Deformation Theory of Algebras and Their Diagrams 0821889796, 9780821889794

Table of contents :
Cover
Title page
Contents
Preface
Basic notions
Deformations and cohomology
Finer structures of cohomology
The gauge group
The simplicial Maurer-Cartan space
Strongly homotopy Lie algebras
Homotopy invariance and quantization
Brief introduction to operads
𝐿_{∞}-algebras governing deformations
Examples
Index
Bibliography
Back Cover

Citation preview

Conference Board of the Mathematical Sciences

CBMS Regional Conference Series in Mathematics Number 116

Deformation Theory of Algebras and Their Diagrams Martin Markl

American Mathematical Society with support from the National Science Foundation Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

Deformation Theory of Algebras and Their Diagrams

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

http://dx.doi.org/10.1090/cbms/116

Conference Board of the Mathematical Sciences

CBM S

Regional Conference Series in Mathematics Number 116

Deformation Theory of Algebras and Their Diagrams Martin Markl

Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island with support from the National Science Foundation Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

NSF-CBMS Regional Research Conference in the Mathematical Sciences on Deformation Theory of Algebras and Modules held at North Carolina State University, Raleigh, NC, May 16–20, 2011 Partially supported by the National Science Foundation. The author acknowledges support from the Conference Board of the Mathematical Sciences and NSF grant DMS-1040647; ˇ the Eduard Cech Institute P201/12/G028; and RVO: 67985840 2010 Mathematics Subject Classification. Primary 13D10, 14D15; Secondary 53D55, 55N35.

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Library of Congress Cataloging-in-Publication Data Markl, Martin, 1960-author. [Lectures. Selections] Deformation theory of algebras and their diagrams / Martin Markl. p. cm. — (Regional conference series in mathematics, ISSN 0160-7642 ; number 116) Covers ten lectures given by the author at the NSF-CBMS Regional Conference in the Mathematical Sciences on Deformation Theory of Algebras and Modules held at North Carolina State University, Raleigh, NC, May 16–20, 2011. Includes bibliographical references and index. ISBN 978-0-8218-8979-4 (alk. paper) 1. Algebra, Homological–Congresses. I. Conference Board of the Mathematical Sciences, sponsoring body. II. NSF-CBMS Regional Conference in the Mathematical Sciences on Deformation Theory of Algebras and Modules (2011 : Raleigh, NC) III. Title. QA169.M355 2012 512.64—dc23

2012025195

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Contents Preface Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Index

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Basic notions Deformations and cohomology Finer structures of cohomology The gauge group The simplicial Maurer-Cartan space Strongly homotopy Lie algebras Homotopy invariance and quantization Brief introduction to operads L∞ -algebras governing deformations Examples

Bibliography

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vii 1 19 25 45 55 69 75 85 99 113 121

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Preface This monograph covers ten lectures given by the author at the North Carolina State University at Raleigh, NC, during the week May 16–20, 2011. The choice of topics was a result of a compromise, given by the fact that the audience consisted of both graduate students and specialists in the field. The author resisted the temptation to devote the talks to his own results and attempted to present the material scattered in the literature in a compact fashion. Also, the level of generality was determined by the purpose of the book. We decided to focus on deformations over local complete Noetherian rings (which of course includes the Artin case), though more general bases, as formal dg-commutative algebras or formal dg-schemes, can be considered. The complete Noetherian case covers most types of deformations of algebraic structures a working mathematician meets in his/her professional life. The place for the conference was sensibly chosen, because the birth of deformation theory as we understand it today is related to this part of the globe. Mike Schlessinger and Jim Stasheff worked at Chapel Hill, only 28 miles from Raleigh—Mike contributed the Artin ring approach to deformation theory and, together with Jim, introduced the intrinsic bracket. They also wrote a seminal paper that predated modern deformation theory with its emphasis on the moduli space of the Maurer-Cartan equation. Tom Lada, who has spent most of his professional career at Raleigh, together with Jim, introduced L∞ -algebras. And, of course, the life of Murray Gerstenhaber, the founding father of algebraic deformation theory, is connected to Philadelphia, a 7 hour drive from Raleigh. Moreover, the author of this monograph worked as a Fulbright fellow for two fruitful semesters in Chapel Hill with Jim and Tom. The book consists of 10 chapters which more or less correspond to the material of the respective talks. Chapters 1–3 review classical Gerstenhaber’s deformation theory of associative algebras over a local complete Noetherian ring. In chapters 4 and 5, which are devoted to Maurer-Cartan elements in differential graded Lie algebras, the moduli space point of view begins to prevail. In chapter 6 we recall L∞ -algebras and, in chapter 7, the related simplicial version of the Maurer-Cartan moduli space, and prove its homotopy invariance. As an application, we review the main features of Kontsevich’s approach to deformation quantization of Poisson manifolds. In chapters 8–9 we describe a construction of an L∞ -algebra governing deformations of a given class of (diagrams of) algebras. The last chapter contains a couple of explicit examples and indicates possible generalizations. Our intention was to make the presentation self-contained, assuming only basic knowledge of commutative algebra, homological algebra and category theory. Suitable references are [AM69, HS71, ML63a, ML71]. We sometimes omitted vii Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

viii

PREFACE

technically complicated proofs when a suitable reference was available. Namely, we did not prove Theorem 6.13 about the Kan property of the induced map between the simplicial Maurer-Cartan spaces. Since operads are not the central topic of this book, we also omitted proofs of statements from operad theory used in chapters 8 and 9. On the other hand, we explained in detail the relation between the uniform continuity of algebraic maps and topologized tensor products and included proofs of the related statements, as this subject does not seem to be commonly known and the literature is scarce. This monograph is not the first text attempting to present algebraic deformation theory. The classical theory is the subject of [GS88], there are lecture notes [DMZ07] and very recent shorter accounts [Fia08, Gia11]. Useful historical remarks can be found in [Pfl06]; an annotated historical bibliography is contained in [DW], perturbations, deformations, variations and “near misses” are treated in [Maz04]. There is also the influential though still unfinished book [KS]. Acknowledgments. I would like to express my thanks to the organizers of the conference, namely to Tom Lada and Kailash Mishra, for the gigantic work they have done. I am indebted also to the audience, which demonstrated striking tolerance to my halting English. During my work on the manuscript, I enjoyed the stimulating atmosphere of the Universidad de Talca and of the Max-Planck-Institut f¨ ur Mathematik in Bonn. In formulating my definition of A∞∞ -algebras I profited from conversations with M. Doubek and M. Livernet. Also, comments and suggestions from T. Giaquinto, A. Lazarev, M. Manetti, J. Stasheff and D. Yau were very helpful. I wish to thank, in particular, Tom Lada for reading the drafts of the manuscript and correcting typos and the worst of my language insufficiency. Martin Markl

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PREFACE

ix

Conventions. Most of the algebraic objects will be considered over a fixed field of characteristic zero, although some results remain true in arbitrary characteristic, or even over the ring of integers. The symbol ⊗ will denote the tensor product over and Lin(−, −) the space of -linear maps. By Span(S) we denote the -vector space spanned by the set S. The arity of a multilinear map is the number of its arguments. For instance, a bilinear map has arity two. We will denote by X or simply by  when X is understood, the identity endomorphism of an object X (set, vector space, algebra, &c.). The symbol Sn will refer to the symmetric group on n elements. We will observe the usual convention of calling a commutative associative algebra simply a commutative algebra. We denote by  the ring of integers and by  the set {1, 2, 3, . . .} of natural numbers. The adjective “graded” will usually mean -graded, though we will also consider non-negatively or non-positively graded objects; the actual meaning will always be clear from the context. The degree of an homogeneous element a will be denoted by deg(a) or by |a|. The grading will sometimes be indicated by ∗ in sub- or superscript, the simplicial and cosimplicial degrees by •. If we write v ∈ V ∗ for a graded vector space V ∗ , we automatically assume that v is homogeneous, i.e. it belongs to a specific component of V . The abbreviation ‘dg’ will mean ‘differential graded.’ Since we decided to require that the Maurer-Cartan elements are placed in degree +1, our preferred degree of differentials is +1. As a consequence, resolutions are non-positively graded. An unpleasant feature of the graded word is the necessity to keep track of complicated signs. We will always use the Koszul sign convention requiring that, whenever we interchange two graded objects of degrees p and q, respectively, we change the overall sign by (−1)pq . This rule however does not determine the signs uniquely. For instance, in Remark 3.51 we explain that the signs in the definition of strongly homotopy algebras depend on the preference for the inversion of the tensor power of the suspension. We will use the sign convention determined by requiring that (i) all terms in the L∞ -Maurer-Cartan equation come with the + sign,1 and that (ii) the intrinsic bracket (9.22) agrees, in the associative algebra case, with the one of [Ger63]. Requirement (i) fixes the signs in L∞ -algebras. Requirement (ii) introduces the correction (−1)k+1 to formula (9.4) and affects the sign in (9.13). Since A∞ -algebras are Maurer-Cartan elements in the extended Gerstenhaber-Hochschild dg-Lie algebra (3.24), (ii) in turn determines the convention for A∞ -algebras. An unfortunate but necessary consequence is the minus sign in the expression (9.15) for the curvature and in the related formulas.

1 This

convention is used in [Get09a].

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http://dx.doi.org/10.1090/cbms/116/01

CHAPTER 1

Basic notions Augmented rings. By a ring we understand a commutative associative unital algebra1 over a basic field , that is, the word “ring” shall mean a ring in the sense of Chapter 1 in [AM69] which is simultaneously a -vector space and whose structure operations are compatible with the scalar multiplication, i.e. α(r  + r  ) = αr  + αr  , and α(r  r  ) = (αr  )r  = r  (αr  ), for α ∈ , r  , r  ∈ R. Definition 1.1. Let R be a ring with unit e and ω : → R the morphism given by ω(1) := e. A morphism  : R → is an augmentation of R if  ω =  or, diagrammatically,  R 3  ω 6    A ring with an augmentation is an augmented ring. The subspace R := Ker  is called the augmentation ideal of R. Since the quotient R/R is isomorphic to the field , the augmentation ideal is always maximal. In this way, each augmentation determines a maximal ideal in R. Vice versa, each maximal ideal m ⊂ R defines an augmentation R → R/m over the field R/m which however, as we will see in Example 1.3 below, need not be isomorphic to the basic field . Example 1.2. The unital ring [[t]] of formal power series with coefficients in is augmented, with the augmentation  : [[t]] → given by   i  ai t := a0 . i≥0

It turns out that [[t]] is a local Noetherian ring, with the unique maximal ideal (t) and residue field , see [AM69, Chapter 1] for the terminology. Example 1.3. Every α ∈ determines an augmentation α : [t] → of the polynomial ring [t] given by α (f ) := f (α), for f ∈ [t]. On the other hand, given an augmentation  : [t] → , take α := (t). It is clear that, for this α,  = α . There is therefore a one-to-one correspondence between augmentations of [t] and points in the affine plane . The augmentation ideal of α : [t] → is the maximal ideal generated by (t − α). If is algebraically closed, then this assignment is one-to-one, i.e. there 1 See

Definition 1.15 below. 1

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2

1. BASIC NOTIONS

is a correspondence between augmentations of [t] and maximal ideals in [t], see [Har77, Example 1.4.4]. For a general field , there may be more maximal ideals in [t] than augmentations  : [t] → . For instance, the ring [t] of polynomials with real coefficients contains the maximal principal ideal generated by (1 + t2 ), but the quotient [t]/(1 + t2 ) is isomorphic to the field of complex numbers . This isomorphism is induced by the ring √ morphism (augmentation over )  : [t] →  given by (t) = −i, where i := −1 is the imaginary unit. Example 1.4. The truncated polynomial ring [t]/(tn+1 ), n ≥ 1, is augmented, with the augmentation     ai ti := a0 . 0≤i≤n n+1

It turns out that [t]/(t ) is a local Artin ring, with the unique maximal ideal (t) and residue field – for the terminology see again [AM69, Chapter 1]. The particular case n = 1 leads to the ring D := [t]/(t2 ) of dual numbers. In the rest of this chapter, R will be an augmented ring, with the augmentation  : R → and the unit map ω : → R. Modules over augmented rings. By an R-module we will understand a left R-module, i.e. a module in the sense of [AM69, Chapter 2] or [ML63a, §I.1]. As usual, a vector space is a module over a field, in most cases over our basic field . Bimodules are defined in [ML63a, §V3]. Let us formulate a couple of useful remarks. A unital augmented ring R is a - -bimodule (that is, left - right - bimodule), with the bimodule structure induced by the unit map ω in the obvious manner. Likewise, is an R-R bimodule, with the structure induced by the augmentation . For a -vector space V and a unital augmented ring R we denote by RV  the tensor product R ⊗ V , with the left R-module action r  (r  ⊗ v) := r  r  ⊗ v, for r  , r  ∈ R and v ∈ V . It is clear that RV  together with the natural -linear inclusion ι : V ∼ = 1 ⊗V → RV  is the free R-module generated by V . This means that, for every R-module M and a -linear map φ : V → M , there exists a unique R-module morphism Φ : RV  → M making the diagram ι - RV  V HH Φ φ HH j ? H M commutative. We will use both notations for RV  = R ⊗ V . The advantage of RV  is that it is shorter and that it emphasizes the left R-module action, while R ⊗ V refers directly to the tensor product structure. Topologies and completions. To include formal deformations2 into our general setup, it will be necessary to introduce a completed version of the free Rmodule RV . To this end we recall some basic facts from Chapter 10 of [AM69], which, along with [Lef42, Chapter II], should serve as the basic reference for this 2 See

Definition 1.24 on page 14.

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1. BASIC NOTIONS

3

subsection. Some relevant notes on topologies can also be found in Section 4.1 of [CP94]. Suppose we are given a descending sequence G1 ⊃ G2 ⊃ G3 ⊃ · · · of subgroups of an Abelian group G. Then G has the unique linear topology with {Gi }i≥1 a fundamental system of neighborhoods of 0. The completion with respect to this topology equals the inverse limit  := lim G/Gi , G ←− i

 i }i≥1 , where with the topology given by the fundamental system {G  i := lim Gi /(Gi ∩ Gj ). G ←− j

 can also be described explicitly as The completion G    = (g1 , g2 , g3 , . . .); gi ∈ G/Gi , gi = πij (gj ), ∀ i ≤ j , (1.1) G  consists where πij : G/Gj → G/Gi is the canonical projection. So the completion G of sequences (g1 , g2 , g3 , . . .) of elements which are compatible in that gi = πij (gj ),  for ∞all i ≤ j. In this description, G appears a subspace of the Cartesian product i=1 G/Gi of discrete spaces, with the induced topology. It is a standard fact that  is a Hausdorff space. We will often tacitly use the isomorphism the completion G of the (discrete) quotients [AM69, Corollary 10.4]: (1.2)

 G  i , i ≥ 1, G/Gi ∼ = G/

An important special case of the above situation is provided by a ring R with a distinguished ideal a, together with an R-module M . The descending sequence Mn := an M , n ≥ 1, determines the a-adic topology of M and one can form the with respect to this topology. In particular, one can consider a local completion M  with respect to ring R = (R, m) as a module over itself, and take its completion R  is an the m-adic topology. Recall that R is complete if the canonical map R → R isomorphism. Example 1.5. It is easy to prove that the completion of the ring of polynomials [t] with respect to the (t)-adic topology equals the local ring [[t]] of formal power series. Since the completion of any ring is complete, [[t]] is a complete ring. Moreover, the completion of a Noetherian ring is Noetherian and [t] is Noetherian by the Hilbert basis theorem, so [[t]] is also Noetherian [AM69, Corollary 10.27]. Example 1.6. In an Artin local ring (R, m), mn = 0 for n sufficiently large.  = R and R is complete. It is Therefore the m-adic topology of R is discrete, so R a standard fact that each Artin ring is Noetherian. Suppose that R = (R, m) is a compete local Noetherian ring with residue field .  = R ⊗  V the m-adic completion of the R-module RV , Denote by RV  := lim R/mi ⊗ V. (1.3) RV ←− i

 is given by by the fundamental system The linear topology of RV  = R ⊗  V ⊃ m⊗  R ⊃ m2 ⊗  R ⊃ m3 ⊗  R ⊃ · · · ⊃ {0} RV where

 V := lim mi /mj ⊗ V, for i ≥ 0. mi ⊗ ←− j

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4

1. BASIC NOTIONS

Isomorphism (1.2) describes the quotients as /(mi ⊗  V)∼ RV = R/mi ⊗ V, for each i ≥ 0. By (1.1), the inverse limit in (1.3) equals    = (x1 , x2 , x3 , . . .); xi ∈ R/mi ⊗ V, πij (xj ) = xi , ∀ i ≤ j (1.4) RV with the component-wise R-module structure. In this description, the sub-module  V of R ⊗  V consists of sequences (x1 , x2 , x3 , . . .) such that xj = 0 for j ≤ i. mi ⊗ One has, for each i ≥ 0, the inclusion (1.5)

 V ) ⊂ mi ⊗ V mi (R ⊗

V which may, in general, be a proper one. There is a natural map i : R ⊗ V → R ⊗ given by i(a) := ([a]1 , [a]2 , [a]3 , . . .), a ∈ R ⊗ V, where [a]n is, for n ≥ 1, the equivalence class

of a in R/mn ⊗ V . Clearly, i(a) = 0 n means that a ∈ m ⊗ V for each n ≥ 1. Since n≥1 (mn ⊗ V ) = ∅ (we assume that R is complete), the map i is a monomorphism. We may use it to identify R ⊗ V  V . It is a standard fact that R ⊗ V is dense in R ⊗  V. with a subspace of R ⊗ Finally, one has the composed inclusion of -vector spaces V ι : V → R ⊗ V → R ⊗ given, in the language of (1.4), by ι(v) := (1 ⊗ v, 1 ⊗ v, 1 ⊗ v, . . .), for v ∈ V . ι  is the free complete topological RIt is easy to show that the object V → RV module generated by V – it has a universal property in the category of complete topological R-modules similar to that of RV .

 is best explained when Example 1.7. The difference between RV  and RV we take as R the power series ring [[t]] recalled in Example 1.2. The module  = [[t]] ⊗  V then consists of expressions RV (1.6)

v0 + v1 t + v2 t2 + v3 t3 + · · · , v0 , v1 , v2 , . . . ∈ V,

which can be understood as power series with coefficients in V . For this reason, one is, up to isomorphism,  V by V [[t]]. The [t]-module RV sometimes denotes [[t]] ⊗ characterized by the property that it is flat, (t)-adically complete, and ∼ V. /tRV  = RV  V consisting of The uncompleted RV  = [[t]] ⊗ V is the subspace of [[t]] ⊗ expressions (1.6) such that the coefficients v0 , v1 , v2 , . . . span a finite-dimensional subspace of V . In particular, for V finite-dimensional, one has a [[t]]-module V ∼ isomorphism [[t]] ⊗ = [[t]] ⊗ V . The observation made in Example 1.7 is a particular case of: Proposition 1.8. Suppose that either V is a finite dimensional -vector space and R a local complete Noetherian ring, or V is arbitrary and R is Artin. Then one has an isomorphism  ∼ RV = RV  of R-modules.

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1. BASIC NOTIONS

5

Proof. For V finite dimensional and R complete, the proposition follows from [AM69, Proposition 10.13]. If R is Artin, mi = 0 for i sufficiently large, so the  is therefore discrete and inverse limit in (1.3) stabilizes. The topology of RV  ∼ RV  = RV  as (discrete) topological R-modules. Lemma 1.9. Let U and V be (discrete) vector spaces, and R = (R, m) a local complete Noetherian ring with residue field . Then there is a natural one-to-one correspondence   ∼   =  ←→ , RV  : Ψ, (1.7) Φ : Lin U, RV Lin cR RU where Lin(−, −) denotes, as usual, the space of -linear maps and Lin cR (−, −) the space of continuous R-linear maps. Moreover, (1.7) restricts, for each k ≥ 0, to the isomorphism    ∼  = , mk ⊗  V ←→  V : Ψk . Lin cR RU Φk : Lin U, mk ⊗  in Proof. The lemma, of course, follows from the universal property of RU the category of complete topological R-modules, but we include a direct proof here. Let us define first the correspondences Φ and Ψ.  denote by φ˜ : R ⊗ U → R ⊗  V its R-linear For a -linear map φ : U → RV ˜ extension given by φ(r ⊗ u) := rφ(u), for u ∈ U and r ∈ R. Clearly, ˜ i ⊗ U ) = mi φ(U ) ⊂ mi (R ⊗  V ), φ(m ˜ i ⊗ U ) ⊂ mi ⊗  V . The map φ˜ therefore induces a map so, by (1.5), φ(m V R⊗ R⊗U −→ i i V m ⊗U m ⊗ of the quotients which, combined with the isomorphisms V R⊗U ∼ R⊗ ∼ = R/mi ⊗ U and = R/mi ⊗ V i i V m ⊗U m ⊗ gives a map φ˜i : R/mi ⊗ U → R/mi ⊗ V . Define finally, in the notation (1.4) for elements of the inverse limits, Φ(φ)(x1 , x2 , x3 , . . .) := (φ˜1 (x1 ), φ˜2 (x2 ), φ˜3 (x3 ), . . .).  → RV  is a well-defined continuous map. It is easy to verify that Φ(φ) : RU The definition of the inverse correspondence Ψ is even simpler. Given an R → RV , Ψ(f ) : U → RV  is the composition linear map f : RU  −→ RV . Ψ(f ) : U → RU ι

f

It is clear that Ψ ◦ Φ = . It therefore remains to prove that Ψ is a monomorphism.  → RV  and prove that So assume that Ψ(f ) = 0 for a continuous f : RU  then f = 0. Since R ⊗ V is dense in RV , it is enough to show that f (r ⊗ v) = 0 for r ∈ R and v ∈ V . But this is obvious, since f (r ⊗ v) = rf (1 ⊗ v) = rf (ι(v)) = rΨ(f )(v) = 0. We leave the proof that the isomorphisms Φ resp. Ψ restrict to Φk resp. Ψk as an exercise. 

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6

1. BASIC NOTIONS

Example 1.10. Let [[t]] be the formal power series ring and U , V (discrete)  U → [[t]] ⊗ V -vector spaces. Let us show that each [[t]]-linear map f : [[t]] ⊗ is automatically continuous. We verify this fact by proving for each sequence {an }∞ 1 converging to  that, ∞  U , the sequence f (an )  V . The conconverges to f (a) in [[t]] ⊗ a ∈ [[t]] ⊗ 1 ∞ vergence {an }0 → a means that, for each n ≥ 0 there exists k ≥ 1 such that  U . It is obvious from the description of [[t]] ⊗  U in terms of power a − ak ∈ (tn ) ⊗ series with coefficients in U given in Example 1.7 that the last condition in fact says that a − ak is divisible by tn :  U. a − ak = tn · unk , for some unk ∈ [[t]] ⊗    V , which shows that f (an ) ∞ converges We conclude that f (a) − f (ak ) ∈ (t)n ⊗ 1  V. to f (a) in the topology of [[t]] ⊗ We leave as an exercise based on Theorem 11.22 of [AM69] to prove the following generalization of Example 1.10. Proposition 1.11. Let R be a regular3 local complete Noetherian ring and U ,  → RV  is continuous. V discrete vector spaces. Then each R-linear map f : RU Topologized tensor products. Suppose we are given a ring S and topological S-modules M and N . One may topologize the tensor product M ⊗S N by requiring that the subspaces (1.8)

M ⊗S V + U ⊗S N ⊂ M ⊗S N,

where U (resp. V) are open S-submodules of M (resp. N ), form a basis of open neighborhoods of zero in M ⊗S N . This topology has a certain universal property with respect to uniformly continuous maps which we formulate below. Let us recall some necessary definitions. A uniformity on a set X is a system  of neighborhoods of the diagonal   Δ(X) := (x, x) | x ∈ X ⊂ X × X satisfying suitable axioms [Kel55, Chapter 6]. A set with a uniformity is called a uniform space. Each uniformity induces a topology on X, with a basis of open neighborhoods of x ∈ X given by the sets Ux := {x ∈ X | (x , x) ∈ U}, where U ∈ . A map f : (X, ) → (Y, ) is uniformly continuous if, for each V ∈ , there exists U ∈  such that   (x1 , x2 ) ∈ U =⇒ f (x1 ), f (x2 ) ∈ V. Each uniformly continuous map is continuous with respect to the induced topologies. The cartesian product X1 × X2 of uniform spaces (Xi , i ), i = 1, 2, has a uniformity 1 × 2 given by the subsets U1 × U2 ⊂ (X1 × X1 ) × (X2 × X2 ) ∼ = (X1 × X2 ) × (X1 × X2 ) = Δ(X1 × X2 ), where Ui ∈ i . The induced topology on X1 × X2 is the product of the induced topologies. 3 See

[AM69, Theorem 11.22] for a definition of regular local rings.

