with pro-representing hull H = kr . When D = Def X is a rigid deformation functor, the family X has no nontrivial deformations.

3.2.5 Swarms We consider a family X = {X1 , X2 , . . . , Xr } of r objects in an Abelian k-category A, and assume that the noncommutative deformation functor Def X : ar → Sets of X in A has an obstruction theory with cohomology {H p }.

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Noncommutative Deformation Theory

Definition 3.6. We say that X is a swarm in A if H1 is a finitely generated kr -bimodule. This condition is equivalent to the condition that dimk H1i j < ∞ for 1 ≤ i, j ≤ r. If Def X has an obstruction theory with cohomology {H p} such that r k -bimodule, and if there is another obstruction theory for Def X with there are isomorphisms

H1 is a finitely generated cohomology {′ H p }, then

H1 ∼ = t(Def X ) ∼ = ′ H1

of kr -bimodules by Remark 3.1. Hence, the definition of a swarm is independent of the choice of obstruction theory. We do not have a similar invariance result for H2 . However, when (u : R → S, XS ) is a small lifting situation for Def X , we have that o(u, XS ) = 0 in (Ki j ⊗k H2i j ) if and only if o(u, XS ) = 0 in (Ki j ⊗k ′ H2i j ). If X is a swarm, we may therefore replace H2 with a finitely generated kr -bimodule ′ H2 ⊆ H2 , if necessary. Proposition 3.7. Let X = {X1 , X2 , . . . , Xr } be a finite family in an Abelian k-category A. If X is a swarm, then the noncommutative deformation functor Def X : ar → Sets of X in A has a pro-repressenting hull H, with versal family XH ∈ Def X (H).

3.2.6 Relations with commutative deformation functors Let D : ar → Sets be a functor of noncommutative Artin rings. For 1 ≤ i ≤ r, there is a natural inclusion of categories a1 → ar , induced by the ring homomorphism k → kr given by 1 7→ ei . We write Di : a1 → Sets for the restriction of D to a1 using this inclusion of categories, and Dci : l → Sets for the restriction of Di to the full subcategory l of a1 . Let X = {X1 , X2 , . . . , Xr } be a swarm in an Abelian k-category A, and let D = Def X : ar → Sets be the noncommutative deformation functor of X in A. Then Di = Def Xi : a1 → Sets is the noncommutative deformation functor of Xi , and Dci = Def cXi : l → Sets is the commutative deformation functor of Xi . Since X is a swarm, there is an obstruction theory of Def X with finite dimensional cohomology {H p }, and Def X has a pro-representing hull H = H(X ) by Proposition 3.7. For 1 ≤ i ≤ r, we notice that the functor Def Xi : a1 → Sets has an obstruction theory with cohomology {Hiip }. Hence Def Xi : a1 → Sets has a pro-representing hull H(Xi ), and H(Xi ) is the subalgebra of Hii generated by (H1ii )∗ . Moreover, the functor Def cXi : l → Sets has a (commutative) obstruction theory with cohomology {Hiip }. It follows that Def cXi : l → Sets has a (commutative) prorepresenting hull H c (Xi ), and since the full subcategory l of a1 consists of exactly those algebras R in a1 that are commutative, H c (Xi ) = H(Xi )(1) = H(1)ii , where we write R(1) = R/I(1) = R/({rs − sr : r, s ∈ R}) for the commutativization of an associative algebra R.

3.3 Examples of noncommutative deformation functors In this section, we consider noncommutative deformation functors of algebraic objects of special interest to us. For each Abelian k-category A, we describe the noncommutative deformation functor in concrete terms, and use this to describe an obstruction theory for the deformation functor. In particular, we obtain a concrete description of the tangent space, and a method for computing the pro-representing hull explicitly.

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47

3.3.1 Modules Let ModA be the category of right modules over an associative k-algebra A. We consider the noncommutative deformation functor Def M : ar → Sets of a family M = {M1 , M2 , . . . , Mr } of r right A-modules in the category ModA . Let R be an object of ar . By Definition 3.3, Def M (R) is the set of equivalence classes of liftings {(MR , ηi )} of M to R, where MR is an R-A bimodule (on which k acts centrally) that is R-flat, and ηi : ki ⊗R MR → Mi is an isomorphism of right A-modules for 1 ≤ i ≤ r. By Lemma 3.2, we know that MR is R-flat if and only if MR ∼ = (Ri j ⊗k M j ) considered as left R-modules. Finally, the liftings (MR , ηi ) and (MR′ , ηi′ ) are equivalent if and only if there is an isomorphism τ : MR → MR′ of R-A bimodules such that ηi′ ◦ (id ⊗R τ ) = ηi for 1 ≤ i ≤ r. Description using Hochschild cohomology Let us identify the right A-module Mi with the representation ρi : A → Endk (Vi ) for 1 ≤ i ≤ r, where Vi is the underlying vector space over k and ρi is the algebra homomorphism defined by the right multiplication of A. We write V = ⊕i Vi for the direct sum of vector spaces, and ρ for the direct sum representation ρ : A → Endk (V ), corresponding to the right A-module M = ⊕i Mi . For any algebra R in ar , we consider the left R-module VR = (Ri j ⊗k V j ), and notice that we have isomorphisms EndR (VR ) ∼ = (Homk (Vi , Ri j ⊗k V j )) ∼ = (Ri j ⊗k Homk (Vi ,V j )) as vector spaces over k. The multiplication in (Ri j ⊗k Homk (Vi ,V j )) induced by the product in EndR (VR ) is given by (ri j ⊗ φi j ) · (r′jk ⊗ φ ′jk ) = (ri j · r′jk ) ⊗ (φ ′jk ◦ φi j ) for all ri j ∈ Ri j , r′jk ∈ R jk , φi j ∈ Homk (Vi ,V j ) and φ ′jk ∈ Homk (V j ,Vk ). It is evident that the structural homomorphism π : R → kr induces an algebra homomorphism

π : EndR (VR ) → Endkr (Vkr ) We notice that Endkr (Vkr ) = ⊕i Endk (Vi ) ⊆ Endk (V ). It follows that the direct sum representation ρ : A → Endk (V ) can be considered as an algebra homomorphism

ρ : A → Endkr (Vkr ) ⊆ Endk (V ) Therefore, a lifting of the family M to R is an algebra homomorphism ρR making the following diagram commutative: ρR A ❴PPP❴ ❴ ❴ ❴ ❴/ EndR (VR ) PPP PPP P π ρ PPPP '  Endkr (Vkr ) Furthermore, two such algebra homomorphisms ρR , ρR′ correspond to equivalent liftings of M to R if and only if there is an R-linear automorphism τ : VR → VR lifting the identity (that is, with π (τ ) = id) such that ρR′ (a) = τ ◦ ρR(a) ◦ τ −1 for all a ∈ A. Let R be an algebra in ar with di j = dimk I(R)i j for 1 ≤ i, j ≤ r. We may choose a k-linear base {rii (l) : 0 ≤ l ≤ dii } for Rii with rii (0) = ei when 1 ≤ i ≤ r, and a k-linear base {ri j (l) : 1 ≤ l ≤ di j } for Ri j when 1 ≤ i, j ≤ r with i 6= j. Then ρR is explicitly given by

ρR (a) = ∑ ei ⊗ ρi(a) + ∑ ri j (l) ⊗ δi j (l)(a) i

i, j,l

where δ = {δi j (l) : 1 ≤ i, j ≤ r, 1 ≤ l ≤ di j } is a family of k-linear maps δi j (l) : A → Homk (Vi ,V j )

48

Noncommutative Deformation Theory

such that δi j (l)(1) = 0. The last condition comes from the fact that the lifting MR of M to R is a unitary right A-module. Conversely, such a family δ = {δi j (l) : 1 ≤ i, j ≤ r, 1 ≤ l ≤ di j } of k-linear homomorphisms δi j (l) : A → Homk (Vi ,V j ) such that δi j (l)(1) = 0 defines a lifting of M to R if and only if the associated k-linear map ρR : A → EndR (VR ), given by

ρR (a) = ∑ ei ⊗ ρi(a) + ∑ ri j (l) ⊗ δi j (l)(a) i

i, j,l

is an algebra homomorphism. This is equivalent to the associativity condition, which can be written ρR (a)ρR (b) − ρR (ab) = 0 for all a, b ∈ A. Explicitly, the associativity condition can be expressed in terms of δ as

∑ ri j (l) ⊗ [δi j (l)(b)ρi (a) − δi j (l)(ab) + ρ j (b)δi j (l)(a)]

i, j,l

+

∑

ri j (l)r jk (l ′ ) ⊗ δ jk (l ′ )(b)δi j (l)(a) = 0

i, j,k,l,l ′

Definition 3.7. We recall the definition of the Hochschild cohomology HH• (A, Q) of an associative k-algebra A with values in an A-A bimodule Q. It is defined as the cohomology of the Hochschild complex HC• (A, Q), where HCn (A, Q) = Homk (⊗nk A, Q) for all n ≥ 0, and where the differential d n : HCn (A, Q) → HCn+1 (A, Q) is given by d n (φ )(a1 ⊗ · · · ⊗ an+1) = a1 · φ (a2 ⊗ · · · ⊗ an+1) n

+ ∑ (−1)i φ (a1 ⊗ · · · ⊗ ai · ai+1 ⊗ · · · ⊗ an+1) i=1

+ (−1)n+1φ (a1 ⊗ · · · ⊗ an) · an+1 for all φ ∈ HCn (A, Q) and a1 , . . . , an+1 ∈ A. For any right A-modules M, N, it is wellknown that HHn (A, Homk (M, N)) ∼ = ExtnA (M, N) for n ≥ 0; see Weibel [49], Theorem 8.7.10 and Lemma 9.1.9. In particular, there is an exact sequence Homk (M, N) → Derk (A, Homk (M, N)) → Ext1A (M, N) → 0 induced by the isomorphism Ext1A (M, N) ∼ = HH1 (A, Homk (M, N)). The image of the map Homk (M, N) → Derk (A, Homk (M, N)) is called the module of inner derivations, which we denote by IDerk (A, Homk (M, N)) = {d 0 (φ ) = [−, φ ] : φ ∈ Homk (M, N)} The inner derivation corresponding to φ is explicitly given by [a, φ ](m) = φ (ma) − φ (m)a for all a ∈ A, m ∈ M. In particular, Ext1A (M, N) ∼ = Derk (A, Homk (M, N))/ IDerk (A, Homk (M, N)). Let u : R → S be a small surjection in ar , and let MS ∈ Def M (S) be a lifting of M to S. Then there is an algebra homomorphism ρS : A → EndS (VS ) representing MS . To lift MS to R, we must find an algebra homomorphism ρR : A → EndR (VR ) such that the following diagram commutes: ρR / EndR (VR ) A PPP PPP PPP P u ρS PPPP '  EndS (VS )

Noncommutative Deformation Theory

49

By abuse of notation, we write u for the vertical map induced by u : R → S. Let us choose a k-linear section σ : S → R such that σ (ei ) = ei and σ (Si j ) ⊆ Ri j for 1 ≤ i, j ≤ r, and consider the map ρ ′ : A → EndR (VR ) given by ρ ′ = σ ◦ ρS . Let Q = (Ki j ⊗k Homk (Vi ,V j )) ⊆ EndR (VR ), where we write K = ker(u). We remark that Q = ker(u) ⊆ EndR (VR ) is an ideal, hence ρ ′ gives Q a natural A-bimodule structure. Moreover, we have that Q2 = 0 since K 2 = 0. It follows that the A-bimodule structure on Q, and therefore the cohomology HH∗ (A, Q), is independent of the chosen section σ . We define the map o′ : A ⊗k A → Q by o′ (a ⊗ b) = ρ ′ (a)ρ ′ (b) − ρ ′ (ab) for a, b ∈ A. A calculation shows that o′ is a 2-cocyle in HC• (A, Q), and that its cohomology class o(u, MS ) = [o′ ] ∈ HH2 (A, Q) is independent of the choice of k-linear section σ . By definition, o(u, MS ) = 0 if and only if o′ = d 1 (τ ) for some τ ∈ HC1 (A, Q), or equivalently that ρR = ρ ′ − τ is an algebra homomorphism lifting ρS to R. This proves that o(u, MS ) = 0 if and only if there is an algebra homomorphism ρR lifting ρS to R. Assume that ρR : A → EndR (VR ) is an algebra homomorphism lifting ρS to R. Let σ : S → R be a k-linear section of u as above, and let ρ ′ : A → EndR (VR ) be the map given by ρ ′ = σ ◦ ρS . Then ρ ′ is an algebra homomorphism if and only if τ = ρ ′ − ρR ∈ HC1 (A, Q) is a 1-cocyle. Moreover, we see that ρ ′ and ρR give rise to equivalent liftings to R if and only if τ is a 1-coboundary. This proves that there is a transitive and free action of HH1 (A, Q) on the set of liftings of ρS to R. Proposition 3.8. The noncommutative deformation functor Def M has an obstruction theory with cohomology {HH p (A, Endk (M))}, the Hochschild cohomology of A with values in Endk (M). Proof. Since I(R) · K = K · I(R) = 0 for any small surjection u : R → S in ar with kernel K, it follows that HH p (A, Q) ∼ = (Ki j ⊗k HH p (A, Homk (Mi , M j ))) in the construction above. The functorial nature of the construction above implies that Def M has an obstruction theory with cohomology {H p }, where H p = (HH p (A, Homk (Mi , M j ))) ∼ = HH p (A, Endk (M)) as a bimodule over kr . Example 3.1. Let A = k[x, y] be the commutative polynomial ring, with generators {x, y} and the relation [x, y] = xy − yx = 0, and let M be the simple right A-module M = A/(x, y)A. We consider the noncommutative deformation functor Def M : a1 → Sets of M = A/(x, y)A ∼ = k considered as a right A-module. We identify Endk (M) ∼ = k and Ext1A (M, M) ∼ = Derk (A, k)/ IDerk (A, k), and obtain the tangent space t(Def M ) = k t1 + k t2 where ti∗ ∈ Ext1A (M, M) is represented by Di ∈ Derk (A, k) for i = 1, 2 given by D1 (x) = 1, D1 (y) = 0 and D2 (x) = 0, D2 (y) = 1. This implies that T1 = khht1 ,t2 ii with radical I = I(T1 ) = (t1 ,t2 ), and the versal family at the tangent level H2 = T1 /I 2 is given by MH2 = H2 ⊗k M ∼ = H2 , with right multiplication of A given by ρ2 : A → H2 with  1,    t , 1 ρ2 (a) =  t 2,    0,

a=1 a=x a=y a ∈ (x, y)2

50

Noncommutative Deformation Theory

Let us consider the assignment o2 , given by o2 (a, b) = ρ2 (a)ρ2 (b) − ρ2 (ab). By construction, o2 vanishes. We may check that this is the case: A computation shows that o2 (a, b) = 0 for all a, b ∈ A except o2 (x, x) = t12 , o2 (x, y) = t1t2 , o2 (y, x) = t2t1 , o2 (y, y) = t22 and these elements vanish in H2 = T1 /(t1 ,t2 )2 . To lift ρ2 to ρ3 : A → H3 , we let  1, a=1      t1 , a=x      t , a=y 2 2 ρ3 (a) = t1 , a = x2    t1t2 , a = xy     t22 , a = y2    0, a ∈ (x, y)3

We see that the corresponding assignment o3 (a, b) = ρ3 (a)ρ3 (b) − ρ3 (ab) is given by o3 (a, b) = 0 for all a, b ∈ A except o3 (y, x) = t2t1 − t1t2 This implies that [t2 ,t1 ] = t2t1 − t1t2 is an obstruction for lifting the deformation ξ2 = (MH2 , ρ2 ) to T13 = T1 /I 3 . Hence H3 = T13 /(t2t1 − t1t2 ) and it follows that the obstruction o3 : A ⊗k A → H3 vanishes. Hence, there is a lifting ξ3 ∈ Def M (H3 ) of ξ2 to H3 given by ρ3 . In fact, we may lift ξ3 to H = T1 /(t2t1 − t1t2 ) ∼ = k[[t1 ,t2 ]] since there are no more obstructions. The versal family ξ ∈ Def M (H) is given by ρ : A → H with

ρ (xn ym ) = t1nt2m ,

n, m ≥ 0

since ρ (a)ρ (b)− ρ (ab) = 0 in H for all a, b ∈ A. In other words, H = k[[t1 ,t2 ]] is the pro-representing b kM ∼ hull for Def M , and the versal family ξ ∈ Def M (H) is given by MH = H ⊗ = H with right multiplication of A given by ρ , so that 1 · x = t1 , 1 · y = t2 There is an algebraization of H and its versal family, given by H = k[t1 ,t2 ] and the versal family MH = H ⊗k M ∼ = k[t1 ,t2 ] with right multiplication of A given by 1 · x = t1 , 1 · y = t2 . The space X(k) of simple k-rational right H-modules in X = Simp(H) is given by X(k) = {k[t1 ,t2 ]/(t1 − τ1 ,t2 − τ2 ) : (τ1 , τ2 ) ∈ A2k } and the family of deformations parametrized by X(k) is given by {MH (τ1 , τ2 ) : (τ1 , τ2 ) ∈ A2k } with MH (τ1 , τ2 ) ∼ = k[x, y]/(x − τ1 , y − τ2 ). This is the family of simple one-dimensional right Amodules. Example 3.2. Let V be the bimodule over k4 generated by {x12 , x13 , x24 , x34 }, with xi j ∈ Vi j for 1 ≤ i, j ≤ 4, and let A = T(V )/(x12 x24 − x13x34 ). Then A ⊆ M4 (k) is the algebra     k k k k k e1 k x12 k x13 k x14 0 k 0 k   0 k e2 0 k x24     A= 0 0 k k  =  0 0 k e3 k x34  0 0 0 k 0 0 0 k e4

Noncommutative Deformation Theory

51

where we write x14 = x12 x24 = x13 x34 . Clearly A is an algebra in a4 , and we consider the noncommutative deformation functor Def M : a4 → Sets of the family M = {k1 , k2 , k3 , k4 } of simple right A-modules. Using Ext1A (ki , k j ) ∼ = Derk (A, Homk (ki , k j ))/ IDerk (A, Homk (ki , k j )), we obtain the tangent space   0 k t12 k t13 0 0 0 0 k t24   t(Def M ) =  0 0 0 k t34  0 0 0 0 where ti∗j ∈ Ext1A (ki , k j ) is represented by the generators x ∈ A. This implies that  k k t12 0 k 1 T = 0 0 0 0

derivation given by xi j 7→ 1 and x 7→ 0 for all other k t13 0 k 0

 kt12t24 + kt13t34  k t24   k t34 k

with radical I = I(T1 ) = (t12 ,t13 ,t24 ,t34 ), and the versal family at the tangent level H2 = T1 /I 2 is given by MH2 = ((H2 )i j ⊗k k j ) ∼ = H2 , with right multiplication of A given by ρ2 : A → H2 with    ei , a = ei ρ2 (a) = ti j , a = xi j with (i, j) = (1, 3), (1, 4), (2, 4), (3, 4)   0, a = x14

Let us consider the assignment o2 , given by o2 (a, b) = ρ2 (a)ρ2 (b) − ρ2 (ab) for all a, b ∈ A. By construction, o2 : A ⊗k A → H2 vanishes. We may check that this is the case: A computation shows that o2 (a, b) = 0 for all a, b ∈ A except o2 (x12 , x24 ) = t12t24 ,

o2 (x13 x34 ) = t13t34

and these elements vanish in H2 = T1 /I 2 . To lift ρ2 to ρ3 : A → H3 , we let   a = ei  ei , ρ3 (a) = ti j , a = xi j with (i, j) = (1, 3), (1, 4), (2, 4), (3, 4)   t12t24 , a = x14

We see that the corresponding assignment o3 (a, b) = ρ3 (a)ρ3 (b) − ρ3 (ab) is given by o3 (a, b) = 0 for all a, b ∈ A except o3 (x13 , x34 ) = t13t34 − t12t24 Hence t13t34 − t12t24 is an obstruction for lifting the deformation ξ2 = (MH2 , ρ2 ) to T13 = T1 /I 3 . This implies that H3 = T13 /(t13t34 − t12t24 ) and it follows that the obstruction o3 : A ⊗k A → H3 vanishes. In particular, there is a lifting ξ3 ∈ Def M (H3 ) of ξ2 to H3 given by ρ3 . In fact, we may lift ξ3 to H = T1 /(t13t34 − t12t24 ) since there are no more obstructions. The versal family ξ ∈ Def M (H) is given by ρ : A → H with   a = ei  ei , ρ (a) = ti j , a = xi j with (i, j) = (1, 3), (1, 4), (2, 4), (3, 4)   t12t24 , a = x14

since ρ (a)ρ (b) − ρ (ab) = 0 in H for all a, b ∈ A. The pro-representing hull of Def M is therefore b kk j) ∼ given by H = T1 /(t13t34 − t12t24 ), with versal family ξ ∈ Def M (H) given by MH = (Hi j ⊗ =H with right multiplication of A given by ρ .

52

Noncommutative Deformation Theory

In this case, the pro-representing hull H is already algebraic since it has finite dimension over k. In fact, we see that ρ : A → H is an isomorphism, and the family of deformations of M parametrized by the k-rational points X(k) = {k1 , k2 , k3 , k4 } in X = Simp(H) is exactly the simple right A-modules. Description using Yoneda cohomology Let us choose a free resolution (Li• , di• ) of the right A-module Mi for 1 ≤ i ≤ r. We write M = ⊕i Mi for the direct sum of right A-modules and (L• , d• ) = ⊕i (Li• , di• ) for the direct sum of the free resolutions. Explicitly, (L• , d• ) is given by       L10 0 . . . 0 L11 0 . . . 0 L12 0 . . . 0  0 L20 . . .   0 0 0   d0  0 L21 . . .  d1  0 L22 . . .  d2 ←−  . ←−  .  ..   .. . . . . ..  ←− . . . . . . . . . . . . . . .  .     . . . . . . . . . . .  0 0 . . . Lr0 0 0 . . . Lr1 0 0 . . . Lr2

and it is a free resolution of M. For any algebra R in ar and any family L = {L1 , . . . , Lr } of free right A-modules, we consider the trivial lifting LR of the family L to R. In concrete terms, LR = (Ri j ⊗k L j ) is the R-flat R-A bimodule with trivial right A-module structure, given by (ri j ⊗ l j ) · a = ri j ⊗ (l j · a) for all ri j ∈ Ri j , l j ∈ L j and a ∈ A. We remark that LR is a projective object in the category of R-A bimodules, and we will describe deformations MR ∈ Def M (R) in terms of projective resolutions (LR• , d•R ) of MR where LRm is a projective object of the above type for m ≥ 0. The starting point is that M = ⊕i Mi represents the unique element ∗ ∈ Def M (kr ) and that (L• , d• ) = ⊕i (Li• , di• ) is a projective resolution of M considered as a kr -A bimodule. A lifting of complexes of (L• , d• ) to R is given by a commutative diagram LR0 o  L0 o

d0R

d0

LR1 o  L1 o

d1R

d1

LR2 o

...

