Hochschild

Sarah Witherspoon

Introduction At the end of the nineteenth century, Poincar´e created invariants to distinguish different topological spaces and their features, foreshadowing the and cohomol- ogy groups that would appear later. Towards the middle of the twentieth century, these notions were imported from to algebra: The subject of homology and cohomology was founded by Eilenberg and Mac Lane, and the subject of and cohomology by Hochschild. The uses of homological techniques contin- ued to grow and spread, spilling out of algebra and topol- ogy into many other fields. In this article we will focus on Hochschild cohomol- ogy, which now appears in the settings of algebraic geom- etry, theory, functional analysis, topology, and Gerhard Hochschild, 2003. beyond. There are strong connections to and K-theory. Many mathematicians use Hochschild co- We will begin this story by setting the scene: We are in- homology in their research, and many continue to develop terested here in a 퐴 that is also a over a theoretical and computational techniques for better under- 푘 such as ℝ or ℂ. We require the multiplication standing. Hochschild cohomology is a broad and growing on 퐴 to be bilinear; that is, the map 퐴 × 퐴 → 퐴 given by field, with connections to diverse parts of mathematics. (푎, 푏) ↦ 푎푏 for 푎, 푏 in 퐴 is bilinear (over 푘). A ring with Our scope here is exclusively Hochschild cohomology this additional structure is called an algebra over 푘. Some for algebras, including Hochschild’s original design [3] examples are polynomial rings 퐴 = 푘[푥1, … , 푥푛], which are and just a few of the many important uses and recent de- commutative, and matrix rings 퐴 = 푀푛(푘), which are non- velopments in algebra. Some details and further references commutative when 푛 > 1. may be found in [5, 12, 13]. Our focus will be on multilinear maps

Sarah Witherspoon is a professor and head of the Department of Mathematics 퐴 × ⋯ × 퐴 → 퐴 at Texas A&M University. Her email address is [email protected]. and what they can tell us about the structure and the rep- Communicated by Notices Associate Editor Daniel Krashen. resentations, that is, modules, of 퐴. Our story has two For permission to reprint this article, please contact: threads: One starts with particular functions on 퐴 called [email protected]. derivations. The other starts with the 푍(퐴) of 퐴, that DOI: https://doi.org/10.1090/noti2099 is, the subalgebra of all elements commuting with every