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1. BASIC NOTIONS

7

Important examples of uniform spaces are provided by topological linear spaces. Such a space X is uniform, with the uniformity  given by the sets   (x1 , x2 ) ∈ X × X | x1 − x2 ∈ U , where U runs over a basis of open neighborhoods of 0 in X. It is clear that the topology induced by  is the original one. A linear map between linear topological spaces is continuous if and only if it is uniformly continuous. Let us finally formulate the universal property of the topology (1.8). Proposition 1.12. Suppose that M , N and Z are topological S-modules. The map F : M ⊗S N → Z induced by an S-bilinear map F : M × N → Z is continuous if and only if F is uniformly continuous.4 Proof. The continuity of F means by definition that, for each open neighborhood W of 0 in Z, there exist open S-linear subspaces U ⊂ M and V ⊂ N such that (1.9) F(M ⊗S V + U ⊗S N ) ⊂ W. By the definition of F and its linearity one has F(M ⊗S V + U ⊗S N ) = F(M ⊗S V) + F(U ⊗S N )     = SpanS F (M, V) + SpanS F (U, N ) , where SpanS (−) denotes the S-linear envelope. We see that (1.9) is equivalent to F (M, V) ⊂ W & F (U, N ) ⊂ W.

(1.10)

On the other hand, the uniform continuity of F means that, for each open neighborhood W of 0 in Z, there exist neighborhoods U ⊂ M and V ⊂ N such that (1.11)

x − x ∈ U & y  − y  ∈ V =⇒ F (x , y  ) − F (x , y  ) ∈ W.

One has, by bilinearity, F (x , y  ) − F (x , y  ) = F (x − x , y  ) + F (x , y  − y  ). Assume the inclusions (1.10). Then F (x − x , y  ) ∈ W because F (U, N ) ⊂ W, and F (x , y  − y  ) ∈ W because F (M, V) ⊂ W. So (1.10) implies (1.11). Putting y  = y  = y in (1.11) we get that F (x , y) − F (x , y) = F (x − x , y) ∈ W, whenever x − x ∈ U, therefore F (U, N ) ⊂ W. The same argument establishes the inclusion F (M, V) ⊂ W, thus (1.11) implies (1.10). The lemma is proved.  The tensor product − ⊗S − with the topology (1.8) is suited for studying multilinear maps and adequate to our purposes. It makes the category of topological S-modules a symmetric monoidal category with the unit object S. This, by definition, means the existence of natural isomorphisms of topological S-modules S ⊗S M ∼ = M ⊗S S ∼ =M M ⊗S N ∼ = N ⊗S M M ⊗S (N ⊗S O) ∼ = (M ⊗S N ) ⊗S O 4 Bilinear

(unitality), (commutativity), (associativity),

maps are not linear, so their continuity does not imply their uniform continuity.

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8

1. BASIC NOTIONS

satisfying appropriate coherence relations, see [ML63b]. Also the quotients by open submodules are easy to describe; by standard linear algebra, M ⊗S N ∼ = M/U ⊗S N/V. M ⊗S V + U ⊗S N Therefore the completion of the tensor product M ⊗S N with respect to topology (1.8) equals (1.12)

 S N := lim M⊗

←− U, V

M/U ⊗S N/V.

We say that an algebraic structure is topological if all its structure operations are uniformly continuous. There exits another topology on M ⊗S N having the universal property of Proposition 1.12 with respect to all continuous S-bilinear maps. This topology, however, does not have nice properties and it does not seem to play any rˆ ole in multilinear algebra, see the discussion in [BH96, §24]. There are yet some important cases where R-bilinear continuous maps are automatically uniformly continuous. Let us prove the following Lemma 1.13. Let R be a local complete Noetherian ring with residue field and V1 , V2 , W discrete -vector spaces. Any continuous R-bilinear map  V1 ) × (R ⊗  V2 ) → R ⊗ W F : (R ⊗ is uniformly continuous. There is thus an one-to-one correspondence between continuous R-bilinear maps  V1 ) × (R ⊗  V2 ) → R ⊗ W (R ⊗ and R-linear continuous maps  V1 ) ⊗R (R ⊗  V2 ) → R ⊗  W. (R ⊗  V1 (resp. mk ⊗  V2 , resp. mk ⊗  W ), k ≥ 0, Proof. The R-submodules mk ⊗  V2 , resp. R ⊗  W ).  V1 , (resp. R ⊗ form a basis of open neighborhoods of 0 in R ⊗ By (1.10), the uniform continuity of F will therefore be established if we prove that, for each k ≥ 0, there exist k1 , k2 ≥ 0 such that (1.13)

 V1 , mk1 ⊗  V2 ) ⊂ mk ⊗  V2 ) ⊂ mk ⊗  W and F (mk2 ⊗  V1 , R ⊗  W. F (R ⊗

Let us prove that the first inclusion is satisfied with k1 := k. Since F is  W is complete, it is enough to continuous separately in each variable and mk ⊗ verify the inclusion (1.14)

 V1 , mk ⊗ V2 ) ⊂ mk ⊗  W. F (R ⊗

The R-bilinearity of F implies that (1.15)

 V1 , mk ⊗ V2 ) = mk F (R ⊗  V1 , V2 ). F (R ⊗

 V1 , V2 ) ⊂ R ⊗  W and mk (R ⊗  W ) ⊂ mk ⊗  W , (1.15) implies (1.14) Since F (R ⊗ and thus also the first inclusion of (1.13) with k1 = k. The second inclusion can be treated analogously.  Examples 1.14. Let R be a complete Noetherian local ring and V a discrete  = R ⊗  V of (1.3) is a particular instance of k vector space. The completed RV the completed tensor product (1.12), with S = , M = R, and discrete N = V .

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1. BASIC NOTIONS

9

∼ V , where  : R → is the Consider the map  ⊗ V : R ⊗ V → ⊗ V = augmentation. There are two open subspaces of , {0} and the whole . It is clear that both subspaces m ⊗ V = ( ⊗ V )−1 (0) and R ⊗ V = ( ⊗ V )−1 ( ) are open in the topology (1.8), so ( ⊗ V ) is continuous. The space V is, as each discrete space, complete, so  ⊗ V uniquely extends into a continuous map  V : R ⊗  V → V. ⊗

(1.16)

  Another important particular case is S = R, M = RV 1  and N = RV2  for some discrete -vector spaces V1 , V2 . The completed tensor product then equals    R RV RV 1 ⊗ 2  = lim

←− a, b

(R/ma ⊗ V1 ) ⊗R (R/mb ⊗ V2 ).

From the obvious isomorphism (R/ma ⊗ V1 ) ⊗R (R/mb ⊗ V2 ) ∼ = R/mmax{a,b} ⊗ V1 ⊗ V2 we obtain (1.17)

i ∼ ∼     R RV RV 1 ⊗ 2  = lim R/m ⊗ V1 ⊗ V2 = RV1 ⊗V2 . ←− i

Iterating (1.17) gives a natural isomorphisms k k  ∼  (1.18) V , for each k ≥ 0. R RV  = R As the last example of the completed tensor product, consider the situation  V . Since ⊗R (R/mn ⊗ V ) ∼ S = R, M = and N = R ⊗ = V for each n ≥ 1, one has (1.19)

 V ) = lim  R (R ⊗ ⊗

←− n

⊗R (R/mn ⊗ V ) ∼ = lim V ∼ = V. ←− n

 → RV , the This isomorphism induces, for each R-linear continuous φ : RV -linear map φ : V → V via the commutativity of the diagram  V)  R (R ⊗ ⊗ (1.20)

6∼ = ? V

 Rφ ⊗

-

φ

 V)  R (R ⊗ ⊗ 6∼ = ? - V.

The map (1.16) then fits into the diagram V R⊗ (1.21)

φ

 V ⊗

? V

- R⊗ V  V ⊗

φ

? - V.

Deformations of associative algebras. We will illustrate basic notions of deformation theory on the particular example of associative algebras. We will see in chapters 9 and 10 that most of the material extends to a broad class of equationally given structures as Lie, commutative associative, Poisson, Leibniz algebras, various bialgebras, and their diagrams. Recall

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Definition 1.15. An associative R-algebra 5 is a couple B = (M, μ) consisting of an R-module M with an R-bilinear multiplication μ : M × M → M satisfying a(bc) = (ab)c,

for all a, b, c ∈ M,

where we abbreviate, as usual, μ(a, b) := ab, &c. The algebra B = (M, μ) is commutative, if ab = ba,

for all a, b ∈ M.

It is unital, if there exists a unit e ∈ M satisfying ae = ea,

for each a ∈ M.

If M is a topological R-module, we assume that μ is uniformly 6 continuous. If R is the basic field , we sometimes call an R-algebra simply an algebra. The R-module M is the underlying module of the R-algebra B. A morphism f : B  → B  from the associative algebra B  = (M  , μ ) to the associative algebra B  = (M  , μ ) is a morphism f : M  → M  of the underlying modules commuting with the multiplications, that is, satisfying f μ = μ (f × f ). We remind the reader that the R-bilinearity of the multiplication of B means that, for each a , a , b , b ∈ M and r  , r  , s , s ∈ R, (r  a + r  a )(s b + s b ) = (r  s )(a b ) + (r  s )(a b ) + (r  s )(a b ) + (r  s )(a b ). The universal property of the tensor product implies that each R-bilinear map μ : M × M → M gives rise to an R-module morphism (denoted by the same symbol) μ : M ⊗R M → M . We will usually use this tensor-product notation for structure operations of algebraic systems. The associativity of μ can then be expressed as the equality μ(M ⊗R μ) = μ(μ ⊗R M ) of R-linear maps M ⊗R M ⊗R M → M . If M is topological and the multiplication uniformly continuous, then μ : M ⊗R M → M is continuous in the topology (1.8). If M is, moreover, complete, μ uniquely extends into a continuous map (denoted  R M → M from the completed tensor product. again by the same symbol) μ : M ⊗ The central definition of this chapter reads: Definition 1.16. Let A be an associative -algebra with the underlying vector space V , and R a local complete Noetherian ring with residue field . An R-deformation of A is an associative continuous7 R-algebra structure on the topo V such that the map logical R-module RV  = R ⊗ (1.22)

 V : R ⊗ V →V ⊗

induced by the augmentation  : R → is a morphism of associative -algebras. The trivial R-deformation of A is the one given by the R-linear extension of the original  V . Deformations in the above sense will sometimes be multiplication of A to R ⊗ called deformations with the base R or deformations over R. 5 We

sometimes say also an associative algebra over R. the discussion on page 8. 7 By Lemma 1.13, such a structure is automatically uniformly continuous. 6 See

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1. BASIC NOTIONS

11

Remark 1.17. We need different symbols for algebras and for its underlying modules. This will make our notation somehow heavy but, since we will sometimes consider several algebras with the same underlying module, such a distinction is necessary. We will, however, try to simplify the notation if no confusion may occur. For instance, the trivial deformation of an associative algebra A = (V, μ) will be  A. denoted by R ⊗ In the following definition, φ : V → V denotes the -linear map induced by  V → R⊗  V as in diagram (1.20). a continuous R-linear endomorphism φ : R ⊗  V, μ ) of an asso V, μ ) and (R ⊗ Definition 1.18. Two R-deformations (R ⊗ ciative algebra A with the underlying vector space V are equivalent if there exists a continuous R-algebra isomorphism ∼ =

 V, μ )  V, μ ) −→ (R ⊗ φ : (R ⊗ such that φ : V → V is the identity automorphism V of V . We denote by DefA (R) the set of equivalence classes of deformations of A with the base R. In the important particular case when R is Artin,8 all topologies involved in Definitions 1.16 and 1.18 are discrete, so we can omit the completions . If R is regular local complete Noetherian, then the continuity of all maps follows from their R-linearity, see Proposition 1.11. We can therefore avoid the topologies also in this case and work in the realm of ‘standard’ algebra. As a matter of fact, all base rings in this monograph will be of one of the above two types. Let us show that the set DefA (R) behaves functorially in R. Assume that  V, μ ) an R -deformation f : R → R is a morphism of augmented rings and (R ⊗ of A. One can easily check that μ induces an associative multiplication f! (μ) on A∼  R (R ⊗  V) R ⊗ = R ⊗ which is an R -deformation of A. We will call f! (μ ) the push-forward of the deformation μ . This construction induces a natural map (denoted by the same symbol) f! : DefA (R ) → DefA (R ). The sets DefA (R) therefore assemble into a covariant functor DefA (−) from the category of complete local Noetherian rings with a given residue field, into the category of sets. This point of view was, in the Artin case, pioneered by M. Artin and M. Schlessinger [Sch68]. We include a brief subsection devoted to this approach at the end of Chapter 4. We denote by Def A (R) the set of R-deformations of an associative algebra A as in Definition 1.16. Denote also by GA (R) the group of R-module automorphisms  V → R⊗  V such that φ = V , with the group structure given by the φ : R⊗ composition. We will call GA (R) the gauge group. An automorphism φ ∈ GA (R) acts on μ ∈ Def A (R) by φ · μ = μ , where   (1.23) μ (a, b) := φ ◦ μ φ−1 (a), φ−1 (b) , a, b ∈ R ⊗ V. The next proposition follows immediately from definitions. 8 Recall [AM69, Theorem 8.5] that Artin local rings are precisely complete Noetherian rings of global dimension zero.

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1. BASIC NOTIONS

Proposition 1.19. The set of equivalence classes DefA (R) of R-deformations of an associative algebra A is the quotient (1.24)

DefA (R) ∼ = Def A (R)/GA (R).

Variants. One can modify Definition 1.16 in several ways. For instance, one can take as R an arbitrary ring augmented over and consider deformations with the (uncompleted) R ⊗ V as the underlying space. In [Fia08], these deformations are called global. For deformations in this sense one however loses the cohomology as a tool and several other statements, as the invertibility of Proposition 1.21, cease to hold. -algebra Another modification is to define an R-deformation of an associative ∼ =  RB → A. There A as an associative R-algebra B with a -algebra isomorphism ⊗ is, however, not much to be said about R-deformations without some additional assumptions on the underlying R-module M of B. In our Definition 1.16 we assumed that it was a free complete R-module. Another assumption frequently used in algebraic geometry [Har77, Section III.§9] is that M is flat which, by definition, means that the functor M ⊗R − is left exact. One then speaks about flat deformations. If R is a local Noetherian ring, a finitely generated R-module is flat if and only if it is free (see Exercise 7.15, Corollary 10.16 and Corollary 10.27 of [AM69]). Therefore, for A with a finite-dimensional underlying vector space, free deformations are the same as the flat ones. Our restriction to free deformations includes most of instances of algebraic deformation theory, including the (mini)versal deformations recalled on page 16 and, of course, also the classical setup of [Ger64]. The R-linearity built in Definitions 1.16 and 1.18 implies the following lemma.  V, μ) be an R-deformation of A as in Definition 1.16. Lemma 1.20. Let (R ⊗ Then the multiplication μ is determined by its restriction to  V ).  V) ⊗  R (R ⊗ V ⊗ V ⊂ (R ⊗  V, μ ) is deter V, μ ) → (R ⊗ Likewise, every equivalence of deformations φ : (R ⊗  V. mined by its restriction to V ⊂ R ⊗ Proof. To prove the first part of the lemma, we invoke the isomorphism  V) ⊗  R (R ⊗  V)∼ (R ⊗ = RV ⊗ V  of (1.17) and apply Lemma 1.9 to the case U = V ⊗ V . The second part of the proposition follows from the same lemma with U = V .  The following proposition will also be useful. Proposition 1.21. Assume that V is a -vector space and R a local complete Noetherian ring with residue field .9 Then every R-linear continuous morphism  V ) → (R ⊗  V) φ : (R ⊗ of R-modules such that φ = V is invertible. The proposition will follow from: 9 This

of course includes also the Artin case.

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13

Lemma 1.22. Let (U, · , e) be an associative unital -algebra with a complete linear topology given by a descending sequence {Ui }i≥1 of ideals such that (1.25)

Ua · Ub ⊂ Ua+b , for each a, b ≥ 1.

Then each element v ∈ U such that v − e ∈ U1 is invertible. In other words, the subset (U1 + e) := {v ∈ V | v − e ∈ U1 } of U is a group. Proof. Let α := v − e so that v = e + α. For any k ≥ 1 consider the element vk−1 := e − α + α2 − α3 + · · · + (−1)k αk . By (1.25), αi ∈ Ui for each i ≥ 1 and, since U is complete, the sequence {vk−1 }k≥1 converges to an element v −1 ∈ U . One clearly has, for each k ≥ 1, e − vk−1 v = e − vvk−1 ∈ Uk+1 ,

which implies that e − v −1 v = e − vv −1 ∈ i≥1 Ui . Since complete linear spaces

are Hausdorff, i≥1 Ui = {0}, therefore v −1 v = vv −1 = e.  By (1.10), condition (1.25) implies the uniform continuity of the multiplication · : U × U → U . Notice also that all ideals Ui , i ≥ 1, must be proper. Indeed, if Ui = U for some i, then certainly U = U1 for the filtration is descending. Thus, by (1.25), Ua · U1 = Ua · U = Ua ⊂ Ua+1 .

This implies that Ua = U for each a ≥ 1, so i≥1 Ui = U = {0}, which contradicts the completeness of U . Proof of Proposition 1.21. We apply Lemma 1.22 to the unital associative  V, R ⊗  V ) of continuous R-linear maps, with the multiplication algebra Lin cR (R ⊗  V → R⊗ V given by the (point-wise) composition, and the identity map  : R ⊗ as the unit. It is easy to see that the descending filtration  V, R ⊗  V )i := Lin cR (R ⊗  V, mi ⊗  V ), i ≥ 1, (1.26) Lin cR (R ⊗  V, R ⊗  V ), the induced map φ : V → V satisfies (1.25) and that, for φ ∈ Lin cR (R ⊗ is the identity if and only if  V, R ⊗  V )1 . φ −  ∈ Lin cR (R ⊗ Lemma 1.22 now produces an inverse map φ−1 .



Another useful consequence of Lemma 1.22 is the following standard Lemma 1.23. Let R = (R, m) be a local, not necessary complete, Noetherian ring with residue field . For each n ≥ 0, the quotient an := R/mn+1 is a local Artin ring with the maximal ideal m/mn+1 . Proof. Since the completion of a local Noetherian ring is again local Noetherian [AM69, Proposition 10.16] and since  m,  R/m ∼ = R/ we may assume that R is complete. It is clear that m/mn+1 is an ideal in R/mn+1 . Since R/mn+1 ∼ R ∼ = , = m/mn+1 m

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1. BASIC NOTIONS

m/mn+1 is maximal. Let us show that any proper ideal I ⊂ a is contained in m/mn+1 . If I ⊂ m/mn+1 , then there exists α ∈ I represented by some x ∈ R satisfying (x) = 0, where  : R → R/m ∼ is the augmentation map. By = multiplying with a scalar if necessary, we may clearly achieve that (x) = 1 that is, x − 1 ∈ m. By Lemma 1.22, there exists y ∈ R such that xy = 1. Then, for the equivalence classes in a = R/mn+1 we have 1 = [x][y] = α[y], therefore 1 ∈ I, so I = a. Since m/mn+1 contains all proper ideals, it is the unique maximal proper ideal, which proves that a is local. The Artin property of a is also clear: any proper ideal  I ⊂ a is contained in m/mn+1 , so I m+1 ⊂ (m/mn+1 )m+1 = 0. Let us review three most important types of deformations. As usual, for elements a, b of an associative algebra A we denote by ab their product. Also the notation  := {1, 2, 3, . . .} of the set of natural numbers is standard. Definition 1.24. A formal deformation is a deformation over the complete local augmented ring [[t]]. Proposition 1.25. A formal deformation of A = (V, · ) is given by a family (1.27)

{μi : V ⊗ V → V | i ∈ }

satisfying, for each k ≥ 1 and a, b, c ∈ V , μk (a, b)c + μk (ab, c) +



μi (μj (a, b), c) =

i+j=k

(Dk )

aμk (b, c) + μk (a, bc) +



μi (a, μj (b, c)).

i+j=k

Proof. By Lemma 1.20, the deformed multiplication μ is determined by its re V) ⊗  [[t]] ( [[t]] ⊗  V ). Now expand μ(a, b), for a, b ∈ V , striction to V ⊗ V ⊂ ( [[t]] ⊗ into a power series μ(a, b) = μ0 (a, b) + tμ1 (a, b) + t2 μ2 (a, b) + t3 μ3 (a, b) + · · · with some -bilinear functions μi : V ⊗ V → V , i ≥ 0. Clearly, μ0 must be the original multiplication in A so, in fact, (1.28)

μ(a, b) = ab + tμ1 (a, b) + t2 μ2 (a, b) + t3 μ3 (a, b) + · · · .

It is easy to see that μ is associative if and only if (Dk ) is satisfied for each k ≥ 1.  Proposition 1.25 shows that the set Def A ( [[t]]) of formal deformations of A consists of families (1.27) satisfying the infinite system (Dk ), k ≥ 1, of quadratic equations. The trivial deformation is the one with μi = 0 for each i ≥ 1. It is also clear that     (1.29) GA [[t]] ∼ = u = V + φ1 t + φ2 t2 + φ3 t3 + · · · | φi ∈ Lin(V, V ) , with the group multiplication (V +φ1 t + φ2 t2 + φ3 t3 + · · · )(V + φ1 t + φ2 t2 + φ3 t3 + · · · ) :=

V + (φ1 + φ1 )t + (φ2 + φ1 φ1 + φ2 )t2 + (φ3 + φ2 φ1 + φ1 φ2 + φ3 )t3 + · · · ,

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1. BASIC NOTIONS

15

where φi φj denotes the standard composition of linear maps. Observe that, by Proposition 1.21, each u as in (1.29) indeed induces an invertible  V → [[t]] ⊗  V. φ : [[t]] ⊗ By (1.24),

      DefA [[t]] ∼ = Def A [[t]] /GA [[t]] . It is the quotient of an infinite dimensional affine quadratic algebraic variety, modulo an action of a pro-unipotent group. From the point of view of singularity theory, this is the worst situation. Expansion (1.28) exhibits μ as a one-dimensional family, depending on the parameter t, of associative products whose value at t = 0 is the original undeformed multiplication. Its ‘formality’ means that no kind of convergence is required, so the series (1.28) has only a ‘formal’ meaning. In [Fia08], all deformations with a complete local base are called formal. Let [t] be the polynomial ring as in Example 1.3, with the augmentation 0 : [t] → defined by 0 (f ) := f (0) ∈ , for f ∈ [t]. Associative [t]-algebra structures on the (uncompleted) [t] ⊗ V such that  ⊗ V : [t] ⊗ V → V is a morphism of associative algebras are examples of global deformations of A = (V, · ) in the sense of [FP02]. It is easy to verify that these deformations are precisely finite expressions (1.28). Definition 1.26. An infinitesimal deformation, sometimes also called a first order deformation, of an algebra A is a deformation over the local Antin ring D := [t]/(t2 ) of dual numbers. Notice that in [Fia08], all deformations over a local base (R, m) with m2 = 0 are called infinitesimal. We leave the proof of the following version of Proposition 1.25 as an exercise. Proposition 1.27. An infinitesimal deformation of A = (V, · ) is given by a linear map μ1 : V ⊗ V → V fulfilling (1.30)

aμ1 (b, c) − μ1 (ab, c) + μ1 (a, bc) − μ1 (a, b)c = 0

for each a, b, c ∈ V . Therefore Def A (D) consists of linear maps μ1 : V ⊗ V → V satisfying (1.30). It is easy to see that   GA (D) ∼ = u = V + φ1 t | φ1 ∈ Lin(V, V ) ∼ = Lin(V, V ), with the abelian group structure of point-wise addition of linear maps, and the action on μ1 ∈ Def A (D) given by (1.31)

φ1 (μ1 )(a, b) := μ1 (a, b) + φ1 (ab) − φ1 (a)b − aφ1 (b).

Not very surprisingly, the set (1.32)

DefA (D) = Def A (D)/GA (D)

of isomorphism classes of infinitesimal deformations of A is a vector space that equals the second Hochschild cohomology group HH 2 (A, A) recalled in the next chapter, see Theorem 2.3. Definition 1.28. Let n ≥ 1. An n-deformation of an algebra A is a deformation over the local Artin ring [t]/(tn+1 ).