 L2 o

...

where LRm = (Ri j ⊗k L jm ) is a projective R-A bimodule and dmR : LRm+1 → LRm is a homomorphism of R R-A bimodules such that dmR dm+1 = 0 for all m ≥ 0. The vertical maps in the diagram are the natural r maps induced by π : R → k . We say that two liftings of complexes (LR• , d•R ) and (′ LR• , ′ d•R ) are equivalent if there is an isomorphism (LR• , d•R ) ∼ = (′ LR• , ′ d•R ) of complexes of R-A bimodules inducing the identity on (L• , d• ). Let (LR• , d•R ) be a lifting of complexes of (L• , d• ) to R. We shall give a description in concrete terms of the differential dmR : LRm+1 = (Ri j ⊗k L j,m+1 ) → LRm = (Ri j ⊗k L jm ) We fix a k-linear base {ri j (l) : 1 ≤ l ≤ dimk Ri j } of Ri j for 1 ≤ i, j ≤ r, such that the base {rii (l)} contains ei . Since dmR is R-linear, it is determined by its values on ei ⊗ Li,m+1 ⊆ LRm+1 for 1 ≤ i ≤ r. We may therefore express the differential dmR uniquely as dmR =

∑ ri j (l) ⊗ α (ri j (l))m

i, j,l

(3.1)

53

Noncommutative Deformation Theory

for all m ≥ 0, where α = {α (ri j (l))m : m ≥ 0, 1 ≤ i, j ≤ r, 1 ≤ l ≤ dimk Ri j } is a family of A-linear homomorphisms α (ri j (l))m : Li,m+1 → L jm with α (ei )m = dim . Conversely, such a family α of AR linear homomorphisms represents a lifting of complexes of (L• , d• ) to R if and only if dmR ◦ dm+1 =0 for all m ≥ 0. Lemma 3.9. Let R be an algebra in ar , and let (LR• , d•R ) be a lifting of complexes of (L• , d• ) to R. Then we have: 1. Hm (LR• , d•R ) = 0 for all m > 0, 2. H0 (LR• , d•R ) is a lifting of the family M to R. Proof. The lemma holds for R = kr . We consider a small surjection u : R → S in ar and liftings of complexes (LU• , d•U ) of (L• , d• ) to U for U = R and U = S such that the diagram LR0 o  LS0 o

d0R

d0S

LR1 o  LS1 o

d1R

d1S

LR2 o

...

 LS2 o

...

commutes, where the vertical maps are the natural maps induced by u : R → S. In this situation, we shall prove that if the conclusion of the lemma holds for S, then it holds for R as well. Notice that this is enough to prove the lemma. Let K = ker(u), and define LKm = ker(LRm → LSm ). We see that LKm = (Ki j ⊗k L jm ) with the trivial right A-action for all m ≥ 0, and that (LK• , d•K ) is a complex of R-A bimodules, where d•K is the restriction of d•R . Moreover, it is clear that LRm → LSm is surjective for all m ≥ 0. Define MU = H0 (LU• , d•U ) for U = R and U = S, let v : MR → MS be the induced map, and denote its kernel by MK = ker(v). Then v is surjective, and we have the following commutative diagram: 0

0

0o

 MK o

 LK0 o

d0K

 LK1 o

d1K

 LK2 o

...

0o

 MR o

 LR0 o

d0R

 LR1 o

d1R

 LR2 o

...

 LS2 o

...

0

0

v

0o

 MS o

 LS0 o

 0

 0

d0S

 LS1 o  0

d1S

 0

All columns are exact, so the diagram gives a short exact sequence of complexes. By assumption, the bottom row is exact and MS = (Si j ⊗k M j ) is a lifting of the family M to S. Since u is small, we have K · I(R) = 0, and this implies that (LK• , d•K ) is the restriction of the trivial lifting of (L• , d• ) to R. Hence Hm (LK• , d•K ) = 0 for all m > 0. It follows from the long exact sequence of cohomology of 0 → LK• → LR• → LS• → 0 that Hm (LR• , d•R ) = 0 for all m > 0, and that there is a short exact sequence 0 → H0 (LK• , d•K ) → MR → MS → 0

54

Noncommutative Deformation Theory

∼ H0 (LK , d K ). Finally, since (LK , d K ) is the restriction of the trivial lifting of In particular, MK = • • • • (L• , d• ) to R, it follows that MK ∼ = H0 (LK• , d•K ) ∼ = (Ki j ⊗k H0 (L j• , d j• )) ∼ = (Ki j ⊗k M j ) This implies that MR ∼ = (Ri j ⊗k M j ) is an R-flat R-A bimodule, hence MR ∈ Def M (R) is a lifting of the family M to R. Lemma 3.10. Let R be an algebra in ar , and let MR be a lifting of the family M to R. Then there exists a lifting (LR• , d•R ) of complexes of (L• , d• ) to R which is a projective resolution of MR . Proof. The lemma holds for R = kr . We consider a small surjection u : R → S in ar , deformations MR ∈ Def M (R) and MS ∈ Def M (S) such that MR lifts MS to R, and a lifting (LS• , d•S ) of complexes of (L• , d• ) to S which is a projective resolution of MS . In this situation, we shall prove that there exists a lifting (LR• , d•R ) of complexes of (LS• , d•S ) to R which is a projective resolution of MR . Notice that this is enough to prove the lemma. Let v : MR → MS be the natural surjective map induced by u : R → S, and define K = ker(u) and MK = ker(v) = (Ki j ⊗k M j ). Let us also write LRm = (Ri j ⊗k L jm ) and LKm = (Ki j ⊗k L jm ) for all m ≥ 0. We consider the diagram 0

0

0

0

    MK o❴ ❴ ❴ LK0 o❴ ❴ ❴ LK1 o❴ ❴ ❴ LK2     MR o❴ ❴ ❴ LR0 o❴ ❴ ❴ LR1 o❴ ❴ ❴ LR2 v

0o

 MS o  0

ρS

 LS0 o  0

d0S

 LS1 o  0

d1S

 LS2 o

...

 0

where the vertical maps are induced by u : R → S and all columns are exact. We must find differentials dmR : LRm+1 → LRm for m ≥ 0 and an augmentation map ρ R : LR0 → MR such that (LR• , d•R ) is a projective resolution of MR commuting with (LS• , d•S ). Since LR0 is a projective R-A bimodule, we can find a map ρ R : LR0 → MR lifting ρ S . Let us write ρ K : LK0 → MK for its restriction. Since u is small, we have K · I(R) = 0, and this implies that ρ K is the restriction of the trivial lifting of ρ to R, where ρ : L0 → M is the augmentation map of (L• , d• ). In particular, this implies that ρ K is surjective, hence the induced map ker(ρ R ) → ker(ρ S ) is surjective by the Snake Lemma. Using induction, we can find a lifting (LR• , d•R ) of complexes of (LS• , d•S ) to R, and it follows from the proof of Lemma 3.9 that (LR• , d•R ) is a projective resolution of MR with augmentation map ρ R . Let R be any algebra in ar . It follows from Lemma 3.9 and Lemma 3.10 that there is a bijective correspondence between the set of liftings of the family M to R and the set of liftings (LR• , d•R ) of complexes of (L• , d• ) to R. Moreover, two liftings of complexes (LR• , d•R ) and (′ LR• , ′ d•R ) are equivalent if and only if the corresponding liftings MR and ′ MR of the family M to R are equivalent. Definition 3.8. We recall the definition of the Yoneda complex and the Yoneda cohomology. Let

Noncommutative Deformation Theory

55

M N N M, N be right A-modules with free resolutions (LM • , d• ) and (L• , d• ). The Yoneda cohomology • N YH (M, N) is defined as the cohomology of the Yoneda complex YC• (LM • , L• ), where a N N YCn (LM HomA (LM • , L• ) = m+n , Lm ) m≥0

n+1 M N N for all n ≥ 0, and where the differentials d n : YCn (LM (L• , L• ) are given by • , L• ) → YC M + (−1)n+1dmN φm+1 d n (φ )m = φm dn+m

for m ≥ 0

n n N ∼ for all φ = (φm )m≥0 ∈ YCn (LM • , L• ). It is easy to see that YH (M, N) = ExtA (M, N) for all n ≥ 0. In particular, the Yoneda cohomology is indepedent of the choice of free resolutions.

Let u : R → S be a small surjection in ar , and let MS ∈ Def M (S) be a lifting of the family M to S. Then there is a lifting (LS• , d•S ) of complexes of (L• , d• ) to S which is a projective resolution of MS . To lift MS to R, we must find a homomorphism dmR : LRm+1 → LRm of R-A bimodules lifting dmS to R for all m ≥ 0 such that (LR• , d•R ) is a complex. Let us choose a k-linear section σ : S → R such that σ (ei ) = ei and σ (Si j ) ⊆ Ri j for 1 ≤ i, j ≤ r. We consider dmS as an element dmS ∈ (Si j ⊗k HomA (Li,m+1 , L jm )) for all m ≥ 0, and use the σ to define ′ R dm = (σ ⊗ id)(dmS ) ∈ (Ri j ⊗k HomA (Li,m+1 , L jm )). Hence ′ dmR : LRm+1 → LRm is a homomorphism of R-A bimodules lifting dmS to R for all m ≥ 0. R We define o′m = ′ dmR ◦ ′ dm+1 for all m ≥ 0, and let K = ker(u) and LKm = (Ki j ⊗k L jm ). Since S S (L• , d• ) is a complex, it follows that o′m : LRm+2 → LKm . We may therefore consider o′ as an element o′ ∈ (Ki j ⊗k YC2 (Li• , L j• )). A calculation shows that o′ is a 2-cocycle in (Ki j ⊗k YC• (Li• , L j• )), and that its cohomology class o(u, MS ) = [o′ ] ∈ (Ki j ⊗k YH2 (Mi , M j )) is independent of the choice of k-linear section σ . By definition, we have that o(u, MS ) = 0 if and only if o′ = d 1 (τ ) for some τ ∈ (Ki j ⊗k YC1 (Li• , L j• )), or equivalently that dmR = ′ dmR + τ is a homomorphism of R-A bimodules lifting dmS to R for all m ≥ 0 such that (LR• , d•R ) is a complex. This proves that o(u, MS ) = 0 if and only if there is a lifting MR ∈ Def M (R) of MS to R. Assume that (LR• , d•R ) is a lifting of complexes of (L• , d• ) to R, and let ′ dmR be the R-A linear map ′ d R = (σ ⊗id)(d S ) ∈ (R ⊗ Hom (L ij k A i,m+1 , L jm )) for m ≥ 0, defined by the k-linear section σ : S → R m m ′ R ′ R of u as above. Then ( L• , d• ) is a complex if and only if τ = ′ d R − d R ∈ (Ki j ⊗k YC1 (Li• , L j• )) is a 1-cocyle. Moreover, we see that ′ d R and d R give rise to equivalent liftings to R if and only if τ is a 1-coboundary. This proves that there is a transitive and free action of (Ki j ⊗k YH1 (Mi , M j )) on the set of liftings of MS ∈ Def M (S) to R. Proposition 3.11. The deformation functor Def M : ar → Sets has an obstruction theory with cohomology {YH p (M, M)}, the Yoneda cohomology of M. Example 3.3. Let A = khx, yi/(xy−yx−1), and let M = {M1 , M2 } be the family of right A-modules given by M1 = A/yA and M2 = A/xA, with free resolutions y·

0 ← M1 ← A ← − A ← 0,

x·

0 ← M2 ← A ← −A←0

Then, the direct sum M = M1 ⊕ M2 has free resolution (L• , d• ) given by !

y 0 · 0 x 2 0 ← M ← A ←−−−−−− A2 ← 0 We shall consider the noncommutative deformation functor Def M : a2 → Sets of the family M ,

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Noncommutative Deformation Theory

and compute its pro-representing hull and versal family. We will use Yoneda cohomology and the obstruction calculus for liftings of complexes. p To compute the Yoneda cohomology groups YH p (Mi , M j ) ∼ = ExtA (Mi , M j ) for 1 ≤ i, j ≤ 2 and 1 ≤ p ≤ 2, we consider the complex HomA (Li• , M j ): ·y

i=1:

Mj − → M j → 0,

i=2:

·x

Mj − → Mj → 0

We immediately see that YH2 (Mi , M j ) = Ext2A (Mi , M j ) = 0 for all i, j. We also note that M1 ∼ = k[x] and M2 ∼ = k[y] as k-linear spaces, with right action given by xn · x = xn+1 ,

xn · y = nxn−1,

yn · x = −nyn−1,

for all n ≥ 0. Therefore, we have that 1

YH (Mi , M j ) ∼ = Ext1A (Mi , M j ) ∼ =

(

yn · y = yn+1

0, i= j k · 1, i = 6 j

It follows that the noncommutative deformation functor is unobstructed, and that its prorepresenting hull H ∼ = T1 is given by     H11 H12 k[[t12t21 ]] ht12 i H= = H21 H22 ht21 i k[[t21t12 ]] where H11 = k[[t12t21 ]] and H22 = k[[t21t12 ]] are formal power series algebras in one variable, and H12 and H21 are cyclic bimodules generated by t12 and t21 . At the tangent level, H2 = H/(t12 ,t21 )2 and the versal family ξ2 ∈ Def M (H2 ) is given by a lifting H2 H2 H 2 of complexes (LH • , d• ) of (L• , d• ) to H2 . In concrete terms, the differential d 2 = (dm ) is given by dmH2 = 1 ⊗ dm + ∑ ti j (l) ⊗ α (ti j (l))m i, j,l

for m ≥ 0, where

dmH2

d0H2

d0H2

= 0 for m ≥ 1 and is explicitly given by        y 0 0 1 0 0 y = 1⊗ + t12 ⊗ + t21 ⊗ = 0 x 0 0 1 0 t21

t12 x



The versal family ξ2 ∈ Def M (H2 ) can clearly be lifted to H, and the differential d0H is given by   y t12 H d0 = t21 x This means that the versal family ξ ∈ Def M (H) is the H-A bimodule MH given by   y t12 MH = coker(d0H ) = Hhx, yi2 / t21 x We see that there is an algebraization of H and its versal family, given by     k[t12t21 ] ht12 i y t12 H= , MH = Hhx, yi2 / ht21 i k[t21t12 ] t21 x The space X(k) of simple k-rational right H-modules in X = Simp(h) contains the one-dimensional simple modules X1 (k) = {k1 , k2 } since H(1) = k2 , and the family of deformations parametrized by X1 (k) is given by {MH (i) : i = 1, 2} ∼ with MH (i) = Mi . However, X(k) 6= X1 (k) in this case, and there are many k-rational simple modules of higher dimension.

Noncommutative Deformation Theory

57

3.3.2 Modules with group action Let A be an associative k-algebra, and let G be a group with a right action on A, given by a group homomorphism σ : G → Autk (A). We recall that a right A-G module is a right A-module M with a k-linear right action of G, given by a group homomorphism G → Autk (M), such that (ma) • g = (m • g) · (a • g) for all m ∈ M, a ∈ A, g ∈ G, where we write a • g for the right action of g on a ∈ A and m • g for the right action of g on m ∈ M. A morphism φ : M → N of right A-G modules is a homomorphism of right A-modules such that φ (m • g) = φ (m) • g for all m ∈ M, g ∈ G. We remark that the category ModA−G of right A-G modules is a module category, canonically isomorphic to the category ModA[G] of right modules over the skew group algebra A[G] associated with the right action of G on A. Recall that A[G] is the free right A-module with base {g : g ∈ G} and with multiplication given by a · g = g · (a • g) for all a ∈ A, g ∈ G. We consider the noncommutative deformation functor Def M of a family M = {M1 , M2 , . . . , Mr } of right A-G modules in the category ModA−G . We may think of ModA−G as the category of right modules over the associative algebra A[G]. Hence, the noncommutative deformation functor Def M has an obstruction theory with Hochschild cohomology {HH p (A[G], Homk (M, M))}, and also an obstruction theory with Yoneda cohomology {YH p (M, M)}, where we have p HH p (A[G], Homk (M, M)) ∼ = YH p (M, M) ∼ = ExtA[G] (M, M)

when we consider M = ⊕i Mi as a right A[G]-module; see Subsection 3.3.1 for details. But it can be difficult to work with the skew group algebra A[G] directly, and we shall consider another point of view. We note that for any right A-G modules M and N, the vector space HomA (M, N) has an induced right k-G module structure, with right G-action given by

φ • g = {m 7→ φ (m • g−1) • g} for all g ∈ G, φ ∈ HomA (M, N), m ∈ M. We see that (φ • g)(m • g) = φ (m) • g, and it follows that HomA[G] (M, N) = HomA (M, N)G . Therefore, the cohomology p ExtA[G] (M, N) ∼ = HH p (A[G], Homk (M, N)) ∼ = YH p (M, N)

is the p’th derived functor of the composition M 7→ HomA (M, N) 7→ HomA (M, N)G . Let G be a linear algebraic group over k. We say that a right k-G module V is rational if the following condition holds: For any v ∈ V , there is a finite dimensional G-invariant subspace W ⊆ V containing v such that G → GL(W ) is a morphism of algebraic groups. The following facts are wellknown from the representation theory of algebraic groups: If G is a finite group, then any right k-G module V is rational. Moreover, if X is an affine algebraic variety over k with a right action of G such that X × G → X is a morphism of algebraic varieties, then the induced right action of G on A = O(X) is rational. It is explicitly given by ( f • g)(x) = f (x • g−1 ) for f ∈ O(X), g ∈ G, x ∈ X. In the rest of this subsection, we assume that G is a linear algebraic group over k with a rational right action on a commutative algebra A. We consider the full subcategory Modrat A−G of ModA−G consisting of rational right A-G modules (that is, A-G modules that are rational considered as k-G

58

Noncommutative Deformation Theory

modules). It is wellknown that Modrat A−G is an exact Abelian subcategory, closed under subobjects, quotient objects and direct sums. If G is linearly reductive, then the functor M 7→ M G is exact on Modrat A−G . Proposition 3.12. Let G be a linear algebraic group with a rational action on A. 1. Let W be a rational right k-G module of dimension n. Then, L = W ⊗k A is a rational right A-G module, free of rank n considered as an A-module. 2. If G is a linearly reductive group, then L = W ⊗k A is projective in the category Modrat A−G for any rational right k-G module W of finite dimension. 3. Let M be a rational right A-G module that is finitely generated as an A-module. If G is linearly reductive and A is Noetherian, then M has a projective resolution (L• , d• ) in Modrat A−G such that Li = Wi ⊗k A for a finite dimensional rational right k-G module Wi . Proof. Let L = W ⊗k A for a rational right k-G module W of dimension n. Then L has an obvious right A-module structure, and L has a right action of G given by (w ⊗ a) • g = (w • g) ⊗ (a • g) for all a ∈ A, w ∈ W, g ∈ G. Since W and A are rational right k-G modules, it follows that L is a rational right A-G module, and L is free of rank n considered as an A-module. For the second part, let φ : M → N be a surjective homomorphism of rational right A-G modules, and consider the induced linear map HomA (L, φ )G : HomA (L, M)G → HomA (L, N)G We have that HomA (L, M) and HomA (L, N) are rational right k-G modules, that L is free as a right A-module, and that G is linearly reductive. It follows that HomA (L, φ )G is surjective, and therefore that L is projective in Modrat A−G . For the last part, let M be a rational right A-G module that is generated by {m1 , . . . , mn } as an A-module. Then, there is a finite dimensional G-invariant linear subspace W ⊆ M containing {m1 , . . . , mn } such that W is a rational right k-G module. Hence, there is a well-defined surjection L = W ⊗k A → M of right A-G modules given by w ⊗ a 7→ wa for all w ∈ W, a ∈ A. It follows by induction that there is a projective resolution of M of the required type, since ker(W ⊗k A → M) is finitely generated as an A-module when A is Noetherian. Assume that A is Noetherian, that G is linearly reductive and that M, N are rational right A-G modules that are finitely generated considered as A-modules. Then it follows from the comments above that there is a projective resolution (L• , d• ) of M in the category of right A-G modules such that Li = Wi ⊗k A where Wi is a rational k-G module of finite dimension. Since di is a map of right A-G modules, the induced complex HomA (Li , N) is a complex of right A-G modules. In particular, there is an induced right A-G module structure on ExtAp (M, N). Corollary 3.13. Let G be a linear algebraic group with a rational right action on a commutative Noetherian algebra A, and let M, N be rational right A-G modules such that M is finitely generated p p as an A-module. If G is linearly reductive, then ExtA[G] (M, N) ∼ = ExtA (M, N)G for all p ≥ 0. Proof. By Proposition 3.12, there is a projective resolution 0 ← M ← L• of the right A-G module M such that Li = Wi ⊗k A and Wi is a rational right k-G module of finite dimension for i ≥ 0. We consider the complex HomA−G (L• , N) = HomA (L• , N)G p By definition, H p (HomA−G (L• , N)) = ExtA−G (M, N). On the other hand, we have that p p H (HomA (L• , N)) = ExtA (M, N) since Li is free considered as a right A-module for i ≥ 0. Moreover, since G is linearly reductive and HomA (Li , N) is rational for i ≥ 0, it follows that H p (HomA (L• , N)G ) = ExtAp (M, N)G .

Noncommutative Deformation Theory

59

We often consider rational A-G modules of the form M = A/p and N = A/q, where A is a commutative Noetherian ring and p, q ⊆ A are G-invariant ideals in A. In this situation, we have the following results: Lemma 3.14. If p ⊆ q are G-invariant ideals in A, then there is an isomorphism of right A-G modules Ext1A (A/p, A/q) ∼ = HomA (p, A/q). Proof. Let us choose a free resolution (L• , d• ) of the right A-G module A/p such that L0 = A. Then Ext1A (A/p, A/q) is the first cohomology group ker(d 1 )/ im(d 0 ) of the complex HomA (L• , A/q) with differentials d i , and ker(d 1 ) = {ξ ∈ HomA (L1 , A/q) : ξ d1 = 0}. Since any such homomorphism factors through A/ im(d 1 ) = A/ ker(d0 ) ∼ = im(d0 ) = p, it follows that the A-linear map HomA (p, A/q) → ker(d 1 ) given by φ 7→ φ d0 is an isomorphism. Since d0 is A-G linear, it is also an isomorphism of right A-G modules. Finally, we see that im(d 0 ) corresponds to the A-linear maps p → A/q that can be extended to an A-linear map A → A/q. When p ⊆ q, it follows that im(d 0 ) = 0, and this completes the proof. Lemma 3.15. If p ⊇ q are G-invariant ideals in A with p 6= q, then there is a surjection of right A-G modules HomA (p, A/q) → Ext1A (A/p, A/q). In particular, the induced homomorphism HomA (p, A/q)G → Ext1A (A/p, A/q)G is surjective when G is linearly reductive. Proof. There is an isomorphism ker(d 1 ) ∼ = HomA (p, A/q) of right A-G modules by the proof of Lemma 3.14, so there is a surjection HomA (p, A/q) → Ext1A (A/p, A/q) of right A-G modules with kernel im(d 0 ). We consider the noncommutative deformation functor Def M of a family M = {M1 , M2 , . . . , Mr } of r rational right A-G modules in the category Modrat A−G . It is useful to compare this deformation functor with the noncommutative deformation functor of M considered as a family of right AA modules in the category ModA . We denote these deformation functors by Def A−G M and Def M . A−G A There is a forgetful natural transformation Def M → Def M of noncommutative deformation functors. In fact, for any algebra R in ar , we have that Def A−G M (R) = {(MR , ∇, ηi )}/ ∼ , where (MR , ηi ) ∈ Def AM (R), ∇ : G → EndR (MR ) is a rational right action of G on MR such that ηi is G-linear for 1 ≤ i ≤ r, and (MR , ∇, ηi ) ∼ (MR′ , ∇′ , ηi′ ) if there is an equivalence (MR , ηi ) ∼ (MR′ , ηi′ ) in Def AM (R) given by a G-linear isomorphism τ : MR → MR′ of R-A bimodules. We shall describe an obstruction theory for Def M = Def A−G based on Yoneda cohomology. M Assume that G is linearly reductive, that A is Noetherian and that Mi is finitely generated as an A-module for 1 ≤ i ≤ r. By Proposition 3.12, we may choose a projective resolution 0 ← Mi ← Li0 ← · · · ← Lim ← . . . of Mi in Modrat A−G for 1 ≤ i ≤ r, where Lim = Wim ⊗k A for some finite dimensional rational right k-G module Wim . We write L• = ⊕i Li• for the direct sum projective resolution of M = ⊕i Mi in Modrat A−G . For all algebras R in ar , we write LR• = (Ri j ⊗k L j• ) for the trivial lifting of L• to R, i.e., the projective R-A bimodule with right G-action induced by the G-action on L j• for 1 ≤ j ≤ r. Using the characterization of Def A−G M given above and the characterization of noncommutative deformations of modules via lifting of complexes in Subsection 3.3.1, we see that there is a bijective correspondence between the set Def A−G M (R) of deformations of the family M to R and the set of liftings (LR• , d•R ) of complexes of (L• , d• ) to R such that d•R is G-linear. We remark that the right G-action on HomA (Li,m+n , L jm ) for 1 ≤ i, j ≤ r, m, n ≥ 0 makes the Yoneda complex YC• (Li• , L j• ) a complex of rational right A-G modules for 1 ≤ i, j ≤ r. Hence