780 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6 element of 퐴. Both are then viewed in a wider, multilinear for all 훿, 훿′, 훿″ in Der(퐴). That is, the bracket operation [,] context. These two threads lead in the same direction and is alternating and the Jacobi identity holds. Put another way, combine in the theory of Hochschild cohomology and Der(퐴) is a Lie algebra. in its two important binary operations, the Gerstenhaber For example, Der(푀2(푘)) may be identified with the bracket and the , as we will see. Lie algebra 픰픩2(푘) of 2 × 2 matrices of trace 0, called the special linear algebra, provided the characteristic is not 2. Derivations Specifically, each matrix 푋 of trace 0 determines a deriva- Every calculus student has seen the Leibniz rule (also tion 훿푋 given by a commutator: For all 푌 in 푀2(푘), set known as the product rule), 훿푋 (푌) = 푋푌 − 푌푋. All derivations are of the form 훿푋 for ′ 푑 푑 푑 some 푋 in 픰픩2(푘), and distinct matrices 푋, 푋 in 픰픩2(푘) de- (푓푔) = (푓)푔 + 푓 (푔), 푑푥 푑푥 푑푥 termine distinct derivations 훿푋 , 훿푋′ . From another vantage point, we see Lie algebras acting for differentiable functions 푓, 푔. The set of all differen- on an algebra 퐴 by derivations, that is, through Lie algebra tiable functions from ℝ to ℝ forms a ring 퐴 that is also homomorphisms to Der(퐴). Such actions are ubiquitous a vector space over ℝ, and in fact is an algebra over ℝ. and uncover important structural information about 퐴. Generalizing the Leibniz rule to other algebras over our field 푘, we define a derivation on an algebra 퐴 to be a 푘- Multilinear Functions 훿 ∶ 퐴 → 퐴 linear such that We wish to consider more generally multilinear functions 훿(푎푏) = 훿(푎)푏 + 푎훿(푏) 훿 ∶ 퐴푛 → 퐴 for all 푎, 푏 in 퐴. We will be interested in the set of all deriva- tions, where 퐴푛 = 퐴 × ⋯ × 퐴 (푛 factors). Are there useful multi- Der(퐴) = {훿 ∶ 퐴 → 퐴 ∣ 훿 is a derivation}, linear analogs of derivations on 퐴? The answer is yes, and this is where we start to see the ideas of Hochschild from which is itself a vector space under addition and scalar mul- the 1940s. tiplication of functions. From now on we will use some homological terminol- As a small example, let 퐴 = 푘[푥], that is, polynomials ogy: A derivation is also called a Hochschild 1-cocycle. Now in one indeterminate 푥 with coefficients in 푘. Define the suppose 훿 ∶ 퐴2 → 퐴 is a bilinear function. It is a Hochschild 푑 derivation as usual, that is, 푑푥 2-cocycle if 푑 (푥푛) = 푛푥푛−1 훿(푎푏, 푐) − 훿(푎, 푏푐) = 푎훿(푏, 푐) − 훿(푎, 푏)푐 푑푥 푑 푛 for all 푛 ≥ 1 and (1) = 0, and extend linearly so that for all 푎, 푏, 푐 in 퐴. More generally, suppose 훿 ∶ 퐴 → 퐴 is 푑푥 푑 an 푛-linear function. It is a Hochschild 푛-cocycle if (푝) is defined for each polynomial 푝 = 푝(푥). A calcula- 푑푥 푑 푛 tion shows that Der(퐴) = 푘[푥] ; that is, the derivations 푖 푑푥 ∑(−1) 훿(푎1, … , 푎푖푎푖+1, … , 푎푛+1) 푑 are precisely the 푘[푥]-multiples of , where, for polyno- 푖=1 푑푥 푛 푑 = −푎1훿(푎2, … , 푎푛+1) + (−1) 훿(푎1, … , 푎푛)푎푛+1 mials 푝 and 푞, the function 푞 takes 푝 to 푞 times its de- 푑푥 푑푝 푛 rivative . for all 푎1, … , 푎푛+1 ∈ 퐴. Let 퐶 (퐴) be the vector space of all 푑푥 multilinear functions 훿 ∶ 퐴푛 → 퐴, and set Is the space Der(퐴) of derivations closed under compo- sition of functions? The answer is no, and a counterexam- 푛 푛 푑 푍 (퐴) = {훿 ∈ 퐶 (퐴) ∣ 훿 is a Hochschild 푛-cocycle}, ple is given by composing with itself as a function on 푑푥 푘[푥], resulting in a function that is not itself a derivation. so that 푍1(퐴) = Der(퐴) as before. These historical defini- However, Der(퐴) is closed under a commutator, or bracket: tions contain remnants of a template from algebraic topol- Let ogy. We will see that these definitions are meaningful in [훿, 훿′] = 훿 ∘ 훿′ − 훿′ ∘ 훿 (1) algebra in a number of ways. We saw that 푍1(퐴) = Der(퐴) is a Lie algebra. More gen- for all 훿, 훿′ in Der(퐴). Calculations show that [훿, 훿′] is in- erally, is 푍푛(퐴) a Lie algebra? The answer is no, at least not deed a derivation, that this commutator is bilinear in its in a meaningful way. (It is always possible to define the two arguments, and that it has the following additional bracket of two vector space elements to be 0, but we are properties: [훿, 훿] = 0 and looking for more interesting answers.) Gerstenhaber [2] [훿, [훿′, 훿″]] + [훿′, [훿″, 훿]] + [훿″, [훿, 훿′]] = 0 had an insightful idea in the 1960s.

JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 781 general, the bracket is a function [ , ] ∶ 푍푚(퐴) × 푍푛(퐴) → 푍푚+푛−1(퐴) for all 푚, 푛, called the Gerstenhaber bracket, named in honor of Gerstenhaber [2], who first defined it. It can be shown that 푍∗(퐴) is indeed a graded Lie algebra under [,]. That is, it satisfies modified versions of the alternating property and the Jacobi identity that include signs depending on the degrees of elements. Hochschild Cohomology Vector Space We will define the Hochschild cohomology vector space in degree 푛 as a quotient of the vector space 푍푛(퐴) of Hochschild 푛-cocycles as follows. Let 훿′ ∶ 퐴푛−1 → 퐴 be any (푛 − 1)-linear function. Define 훿 ∶ 퐴푛 → 퐴 by ′ 훿(푎1, … , 푎푛) = 푎1훿 (푎2, … , 푎푛) 푛−1 푖 ′ + ∑ (−1) 훿 (푎1, … , 푎푖푎푖+1, … , 푎푛) Murray Gerstenhaber, 2017. 푖=1 +(−1)푛훿′(푎 , … , 푎 )푎 His idea was this: Consider all of the vector spaces 1 푛−1 푛 푛 푍 (퐴) together, setting for 푎1, … , 푎푛 in 퐴. Call such a function 훿 a Hochschild 푛- coboundary. Recall that 퐶푛(퐴) is the vector space of all mul- 푍∗(퐴) = 푍푛(퐴), 푛 ⨁ tilinear functions from 퐴 to 퐴. Let 푛≥0 퐵푛(퐴) 0 ∗ with 푍 (퐴) = 푍(퐴) the center of the algebra 퐴. Then 푍 (퐴) = {훿 ∈ 퐶푛(퐴) ∣ 훿 is a Hochschild 푛-coboundary} is in fact a graded version of a Lie algebra for which the 0 푛 bracket of two elements in 푍1(퐴) is as before. If 훿 is in for 푛 > 0 and 퐵 (퐴) = 0. A calculation shows that 퐵 (퐴) 푛 푍푚(퐴) and 훿′ is in 푍푛(퐴), their bracket [훿, 훿′] is defined is a subspace of 푍 (퐴) for each 푛. Now let similarly to the case 푚 = 1, 푛 = 1 as in equation (1), HH푛(퐴) = 푍푛(퐴)/퐵푛(퐴) but we replace composition of functions with a general- ∗ 푛 and HH (퐴) = HH (퐴). ized version of composition. We illustrate for small values ⨁ 푛≥0 of 푚 and 푛: If 훿 is in 푍1(퐴) and 훿′ is in 푍2(퐴), define [훿, 훿′] in 푍2(퐴) by The latter is named the Hochschild cohomology vector space of 퐴 in honor of Hochschild [3], who first defined it. It can [훿, 훿′](푎, 푏) = 훿(훿′(푎, 푏)) − 훿′(훿(푎), 푏) + 훿′(푎, 훿(푏)), be shown that the Gerstenhaber bracket [,] as defined on 푍∗(퐴) passes to this quotient, making HH∗(퐴) a graded Lie and if 훿, 훿′ are both in 푍2(퐴), define [훿, 훿′] in 푍3(퐴) by algebra. (푎, 푏, 푐) = 훿(훿′(푎, 푏), 푐) − 훿(푎, 훿′(푏, 푐)) There is an important dual notion, Hochschild homol- + 훿′(훿(푎, 푏), 푐) − 훿′(푎, 훿(푏, 푐)) ogy: Vector spaces HH푛(퐴) are defined analogously as quo- tients of subspaces of some spaces constructed from 푛 for all 푎, 푏, 푐 in 퐴. These bracket operations can be visual- copies of 퐴. There are also more general versions of both ized with the aid of tree diagrams. For example, the four Hochschild cohomology and homology, defined for 퐴- ∗ terms on the right side of the last equation above corre- 푀, denoted HH (퐴, 푀) and HH∗(퐴, 푀). We spond to the four trees will not need the details of these constructions in this arti- cle. As a small Hochschild cohomology example, consider 훿′ 훿′ 훿 훿 HH푛(푘[푥]). This space turns out to be nonzero only when 푛 = 0 and 푛 = 1. When 푛 = 0, it is simply the algebra 훿 훿 훿′ 훿′ 푘[푥]. When 푛 = 1, it is essentially the space of derivations 푑 푘[푥] that we saw before since 퐵1(푘[푥]) turns out to be 0. Each tree indicates on which two arguments to evaluate 훿′ 푑푥 ′ Thus (respectively, 훿) first, then 훿 (respectively, 훿 ) is evaluated 푑 HH∗(푘[푥]) = 푘[푥] ⊕ 푘[푥] . on that output together with the remaining argument. In 푑푥

782 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6 As another small example, take a in two diagram. For 푚 = 2, 푛 = 2, the tree is indeterminates, 퐴 = 푘[푥, 푦]. Then