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1. BASIC NOTIONS

1-deformations are infinitesimal (= first-order) deformations of Definition 1.26. We have the following version of Proposition 1.25 which generalizes Proposition 1.27. Proposition 1.29. An n-deformation of A = (V, · ) is given by a family {μi : V ⊗ V → V | 1 ≤ i ≤ n} of -linear maps satisfying condition (Dk ) of Theorem 1.25 for 1 ≤ k ≤ n. The proof is obvious, as well as the description     GA [t]/(tn+1 ) ∼ = u = V + φ1 t + φ2 t2 + · · · + φn tn | φ1 , . . . , φn ∈ Lin(V, V ) of the gauge group. We leave both as an exercise. (Mini)versal deformations. A deformation ω of an associative algebra A with a base S would be universal , if for any other deformation μ with the base R there exists a unique ring morphism f : S → R such that the push-forward f! (ω) of ω along f is equivalent to μ. Unfortunately, as a consequence of the fact that the category of algebraic varieties is not closed under quotients, universal deformations seldom exist, the uniqueness of f is too much to ask. Under some mild assumptions, there however exist miniversal deformations. Recall that a deformation ω of an algebra A with the base S is miniversal, if (i) for any deformation μ of A with the base R there exists a, not necessarily unique, ring morphism f : S → R such that f! (ω) is equivalent to μ, and (ii) if the maximal ideal of R satisfies m2 = 0, then f is unique. Deformations satisfying (i) only are called versal . The existence of miniversal deformations for a large class of algebras was proved in [FP02] and the citations therein; see also [SS84]. Deformations in algebraic geometry. Let us mention briefly how deformations are treated in algebraic geometry. Since we are not going to follow this direction in the sequel, we will be very telegraphic here, referring to [Har10, Ser06] for terminology and details. For a point α of a scheme Y , let (Oα , mα ) denote the local ring of α and k(α) := Oα /mα its residue field. The inclusion {α} → Y induces a morphism Spec k(α) → Y . Let E be another scheme and p : E → B a proper flat morphism. The fiber of p over α ∈ Y is the pull-back - E Fα (1.33)

? Spec k(α)

p

? - Y.

It turns out that Fα is a scheme over k(α) whose underlying topological space equals p−1 (b) ⊂ E. Let X be a scheme over k(b) isomorphic to Fb , via a fixed isomorphism which is considered to be a part of the structure. The morphism p : E → Y can be viewed as a family of deformations of the scheme X parametrized by the points of Y . The flatness and properness of the morphism p : E → Y guarantee that the fibers vary in a ‘controlled’ way, and that the above concept is invariant under the base change. An R-deformation B of an associative -algebra A as in Definition 1.16 should then ‘ideologically’ be the same as a morphism Spec B → Spec R of spectra whose

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1. BASIC NOTIONS

17

fiber F0 is isomorphic to Spec A. Such an interpretation is, however, very superficial. Besides the non-commutativity of A and B, we do not assume B to be unital, so there is no natural algebra morphism R → B that would induce the above map of spectra.

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http://dx.doi.org/10.1090/cbms/116/02

CHAPTER 2

Deformations and cohomology In this chapter we explore the rˆole of cohomology in analysis of deformations of associative algebras; the basic reference is [Ger64] and [ML63a, Chapter X]. Later, in Chapter 9, we explain how to construct a cohomology theory related to deformations of a given general type of algebras. For associative algebras, the relevant cohomology is the Hochschild cohomology whose definition we recall first. Hochschild cohomology. Let B = (M, μ) be an associative R-algebra as in Definition 1.15 and K = (N, r, l) a B-bimodule over R, i.e. an R-module N equipped with R-linear maps r : M ⊗R N → N and l : N ⊗R M → N (the right and left actions, respectively) such that a (a m) = (a a )m, a (ma ) = (a m)a and m(a a ) = (ma )a , for a , a ∈ M and m ∈ N . In the above display we abbreviated, as usual, am := l(a ⊗R m) and ma := r(m ⊗R a), a ∈ M, m ∈ N. For us, the most important bimodule will be the algebra B = (M, μ) itself, with the right and left actions given by the multiplication μ. In the following definition taken from [ML63a, X.3], kR M denotes the kth iterate of the tensor product of an R-module M , k ⊗R · · · ⊗R M . R M := M

  k times

Definition 2.1. The Hochschild cohomology of an associative R-algebra B = (M, μ) with coefficients in a B-bimodule K = (N, r, l) is the cohomology of the Hochschild complex δB,K

δB,K

δB,K

δB,K

0 −→ M −−→ CH 1R (M, N ) −−→ · · · −−→ CH kR (M, N ) −−→ · · ·  k  is the space of R-multilinear maps in which CH kR (M, N ) := Lin R R M, N ×k M → N . The coboundary operator

(2.1)

δB,K : CH kR (M, N ) → CH k+1 R (M, N ) is defined by δB,K f (a0 ⊗R · · · ⊗R ak ) := (2.2)

(−1)k+1 a0 f (a1 ⊗R · · · ⊗R ak ) + f (a0 ⊗R · · · ⊗R ak−1 )ak  − (−1)i+k f (a0 ⊗R · · · ⊗R ai+1 ai+2 ⊗R · · · ⊗R ak ), 1≤i≤k 19

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20

2. DEFORMATIONS AND COHOMOLOGY

  for a0 , . . . , ak ∈ M . Denote HH kR (B, K) := H k CH ∗R (M, N ), δB,K . If R is the ground field , we omit the subscript indicating the ground ring from the notation. As we noted in Remark 1.17, we need to distinguish between algebras (resp. bimodules) and their underlying modules. Since the graded space of Hochschild cochains depends only on the underlying modules, we denoted it CH ∗R (M, N ), not CH ∗R (B, K) as the tradition demands. On the other hand, the differential δB,K bears B and K as the subscript, because it depends also on the structure operations of the algebra B and the B-bimodule K. If K = B, we simplify the notation and write δB instead of δB,K . 2 = 0. Expanding the definition of δB,K , show Exercise 2.2. Prove that δB,K 1 that, for f ∈ CH R (M, N ), g ∈ CH 2R (M, N ) and a, b, c ∈ M ,

(2.3a) (2.3b)

δB,K (f )(a, b) = af (b) − f (ab) + f (a)b, and δB,K (g)(a, b, c) = −ag(b, c) + g(ab, c) − g(a, bc) + g(a, b)c.

When B = (M, μ) is a topological R-algebra and K = (N, r, l) a topological B-bimodule, we equip the tensor power kR M with the iterated topology (1.9) and consider also the subspace of the continuous Hochschild cochains  k  k   k (2.4) CH kR (M, N )cnt := Lin cR R M, N ⊂ Lin R R M, N = CH R (M, N ). The continuity of the multiplication in B implies that CH ∗R (M, N )cnt is a subcomplex of CH ∗R (M, N ). In the rest of this chapter we use Definition 2.1 only in the particular case when R is the ground field and B a discrete -algebra A. The Hochschild cochains of continuous algebras over general rings will resurface in the following chapter. Formulas in Exercise 2.2 offer a direct description of the moduli space (1.32) of infinitesimal deformations of an associative algebra A = (V, · ) via its Hochschild cochains. Indeed, (1.30) means by (2.3b) that μ1 ∈ Lin(V ⊗2 , V ) is a Hochschild cocycle, δA (μ1 ) = 0. If we interpret φ1 ∈ Lin(V, V ) as a cochain in CH 1 (V, V ), then (1.31) according to (2.3a) with B = K = A reads φ1 (μ1 ) = μ1 − δA (φ1 ). Thus the description (1.32) of the moduli space of infinitesimal deformations combined with the definition of the Hochschild cohomology immediately gives: Theorem 2.3. There is an isomorphism between the set of equivalence classes of infinitesimal deformations of A and the second Hochschild cohomology HH 2 (A, A) of A with coefficients in itself. Let us quote another classical result, this time concerning infinitesimal deformations. Theorem 2.4. Let A be an associative algebra such that HH 2 (A, A) = 0. Then all formal deformations of A are equivalent to the trivial deformation. Proof. Although the proof is straightforward, we give it here as an illustration of the interplay between cohomology and deformations. We use the description of formal deformations provided by Proposition 1.25, and of the corresponding gauge group given in (1.29). Let us consider a formal deformation ν of A = (V, · ) that has the form ν(a, b) = ab + νk (a, b)tk + νk+1 (a, b)tk+1 + · · · = ab + νk (a, b)tk + h.o.t., a, b ∈ V,

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2. DEFORMATIONS AND COHOMOLOGY

21

for some k ≥ 1, where h.o.t. abbreviates ‘higher order terms.’ We say that ν has the gap of length k −1. We say that ν has infinite gap, if it has the gaps of arbitrary lengths. This clearly happens if and only if ν is the trivial deformation. Condition (Dk ) for ν has only four terms: νk (a, b)c + νk (ab, c) = aνk (b, c) + νk (a, bc), ∀a, b, c ∈ V. It says, by (2.3b), that νk : V ⊗ V → V interpreted as a cochain in CH 2 (V,  V ) is a cocycle, δA (νk ) = 0. Consider now an element of the gauge group ψ ∈ GA [[t]] which has, in the presentation (1.29), the form Its inverse in GA



ψ = V + ψk tk , for some ψk ∈ Lin(V, V ).  [[t]] clearly equals

ψ −1 = V − ψk tk + ψk ψk t2k − ψk ψk ψk t3k + · · · , so it is the identity modulo tk . One sees that ψ(ν) given by formula (1.23) is of the form   ψ(ν)(a, b) = ab + νk (a, b) + ψk (ab) − aψk (b) − ψk (a)b tk + h.o.t.   = ab + νk (a, b) − δA (ψk ) tk + h.o.t., a, b ∈ V, Since νk ∈ CH 2 (V, V ) is a cocycle and HH 2 (A, A) = 0 by assumption, one can choose ψk such that νk = δA (ψk ). The formal deformation φ(ν) then has the gap of length k. Thus, if HH 2 (A, A) = 0, each formal deformation of A having the gap of length k − 1 is equivalent, via an element of the gauge group which is the identity modulo tk , to a formal deformation with the gap of length k. Let μ be an arbitrary formal deformation of A. By definition, it has the gap of length 0. Applying inductively the above modifications, we construct a sequence of formal deformations {μ(k) }k≥1 of A and a sequence of elements of the gauge group {φ(k) }k≥2 such that (i) μ(1) = μ and μ(k) has, for each k ≥ 1, the gap of length k − 1, (ii) μ(k+1) = φ(k+1) (μ(k) ) for each k ≥ 1, and (iii) φ(k) is the identity modulo tk−1 , for each k ≥ 2. One can easily verify that the sequence {φ(k) · · · φ(2)}k≥2 of products in the gauge group (t)-adically converges to an element φ ∈ GA [[t]] with the property that φ(μ) has the gap of infinite length, i.e. φ(μ) is the trivial deformation of A.  Example 2.5. Algebras satisfying HH 2 (A, A) = 0 are called absolutely rigid . The following examples of these algebras were taken from Section 4 of [Gia11]. Any separable algebra A is absolutely rigid, as it is characterized by HH n (A, K) = 0 for each A-bimodule K and each n ≥ 0. The universal enveloping algebra Ug of a finitedimensional semisimple Lie algebra g is absolutely rigid, since HH n (Ug, Ug) = 0 for n ≥ 1. For the same reason, the mth Weyl (Heisenberg) algebra Am is absolutely rigid. The next definition refers to n-deformations introduced in Definition 1.28, expressed as in Proposition 1.29. Definition 2.6. Let μ be an (n + 1)-deformation of an associative algebra A = (V, · ) given by a sequence {μ1 , . . . , μn+1 } of maps V ⊗ V → V . Then clearly μ := {μ1 , . . . , μn } is an n-deformation. It is called the restriction of μ. In this context we also say that μ is an extension of the n-deformation μ.

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22

2. DEFORMATIONS AND COHOMOLOGY

We are going to formulate another classical result by which the third Hochschild cohomology group HH 3 (A, A) contains obstructions to extension of n-deformations. Let us rearrange (Dn+1 ) of Proposition 1.24 into −aμn+1 (b, c) + μn+1 (ab, c) − μn+1 (a, bc) + μn+1 (a, b)c =    μi (a, μj (b, c)) − μi (μj (a, b), c) . =

(2.5)

i+j=n+1

Denote the function in the right-hand side by On (μ) : V ⊗3 → V and interpret it as an element of CH 3 (V, V ),    μi (a, μj (b, c)) − μi (μj (a, b), c) ∈ CH 3 (V, V ). (2.6) On (μ) := i+j=n+1

The notation is justified by the obvious fact that On (μ) depends only on μ1 , . . . , μn , i.e. on the restriction μ of μ. Using the Hochschild differential recalled in (2.3b), one can rewrite (2.5) as δA (μn+1 ) = On (μ). We conclude that, if an n-deformation μ extends to an (n + 1)-deformation μ, then On (μ) is a Hochschild coboundary. In fact, one can prove: Theorem 2.7. For an arbitrary n-deformation μ of an algebra A = (V, · ), the Hochschild cochain On (μ) ∈ CH 3 (V, V ) defined in (2.6) is a cocycle, δA (On ) = 0. Moreover, [On (μ)] = 0 in HH 3 (A, A) if and only if the n-deformation μ extends into some (n + 1)-deformation. The proof of the above theorem is tedious but straightforward and we leave it as an exercise. Theorem 2.7 has an immediate Corollary 2.8. Assume that HH 3 (A, A) = 0. Then each n-deformation of A extends into a formal deformation. In particular, every infinitesimal deformation of A extends into a formal one. Geometric deformation theory. Let V = Span{e1 , . . . , ed } be the d-dimensional -vector space with the basis e1 , . . . , ed . Denote by Ass(V ) the set of all associative algebras with the underlying space V . Such a structure is given by a linear map μ : V ⊗ V → V which is, in turn, determined by its structure constants, i.e. scalars Γlij ∈ , 1 ≤ i, j, l ≤ d such that  μ(ei , ej ) = Γlij el . 1≤l≤d

  The associativity μ ei , μ(ej , ek ) = μ μ(ei , ej ), ek of μ in terms of the structure constants reads   Γril Γljk = Γlij Γrlk , ∀ i, j, k, r ∈ {1, . . . , d}. 

1≤l≤d



1≤l≤d

4

These d polynomial equations define an affine algebraic variety, which is just another way to view Ass(V ), since every point of this variety corresponds to an associative algebra structure on V . We call Ass(V ) the variety of structure constants of associative algebras. The group GL(V ) of linear automorphisms of V acts on Ass(V ) via an action μ → μφ given by   (2.7) μφ (a, b) := φ ◦ μ φ−1 (a), φ−1 (b) ,

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2. DEFORMATIONS AND COHOMOLOGY

23

for a, b ∈ V and φ ∈ GL(V ). In geometric deformation theory we interpret an algebra A as a point of Ass(V ) and its deformations as points of some ‘small’ neigborhood of A. Definition 2.9. Let A be an algebra with the finite-dimensional underlying vector space V , interpreted as a point in the variety of structure constants Ass(V ). The algebra A is called geometrically rigid or simply rigid if the GL(V )-orbit of A in Ass(V ) is Zarisky-open. Let us remark that, if =  or , then, by [NR66, Proposition 17.1], the GL(V )-orbit of A in Ass(V ) is Zarisky-open if and only if it is (classically) open. The following statement whose proof can be found in [NR66, §5] specifies the relation between the Hochschild cohomology and geometric rigidity, compare also Propositions 1 and 2 of [F´ el81]. Theorem 2.10. Suppose that the ground field is algebraically closed. (i) If HH 2 (A, A) = 0 then A is rigid, and (ii) if HH 3 (A, A) = 0 then A is rigid if and only if HH 2 (A, A) = 0. Three concepts of rigidity. One says that a finite-dimensional associative -algebra is infinitesimally rigid (resp. analytically rigid ) if all its infinitesimal (resp. formal) deformations are equivalent to the trivial one. By Theorem 2.3, A is infinitesimally rigid if and only if HH 2 (A, A) = 0. Together with Theorem 2.4 this establishes the first implication in the following display which in fact holds over fields of arbitrary characteristic infinitesimal rigidity =⇒ analytic rigidity =⇒ geometric rigidity. The second implication above is [GS86, Theorem 3.2]. Theorem 7.1 of the same paper then says that in characteristic zero, the analytic and geometric rigidity are equivalent concepts: char. 0

analytic rigidity ⇐⇒ geometric rigidity. We saw a close relationship between the geometric properties of the GL(V )group action on the variety of structure constants Ass(V ), and the Hochschild cohomology of an associative algebra A interpreted as a point in Ass(V ). To explain this fact, consider the moduli space of associative algebra structures on V defined as the quotient Ass(V ) := Ass(V )/GL(V ). It is no longer an affine variety, but only a (possibly singular) algebraic stack (in the sense of Grothendieck). One can remove the singularities by replacing Ass(V ) by a smooth dg-scheme M. The Hochschild cohomology HH ∗ (A, A) is then the cohomology of the tangent space of this dg-scheme at the point corresponding to A, see [CFK01, CFK02]. Valued deformations. The authors of [GR04] studied R-deformations of finite dimensional algebras in the case when R was a valuation ring [AM69, Chapter 5]. In particular, they considered deformations over the non-standard extension ∗ of the field of complex numbers, and called these ∗ -deformations perturbations. They proved in [GR04, Theorem 4] that an algebra A admits only trivial perturbations if and only if it is geometrically rigid.

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http://dx.doi.org/10.1090/cbms/116/03

CHAPTER 3

Finer structures of cohomology In the previous chapter we investigated the importance of the second and the third Hochschild cohomology for deformation theory. The necessary step in understanding the rˆole of higher cohomology groups is to observe that the Hochschild cochain complex bears the structure of a dg-Lie algebra with the property that deformations are solutions of the corresponding Maurer-Cartan equation. Most of the material of this chapter is taken from [Ger63] or inspired by that paper. k Recall that, for an R-module M , CH kR (M, M ) denotes the space Lin R R M, M of R-linear maps acting on k elements from M , with values in M . (M, M ), g ∈ CH n+1 Definition 3.1. For cochains f ∈ CH m+1 R R (M, M ) and m+n+1 i ∈ {1, . . . , m + 1} define f ◦i g ∈ CH R (M, M ) as the composed map  ⊗R (i−1) ⊗ (m−i+1)  (3.1a) f ◦i g := f V ⊗R g ⊗R V R obtained by inserting g into the ith input of f . Define also  (3.1b) f ◦ g := (−1)n(i+1) f ◦i g 1≤i≤m+1

and, finally, (3.1c)

[f, g] := f ◦ g − (−1)mn g ◦ f.

The operation [−, −] is called the Gerstenhaber bracket. Remark 3.2. Formulas (3.1a)–(3.1c) are R-linear versions of the original definitions given by Gerstenhaber. The only difference is that what we denoted by f ◦i g in (3.1a) was in [Ger63] denoted f ◦i−1 g. Our notation is the ‘operadic one.’ If we consider an abstract ‘operation’ f with multiple inputs numbered from the left to the right, then f ◦i g is an obvious notation for the ‘operation’ obtained by inserting ginto the input  number i of f . Our notation also emphasizes that the collection CH kR (M, M ) k≥1 consists of the pieces of the endomorphism operad EndM of the R-module M recalled in Example 8.4. This fact has far-reaching implications for the analysis of natural operations on the Hochschild cochain complex, see [MS02, BF04]. Example 3.3. The product ϕ ◦ ϕ of ϕ , ϕ ∈ CH 1R (M, M ) is the ‘point-wise’ composition ϕ (ϕ ) of linear maps. The identity map  : M → M ∈ CH 1R (M, M ) is the two-sided unit for ◦. The ◦-operation therefore extends the standard unital associative structure on the space Lin R (M, M ) = CH 1R (M, M ) of R-linear endomorphisms of M . Before coming to the main result of this chapter, we recall the following variation on a classical definition. 25 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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3. FINER STRUCTURES OF COHOMOLOGY

  Definition 3.4. A graded Lie R-algebra is a couple L = X, [−, −] consisting of a graded R-module X with a degree 0 R-linear map [−, −] : X ⊗R X → X (the bracket) which is skew-symmetric: (3.2a)

[a, b] = −(−1)|a||b| [b, a], ∀a, b ∈ X,

and fulfills, for each a, b, c ∈ X, the graded Jacobi identity:       (3.2b) (−1)|a||c| [a, b], c + (−1)|b||a| [b, c], a + (−1)|c||b| [c, a], b = 0. A differential graded Lie R-algebra (dg-Lie R-algebra for short) is a graded Lie R-algebra L whose underlying R-module X bears a degree +1 R-linear differential d : X → X which is a degree +1 derivation with respect to the bracket: (3.2c)

d[a, b] = [da, b] + (−1)|a| [a, db], ∀a, b ∈ X.

The word ‘algebra’ in the phrase ‘Lie algebra’ usually indicates that R is a field. Lie ‘algebras’ over a general ring should then be called Lie modules. Instead of the antisymmetry of the bracket [−, −] , in an (un-graded) Lie module one sometimes demands the stronger condition [x, x] = 0 for each x ∈ X.1 However, as we agreed to assume that R is an algebra over the ground field, X will automatically be a vector space and these terminological subtleties will not be relevant.   Proposition 3.5. The shifted Hochschild cochain complex CH ∗+1 R (M, M ), δB of an associative R-algebra B = (M, μ) with the Gerstenhaber bracket (3.1c) forms a dg-Lie R-algebra. We will call it the Gerstenhaber-Hochschild algebra of B. The shift (regrading) of CH ∗+1 R (M, M ) expressed by the superscript ∗+1 means that the component CH kR (M, M ) is placed in degree k − 1. This is necessary for the Gerstenhaber bracket to be a degree 0 operation. The proof of Proposition 3.5 is a direct verification whose details can be found in [Ger63]. The first step is to realize that the ◦-operation of (3.1b) is R-linear and o+1 (M, M ), g ∈ CH n+1 satisfies, for each f ∈ CH m+1 R R (M, M ) and h ∈ CH R (M, M ),   (3.3) f ◦ (g ◦ h) − (f ◦ g) ◦ h = (−1)no f ◦ (h ◦ g) − (f ◦ h) ◦ g . In other words, the associator Φ(f, g, h) := f ◦ (g ◦ h) − (f ◦ g) ◦ h is graded symmetric in the last two variables. Objects with a bilinear operation sharing this property are known as pre-Lie R-algebras. They appear also under different names as right symmetric or Gerstenhaber algebras. Their opposite versions are known as left symmetric, Vinberg, Koszul or quasi-associative algebras,   (M, M ), ◦ is see the overviews [Bur06, Man11]. Once we establish that CH ∗+1 R a pre-Lie R-algebra, we use the standard and elementary 3.6. Let (X, ◦) be a graded pre-Lie R-algebra. Then the couple  Proposition  X, [−, −] with [−, −] : X ⊗R X → X the graded commutator [f, g] := f ◦ g − (−1)|f ||g| · g ◦ f is a graded Lie R-algebra. 1 Over

a field of characteristic = 2, [x, x] = 0 ∀x is equivalent to the antisymmetry of [−, −].

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Pre-Lie algebras are therefore instances of Lie admissible algebras which are, by definition, algebras with an operation whose commutator is a Lie bracket. Now we need only to notice that the Gerstenhaber bracket (3.1c) is the graded commutator of ◦. The Leibniz rule for δB can be established directly, which finishes the proof of Proposition 3.5. Let us formulate an elementary but useful   Lemma 3.7. Let L = X, [−, −] be a graded Lie R-algebra and χ ∈ X 1 such that [χ, χ] belongs to the center of L. Then the degree +1 map d : X → X defined by d(θ) := [χ, θ] for θ ∈ X

(3.4)

is a differential that makes L a dg-Lie R-algebra. Proof. The graded Jacobi identity (3.2b) implies that, for each homogeneous θ,       χ, [χ, θ] = −(−1)|θ|+1 χ, [θ, χ] − θ, [χ, χ] . Since [χ, χ] belongs, by assumption, to the center, the rightmost term vanishes. Now we use the graded antisymmetry [θ, χ] = (−1)|θ|+1 [χ, θ] to conclude from the above display that     χ, [χ, θ] = − χ, [χ, θ] . Since the characteristic of the ground field is zero, d2 (θ) = [χ, [χ, θ]] = 0, so d is a differential. The derivation property of d with respect to the bracket can be verified in the same way and we leave it as an exercise.  In most application of the above lemma, the expression [χ, χ] = 0 actually vanishes, but there are naturally appearing situations where [χ, χ] is a nontrivial central element. This happens for instance in the graded Lie algebra related to deformations of isomorphisms which we describe in Chapter 10. Let us show that the Hochschild differential has the form (3.4). Proposition 3.8. Let μ : M ⊗R M → M be an R-linear associative multiplication of an algebra B = (M, μ) interpreted as an element of CH 2R (M, M ), and f ∈ CH n+1 R (M, M ) be arbitrary. Then the Hochschild differential of f equals δB (f ) = [μ, f ]. Proof. By (3.1b), for f ∈ CH n+1 R (M, M ) one has  f ◦μ= (−1)i+1 f ◦i μ and μ ◦ f = μ ◦1 f + (−1)n μ ◦2 f 1≤i≤n+1

therefore, by (3.1c), [f, μ] = (−1)n μ ◦2 f + μ ◦1 f −



(−1)i+n f ◦i μ.