60

Noncommutative Deformation Theory

there is a natural right G-action on the Yoneda cohomology YH• (M, M). Using Corollary 3.13, it follows that YH• (M, M)G ∼ = Ext•A (M, M)G ∼ = Ext•A−G (M, M). Let u : R → S be a small surjection in ar , and let (MS , ∇, ηi ) ∈ Def A−G M (S) be a lifting of the family M to S. By the remarks above, we can find a G-linear lifting (LS• , d•S ) of complexes of (L• , d• ) to S corresponding to this deformation, and we have that dmS ∈ (Si j ⊗k HomA (Li,m+1 , L jm )G ) for m ≥ 0. We choose a k-linear section σ : S → R of u such that σ (ei ) = ei , and use the construction in Subsection 3.3.1 to find a G-linear lifting dmR ∈ (Ri j ⊗k HomA (Li,m+1 , L jm )G ) of dmS to R. Hence we obtain a G-invariant 2-cocycle R o′m = dmR ◦ dm+1 ∈ (Ki j ⊗k YC2 (Li,m+2 , L j,m )G )

which defines an obstruction o(u, MS ) ∈ (Ki j ⊗k YH2 (Mi , M j )G ) for the existence of a deformation (MR , ∇, ηi ) ∈ Def M (R) lifting MS to R. Proposition 3.16. Let G be a linear algebraic group with a rational right action on a commutative Noetherian algebra A, and let M = {M1 , . . . , Mr } be a family of r rational right A-G modules that are finitely generated as A-modules. If G is linearly reductive, then there is an obstruction theory p for Def M with cohomology {YH p (M, M)G ∼ = ExtA (M, M)G }. Example 3.4. Let G = k∗ , and consider the right action of G on the affine plane X = A2 given by (a, b) • g = (ag−1 , bg−1 ) for all a, b ∈ k, g ∈ k∗ . It is wellknown that G = GL(1, k) is a linearly reductive algebraic group, that X is an affine algebraic variety with coordinate ring A = k[x, y], and that the action X × G → X is a morphism of algebraic varieties. Hence, it follows that there is an induced rational right action of G on A, and it is given by fi (x, y) • g = fi (xg−1 , yg−1 ) = fi (x, y) · g−i for any homogeneous polynomial fi ∈ k[x, y] of degree i and for any g ∈ G. Let a1 = (x) ⊆ A and a2 = (x, y) ⊆ A be the G-invariant ideals corresponding to the line with equation x = 0 and the point with coordinates (0, 0) in the affine plane. We shall consider the noncommutative deformation functor Def M : a2 → Sets of the family M = {M1 , M2 } in Modrat A−G , where M1 = A/a1 and M2 = A/a2 , and compute its pro-representing hull and versal family. By the construction in Proposition 3.12, there are projective resolutions (Li• , di• ) of Mi in Modrat A−G for i = 1, 2 such that Li• is a free right A-module of finite rank. We fix projective resolutions of this form, given by

x·

0 ← M1 ← A ← − A ← 0,

!

y · x −x 2 0 ← M2 ← A ←−−−−− A ←−−−− A ← 0 

 y ·

Then, the direct sum M = M1 ⊕ M2 has projective resolution (L• , d• ), given by 



0 !    y · x 0 0   · 0 x y −x 0 ← M ← A2 ←−−−−−−−−− A3 ←−−−−− A ← 0 Notice that the right action of G on Li• = Wi• ⊗A A ∼ = An is the action induced by the right action of G on Wi• . By Proposition 3.16, there is an obstruction theory of Def M with cohomology {YH p (M, M)G }. To compute YH p (Mi , M j ), we consider the complex HomA (Li• , M j ) for 1 ≤ i, j ≤ 2:

i=1:

·x

Mj − → M j → 0,

i=2:

!

y y −x M j −−−−−→ M 2j −−−−→ M j → 0  · x



·

Noncommutative Deformation Theory

61

We notice that M1 ∼ = k[y] and M2 ∼ = k as k-linear spaces, and see that we have   k[y], (i, j) = (1, 1) 0, (i, j) = (1, 1)       k,  (i, j) = (1, 2) 0, (i, j) = (1, 2) YH1 (Mi , M j ) ∼ YH2 (Mi , M j ) ∼ = =   k, (i, j) = (2, 1)    k, (i, j) = (2, 1)   2 k , (i, j) = (2, 2) k, (i, j) = (2, 2)

To compute YH1 (Mi , M j )G , we use Lemma 3.14 and Lemma 3.15 to represent elements in YH1 (Mi , M j ) by homomorphisms in HomA (ai , A/a j ), and find that   k, (i, j) = (1, 1) 0, (i, j) = (1, 1)       0, (i, j) = (1, 2) 0, (i, j) = (1, 2) YH1 (Mi , M j )G ∼ YH2 (Mi , M j )G ∼ = =   0, (i, j) = (2, 1) ∗, (i, j) = (2, 1)       0, (i, j) = (2, 2) ∗, (i, j) = (2, 2)

It follows that the noncommutative deformation functor is unobstructed, and that its prorepresenting hull H ∼ = T1 is given by     H11 H12 k[[t11 ]] 0 H= = H21 H22 0 k

∗ ∈ YH1 (M , M )G is represented by the element y· ∈ Hom (L , L ). where t11 1 1 A 1,1 1,0 2 ) and the versal family ξ ∈ Def (H ) is given by a lifting of At the tangent level, H2 = H/(t11 2 2 M H2 H2 H 2 complexes (LH • , d• ) of (L• , d• ) to H2 . In concrete terms, the differential d 2 = (dm ) is given by

dmH2 = 1 ⊗ dm + ∑ ti j (l) ⊗ α (ti j (l))m i, j,l

for m ≥ 0. Explicitly, dmH2 = 0 for m ≥ 2 and dmH2 for m = 0, 1 is given by     0 x + t11y 0 0 H2 H2  , d1 = y  d0 = 0 x y −x

The versal family ξ2 ∈ Def M (H2 ) can clearly be lifted to H, and the differential d0H is given by   x + t11 y 0 0 H d0 = 0 x y This means that the versal family ξ ∈ Def M (H) is the H-A bimodule MH given by   x + t11y 0 0 MH = coker(d0H ) = H[x, y]2 / 0 x y

with the induced right G-action. We see that there is an algebraization of H and its versal family, given by     x + t11y 0 0 k[t11 ] 0 2 H= , MH = H[x, y] / 0 k 0 x y

The space X(k) of simple k-rational right H-modules in X = Simp(H) is given by the disjoint union X(k) = X1 (k) = A1 ∐A0 = {τ : τ ∈ k} ∐{∗} since H is commutative, and the family of deformations parametrized by X(k) is given by {MH (τ ) : τ ∈ k} ∐ {MH (∗)} with MH (τ ) ∼ = k[x, y]/(x + τ y) for τ ∈ k and MH (∗) = k[x, y]/(x, y). That is, MH (τ ) corresponds to the line with equation x + τ y = 0 and MH (∗) corresponds to the point with coordinates (0, 0).

62

Noncommutative Deformation Theory

Example 3.5. Assume that k is an algebraically closed field of characteristic 0, and let V be a vector space over k of dimension two. We consider the right action of the group G = GL(V ) on X = Endk (V ) given by the conjugation M • g = g−1 Mg for g ∈ G, M ∈ X. It is wellknown that G is a linearly reductive algebraic group, that X is an affine algebraic variety with coordinate ring A = k[x11 , x12 , x21 , x22 ], and that the action X × G → X is a morphism of algebraic varieties. Hence, it follows that there is an induced rational right action of G on A, and it is given by     x11 x12 x11 x12 •g = g· · g−1 x21 x22 x21 x22

for all xi j ∈ k, g ∈ G = GL(2, k). It is wellknown that the trace s1 = x11 + x22 and determinant s2 = x11 x22 − x12x21 generate the ring AG = k[s1 , s2 ] ⊆ A of invariants. Let a1 = (s1 , s2 ) ⊆ A and a2 = (x11 , x12 , x21 , x22 ) ⊆ A be the G-invariant ideals corresponding to the endomorphisms with vanishing trace and determinant and the endomorphism with coordinates
0 0 in X = End (V ). We consider the noncommutative deformation functor Def k M : a2 → Sets 00 of the family M = {M1 , M2 } in Modrat , where M = A/a and M = A/a , and compute its 1 1 2 2 A−G pro-representing hull and versal family. By the construction in Proposition 3.12, there are projective resolutions (Li• , di• ) of Mi in Modrat A−G for i = 1, 2 such that Li• is a free right A-module of finite rank. We fix projective resolutions of this form, and see that M1 has the resolution !

s2 · s1 s2 · 2 −s1 0 ← M1 ← A ←−−−−−− A ←−−−−− A ← 0 



where G acts trivially on L1m for m ≥ 0, and M2 has the resolution 



x12 x21 x22 0 0 0   −x11 0 0 x x 0  21 22  ·  −x11 0 −x12 0 x22   0    x11 x12 x21 x22 · 4 0 0 −x11 0 −x12 −x21 0 ← M2 ← A ←−−−−−−−−−−−−−−−− A ←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− A6 ← . . . where G acts trivially on L20 = A and where G has the induced action on L2m for m ≥ 1. Then, the direct sum M = M1 ⊕ M2 has the direct sum projective resolution (L• , d• ). By Proposition 3.16, there is an obstruction theory of Def M with cohomology {YH p (M, M)G }. To compute YH p (Mi , M j ), we consider the complex HomA (Li• , M j ) for 1 ≤ i, j ≤ 2. We notice that M2 ∼ = k as k-linear spaces, and see that  2  M1 , (i, j) = (1, 1) M1 , (i, j) = (1, 1)       k2 , (i, j) = (1, 2) k, (i, j) = (1, 2) YH2 (Mi , M j ) ∼ YH1 (Mi , M j ) ∼ = =   (i, j) = (2, 1) (i, j) = (2, 1)   0, ∗,   k4 , (i, j) = (2, 2) k6 , (i, j) = (2, 2) To compute YH1 (Mi , M j )G , we use the action of G on Lim and find that   2 k, k , (i, j) = (1, 1)       k, k2 , (i, j) = (1, 2) YH2 (Mi , M j )G ∼ YH1 (Mi , M j )G ∼ = =   ∗, 0, (i, j) = (2, 1)       ∗, k, (i, j) = (2, 2)

(i, j) = (1, 1) (i, j) = (1, 2) (i, j) = (2, 1) (i, j) = (2, 2)

63

Noncommutative Deformation Theory It follows that the noncommutative deformation functor has pro-representing hull given by     H11 H12 khht11 (1),t11 (2)ii/( f11 ) ht12 (1),t12 (2)i/( f12 ) H= = H21 H22 0 k[[t22 ]]/I22

where fi j = o(s∗i j ) is a formal power series that is determined by the obstruction morphism for (i, j) = (1, 1), (1, 2) and where I22 is an ideal generated by such power series. We choose a representative α (ti j (l)) = {α (ti j (l))0 : Li1 → L j0 , α (ti j (l))1 : Li2 → L j1 } of ti∗j (l) ∈ YH1 (Mi , M j )G , given by      

0 1 α11 (1) = 1 0 ·, α11 (2) = 0 1 ·, · · −1 0       −x22  1         
 x21  
0      1 0 ·,  0 1 ·,   · · α12 (1) = α12 (2) = 0  0            0 1    0 0 1 0 0 0      
−1 0 0 0 1 0   1 0 0 1 ·,  α22 = · 0 −1 0 0 0 1       0 0 −1 0 0 0

At the tangent level, H2 = H/ I(H)2 and the versal family ξ2 ∈ Def M (H2 ) is given by a lifting of H2 H2 H 2 complexes (LH • , d• ) of (L• , d• ) to H2 . In concrete terms, the differential d 2 = (dm ) is given by dmH2 = 1 ⊗ dm + ∑ ti j (l) ⊗ α (ti j (l))m i, j,l

for m ≥ 0. Explicitly, dmH2 for m = 0, 1 is given by  0 s + t (1) s2 + t11(2) H2 d0 = 1 11 t12 (1) t12 (2) x11 + t22

0 x12

0 x21

0 x22 + t22



and 

s2 + t11 (2)  −s1 − t11 (1)  −t (1)x22 + t12 (2) d1H2 =  12 t12 (1)x21   0 t12 (2)

0 0 x12 −x11 − t22 0 0

0 0 x21 0 −x11 − t22 0

0 0 x22 + t22 0 0 −x11 − t22

0 0 0 x21 −x12 0

0 0 0 x22 + t22 0 −x12

By construction, this defines a complex over H2 . We may check that this is the case:  t11 (2)t11 (1) − t11(1)t11 (2) 0 0 0 0 d0H2 ◦ d1H2 = t11 (2)t12 (1) − t11(1)t12 (2) + 2t12(2)t22 − t12 (1)t22 x22 0 0 0 0



0 0   0   0  x22 + t22  −x21

0 0 0 0



In this case, we use that I(H)2 = 0 in H2 to show that the obstruction above vanishes. In fact, the obstruction is given by [t11 (2),t11 (1)]s∗11 + (t11 (2)t12 (1) − t11(1)t12 (2) + 2t12(2)t22 ) s∗12 − t12 (1)t22 x22 s∗12 This implies that the second-order approximations of the power series fi j and of the ideal I22 are given by 2 2 f11 = [t11 (2),t11 (1)], f12 = t11 (2)t12 (1) − t11(1)t12 (2) + 2t12(2)t22

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2 = 0, and since x s∗ and I22 22 12 is a coboundary, there is an element α (t12 (1)t22 ) with 1 d (α (t12 (1)t22 )) = x22 s∗12 , given by    0     
0    0 0 · , · α (t12 (1)t22 ) = 0       1

Explicitly, the lifting of the versal family to H3 is given by the differentials   0 0 0 0 s + t (1) s2 + t11(2) d0H3 = 1 11 t12 (1) t12 (2) x11 + t22 x12 x21 x22 + t22 and



s2 + t11 (2)  −s1 − t11 (1)  −t (1)x22 + t12 (2) H d1 3 =  12 t12 (1)x21   0 t12 (2) + t12 (1)t22

0 0 x12 −x11 − t22 0 0

0 0 x21 0 −x11 − t22 0

0 0 x22 + t22 0 0 −x11 − t22

0 0 0 x21 −x12 0

0 0 0 x22 + t22 0 −x12

By construction, this defines a complex over H3 . We may check that this is the case:  t11 (2)t11 (1) − t11(1)t11 (2) 0 0 0 0 H H d0 3 ◦ d1 3 = 2 t11 (2)t12 (1) − t11(1)t12 (2) + 2t12(2)t22 + t12 (1)t22 0 0 0 0



0 0   0   0  x22 + t22  −x21

0 0 0 0



This implies that the third-order approximations of the power series fi j are given by 3 f11 = [t11 (2),t11 (1)],

3 2 f12 = t11 (2)t12 (1) − t11(1)t12 (2) + 2t12(2)t22 + t12(1)t22

3 = 0. We see that the versal family defined over H can be lifted to H defined by the power and I22 3 series

f11 = [t11 (2),t11 (1)],

2 f12 = t11 (2)t12 (1) − t11(1)t12 (2) + 2t12(2)t22 + t12(1)t22

and I22 = 0. Therefore, the pro-representing hull of Def M is given by   k[[t11 (1),t11 (2)]] ht12 (1),t12 (2)i/( f12 ) H= 0 k[[t22 ]] and the versal family over H is given by the differentials  s + t (1) s2 + t11 (2) 0 H d0 = 1 11 t12 (1) t12 (2) x11 + t22 and



s2 + t11 (2)  −s1 − t11 (1)  −t (1)x22 + t12 (2) d1H =  12 t12 (1)x21   0 t12 (2) + t12 (1)t22

0 0 x12 −x11 − t22 0 0

0 0 x21 0 −x11 − t22 0

0 x12

0 0 x22 + t22 0 0 −x11 − t22

0 x21

0 x22 + t22

0 0 0 x21 −x12 0



0 0 0 x22 + t22 0 −x12

This means that the versal family ξ ∈ Def M (H) is the H-A bimodule MH given by MH = coker(d0H )



0 0   0   0  x22 + t22  −x21

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65

with the induced right G-action. There is an obvious algebraization of H and its versal family, given by   k[t11 (1),t11 (2)] ht12 (1),t12 (2)i/( f12 ) H= 0 k[t22 ] and the corresponding versal family MH . The space X(k) of simple k-rational right H-modules in X = Simp(H) is given by the disjoint union X(k) = X1 (k) = A2 ∐ A1 = {(τ1 , τ2 ) ∈ k2 } ∐ {τ ∈ k} This follows from Proposition 2.17 since H11 , H22 are commutative and H21 = 0. The family of deformations parametrized by X(k) is given by {MH (τ1 , τ2 ) : (τ1 , τ2 ) ∈ k2 } ∐ {MH (τ ) : τ ∈ k} with MH (τ1 , τ2 ) ∼ = A/(s1 + τ1 , s2 + τ2 ),

MH (τ ) = A/(x11 + τ , x12 , x21 , x22 + τ )

The deformation MH (τ1 , τ2 ) corresponds to the orbit with Jordan form   λ1 1 with s1 = λ1 + λ2 = −τ1 and s2 = λ1 λ2 = −τ2 0 λ2 and the deformation MH (τ ) corresponds to the orbit with Jordan form   λ 0 with λ = −τ 0 λ From the relation f12 it follows that dimk Ext1H (MH (τ1 , τ2 ), MH (τ )) = 2 if and only if τ 2 + τ2 = 0 and 2τ − τ1 = 0. This is the case if and only if s21 = 4s2 and λ = s1 /2, or equivalently if the Jordan forms are given by     λ 1 λ 0 and 0 λ 0 λ for some λ ∈ k.

3.4 Noncommutative deformations of sheaves and presheaves In this section, we consider the noncommutative deformation functor Def F : ar → Sets of a family F = {F1 , F2 , . . . , Fr } of objects in an Abelian k-category A of presheaves or sheaves of modules. We describe the noncommutative deformation functor in concrete terms, and use this to describe an obstruction theory for the deformation functor using global Hochschild cohomology. We obtain a concrete description of the tangent space, and a method for computing the pro-representing hull explicitly; see also Eriksen [10].

3.4.1 Deformations of presheaves of modules Let A be a presheaf of associative algebras on a small category c. That is, A : c → Algk is a covariant functor. Then, the category A = PreSh(c, A ) of presheaves of right A -modules on c is an Abelian k-category, and we shall consider noncommutative deformations in this category. For any finite family F = {F1 , . . . , Fr } of presheaves of right A -modules on c, we consider the noncommutative deformation functor Def F : ar → Sets of the family F in the Abelian k-category A = PreSh(c, A ). We shall describe this functor in concrete terms.

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Noncommutative Deformation Theory

Let R ∈ ar , and consider a lifting FR of the family F to R. Without loss of generality, we may assume that FR (c) = (Ri j ⊗k F j (c)) with the natural left R-module structure for all c ∈ c. To describe the lifting completely, we must specify the right action of A (c) on (Ri j ⊗k F j (c)) for any object c ∈ c, and the restriction map FR (φ ) : (Ri j ⊗k F j (c)) → (Ri j ⊗k F j (c′ )) for any morphism φ : c → c′ in c. It is enough to specify the action of a ∈ A (c) on elements of the form ei ⊗ fi in Rii ⊗k Fi (c), and we must have (ei ⊗ fi ) · a = ei ⊗ ( fi · a) + ∑ ri j ⊗ f j′

(3.2)

f j′

with ∈ F j (c), ri j ∈ I(R)i j for all objects c ∈ c. Similarly, it is enough to specify the restriction map FR (φ ) on elements of the form ei ⊗ fi in Rii ⊗k Fi (c), and we must have FR (φ )(ei ⊗ fi ) = ei ⊗ Fi (φ )( fi ) + ∑ ri j ⊗ f j′ f j′

(c′ ),

(3.3)

→ c′

with ∈ F j ri j ∈ I(R)i j for all morphisms φ : c in c. R ′ ′ Let Q (c, c ) = (Homk (Fi (c), Ri j ⊗k F j (c ))) for all objects c, c′ ∈ c. To simplify notation, we write QR (c) = QR (c, c). There is a natural product QR (c, c′ ) ⊗k QR (c′ , c′′ ) → QR (c, c′′ ) for all objects c, c′ , c′′ ∈ c, given by composition of maps and multiplication in R, such that QR (c) is an associative algebra and QR (c, c′ ) is an QR (c)-QR (c′ ) bimodule in a natural way. Lemma 3.17. For any R ∈ ar , there is a bijective correspondence between the following data, up to equivalence, and Def F (R): 1. For any c ∈ c, an algebra homomorphism L(c) : A (c) → QR (c) that satisfies Equation (3.2). 2. For any morphism φ : c → c′ in c, an element L(φ ) ∈ QR (c, c′ ) that satisfies Equation (3.3) and L(φ )L(c) = L(c′ )L(φ ), 3. We have L(id) = id and L(φ )L(φ ′ ) = L(φ ′ ◦ φ ) for all morphisms φ : c → c′ , φ ′ : c′ → c′′ in c. For any R ∈ ar , there is a trivial deformation ∗R ∈ Def F (R) given by the following data: For any c ∈ c, we let L(c) : A (c) → QR (c) be given by L(c)(a)( fi ) = ei ⊗ f j a for all a ∈ A (c), fi ∈ Fi (c), and for any morphism φ : c → c′ in c, we let L(φ ) ∈ QR (c, c′ ) be given by L(φ )( fi ) = ei ⊗ Fi (φ )( fi ) for all fi ∈ Fi (c). Lemma 3.18. There is a bijection φi j : t(Def F )i j → Ext1A (Fi , F j ) for 1 ≤ i, j ≤ r that maps trivial deformations to split extensions. Proof. Let V = Mr (k) be the bimodule over kr with dimk Vi j = 1 for all i, j, and let {εi j } be a set of generators of this bimodule. For 1 ≤ i, j ≤ r, the tangent space t(Def F )i j = Def F (R), where R = T(V (i, j))2 with V (i, j) = k εi j ⊆ V . For any lifting FR of the family F to R, we consider the i’th row FRi of FR , given by c 7→ ei FR (c), which is a subpresheaf of FR of right A -modules on c since FRi (c) is invariant under L(c) and L(φ ) for any c ∈ c and any φ : c → c′ in c. Moreover, there is a natural exact sequence 0 → F j → FRi → Fi → 0 in PreSh(c, A ), since FRi (c) = Fi (c) ⊕ εi j F j (c) for all c ∈ c. Clearly, equivalent liftings of F to R give equivalent extensions in PreSh(c, A ), so FR 7→ FRi defines a map

φi j : t(Def F )i j → Ext1A (Fi , F j ) that maps trivial deformations to split extensions. To construct an inverse of φi j , we consider an extension 0 → F j → E → Fi → 0 in PreSh(c, A ), and define a lifting FR of the family F to R: We let ci ⊕ . . . ⊕F [j ⊕ · · · ⊕ Fr FR = E ⊕ F1 · · · ⊕ F

Then FR is a presheaf of right A -modules on c, and it is easy to see that it defines a lifting of Def F to R since E (c) ∼ = Fi (c) ⊕ F j (c) as k-linear vector spaces for any c ∈ c. It follows that the assignment E 7→ FR defines an inverse of φi j .