1≤i≤n+1

Substituting n = k−1 we recognize formula (2.2) for the Hochschild differential.



Proposition 3.9. A linear map κ : M ⊗R M → M is associative if and only if [κ, κ] = 0.

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3. FINER STRUCTURES OF COHOMOLOGY

Proof. By Definition 3.1 (with m = n = 1),  1 1 [κ, κ] = κ ◦ κ − (−1)mn κ ◦ κ = κ ◦ κ = κ ◦1 κ − κ ◦2 κ 2 2 = κ(κ ⊗ M ) − κ(M ⊗ κ), therefore [κ, κ] = 0 is equivalent to the associativity κ(κ ⊗R M ) = κ(M ⊗R κ).  Proposition 3.10. Assume that B = (M, μ) is an associative R-algebra and ν : M ⊗R M → M a map. Then μ + ν is associative if and only if (3.5)

1 δ(ν) + [ν, ν] = 0. 2

Proof. By Proposition 3.9, μ + ν is associative if and only if  1 1 1 0 = [μ + ν, μ + ν] = [μ, μ] + [ν, ν] + [μ, ν] + [ν, μ] = δB (ν) + [ν, ν], 2 2 2 which proves the statement. We used the fact that, since μ is associative, [μ, μ] = 0 by Proposition 3.9. We also observed that [μ, ν] = [ν, μ] = δB (ν) 

by Proposition 3.8.

Equation (3.5) is a particular case of the Maurer-Cartan equation in a arbitrary dg-Lie algebra:   Definition 3.11. Let L = X, [−, −], d be a dg-Lie algebra. A degree 1 element s ∈ X 1 is Maurer-Cartan if it satisfies the Maurer-Cartan equation (3.6)

1 ds + [s, s] = 0. 2

We denote by MC(L) the set of all Maurer-Cartan elements in L. Remark 3.12. The Maurer-Cartan equation is one of the most important equations in mathematics and physics. In geometry, 1 dθ + [θ, θ] = 0 2 expresses the flatness of a principal connection θ, see for instance [KN63]. In physics, it is known as the ’master equation,’ sometimes with the adjective ’classical’ or ’quantum.’ k Recall that, for a topological R-algebra B = (M, μ),  CH R (M, M )cnt denotes k the space of continuous R-linear maps Lin R R M, M . The subcomplex     ∗+1 CH ∗+1 R (M, M )cnt , δB ⊂ CH R (M, M ), δB

is obviously a dg-Lie sub-algebra of the Gerstenhaber-Hochschild algebra of Proposition 3.5. We denote this dg-Lie R-algebra of continuous Hochschild cochains by gR (B). If R is the ground field , we will as always drop the subscript specifying the ground ring. In particular, since every Hochschild cochain of a discrete -algebra A is continuous, g(A) will denote the Gerstenhaber-Hochschild algebra of all cochains.

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3. FINER STRUCTURES OF COHOMOLOGY

29

Example 3.13. It follows from the description of the ◦-product restricted to CH 1R (M, M ) given in Example 3.3 that the Lie subalgebra g0R (B) of degree 0 elements in g∗R (B) equals the standard Lie algebra endcR (M ) of continuous R-linear endomorphisms of the R-module M . Remark 3.14. Each R-module M admits the trivial associative R-algebra structure given by the null map 0 : M ⊗R M → 0 → M . Abusing the notation slightly, we denote this associative algebra by M . The differential of the related Gerstenhaber-Hochschild algebra vanishes, i.e. gR (M ) is just a graded Lie algebra. Since the Gerstenhaber bracket does not involve the algebra multiplication, gR (M ) is canonically isomorphic, as a graded non-differential Lie algebra, to gR (B) for any associative R-algebra B with the underlying R-module M . Analogously, g(V ) will denote the Gerstenhaber-Hochschild Lie algebra of the vector space V with the trivial multiplication.   Let L = U, [−, −], d be an arbitrary dg-Lie -algebra with the underlying vector space U , and a a commutative associative -algebra. It is a standard fact that the tensor product a ⊗ U carries a natural dg-Lie a-algebra structure, with the differential  ⊗ d given by (3.7a)

( ⊗ d)(a ⊗ g) := a ⊗ dg, for a ∈ a and g ∈ U,

and the bracket defined as (3.7b)

[a ⊗ g  , a ⊗ g  ] := a a ⊗ [g  , g  ], for a, a , a ∈ a and g, g  , g  ∈ U.

Let  R = (R,m) be a complete local Noetherian ring with residue field and L = U, [−, −], d a (discrete) -Lie algebra. Formulas (3.7a)–(3.7b) make R ⊗ U a dg-Lie R-algebra with continuous structure operations

 ⊗ d : R ⊗ U → R ⊗ U and [−, −] : (R ⊗ U ) ⊗R (R ⊗ U ) → R ⊗ U.  U imply The continuity of these operations together with the completeness of R ⊗ that they uniquely extend into continuous R-linear maps  U) → R ⊗  U.  d : R⊗  U → R⊗  U and [−, −] : (R ⊗  U ) ⊗R (R ⊗ ⊗

   L := R ⊗  U, [−, −],  ⊗  d is a dg-Lie R-algebra. Lemma 3.15. The object R ⊗ Proof. Let us verify the graded Jacobi identity (3.2b). It is fulfilled if and only if the ‘Jacobiator’  U ) ⊗R (R ⊗  U ) ⊗R (R ⊗  U ) → (R ⊗  U) J : (R ⊗  U , as defined, for a, b, c ∈ R ⊗       (3.8) J(a, b, c) := (−1)|a||c| [a, b], c + (−1)|b||a| [b, c], a + (−1)|c||b| [c, a], b , vanishes. The map J is, as a linear combination of compositions of continuous functions, continuous as well. Because the uncompleted R ⊗ U is a dg-Lie algebra,  U . The vanishing of the Jacobiator J vanishes on triples a, b, c ∈ R ⊗ U ⊂ R ⊗  U follows from the fact that 3R (R ⊗ U ) is dense on all triples of elements of R ⊗ 3  in R (R ⊗ U ) and from its continuity. The remaining axioms can be verified similarly. 

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Remark 3.16. Lemma 3.15 follows from a more general principle. Assume that a is a commutative associative -algebra and U an algebra of a specific type (Lie, associative, Poisson, &c). Then a ⊗ U is an algebra of the same type as U . This can be formulated more precisely using the language of operads recalled in Chapter 8. If U is an algebra over a -linear operad P, then a ⊗U is also a P-algebra. The explanation is that the operad Com governing commutative associative algebras is the unit for the tensor product of operads, that is P ⊗ Com ∼ = Com ⊗ P ∼ =P for each -linear operad P. In particular, for a local complete Noetherian ring R with residue field and a P-algebra U , R ⊗ U is also a P-algebra. The obvious continuity and completeness arguments extend this P-algebra structure to the  U. completed R ⊗ Let R be, as before, a complete local Noetherian ring with residue field and and L = U, [−, −], d a (discrete) -Lie algebra with the underlying -vector space U . Denote  L), MCL (R) := MC(m ⊗  L. Let us prove that the set of Maurer-Cartan elements in the dg-Lie algebra m ⊗ the functor MCL (−) behaves well with respect to the inverse limits. By Lemma 1.23, the quotients an := R/mn+1 are, for each n ≥ 0, local Artin  U → m/mn+1 ⊗ U rings with the maximal ideal m/mn+1 . The projections πn : m ⊗ induce the maps2 (3.9)

MCL (πn ) : MCL (R) → MCL (R/mn+1 ), n ∈ .

Proposition 3.17. The system (3.9) gives rise to an isomorphism MCL (R) ∼ = lim MCL (R/mn+1 ). ←− n

 U is the space of infinite sequences Proof. The completion m ⊗ 1

s = (s1 , s2 , s3 , . . .), sn ∈ m/mn+1 ⊗ U 1 , such that si = πij (sj ) for each i ≤ j ∈ , where πij : m/mj+1 ⊗ U 1 → m/mi+1 ⊗ U 1  U, are the natural projections. By the definition of the dg-Lie structure on m ⊗   [s, s] = [s1 , s1 ], [s2 , s2 ], [s3 , s3 ], . . . and

   d)(s) = ( ⊗  d)(s1 ), ( ⊗  d)(s2 ), ( ⊗  d)(s3 ), . . . . ( ⊗  U if and only if sn is, for each n ∈ , This implies that s is Maurer-Cartan in m ⊗ n+1 Maurer-Cartan in m/m ⊗ U . The obvious fact that MCL (πn )(s) = sn finishes the proof.  Recall that, in the following central statement of this chapter, g(A) denotes the dg-Lie algebra of the Hochschild cochains of a (discrete) associative -algebra A = (V, · ). 2 Since

 U. m/mn+1 is Artin, m/mn+1 ⊗ U ∼ = m/mn+1 ⊗

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31

 g(A) is isomorProposition 3.18. The set of Maurer-Cartan elements in m ⊗ phic to the set of all R-deformations of the algebra A, MCg(A) (R) ∼ = Def A (R).

(3.10)

The proof of the the proposition will be based on Lemma 3.19 below which re g(A) with the Gerstenhaber-Hochschild dg-Lie algebra lates the dg-Lie algebra R ⊗  A) of continuous R-linear cochains of the associative algebra R ⊗  A. gR (R ⊗  A) of the Lie algebra g∗R (R ⊗  A) is defined Recall that the degree n part gnR (R ⊗  n+1    as the space Lin cR R (R ⊗ V ), R ⊗ V . The formula  n+1  i  A)i := Lin cR   (3.11) gnR (R ⊗ R (R ⊗ V , m ⊗ V ), for i ≥ 1, which extends (1.26) defines a descending sequence (3.12)

 A) ⊃ gnR (R ⊗  A)1 ⊃ gnR (R ⊗  A)2 ⊃ gnR (R ⊗  A)3 ⊃ · · · ⊃ {0} gnR (R ⊗

of linear subspaces which clearly satisfies, for each a, b ≥ 1 and m, n ≥ 0,  m   A)a , gnR (R ⊗  A)b ⊂ gm+n  A)a+b and gR (R ⊗ (R ⊗ R (3.13)  A)a ⊂ gn+1  δgnR (R ⊗ R (R ⊗ A)a . Filtration (3.12) determines a linear topology on each component of the dg-Lie  A). By (3.13), both the bracket and the differential are uniformly algebra gR (R ⊗ continuous maps. Lemma 3.19. There exists a canonical continuous isomorphism ∼

=  A)  g(A) −→ gR (R ⊗ ρ : R⊗

of graded Lie algebras such that, for each i ≥ 1,    A)i .  g(A) = gR (R ⊗ (3.14) ρ mi ⊗ Proof. The degree n component of ρ is defined as the following chain of three canonical isomorphisms ∼ =1 , ∼ =2 and ∼ =3 :  gn (A) = R ⊗  Lin(V ⊗(n+1) , V ) ∼  V) R⊗ =1 Lin(V ⊗(n+1) , R ⊗  n+1  c c ⊗(n+1) ∼2 Lin (R ⊗ V  V)∼  V ), R ⊗ V ,R⊗ (R ⊗ =3 Lin = =

R n  A). gR (R ⊗

R

R

The first isomorphism ∼ =1 is the inverse limit of the standard isomorphisms R/mk ⊗Lin(V ⊗(n+1) , V ) ∼ = Lin(V ⊗(n+1) , R/mk ⊗V ), k ≥ 0, the isomorphism ∼ =2 follows from Lemma 1.9 and the isomorphism ∼ =3 uses (1.18) with k = n + 1. We leave as an exercise to verify that ρ commutes with the Lie  A) are, by Proposi g(A) and gR (R ⊗ brackets. Since the differentials of both R ⊗ tion 3.8, the brackets with appropriate elements, ρ commutes also with the differentials. It is simple to check that ρ fulfills (3.14) from which its continuity follows immediately.  Proof of Proposition 3.18. The multiplication · : V ⊗ V → V in the algebra A = (V, · ) is, by Proposition 3.9 taken with M = V and R = , the same as

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32

3. FINER STRUCTURES OF COHOMOLOGY

 μ0 ∈ R ⊗  g1 (V ) fulfills an element μ0 ∈ g1 (V ) satisfying [μ0 , μ0 ] = 0,3 therefore 1 ⊗  μ0 , 1 ⊗  μ0 ] = 0. (3.15) [1 ⊗ The key ingredient of the proof is the non-linear monomorphism  g1 (A) → g1R (R ⊗  V) Υ : m⊗  m, using the isomorphism ρ of Lemma 3.19 as defined, for t ∈ g1 (A) ⊗  μ0 ). (3.16) Υ(t) := ρ(t) + ρ(1 ⊗  V ) and gR (R ⊗  A) have the same underlying Since the dg-Lie algebras gR (R ⊗ spaces, the sum in (3.16) makes sense. As ρ is a Lie algebra morphisms, one has  1 1  μ0 ), ρ(t) + ρ(1 ⊗  μ0 ) [Υ(t), Υ(t)] = ρ(t) + ρ(1 ⊗ 2 2 (3.17) 1 1  μ0 , 1 ⊗  μ0 ] + ρ[1 ⊗  μ0 , t ]. = ρ[ t , t ] + ρ[1 ⊗ 2 2 By (3.15),  term in the second line vanishes. The last term obviously  the middle  δA )(t) , where δA is the differential of g(A). Assembling the above equals ρ ( ⊗ facts, we conclude that (3.17) implies 1  1  δA )(t) . [Υ(t), Υ(t)] = ρ [ t , t ] + ( ⊗ (3.18) 2 2  V ) satisfying [  By Proposition 3.9, elements  t ∈ g(R ⊗ t,  t ] = 0 are continuous  V . It is clear that such a  associative R-linear multiplications on R ⊗ t belongs    g1 (V ) precisely when (1.22) is a morphism of algebras, that is, when to Υ m ⊗  t is an R-deformation of A = (V, · ). Since Υ is a monomorphism, (3.18) shows that these  t ’s are in one-to one correspondence, via Υ, with the solutions t of the  g(A). This finishes the proof. Maurer-Cartan equation in m ⊗  Proposition 3.17 combined with Proposition 3.18 leads to Corollary 3.20. For R a local complete Noetherian ring with residue field and a discrete associative -algebra A, Def A (R) ∼ = lim Def A (R/mk ). ←− k

In other words, deformations over complete local Noetherian rings are inverse limits of deformations over local Artin rings.   consists of formal deformations of an asExample 3.21. The set Def  A [[t]]  sociative algebra A while Def A [t]/(tn+1 ) is the set of n-deformations of A in the sense of Definition 1.28. The canonical map     (3.19) Def A (πn ) : Def A [[t]] → Def A [t]/(tn+1 ) assigns to a formal deformation (μ1 , μ2 , μ3 , . . .) its n-truncation (μ1 , μ2 , . . . , μn ). Observe that the map (3.19) need not be an epimorphism, since not every ndeformation extends to a formal one. Yet, by Corollary 3.20,     Def A [[t]] ∼ = lim Def A [t]/(tn+1 ) . ←− n

We conclude that formal deformations are the same as compatible families of partial deformations. 3 Recall that g(V ) denotes both the underlying vector space of g(A) and the GerstenhaberHochschild algebra of the trivial associative algebra V .

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33

Exercise 3.22. We suggest to work out the details of the Lie version of deformation theory for associative algebras, with the Chevalley-Eilenberg cohomology [Car55, Section I.4] replacing the Hochschild cohomology. That is to define, following Definition 1.16, R-deformations of Lie algebras, to construct a dg-Lie algebra structure on the Chevalley-Eilenberg complex, and to prove an analog of Proposition 3.18 for Lie algebras. There is a rich literature devoted specifically to deformations of Lie algebras, see [FdM06] and the citations therein. The dg-Lie algebra structure on the Chevalley-Eilenberg complex is addressed also in [LM05]. Quite analogously, one can develop deformation theory for associative commutative algebras. Instead of the Hochschild cohomology, one needs to use the Harrison cohomology [Har62]. Deformation theory for associative algebras as well as the constructions mentioned in Example 3.22 are particular cases of deformation theory for algebras over quadratic Koszul operads worked out in [Bal96, Bal97]. This theory was further generalized to deformations of algebras over general (not necessarily Koszul quadratic) operads and PROPs in [Mar10, MV09a, MV09b]. A several-objects version of this theory governs deformations of diagrams of (bi)algebras, see [FMY09, Dou11a, Dou11b]. We will recall some particular instances of this general theory in chapters 9 and 10. Gradings and A∞ -algebras. Let us return to the interpretation of the higher graded components of the Hochschild cochain complex. Recall that as a graded n M whose sociative R-algebra B = (M, μ) is a graded R-module M = n∈ associative multiplication μ satisfies μ(M m , M n ) ⊂ M m+n for allm, n ∈ . A graded B-bimodule K = (N, r, l) over R is a graded R-module N = n∈ N n with an R-linear B-bimodule actions that satisfy r(N m , M n ) ⊂ N m+n and l(M m , N n ) ⊂ N m+n for all m, n ∈ . A graded R-algebra forms a graded bimodule over itself. The Hochschild complex (2.1) of a graded associative R-algebra B = (M, μ) with coefficients in a graded B-bimodule K = (N, r, l) over R acquires a second grading by setting  r  CH r,d R (M, N ) := f : R M → N | deg(f ) = d . We put CH kR (M, N ) := CH k,0 R (M, N ) and   kR (M, N ) := (3.20) CH

CH r,d R (M, N ).

r+d=k, r≥1

r  kR (M, N ) are sequences (f1 , f2 , f3 , . . .), where fr : Elements of CH RM → N is, for r ≥ 1, an R-linear map of degree k − r. When M and N are ungraded r (i.e. concentrated in degree 0), all maps R M → N are of degree 0, thus   r  Lin M, N , if d = 0, and r,d R CH R (M, N ) = 0, otherwise.  kR (M, N ) and CH kR (M, N ) coincide with the So, in the ungraded case, both CH space of -linear Hochschild cochains recalled in Definition 2.1. Formula (2.2) for

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34

3. FINER STRUCTURES OF COHOMOLOGY

the Hochschild differential extends by δB,K (f1 , f2 , f3 , . . .) = (0, δ(f1 )2 , δ(f2 )3 , δ(f3 )4 , . . .),  kR (M, N ), where, for f = (f1 , f2 , f3 , . . .) ∈ CH δ(fr )r+1 (a0 ⊗R · · · ⊗R ar ) := (−1)k+1+|a0 |(k+r+1) a0 fk (a1 ⊗R · · · ⊗R ar ) + (−1)k+r fr (a0 ⊗R · · · ⊗R ar−1 )ar  − (−1)i+r fr (a0 ⊗R · · · ⊗R ai+1 ai+2 ⊗R · · · ⊗R ar ),

(3.21)

1≤i≤r

for a1 , . . . , ar ∈ M . We leave as an exercise to prove that, with this differential,  ∗R (M, N ) is a cochain complex containing CH ∗R (M, N ) as a subcomplex. In the CH un-graded case, fr = 0 only if r = k and the above formula agrees with (2.2). Let us describe an extension of the ◦-product. For  m+1  n+1 (M, M ) and g = (g1 , g2 , g3 , . . .) ∈ CH f = (f1 , f2 , f3 , . . .) ∈ CH R R (M, M )   m+n+1 R define f ◦ g = (f ◦ g)1 , (f ◦ g)2 , (f ◦ g)3 , . . . ∈ CH (M, M ) by   (3.22) (f ◦ g)u := (−1)n(r+1)+(s+1)(r+i) fr ◦i gs , r+s=u+1 1≤i≤r

where ◦i means, as in (3.1a), inserting gs into the ith input of fr . The extended Gerstenhaber bracket [f, g] is then, as in (3.1c), the graded commutator of the above ◦-product. In the ungraded case, r = m + 1, s = n + 1, thus, modulo 2 n(r + 1) + (s + 1)(r + i) ≡ nm + n(m + 1 + i) ≡ n(i + 1) so (3.22) acquires the sign of (3.1b). The proof of Proposition 3.5 translates verbatim, showing that the (shifted)   ∗+1 of an associative R-algebra B = (M, μ) with (M, M ), δ cochain complex CH B R the above bracket and differential (3.21) is a dg-Lie R-algebra. It is also clear that  ∗+1 CH ∗+1 R (M, M ) ⊂ CH R (M, M ) is its dg-Lie subalgebra. Exercise 3.23. Let B = (M, μ) be a graded associative R-algebra. Its multiplication μ is an element of CH 2R (M, M ). We denote by the same symbol the  2R (M, M ), i.e. multiplication considered, in the obvious way, as an element of CH  2R (M, M ). μ = (0, μ, 0, 0, . . .) ∈ CH Prove an analog of Proposition 3.8 for the extended Hochschild differential (3.21),  n+1 i.e. show that, for f ∈ CH R (M, M ) and μ as above, δB (f ) = [μ, f ].  2R (M, M ) satisfying [κ, κ] = 0, we need Before we characterize elements κ of CH to recall the following classical definition [Sta63, page 294]. Definition 3.24. An A∞ -algebra (also called a strongly homotopy associative or sh associative algebra) over R is a graded R-module M together with a system u mu : R M → M, u ∈ , of R-linear maps such that mu is of degree 2−u and, for every u ∈ , v1 , . . . , vu ∈ M ,    (−1)(u,r,s,i) · mr v1 , . . ., vi−1 , ms (vi , . . ., vi+s−1 ), vi+s , . . ., vu = 0, (Au ) r+s=u+1 1≤i≤r

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3. FINER STRUCTURES OF COHOMOLOGY

35

where (3.23)

(u, r, s, i) := (s + 1)i + su + s(|v1 | + · · · + |vi−1 |).

We call A∞ -algebras over the ground field simply A∞ -algebras. A strict morphism of A∞ -algebras over R is a degree-zero R-linear map of the underlying R-modules which commutes with all structure operations.4 Exercise 3.25. Axiom (A1 ) means m1 ◦ m1 = 0, i.e. d := m1 is a degree +1 differential. Axiom (A2 ) says that dm2 (a, b) = m2 (da, b) + (−1)|a| m2 (a, db), a, b ∈ M, thus d is a degree +1 derivation with respect to the product m2 : M ⊗R M → M . Verify that (A3 ) expands into     m2 a, m2 (b, c) − m2 m2 (a, b), c = = dm3 (a, b, c) + m3 (da, b, c) + (−1)|a| m3 (a, db, c) + (−1)|a|+|b| m3 (a, b, dc), for a, b, c ∈ M . The multiplication m2 : M ⊗R M → M of an A∞ -algebra is therefore associative up to the cochain homotopy m3 . Example 3.26. Graded associative R-algebras can be interpreted as A∞ -algebras (M, m1 , m2 , m3 , . . .) over R such that mk = 0 for all k = 2. One therefore has a full embedding of the category of graded associative algebras to the category of A∞ -algebras and their strict morphisms. Similarly, a dg R-module (M, d) is an A∞ -algebra over R with m1 := d and mu := 0 for u ≥ 2. In his seminal work [Sta63] where A∞ -algebras were introduced, J. Stasheff used the opposite grading; his operations mu were of degree u − 2, thus in his case the differential d = m1 was of degree −1. The reason was that in Stasheff’s work A∞ -algebras appeared as algebras over the cellular chain (not co-chain!) complex of a certain non-Σ operad K = {Ku }u≥1 . The arity u piece of this operad was a (u − 2)-dimensional convex polytope Ku called today the associahedron. The operation mu corresponded in [Sta63] to the top-dimensional cell of Ku so it had to be of degree u − 2. The degree convention used in our Definition 3.24 is dictated by the requirement that the Maurer-Cartan elements are placed in degree +1. The polytopes Ku have interesting combinatorial properties. While K2 is the point and K3 the closed interval, K4 is the pentagon appearing in Mac Lane’s theory of monoidal categories [ML63b]. A portrait of K5 due to Masahico Saito is in Figure 1. More information about the associahedra can be found in [MSS02, Section II.1.6]. A neat explicit convex realizations of Ku ’s are described in [Lod04]. There are two main sign conventions for A∞ -algebras. While we accept the original Stasheff’s convention, the other one, used for instance in [Mar92], differs by the substitution u(u−1)

mu → (−1) 2 · mu , u ∈ . The origin of the difference between these two conventions is explained in Remark 3.51. The advantage of the convention used here is its compatibility with the convention for L∞ -algebras in which all terms of the L∞ -Maurer-Cartan equation (6.5) come with the + sign. See also related comments in Remark 6.3. 4 Strict morphisms are particular cases of general morphisms of A -algebras recalled in Ex∞ ercise 4.10.

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3. FINER STRUCTURES OF COHOMOLOGY

• XX X X• •
B AA
B  A B• •
LL A• • •
HH C @   H @• C H• L  @  CC @• •a !L• ! aa !! a a•! Figure 1. Saito’s portrait of K5 . Proposition 3.27. There is a one-to-one correspondence between cochains κ ∈  2R (M, M ) such that [κ, κ] = 0 and A∞ -algebras over R with the underlying RCH module M . The A∞ -algebra corresponding to κ is a graded associative R-algebra5 if and only if κ belongs to the subspace CH 2R (M, M ). Proof. By definition, κ is the same as a sequence (m1 , m2 , m3 , . . .) whose uth the definition of factor mu : uR M → M is a map of degree u − 2. Expanding  the bracket, we get that the uth component of [κ, κ] = [κ, κ]1 , [κ, κ]2 , [κ, κ]3 , . . . equals   2· (−1)(r+1)+(s+1)(r+i) · mr ◦i ms . r+s=u+1 1≤i≤r

As r + s = u + 1, one sees that, modulo 2, r + 1 + (s + 1)(r + i) ≡ (s + 1)i + su + 1. Therefore [κ, κ] = 0 is equivalent to the infinite set of conditions   (−1)(s+1)i+su · mr ◦i ms = 0 (Au ) r+s=u+1 1≤i≤r

that have to be satisfied for each u ≥ 1. The A∞ -axiom (Au ) of Definition 3.24 is obtained by evaluating (Au ) at v1 , . . . , vu ∈ M . The missing part s(|v1 |+· · ·+|vi−1 |) of the sign (u, r, s, i) in (Au ) enters from the Koszul sign convention as the result of moving the variables v1 , . . . , vi−1 over the ‘object’ ms of degree s modulo 2. Therefore (m1 , m2 , m3 , . . .) are structure operations of an A∞ -algebra. It is also clear that κ ∈ CH 2R (M, M ) if and only if mu = 0 for all u = 2.  For a topological graded R-algebra B = (M, μ) we have, as in the ungraded case, the dg-Lie subalgebra    ∗+1  (3.24) g∗R (B) ⊂ CH R (M, M ), [−, −], δB  ∗+1 consisting of sequences f = (f1 , f2 , f3 , . . .) ∈ CH R (B, B) with continuous components. We call  gR (B) the extended Gerstenhaber-Hochschild algebra. It contains the dg-Lie subalgebra g∗R (B) ∩ CH ∗+1 g∗R (B) :=  R (M, M ). If B is ungraded,  gR (B) equals gR (B) which in turn coincides with gR (B) introduced on page 28. As before, when R is the ground field, we drop the subscript and write  g(A) (resp. g(A)) instead of  g (A) (resp. g (A)). We have the following analog of Proposition 3.18. 5 In

Example 3.26 we saw that graded associative algebras are particular cases of A∞ -algebras.

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3. FINER STRUCTURES OF COHOMOLOGY

Proposition 3.28. Let R = (R, m) be a complete local Noetherian and A = (V, · ) a discrete graded associative -algebra. residue field  g(A) is isomorphic to set MCg(A) (R) of Maurer-Cartan elements in m ⊗ V all continuous A∞ -algebras over R with the underlying R-module R ⊗ the map (3.25)

37

ring with Then the the set of for which

V →V  V : R ⊗ ⊗

is a strict morphism of A∞ -algebras.6 In other words, MCg(A) (R) is the set of R-deformations of the associative algebra A inside the category of A∞ -algebras. The subset MCg(A) (R) ⊂ MCg(A) (R) consists of deformations that stay in the category of graded associative algebras. Proof. It is easy to verify that the definition of the morphism ρ of Lemma 3.19 extends to the graded case, giving rise to the isomorphism (denoted by the same symbol) ∼ =  A)  gR (R ⊗ ρ : R⊗ g(A) −→  ∼   A). One then defines Υ which restricts to an isomorphism R ⊗ g(A) = gR (R ⊗ precisely as in (3.16) and proves, in the same way as in the ungraded case, rela V ) satisfies [  t, t ] = 0 if and only if it tion (3.18). By Proposition 3.27,  t∈ g1R (R ⊗  V . Such a  t is in the image of Υ if represents a continuous A∞ -structure on R ⊗  and only if (3.25) is a strict morphism of A∞ -algebras. Exercise 3.29. Our aim is to further extend the definition of  g(A) to the case g(V ) of an A∞ -algebra A = (V, m1 , m2 , m3 , . . .) and investigate its properties. Let  be the graded Lie algebra (3.24) associated, as in Remark 3.14, to the graded g1 (V ) vector space V with the trivial multiplication, and κ = (m1 , m2 , m3 , . . .) ∈  the element given by the structure operations of A. Since, according to Proposition 3.27, [κ, κ] = 0, the formula (3.26)

g(V ) δA (f ) := [κ, f ] for f ∈ 

defines a degree +1 differential which makes, by Lemma 3.7,  g(V ) with its Gerstenhaber bracket a dg-Lie algebra denoted  g(A). Let R = (R, m) be a local complete Noetherian ring with residue field . In g(A) as the set of terpret the set MCg(A) (R) of Maurer-Cartan elements in m ⊗ R-deformations of the A∞ -algebra A in the category of A∞ -algebras. Notice that it does not make sense to extend in the same manner the definition of g(A), since g(V ) is not, for a general A∞ -algebra A, closed under the differential δA . One can expand (3.26) into an explicit formula similar to (3.21). Such a formula can be found for instance in [Mar92] which, however, uses a different sign convention. Opetopic principle. For an ungraded (i.e. concentrated in degree zero) associative algebra A, the Maurer-Cartan equation in g(A) uses only the Hochschild cochains of arities 2 and 3. To interpret higher arities, we moved to the graded world and studied the extended Gerstenhaber-Hochschild dg-Lie algebra  g(A). The MaurerCartan equation in this extended  g(A) sees cochains of arbitrary arities. Virtually the same result can be achieved by considering deformation over graded base rings. 6 In

(3.25) we consider the associative algebra A as an A∞ -algebra, see Example 3.26.

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38

3. FINER STRUCTURES OF COHOMOLOGY

We were lead to rediscover Stasheff’s A∞ -algebras that emerged ‘for free’ as solutions of the Maurer-Cartan equation in the extended dg-Lie algebra  g(A). This process can be iterated: understanding deformations of A∞ -algebras leads to objects which we call A∞∞ -algebras, from A∞∞ -algebras we can pass to A∞∞∞ algebras, &. We get the opetopic sequence for associative algebras (3.27) assoc. algebras ⊂ A∞ -algebras ⊂ A∞∞ -algebras ⊂ A∞∞∞ -algebras ⊂ · · · We call it this way for its feature of self-replication which it shares with opetopes in category theory, see [BD98, BJKM10]. As a curiosity which will not be needed in the rest of the book, we give Definition 3.30. An A∞∞ -algebra over R is a graded R-module M equipped with R-linear maps (3.28) mua : uR M → M, (u, a) ∈ ( × ) ∪ {(1, 0)} such that mua has degree u−2 and, for every (u, a) as in (3.28) and v1 , . . . , vu ∈ M , (Aua )



  (−1)(u,r,s,i) · mrp v1 , . . ., vi−1 , msq (vi , . . ., vi+s−1 ), vi+s , . . ., vu = 0,

r+s=u+1 1≤i≤r p+q=a

with (u, r, s, i) as in (3.23). It is convenient to denote d := m00 and da := m1a for a ≥ 1. The structure operations of A∞∞ -algebras can then be organized into the table whose uth column contains operations of arity u and degree 2 − u : d d1 d2 d3 d4 .. .

m21 m22 m23 m24 .. .

m31 m32 m33 m34 .. .

m41 m42 m43 m44 .. .

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

Example 3.31. A∞∞ -algebras with d = 0, mua = 0 and da = 0 for all u, a ≥ 2, are precisely A∞ -algebras. In this way, A∞ -algebras form a subcategory of the category of A∞∞ -algebras. This explains the second inclusion in (3.27). Example 3.32. Axiom (A10 ) says that d is a degree +1 differential. Axiom (A11 ) reads dd1 + d1 d = 0 so d commutes, in the graded sense, with d1 . Axiom (A12 ) requires d21 = −dd2 − d2 d so d1 is a differential up to the homotopy −d2 . Axiom (A1a ) for a general a ≥ 1 demands  da db = −dda − da d. p+q=a

We see that d is an R-linear degree +1 operation which squares to zero up to a coherent system of cochain homotopies d1 , d2 , d3 , . . . The structure (M, d, d1 , d2 , d3 , · · · ), which is a part of each A∞∞ -algebra, can therefore be called a strongly homotopy dg R-module.

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39

Axiom (A21 ) requires d to be a derivation with respect to the product m12 : dm21 (a, b) = m21 (da, b) + (−1)|a| m21 (a, db), a, b ∈ M. The meaning of axioms (Au1 ) for u ≥ 2 is similar: d is a derivation with respect to each mu1 . The interpretation of axiom (A32 ) which requires, for each a, b, c ∈ M , dm32 (a, b, c) + m32 (da, b, c) + (−1)|a| m32 (a, db, c) + (−1)|a|+|b| m32 (a, b, dc) = m21 (a, m21 (b, c)) − m21 (m21 (a, b), c) − d1 m31 (a, b, c) − m31 (d1 a, b, c) − (−1)|a| m31 (a, d1 b, c) + (−1)|a|+|b| m31 (a, b, d1 c) is that m21 is associative up to the homotopy m31 , up to the homotopy m32 . A∞∞ -algebras are therefore indeed strongly homotopy A∞ -algebras, which explains our notation. Exercise 3.33. Show that A∞∞ -algebras are the same as formal deformations of dg-vector spaces in the category of A∞ -algebras. Hint: Each structure operation mu of an A∞ -algebra over [[t]] with the un V can be expanded as derlying [[t]]-module V [[t]] = [[t]] ⊗ mu = mu0 + tmu1 + t2 mu2 + t3 mu3 + · · · , with some linear degree 2 − u maps mua : V ⊗u → V , a ≥ 0. Such a structure is an A∞∞ -algebra precisely when mu0 = 0 for all u ≥ 2. Remark 3.34. S. Sagave introduced in [Sag10] derived A∞ -algebras as objects retaining homotopy transfer properties of A∞ -algebras when the underlying spaces are modules over a general ring, see also [LRW11] and the citations therein. They are related to our A∞∞ -algebras as follows. The underlying space of a derived A∞ -algebra is, by Sagave’s definition, a bigraded module with a vertical differential. In particular, given a dg-vector space (V, d) and a formal variable t, one may convert V [[t]] into the bicomplex V ∗∗ with V na := ta V n and the vertical differential induced by d. It turns out that derived A∞ -algebras with the underlying bicomplex V ∗∗ are exactly A∞ -algebras over [[t]] with the underlying module V [[t]]. In the light of Exercise 3.33, such an object is an A∞∞ -algebra if and only if all its structure operations except the linear one are zero mod t. Exercise 3.35. While associative R-algebras with the underlying R-module M are solutions of the Maurer-Cartan equation in the classical GerstenhabergR (M ) describes A∞ -algebras.7 Hochschild graded Lie algebra gR (M ), its extension  This illustrates a common principle of generalizing algebraic structures by enlarging the corresponding Lie algebra. Show that the graded Lie algebra  gR (B) of (3.24) with B the trivial algebra M can further be enlarged into gR (M ) :=  gR (M ) ⊕ ↓M d by allowing r = 0 in (3.20), with CH 0,d R (M, M ) := M . The Gerstenhaber bracket is defined as an obvious formal extension of the formulas on page 34, interpreting elements of M d as degree d R-linear functions of arity 0. Prove that the MaurerCartan elements in gR (M ) are curved A∞ -algebras. They are A∞ -algebras over 7 As

in Remark 3.14, we consider M as an associative R-algebra with the trivial multiplication.

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40

3. FINER STRUCTURES OF COHOMOLOGY

R as in Definition 3.24 having also the constant operation m0 ∈ M 2 called the curvature, which measures the deviation of d = m1 to be a differential via d2 (a) = m2 (a, m0 ) − m2 (m0 , a), a ∈ M, which is a ‘curved’ version of (A1 ). Hochschild cochains as coderivations. In this subsection we give, following [SS85], an alternative description of the Gerstenhaber dg-Lie algebra  g(A) of a discrete graded associative -algebra A, based on an isomorphism between the Hochschild cochain complex and the graded Lie algebra of coderivations of a certain tensor coalgebra. Our restriction to discrete associative algebras avoids topologies and fully suffices for our purposes. Remark 3.36. The psychological drawback of this approach is that it uses coderivations of coalgebras, while human minds are better suited for derivations of algebras. There exists a dual description in terms of continuous derivations of free complete topological algebras. The ubiquity of topologies stems from the need to dualize not necessarily finite-dimensional vector spaces. As explained in [Kon93, Kon03], A∞ -algebras can then be viewed as formal non-commutative pointed manifolds equipped with odd vector fields that square to zero. Their A∞ morphisms (see Example 4.10) are then maps of formal pointed manifolds preserving these odd vector fields. L∞ -algebras and their morphism recalled in chapters 6 and 7 can be treated similarly, via (ordinary, commutative) formal manifolds and their morphisms. This revolutionary change of view, systematically used for instance in [HL09, HL08, CL11], opens homotopy algebras to geometric intuition. For instance, the components of an A∞ -morphism appear as Taylor coefficients at zero, and the statement formulated in the last paragraph of Example 4.10 as well as its L∞ analog, Proposition 7.5, as the inverse function theorem. More on the topological ˇ background of this approach can be found in the diploma thesis [Cer11]. The possibility to describe dg-Lie algebras controlling deformations via spaces of (co)derivations is typical for algebras over Koszul quadratic operads. Although for more general (bi)algebras or their diagrams such a simple description need not exist, we recall it here since it is part of the classical picture. A bonus is a compact definition of A∞ -algebras and their morphisms, which can be easily modified to L∞ -algebras. Let us start by some classical definitions. Definition 3.37. A graded coassociative coalgebra is a graded -vector space C with a linear degree 0 map Δ : C → C ⊗ C satisfying the coassociativity (Δ ⊗ )Δ = ( ⊗ Δ)Δ. Example 3.38. For a graded vector space W , consider the space  T (W ) = W ⊗u u≥1

graded by |w1 ⊗ . . . ⊗ wu | := |w1 | + · · · + |wu |, for w1 , . . . , wu ∈ W. The graded vector space T (W ) with the comultiplication Δ : T (W ) → T (W )⊗T (W )

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3. FINER STRUCTURES OF COHOMOLOGY

given by the de-concatenation Δ(w1 ⊗ . . . ⊗ wu ) :=



41

(w1 ⊗ . . . ⊗ wi ) ⊗ (wi+1 ⊗ . . . ⊗ wu )

1≤i≤u−1

is a graded coassociative coalgebra. We will denote this coalgebra by c T (W ) and call it the tensor coalgebra. Warning. Contrary to general belief, the coalgebra c T (W ) is not cofree in the category of coassociative coalgebras. Cofree coalgebras are surprisingly complicated objects [Fox93a, Haz03]. The coalgebra c T (W ) is, however, cofree in the subcategory of coaugmented nilpotent coalgebras [MSS02, Section II.3.7]. In (3.29) below we use  Sweedler’s convention expressing the comultiplication in a coalgebra C as Δ(c) = c(1) ⊗ c(2) , c ∈ C. Definition 3.39. A degree d coderivation of a -graded coalgebra C is a linear degree d map θ : C → C satisfying the dual Leibniz rule   (−1)d|c(1) | c(1) ⊗ θ(c(2) ), (3.29) Δθ(c) = θ(c(1) ) ⊗ c(2) + for every c ∈ C. We denote by Der d (C) the space of all degree d coderivations of C and  Der d (C). Der ∗ (C) := d∈

As in [SS85], we decided to use Der (C) for the space of coderivation instead of more descriptive but clumsy coDer (C), believing that the fact that we mean coderivations is sufficiently encoded by C being a coalgebra. Exercise 3.40. Show that (3.29) is equivalent to Δθ = (θ ⊗ )Δ + ( ⊗ θ)Δ by noticing that the nontrivial sign of the rightmost term in (3.29) is dictated by the Koszul sign rule. Proposition 3.41. Let W be a graded -vector space. For any d ∈ , one has a natural -linear isomorphism     (3.30) Der d c T (W ) ∼ = Lin d T (W ), W . Proof. Let T s (W ) be, for s ≥ 1, the subspace of c T (W ) spanned by  the tensor  products of s elements of W , T s (W ) := W ⊗s ⊂ c T (W ). For θ ∈ Der d T (W ) and   s ∈  denote θs ∈ Lin d T s (W ), W the composition θ|T s (W )

proj.

θs : T s (W ) −−−→ c T (W ) −−→ W.

(3.31)

The dual Leibniz rule (3.29) implies that, for w1 , . . . , wu ∈ W and u ∈ , θ(w1 ⊗ · · · ⊗ wu ) :=  (−1)d(|w1 |+···+|wi−1 |) w1 ⊗ · · · ⊗ θs (wi ⊗ · · · ⊗ wi+s−1 ) ⊗ · · · ⊗ wu , 1≤i≤u−s+1 s≥1

which shows that θ is uniquely determined by

  f := (θ1 , θ2 , θ3 , . . .) ∈ Lin d T (W ), W .

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42

3. FINER STRUCTURES OF COHOMOLOGY

  On the other hand, it is easy to verify that, for any map f ∈ Lin d T (W ), W decomposed into the product (θ1 , θ2 , θ3 , . . .) of its homogeneous components, the   above formula determines a coderivation θ ∈ Der d T (W ) . Definition 3.42. The linear map θs : T s (W ) → W defined in (3.31) is called the sth corestriction of the coderivation θ. A coderivation θ ∈ Der d (T (W )) is quadratic if its sth corestriction is non-zero only for s = 2. A degree 1 coderivation θ is a differential if θ 2 = 0. Proposition 3.43. For coderivations θ  , θ  ∈ Der (C) of a graded C define   [θ  , θ  ] := θ  ◦ θ  − (−1)|θ ||θ | · θ  ◦ θ  ,

-coalgebra

where ◦ is the ordinary composition of linear functions. The bracket [−, −] makes Der ∗ (C) a graded Lie -algebra. Proof. The crucial observation is that [θ  , θ  ] is a coderivation – note that neither θ  ◦ θ  nor θ  ◦ θ  are coderivations in general! Verifying this and the Lie algebra axioms of the commutator bracket is straightforward and we leave it as an exercise.  Exercise 3.44. A coderivation θ ∈ Der 1 (C) is a differential in the sense of Definition 3.42 if and only if [θ, θ] = 0.  Definition 3.45. Assume that V ∗ = i∈ V i is a graded vector space. The suspension operator ↑ assigns to V the graded vector space ↑V with the -grading (↑V )i := V i−1 . It comes with a natural degree +1 map ↑ : V → ↑V that sends v ∈ V into its suspended copy ↑v ∈ ↑V . Likewise, the desuspension operator ↓ changes the grading of V according to the rule (↓V )i := V i+1 . The related degree −1 map ↓ : V → ↓V is defined in the obvious way. The suspension (resp. the desuspension) of V is sometimes also denoted sV or V [−1] (resp. s−1 V or V [1]). Example 3.46. If V is an un-graded vector space, then ↑V is V placed in degree +1 and ↓V is V placed in degree −1. Exercise 3.47. Show that the Koszul sign convention implies the equality (↓ ⊗ ↓) ◦ (↑ ⊗ ↑) = −V ⊗V . More generally, ↓⊗u ◦ ↑⊗u = ↑⊗u ◦ ↓⊗u = (−1)

u(u−1) 2

· V ⊗u

⊗u

for an arbitrary u ≥ 1. This means that the maps ↑ and ↓⊗u are inverse to u(u−1) each other only modulo the sign factor (−1) 2 . We will explain in Remark 3.51 below that this unexpected feature lies behind the difference between two major sign conventions used in the definition of A∞ -algebras. In the following main result of this subsection,  g(V ) denotes the extended Gerstenhaber-Hochschild graded Lie algebra of a graded discrete -vector space V with the trivial multiplication. Theorem 3.48. There is an isomorphism of graded Lie -algebras   ∼ = ξ: g(V ) −→ Der c T (↓V ) .

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3. FINER STRUCTURES OF COHOMOLOGY

43

Proof. Given f = (f1 , f2 , f3 , . . .) ∈  g(V )n , let θr : (↓V )⊗r → ↓V be, for r ∈ , the linear map defined by the diagram (↓V )⊗r ↑⊗r

(3.32)

θr

- ↓V 6 ↑

? fr - V. V We remind the reader that, due to the Koszul sign convention, the left vertical map acts, for v1 , . . . , vr ∈ V , by ↑⊗r (↓ v1 ⊗ · · · ⊗ ↓ vr ) = (−1)η · (↑ ↓ v1 ⊗ · · · ⊗ ↑ ↓ vr ) = (−1)η · (v1 ⊗ · · · ⊗ vr ), with

r(r − 1) + (r − 1)|v1 | + (r − 2)|v2 | + · · · + |vr−1 |. 2 Since deg(fr ) = n+1−r, deg(θr ) = n for each  r ∈ . By Proposition 3.41, there n c exists a unique coderivation θ ∈ Der T (↓V ) whose rth corestriction is θr . The   map ξ : g∗ (V ) → Der ∗ (c T ↓V ) defined by ξ(f ) := θ is clearly an isomorphism. The verification that ξ commutes with the brackets is a straightforward though involved exercise on the Koszul sign convention which we leave to the reader.  η :=

Let A∞ (V ) be the set of all A∞ -algebras with the underlying graded vector space V . Since each associative algebra is, in a natural manner, also an A∞ -algebra, A∞ (V ) contains the set Ass(V ) of graded associative algebras on V .   Corollary 3.49. For a graded vector space V denote by Diff 1 c T(↓V ) the 1 set of all differentials on the tensor coalgebra c T (↓V ), and Diff 2 c T (↓V ) its subset consisting of quadratic differentials. One then has an isomorphism   (3.33a) A∞ (V ) ∼ = Diff 1 c T (↓V ) that restricts to the bijection (3.33b)

  Ass(V ) ∼ = Diff 12 (c T ↓V ) .

Proof. Since the isomorphism ξ of Theorem 3.48 commutes with the brackets, it induces a one-to-one correspondence between elements κ ∈ g1 (V ) satisfying   1 c [κ, κ] = 0 and coderivations θ ∈ Der T (↓V ) such that [θ, θ] = 0, which are, by Exercise 3.44, precisely the differentials. We thus get (3.33a) as the composition   ∼  ∼ =  = g1 (V ) | [κ, κ] = 0 −→ Diff 1 c T (↓V ) , Ξ : A∞ (V ) −→ κ ∈  where the first isomorphism is the content of Proposition 3.27 (with M = V and R = ) and the second one is given by ξ. It is clear that Ξ restricts to (3.33b).  Corollary 3.50. Let A =  (V, · ) be a graded associative -algebra and assume that χ ∈ Diff 12 (c T ↓V ) is the corresponding differential. The extended   Gerstenhaber-Hochschild dg-Lie algebra  g(A) is then isomorphic to Diff 12 (c T ↓V ) with the commutator bracket and the differential [χ, −]. If A is ungraded, the above corollary whose proof is obvious offers an alternative description of the classical Gerstenhaber-Hochschild algebra g(A) via an algebra of coderivations.

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44

3. FINER STRUCTURES OF COHOMOLOGY

Remark 3.51. Corollary 3.49 gives a compact way to define A∞ -algebras simply as degree +1 differentials on the tensor coalgebra c T (↓V ). One however needs to keep in mind that this translation of Definition 3.24 involves the isomorphism ξ of Theorem 3.48 which depends on the choices of isomorphisms between (↓ V )⊗r and V ⊗r in diagram (3.32), r ∈ . Our choice ↑⊗r : (↓ V )⊗r → V leads to the original Stasheff’s convention, while the choice  ⊗r −1 : (↓ V )⊗r → V ↓ gives the convention used for instance in [Mar92] or [DMZ07]. As we saw in Example 3.47,  ⊗r −1 r(r−1) = (−1) 2 ↑⊗r , ↓ so the choices are different.