Noncommutative Deformation Theory

67

Global Hochschild cohomology For any presheaves F , G of right A -modules on c, we may consider the Hochschild complex HH• (A (c), Homk (F (c), G (c))) of A (c) with values in the bimodule Homk (F (c), G (c)) for any object c ∈ c. Unfortunately, this construction is not functorial in c. We shall consider a variation of this construction that is functorial, and use this to define a global Hochschild cohomology theory. Let Mor c denote the category of morphisms in c defined in the following way: An object in Mor c is a morphism in c, and given two objects f : c → c′ and g : d → d ′ in Mor c, a morphism (α , β ) : f → g in Mor c is a couple of morphisms α : d → c and β : c′ → d ′ in c such that β f α = g. Clearly, Morc is a small category. Let H omk (F , G ) be the presheaf on Mor c given by H omk (F , G )(φ ) = Homk (F (c), G (c′ )) for any morphism φ : c → c′ in c. Then, H omk (F , G ) is a presheaf of bimodules over A since Homk (F (c), F (c′ )) is a bimodule over A (c) via A (c) → A (c′ ) for any morphism φ : c → c′ in c. The Hochschild complex of A with values in H omk (F , G ) is defined to be the complex HC• (A , H omk (F , G )) of presheaves on Mor c given by HC p (A , H omk (F , G ))(φ ) = Homk (⊗kp A (c), Homk (F (c), G (c′ ))) for any morphism φ : c → c′ in c and any integer p ≥ 0, with differential given by d p (φ )( f )(a1 ⊗ · · · ⊗ a p+1) = a1 f (a2 ⊗ · · · ⊗ a p+1) p

+ ∑ (−1)i f (a1 ⊗ · · · ⊗ ai ai+1 ⊗ · · · ⊗ a p+1) i=1

+ (−1) p+1 f (a1 ⊗ · · · ⊗ a p ) a p+1 for any f ∈ HC p (A , H omk (F , G ))(φ ) and a1 , . . . , a p+1 ∈ A (c). For a fixed morphism φ : c → c′ in c, HC• (A , H omk (F , G ))(φ ) is the Hochschild complex of A (c) with values in the bimodule Homk (F (c), G (c′ )), and the definition of Mor c ensures that φ 7→ HH• (A , H omk (F , G ))(φ ) is functorial. n (F , G ) : Mor c → Mod for the composition of functors For any integer n ≥ 0, we write E xtA k • n given by φ 7→ H (HC (A , H omk (F , G ))(φ )). We see that n E xtA (F , G )(φ ) ∼ = ExtnA (c) (F (c), G (c′ ))

for any morphism φ : c → c′ in c. For any functor G : Mor c → Modk , we may consider the resolving complex D• (c, G) in Modk of the projective limit functor of G, see Laudal [27]. We recall that for any integer p ≥ 0, we have that D p (c, G) = ∏ G(φ p ◦ · · · ◦ φ1 ), c0 →···→c p

where the product is taken over all p-tuples (φ1 , . . . , φ p ) of composable morphisms φi : ci−1 → ci in c. The differential d p : D p (c, G) → D p+1 (c, G) is given by (d p g)(φ1 , . . . , φ p+1 ) = G(φ1 , id)(g(φ2 , . . . , φ p+1 )) p

+ ∑ (−1)i g(φ1 , . . . , φi+1 ◦ φi , . . . , φ p+1 ) i=1

+ (−1) p+1 G(id, φ p+1 )(g(φ1 , . . . , φ p )) for all g ∈ D p (c, G) and for all (p + 1)-tuples (φ1 , . . . , φ p+1 ) of composable morphisms φi : ci−1 → ci in c. We denote the cohomology of D• (c, G) by H• (c, G), and recall the following standard result:

68

Noncommutative Deformation Theory

Proposition 3.19. Let c be a small category and let G ∈ PreSh(Mor c, k) be presheaf of modules over the constant sheaf k. Then, the resolving complex D• (c, G) has the following properties: 1. D• (c, −) : PreSh(Mor c, k) → Compl(k) is exact (p) 2. H p (c, G) ∼ = lim ←− G for all G ∈ PreSh(Mor c, k) and for all p ≥ 0

In particular, H• (c, −) : PreSh(Mor c, k) → Modk is an exact δ -functor. For any functor C• : Mor c → Compl(k), we may consider the double complex D•• = D• (c, C∗ ) of vector spaces over k. Explicitly, we have that D pq = D p (c, Cq ) for all integers p, q with p ≥ 0, that pq pq dI : D pq → D p+1,q is the differential d p in D• (c, Cq ), and that dII : D pq → D p,q+1 is the differential pq p p q q q q+1 given by dII = (−1) D (c, d ), where d : C → C is the differential in C• . Note that if Cq = 0 •• for all q < 0, then D lies in the first quadrant. We define the global Hochschild complex of A with values in H omk (F , G ) on c to be the total complex of the double complex D•• = D• (c, HC• (A , H omk (F , G ))), and we denote this complex by HC• (c, A , H omk (F , G )). Moreover, we define the global Hochschild cohomology HH• (c, A , H omk (F , G )) of A with values in H omk (F , G ) on c to be the cohomology of the global Hochschild complex HC• (c, A , H omk (F , G )). q Proposition 3.20. There is a spectral sequence for which E2pq = H p (c, E xtA (F , G )), such that E∞ = gr HH• (c, A , H omk (F , G )), the associated graded vector space over k with respect to a suitable filtration of HH• (c, A , H omk (F , G )).

We remark that HH• (c, A , H omk (F , G )) can be calculated in concrete terms in many situations, using the above spectral sequence. We shall give examples of such computations in the end of this section. Obstruction theory for presheaves of modules We consider the noncommutative deformation functor Def F of a family F = {F1 , . . . , Fr } of presheaves of right A -modules on c. Using global Hochschild cohomology, we shall construct an obstruction theory for Def F with cohomology (HHn (c, A , H omk (Fi , F j ))). Proposition 3.21. Let u : R → S be a small surjection in ar with kernel K, and let FS ∈ Def F (S) be a deformation. Then, there exists a canonical obstruction o(u, FS ) ∈ (Ki j ⊗k HH2 (c, A , H omk (Fi , F j ))) such that o(u, FS ) = 0 if and only if there exists a deformation FR ∈ Def F (R) lifting FS to R. Moreover, if o(u, FS ) = 0, then there is a transitive and effective action of (Ki j ⊗k HH1 (c, A , H omk (Fi , F j ))) on the set of liftings of FS to R. Proof. Let FS ∈ Def F (S) be given. By Lemma 3.17, this deformation corresponds to the following data: A k-algebra homomorphism LS (c) : A (c) → QS (c) for each object c ∈ c and an element LS (φ ) ∈ QS (c, c′ ) for each morphism φ : c → c′ in c, such that the conditions of Lemma 3.17 are satisfied. Moreover, to lift FS to R is the same as to lift these data to R. Choose a k-linear section σ : S → R such that σ (I(S)i j ) ⊆ I(R)i j and σ (ei ) = ei for 1 ≤ i, j ≤ r. Clearly, σ induces a k-linear map Q(σ , c, c′ ) : QS (c, c′ ) → QR (c, c′ ) for all c, c′ in c. We define LR (c) = Q(σ , c, c) ◦ LS (c) for all objects c ∈ c and LR (φ ) = Q(σ , c, c′ )(LS (c, c′ )) for all morphisms φ : c → c′ in c. Then, LR (c) : A (c) → QR (c) is a k-linear map which lifts LS (c) to R for all c ∈ c, and we define the obstruction o(0, 2)(c) : A (c) ⊗k A (c) → QK (c)

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69

by o(0, 2)(c)(a ⊗ b) = LR (c)(ab) − LR (c)(a) LR (c)(b) for all a, b ∈ A (c). It is clear that LR (c) is a k-algebra homomorphism if and only if o(0, 2)(c) = 0. Moreover, LR (φ ) lifts LS (φ ) to R for all morphisms φ : c → c′ in c, and we define the obstruction o(1, 1)(φ ) : A (c) → QK (c, c′ ) by o(1, 1)(φ )(a) = LR (φ ) ◦ LR (c)(a) − LR (c′ )(A (φ )(a)) ◦ LR (φ ) for all a ∈ A (c). It is clear that LR (φ ) is A (φ )-linear if and only if o(1, 1)(φ ) = 0. Finally, we define the obstruction o(2, 0)(φ , φ ′ ) ∈ QK (c, c′′ ) by o(2, 0)(φ , φ ′ ) = LR (φ ′ ) LR (φ ) − LR (φ ′ ◦ φ ) for all morphisms φ : c → c′ and φ ′ : c′ → c′′ in c. It is clear that LR satisfies the cocycle condition if and only if o(2, 0) = 0. We see that o = (o(0, 2), o(1, 1), o(2, 0)) is a 2-cochain in (Ki j ⊗k HC• (c, A , H omk (Fi , F j ))). A calculation shows that o is a 2-cocycle, and that its cohomology class o(u, FS ) ∈ (Ki j ⊗k HH2 (c, A , H omk (Fi , F j ))) is independent of the choice of LR (c) and LR (φ ). It is clear that if there is a lifting of FS to R, we may choose LR (c) and LR (φ ) such that o = 0, hence o(u, FS ) = 0. Conversely, assume that o(u, FS ) = 0. Then, there exists a 1-cochain of the form (ε , ∆) with ε ∈ D01 and ∆ ∈ D10 such that d(ε , ∆) = o. Let L′ (c) = LR (c) + ε (c) and L′ (φ ) = LR (φ ) + ∆(φ ). Then, L′ (c) is another lifting of LS (c) to R, L′ (φ ) is another lifting of LS (φ ) to R, and essentially the same calculation as above shows that the corresponding 2-cocycle o′ = 0. Hence, there is a lifting of FS to R, and this proves the first part of the proposition. For the second part, assume that FR is a lifting of F to R. Then, FR is defined by liftings LR (c) and LR (φ ) to R such that the corresponding 2-cocycle o = 0. Let us consider a 1-cochain (ε , ∆), and consider the new liftings L′ (c) = LR (c) + ε (c) and L′ (φ ) = LR (φ ) + ∆(φ ). From the previous calculations, it is clear that the new 2-cocycle o′ = 0 if and only if (ε , ∆) is a 1-cocycle. Moreover, if this is the case, the lifting FR′ defined by L′ (c) and L′ (φ ) is equivalent to FR if and only if (ε , ∆) is a 1-coboundary, since an equivalence between FR and FR′ must have the form id +π for some 0-cochain π with d(π ) = (ε , ∆). We see that the obstruction o(u, FS ) is functorial, so it defines an obstruction theory for the noncommutative deformation functor Def F : ar → Sets by definition. If the condition dimk HHn (c, A , H omk (Fi , F j )) < ∞ for 1 ≤ i, j ≤ r, n = 1, 2, holds, it follows that Def F has an obstruction theory with finite dimensional cohomology (HHn (c, A , H omk (Fi , Fi ))). Proposition 3.22. Let c be a small category, let A be a presheaf of k-algebras on c, and let F = {F1 , . . . , Fr } be a finite family of presheaves of right A -modules on c. Then, HH1 (c, A , H omk (Fi , F j )) ∼ = t(Def F )i j ∼ = Ext1A (Fi , F j ) for 1 ≤ i, j ≤ r. Theorem 3.23. Let c be a small category, let A be a presheaf of associative k-algebras on c, and let F = {F1 , . . . , Fr } be a finite family of presheaves of right A -modules on c. If the finiteness condition dimk HHn (c, A , H omk (Fi , F j )) < ∞ for 1 ≤ i, j ≤ r, n = 1, 2 holds, then the noncommutative deformation functor Def F : ar → Sets of F in PreSh(c, A ) has a pro-representing hull H(Def F ), completely determined by the generalised matric Massey products on (HHn (c, A , H omk (Fi , F j ))) induced by the obstruction morphism.

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3.4.2 Deformations of quasi-coherent sheaves of modules Let X be a topological space, and let A be a sheaf of associative k-algebras on X. Then (X, A ) is called a ringed space over k, and we may consider the category Sh(X, A ) of sheaves of right A -modules on X. We recall that a sheaf F of right A -modules on X is called quasi-coherent if for every point x ∈ X, there exists an open neighbourhood U ⊆ X of x, free sheaves L0 , L1 of right A |U -modules on U, and an exact sequence 0 ← F |U ← L0 ← L1 of sheaves of right A |U -modules on U. The category QCoh(X, A ) of quasi-coherent sheaves of right A -modules on X is the full subcategory of Sh(X, A ) consisting of quasi-coherent sheaves. The full subcategory QCoh(X, A ) ⊆ Sh(X, A ) is closed under finite direct sums, but it is not clear if QCoh(X, A ) is closed under kernels and cokernels in general. Hence QCoh(X, A ) is an additive but not necessarily an Abelian k-category. We recall that when U is an open cover of X, we may consider U as a small category, with a morphism U → V if V ⊆ U, and no morphisms otherwise. Then, any sheaf on X can be considered as a presheaf on U. In particular, there is a natural inclusion QCoh(X, A ) ⊆ PreSh(U, A ). We shall give sufficient conditions on U for QCoh(X, A ) to be an exact Abelian subcategory of PreSh(U, A ), and consider noncommutative deformations in the category QCoh(X, A ) in these cases. Let Γ(X, −) : Sh(X, A ) → ModA denote the functor of global sections, with A = Γ(X, A ). This functor is left exact, and we denote its right derived functors by H• (X, −) = R• Γ(X, −). We say that X is A -affine if the following conditions hold: 1. Γ(X, −) induces an equivalence of categories QCoh(X, A ) → ModA 2. Hn (X, F ) = 0 for all F ∈ QCoh(X, A ) and all integers n ≥ 1 Moreover, we say that an open subset U ⊆ X is A -affine if U is A |U -affine, and that an open cover U of X is A -affine if U is A -affine for any U ∈ U. An open cover U of X is good if any finite intersection V = U1 ∩U2 ∩ · · · ∩Un with Ui ∈ U for 1 ≤ i ≤ n can be covered by open subsets W ⊆ V with W ∈ U. In particular, any open cover closed under finite intersections is good. Proposition 3.24. If U is a good A -affine open cover of X , then the natural forgetful functor π : QCoh(X, A ) → PreSh(U, A ) is a full embedding, and π identifies QCoh(X, A ) with an exact Abelian subcategory of PreSh(U, A ) that is closed under extensions. Proof. It is clear that π is a full embedding, and π is exact since H1 (U, F ) = 0 for all U ∈ U, F ∈ QCoh(X, A ). It is therefore enough to show that for any exact sequence 0 → F ′ → F → F ′′ → 0 in PreSh(U, A ) with F ′ = π (G ′ ), F ′′ = π (G ′′ ) for G ′ , G ′′ ∈ QCoh(X, A ), there is a quasicoherent sheaf G ∈ QCoh(X, A ) such that π (G ) = F . Since U is an A -affine open cover, we can find G U ∈ QCoh(U, A |U ) with Γ(U, G U ) = F (U) for all U ∈ U. For any inclusion U ⊇ V in U, there is a natural isomorphism G U |V → G V of quasi-coherent sheaves on V since we have that G ′ , G ′′ ∈ QCoh(X, A ). Hence, it follows from the fact that U is a good cover of X that the quasi-coherent sheaves {G U ∈ QCoh(U, A |U ) : U ∈ U} can be glued to a quasi-coherent sheaf G ∈ QCoh(X, A ) with π (G ) = F . Let U be a good A -affine open cover of X, and let F = {F1 , . . . , Fr } be a finite family of quasicoherent sheaves of right A -modules on X. Since QCoh(X, A ) is an exact Abelian subcategory

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of PreSh(U, A ) by Proposition 3.24, we may consider the noncommutative deformation functor Def qc F : ar → Sets of F as a family of objects in the Abelian k-category A = QCoh(X, A ). We may also consider the noncommutative deformation functor Def U F : ar → Sets of F as a family of objects in the Abelian k-category A = PreSh(U, A ). Proposition 3.25. Let F = {F1 , . . . , Fr } be a family of quasi-coherent sheaves in QCoh(X, A ). If U is a good A -affine open cover of X, then the forgetful functor π : QCoh(X, A ) → PreSh(U, A ) qc induces an isomorphism Def F → Def U F of deformation functors. Proof. Clearly, π induces a morphism of noncommutative deformation functors, and it is enough U to show that the induced map of sets πR∗ : Def qc F (R) → Def F (R) is a bijection for any R ∈ ar . If X is A -affine and U = {X}, then PreSh(U, A ) is naturally equivalent to ModA , so π is an equivalence of categories, and this implies that πR is a bijection for any R ∈ ar . In the general case, let FR ∈ Def U F (R). Then FR (U) is a deformation of the family {F1 (U), . . . , Fr (U)} in ModA (U) to R for any U ∈ U. By the result in the A -affine case, we can find a deformation FRU of the family {F1 |U , . . . , Fr |U } in QCoh(U, A |U ) to R that is compatible with FR (U). We remark that if V ⊆ U is an inclusion in U, then there is a natural isomorphism FRU |V → FRV of sheaves of left A -modules on V , since F is a family of quasi-coherent sheaves of A -modules on X. We must glue the local deformations FRU to a deformation FR of the family F to R in QCoh(A ), and this is clearly possible since U is a good open cover of X. Proposition 3.26. Let (X, A ) be a ringed space over k, and let U be a good A -affine open cover of X . Then, the tangent space qc t(Def F )i j ∼ = HH1 (U, A , H omk (Fi , F j )) ∼ = Ext1QCoh(X,A ) (Fi , F j )

for any finite family F = {F1 , . . . , Fr } of quasi-coherent right A -modules on X . Theorem 3.27. Let (X, A ) be a ringed space over k, let U be a good A -affine open cover of X , and let F = {F1 , . . . , Fr } be a family of quasi-coherent right A -modules in QCoh(X, A ). If the finiteness condition dimk HHn (U, A , H omk (Fi , F j )) < ∞ for 1 ≤ i, j ≤ r, n = 1, 2 holds, then the noncommutative deformation functor Def qc F : ar → Sets of F in QCoh(X, A ) has a qc pro-representing hull H(Def F ), completely determined by the generalised matric Massey products on (HHn (U, A , H omk (Fi , F j ))) induced by the obstruction morphism.

3.4.3 Quasi-coherent ringed schemes Let (X, A ) be a ringed space, and consider the category QCoh(X, A ) of quasi-coherent right A -modules on X. By the results above, this category is an Abelian k-category when there is a good qc A -affine open cover U of X. Moreover, the noncommutative deformation functor Def F : ar → Sets of a family F in QCoh(X, A ) has an obstruction theory in this case. Its cohomology is the global Hochschild cohomology H p = (HH p (U, A , H omk (Fi , F j ))) We are therefore interested in examples of ringed spaces (X, A ) with good A -affine open covers U. We shall give several examples in this section. Example 3.6. Let (X, OX ) be a scheme over k. If U ⊆ X is an open affine subscheme of X, then U is OX -affine by Hartshorne [18], Corollary II.5.5 and Grothendieck [12], Theorem 1.3.1. Hence any open affine cover of X is an OX -affine open cover. If X is separated over k, then any finite intersection of open affine subschemes of X is affine. Hence, if X is quasi-compact and separated over k, then there is a finite OX -affine open cover of X closed under intersections.

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A ringed scheme over k is a ringed space (X, A ) over k defined by a scheme (X, OX ) over k and a morphism i : OX → A of sheaves of associative k-algebras on X. A quasi-coherent ringed scheme over k is a ringed scheme (X, A ) over k such that A is quasi-coherent as a left and right OX module. This notion was introduced in Yekutieli and Zhang [53], and the following result follows from Yekutieli and Zhang [53], Corollary 5.13: Lemma 3.28. A ringed scheme (X, A ) over k is quasi-coherent if and only if the morphism A (U) → A (D( f )) is a ring of fractions with respect to S = { f n : n ≥ 0} for any open affine subscheme U ⊆ X and any f ∈ OX (U). For any quasi-coherent ringed scheme (X, A ) over k, a right A -module F is quasi-coherent if and only if it is quasi-coherent as a right OX -module. This follows from Grothendieck [17], Proposition 9.6.1 when A is a sheaf of commutative rings on X, and the proof can easily be extended to the noncommutative case. Lemma 3.29. Let (X, A ) be a quasi-coherent ringed scheme over k. If U ⊆ X is an open affine subscheme of X, then U is A -affine. Proof. Write A = A (U), and consider Γ(U, −) : QCoh(U, A |U ) → ModA . We claim that Γ(U, −) is an equivalence of categories. By the comments preceding this lemma, QCoh(U, A |U ) can be considered as a subcategory of QCoh(U, OU ), and Γ(U, −) : QCoh(U, OU ) → ModO is an equivalence of categories with O = OX (U). So the claim follows from Lemma 3.28. Finally, Hn (U, F ) = 0 for any integer n ≥ 1 and any F ∈ QCoh(U, A |U ) by the above comments. Let (X, A ) be a quasi-coherent ringed scheme over k. Then, any open affine cover of X is an A -affine open cover of X. If X is quasi-compact and separated over k, then there is a finite A -affine open cover of X closed under intersections. Let (X, A ) be a quasi-coherent ringed scheme over k, and assume that char(k) = 0. We say that A is a D-algebra, and that (X, A ) is a D-scheme, if the following condition holds: For any open subset U ⊆ X and for any section a ∈ A (U), there exists an integer n ≥ 0 such that [. . . [ [a, f1 ], f2 ] . . . , fn ] = 0 for all sections f1 , . . . , fn ∈ OX (U), where [a, f ] = a f − f a is the usual commutator for a ∈ A (U), f ∈ OX (U). The notion of D-schemes was considered in Beilinson and Bernstein [3], and most quasi-coherent ringed schemes that appear naturally are D-schemes. We give some important examples of D-schemes below. Example 3.7. Let (X, OX ) be a scheme over k, and assume that char(k) = 0. For any sheaf F of OX -modules, let Diff(F ) be the sheaf of k-linear differential operators. By definition, Diff(F ) is a sheaf of associative k-algebras on X, equipped with a morphism i : OX → Diff(F ) of sheaves of rings; see Grothendieck [13], Section 16.8, and Diff(F ) is a D-algebra on X if and only if Diff(F ) is quasi-coherent as a left and right OX -module. By Example 1.1.6 in Beilinson and Bernstein [3], this is the case if F is a coherent OX -module. In particular, we may consider the sheaf DX = Diff(OX ) of k-linear differential operators on X when X is locally Noetherian. Since OX is a coherent sheaf of rings in this case, it follows that DX is a D-algebra on X. We remark that there are some examples of schemes over k that are DX -affine but not affine. For instance, this holds for the projective space X = Pn for n ≥ 1, see Beilinson and Bernstein [4]. It also holds for the weighted projective space X = P(a1 , . . . , an ); see Van den Bergh [48].

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Example 3.8. Let (X, OX ) be a separated scheme of finite type over k, and assume that char(k) = 0. A Lie algebroid of X is a quasi-coherent OX -module g with a k-Lie algebra structure, together with a morphism τ : g → Derk (OX ) of sheaves of OX -modules and of k-Lie algebras, such that [g, f · h] = f [g, h] + τg ( f ) · h for any open subset U ⊆ X and any sections g, h ∈ g(U), f ∈ OX (U). The notion of Lie algebroids in algebraic geometry was considered in Beilinson and Bernstein [3]. For any sheaf F of OX -modules, an integrable g-connection on F is a morphism of sheaves of OX -modules and of k-Lie algebras ∇ : g → E ndk (F ) such that ∇U (g)( f m) = f ∇U (g)(m) + g( f ) m for any open subset U ⊆ X and any sections f ∈ OX (U), g ∈ g(U), m ∈ F (U). The quasi-coherent sheaves of OX -modules with integrable g-connections form an Abelian k-category, and there is a universal enveloping D-algebra U(g) of g such that this category is equivalent to QCoh(U, U(g)); see Beilinson and Bernstein [3]. In particular, (X, U(g)) is a D-scheme. The tangent sheaf θX = Derk (OX ) of X is a Lie algebroid of X in a natural way, and U(θX ) is a subsheaf of the sheaf DX of k-linear differential operators on X. If X is a smooth, irreducible quasi-projective variety over k, then U(θX ) = DX .