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http://dx.doi.org/10.1090/cbms/116/04

CHAPTER 4

The gauge group Proposition 3.18 identifies the set Def A (R) of R-deformations of an associative algebra A with the set of Maurer-Cartan elements in the completed product  g(A) of the maximal ideal of R with the Gerstenhaber-Hochschild dg-Lie algem⊗ bra g(A). In this chapter we extend this type of identification to the gauge group GA (R) and its action on Def A (R). The exponential. Let us recall that each Lie -algebra K can be made a group, with the multiplication given by the Hausdorff-Campbell formula:  1 1     (4.1) x · x := x + x + [x , x ] + [x , [x , x ]] + [x , [x , x ]] + · · · 2 12 for x , x , ∈ K, assuming a suitable condition that guarantees that the above potentially infinite sum makes sense in K, see [Ser65, I.IV.§7].1 In our context, K will always be the first member H1 of a fundamental system  {Hi }i≥1  of neighborhoods of zero in a complete topological -Lie algebra H = H, [−, −] 2 satisfying [Ha , Hb ] ⊂ Ha+b for each a, b ≥ 1.

(4.2)

The above condition guarantees that, for x , x ∈ H1 , the terms of (4.1) containing i braces belong to Hi+1 . The sum in (4.1) therefore, by the completeness, converges to an element of H1 . The zero vector 0 ∈ H1 is clearly the unit for (4.1). The -vector space H1 with the above group structure will be denoted exp(H1 ). So H1 and exp(H1 ) are the same vector spaces, but one considered as a Lie algebra with its original Lie algebra structure, and the other as a group with multiplication (4.1). Formula (4.1) can be obtained by expressing the right hand side of the equality of formal power series   (4.3) x · x = log exp(x ) exp(x ) via the the iterated commutators of non-commutative variables x and x . In the above display, exp(−) and log(−) are defined by their formal Taylor expansions  1 1 1 1 xn , and (4.4a) exp(x) := 1 + x + x2 + x3 + x4 + · · · = 1 + 2! 3! 4! n! n≥1

(4.4b)

 (−1)n−1 1 1 (x − 1)n . log(x) := (x − 1) − (x − 1)2 + (x − 1)3 − · · · = 2 3 n n≥1

Assume now that U = (U, · , e) is a unital associative algebra as in Lemma 1.22, i.e. having a complete linear topology determined by a fundamental system {Ui }i≥1 1 An

explicit formula for all terms of (4.1) can be found in [Car55, Section 18.1]. we bend our strict rules and use the same symbol both for a Lie algebra and its underlying space. 2 Here

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46

4. THE GAUGE GROUP

of ideals satisfying (1.25): Ua · Ub ⊂ Ua+b , for each a, b ≥ 1. Formula (4.4a) with e in place of 1 then defines a non-linear map exp : U1 → U ; i.e. we put, for u ∈ U ,  1 1 1 exp(u) := e + u + u2 + u3 + · · · = e + un . 2! 3! n! n≥1

Since, by (1.25), ui ∈ Ui for each u ∈ U1 , this sum converges to an element of U . It is also clear that exp(u) − e ∈ U1 for each u ∈ U . In other words, Im(exp) ⊂ (U1 + e). For U as above, (4.4b) defines a non-linear map log : (U1 + e) → U ; i.e. for v such that v − e ∈ U1 we put  1 1 1 (−1)n−1 · (v − e)n . log(v) := (v − e) − (v − e)2 + (v − e)3 − · · · = 2 3 n n≥1

By (1.25), (v − e) ∈ Ui for each i ≥ 1, so the above sum indeed converges to an element of U . Recall that, by Lemma 1.22, (U1 + e) is a group. There is also another group related to the subalgebra U1 ⊂ U . Consider first the commutator Lie algebra L(U ) of U , i.e. the space U with the bracket i

[u , u ] := u · u − u · u , for u , u ∈ U.   It is clear that L(U ) with the filtration L(Ui ) i≥1 fulfills (4.2), so the Hausdorff  Campbell formula (4.1) gives rise to the group exp L(U1 ) with the underlying space U1 . The following statement was formulated for instance in Appendix A of [Qui69]. Proposition 4.1. Assume that U = (U, ·, e) is a unital associative algebra with a complete linear topology given by a fundamental system {Ui }i≥1 of ideals fulfilling (1.25). Then exp and log are mutually inverse group isomorphisms   ∼ = exp : exp L(U1 ) ←→ (U1 + e) : log . Proof. By high-school mathematics, one derives the equalities of formal power series     log exp(u) = u and exp log(v) = v. As explained above, for u ∈ U1 and v ∈ (U1 +e) both power series actually converge, so exp and log are mutually inverse maps of the underlying sets which preserve the multiplications by the very definition of the Hausdorff-Campbell formula via equation (4.3).  The Hausdorff-Campbell formula offers an alternative description of the gauge group GA (R). Let, as usual, R be a complete local Noetherian ring with residue field and A = (V, · ) a (discrete) associative -algebra. As we saw in the proof  V, R ⊗  V ) with the of Proposition 1.21, the associative unital algebra Lin cR (R ⊗ filtration (1.26) satisfies the assumptions of Proposition 4.1 and, moreover,  V, R ⊗  V )1 +  φ ∈ Lin cR (R ⊗ if and only if φ : V → V is the identity. In other words,  V, R ⊗  V )1 +  = GA (R). Lin cR (R ⊗

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4. THE GAUGE GROUP

47

On the other hand, it follows immediately from  Gersten the definition of the  V, R ⊗  V )1 equals the haber bracket that the commutator Lie algebra L Lin cR (R ⊗  A)1 of the filtration (3.12). Proposition 4.1 therefore gives first piece g0R (R ⊗ Proposition 4.2. The maps exp and log are mutually inverse group isomorphisms  ∼  =  A)1 ←→ exp : exp g0R (R ⊗ GA (R) : log .   The action. Assume that H = H, [−.−], δ is a dg-Lie -algebra such that H 0 (resp. H 1 ) is a complete linear space topologized by a fundamental system {Hi0 }i≥1 (resp. {Hi1 }i≥1 ) of neighborhoods of zero such that, for each a, b ≥ 1, (4.5)

0 1 [Ha0 , Hb0 ] ⊂ Ha+b , [Ha1 , Hb0 ] ⊂ Ha+b and δHa0 ⊂ Ha1 .

The group exp(H10 ) then exists and acts on H 1 by the formula  1  1 adn (s + d) − d, x, [x, s + d] + · · · = (4.6) x · s := s + [x, s + d] + 2! n! x n≥0

with d an auxiliary degree +1 variable such that the expression [x, s + d] is, for x ∈ H10 and s ∈ H 1 , interpreted as [x, s + d] := [x, s] − δ(x). Using this rule, one can rewrite (4.6) into  1  1 1 adnx (s)− adn (δx) (4.7) x·s = s+[x, s]−δ(x)− [x, δ(x)]+· · · = 2! n! (n + 1)! x n≥0

n≥0

which does not involve auxiliary variables. If (4.5) is satisfied, adix (H 1 ) ⊂ Hi1 for each i ≥ 0 and x ∈ H10 , so (4.7) and therefore also (4.6) converges to an element of H 1 . Formula (4.6) can be obtained by expressing the right hand side of the equation x · s = exp(x)(s + d) exp(−x) − d via the adjoint representation of H 0 on H 1 . Exercise 4.3. The action (4.7) is nonlinear. Prove that, for s , s ∈ H 1 and α , α ∈ , 

x · (α s + α s ) = α (x · s ) + α (x · s ) + (1 − α − α )(x · 0) where x·0=−

 n≥0

1 adn (δx). (n + 1)! x

  Let H = H, [−, −], δ be a dg-Lie -algebra such that each H n is a complete topological linear -vector space with fundamental system {Hin }i≥1 of neighborhoods of zero such that, for each a, b ≥ 1 and n, m, (4.8)

m+n , and δHan ⊂ Han+1 . [Ham , Hbn ] ⊂ Ha+b

This in particular means that (4.5) is satisfied, so the group exp(H10 ) is defined and acts on H 1 by action (4.7). H11

Lemma 4.4. Let H be a dg-Lie -algebra as above. The action of exp(H10 ) on preserves the subspace MC(H1 ) ⊂ H11 of Maurer-Cartan elements in H1 .

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48

4. THE GAUGE GROUP

Proof. Let s ∈ H 1 be a Maurer-Cartan element and x ∈ H10 . We need to show that 1 (4.9) δ(x · s) + [x · s, x · s] = 0. 2 Let N be, for i ≥ 1, the quotient of the dg-Lie algebra H1 by the dg-ideal i  n H , and s (resp. x ) be the equivalence class of s (resp. x) in the nilpotent i i i n dg-Lie algebra Ni . It is clear that si is a Maurer-Cartan element in Ni and that 1 (4.10) δ(xi · si ) + [xi · si , xi · si ] ∈ Ni 2 is the equivalence class of the left hand side of (4.9) in Ni . It easily follows from the continuity and completeness of Ni that the lemma will be proved if we show that (4.10) vanishes for each i ≥ 1. It is therefore enough to prove the lemma assuming that H1 is nilpotent. The rest is a modification of the proof of [Man99, Lemma 1.14]. For a given x ∈ H10 and s ∈ H11 , the correspondence  α → (αx) · s ∈ H11 defines, by the assumed nilpotence of H1 , a polynomial function of α with coefficients in H11 . We claim that     ∂ (αx) · s = x, (αx) · s − δ(x). (4.11a) ∂α Due to the -linearity of the structure operations of H, adnαx (s) = αn adnx (s) and (δαx) = (αδx), thus

    ∂ (αx) · s ∂  1 1 = adnαx (s) − adnαx (δαx) ∂α ∂α n! (n + 1)! n≥0

n≥0

n≥0

n≥0

  αn+1 ∂   αn n adx (s) − adnx (δx) = ∂α n! (n + 1)!  αn−1  αn adnx (s) − adn (δx) = (n − 1)! n! x n≥1

n≥0

 αn n+1  αn+1 = adx (s) − adn+1 (δx) − δx n! (n + 1)! x n≥0

n≥0

  αn+1  αn n adx (s) − adnx (δx) − δx = x, n! (n + 1)! n≥0 n≥0    1  1 adnαx (s) − adnαx (δαx) − δx = x, n! (n + 1)! n≥0 n≥0   = x, (αx) · s − δ(x), 

which we had to prove. The next step is to realize that exp(H10 ) acts not only on H11 , but also on H12 by  1 x · t := adn (t), for x ∈ H01 , t ∈ H12 . n! x n≥0

This action is, unlike (4.7), linear in t and will hopefully not be confused with (4.7), because the difference between these two actions is signalized by the degrees of

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4. THE GAUGE GROUP

49

elements they act on. Notice that exp(H10 ) cannot act on H12 by (4.7), as the second summation of this formula represents an element of degree +1. The equality     ∂ (αx) · t = x, (αx) · t (4.11b) ∂α can be verified similarly as (4.11a). For s ∈ H 0 and a fixed x ∈ H 1 , we define the polynomial function Fs : → H 2 by the formula     1 Fs (α) := (−αx) · δ (αx) · s + (αx) · s, (αx) · s] . 2 Using (4.11a) and (4.11b) we obtain        ∂Fs = − x, Fs (α) + (−αx) · δ [x, (αx) · s] − δx + [x, (αx) · s] − δx, (αx) · s] ∂α Observing that Fs (0) = δs + 12 [s, s], we get  1 ∂Fs (0) = −[x, δs] − x, [s, s] + δ[x, s] + [[x, s], s] − [δx, s] = 0. (4.12) ∂α 2 Notice that (4.12) is satisfied for an arbitrary, not necessary Maurer-Cartan, element s ∈ H11 . It is obvious that, for scalars α, β ∈ , Fs (α + β) = (−βx) · F(βx)·s (α) which, combined with (4.12), gives that ∂F(βx)·s ∂Fs (β) = (−βx) · (0) = 0. ∂α ∂α Since is of characteristic zero, the polynomial function Fs : → H 2 whose derivative vanishes at each α ∈ , is constant. In particular, if s is a MaurerCartan element, Fs (1) = Fs (0) = 0. We defined the function Fs in such a way that the equation Fs (1) = 0 which we have just proved implies that x · s = (1x) · s is Maurer-Cartan.    Let L = L, [−, −], d be a (discrete) dg-Lie -algebra and R = (R, m) a local  L is a dgcomplete Noetherian ring with residue field . By Lemma 3.15, R ⊗ Lie algebra whose degree n component is topologized by the fundamental system  Ln }i≥1 . Since {mi ⊗  Lm , mb ⊗  Ln ] ⊂ ma+b ⊗  Lm+n and ( ⊗  d)(ma ⊗  Ln ) ⊂ ma ⊗  Ln+1 (4.13) [ma ⊗  L fulfills (4.8). Lemma 4.4 for each a, b ≥ 0, m, n ≥ 0, the dg-Lie algebra R ⊗ therefore has a  L1 preserves the space  L0 ) on m ⊗ Corollary 4.5. The action of exp(m ⊗  L. MCL (R) of Maurer-Cartan elements in m ⊗  L0 ) and call it the gauge group. We believe Let us denote GL (R) := exp(m ⊗ that it will not be confused with the gauge group GA (R) of an associative algebra introduced on page 11. Thanks to Corollary 4.5, it makes sense to define the moduli  L to be the quotient space of Maurer-Cartan elements in m ⊗ M C L (R) := MCL (R)/GL (R).   The isomorphism. Recall that g∗ (A) = CH ∗+1 (V, V ), [−, −], δA is the Gerstenhaber dg-Lie algebra of the Hochschild cochains of an associative -algebra (4.14)

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50

4. THE GAUGE GROUP

A = (V, · ). The following central statement of this chapter refers to the non-linear map Υ defined in (3.16). Proposition 4.6. Let A be an associative -algebra and R = (R, m) a local  g1 (A) → g1R (R ⊗  V) Noetherian ring with residue field . Then the map Υ : m ⊗ induces an isomorphism M C g(A) (R) ∼ = DefA (R)  g(A) and the set of between the moduli space of Maurer-Cartan elements in m ⊗ equivalence classes of R-deformations of A. Let us formulate a couple of preparatory statements. By (3.13), the dg-Lie  A) with the filtration (3.11) fulfills (4.5), so the exponential of algebra gR (R ⊗ 0  A)1 acts on g1R (R ⊗  A). gR (R ⊗  2   A) is the space Lin cR   Recall that g1R (R ⊗ R R ⊗ V, R ⊗ A of continuous R A)1 = Lin c (R ⊗  V, R ⊗  A) consists of R-linear maps. The linear maps. Also g0R (R ⊗  A)1 on g1 (R ⊗  A) with following lemma relates the iterated adjoint action of g0 (R ⊗ 0 1   the function space nature of g (R ⊗ A)1 and g (R ⊗ A).  A)1 and μ ∈ g1 (R ⊗  A), Lemma 4.7. For y ∈ g0 (R ⊗  1   adny (μ) = exp(y) ◦ μ exp(−y), exp(−y) . n! n≥0

Proof. One has, by the definition (3.1c) of the Gerstenhaber bracket, ady (μ) := [y, μ] = y ◦1 μ − μ ◦1 y − μ ◦2 y which, in the ‘functional’ notation, reads ady (μ) = y ◦ μ(, ) − μ(y, ) − μ(, y). Iterating the above formula, we get ady (μ) = y ◦μ(, ) − μ(y, ) − μ(, y), ad2y (μ) = y 2 ◦ μ(, ) − 2y ◦ μ(y, ) − 2y ◦ μ(, y) + μ(y 2 , ) + 2μ(y, y) + μ(, y 2 ), .. . adny (μ) =



⎞ ⎛  n! s! y n−s ◦ μ ⎝ y i , y s−i ⎠ , (−1)s s!(n − s)! i!(s − i)!

0≤s≤n

0≤i≤s

hence  1 n   ady (μ) = n! n≥0



(−1)s

n≥0 0≤s≤n 0≤i≤s

1 y n−s ◦ μ(y i , y s−i ). i!(n − s)!(s − i)!

The substitution a := n − 1, b := i and c := s − i converts the above expression into   1 n 1 ady (μ) = y a ◦ μ(y b , y c ) (−1)b+c n! a!b!c! n≥0 a,b,c≥0   = exp(y) ◦μ exp(−y), exp(−y) which we wanted to prove.

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4. THE GAUGE GROUP

51

 A)1 and One can generalize Lemma 4.7 and prove that, for each y ∈ g0 (R ⊗  A) with arbitrary k ≥ 0, f ∈ gk (R ⊗  1 n  k+1  ady (f ) = exp(y) ◦ f exp(−y) . R n! n≥0

 g(A) → gR (R ⊗  A) of Lemma 3.19 satisfies Recall that the isomorphism ρ : R ⊗    g0 (A) = g0R (R ⊗  A)1 . ρ m⊗ It therefore induces a group isomorphism      g0 (A) ∼  A)1 . exp(ρ) : Gg(A) (R) = exp m ⊗ = exp g0R (R ⊗ Let us denote by Ξ the composite exp(ρ)

exp

 A)1 ) −−−→ GA (R) Ξ : Gg(A) (R) −−−→ exp(g0R (R ⊗ of exp(ρ) with the group isomorphism exp of Proposition 4.2. The following lemma  V ) is Ξ-equivariant with respect to  g1 (A) → g1R (R ⊗ shows that the map Υ : m ⊗ actions (1.23) and (4.6).  g1 (A), Lemma 4.8. For x ∈ Gg(A) (R) and s ∈ m ⊗ Υ(x · s) = Ξ(x) · Υ(s), where the dot · in the left hand side denotes the action (4.6) while the dot · in the right hand side is the action (1.23). In other words, the diagram   action  g1 (A) Gg(A) (R) × m ⊗

(4.6)

- m⊗  g1 (A)

Ξ×Υ

(4.15)

?  V) GA (R) × g1R (R ⊗

Υ

? - g1 (R ⊗  V) R

action (1.23)

commutes.  μ0 ), where μ0 is the multiplication Proof. Recall that Υ(x·s) = ρ(x·s)+ρ(1 ⊗ in A. Since ρ is a Lie algebra morphism, formula (4.7) for the action (4.6) gives # $  1 n 1 n ρ(x · s) = ρ ad (s) − ad (δx) n! x (n + 1)! x n≥0 n≥0 (4.16)  1 n      1 = adρ(x) ρ(s) − adn ρ(δx) . n! (n + 1)! ρ(x) n≥0

n≥0

 μ0 , x] = −[x, 1 ⊗  μ0 ] = −adx (1 ⊗  μ0 ), we rewrite Remembering that δ(x) = [1 ⊗ the last term in the second line as     1 n   1  μ0 ) . − adnρ(x) ρ(δx) = adρ(x) ρ(1 ⊗ (n + 1)! n! n≥0

n≥1

Substituting it back to (4.16), we obtain  1 n    μ0 ) − ρ(1 ⊗  μ0 ), ρ(x · s) = ρ(s) + ρ(1 ⊗ ad n! ρ(x) n≥0

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52

4. THE GAUGE GROUP

therefore, by the definition of Υ recalled above,  1 n Υ(x · s) = Υ(s). ad n! ρ(x) n≥0

Lemma 4.7 with y = ρ(x) and μ = Υ(s) applied to the right hand side of the above equation gives   Υ(x · s) = exp(ρ(x)) ◦ Υ(s) exp(−ρ(x)), exp(−ρ(x))   = Ξ(x) ◦ Υ(s) Ξ(x)−1 , Ξ(x)−1 . By definition, the rightmost term equals Ξ(x) · Υ(s).



Proof of Proposition 4.6. As we showed in the proof of Proposition 3.18, the map Υ restrict to an isomorphism between MCg(A) (R) and Def A (R). Diagram (4.15) therefore restricts to Gg(A) (R) × MCg(A) (R) Ξ×Υ

action (4.6)

- MCg(A) (R)

∼ =

∼ =

? GA (R) × Def A (R)

Υ

? - Def A (R)

action (1.23)

in which both vertical arrows are isomorphisms. The isomorphism between the quotient M C g(A) (R) = MCg(A) (R)/Gg(A) (R) and

DefA (R) ∼ = Def A (R)/GA (R) 

is now obvious. Exercise 4.9. By Proposition 3.17, MCL (R) ∼ = lim MCL (R/mn+1 ). ←− n

Does the same hold also for the Maurer-Cartan moduli space, i.e. is it true that ∼ lim M C L (R/mn+1 )? M C L (R) = ←− n

In Example 3.3 we noticed that g0 (A) is an associative unital algebra, with the multiplication given by the ◦-operation and the unit V . As in the Lie algebra case of Lemma 3.15, the formula  φ )(r  ⊗  φ ) := r  r  ⊗  φ ◦ φ , r  , r  ∈ R, φ , φ ∈ g0 (A), (r  ⊗  g0 (A) an associative unital algebra with the unit 1 ⊗  V whose filtration makes R ⊗     i  0  g0 (A) equals m ⊗ g (A) i≥1 satisfies (1.25). The associated Lie algebra L m ⊗  g0 (A) with the Lie structure induced by that of g0 (A). Proposition 4.1 provides m⊗ an isomorphism  g0 (A) + 1 ⊗  V exp : Gg(A) (R) ∼ = m⊗  g(A) → gR (R ⊗  A) of Lemma 3.19 restricts to an while the isomorphism ρ : R ⊗  V and gR (R ⊗  A)1 + , where the latter is  g0 (A) + 1 ⊗ isomorphism between m ⊗ the gauge group GA (R). These observations help to organize the maps used in this chapter into the diagram in Figure 1.

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4. THE GAUGE GROUP

g0 (A) 

  ⊗

6

V 

 g0 (A) R⊗ 6

  ⊗

ρ

- g0 (R ⊗  A) R

∼ =

6

ρ

 g0 (A) + 1 ⊗  V m⊗ M

53

∼ =

- GA (R)

M 1   exp log ∼  =   N N  exp(ρ)  0  - exp g (R ⊗  A)1 Gg(A) (R) R

log ∼ = exp

Ξ

∼ =

Figure 1 Exercise 4.10. Let A = (V  , d , m2 , m3 , . . .) and A = (V  , d , m2 , m3 , . . .) be A∞ -algebras represented, as in Corollary 3.49, by degree +1 differentials     θ  ∈ Diff 1 c T (↓V  ) and θ  ∈ Diff 1 c T (↓V  ) , respectively. A morphism F : A → A is defined to be a coalgebra map (4.17)

F : c T (↓V  ) → c T (↓V  )

such that F ◦ θ  = θ  ◦ F . Prove that such an F is given by a sequence (φ1 , φ2 , φ3 , . . .) of linear maps φu : V ⊗u → V  , u ∈ , of degrees 1 − u that satisfy an infinite system of axioms (Mu ), u ≥ 1, where (Mu ) requires the vanishing of a certain linear combination of u-multilinear maps on V  with values in V  which are compositions of φv , mv and mv , with v ≤ u. In particular, (M1 ) requires (4.18)

φ1 : (V  , d ) → (V  , d )

to be a cochain map, (M2 ) says that φ1 : (V  , m2 ) → (V  , m2 ) is a morphism of (non-associative) algebras up to the homotopy φ2 , etc. Morphisms F = (φ1 , φ2 , φ3 , . . .) with φu = 0 for u ≥ 2 are precisely strict morphisms of Definition 3.24. The axioms for A∞ -morphisms are written out explicitly for instance in [Mar92], but one needs to keep in mind that a different sign convention is used there. We say that F : A → A is an isomorphism of A∞ -algebras if (4.17) is an invertible map of dg-coalgebras. Prove that F is an isomorphism if and only if the corresponding linear map φ1 in (4.18) is an isomorphism of cochain complexes. This statement is an A∞ analog of Proposition 7.5 in Chapter 7. Exercise 4.11. Let A be a discrete A∞ -algebra and  g(A) its associated dg-Lie algebra constructed in Example 3.29. Let R be a local complete Noetherian ring with residue field . Interpret the Maurer-Cartan moduli space M C g(A) (R) as the set of equivalence classes of R-deformations of the A∞ -algebra A. The notion of equivalence of deformations shall use continuous R-linear versions of A∞ -morphisms of Example 4.10. Functors of Artin rings. On page 11 we observed that the sets DefA (R) form a covariant functor DefA (−) from a suitably chosen subcategory of augmented rings

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54

4. THE GAUGE GROUP

with a given residue field. There is a rich theory of functors of this type [Sch68]. Our brief exposition is based on the overviews [Man99, Man09] where other references can be found. We will deal with covariant functors F : Art/ → Set from the category Art/ of Artin local rings with residue field to the category of sets. The fibered product - R R ×R R ? α R exists in Art/ and induces a map of sets (4.19)

? -R

η : F (R ×R R ) → F (R ) ×F (R) F (R ).

The functor F is homogeneous if η is an isomorphism whenever α is surjective. It is a deformation functor if (i) η is surjective whenever α is surjective and (ii) η is an isomorphism whenever R = . One says that the functor F is smooth if the induced map F (φ) : F (S) → F (T ) is surjective for an arbitrary surjection φ : S → T in Art/ . Exercise 4.12. Let L be a dg-Lie -algebra. Prove that the gauge group functor GL (−) : Art/ → Set that assigns to each local Artin ring R = (R, m) the gauge group GL (R) = exp(m ⊗ L0 ) is smooth and homogeneous. Prove also that the Maurer-Cartan functor MCL (−) : Art/ → Set whose value at R is the set MCL (R) of Maurer-Cartan elements in m ⊗ L1 is homogeneous and smooth when L is abelian. Finally, prove that Maurer-Cartan moduli space functor M C L (−) = MCL (−)/GL (−) : Art/ → Set is a deformation functor. Since, by Proposition 4.6, for a discrete associative algebra A, the sets DefA (R) and M C g(A) (R) are naturally isomorphic, the functor DefA (−) that assigns to each local Artin ring R the set of R-deformations of A is a deformation functor. This explains the above terminology.

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http://dx.doi.org/10.1090/cbms/116/05

CHAPTER 5

The simplicial Maurer-Cartan space In (4.14) we introduced, for each dg-Lie -algebra L and a local complete Noetherian ring R = (R, m) with residue field , the moduli space M C L (R) of  L as the quotient Maurer-Cartan elements in m ⊗ M C L (R) = MCL (R)/GL (R) of the set of solutions MCL (R) of the Maurer-Cartan equation by the gauge group GL (R). In Theorem 5.12 below we give an alternative description of this moduli space in terms of components of a certain simplicial set. The advantage of this approach is that it generalizes to the L∞ -case when no gauge group is available. Simplicial objects. Let us denote by Δ the category with objects the finite ordered sets [n] := {0 < 1 < · · · < n}, n ≥ 0, and morphisms their non-decreasing maps. A cosimplicial object in a category C is a covariant functor X : Δ → C. Dually, a simplicial object in C is a contravariant functor K : Δ → C. One easily sees that a simplicial object in C amounts to a collection K• = {Kn }n≥0 of objects of C together with morphisms ∂i : Kn → Kn−1 , 0 ≤ i ≤ n, n ≥ 1, and σi : Kn → Kn+1 , 0 ≤ i ≤ n, n ≥ 0, that satisfy the identities: ∂i ∂j = ∂j−1 ∂i if i < j, σi σj = σj+1 σi if i ≤ j, (5.1)

∂i σj = σj−1 ∂i if i < j, ∂j σj = identity = ∂j+1 σi , and ∂i σj = σj ∂i−1 if i > j + 1.

Likewise, a cosimplicial object in C is a collection X • = {X n }n≥0 of objects of C equipped with structure morphisms di : X n−1 → X n , 0 ≤ i ≤ n, n ≥ 1, and si : X n+1 → X n , 0 ≤ i ≤ n, n ≥ 0 that satisfy the identities dual to (5.1) whose explicit form we leave as an exercise. A simplicial set S• is a simplicial object in the category of sets. Elements s ∈ Sn are called n-simplexes, ∂0 s, . . . , ∂n s are the faces of the n-simplex s. In Figure 1 we schematically show the combinatorics of a 2-simplex w, its edges (= 1-faces) Hi = ∂i w, i = 1, 2, 3, and its vertices (= 0-faces) x0 , x1 , x2 . The ith face Hi of w is drawn as an arrow pointing from ∂0 Hi to ∂1 Hi . 55 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

56

5. THE SIMPLICIAL MAURER-CARTAN SPACE

H1

x0

 •

x2 • @ @ H0 @ w @ R• @ x1 H2

Figure 1. A 2-simplex and its faces. Let S• be a simplicial set, n ∈  and 0 ≤ i ≤ n. An n-horn in S• is a configuration (s0 , . . . , si−1 , · , si+1 , . . . , sn )

(5.2)

of (n − 1)-simplexes sj ∈ Sn−1 such that ∂j sk = ∂k−1 sj for all 0 ≤ j < k ≤ n for which sj and sk are defined. A filler of the n-horn (5.2) is an n-simplex w ∈ Sn such that sj = ∂j w for each 0 ≤ j ≤ n, j = i. Let us recall the following classical notion [May67, Definition 1.3]. Definition 5.1. A simplicial set S• satisfies the extension condition if each horn in S• has a filler. In this case S• is called a Kan simplicial set. Notice that the collection of all faces of a simplex except the ith one forms a horn. A simplicial complex S• is Kan if all its horns are of this form. Definition 5.2. A 0-simplex s0 is homotopic to s1 if there exists a 1-simplex H ∈ S1 such that ∂i H = si , i = 0, 1. Such a 1-simplex H is called a homotopy H s1 if we want to specify a homotopy between s0 and s1 . We write s0 ∼ s1 or s0 ∼ that relates s0 to s1 . Proposition 5.3. The relation of homotopy ∼ is reflexive. If S• is Kan, then it is also symmetric and transitive. Proof. For s ∈ S0 take H := σ0 s. It follows from the 4th axiom of (5.1) that H s. This proves that ∼ is reflective. ∂0 σ0 s = ∂1 σ0 s, so s ∼ H Suppose that S• is Kan and prove the transitivity. Assume that s ∼0 s and H 2 s ∼ s . It is easy to check that (H0 , · , H2 ) is a 2-horn. Let w be its filler and put H1 := ∂1 w. The 1st axiom of (5.1) implies that ∂0 H1 = ∂0 ∂1 w = ∂0 ∂0 w = ∂0 H0 = s while, by the same axiom, ∂1 H1 = ∂1 ∂1 w = ∂1 ∂2 w = ∂1 H2 = s , H

so s ∼1 s . H H Let finally s ∼2 s . We leave as an exercise on definitions to show that s ∼0 s  with the homotopy H0 := ∂0 w, where w fills the 2-horn ( · , σ0 s , H2 ). This establishes the symmetry of ∼. Figure 1 helps to keep track of the combinatorics behind the above constructions.  Definition 5.4. The set of connected components π0 (S• ) of a Kan simplicial set S• is the quotient π0 (S• ) := S0 / ∼ .

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5. THE SIMPLICIAL MAURER-CARTAN SPACE

57

The above object is the n = 0 case of the simplicial homotopy group πn (S• , φ) [May67, Definition 3.6]. Simplicial differential forms. Let us recall that a differential graded associative R-algebra (abbreviated dg-associative R-algebra) is an associative R-algebra B = (M, μ) as in Definition 1.15, whose underlying space M is a graded R-module and whose multiplication preserves the degrees, i.e. deg(ab) = deg(a) + deg(b), a, b ∈ M. It is graded commutative if, for all a, b ∈ M , ab = (−1)|a||b| ba. Denote by [z0 , . . . zn , dz0 , . . . , dzn ] the free graded commutative associative unital -algebra generated by degree 0 generators z0 , . . . , zn and degree 1 generators dz0 , . . . , dzn . One clearly has the isomorphism [z0 , . . . zn , dz0 , . . . , dzn ] ∼ = [z0 , . . . zn ] ⊗ ∧(dz0 , . . . , dzn ), where [z0 , . . . zn ] is the algebra of polynomials in z0 , . . . , zn and ∧(dz0 , . . . , dzn ) the exterior (Grassmann) algebra generated by dz0 , . . . , dzn . The formula d(zi ) := dzi , d(dzi ) := 0, 0 ≤ i ≤ n,   extends to a unique degree +1 differential d so that [z0 , . . . zn , dz0 , . . . , dzn ], d is a differential graded commutative associative -algebra. (5.3)

Exercise 5.5. Show that the degree m-component of [z0 , . . . zn , dz0 , . . . , dzn ] is isomorphic to the -vector space of expressions  ωI · dzI , (5.4) ω= I

where I runs over ordered subsets {i1 < · · · < im } of {0 < · · · < n}, ωI is a polynomial from [z0 , . . . zn ] and dzI := dzi1 · · · dzim . Describe the multiplication and the differential (5.3) in terms of the expressions (5.4). For each n ≥ 0 define the graded commutative associative -algebra Ωn as the quotient [z0 , . . . zn , dz0 , . . . , dzn ] Ωn := . (z0 + · · · + zn − 1, dz0 + · · · + dzn ) Since d(z0 + · · · + zn − 1) = dz0 + · · · + dzn and d(dz0 + · · · + dzn ) = 0, the defining ideal of Ωn is d-closed. Formula (5.3) therefore makes Ωn a dgcommutative associative algebra. It is easy to see that Ωn is concentrated in degrees [0, n]. Our next task is to show that the algebras (Ωn , d), n ≥ 0, are pieces of a simplicial dg-algebra Ω• . For 0 ≤ i ≤ n and n ≥ 1 define a dg-algebra morphism ∂i : Ωn → Ωn−1 by ⎧ if j < i, ⎨ zj , 0, if j = i, and (5.5a) ∂i (zj ) := ⎩ zj−1 , if j > i.

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5. THE SIMPLICIAL MAURER-CARTAN SPACE

The map ∂i assigns to the generators z0 , . . . , zn of Ωn the generators z0 , . . . , zn−1 of Ωn−1 according to the diagram ∈ Ωn

z0 , . . . , zi−1 , zi , zi+1 , . . . , zn ↓ ↓ ↓ ↓ ↓ zi , . . . , zn−1 z0 , . . . , zi−1 , 0,

. ∈ Ωn−1

Since ∂i ought to commute with the differentials, formula (5.5a) determines the values of ∂i on the remaining generators dz0 , . . . , dzn of Ωn by ∂i (dzj ) = d∂i (zj ). Clearly ∂i (z0 + · · · + zn ) = z0 + · · · + zn−1 and ∂i (dz0 + · · · + dzn ) = dz0 + · · · + dzn−1 , so the map ∂i preserves the defining ideals of the algebras Ωn and Ωn−1 as required. Likewise, for 0 ≤ i ≤ n and n ≥ 0, we define a dg-algebra map σi : Ωn → Ωn+1 by ⎧ for j < i, ⎨ zj , zi + zi+1 , for j = i, and (5.5b) σi (zj ) := ⎩ for j > i. zj+1 , The map σi acts on the generators z1 , . . . , zn of Ωn according the scheme z0 , . . . , zi−1 , ↓ ↓ z0 , . . . , zi−1 ,

zi , zi+1 , ↓ ↓ zi + zi+1 , zi+2 ,

...,

zn ↓

. . . , zn+1

∈ Ωn . ∈ Ωn+1

One can, as for ∂i , easily check that σi is a well-defined map of dg-algebras. Proposition 5.6. The above system forms a simplicial dg-commutative associative unital -algebra Ω• = {(Ωn , d)}n≥0 , i.e. a simplicial object in the category of dg-commutative associative unital -algebras. Proposition 5.6 can be verified directly by checking the simplicial identities (5.1) for the maps ∂i and σi . There is, however, a conceptual explanation why Ω• is a simplicial dg-algebra. Take the vectors ei := (0, . . . , 0, 1, 0, . . . , 0) ∈

n+1

, 0 ≤ i ≤ n, (1 at the ith-place)

and denote by the affine subspace of n+1 spanned by {e0 , . . . , en } ⊂ the coordinates z0 , . . . , zn : n+1 → defined by  1, if i = j, and zi (ej ) := 0, otherwise, n

n ⊂

n+1

. In

is the zero set of the function z0 + · · · + zn − 1. The system • = { n }n≥0 is a cosimplicial affine algebraic variety, i.e. a cosimplicial object in the category of affine algebraic varieties. The structure maps n+1

di : n−1 → n , 0 ≤ i ≤ n, n ≥ 1, si : n+1 → n , 0 ≤ i ≤ n, n ≥ 0 are given by the formulas (5.6a)

di (ej ) :=



ej , ej+1 ,

for j < i, and for j ≥ i,

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5. THE SIMPLICIAL MAURER-CARTAN SPACE

while



(5.6b)

si (ej ) :=

ej , ej−1 ,

59

for j ≤ i, and for j > i.

Graphically, the above maps are presented by the following assignments of the basis elements of the generating spaces: e0 , e1 , . . . , ei−1 , ei , . . . , en−1 . . . AAU di :
AU 



. . . e0 , e1 , . . . , ei−1 , ei , ei+1 , . . . , en

∈

n

∈

n+1

so di ‘misses ei ,’ and si :

e0 , e1 , . . . , ei , ei+1 , . . . , en+1 AA U AA U . . . AA U

. . .

e0 , e1 , . . . , ei , . . . , en

∈

n+1

∈

n

so si ‘hits ei twice.’ It is clear that (5.5a)-(5.5b) express how the maps di and si defined in (5.6a)-(5.6b) act on the coordinates. The cosimplicial variety • is an algebraic analog of the standard topological cosimplicial simplex. The dg-algebra Ωn is precisely the dg-algebra ΩDR ( n ) of polynomial De Rham differential forms on the variety n and the structure maps ∂i , σi of the simplicial algebra Ω• are induced by the structure cosimplicial maps di , si of • . Therefore Ω• ∼ = ΩDR ( • ) acquires its simplicial structure from the cosimplicial structure • of , via the contravariant functoriality of the functor ΩDR (−), see [BG76, §1] or [Leh77, Section III.2] for details. ∼ , Ω1 is isomorphic to Example 5.7. While clearly Ω0 = isomorphism [z0 , z1 , dz0 , dz1 ] −→ [z, dz] γ : Ω1 = (z0 + z1 − 1, dz0 + dz1 ) given by

[z, dz] via the

γ(z0 ) := z, γ(z1 ) := 1 − z, γ(dz0 ) := dz, and γ(dz1 ) := −dz. Under this isomorphism, the boundary maps ∂i : Ω1 → Ω0 are described by ∂i : Ω1 ∼ = [z, dz]  h(z) + g(z)dz −→ h(i) ∈ ∼ = Ω0 , i = 0, 1. The map σ0 :

→ [z, dz] is given by σ0 (1) := 1.

Exercise 5.8. Example 5.7 can be generalized. Using the defining relations z0 + · · · + zn = 1 and dz0 + · · · + dzn of Ωn , one may eliminate the generators z0 , dz0 and show that Ωn ∼ = [z1 , . . . zn , dz1 , . . . , dzn ] for each n ∈ . Describe, in this presentation, the simplicial structure maps ∂i and σi .   Let L = L, [−, −], d be a dg-Lie -algebra. Then Ln := L ⊗ Ωn carries, for each n ≥ 0, a natural dg-Lie -algebra structure with the bracket (5.7a)

[g1 ⊗ ω1 , g2 ⊗ ω2 ] := (−1)|g2 ||ω1 | [g1 , g2 ] ⊗ [ω1 , ω2 ],

for g1 , g2 ∈ L and ω1 , ω2 ∈ Ωn , and the differential (5.7b)

(d ⊗  +  ⊗ d)(g ⊗ ω) := dg ⊗ ω + (−1)|g| g ⊗ dω,

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60

5. THE SIMPLICIAL MAURER-CARTAN SPACE

for g ∈ L and ω ∈ Ωn . Remark 5.9. Notice that we use the same symbol both for L and its underlying vector space. In the context of the rest of this chapter, this will not lead to confusion but will substantially simplify the notation. By the functoriality of the tensor product, the simplicial structure of the dgalgebra Ω• = {Ωn }n≥0 induces the structure of a simplicial dg-Lie -algebra on the collection L• = {Ln }n≥0 which in turn induces, for any local complete Noetherian ring R = (R, m) with residue field , a simplicial structure on the component-wise  Ln }n≥0 .  L• = {m ⊗ completed tensor product m ⊗ Definition 5.10. For a dg-Lie -algebra L and a local complete Noetherian ring R = (R, m) with residue field , the simplicial Maurer-Cartan space MCR (L)• is the simplicial set  L• ), (5.8) MCR (L)• := MC(m ⊗  L• . with the simplicial structure induced by the one of m ⊗ The nth component MCR (L)  n of the simplicial set MCR (L)• , by definition,   (L ⊗ Ωn ) of Maurer-Cartan elements in the dg-Lie algebra equals the set MC m ⊗  (L ⊗ Ωn ). Observe that MCR (L)0 = MCR (L). The following statement is m⊗ a particular case of the more general Theorem 6.9 which we will prove in Chapter 6. Theorem 5.11. In the situation of Definition 5.10, the simplicial MaurerCartan space MC• (L) is a Kan simplicial set. The rest of this chapter will be devoted to the following version of the Main Homotopy Theorem of [SS84]. Theorem 5.12. Let L be a dg-Lie -algebra and R = (R, m) a local complete Noetherian ring with residue field . Then one has an isomorphism   (5.9) M C L (R) ∼ = π0 MCR (L)• . One can therefore take (5.9) as the definition of the Maurer-Cartan moduli space. It has the advantage that it generalizes to strongly homotopy Lie algebras when the gauge group GL (R) involved in the previous definition of the moduli space M C L (R) need not exist. The right hand side of (5.9) is the set of equivalence classes of Maurer-Cartan  L1 modulo the relation that identifies two such elements s0 , s1 if elements in m ⊗  L11 such that si = ∂i H, i = 0, 1. there exists a Maurer-Cartan element H ∈ m ⊗ To unravel this  description,  we need some notation. For L = L, [−, −], d and R = (R, m) as in Theorem 5.12, let L[z] be the space of polynomials with coefficients in L. We explained on page 3 that an element   L[z] is a sequence X(z) = x1 (z), x2 (z), x3 (z), . . . of X(z) of the completion m ⊗ compatible elements xi (z) ∈ m/mi ⊗L[z]. Since each xi (z) is a polynomial with coefficients in the -vector space m/mi ⊗L, it makes sense to take its evaluation xi (α) ∈ m/mi ⊗ L at a scalar α ∈ and put    L. (5.10a) X(α) := x1 (α), x2 (α), x3 (α), . . . ∈ m ⊗ Each polynomial xi (z) has also its formal z-derivative ∂z xi (z) ∈ m/mi ⊗ L[z] defined as follows. Write xi (z) = a0 + a1 z + a2 z 2 + · · · + aN z N ,

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5. THE SIMPLICIAL MAURER-CARTAN SPACE

61

with some ai ∈ m/mi ⊗ L, 0 ≤ i ≤ N . Then ∂z xi (z) = a1 + 2a2 z + 3a3 z 2 + · · · + N aN −1 z N . We define (5.10b)

   L[z]. ∂z X(z) := ∂z x1 (z), ∂z x2 (z), ∂z x3 (z), . . . ∈ m ⊗

Finally, the differential ( ⊗ d) of m/mi ⊗ L can be applied to the coefficients of the polynomial xi (z). We denote the result dxi (z) ∈ m/mi ⊗ L[z] and    L[z]. (5.10c) dX(z) := dx1 (z), dx2 (z), dx3 (z), . . . ∈ m ⊗ The compatibility of the sequences in the right-hand sides of (5.10a)–(5.10c) is  L[z], obvious. It is also clear that, for X  (z), X  (z) ∈ m ⊗            (5.11) ∂z X (z), X (z) = ∂z X (z), X (z) + X (z), ∂z X  (z) .  L[z] = L[z][[t]], the space of power Example 5.13. If R = [[t]], then R ⊗  L[z] is a formal sum series in t with coefficients in L[z]. An element X(z) of R ⊗  aij ti z j , aij ∈ L, X(z) = i,j≥0

with the property that the set {j | aij = 0} is finite for each fixed i ≥ 0. Such an  L[z] if and only if a0j = 0 for each j ≥ 0. One has X(z) belongs to m ⊗   X(α) = aij αi tj ∈ L[[t]], α ∈ , i

∂z X(z) =



j

jaij ti z j−1 , and

i,j≥0

dX(z) =



daij ti z j .

i,j≥0

The isomorphism γ of Example 5.7 induces an identification L11 ∼ = L[z, dz]1 . 1  Elements of m ⊗ L1 can therefore be written as expressions (5.12)

 L0 [z].  L1 [z] and g(z) ∈ m ⊗ H(z) = h(z) + g(z)dz, with h(z) ∈ m ⊗

 L11 → m ⊗  L10 are the evaluations The simplicial boundaries ∂0 , ∂1 : m ⊗   ∂i h(z) + g(z)dz := h(i), i = 1, 2. Exercise 5.14. Show that, under the identification L11 ∼ = L[z, dz]1 introduced above, the differential (5.7b) of L1 acts on an element (5.12) as   (d ⊗  +  ⊗ d) h(z) + g(z)dz = dh(z) − ∂z h(z)dz + dg(z)dz. Notice in particular the minus sign at ∂z h(z)dz. It follows from the description of the differential given in Example 5.14 that the Maurer-Cartan equation for (5.12) reads  1 dh(z) − ∂z h(z)dz + dg(z)dz + h(z) + g(z)dz, h(z) + g(z)dz = 0, 2 where the meaning of ∂z h(z) (resp. dg(z)) is as in (5.10b) (resp. (5.10c)). By the  L1 , definition of the bracket in m ⊗         h(z) + g(z)dz, h(z) + g(z)dz = h(z), h(z) + 2 h(z), g(z) dz + g(z), g(z) dzdz.

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62

5. THE SIMPLICIAL MAURER-CARTAN SPACE

Since, by the graded commutativity of Ω1 , dzdz = 0, we conclude that the MaurerCartan equation for H(z) = h(z) + g(z)dz is equivalent to  1 (5.13a) dh(z) + h(z), h(z) = 0, and 2   ∂z h(z) = dg(z) + h(z), g(z) . (5.13b) The set π0 (MCL (R)• ) is, by definition, the quotient of the set MCR (L) of  L, by the relation that identifies s0 ∈ MCR (L) Maurer-Cartan elements in m ⊗   with s1 ∈ MCR (L) if there exists a solution h(z), g(z) of (5.13a)-(5.13b) such that si = h(i), for i = 0, 1. Let us prove a couple of auxiliary results. The first one claims the existence and uniqueness of a solution of a linear differential equation. Proposition 5.15. Let H be either a nilpotent graded Lie algebra, or a graded  L, where L is a graded Lie -algebra and m the Lie algebra of the form H = m ⊗ maximal ideal in a local complete Noetherian ring R. Assume that g(z) ∈ H 0 [z] and X0 ∈ H n for some n. Then the equation   (5.14) ∂z X(z) = X(z), g(z) with the boundary condition X(0) = X0 has a unique solution X(z) ∈ H n [z]. Proof. Let us treat the nilpotent case first. Assume g(z) = g0 + g1 z + g2 z 2 + g3 z 3 + · · · + gN z N for some N ≥ 0 and g1 , . . . , gN ∈ H 0 , and seek a solution in the form (5.15)

X(z) = h0 + h1 z + h2 z 2 + h3 z 3 + · · ·

with some h0 , h1 , h2 , . . . ∈ H n . Equation (5.14) expands into h1 + 2h2 z + 3h3 z 2 + · · · =     = [g0 , h0 ] + [g1 , h0 ] + [g0 , h1 ] z + [g2 , h0 ] + [g1 , h1 ] + [g0 , h2 ] z 2 + · · · . Comparing the coefficients at the same powers of z we conclude that X(z) in (5.15) solves (5.14) with the requisite boundary condition if and only if its coefficients satisfy the recursion (5.16)

h0 := X0 and 1  hi := [gr , hs ], i ∈ . i r+s=i−1

It remains to prove that X(z) with the coefficients uniquely determined by (5.16) is a polynomial in z, i.e. that the sum (5.15) terminates after a finite number of terms. Let H(1) := H and H(s) := [H, H(s−1) ] for s ≥ 2. The nilpotence of H means that H(s) = 0 if s is large enough. It is routine to check that (h1 , h2 , h3 , . . .) in (5.16) satisfy h1 , . . . , hN +1 ∈ H(2) hN +2 , . . . , h2N +2 ∈ H(3) .. . h(s−1)N +s , . . . , hsN +s ∈ H(s+1) , s ∈ .