3.4.4 Calculations for D-modules on elliptic curves Let k be an algebraically closed field of characteristic 0, and let X be a smooth irreducible variety over k of dimension d. Then, the sheaf DX of k-linear differential operators on X is a D-algebra on X. We consider the noncommutative deformations of OX as a quasi-coherent left DX -module via the natural left action of DX on OX . As an example, we compute the pro-representing hull of OX as a quasi-coherent left DX -module when X is an elliptic curve; see also Eriksen [9]. Even though we have chosen to consider noncommutative deformations of right modules throughout this book, we shall write this example as a deformation of the quasi-coherent sheaf OX of left DX -modules. We could have transformed OX into a sheaf of right modules over the opposite sheaf of rings, but we shall not carry out this exercise. It is after all more natural to view differential operators as operating from the left. It would also have been possible to develop a noncommutative deformation theory for left modules, see for instance Eriksen [8] and [10], and it is not difficult to see that such a deformation theory would fit with the exposition of this example. Let U be an open affine cover of X. Then U ⊆ X is a smooth, irreducible affine variety over k of dimension d for all U ∈ U. It is wellknown that DX (U) is a simple Noetherian ring of global dimension d and that OX (U) is a simple left DX (U)-module; see Smith and Stafford [46]. Hence q E xtD (OX , OX ) = 0 for q ≥ d + 1 and E ndDX (OX ) = k. If X is a curve, then the spectral sequence X in Proposition 3.20 degenerates, and 1 HHn (U, DX , E ndk (OX )) ∼ (OX , OX )) for n ≥ 1 = Hn−1 (U, E xtD X 0 ∼ HH (U, DX , E ndk (OX )) = k

Let X = V ( f ) ⊆ P2 be the irreducible projective plane curve given by the homogeneous equation f = y2 z− x3 − axz2 − bz3 = 0 for fixed parameters (a, b) ∈ k2 . We assume that ∆ = 4a3 + 27b2 6= 0, so that X is smooth and therefore an elliptic curve over k. We choose an open affine cover U of X closed under intersections, given by U = {U1 ,U2 ,U3 } where U1 = D+ (y), U2 = D+ (z) and U3 = U1 ∩U2 . 1 (O , O ) and Hn−1 (U, E xt 1 (O , O )) for n = 1, 2. We shall compute E xtD X X X X DX X Let Ai = OX (Ui ) and Di = DX (Ui ) for all i. We see that A1 ∼ = k[x, z]/( f1 ) and A2 ∼ = k[x, y]/( f2 ), 3 2 3 2 3 where f1 = z − x − axz − bz and f2 = y − x − ax − b. Moreover, we have that Derk (Ai ) = Ai ∂i

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and Di = Ai h∂i i for i = 1, 2, where

∂1 = (1 − 2axz − 3bz2) ∂ /∂ x + (3x2 + az2) ∂ /∂ z ∂2 = −2y ∂ /∂ x − (3x2 + a) ∂ /∂ y On the intersection U3 = U1 ∩ U2 , we choose an isomorphism A3 ∼ = k[x, y, y−1 ]/( f3 ) with f3 = f2 , and see that Derk (A3 ) = A3 ∂3 and D3 = A3 h∂3 i for ∂3 = ∂2 . The restriction maps of OX and DX , considered as presheaves on U, are given by x 7→ xy−1 , z 7→ y−1 , ∂1 7→ ∂2 for the inclusion U1 ⊇ U3 , and the natural localization map for U2 ⊇ U3 . Finally, we find a free resolution of Ai as a left Di -module for i = 1, 2, 3, given by ·∂

0 ← Ai ← Di ←−i Di ← 0 and use this to compute Ext1Di (Ai , A j ) ∼ = coker(∂i |U j : A j → A j ) for all Ui ⊇ U j in U. We see that Ext1Di (Ai , A3 ) ∼ = coker(∂3 : A3 → A3 ) is independent of i, and find the following k-linear bases for Ext1Di (Ai , A j ):

U1 ⊇ U1 U2 ⊇ U2 U3 ⊇ U3

a 6= 0 : 1, z, z2 , z3 1, y2 x2 y−1 , 1, y−1 , y−2 , y−3

a=0: 1, z, x, xz 1, x x2 y−1 , 1, y−1 , x, xy−1

1 (O , O ) : Mor U → Mod defines the following diagram in Mod , where the The functor E xtD X X k k X maps are induced by the restriction maps on OX :

Ext1D1 (A1 , A1 )  Ext1D1 (A1 , A3 )

Ext1D2 (A2 , A2 )

Ext1D3 (A3 , A3 )

 Ext1D2 (A2 , A3 )

We use that 15y2 = ∆ y−2 in Ext1D3 (A3 , A3 ) when a 6= 0 and that −3b xy−2 = x in Ext1D3 (A3 , A3 ) when a = 0 to describe these maps in the given bases, and compute Hn−1 (U, Ext1DX (OX , OX )) for n = 1, 2 using the resolving complex D• (U, −). We find the following k-linear bases:

a 6= 0 : n = 1 ξ1 = (1, 1, 1), ξ2 = (∆z2 , 15y2 , ∆y−2 ) n = 2 ω = (0, 0, 0, 0, 6ax2 y−1 )

a=0: ξ1 = (1, 1, 1), ξ2 = (−3b xz, x, x) ω = (0, 0, 0, 0, x2 y−1 )

We recall that ξ1 , ξ2 and ω are represented by cocycles of degree p = 0 and p = 1 in the resolving 1 (O , O )), where complex D• (U, E xtD X X X 1 D p (U, E xtD (OX , OX )) = X

∏

U0 ⊇···⊇Up

1 E xtD (OX , OX )(U0 ⊇ U p ) X

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and the product is indexed by the inclusions {U1 ⊇ U1 ,U2 ⊇ U2 ,U3 ⊇ U3 } when p = 0, and by the inclusions {U1 ⊇ U1 ,U2 ⊇ U2 ,U3 ⊇ U3 ,U1 ⊇ U3 ,U2 ⊇ U3 } when p = 1. This proves that the noncommutative deformation functor Def X : a1 → Sets of the quasicoherent left DX -module OX has tangent space HH1 (U, DX , E ndk (OX )) ∼ = k2 and obstruction 2 space HH (U, DX , E ndk (OX )) ∼ = k for any elliptic curve X over k, and a pro-representing hull H = k≪t1 ,t2 ≫/(F) for some noncommutative power series F ∈ k≪t1 ,t2 ≫. We shall compute the noncommutative power series F and the versal family FH ∈ Def OX (H) using the obstruction calculus. We choose base vectors t1∗ ,t2∗ in HH1 (U, A , E ndk (OX )), and representatives (ψl , τl ) ∈ D01 ⊕ D10 of tl∗ for l = 1, 2, where D pq = D p (U, HCq (A , E ndk (OX ))). We may choose ψl (Ui ) to be the derivation defined by ( 0 if Pi ∈ Ai ψl (Ui )(Pi ) = ξl (Ui ) · idAi if Pi = ∂i for l = 1, 2 and i = 1, 2, 3, and τl (Ui ⊇ U j ) to be the multiplication operator in HomAi (Ai , A j ) ∼ = Aj given by τ1 = 0, τ2 (Ui ⊇ Ui ) = 0 for i = 1, 2, 3 and

a 6= 0 : τ2 (U1 ⊇ U3 ) = 0 τ2 (U2 ⊇ U3 ) = −4a2y−1 − 3xy + 9bxy−1 − 6ax2y−1

a=0: τ2 (U1 ⊇ U3 ) = x2 y−1 τ2 (U2 ⊇ U3 ) = 0

The restriction of Def OX to a1 (2) is represented by (H2 , FH2 ), where H2 = kht1 ,t2 i/(t1 ,t2 )2 and the deformation FH2 ∈ Def OX (H2 ) is defined by FH2 (Ui ) = Ai ⊗k H2 as a right H2 -module for i = 1, 2, 3, with left Di -module structure given by Pi (mi ⊗ 1) = Pi (mi ) ⊗ 1 + ψ1(Ui )(Pi )(mi ) ⊗ t1 + ψ2 (Ui )(Pi )(mi ) ⊗ t2 for i = 1, 2, 3 and for all Pi ∈ Di , mi ∈ Ai , and with restriction map for the inclusion Ui ⊇ U j given by mi ⊗ 1 7→ mi |U j ⊗ 1 + τ2(Ui ⊇ U j ) mi |U j ⊗ t2 for i = 1, 2, j = 3 and for all mi ∈ Ai . Let us attempt to lift the family FH2 ∈ Def X (H2 ) to R = k≪t1 ,t2 ≫/(t1 ,t2 )3 . We let FR be the right R-module given by FR (Ui ) = Ai ⊗k R as a right R-module for i = 1, 2, 3, with left Di -module structure given by Pi (mi ⊗ 1) = Pi (mi ) ⊗ 1 + ψ1(Ui )(Pi )(mi ) ⊗ t1 + ψ2 (Ui )(Pi )(mi ) ⊗ t2 for i = 1, 2, 3 and for all Pi ∈ Di , mi ∈ Ai , and with restriction map for the inclusion Ui ⊇ U j given by τ2 (Ui ⊇ U j )2 mi ⊗ 1 7→ mi |U j ⊗ 1 + τ2(Ui ⊇ U j ) mi |U j ⊗ t2 + mi |U j ⊗ t22 2 for i = 1, 2, j = 3 and for all mi ∈ Ai . We see that FR (Ui ) is a left DX (Ui )-module for i = 1, 2, 3, and that t1t2 − t2t1 = 0 is a necessary and sufficient condition for DX -linearity of the restriction maps for the inclusions U1 ⊇ U3 and U2 ⊇ U3 . This implies that FR is not a lifting of FH2 to R. But if we consider the quotient H3 = R/(t1t2 − t2t1 ), we see that the family FH3 ∈ Def OX (H3 ) induced by FR is a lifting of FH2 to H3 . In fact, we claim that the restriction of Def OX : a1 → Sets to a1 (3) is represented by (H3 , FH3 ). One way to prove this is to show that it is not possible to find any lifting FR′ ∈ Def OX (R) of FH2 to

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R. Another approach is to calculate the cup products < ti∗ ,t ∗j > in global Hochschild cohomology for i, j = 1, 2, and this gives < t1∗ ,t2∗ >= o∗ , < t2∗ ,t1∗ >= −o∗ < t1∗ ,t2∗ >= o∗ , < t2∗ ,t1∗ >= −o∗

for a 6= 0 for a = 0

where o∗ ∈ HH2 (U, DX , E ndk (OX )) is the base vector corresponding to ω . All other cup products vanish, and this implies that F = t1t2 − t2t1 mod (t1 ,t2 )3 . Let H = k≪t1 ,t2 ≫/(t1t2 −t2t1 ). We shall show that there is a lifting FH ∈ Def OX (H) of FH3 to b k H as a right H-module for i = 1, 2, 3, with left Di -module structure given H: We let FH (Ui ) = Ai ⊗ by Pi (mi ⊗ 1) = Pi (mi ) ⊗ 1 + ψ1(Ui )(Pi )(mi ) ⊗ t1 + ψ2 (Ui )(Pi )(mi ) ⊗ t2 for i = 1, 2, 3 and for all Pi ∈ Di , mi ∈ Ai , and with restriction map for the inclusion Ui ⊇ U j given by ∞ τ2 (Ui ⊇ U j )n m1 |U j ⊗ t2n = exp(τ2 (Ui ⊇ U j ) ⊗ t2 ) · (m1 |U j ⊗ 1) mi ⊗ 1 7→ ∑ n! n=0 for i = 1, 2, j = 3 and for all mi ∈ Ai . This implies that (H, FH ) is the pro-representing hull of Def OX and that F = t1t2 − t2t1 . We remark that the versal family FH does not admit an algebraization.

3.5 Matric Massey products and A-infinity structures Let Def X be the noncommutative deformation functor of a family X of algebraic objects in an Abelian k-category A. If there is an obstruction theory for Def X with cohomology {H p } and X is a swarm, then we know that Def X has a pro-representing hull H and that it is determined by certain generalised matric Massey products Mno : Dn → H2 induced by the obstruction theory. In this section, we consider the matric Massey products on the cohomology H• (C) of a matric differential graded algebra C. These matric Massey products were introduced in May [36] (see also Kraines [21]). When the multiplication in C is compatible with obstruction theory for Def X and the cohomology H p (C) ∼ = H p , we describe the relationship between the matric Massey products in the sense of May and the generalised matric Massey products Mno : Dn → H2 from Section 3.2. In this setting, we also consider matric A-infinity structures on the matric differential graded algebra H• (C). In particular, we consider the Merkulov models, with nontrivial higher multiplications on H• (C), and describe their relationship with matric Massey products in the sense of May, and therefore also the generalised matric Massey products Mno : Dn → H2 .

3.5.1 Matric Massey products on differential graded algebras We say that C is a matric differential graded algebra over kr if it is a matric algebra over kr with a grading C = ⊕ p C p such that C p ·Cq ⊆ C p+q for all p, q, and a graded differential d : C → C of degree 1 of bimodules over kr such that d 2 = 0 and d(ab) = d(a) · b + (−1)q a · d(b) for all a ∈ C p , b ∈ Cq . A matric differential graded algebra C has cohomology H• (C) = ker(d)/ im(d), and H p (C) = ⊕ H p (Ci j ) = (H p (Ci j )) i, j

for all p ≥ 0 since d(Ci j ) ⊆ Ci j . Moreover, the graded matric multiplication on C induces a graded matric multiplication on the cohomology H• (C).

Noncommutative Deformation Theory

77

Let C be a matric differential graded algebra over kr , and consider its cohomology H• (C). We shall define the n-fold matric Massey products hα1 , α2 , . . . , αn i of the cohomology classes α1 , α2 , . . . , αn ∈ H• (C) for n ≥ 2. In principle, one may think of these matric Massey products as graded homomorphisms of kr -bimodules h−, −, . . . , −i : H• (C) ⊗ H• (C) ⊗ · · · ⊗ H•(C) 99K H• (C) of degree 2 − n, where the tensor products are taken over kr . For n = 2, the two-fold matric Massey product is (up to a sign) the multiplication h−, −i : H p (C) ⊗ Hq(C) → H p+q(C) on cohomology induced by the multiplication in C, and it is well-defined and everywhere defined. However, the n-fold matric Massey products are in general neither well-defined nor everywhere defined for n ≥ 3. We shall assume that there are integers d1 , d2 , . . . , dn and an ordered sequence of integers (i0 , i1 , . . . , in ) in {1, 2, . . . , r} such that αl ∈ H• (C) has degree dl and position (il−1 , il ) for 1 ≤ l ≤ n. Then αl is represented by a cocycle al−1,l ∈ C of the same degree and position for 1 ≤ l ≤ n. The matric Massey product hα1 , α2 , . . . , αn i is defined if the family {al−1,l : 1 ≤ l ≤ n} of cocycles can be extended to a family a = {alm ∈ C : 0 ≤ l < m ≤ n, (l, m) 6= (0, n)} of homogeneous elements alm ∈ C of degree dlm = dl+1 + · · · + dm + 1 − (m − l) and position (il , im ) such that d(alm ) = aelm , where we put m−1

aelm =

∑

s=l+1

m−1

als · asm =

∑

(−1)1+dls als · asm

(3.4)

s=l+1

for 0 ≤ l < m ≤ n. In this case, we call a = {alm : 0 ≤ l < m ≤ n, (l, m) 6= (0, n)} a defining system for the matric Massey product hα1 , α2 , . . . , αn i. It is not difficult to see that if a is a defining system for hα1 , α2 , . . . , αn i, then the elements aelm are cocycles for 0 ≤ l < m ≤ n. In particular, ae0,n is a cocycle of degree d1 + d2 + · · · + dn + 2 − n and position (i0 , in ). We define the matric Massey product hα1 , α2 , . . . , αn i to be the set of cohomology classes hα1 , α2 , . . . , αn i = {[e a0,n ] ∈ H• (C) : a is a defining system} (3.5) whenever there are defining systems for hα1 , α2 , . . . , αn i. Remark 3.5. We often consider matric Massey products of cohomology classes of degree 1. This corresponds to the case d1 = d2 = · · · = dn = 1, and implies that the matric Massey product hα1 , α2 , . . . , αn i ∈ H2 (C). Remark 3.6. The matric Massey product hα1 , α2 i is called the matric cup product, and it is always defined and uniquely defined since it is given by the cocycle (−1)1+d1 a01 · a12 The n-fold matric Massey product hα1 , α2 , . . . , αn i for n ≥ 3 is only defined when all matric Massey products hαl , αl+1 , . . . , αm i of length m − l + 1 < n contain 0. Remark 3.7. The matric Massey products defined above is a special case of the matric Massey products in the sense of May, and we refer to May [36] for details concerning the properties of matric Massey products; see also Lu et al. [33]. However, note that May considers products of matrices with entries in the cohomology algebra H• (C) defined over a commutative ring Λ, while we consider products of elements in the algebra H• (C) = (H• (C)i j ), a matric algebra. In May’s notation, we consider the case where Λ = kr and all matrices have dimension 1 × 1.

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Noncommutative Deformation Theory

3.5.2 Matric Massey products and obstruction calculus Let X = {X1 , . . . , Xr } be a family of objects in an Abelian k-category A, and consider the noncommutative deformation functor Def X : ar → Sets. We assume that Def X has an obstruction theory with cohomology {H p } and that X is a swarm. In that case, it follows from Corollary 3.7 that Def X has a pro-representing hull H, and it is completely determined by the generalised matric Massey products Mno : Dn → H2 induced by the obstruction theory of Def X . We shall furthermore assume that there is a matric differential graded algebra C over kr such that p H (C) ∼ = H p for p = 1, 2 and such that the generalised matric Massey products Mno : Dn → H2 are compatible with the matric Massey products of H• (C) in the sense of May. Under these assumptions, we shall describe an algorithm for computing the pro-representing hull H using the matric Massey products of H• (C). Remark 3.8. When M is a family of right modules over an associative k-algebra A, the Yoneda complex and the Hochschild complex of M = ⊕i Mi are matric differential graded algebras satisfying the assumptions above. To check this, it is enough to verify that the multiplication in the differential graded algebra is compatible with the obstructions for lifting Def M described in Section 3.3. Algorithm for computing the pro-representing hull using matric Massey products It follows from Theorem 3.5 that the pro-representing hull of Def X is given by the obstruction morphism o : T2 → T1 . In fact, H = T1 /a, where a is the ideal a = o(I(T2 )) ⊆ T1 . To describe the ideal a explicitly, we choose k-bases {ti j (l)∗ : 1 ≤ l ≤ di j } for H1i j ,

{si j (l)∗ : 1 ≤ l ≤ ri j } for H2i j

for 1 ≤ i, j ≤ r. Then a is generated by the obstructions fi j (l) = o(si j (l)) ∈ T1 for 1 ≤ i, j ≤ r and 1 ≤ l ≤ ri j . We write t = ti0 ,i1 (l1 ) · ti1 ,i2 (l2 ) · . . . · tin−1 ,in (ln ) for the monomial t ∈ T1 of degree |t| = n and position (i0 , in ) for any n ≥ 1, any sequence i0 , i1 , . . . , in in {1, 2, . . .r} and any sequence l1 , l2 , . . . , ln of natural numbers with 1 ≤ lm ≤ dim−1 ,im . Moreover, we say that another monomial t′ ∈ T1 divides t, and write t′ |t, if t′ has the form t′ = tim ,im+1 (lm+1 ) · . . . · ti p−1 ,i p (l p ) with 0 ≤ m < p ≤ n. Step 1: Tangent level. Let H2 = T1 /a2 = T12 be the pro-representing hull of Def X restricted to ar (2), where a2 = I 2 = I(T1 )2 , and let B1 = {t : |t| = 1} be a k-base of I(H2 ) = I/I 2 . For each t = ti j (l) ∈ B1 , we choose a 1-cocycle αi j (l) in Ci j with cohomology class ti∗j (l), and put α (t) = αi j (l). Then, the versal family ξ2 ∈ Def X (H2 ) is defined by the family α 1 = {α (t) : t ∈ B1 } Step 2: Cup products. Let b3 = Ia2 +a2 I = I 3 and let H3′ = T1 /b3 = T13 . Then, the natural morphism H3′ → H2 is a small surjection with kernel I 2 /I 3 , and B′2 = {t : |t| = 2} is a k-base of I 2 /I 3 . The obstruction for lifting ξ2 to H3′ is given by

∑′

t∈B2

t ⊗ M2o (t∗ ) =

∑′

t∈B2

t ⊗ ht∗i

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Noncommutative Deformation Theory

where M2o : D2 → H2 is the generalised matric Massey product induced by the obstruction morphism o and where ht∗ i = hti∗0 ,i1 (l1 ),ti∗1 ,i2 (l2 )i is the second-order matric Massey product (or cup product) represented by the 2-cocycle s(t) = α (ti0 ,i1 (l1 )) · α (ti1 ,i2 (l2 )) It is uniquely defined, with defining system contained in α 1 . It follows that the image fi j (l)2 of fi j (l) in T13 (the leading term) is given by fi j (l)2 =

∑

si j (l)(ht∗ i) · t

t∈B′2

Let f2 be the ideal in T1 generated by { fi2j (l)}, and let a3 = I 3 + f2 . Then, H3 = T1 /a3 is the prorepresenting hull of Def X restricted to ar (3). We choose a k-base B2 ⊆ B′2 of I 2 /a3 = ker(H3 → H2 ). Then, there is a mapping β : B′2 × B2 → k such that t′ =

∑ β (t′, t) · t

in H3

t∈B2

for all t′ ∈ B′2 . For any t ∈ B2 , we may therefore find a 1-cochain α (t) such that s(t′ )β (t′ , t) ∑ ′ ′

d α (t) = −

t ∈B2

We write B2 = B2 ∪ B1 for the associated k-base for I(H3 ) = I/a3 . The family α 2 = {α (t) : t ∈ B2 } defines the versal family ξ3 ∈ Def X (H3 ). Step 3: Third-order Massey products. Let b4 = Ia3 + a3 I = I 4 + I f2 + f2 I, and let H4′ = T1 /b4 . Then, H4′ → H3 is a small surjection with kernel a3 /b4 , which we may write as a3 I 3 + f2 f2 I3 ∼ = 4 ⊕ 4 = b4 I + If2 + f2 I If2 + f2 I I + If2 + f2 I Let B′3 be a monomial k-base of I 3 /b4 such that for any t ∈ B′3 , there is a monomial t′ ∈ B2 with t′ |t, and let B′3 = B′3 ∪ B2 . For any monomial t′ with |t′ | ≤ 3, we have t′ =

∑ β ′ (t′ , t) · t + ∑ β ′′ (t′ , si j (l)) · fi j (l)2 i, j,l

t∈B′3

in H4′

where β ′ , β ′′ are mappings with values in k. Using β ′ , we can construct a 2-cocyle s(t) ∈ C for any t ∈ B′3 , given by s(t) = ∑ β ′ (t′ , t) · ∑ α (z) · α (z′ ) |t′ |≤3

z,z′ ∈B2 z·z′ =t′

It represents an element in H2 (C) ∼ = H2 that we denote ht∗ i. Although ht∗ i is not necessarily a matric Massey product in the sense of May, we call it a generalised matric Massey product of order three. In fact, we have that ! ht∗ i = M3o

∑ ′

β ′ (t′ , t) t′

∗

|t |≤3

where M3o : D3 → H2 is the generalised third-order matric Massey product induced by the obstruction morphism o. The cohomology class ht∗ i is well-defined, with defining system included in α 2 , and the obstruction for lifting ξ3 to H4′ is given by !

∑′ ht∗ i · t = M3o ∑ ′

t∈B3

|t |≤3

β ′ (t′ , t) t′

∗

·t

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Noncommutative Deformation Theory

It follows that the image fi j (l)3 of fi j (l) in T14 is given by fi j (l)3 = fi j (l)2 +

si j (l)(ht∗ i) · t

∑′

t∈B3

Let f3 be the ideal in T1 generated by { fi j (l)3 }, let a4 = I 4 + f3 and H4 = T1 /a4 . Then H4 is the pro-representing hull of Def X restricted to ar (4). We may choose B3 ⊆ B′3 such that B3 ∪ { fi j (l)2 } is a k-base of a3 /a4 = ker(H4 → H3 ). Then there is a mapping β : B′3 × B3 → k such that t′ =

∑ β (t′, t) · t

in H4

t∈B3

for all t′ ∈ B′3 . For any t ∈ B3 , we may therefore find a 1-cochain α (t) such that d α (t) = −

s(t′ )β (t′ , t)

∑

t′ ∈B′3

We write B3 = B3 ∪ B2 for the associated k-base for I(H4 ) = I/a4 . The family α 3 = {α (t) : t ∈ B3 } then defines the versal family ξ4 ∈ Def X (H4 ). Step 4: Higher order Massey products. Using an inductive process, we can find Hn+1 and ξn+1 for all n ≥ 4: Assume that Hn = T1 /an with an = I n + fn−1 is given, together with the monomial base Bn−1 and the family α n−1 = {α (t) : t ∈ Bn−1 } that represents ξn . We define bn+1 = Ian + an I and ′ ′ Hn+1 = T1 /bn+1. Then Hn+1 → Hn is a small surjection with kernel an /bn+1 , which can be written as an I n + fn−1 fn−1 In ∼ = n+1 ⊕ n+1 = bn+1 I + Ifn−1 + fn−1 I Ifn−1 + fn−1I I + Ifn−1 + fn−1I Let B′n be a monomial k-base of I n /bn+1 such that for any t ∈ B′n , there is a monomial t′ ∈ Bn−1 with t′ |t, and let B′n = B′n ∪ Bn−1 . For any monomial t′ with |t′ | ≤ n, we have t′ =

∑ β ′ (t′ , t) · t + ∑ β ′′(t′ , si j (l)) · fi j (l)n−1 i, j,l

t∈B′n

in Hn′

where β ′ , β ′′ are mappings with values in k. Using β ′ , we can construct a 2-cocyle s(t) ∈ C for any t ∈ B′n , given by s(t) = ∑ β ′ (t′ , t) · ∑ α (z) · α (z′ ) |t′ |≤n

z,z′ ∈Bn−1 z·z′ =t′

It represents an element in H2 (C) ∼ = H2 that we denote ht∗ i. Although ht∗ i is not necessarily a matric Massey product in the sense of May, we call it a generalised matric Massey product of order n. In fact, we have that ! ht∗ i = Mno

∑ ′

β ′ (t′ , t) t′

∗

|t |≤n

where Mno : Dn → H2 is the generalised n’th order matric Massey product induced by the obstruction morphism o. The cohomology class ht∗ i is well-defined, with defining system included in α n−1 , and ′ the obstruction for lifting ξn to Hn+1 is given by !

∑′ ht∗ i · t = Mno ∑ ′

t∈Bn

β ′ (t′ , t) t′

∗

·t

|t |≤n

It follows that the image fi j (l)n of fi j (l) in T1n+1 is given by fi j (l)n = fi j (l)n−1 +

∑′

t∈Bn

si j (l)(ht∗ i) · t

Noncommutative Deformation Theory

81

Let fn be the ideal in T1 generated by { fi j (l)n }, let an+1 = I n+1 + fn and Hn+1 = T1 /an+1. Then Hn+1 is the pro-representing hull of Def X restricted to ar (n + 1). We may choose Bn ⊆ B′n such that Bn ∪ { fi j (l)n−1 } is a k-base of an /an+1 = ker(Hn+1 → Hn ). Then there is a mapping β : B′n × Bn → k such that t′ = ∑ β (t′ , t) · t in Hn+1 t∈Bn

for all

t′

∈

B′n .

For any t ∈ Bn , we may therefore find a 1-cochain α (t) such that d α (t) = −

s(t′ )β (t′ , t) ∑ ′ ′

t ∈Bn

Then Bn = Bn ∪ Bn−1 is a base of I(Hn+1 ) = I/an+1 , and the family α n = {α (t) : t ∈ Bn } defines the versal family ξn+1 ∈ Def X (Hn+1 ). Remark 3.9. In many cases, it is not necessary to compute the generalised matric Massey products of all orders. If there is an algebraization of the pro-representing hull H, it is often possible to stop the process after a finite number of steps and lift ξn ∈ Def X (Hn ) directly to H = T1 /a where a = f is the ideal generated by the noncommutative polynomials fi j (l) = fi j (l)n−1 in T1 . We shall give an example to illustrate both the algorithm and this fact. Example 3.9. Let A = k[x, y]/(x2 − y3 ) with homogeneous maximal ideal (x, y), and consider the graded maximal ideal M = (x, y) as a right A-module. In Example 1.5, we considered the classical deformation functor of the right A-module M. We shall now consider Def M as the noncommutative deformation functor Def M : a1 → Sets of the right A-module M, and compute its pro-representing hull H and versal family MH using the algorithm above. We use the free resolution 

 x y2 · y x



 x −y2 · −y x



 x y2 · y x

0 ← M ← A2 ←−−−− A2 ←−−−−−−− A2 ←−−−− A2 ← . . . of period two, obtained using the generators y and −x. We recall that ExtAp (M, M) is given by Ext1A (M, M) ∼ = Ext2A (M, M) ∼ = k2 where {t1∗ ,t2∗ } is a k-linear base for Ext1A (M, M) and {s∗1 , s∗2 } is a k-linear base for Ext2A (M, M), with         1 0 0 −y 1 0 0 y t1∗ = · t2∗ = · and s∗1 = · s∗2 = · 0 −1 1 0 0 1 1 0 and xti∗ , yti∗ , xs∗i , ys∗i = 0 for i = 1 and i = 2. It follows that H = khht1 ,t2 ii/( f1 , f2 ), and we use the algorithm to compute f1 , f2 and the versal family MH . Step 1: Tangent level. Let H2 = T1 /a2 = T12 , where a2 = (t1 ,t2 )2 = I 2 and let B1 = {t1 ,t2 }. We choose the family α 1 = {α (t1 ), α (t2 )} given by           1 0 −1 0 0 −y 0 y α (t1 ) = ·, · α (t2 ) = ·, · 0 −1 0 1 1 0 −1 0 The versal family ξ2 is explicitly given by a lifting of complexes (LR , d R ) to R = H2 given by       1 0 0 −y x y2 H2 d0 = 1 ⊗ + t1 ⊗ + t2 ⊗ 0 −1 1 0 y x

82

Noncommutative Deformation Theory

and d1H2 = 1 ⊗



     −1 0 0 y x −y2 + t1 ⊗ + t2 ⊗ −y x 0 1 −1 0

Step 2: Cup products. Let b3 = I 3 = (t1 ,t2 )3 , let H3′ = T13 and let B′2 = {t12 ,t1t2 ,t2t1 ,t22 }. We compute the cup products ht∗ i for t ∈ B′2 , represented by s(t∗ ), using defining systems contained in α 1 , and find       1 0 −1 0 −1 0 ∗ ∗ s(t1 t1 ) = α (t1 )0 · α (t1 )1 = · = 0 −1 0 1 0 −1       0 −y −1 0 0 −y ∗ ∗ s(t1 t2 ) = α (t2 )0 · α (t1 )1 = · = 1 0 0 1 −1 0       1 0 0 y 0 y ∗ ∗ · = s(t2 t1 ) = α (t1 )0 · α (t2 )1 = 0 −1 −1 0 1 0       0 −y 0 y y 0 ∗ ∗ s(t2 t2 ) = α (t2 )0 · α (t2 )1 = · = 1 0 −1 0 0 y In cohomology, this gives ht1∗ ,t1∗ i = −s∗1

ht1∗ ,t2∗ i = −s∗2

ht2∗ ,t1∗ i = s∗2

ht2∗ ,t2∗ i = 0

Therefore, we have f12 = −t12 , f22 = t2t1 −t1t2 and a3 = (t1 ,t2 )3 + (−t12 ,t2t1 −t1t2 ), with H3 = T1 /a3 . We choose B2 = {t1t2 ,t22 } and express t2t1 and t22 as linear combinations of base vectors in B2 as t2t1 = t1t2 ,

t22 = 0

in H3

Using these linear combinations, we find the coboundaries    0 0 y ∗ ∗ ∗ ∗ ∗ ∗ s(t1 t2 ) + s(t2 t1 ) = , s(t2 t2 ) = 0 0 0 We may therefore choose  0 α (t1t2 ) = 0

  0 0 , 0 0

 0 , 0

α (t22 ) =

 0 y

    0 1 0 −1 , 0 0 0 0

such that −d(α (t1t2 )) = s(t1∗t2∗ ) + s(t2∗t1∗ ) and −d(α (t22 )) = s(t2∗t2∗ ). Then α 2 = {α (t) : t ∈ B2 } with B2 = {t1 ,t2 ,t1t2 ,t22 } corresponds to the versal family ξ3 ∈ Def M (H3 ). It is explicitly given by a lifting of complexes (LR , d R ) to R = H3 given by         1 0 0 −y 0 1 x y2 H3 2 d0 = 1 ⊗ + t1 ⊗ + t2 ⊗ + t2 ⊗ 0 −1 1 0 0 0 y x and d1H3 = 1 ⊗



x −y

      −y2 −1 0 0 y 0 + t1 ⊗ + t2 ⊗ + t22 ⊗ 0 1 −1 0 0 x

 −1 0

Step 3: Third-order Massey products. Let b4 = Ia3 + a3 I = I 4 + If2 + f2 I and H4′ = T1 /b4 . In I 3 /b4 , we have the relations t13 = t12t2 = t1t2t1 = t2t12 = 0,

t22t1 = t2t1t2 = t1t22

We may therefore choose B′3 = {t1t22 ,t23 } as a base of I 3 /b4 with the property that for any t ∈ B′3 ,

83

Noncommutative Deformation Theory

there is an element t′ ∈ B2 such that t′ |t. We use defining systems contained in α 2 to compute the third order generalised Massey products ht∗ i for t ∈ B′3 , represented by the 2-cocycle s(t∗ ) =

∑ ′

β ′ (t′ , t) ·

|t |≤3

∑ ′

α (z)0 · α (z′ )1

z,z ∈B2 z·z′ =t′

In concrete terms, we find that s(t1∗t2∗t2∗ ) = α (t22 )0 · α (t1 )1 + α (t1 )0 · α (t22 )1           0 1 −1 0 1 0 0 −1 0 0 = · + · = 0 0 0 1 0 −1 0 0 0 0

s(t2∗t2∗t2∗ ) = α (t2 )0 · α (t22 )1 + α (t22 )0 · α (t2 )1           0 −y 0 −1 0 1 0 −y −1 0 = · + · = 1 0 0 0 0 0 −1 0 0 −1 since α (t1 t2 ) = 0. In cohomology, this gives ht1∗ ,t2∗ ,t2∗ i = 0 ht2∗ ,t2∗ ,t2∗ i = −s∗1 Hence f13 = −t12 − t23 , f23 = t2t1 − t1t2 and a4 = (t1 ,t2 )4 + (−t12 − t23 ,t2t1 − t1t2 ), with H4 = T1 /a4 . We choose B3 = {t1t22 }, and let α (t1t22 ) = 0. Then α 3 = {α (t) : t ∈ B3 } with B3 = {t1 ,t2 ,t1t2 ,t22 ,t1t22 } defines the versal family ξ4 ∈ Def M (H4 ). It is explicitly given by a lifting of complexes (LR , d R ) to R = H4 given by         1 0 0 −y 0 1 x y2 H4 2 d0 = 1 ⊗ + t1 ⊗ + t2 ⊗ + t2 ⊗ 0 −1 1 0 0 0 y x and d1H4 = 1 ⊗



x −y

      0 y 0 −y2 −1 0 + t2 ⊗ + t22 ⊗ + t1 ⊗ −1 0 0 0 1 x

 −1 0

Step 4: Higher order Massey products. Let f1 = f13 = −t12 − t23 and f2 = f23 = t2t1 − t1t2 in T1 . We let H ′ = T1 /( f1 , f2 ), and try to lift the versal family ξ4 ∈ Def M (H4 ) to H ′ . We put         1 0 0 −y 0 1 x y2 2 d0 = 1 ⊗ + t1 ⊗ + t2 ⊗ + t2 ⊗ 0 −1 1 0 0 0 y x and d1 = 1 ⊗



x −y

       −y2 −1 0 0 y 0 −1 + t1 ⊗ + t2 ⊗ + t22 ⊗ 0 1 −1 0 0 0 x

and compute that d0 d1 in H ′ . We find that d0 d1 = 0 since t1t22 = t22t1 in H ′ . This implies that we can lift ξ4 to H ′ , and therefore that H = H ′ = T1 /( f1 , f2 ) = khht1 ,t2 ii/(−t12 − t23,t2t1 − t1t2 ) ∼ = k[[t1 ,t2 ]/(t12 + t23 ) is a pro-representing hull of Def M . Its versal family ξ ∈ Def M (H) is explicitly given by a lifting of complexes (LR , d R ) to R = H given by d0H = d0 and d1H = d1 .

3.5.3 Matric A-infinity algebras An A-infinity algebra over a commutative ring S, also called an A∞ -algebra over S, is a Z-graded S-bimodule C = ⊕ p C p endowed with a family of graded morphisms of S-bimodules mn : C⊗n → C,

n≥1

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of degree 2 − n satisfying the Stasheff identities

∑

(−1)r+st mr+1+t (id⊗r ⊗ms ⊗ id⊗t ) = 0

(SI(n))

r+s+t=n r,t≥0, s≥1

for all n ≥ 1, where all tensor products are over S. Note that when these morphisms are applied to elements, additional signs appear due to the Koszul sign rule ( f ⊗ g)(x ⊗ y) = (−1)|g||x| f (x) ⊗ g(y) where f and g are graded morphisms, x and y are homogeneous elements, and where vertical bars denote the degree. In particular, we note that m1 : C → C has degree 1 and m2 : C ⊗S C → C has degree 0, and the first Stasheff identities are given in the following way: SI(1): m1 m1 = 0 SI(2): m1 m2 = m2 (id ⊗m1 + m1 ⊗ id) SI(3): m2 (id ⊗m2 − m2 ⊗ id) = m1 m3 + m3 (m1 ⊗ id⊗2 + id ⊗m1 ⊗ id + id⊗2 ⊗m1 ) It follows that m1 is a differential that acts on C as a graded derivation, and that m2 is a graded (but not necessarily associative) multiplication on C. In fact, the right side of SI(3) is the obstruction for the multiplication m2 on C to be associative; hence, the multiplication on the cohomology H• (C) induced by m2 is associative. The maps {mn : n ≥ 3} are called higher multiplications. If mn = 0 for all n ≥ 3, then the multiplication m2 is associative, and C = (C, m1 , m2 ) is a differential graded algebra over S. A strict unit for C is an element 1 ∈ C0 that is a unit for the multiplication m2 and satisfies mn (c1 ⊗ c2 ⊗ · · · ⊗ cn ) = 0 whenever n 6= 2 and ci = 1 for some i. We assume that any A∞ -algebra C over S has a strict unit 1 such that s · 1 = 1 · s in C0 for all s ∈ S. In that case, there is a canonical ring homomorphism S → C. When S = k is a field, the definition given above is the usual definition of an A∞ -algebra over k; see for instance Keller [20] and Lu et al. [33]. We define a matric A∞ -algebra to be an A∞ -algebra over S = kr for some integer r ≥ 1. Any matric A∞ -algebra over kr has a decomposition C = (Ci j ) = ⊕ Ci j i, j

where Ci j = eiCe j for 1 ≤ i, j ≤ r, and {e1 , e2 , . . . , er } ⊆ C0 is the image of the indecomposable idempotents in kr . It follows that the differential m1 satisfies m1 (Ci j ) ⊆ Ci j for all 1 ≤ i, j ≤ r, and that the (higher) multiplications mn satisfy mn (Ci0 i1 ⊗k Ci1 i2 ⊗k · · · ⊗k Cin−1 in ) ⊆ Ci0 in for all n ≥ 2 and 1 ≤ i0 , i1 , . . . , in ≤ r. A morphism f : (C, mn ) → (C′ , m′n ) of A∞ -algebras over S is a collection of graded S-bimodule homomorphisms fn : C⊗n → C′ , n ≥ 1 of degree 1 − n satisfying the Stasheff morphism identities

∑

(−1)r+st fr+1+t (id⊗r ⊗ms ⊗ id⊗t )

r+s+t=n r,t≥0, s≥1

=

∑

i1 +i2 +···+iq =n i1 ,i2 ,...,iq ≥1

(−1)w m′q ( fi1 ⊗ fi2 ⊗ · · · ⊗ fiq ) (MI(n))

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85

for all n ≥ 1, where the sign on the right-hand side is given by w = (q − 1)(i1 − 1) + (q − 2)(i2 − 1) + · · · + 2(iq−2 − 1) + (iq−1 − 1) When C and C′ has a strict unit, a morphism of A∞ -algebras is also required to satisfy the unital morphism conditions f1 (1) = 1 and fn (c1 ⊗ · · · ⊗ cn ) = 0 when n ≥ 2 and ci = 1 for some i. The morphism f is called a quasi-isomorphism of A∞ -algebras if f1 : (C, m1 ) → (C′ , m′1 ) is a quasiisomorphism of S-bimodule complexes. Theorem 3.30 (Kadeišvili). Let C be a matric A∞ -algebra over kr . Then, H• (C) has a matric A∞ algebra structure with m1 = 0 and m2 induced by the multiplication on C, such that there is a quasi-isomorphism f : H• (C) → C of A∞ -algebras for which H• ( f1 ) = id. Moreover, this A∞ -algebra structure on H• (C) is unique up to (noncanonical) isomorphism of A∞ -algebras. Proof. Since S = kr is semisimple, this follows from Theorem 1 in Kadeišvili [19] and the comments following the proof. A minimal model for a matric A∞ -algebra C over k is an A∞ -algebra structure on H• (C) satisfying the conditions of the theorem. We shall give an explicit description of a minimal model for a matric differential graded algebra C over kr with differential d; see Merkulov [37] and Lu et al. [33] for further details: Let Z n = {x ∈ Cn : d(x) = 0} ⊆ Cn denote the cocycles and Bn = d(Cn−1 ) ⊆ Cn denote the coboundaries in (C, d). It is clear that the inclusions Bn ⊆ Z n ⊆ Cn of bimodules over kr are split injections; hence, we can find kr -bimodules H n ⊆ Z n and Ln ⊆ Cn such that Bn ⊕ H n = Z n , 1 ∈ H 0 and Z n ⊕ Ln = Cn . We shall identify H• (C) with H = ⊕n H n ⊆ C, and write p : C → C for the projection onto H. There is a graded homomorphism of kr -bimodules G : C → C of degree −1 such that idC −p = dG − Gd. In fact, we may choose G such that G(Ln ) = G(H n ) = 0 and G|Bn = d −1 , where we consider d : Ln−1 → Bn as an isomorphism of kr -bimodules. We define graded homomorphisms of kr -bimodules λn : C⊗n → C of degree 2 − n for all n ≥ 2 by the formula

λn =

∑

s+t=n s,t≥1

(−1)s+1 λ2 (Gλs ⊗ Gλt )

where λ2 is the multiplication on C and Gλ1 = − idA . Then, mn = pλn for n ≥ 2 give H• (C) a matric A∞ -algebra structure, and f : H• (C) → C with fn = −Gλn for n ≥ 1 is a quasi-isomorphism of A∞ -algebras such that H• ( f1 ) = id. We call the minimal model of C constructed above a Merkulov model of C; it is a special case of the general construction in Kadeišvili [19]. It is clear from the explicit construction that all Merkulov models of C are quasi-isomorphic A∞ -algebras. Remark 3.10. Even if C is a matric differential graded algebra (that is, a matric A∞ algebra with vanishing higher multiplications mn = 0 for n ≥ 3), it is in general not true that the higher multiplications of its Merkulov model H• (C) vanish. Remark 3.11. Let Def X be the noncommutative deformation functor of a family X in an Abelian k-category A, and assume that Def X has an obstruction theory with cohomology {H p } and that there is a matric differential graded algebra C such that H p (C) ∼ = H p . In this case, the Merkulov model of C has a multiplication that is the matric Massey cup product on H• (C), and higher multiplications that are related to the matric Massey products of H• (C) of higher order: Theorem 3.31. Let C be a matric differential graded algebra over kr , and let (H• (C), m2 , m3 , . . . ) be a Merkulov model of C. If n ≥ 3 and α1 , . . . , αn ∈ H• (C) are homogeneous elements such that the matric Massey product hα1 , α2 , . . . αn i is defined, then (−1)b mn (α1 ⊗ α2 ⊗ · · · ⊗ αn ) ∈ hα1 , α2 , . . . αn i where b = (|αn−1 | + |αn−3 | + . . .) + 1 = (dn−1 + dn−3 + . . .) + 1.

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Proof. See Theorem 3.1 in Lu et al. [33]. Remark 3.12. For a given matric differential graded algebra C with cohomology H• (C), there may be several Merkulov models with different higher multiplications. We know that different Merkulov models are isomorphic as A∞ -algebras, and that the higher multiplications must satisfy (−1)b mn (α1 ⊗ α2 ⊗ · · · ⊗ αn ) ∈ hα1 , α2 , . . . αn i However, this is not enough to ensure that the values of the higher multiplications are unique. For instance, we may consider Example 6.5 in Lu et al. [33] to illustrate this: Let A = kha, b, c, x, yi be a differential graded algebra over k, where all generators have degree one and where the differential d is given by d(a) = d(b) = d(c) = 0, d(x) = ab, d(y) = bc Let α , β and γ be the cohomology classes of a, b and c. Then, the matric Massey product hα , β , γ i is defined and given by hα , β , γ i = {θ + rαγ + sγ 2 + t α 2 : r, s,t ∈ k} where θ is the cohomology class of xc + ay. All cohomology classes in this matric Massey product have the form m3 (α , β , γ ) for some Merkulov model of H• (A).

3.6 The Generalised Burnside Theorem Let A be an associative algebra, and let M = {M1 , . . . , Mr } be a family in ModA , the category of right A-modules. For any algebra R in ar and any deformation MR ∈ Def M (R), there is an algebra homomorphism ηR : A → EndR (MR ) ∼ = (Ri j ⊗k Homk (Mi , M j )) given by right multiplication of a ∈ A on MR . See also Subsection 3.3.1, where we described Def M using Hochschild cohomology and wrote VR for MR and ρR for ηR . Explicitly, the element ηR (a)(ei ⊗ mi ) = (ei ⊗ mi )a can be written in the form (ei ⊗ mi )a = ei ⊗ (mi a) + ∑ ri j (l) ⊗ φ i j (l)a (mi ) i, j,l

where {ri j (l) : l} is a base of I(R)i j , and φ i j (l) : A → Endk (M) are 1-cochains in the Hochschild complex of A with values in Endk (M) with M = ⊕i Mi . If M is a swarm, then Def M has a pro-representing hull H and a corresponding versal family MH ∈ Def M (H) by Proposition 3.7. We define the algebra of observables of the swarm M to be the algebra b k Homk (Mi , M j )) O(M ) = EndH (MH ) ∼ = (Hi j ⊗

and let η : A → O(M ) be the algebra homomorphism given by right multiplication by A on MH . We call η the miniversal morphism of the swarm M since it determines the miniversal family MH ∈ Def M (H). In this section, we study the algebra of observables O(M ) and the miniversal morphism η : A → O(M ) when M is a swarm in ModA . In particular, we prove the Generalised Burnside Theorem. This key result states that when A has finite dimension and M is the family of simple right A-modules, then η is an isomorphism if the division algebra EndA (Mi ) = k for 1 ≤ i ≤ r, or equivalently, if M is a family of k-rational points in X = Simp(A).