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5. THE SIMPLICIAL MAURER-CARTAN SPACE

63

It follows from the nilpotence of H that hi = 0 for i sufficiently large, therefore X(z) in (5.15) indeed belongs to H n [z]. This finishes the nilpotent case.  L and express X(z), g(z) and X0 as compatible Assume now that H = m ⊗ sequences X(z) = (x1 (z), x2 (z), x3 (z), . . .), xi (z) ∈ m/mi ⊗ Ln [z], g(z) = (g1 (z), g2 (z), g3 (z), . . .), gi (z) ∈ m/mi ⊗ L0 [z], and X0 = (x01 , x02 , x03 , . . .), xi ∈ m/mi ⊗ Ln . Equation (5.14) is equivalent to the infinite set of equations   (5.17a) ∂z xk (z) = xk (z), gk (z) , k ≥ 1, and the boundary condition X(0) = X0 to the infinite set of boundary conditions xk (0) = x0k , k ≥ 1.

(5.17b)

Notice that the Lie algebra m/mk ⊗ L is nilpotent for each k ≥ 0. Therefore, by the already proved nilpotent case taken with H = m/mk ⊗ L, equation (5.17a) with the boundary condition (5.17b) has, for each fixed k, a unique solution xk (z) ∈ m/mk ⊗ Ln [z]. It remains to prove that  X(z)  = (x1 (z), x2 (z), x3 (z), . . .) is a compatible sequence, that is, xi (z) = πij xj (z) , for all i ≤ j, where πij : m/mj ⊗ L[z] → m/mi ⊗ L[z] is the map induced by the canonical projection m/mj  m/mi . Since   πij gj (z) = gi (z) and since πij is a Lie algebra morphism that commutes   with ∂z , by applying πij to (5.17a)-(5.17b) with k = j we verify that πij xj (z) satisfies       (5.18a) ∂z πij xj (z) = πij xj (z) , gk (z) with the boundary condition

  πij xj (0) = πij (x0j ) = x0i .

(5.18b)

  Since xi (z) is another solution to (5.17a)-(5.17b), xi (z) = πij xj (z) by the uniqueness. This finishes the proof in the completed case.  Lemma 5.16. Equation (5.13a) follows from (5.13b) and the requirement that  L. h(0) is a Maurer-Cartan element in m ⊗ Proof. Let us denote, only for the purpose of this proof, the left hand side  L2 [z]. Clearly, of (5.13a) by X(z) ∈ m ⊗   (5.19) ∂z X(z) = d∂z h(z) + ∂z h(z), h(z) . Equation (5.13b) implies

  ∂z h(z) = dg(z) + h(z), g(z) and     d∂z h(z) = dh(z), g(z) − h(z), dg(z) .

Substituting the right hand sides of the above equations for ∂z h(z) and d∂z h(z) in (5.19), we obtain         ∂z X(z) = dh(z), g(z) − h(z), dg(z) + dg(z), h(z) + [h(z), g(z)], h(z) .

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5. THE SIMPLICIAL MAURER-CARTAN SPACE

Taking into account that, by the graded Jacobi identity and the graded anticommutativity,    1 [h(z), g(z)], h(z) = [h(z), h(z)], g(z) 2 we get    1 1 ∂z X(z) = [dh(z), g(z)] + [h(z), h(z)], g(z) = dh(z) + [h(z), h(z)], g(z) , 2 2 so X(z) satisfies the linear differential equation   (5.20) ∂z X(z) = X(z), g(z) with the boundary condition  1 h(0), h(0) = 0 2 expressing our assumption that h(0) is a Maurer-Cartan element.  L2 [z] also solves (5.20) with the same boundary condition, by Since 0 ∈ m ⊗ the uniqueness of Proposition 5.15, X(z) = 0. This finishes the proof.  X(0) = dh(0) +

 L1 Corollary 5.17. Let h(z) and g(z) satisfy (5.13a)-(5.13b) and l0 ∈ m ⊗ be a Maurer-Cartan element. Then h(z) is uniquely determined by g(z) and by the boundary condition h(0) = l0 .     Proof. Let h (z), g(z) and h (z), g(z) be two solutions of (5.13a)-(5.13b) the same boundary condition h(0) = h (0) = l0 . Subtracting (5.13b) for fulfilling  h (z), g(z) from (5.13b) for h (z), g(z) we verify that the difference h (z)−h (z) solves the linear differential equation     ∂z h (z) − h (z) = h(z) − h (z), g(z) with the boundary condition h (0) − h (0) = 0. By the uniqueness of Proposi tion 5.15, h (z) − h (z) = 0.  L1 . Then, for each n ∈ ,  L0 [z] and x ∈ m ⊗ Lemma 5.18. Let φ(z) ∈ m ⊗   (5.21a) ∂z adnφ(z) (x) = n ∂z φ(z), adn−1 φ(z) (x) , and     adnφ(z) ∂z dφ(z) = ∂z φ(z), adn−1 (5.21b) φ(z) (dφ(z)) . Proof. We prove both equations by induction. For n = 1, (5.21a) is an immediate consequence of (5.11). Assume we have proved (5.21a) for some n ≥ 1. Then, by (5.11) and the inductive assumption,   n ∂z adn+1 φ(z) (x) = ∂z φ(z), adφ(z) (x)     = ∂z φ(z), adnφ(z) (x) + φ(z), ∂z adnφ(z) (x) (5.22)     = ∂z φ(z), adnφ(z) (x) + n φ(z), [∂z φ(z), adn−1 φ(z) (x)] . The graded Jacobi identity gives   φ(z), [∂z φ(z),adn−1 φ(z) (x)] = (5.23)    n−1  ∂z φ(z), [φ(z), adn−1 φ(z) (x)] + adφ(z) (x), [∂z φ(z), φ(z)] .   Since deg φ(z) = 0, the graded skew-symmetry of the bracket implies   φ(z), φ(z) = 0,

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5. THE SIMPLICIAL MAURER-CARTAN SPACE

65

thus

  1   ∂z φ(z), φ(z) = ∂z φ(z), φ(z) = 0. 2 Combining it with the observation that   n φ(z), adn−1 φ(z) (x) = adφ(z) (x),

(5.24)

we rewrite (5.23) into    n φ(z), [∂z φ(z), adn−1 φ(z) (x)] = ∂z φ(z), adφ(z) (x)]. Replacing the last term of (5.22) by the right hand side of the above equation, we obtain   n ∂z adn+1 φ(z) (x) = (n + 1) ∂z φ(z), adφ(z) (x) , which finishes the induction step. Equation (5.21b) with n = 1 says that [φ(z), ∂z dφ(z)] = [∂z φ(z), dφ(z)], which is the result of applying the differential d to (5.24). Assume we have proved (5.21b) for some n ≥ 1. Then, by the definition of the adjoint action and induction,       n n−1 adn+1 φ(z) ∂z dφ(z) = φ(z), adφ(z) (∂z dφ(z)) = φ(z), [∂z φ(z), adφ(z) (dφ(z))] . The graded Jacobi identity gives   φ(z), [∂z φ(z), adn−1 φ(z) (dφ(z))] =     n−1 = adn−1 φ(z) (dφ(z)), [∂z φ(z), φ(z)] + ∂z φ(z), [φ(z), adφ(z) (dφ(z))] . By (5.24), the first term in the right hand side equals zero, while clearly     n ∂z φ(z), [φ(z), adn−1 φ(z) (dφ(z))] = ∂z φ(z), adφ(z) (dφ(z)) . Assembling the above formulas, we get     n adn+1 φ(z) ∂z dφ(z) = ∂z φ(z), adφ(z) (dφ(z)) , 

which establishes the induction step.  L1 , Corollary 5.19. For φ(z) ∈ GL[z] (R) and l ∈ m ⊗     ∂z φ(z) · l + φ(z) · l, ∂z φ(z) + ∂z dφ(z) = 0. Proof. By the definition (4.7) of the action  1      1 adnφ(z) (l) − ∂z adnφ(z) (dφ(z)) ∂z φ(z) · l = ∂z n! (n + 1)! n≥0 n≥0 (5.25)   1   1 ∂z adnφ(z) (l) − ∂z adnφ(z) dφ(z) . = n! (n + 1)! n≥0

n≥0

Using (5.21a) we obtain that, for n ≥ 1,   1 1 ∂z adnφ(z) (l) = ∂z φ(z), adn−1 φ(z) (l) n! (n − 1)! while, for n = 0, 1 ∂z adnφ(z) (l) = ∂z (l) = 0. n! Similarly, by (5.21a) and (5.21b), for n ≥ 1,     n ∂z adnφ(z) (dφ(z)) = n ∂z φ(z), adn−1 φ(z) (dφ(z)) + adφ(z) ∂z dφ(z)

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66

5. THE SIMPLICIAL MAURER-CARTAN SPACE

    n−1 = n ∂z φ(z), adn−1 φ(z) (dφ(z)) + ∂z φ(z), adφ(z) (dφ(z))   = (n + 1) ∂z φ(z), adn−1 φ(z) (dφ(z)) , therefore

   1 1 ∂z adnφ(z) dφ(z) = ∂z φ(z), adn−1 φ(z) (dφ(z)) , (n + 1)! n! while, for n = 0, 1 ∂z adnφ(z) (dφ(z)) = ∂z dφ(z). (n + 1)! Substituting the above results into (5.25) gives     1 1 n−1 ∂z (φ(z) · l) = ∂z φ(z), (l) − (dφ(z)) − ∂z dφ(z) adn−1 ad (n − 1)! φ(z) n! φ(z) n≥1 n≥1    1  1 = ∂z φ(z), adnφ(z) (l) − adnφ(z) (dφ(z)) − ∂z dφ(z) n! (n + 1)! n≥0 n≥0   = ∂z φ(z), φ(z) · l − ∂z dφ(z). The graded commutativity     ∂z φ(z), φ(z) · l = − φ(z) · l, ∂z φ(z) of the bracket in the first term of the bottom line converts the above equation into the requisite result.  We say that Maurer-Cartan elements l0 , l1 ∈ MCL (R) are gauge equivalent if there exists g ∈ GL (R)  such that  l1 = g · l0 . They are homotopic if there exists a homotopy H(z) = h(z), g(z) as in (5.13a)-(5.13b) such that li = h(i), i = 1, 2. Theorem 5.12 follows from Theorem 5.20. Maurer-Cartan elements l0 and l1 are gauge equivalent if and only if they are homotopic. Proof. We will follow the scheme of the proof of [CL10, Theorem 4.4]. Suppose that l0 and l1 are gauge equivalent which, by definition, means   that l1 = g · l0 for some g ∈ GL (R) ⊂ GL[z] (R). We claim that then (gz) · l0 , −g is a homotopy between l0 and l1 .1 The boundary conditions (gi) · l0 = li for i = 1, 2 are clearly satisfied. By  L[z], so h(z) := (gz) · l0 element in m ⊗ Corollary 4.5, (gz) · l0 is the Maurer-Cartan  satisfies (5.13a). Equation (5.13b) for (gz) · l0 , −g amounts to     (5.26) ∂z (gz) · l0 + dg + (gz) · l0 , g = 0 which immediately follows from Corollary 5.19 taken with φ(z) = gz. Let us prove the opposite  implication. Suppose that l0 and l1 are homotopic, via a homotopy h(z), g(z) . By Corollary 5.17, h(z) is determined by g(z) and  L0 [z] the boundary condition h(0) = l0 . Suppose we found φ(z) ∈ GL[z] (R) = m ⊗ solving the equation (5.27)

∂z φ(z) = −g(z),

with the boundary condition φ(0) = 0. By Corollary 5.19,         ∂z φ(z) · l + φ(z) · l, ∂z φ(z) + ∂z dφ(z) = ∂z φ(z) · l − φ(z) · l, g − dg = 0, 1 The

  unexpected minus sign at g in the homotopy (gz) · l0 , −g is explained in Remark 5.21.

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5. THE SIMPLICIAL MAURER-CARTAN SPACE

L∗ :

···

4

3

2

•1

0

−1

−2

−3

−4

67

···

simply connected homotopy types connected homotopy types

disconnected homotopy types Figure 2. Connectivity of L and the corresponding homotopy types. The bullet • marks the degree of Maurer-Cartan elements. therefore

    ∂z φ(z) · l0 = dg(z) + φ(z) · l0 , g(z) .   Thus the pair φ(z)·l0 , g(z) also solves (5.13a)-(5.13b) with the boundary condition φ(0) · l0 = 0 · l0 = l0 , therefore φ(z) · l0 = h(z) by Corollary 5.17. In particular, φ(1) · l0 = h(1) = l1 . So l0 is gauge equivalent to l1 via φ(1) ∈ GL (R). To finish the proof, it remains to notice that equation (5.27) can be solved by an obvious formal integration, over z, of the polynomial −g(z). This is easy and we leave it as an exercise.    Remark 5.21. The unpleasant minus sign in the homotopy (gz) · l0 , −g and in the right hand side of (5.27) stems from the fact that (4.6) defines a left action of the gauge group. This is dictated by the standard definition of the adjoint action x, s → [x, s], which is the infinitesimal of a left, not right, action. To get a proof of Theorem 5.20 without the disturbing minus sign, one needs to replace (4.6) by the right action x · s := (−x) · s. Let us close this chapter with a couple of heuristic remarks. The geometric realization [May67, §14] applied to the simplicial Maurer-Cartan space gives the functor L → XL := |MC• (L)| from the category of dg-Lie algebras to the category of topological spaces. Assuming that is the field of rational numbers, and restricting to dg-Lie algebras such that Ln = 0 for n ≥ 0, one gets precisely the functor that in rational homotopy theory relates dg-Lie algebras to simply connected rational homotopy types, see [Qui69, Tan83]. One then has, for each n ≥ 2, an isomorphism ∼ H 1−n (L, d) πn (XL ) ⊗ = of -vector spaces. The unusual 1 − n instead of the expected n − 1 is given by our convention that places Maurer-Cartan elements in degree +1. The degree of differentials in dg-Lie algebras then must be +1, not −1 as in rational homotopy theory. Dg-Lie algebras satisfying the weaker condition Ln = 0 for n ≥ 1 should therefore “ideologically” correspond to connected homotopy types. The Maurer-Cartan moduli space functor acting on dg-Lie algebras with no degree restriction can be, at least on the conceptual level, interpreted as disconnected rational homotopy theory, see Figure 2.

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http://dx.doi.org/10.1090/cbms/116/06

CHAPTER 6

Strongly homotopy Lie algebras In this chapter we extend the definition of the Maurer-Cartan moduli space from dg-Lie algebras to their homotopy versions. Let us start by recalling some necessary notions. For a -graded vector space W , we denote by S(W ) the free graded commutative associative algebra generated by W . It is characterized by the standard universal property with respect to graded commutative associative algebras. We identify S(W ) with the quotient of the tensor algebra T (W ) by the ideal generated by 



w ⊗ w − (−1)|w ||w | w ⊗ w , w , w ∈ W. Decomposing W = W even ⊕ W odd , where   W k and W odd := Wk W even := k even

k odd

are the even and odd parts of W , respectively, then S(W ) is isomorphic to the product S(W ) ∼ = [W even ] ⊗ ∧(W odd ), of the polynomial algebra generated by W even with the the exterior (Grassmann) algebra generated by W odd . Various different notations for S(W ) are used in the literature. In [LM95] it was denoted ∧(W ), which is the notation accepted in rational homotopy theory [FHT01]. In Chapter 5 we denoted the free graded commutative associative algebra generated by W := Span (z0 , . . . , zn , dz0 , . . . , dzn ) as [W ], following the convention traditionally used in the context of polynomial differential forms. For a permutation σ ∈ Su and linearly independent homogeneous elements w1 , . . . , wu ∈ W we define the Koszul sign ε(σ) ∈ {−1, +1} by the equality (6.1a)

w1 · · · wu = ε(σ)wσ(1) · · · wσ(u)

of elements of S(W ). The skew-symmetric Koszul sign χ(σ) ∈ {−1, +1} is given by (6.1b)

χ(σ) := sgn(σ)ε(σ).

Both ε(σ) and χ(σ) clearly depend only on the permutation σ and the degrees of the elements w1 , . . . , wu . Exercise 6.1. Express (σ) and χ(σ) in terms of the permutation σ and the degrees |w1 |,. . . ,|wn |. Recall that a permutation σ ∈ Su is an (s, u−s)-unshuffle with some 1 ≤ s ≤ u if σ(1) < . . . < σ(s) and σ(s + 1) < . . . < σ(u). The set of all (s, u − s)-unshuffles will be denoted Ss,u−s . 69 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

70

6. STRONGLY HOMOTOPY LIE ALGEBRAS

Definition 6.2. An L∞ -algebra (also called a strongly homotopy Lie or sh Lie algebra) over a commutative ring R is a graded R-module X together with a system lu : uR X → X, u ∈ , of linear maps of degrees 2 − u satisfying the following axioms. – Antisymmetry: For every u ∈ , σ ∈ Su and v1 , . . . , vu ∈ X, lu (vσ(1) , . . . , vσ(u) ) = χ(σ)lu (v1 , . . . , vu ).

(6.2)

– For every u ∈  and v1 , . . . , vu ∈ X,   (−1)s χ(σ)lr (ls (vσ(1) , . . . , vσ(s) ), vσ(s+1) , . . . , vσ(u) ) = 0. (Lu ) r+s=u+1

σ∈Ss,u−s

A strict morphism of L∞ -algebras over R is a degree-zero R-linear map of the underlying R-modules which commutes with all structure operations.1 Remark 6.3. The sign in (Lu ) was taken from [Get09a]. With this sign convention, all terms of the (generalized) Maurer-Cartan equation (6.5) below bear +1 signs. Our sign convention is related to the original one of [LS93, LM95] via the transformation u(u+1) lu → (−1) 2 lu . We also use the opposite grading, better suited for our purposes – the operations lu as introduced in [LS93, LM95] have degrees u − 2. Example 6.4. Axiom (L1 ) means l1 ◦ l1 = 0, so l1 : X → X is a degree +1 differential which we denote, as in the A∞ -case, by d. The antisymmetry (6.2) demands that l2 : X ⊗R X → X is a linear degree 0 graded anticommutative operation, l2 (b, a) = −(−1)|a||b| l2 (a, b), a, b ∈ X. Axiom (L2 ) gives dl2 (a, b) = l2 (da, b) + (−1)|a| l2 (a, db), a, b ∈ X, meaning that d is a degree +1 derivation with respect to the multiplication l2 . Writing [u, v] := l2 (u, v), (L2 ) takes more usual form d[a, b] = [da, b] + (−1)|a| [a, db]. The degree −1 graded skew-symmetric map l3 : X ⊗R X ⊗R X → X satisfies (L3 ), which, for a, b, c ∈ X reads (6.3)

[[a, b], c] + (−1)|b||c|+1 [[a, c], b] + (−1)|a|(|b|+|c|) [[b, c], a] = dl3 (a, b, c) + l3 (da, b, c) + (−1)|a| l3 (a, db, c) + (−1)|a|+|b| l3 (a, b, dc).

Notice that the left hand side of the above equation equals        (−1)|a||c| (−1)|a||c| [a, b], c + (−1)|b||a| [b, c], a + (−1)|c||b| [c, a], b which is the Jacobiator (3.8) multiplied by (−1)|a||c| , while in the right hand side we see the d-boundary of the trilinear map l3 . Axiom (L3 ) therefore requires that the bracket [−, −] satisfies the graded Jacobi identity modulo the homotopy (−1)|a||c| · l3 . 1 Strict

morphisms are particular cases of morphisms of L∞ -algebras recalled in Definition 7.1.

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6. STRONGLY HOMOTOPY LIE ALGEBRAS

71

Example 6.5. An L∞ -algebra L = (X, l1 , l2 , l3 , . . .) over R whose structure operations all vanish except for l1 is called abelian. It is obviously the same as a graded R-module X with the differential d = l1 . If the only nontrivial operations are l1 and l2 , then L is our familiar dg-Lie R-algebra from Definition 3.4 with d = l1 and the Lie bracket [−, −] = l2 . L∞ -algebras thus are generalizations of dg-Lie algebras. Example 6.6. It is classical that each graded associative algebra B = (M, · ) defines a dg-Lie algebra with the same underlying space and the bracket [a, b] := ab − (−1)|a||b| ba, for a, b ∈ M. As shown in [LM95, Theorem 3.1], an analogous construction holds also for strongly homotopy algebras, i.e. the antisymmetrization of the structure operations of an A∞ -algebra leads to an L∞ -algebra. Generalized M.-C. equation. Let H be an L∞ -algebra whose each component H n is a complete -vector space with a fundamental system {Hin }i≥1 of neighborhoods of zero such that the following L∞ -analog of (4.8) is fulfilled:2 (6.4)

+···+nu +2−u lu (Han11 , . . . , Hanuu ) ⊂ Han11+···+a u

for each a1 , . . . , au ≥ 1, n1 , . . . , nu , and u ≥ 1. Then, for each s ∈ H11 and u ≥ 1, lu (s, . . . , s) ⊂ Hu2 , so it makes sense to consider the L∞ -Maurer-Cartan equation 1 1 1 (6.5) l1 (s) + l2 (s, s) + l3 (s, s, s) + · · · + ln (s, . . . , s) + · · · = 0, 2 3! n! whose left hand side converges to an element of H 2 . We denote MC(H1 ) ⊂ H11 the set of solutions of (6.5). When H is a dg-Lie algebra L, one recognizes the ordinary Maurer-Cartan equation (3.6). Remark 6.7. While the Maurer-Cartan equation (3.6) is defined in an arbitrary dg-Lie algebra, its L∞ -version requires additional assumptions that guarantee the convergence of the infinite sum it contains. This can be achieved either by assuming the completeness of H along with a condition similar to (6.4), or by a certain form of nilpotence as in Definition 6.10. Another kind of condition under which the L∞ Maurer-Cartan equation makes sense will be the local finiteness of Proposition 9.17. Let L be an L∞ -algebra over . The following generalization of formulas (5.7a)(5.7b) equip, for each n ≥ 0, the tensor product Ln := L ⊗ Ωn of L and the nth piece Ωn of the simplicial dg-algebra Ω• recalled in Chapter 5, with a natural L∞ structure, with the differential (5.7b) and the higher brackets lu (g1 ⊗ ω1 , . . . , gu ⊗ ωu ) := (−1)



i>j

|gi ||ωj |

· lu (g1 , · · · , gu ) ⊗ (ω1 · · · ωu ), u ≥ 2,

where g1 , . . . , gu ∈ L and ω1 , . . . , ωu ∈ Ωn . Likewise, for any L∞ -algebra H and a commutative associative -algebra a, the tensor product a ⊗ H is an L∞ -algebra, with the structure operations lu (a1 ⊗ h1 , . . . , au ⊗ hu ) := (a1 · · · au ) ⊗ lu (h1 , · · · , hu ), u ≥ 1, where h1 , . . . , hu ∈ H and a1 , . . . , au ∈ a. As a consequence of the above observations, for a local complete Noetherian ring R = (R, m) with residue field and n ≥ 0, the tensor product m ⊗ Ln is 2 As no confusion may occur here, we denote by the same symbol both the L -algebra and ∞ its underlying space.

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72

6. STRONGLY HOMOTOPY LIE ALGEBRAS

an L∞ -algebra. It can moreover be, as in the proof of Lemma 3.15, easily shown that its L∞ -structure extends, by the continuity and completeness, to an L∞  Ln which fulfills (6.4). By the naturality of above constructions, structure on m ⊗  Ln }n≥0 is a simplicial object in the category of L∞ -algebras and  L• = {m ⊗ m⊗ their strict morphisms. Definition 6.8. For an L∞ -algebra L over and a complete local Noetherian ring R = (R, m) with residue field , the simplicial L∞ -Maurer-Cartan space is the simplicial set  L• ) MCR (L)• := MC(m ⊗  with the simplicial structure induced by m ⊗ L• . Theorem 6.9. Let R be a local complete Noetherian ring with residue field and L an arbitrary L∞ -algebra over . Then the simplicial Maurer-Cartan space MCL (R)• of L is a Kan simplicial set. Since dg-Lie algebras are particular cases of L∞ -algebras, the above statement implies Theorem 5.11 of Chapter 5. Thanks to Theorem 6.9, it makes sense to define the L∞ -Maurer-Cartan moduli space as the the set of connected components of the simplicial space MCL (R)• :   M C L (R) := π0 MCR (L)• . The rest of this chapter will be devoted to the proof of Theorem 6.9. We will use the fact that MCL (R)• is the inverse limit of the simplicial Maurer-Cartan spaces of L∞ -algebras which are nilpotent in the sense specified in Definition 6.10 below. We will also need Theorem 6.13 about maps between the Maurer-Cartan simplicial spaces induced by epimorphisms of nilpotent L∞ -algebras. Let us recall the necessary terminology. The lower central series of an L∞ algebra H is defined inductively by H(1) := H and, for s ≥ 2,   lu (H(s1 ) , . . . , H(su ) ). (6.6) H(s) := u≥2 s1 +···+su ≥s s1 ,...,su