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87

3.6.1 The algebra of observables Let M = {M1 , M2 , . . . , Mr } be a swarm of right A-modules, let H be the pro-representing hull of the noncommutative deformation functor Def M : ar → Sets, and let MH ∈ Def M (H) be its versal family. The algebra of observables of the swarm M is the associative k-algebra b k Homk (Mi , M j )) O(M ) = EndH (MH ) ∼ = (Hi j ⊗

It comes equipped with an algebra homomorphism η : A → O(M ), given by right multiplication of A on the versal family MH , and it fits into the natural commutative diagram η

/ O(M ) A❋ ❋❋ ❋❋ ❋ π ρ ❋❋ ❋#  Endk (M) where ρ is the algebra homomorphism given by right multiplication of A on M = ⊕i Mi , and π is the algebra homomorphism determined by the structure morphism H → kr . In fact, π induces a natural algebra homomorphism π ∗ : EndH (MH ) → Endkr (Mkr ), and we have that Endkr (Mkr ) = ((kr )i j ⊗k Homk (Mi , M j )) ∼ = ⊕ Endk (Mi ) ⊆ Endk (M) i

Hence, there is a natural algebra homomorphism O(M ) = EndH (MH ) → Endk (M) induced by π . Remark 3.13. For any algebra R in ar and any deformation MR ∈ Def M (R), there is a morphism u : H → R in aˆ r such that Def M (u)(MH ) = MR by the miniversal property, and the deformation MR is therefore given by the composition ηR = u∗ ◦ η in the diagram η / O(M ) A PPP PPP PPP P u∗ =u⊗id ηR PPPP '  EndR (MR )

In this sense, η : A → O(M ) determines all noncommutative deformations of the family M , and we call it the miniversal morphism of the swarm M . The algebra O(M ) of observables has an induced right action on the family M via π , which extends the right action of A on M . We may therefore consider any swarm M of right A-modules as a family of right B-modules, where B = O(M ), via the algebra homomorphism η : A → B = O(M ). In fact, M is the family of simple right B-modules. This follows from the fact that π can be identified with the quotient B → B/J(B). If M is a swarm of right B-modules, we can also iterate the process and compute O B (M ). It is given by the pro-representing hull H B of Def M , the noncommutative deformation functor of M considered as a family of right B-modules, and its versal family.

3.6.2 The kernel of the miniversal morphism Let η : A → O(M ) be the miniversal morphism of the swarm M , and consider the kernel K = ker(η ) of η . Explicitly, we have that a ∈ K if and only if MH · a = 0. In concrete terms, using the notation in Subsection 3.5.2, this means that (ei ⊗ mi ) a = ei ⊗ (mi a) +

∑

|t|=1

t ⊗ α (t)a(mi ) + ∑

∑ t ⊗ α (t)a(mi ) = 0

l≥2 t∈Bl

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Noncommutative Deformation Theory

In other words, we have that a ∈ K if and only if Ma = 0 and α (t)a = 0 for all n ≥ 1 and all t ∈ Bn . Let K1 = {a ∈ A : Ma = 0} be the annihilator of M = ⊕i Mi . Any extension ξ ∈ Ext1A (M, M) can be represented by a derivation ψ ∈ Derk (A, Endk (M)) using Hochschild cohomology, and we define ξ (a) = ψ (a) when a ∈ K1 . This gives a well-defined map ξ : K1 → Endk (M) since the inner derivation a 7→ aφ − φ a = 0 for any a ∈ K1 and any φ ∈ Endk (M). We define K2 = {a ∈ K1 : ξ (a) = 0 for all ξ ∈ Ext1A (M, M)} and remark that a ∈ K2 if and only if Ma = 0 and α (t)a = 0 for all monomials t of degree one. In other words, K2 = ker(ηH2 ), where ηH2 is the algebra homomorphism given by the versal family MH2 ∈ Def M (H2 ). Let Hn be the pro-representing hull of the restriction of Def M to ar (n), and let MHn be its versal family. We define Kn = ker(ηHn ) to be the kernel of the associated algebra homomorphism. It is clear that K is the ideal \ Kn K= n≥1

Moreover, for any n ≥ 3, we have that a ∈ Kn if and only if a ∈ Kn−1 and α (t)a = 0 for all t ∈ Bn . This follows from the construction in Subsection 3.5.2.

3.6.3 Iterated extensions and matric Massey products Let E be a right A-module and let r ≥ 1 be a positive integer. If E has a cofiltration of length r, given by a sequence fr

f

f

2 1 E = Er − → Er−1 → · · · → E2 − → E1 − → E0 = 0

of surjective right A-module homomorphisms fi : Ei → Ei−1 , then we call E an iterated extension of the right A-modules M1 , M2 , . . . Mr , where Mi = ker( fi ). In fact, the cofiltration induces short exact sequences f

i 0 → Mi → Ei − → Ei−1 → 0

for 1 ≤ i ≤ r. Hence, E1 ∼ = M1 , E2 is an extension of E1 by M2 , and in general, Ei is an extension of Ei−1 by Mi . We write ξi−1,i ∈ Ext1A (Mi−1 , Mi ) for the image of the extension above under the induced map Ext1A (Ei−1 , Mi ) → Ext1A (Mi−1 , Mi ). When the modules M1 , M2 , . . . , Mr and the extensions ξ12 , ξ23 , . . . , ξr−1,r are given, we claim that a cofiltration of the above type exists if and only if the matric Massey product hξ12 , ξ23 , . . . , ξr−1,r i is defined and contains zero. This is a matric Massey product in the sense of May (see Section 3.5). We shall prove the claim using an explicit construction of the module E and its cofiltration. We consider the matric differential graded algebra HC• (A, Endk (M)) over kr , the Hochschild complex of A with values in Endk (M), with M = M1 ⊕ · · · ⊕ Mr . It has cohomology p H p (A, Endk (M)) ∼ = (H p (A, Homk (Mi , M j ))) ∼ = (ExtA (Mi , M j ))

Hence, ξi−1,i ∈ Ext1A (Mi−1 , Mi ) can be represented by a 1-cocyle αi−1,i : A → Homk (Mi−1 , Mi ) in the Hochschild complex HC• (A, Endk (M)) for 2 ≤ i ≤ r. This is the starting point for the construction. Let E2 be a right A-module that is an extension of M1 by M2 , such that there is a short exact sequence 0 → M2 → E2 → M1 → 0. Then it is wellknown that E2 ∼ = M2 ⊕ M1 considered as a vector space over k, and that the right action of A is given by (m2 , m1 )a = (m2 · a + ψa12(m1 ), m1 · a)

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89

where ψ 12 : A → Homk (M1 , M2 ) is a k-linear map. Since the action of A is associative, ψ 12 is a derivation. In fact, we may choose ψ 12 = α12 , the 1-cocycle that represents the extension ξ12 . Let E3 be a right A-module that is an extension of E2 by M3 , such that there is a short exact sequence 0 → M3 → E3 → E2 → 0. Then, E3 ∼ = M3 ⊕ E2 ∼ = M3 ⊕ M2 ⊕ M1 considered as a vector space over k, and the right action of A is given by (m3 , m2 , m1 )a = (m3 · a + ψa23(m2 ) + ψa13(m1 ), m2 · a + ψa12(m1 ), m1 · a) where ψ i3 : A → Homk (Mi , M3 ) is a k-linear map for i = 1, 2. Since the action of A is associative, ψ 23 is a derivation. In fact, we may choose ψ 23 = α23 , the 1-cocycle that represents the extension ξ23 . Moreover, ψ 13 satisfies e13 , −d(ψ 13 ) = ψ

e13 = ψ 12 · ψ 23 = α12 · α23 with ψ

such that the cup product hξ12 , ξ23 i = 0. In fact, we may choose α13 = −ψ 13 so that {α12 , α13 , α23 } is a defining system for the matric Massey product hξ12 , ξ23 , ξ34 i. It follows by an inductive argument that in the cofiltration fr

f

f

2 1 → Er−1 → · · · → E2 − → E1 − → E0 = 0 E = Er −

we have that E = Er ∼ = Mr ⊕ · · · ⊕ M2 ⊕ M1 considered as a vector space over k, with right action of A given by r−1

(mr , . . . , m2 , m1 )a = (mr · a + ∑ ψair (mi ), . . . , m2 · a + ψa12(m1 ), m1 · a) i=1

where ψ i j : A → Homk (Mi , M j ) is a 1-cochain for 1 ≤ i < j ≤ r. The conditions that these cochains must satisfy for the action of A to be associative, is that ei j , −d(ψ i j ) = ψ

j−1

ei j = with ψ

∑

ψ il · ψ l j

l=i+1

In other words, the family α = {αi j : 1 ≤ i < j ≤ r, (i, j) 6= (1, r)} given by αi j = (−1) j−i+1ψ i j is a defining system for the matric Massey product hξ12 , ξ23 , . . . , ξr−1,r i e1r , is zero. This proves the following result: such that its value, given by the cohomology class of α

Proposition 3.32. Let M1 , . . . , Mr be right A-modules, and let ξi−1,i ∈ Ext1A (Mi−1 , Mi ) for 2 ≤ i ≤ r. There is an iterated extension E of M1 , . . . , Mr with induced extensions ξ12 , ξ23 , . . . , ξr−1,r if and only if the matric Massey product hξ12 , ξ23 , . . . , ξr−1,r i is defined and contains zero. Let us consider the case when M is a swarm of right A-modules and E is an iterated extension of the right A-modules M1 , M2 , . . . , Mr in the family M . Proposition 3.33. Let M = {M1 , . . . , Mr } be a swarm of right A-modules, and let E be an iterated extension of the right A-modules M1 , M2 , . . . , Mr in M . Then, E · K = 0, where K = ker(η ) is the kernel of the miniversal morphism of the swarm M .

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Proof. For any a ∈ K, it is clear that Ma = 0 and that ξ (a) = 0 for any ξ ∈ Ext1A (M, M), with M = ⊕i Mi . In particular, we have that ξi−1,i (a) = 0 for 2 ≤ i ≤ r, where ξ12 , ξ23 , . . . , ξr−1,r are the extensions induced by a cofiltration of E. To show that E · K = 0, it is enough to show that αi j (a) = 0 for all αi j in the defining system α for the matric Massey product hξ12 , ξ23 , . . . , ξr−1,r i with associated value 0. We clearly have that

ξi,i+1 ⊗ ξi+1,i+2 ⊗ · · · ⊗ ξ j−1, j ∈ D j−i and M oj−i (ξi,i+1 ⊗ ξi+1,i+2 ⊗ · · · ⊗ ξ j−1, j ) = hξi,i+1 , ξi+1,i+2 , . . . , ξ j−1, j i, the matric Massey product defined by α . Using the notation of Subsection 3.5.2, the fact that α (t)a = 0 for all t ∈ Bn implies that αi j (a) = 0 for a ∈ K. Corollary 3.34. If A considered as a right A-module is an iterated extension of a swarm M , then the miniversal morphism η : A → O(M ) of the swarm M is injective. In particular, η is injective when A has finite dimension and M is the family of simple right A-modules. Proof. If A is an iterated extension of M , then 1 · a = 0 for all a ∈ K by Proposition 3.33, and this implies that K = 0. If A is finite dimensional, then A is a right A-module of finite length, and it is therefore an iterated extension of the family M of simple right A-modules, which is a swarm.

3.6.4 The Generalised Burnside Theorem Let A be a finite dimensional algebra, and let M = {M1 , . . . , Mr } be the family of simple right A-modules (up to isomorphism). Then M is a swarm, and the versal morphism η : A → O(M ) is injective by Corollary 3.34. We recall that η fits into the natural commutative diagram η

/ O(M ) A ❍❍ ❍❍ ❍❍ ❍ π ρ ❍❍ ❍$  Endkr (Mkr ) where Endkr (Mkr ) ∼ = ⊕i Endk (Mi ) ⊆ Endk (M). If EndA (Mi ) = k for 1 ≤ i ≤ r, then ρ is surjective by the classical Burnside Theorem; see Theorem 2.12. Notice that this conlusion implies that the induced map ρ : A/J(A) → ⊕i Endk (Mi ) is an isomorphism, and that the first-order approximation gr0 (η ) : A/J(A) → O(M )/J(O(M )) b k Homk (Mi , M j )). We shall prove the of η coincides with ρ since η (J(A)) ⊆ J(O(M )) = (I(H)i j ⊗ following generalization of the classical Burnside Theorem:

Theorem 3.35 (Generalised Burnside Theorem). Let A be a k-algebra of finite dimension, and let M = {M1 , . . . , Mr } be the family of right A-modules (up to isomorphism). If EndA (Mi ) = k for 1 ≤ i ≤ r, then η : A → O(M ) is an isomorphism. In particular, η is an isomorphism if k is algebraically closed.

Proof. It follows from Corollary 3.34 that η is injective. We shall prove that η is also surjective. Since A is Artinian, it follows that J(A) is a nilpotent ideal, and therefore A is complete. Moreover, b k Homk (Mi , M j )) is the ideal the Jacobson radical of O(M ) = (Hi j ⊗ b k Homk (Mi , M j )) J = (I(H)i j ⊗

Since H is complete, it follows that O(M ) is also complete. By the classical Burnside Theorem, it follows that the map gr0 (η ) : A/J(A) → O(M )/J is surjective; see the comments preceeding the

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Generalised Burnside Theorem. By a standard result for filtered algebras, it is therefore sufficient to show that gr1 (η ) : J(A)/J(A)2 → J/J 2 is surjective. We notice that J/J 2 ∼ = ((I(H)/ I(H)2 )i j ⊗k Homk (Mi , M j )) ∼ = (Ext1A (Mi , M j )∗ ⊗k Homk (Mi , M j )) since I(H)/ I(H)2 is the dual of the tangent space (Ext1A (Mi , M j )) of Def M . By Theorem 2.12, we have that A/J(A) ∼ = ⊕ Endk (Mi ) 1≤i≤r

and therefore A/J(A) is separable. By Wedderburn-Malcev Theorem (Theorem 2.13), it therefore follows that there is a semisimple subalgebra S ⊆ A such that A = J(A) ⊕ S. In particular, any a ∈ A can be written as a = j + s with j ∈ J(A), s ∈ S, and this decomposition is unique. We claim that Ext1A (Mi , M j ) ∼ = HomA−A (J(A)/J(A)2 , Homk (Mi , M j )) for all i, j. We shall use that Ext1A (Mi , M j ) ∼ = Derk (A, Homk (Mi , M j ))/ im(d 0 ), where im(d 0 ) is the 1-coboundaries in Hochschild cohomology (or the inner derivations) to prove the claim above. We define a map u : Derk (A, Homk (Mi , M j )) → HomA−A (J(A)/J(A)2 , Homk (Mi , M j )) where u(D)(x) = D(x) for all x ∈ J(A). Since D(J(A)2 ) = 0, we see that u(D) is well-defined and A-A bilinear. We shall prove that im(d 0 ) = ker(u): Any inner derivation D = d 0 (φ ) clearly gives u(D) = 0. Conversely, if u(D) = 0, then there is an induced derivation D : A/J(A) → Homk (Mi , M j ), and D = d 0 (φ ) is inner since A/J(A) is semisimple. This means that D is also inner. Finally, we show that u is surjective: For any A-A bilinear map φ : J(A)/J(A)2 → Homk (Mi , M j ), we define a map D by D(a) = φ ( j), where a = j + s is the decomposition mentioned above. If b = j′ + s′ is the decomposition of another element b ∈ A, then ab = ( j + s)( j′ + s′ ) = j j′ + js′ + s j′ + ss′ is the decomposition of ab, with j j′ + js′ + s j′ ∈ J(A), ss′ ∈ S. It follows that D : A → Homk (Mi , M j ) is a derivation. In fact, we have that aD(b) + D(a)b = aφ ( j′ ) + φ ( j)b = ( j + s)φ ( j′ ) + φ ( j)( j′ + s′ ) = φ (s j′ + js′ ) = D(ab) It is clear that u(D) = φ , and therefore u is surjective. When we write A = A/J(A), this implies that we have Ext1A (Mi , M j ) ∼ = HomA−A (J(A)/J(A)2 , Homk (Mi , M j )) ∼ (J(A)/J(A)2 , Homk (Mi , M j )) = Hom A−A

Since S ∼ = A/J(A) ∼ = ⊕i Endk (Mi ), there are idempotents e1 , . . . , er in S ⊆ A corresponding to the identities on M1 , M2 , . . . , Mr , and we have that ei e j = 0 for i 6= j and 1 = e1 + · · · + er . Let A = (Ai j ) be the corresponding decomposition of A, with Aii /J(A)ii ∼ = Endk (Mi ). Let E = J(A)/J(A)2 and 2 Ei j = ei Ee j . Then J(A)/J(A) = (Ei j ), where Ei j is an Endk (Mi )-Endk (M j ) bimodule, and therefore there is a vector space Wi j of finite dimension over k such that Ei j ∼ = Mi∗ ⊗ Wi j ⊗ M j ∼ = Wi j ⊗k Homk (Mi , M j )

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Let ni j = dimk Wi j . Then, we have isomorphisms Ext1A (Mi , M j ) ∼ = HomA−A (J(A)/J(A)2 , Homk (Mi , M j )) ∼ (Ei j , Homk (Mi , M j )) = Hom A−A

∼ = HomEndk (Mi )−Endk (M j ) (Wi j ⊗k Homk (Mi , M j ), Homk (Mi , M j )) ∼ W ∗ ⊗k EndEnd (M )−End (M ) (Homk (Mi , M j )) = ij

k

i

k

j

∼ = Wi∗j ⊗k k ∼ = Wi∗j This implies that J(A)/J(A)2 = (Ei j ) ∼ = (Ext1A (Mi , M j )∗ ⊗k Homk (Mi , M j )) ∼ = J/J 2 Since gr1 (η ) : J(A)/J(A)2 → J/J 2 is injective, it follows that gr1 (η ) is an isomorphism. By the comments above, this proves that η is surjective.

3.6.5 Properties of the algebra of observables Let A be a finite-dimensional algebra, and let M = {M1 , . . . , Mr } be a family of right Amodules of finite dimension. Then, M is a swarm, and we denote its algebra of observables by B = O A (M ). Let J = J(B) be the Jacobson radical of B. As remarked above, we have that J = (I(H)i j ⊗k Homk (Mi , M j )). This implies that M , considered as a family of right B-modules, is the family of simple B-modules since B and B/J ∼ = ⊕i Endk (Mi ) have the same simple modules. Lemma 3.36. The family M is a swarm of right B-modules when B = O A (M ). Proof. Hochschild cohomology gives that Ext1B (Mi , M j ) ∼ = Derk (B, Homk (Mi , M j ))/ im(d 0 ), and Homk (Mi , M j ) is finite dimensional. Moreover, any derivation D : B → Homk (Mi , M j ) satisfies D(J 2 ) = JD(J) + D(J)J = 0 since J = J(B) and M is the family of simple B-modules. Since B/J 2 ∼ = ((H/ I(H)2 )i j ⊗k Homk (Mi , M j )) is finite dimensional, it follows that M is a swarm. Since M is a swarm of right B-modules with B = O A (M ), we can iterate the process. We consider the pro-representing hull H B of the noncommutative deformation functor Def M of the family M of right B-modules, and its versal family MHB . The versal family has an induced versal morphism ηB : B → O B (M ) = EndH B (MHB ) given by right multiplication by B. Proposition 3.37. If EndA (Mi ) = k for 1 ≤ i ≤ r, then the versal morphism ηB : B → O B (M ) is an isomorphism. In particular, ηB is an isomorphism if k is algebraically closed. Proof. Since M is a swarm of A-modules and of B-modules, we may consider the commutative diagram ηA

ηB

/ B = O A (M ) / C = O B (M ) A ■■ ■■ ■■ ♦♦♦ ■ ♦♦♦ ♦ ♦ ρ ■■■ ♦ ■$  w♦♦♦ Endkr (Mkr ) The algebra homomorphism η B induces maps B/ I(B)n → C/ I(C)n for all n ≥ 1, and it is enough to show that each of these induced maps is an isomorphism. For n = 1, we have B/J(B) ∼ = C/J(C) ∼ = Endkr (Mkr ∼ = ⊕ Endk (Mi ) i

so it is clearly an isomorphism for n = 1. For n ≥ 2, we have Bn = B/J(B)n is an finite dimensional

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algebra with the same simple modules as B since Mi J n = 0. We may therefore consider the versal morphism of the swarm M of right Bn -modules, which is an isomorphism by the Generalised Burnside Theorem. Finally, any derivation D : B → Homk (Mi , M j ) satisfies D(J n ) = 0 when n ≥ 2. Therefore, we have that Ext1Bn (Mi , M j ) ∼ = Ext1B (Mi , M j ) This implies that B/J(B)n → C/J(C)n coincides with the versal morphism of the swarm M of right Bn -modules, and therefore it is an isomorphism. Proposition 3.37 implies that the assignment (A, M ) 7→ (B, M ) is a closure operation when A is a finite dimensional k-algebra and M = {M1 , . . . , Mr } is a family of finite dimensional right A-modules such that Endk (Mi ) = k for 1 ≤ i ≤ r. In other words, the algebra B = O A (M ) has the following properties: 1. The family M is the family of simple right B-modules. 2. The family M has exactly the same module-theoretic properties, in terms of extensions and matric Massey products, considered as a family of modules over B as over A. Moreover, these properties characterize the algebra of observables B = O A (M ).

3.7 Iterated extension Let M be a swarm of right A-modules, with pro-representing hull H and versal family MH . In this section, we assume that the swarm M consists of non-zero and pairwise non-isomorphic modules, and we shall describe the full subcategory ModA (M ) ⊆ ModA of right A-modules that are iterated extensions of modules in the family M . By abuse of notation, we define an iterated extension of the swarm M = {M1 , . . . , Mr } to be a pair (E,C), where E is a right A-module and C is a cofiltration of length n, given by a sequence fn

f

f

2 1 → Cn−1 → · · · → C2 − → C1 − → C0 = 0 E = Cn −

of surjective right A-module homomorphisms fi : Ci → Ci−1 with kernel Ki = ker( fi ) for 1 ≤ i ≤ n such that Ki ∼ = Mli with 1 ≤ li ≤ r (see also Subsection 3.6.3). We say that two iterated extensions (E,C) and (E ′ ,C′ ) of the swarm M of length n are equivalent if there is an isomorphism u : E → E ′ of right A-modules such that u(Ci ) = Ci′ for 1 ≤ i ≤ n. Let ModA (M ) ⊆ ModA be the minimal full subcategory that contains M and is closed under extensions. We call ModA (M ) the category of iterated extensions of M since a right A-module E is an object of this subcategory if and only if there is a cofiltration C of E such that (E,C) is an iterated extension of the swarm M in the above sense. Equivalently, a right A-module E is an object of ModA (M ) if and only if there is a filtration 0 = Fn ⊆ · · · ⊆ F1 ⊆ F0 = E of right A-modules such that the quotient Fi−1 /Fi ∼ = Mli for 1 ≤ i ≤ n. In fact, filtrations and cofiltrations are dual notions: If C is a cofiltration of E, then Fi = ker(E → Ci ) together with the natural inclusions defines a filtration of E. Conversely, if F is a filtration of E, then Ci = F0 /Fi together with the natural surjections fi : Ci → Ci−1 defines a cofiltration of E.

3.7.1 Moduli of iterated extensions The order vector of the iterated extension (E,C) of the swarm M is the vector (l1 , l2 , . . . , ln ) defined by Ki = ker( fi ) ∼ = Mli . It can be represented by an ordered quiver Γ, with nodes {1, 2, . . . , r}

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and arrows γi−1,i : li−1 → li from node li−1 to node li for 2 ≤ i ≤ n, and with a total ordering given by γ12 < γ23 < · · · < γn−1,n . We call Γ the extension type of the iterated extension (E,C). Clearly, equivalent iterated extensions have the same extension type. We let E (M , Γ) denote the set of equivalence classes of iterated extensions of the family M with extension type Γ. The path algebra k[Γ] of the ordered quiver Γ is the k-algebra with base consisting of paths γi−1,i · γi,i+1 · · · · · γ j−1, j of length j − i + 1 for 2 ≤ i ≤ j ≤ n. The product of two paths γ · γ ′ is given by juxtaposition when the last arrow γ j−1, j in the first path γ is the predecessor of the first arrow γ j, j+1 in the second path γ ′ in the total ordering, and otherwise the product γ · γ ′ = 0. We consider ei as a path of length 0 for 1 ≤ i ≤ r. It is clear that the path algebra k[Γ] is a matric algebra in ar (n). Proposition 3.38. There is a bijective correspondence between the noncommutative deformations in Def M (k[Γ]) and the set E (M , Γ) of equivalence classes of iterated extensions of the family M with extension type Γ. Proof. Let Γ be an ordered quiver, with corresponding order vector (l1 , . . . , ln ), and let α = l1 . For any noncommutative deformation MΓ ∈ Def M (k[Γ]), we may assume that MΓ = (k[Γ]i j ⊗k M j ) considered as a left k[Γ]-module, with a right multiplication of A. Let MΓ (α ) = eα · MΓ ⊆ MΓ , which is closed under right multiplication with A. A path in eα · k[Γ] is called leading if it has the form γ12 γ23 · γi−1,i and nonleading otherwise. By convention, we consider the path eα as leading. We let MΓNL (α ) = ⊕ γ · MΓ (α ) γ

where the sum is taken over all nonleading paths γ in eα · k[Γ], and notice that MΓNL (α ) ⊆ MΓ (α ) is closed under right multiplication by A. We define E = MΓ (α )/MΓNL (α ) which has an induced right A-module structure. As a k-linear space, we have that E∼ = ⊕ (γ12 γ23 . . . γi−1,i ) ⊗k Mli 1≤i≤n

We claim that there is a cofiltration C of E such that (E,C) is an iterated extension of M with extension type Γ. In fact, we may choose the cofiltration C dual to the filtration F given by Fl =

⊕

l+1≤i≤n

(γ12 γ23 . . . γi−1,i ) ⊗k Mli

for 0 ≤ l ≤ n, where Fl ⊆ E is closed under right multiplication with A. Conversely, if (E,C) is an iterated extension of M with extension type Γ, then it follows from the construction in Subsection 3.6.3 that E∼ = Kn ⊕ Kn−1 ⊕ · · · ⊕ K2 ⊕ K1 as a k-linear vector space, with Ki ∼ = Mli and with right multiplication of A given by n−1

(mn , . . . , m2 , m1 )a = (mn · a + ∑ ψain (mi ), . . . , m2 · a + ψa12(m1 ), m1 · a) i=1

for mi ∈ Ki , a ∈ A. Let Il = {i : li = l}. Then, the right multiplication of A on MΓ = (k[Γ]i j ⊗k M j ) given by
(el ⊗ ml ) · a = el ⊗ (ml · a) + ∑ ∑ γi,i+1 γi+1,i+2 . . . γ j−1, j ⊗ ψai j (m) i∈Il i+1≤ j≤n

for 1 ≤ l ≤ r, a ∈ A, ml ∈ Ml defines a noncommutative deformation MΓ ∈ Def M (k[Γ]).

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Let us write X(M , Γ) = Mor(H, k[Γ]) for the set of morphisms φ : H → k[Γ] in aˆ r . There is a natural map X(M , Γ) → Def M (k[Γ]) given by φ 7→ Mφ = Def M (φ )(MH ), and this map is surjective by the versal property. Lemma 3.39. The set X(M , Γ) = Mor(H, k[Γ]) is an affine algebraic variety. Proof. Note that k[Γ] is an algebra in ar (n). This implies that any morphism φ : H → k[Γ] in aˆ r can be identified with φn : Hn → k[Γ] since φ (I(H)n ) = 0. To prove that X(M , Γ) = Mor(Hn , k[Γ]) is an affine algebraic variety, it is enough to notice that Hn is a quotient of T1n , that Mor(T1n , k[Γ]) is isomorphic to affine space AN , where
N = ∑ dimk Ext1A (Mi , M j ) · dimk I(k[Γ])/ I(k[Γ])2 i j i, j

and that Mor(Hn , k[Γ]) ⊆ Mor(T1n , k[Γ]) is a closed subset in the Zariski topology, with equations given by the obstructions fi j (l)n ∈ T1n . Corollary 3.40. The set E (M , Γ) of equivalence classes of iterated extensions of the family M with extension type Γ is a quotient of the affine algebraic variety X(M , Γ).

3.7.2 The category of iterated extensions Let modH be the category of finite dimensional right H-modules. Then, the category of iterated extensions of a given family M of right A-modules is closely related to finitely generated H-modules: Theorem 3.41. Let A be an algebra and let M be a swarm of right A-modules. Then, there is a faithful functor i(M ) : modH → ModA (M ) that is essentially surjective. Moreover, i(M ) is an equivalence of categories if and only if M satisfies the conditions that EndA (Mi ) = k and that HomA (Mi , M j ) = 0 for 1 ≤ i, j ≤ r with i 6= j. Proof. Let N be a finite dimensional right H-module, and let Ni = Nei for 1 ≤ i ≤ r. Then, N = ⊕i Ni and the right H-module structure on N defines an algebra homomorphism (Hi j ) → (Homk (Ni , N j )) in Ar . We let E = ⊕i (Ni ⊗k Mi ). Then η : A → O(M ) defines an algebra homomorphism A → (Hi j ⊗k Homk (Mi , M j )) → (Homk (Ni , N j ) ⊗k Homk (Mi , M j )) = Endk (E) ∼ N ⊗H MH . It is clear that and N 7→ E defines a faithful functor i(M ) : modH → ModA , with E = i(M )(ki ) = Mi for each of the simple right H-modules ki . Since any finite dimensional right Hmodule is an iterated extension of the simple modules {k1 , . . . , kr }, it follows that i(M )(N) is an iterated extension in ModA (M ), and i(M ) defines a faithful functor i(M ) : modH → ModA (M ). It follows from the construction in Proposition 3.38 that i(M ) is essentially surjective: Let E be an iterated extension of M with extension type Γ. Then M corresponds to a noncommutative deformation in Def M (k[Γ]) induced by a morphism φ : H → k[Γ]. Let N be the quotient of eα · k[Γ] by the submodule with nonleading paths as base, where α = l1 . This quotient is a finite dimensional right H-module via φ , and i(M )(N) ∼ = E. To prove the last part, the conditions of the theorem are clearly necessary since the corresponding property hold for the right H-modules {k1 , . . . , kr } in modH . It is therefore enough to notice that M is the family of simple objects in ModA (M ) when EndA (Mi ) = k and HomA (Mi , M j ) = 0 for 1 ≤ i, j ≤ r with i 6= j.

Chapter 4 The Noncommutative Phase Space

In this chapter, we introduce the noncommutative phase space functor Ph : Algk → Algk , defined for associative k-algebras, and the cosimplicial structure of the infinitely iterated phase space functor Ph∗ . We study the inductive limit Ph∞ (A) of this cosimplicial object, and the induced universal derivation δ : Ph∞ (A) → Ph∞ (A), which is called the Dirac derivation. The corresponding relations to de Rham theory is treated in this general context. We also study the actions of global and local gauge groups, and show that our general models for general relativity (GR) and quantum field theory (QFT) are tightly related to invariant theory in noncommutative algebraic geometry.

4.1 Introduction to noncommutative phase spaces Let A be a finitely generated associative k-algebra, and consider the category AlgA/k . The objects in this category are the morphisms κ : A → R of associative k-algebras, and the morphisms from κ : A → R to κ ′ : A → R′ are algebra homomorphisms ψ : R → R′ such that the diagram

κ

R



A❄ ❄❄ ′ ❄❄κ ❄❄  / R′ ψ

commutes. We shall often write R for an object in AlgA/k when the morphism κ : A → R is understood from the context. Given objects R, R′ in AlgA/k , we denote the set of morphisms R → R′ in AlgA/k by Mor(R, R′ ). In other words, Mor(R, R′ ) is the set of algebra homomorphisms R → R′ that commutes with the structural algebra homomorphisms κ : A → R and κ ′ : A → R′ . Moreover, Aut(R) ⊂ Mor(R, R) denotes the subset of automorphisms. For any object R in AlgA/k , we define a k-derivation ξ : A → R to be a k-linear map ξ : A → R such that ξ (ab) = κ (a)ξ (b) + ξ (a)κ (b) for all a, b ∈ A. We often write ξ (ab) = aξ (b) + ξ (a)b when κ is understood from the context. The set of k-derivations ξ : A → R is denoted Derk (A, R) and it is a k-linear vector space. Lemma 4.1. The functor Derk (A, −) : AlgA/k → Modk is represented by an object Ph(A) in AlgA/k with structural homomorphism i0 : A → Ph(A) and universal derivation d : A → Ph(A). Proof. Since A is a finitely generated algebra, there is a free associative k-algebra F = kht1 , . . . ,tn i and a surjective algebra homomorphism u : F → A. We consider the free associative algebra Ph(F) = kht1 , . . . ,tn ,t1 , . . . ,tn i, and define an algebra homomorphism F → Ph(F) by ti 7→ ti , and a derivation d : F → Ph(F) by ti 7→ dti . It is clear that Derk (F, −) is represented by the object Ph(F) with universal derivation d : F → Ph(F). We define Ph(A) to be the quotient Ph(A) = Ph(F)/(I, dI) 97

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where I = ker(u) ⊆ A. Then Derk (A, −) is represented by the object Ph(A) with induced structural homomorphism i0 : A → Ph(A) and induced universal derivation d : A → Ph(A). We define the noncommutative phase space of a finitely generated k-algebra A to be the universal object Ph(A) in AlgA/k . It is clear that the assignment A 7→ Ph(A) is functorial, and we call the associated functor the phase space functor. With this notation, we clearly have identifications d∗ : Derk (A) → MorA (Ph(A), A) and d ∗ : Derk (A, Ph(A)) → EndA (Ph(A)) by the universal property. Corollary 4.2. The zero derivation in Derk (A) corresponds to a cosection o : Ph(A) → A of the structural morphism i0 : A → Ph(A) in AlgA/k .

4.1.1 The noncommutative Kodaira-Spencer map For any right A-module M, we may identify Ext1A (M, M ⊗A Ph(A)) with the first Hochschild cohomology group HH1 (A, Homk (M, M ⊗A Ph(A))). Therefore, there is an exact sequence Homk (M, M ⊗A Ph(A)) → Derk (A, Homk (M, M ⊗A Ph(A))) → Ext1A (M, M ⊗A Ph(A)) → 0 see Definition 3.7. We consider the natural derivation µM : A → Homk (M, M ⊗A Ph(A)) given by µM (a) = {m 7→ m ⊗ d(a)}, and define the noncommutative Kodaira-Spencer class of M to be its image c(M) ∈ Ext1A (M, M ⊗A Ph(A)). The noncommutative Kodaira-Spencer map of M is the klinear map gM : Derk (A) → Ext1A (M, M) induced by c(M). It is defined by ξ 7→ Ext1A (M, idM ⊗d∗ (ξ ))(c(M)), where ξ is identified with the morphism d∗ (ξ ) : Ph(A) → A in AlgA/k . This means that gM (ξ ) is represented by the derivation δξ : A → Endk (M) given by δξ (a) = {m 7→ m · ξ (a)}. Notice that in general, neither Derk (A) nor Ext1A (M, M) has a natural A-module structure when A is a noncommutative k-algebra. For any right A-module M, we define a noncommutative connection on M to be a k-linear map ∇ : M → M ⊗A Ph(A) such that the derivation property ∇(ma) = ∇(m)a + m ⊗ d(a) holds for all a ∈ A, m ∈ M. The Kodaira-Spencer class c(M) is the obstruction for the existence of a noncommutative connection on M: Lemma 4.3. There is a noncommutative connection on M if and only if c(M) = 0. Proof. We have that c(M) = 0 if and only if there is a k-linear map ∇ ∈ Homk (M, M ⊗A Ph(A)) such that d(∇) = µM . This means that ∇ : M → M ⊗A Ph(A) satisfies [a, ∇](m) = µM (a)(m) = m ⊗ d(a) and this is the derivation property of ∇ since it can be written as ∇(ma) − ∇(m)a = m ⊗ d(a). A noncommutative covariant derivative is a k-linear map ∇ : Derk (A) → Endk (M) such that ∇ξ (ma) = ∇ξ (m)a + m · ξ (a) for all ξ ∈ Derk (A), a ∈ A, m ∈ M. Any noncommutative connection ∇ on M induces a noncommutative covariant derivative, which by abuse of notation we also denote ∇, and by abuse of language call a connection. The induced covariant derivative is given by ∇ξ = (idM ⊗A d∗ (ξ )) ◦ ∇ where ∇ on the right-hand side refers to the noncommutative connection ∇ : M → M ⊗A Ph(A). We define the noncommutative Kodaira-Spencer kernel VM = ker(gM ) to be the kernel of the Kodaira-Spencer map. In turns out that VM ⊆ Derk (A) is a Lie-subalgebra:

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Lemma 4.4. For a derivation ξ ∈ Derk (A), we have that ξ ∈ VM if and only if there is a k-linear map ∇ξ : M → M such that the derivation property ∇ξ (ma) = ∇ξ (m)a + m · ξ (a) holds for all a ∈ A, m ∈ M. In particular, there is a covariant derivation ∇ : VM → Endk (M). Moreover, VM ⊆ Derk (A) is a Lie-subalgebra. Proof. The first part is clear. For the last part, assume that ξ , ξ ′ ∈ VM and let ∇ξ , ∇ξ ′ ∈ Endk (M) be the corresponding k-linear maps in Endk (M). We consider the Lie-product [∇ξ , ∇ξ ′ ] in the Lie algebra Endk (M). Its image in Derk (A, Endk (M)) is given by d([∇ξ , ∇ξ ′ ])(a)(m) = [∇ξ , ∇ξ ′ ](ma) − [∇ξ , ∇ξ ′ ](m) · a = m · [ξ , ξ ′](a) This is a derivation that represents gM ([ξ , ξ ′ ]). Hence, gM ([ξ , ξ ′ ]) = 0. It is clear that if c(M) = 0, then VM = Derk (A) and therefore, there is a covariant derivative ∇ : Derk (A) → Endk (M). In general, we see that if a covariant derivative ∇ : VM → Endk (M) is given, then ∇′ = ∇ + φ is a covariant derivative if and only if φ ∈ Homk (VM , EndA (M)). In this case, φ is called a potential.

4.1.2 Generalised momenta Let M be a right Ph(A)-module, and let ρM : Ph(A) → Endk (M) be the algebra homomorphism given by ρM (x) = {m 7→ mx} for any x ∈ Ph(A). Then M is also a right A-module via i0 : A → Ph(A), and we have a commutative diagram i0

/ Ph(A) A❋ ❋❋ ❋❋ ❋❋ ❋❋ ρM #  Endk (M) There is a derivation δM : A → Endk (M) associated with ρM , given by δM = d ◦ ρM . Via Hochschild cohomology, δM represents an extension ξM ∈ Ext1A (M, M). Conversely, if M is a right A-module and δM : A → Endk (M) is a derivation, then there is an algebra homomorphism ρM : Ph(A) → Endk (M) such that δM = d ◦ ρM by the universal property of Ph(A). Thus, we have the following: Lemma 4.5. There is a bijective correspondence between right Ph(A)-modules and pairs (M, δM ), where M is a right A-module and δM : A → Endk (M) is a derivation. We may think of the right A-module M as a point in the moduli space of right A-modules, and the extension ξM as a tangent at M. We therefore call the pairs (M, ξM ) a generalised momentum. It follows from the construction above that all generalised momenta are given by Ph(A)-modules. If we fix a right A-module M and ξM ∈ Ext1A (M, M), then the set of right Ph(A)-module structures on M with generalised momentum (M, ξM ) is given by Endk (M)/ EndA (M) ∼ = ker(Derk (A, Endk (M)) → Ext1A (M, M)) via d : Endk (M) → Homk (A, Endk (M)) in the Hochschild complex. Example 4.1. Let A = k[t], with phase space Ph(A) = kht, dti. A right A-module M of dimension two is given by a vector space V of dimension two and a matrix X ∈ M2 (k) giving the right action of t on V . To equip M = (V, X) with a derivation δM : A → Endk (M) is the same as to specify the

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matrix Y = δM (t) ∈ M2 (k). This corresponds to the right Ph(A)-module structure on V , where X gives the right action of t and Y gives the right action of dt. We find that dimk Ext1A (M, M) = dimk EndA (M) = dimk {Z ∈ M2 (k) : [Z, X] = 0} since Ext1A (M, M) ∼ = Derk (A, Endk (M))/ im(d), and im(d) ∼ = Endk (M)/ EndA (M). Similarly, we find that dimk Ext1Ph(A) (M, M) = 8 − 4 + dimk EndPh(A) (M) = 4 + dimk {Z ∈ M2 (k) : [Z, X] = [Z,Y ] = 0} since Ext1Ph(A) (M, M) ∼ = Derk (Ph(A), Endk (M))/ im(d), and im(d) ∼ = Endk (M)/ EndPh(A) (M). Hence, we have the inequalities 2 ≤ dimk Ext1A (M, M) ≤ 4 and 4 ≤ Ext1Ph(A) (M, M) ≤ 8 Example 4.2. Let A = k[t1 ,t2 , . . . ,tn ] be a commutative polynomial algebra in n variables. Since A = kht1 ,t2 , . . . ,tn i/I, where I is the ideal generated by the commutators [ti ,t j ], its phase space Ph(A) = kht1 , . . . ,tn , dt1 , . . . , dtn i/(I, dI). For instance, when n = 2, this gives us Ph(A) = kht1 ,t2 , dt1 , dt2 i/([t1 ,t2 ], [dt1 ,t2 ] + [t1 , dt2 ]) since d([t1 ,t2 ]) = [dt1 ,t2 ] + [t1, dt2 ]. We recall that when k is a field of characteristic 0, the n’th Weyl algebra over k is the associative k-algebra An (k) = kht1 , . . . ,tn , dt1 , . . . , dtn i/([ti ,t j ], [dti , dt j ], [dti ,t j ] − δi j ) In particular, we have that [dti ,t j ] = 0 in An (k) when i 6= j. It follows that there is a natural surjective algebra homomorphism Ph(A) → An (k) when A = k[t1 ,t2 , . . . ,tn ]. Furthermore, it is wellknown that the Weyl algebra An (k) has an order filtration {Fn }: An element of Fn is a differential operator of the form β ∑ cα ,β · t1α1 · · ·tnαn dt1 1 · · · dtnβn |β |≤n

of order |β | = β1 + · · · + βn ≤ n. The associated graded ring of An (k) with respect to the order filtration is the classical phase space of A = k[x1 , . . . , xn ], and it is given by gr An (k) ∼ = k[t1 , . . . ,tn , ξ1 , . . . , ξn ] where ξi is the image of dti . There is a natural surjective algebra homomorphism Ph(A) → gr An (k) into the classical phase space of A = k[t1 ,t2 , . . . ,tn ]. Example 4.3. Let A = k[t1 ,t2 , . . . ,tn ] be a commutative polynomial algebra in n variables, and let Ph(A) = kht1 , . . . ,tn , dt1 , . . . , dtn i/([ti ,t j ], [dti ,t j ] + [ti , dt j ]) be its phase space. A right Ph(A)-module M = M(q, p) of dimension one is given by a vector space V = k of dimension one and a right Ph(A)-module structure given by v · ti = v · qi,

v · dti = v · qi

for n-tuples q = (q1 , . . . , qn ) and p = (p1 , . . . , pn ). We can think of (q, p) as a point q ∈ An and a tangent vector p at the point q. For any right Ph(A)-modules M = M(q, p), M ′ = M(q′ , p′ ), we have that  ′ ′  2n, (q, p) = (q , p ) dimk Ext1Ph(A) (M, M ′ ) = n, q = q′ , p 6= p′   1, q 6= q′

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In fact, a derivation δ : Ph(A) → Homk (M, M ′ ) is determined by its values δ (ti ) = xi and δ (dti ) = yi , and since Homk (M, M ′ ) ∼ = k, we may assume that xi , yi ∈ k. In Ph(A), the relations [ti ,t j ] = 0 and [dti ,t j ] + [ti , dt j ] = 0 implies that [δ (ti ),t j ] + [ti , δ (t j )] = 0 [δ (dti ),t j ] + [dti , δ (t j )] + [δ (ti ), dt j ] + [ti , δ (dt j )] = 0 Let us define αi = qi − q′i and βi = pi − p′i . Then, the relations above imply that the n-tuple (x1 , . . . , xn , y1 , . . . , yn ) must satisfy αi x j − α j xi = 0 and βi x j − β j xi + αi y j − α j yi = 0 for 1 ≤ i < j ≤ n. This is a homogeneous linear system with coefficient matrix of rank n − 1 when q = q′ and p 6= p′ , and rank 2n − 2 when q 6= q′ . For instance, when n = 3 the coefficient matrix is given by   −α2 α1 0 0 0 0 −α3 α1 0 0 0 0    0 α α − 0 0 0 3 2    −β2 β1 0 −α2 α1 0    −β3 β1 −α3 α1  0 0 0 −α3 α2 0 −β3 β2 This implies that

  (q, p) = (q′ , p′ ) 2n, dimk Derk (Ph(A), Homk (M, M ′ )) = n + 1, q = q′ , p 6= p′   2, q 6= q′

We have that Ext1Ph(A) (M, M ′ ) ∼ = Derk (Ph(A), Homk (M, M ′ ))/ im(d) where d is the differential in the Hochschild complex of Ph(A) with values in Homk (M, M ′ ). The image of d is the one-dimensional subspace of derivations given by xi = αi and yi = βi when (q, p) 6= (q′ , p′ ), and this proves the result. We remark that when q 6= q′ , then the derivation δ : A → Homk (M, M ′ ) given by xi = δ (ti ) = 0,

yi = δ (dti ) = αi

for 1 ≤ i ≤ n represents a base of Ext1Ph(A) (M, M ′ ) ∼ = k, and we may think of Ext1Ph(A) (M, M ′ ) as the ′ line spanned by the tangent vector from q to q since αi = qi − q′i .

4.2 The iterated phase space functor and the Dirac derivation The phase-space construction may be iterated. Given the k-algebra A, we may form the sequence Ph∗ (A) := {Phn (A)}n≥0 , defined inductively by Ph0 (A) = A, Ph1 (A) = Ph(A), and Phn+1 (A) = Ph(Phn (A)) for n ≥ 1. Let in0 : Phn (A) → Phn+1 (A) be the canonical imbedding, and let dn : Phn (A) → Phn+1 (A) be the corresponding derivation. Since the composition of in0 and the derivation dn+1 is a derivation Phn (A) → Phn+2 (A), corresponding to the composition of homomorphisms in

in+1

0 0 Phn (A) −→ Phn+1 (A) −→ Phn+2 (A)

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there exists by universality a homomorphism in+1 : Phn+1 (A) → Phn+2 (A) such that 1 in0 ◦ i1n+1 = in0 ◦ in+1 0

and dn ◦ in+1 = in0 ◦ dn+1 1

Notice that here, and in the rest of Chapter 4 and 5, we compose functions and functors from “left to right”. Clearly, we may continue this process to construct new homomorphisms inj : Phn (A) → Phn+1 (A) for 0 ≤ j ≤ n, such that inp ◦ in+1 = in0 ◦ in+1 0 p+1

n and dn ◦ in+1 j+1 = i j ◦ dn+1

We find the following identities: inp in+1 = inq−1 in+1 q p , p