훿 훿′ 휕 휕 휕 휕 HH∗(퐴) = 퐴 ⊕ (퐴 ⊕ 퐴 ) ⊕ 퐴 ∧ . 휕푥 휕푦 휕푥 휕푦 휇

1 where 휇 is the multiplication map. This cup product on The two middle summands above constitute HH (퐴) and cocycles induces an associative product together correspond to all 퐴-linear combinations of the 휕 휕 푚 푛 푚+푛 standard derivations and . The first summand above ⌣∶ HH (퐴) × HH (퐴) → HH (퐴). 휕푥 휕푦 0 is HH (퐴), corresponding to the algebra 퐴 itself. The It can be shown that HH∗(퐴), under cup product, is a last summand above is HH2(퐴), corresponding to all 퐴- graded commutative algebra, specifically, 휕 휕 multiples of the ∧ given by applying ′ 푚푛 ′ 휕푥 휕푦 훿̄ ⌣ 훿̄ = (−1) 훿̄ ⌣ 훿,̄ 휕 휕 and to the first and second arguments, respectively, ′ ′ 푚 푛 휕푥 휕푦 where 훿,̄ 훿̄ are the images of 훿, 훿 in HH (퐴), HH (퐴) then multiplying. The dual notion of Hochschild homol- under the respective quotient maps. ∗ ogy, HH푛(퐴), turns out to be the same vector space as The two binary operations on HH (퐴), namely [,] and 푛 Hochschild cohomology HH (퐴) in each degree 푛 for this ⌣, can be seen to work well together in the sense that the example as well as for the previous one. functions [훿,̄ −] from HH∗(퐴) to HH∗(퐴) are themselves These polynomial ring examples can be viewed as graded derivations with respect to cup product, specifi- special cases of the Hochschild–Kostant–Rosenberg the- cally, orem [4]. The theorem states that for a smooth finitely ̄ ′̄ ″̄ ̄ ′̄ ″̄ 푚(푝−1) ′̄ ̄ ″̄ generated commutative algebra 퐴, the Hochschild coho- [훿, 훿 ⌣ 훿 ] = [훿, 훿 ] ⌣ 훿 + (−1) 훿 ⌣ [훿, 훿 ] ∗ (퐴) 푚 푛 푝 mology vector space HH is isomorphic to the space for all 훿,̄ 훿′̄ , 훿″̄ in HH (퐴), HH (퐴), HH (퐴), respectively. of polyvector fields, and there is a dual version for We thus say that HH∗(퐴) is a Gerstenhaber algebra in honor 퐴 Hochschild homology. The hypothesis that is smooth of Gerstenhaber, who first discovered this structure. and finitely generated translates in a precise way tothe Take as an example 퐴 = 푘[푥, 푦]: The cup product is the 퐴 statement that is a ring of functions on a space that is product on 퐴 combined with that on the smooth, that is, has no cusps or other singularities. The 휕 휕 on the derivations , , for example, cohomology isomorphism of the Hochschild–Kostant– 휕푥 휕푦 Rosenberg theorem is in fact an isomorphism of graded 휕 휕 휕 휕 Lie algebras, and even further of Gerstenhaber algebras. (푝 ) ⋅ (푞 ) = 푝푞 ∧ , 휕푥 휕푦 휕푥 휕푦 Gerstenhaber Algebra Structure and the Gerstenhaber bracket is the Schouten bracket on The bracket operation on the Hochschild cohomology polyvector fields, for example, space HH∗(퐴) was Gerstenhaber’s idea, as described above. 휕 휕 휕 휕 휕 휕 휕 It extends the Lie algebra structure on derivations of 퐴 and [푝 , 푞 ∧ ] = (푝 (푞) − 푞 (푝)) ∧ 휕푥 휕푥 휕푦 휕푥 휕푥 휕푥 휕푦 on HH1(퐴), as we have seen. There is another, simpler, binary operation on the Hochschild cohomology vector for all 푝, 푞 in 푘[푥, 푦]. These structures extend to any poly- space HH∗(퐴) that instead extends the nomial ring in finitely many indeterminates. structure on the center 푍(퐴) = 푍0(퐴) = HH0(퐴) of the alge- bra 퐴. We turn to this structure, given by a cup product, next. Appearances of Hochschild Cohomology Letting 훿 be a Hochschild 푚-cocycle and 훿′ a Hochschild Hochschild cohomology has found many uses since its 푛-cocycle, define inception. We mention here only a few places where Hochschild cohomology appears in order to show how widespread and diverse its uses can be. ′ (훿 ⌣ 훿 )(푎1, … , 푎푚+푛) Noncommutative . There are many notions 푚푛 ′ = (−1) 훿(푎1, … , 푎푚)훿 (푎푚+1, … , 푎푚+푛) from classical geometry that can be defined homologically, and Hochschild cohomology plays a role. In classical geometry, we glean information about geometric spaces ′ for all 푎1, … , 푎푚+푛 in 퐴. Then 훿 ⌣ 훿 is a Hochschild from commutative rings of functions on the spaces. For ex- (푚 + 푛)-cocycle. This product can be visualized with a tree ample, the polynomial ring ℝ[푥, 푦] may be considered to

JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 783 be a ring of functions from the plane ℝ2 to ℝ, with 푥, 푦 cor- responding to the coordinate functions. In noncommuta- tive geometry, we replace such commutative rings of func- tions by noncommutative rings and ask analogous ques- tions about their structure. For example, the Weyl algebra is an algebra that looks very much like a polynomial ring, but we replace commutativity of indeterminates 푥 and 푦 with the relation 푥푦 = 푦푥 − 1. Notions such as , smoothness, and differential forms can be defined in the setting of via Hochschild co- homology of noncommutative algebras. Such definitions may be viewed as generalizing those in commutative alge- bra. Algebraic deformation theory. Some algebras are re- garded as deformations of simpler ones. For example, the Weyl algebra is a deformation of a polynomial ring. A deformation has associated to it a Hochschild 2-cocycle, technically by first routing through formal power series James Stasheff, 2016. over the original algebra. For example, the cocycle for the 휕 휕 휕 휕 Weyl algebra is − ∧ : Applying − to 푥, to 푦, and 휕푥 휕푦 휕푥 휕푦 general than the classical theory of support varieties for multiplying yields −1, the scalar in the defining relation modules of finite groups. 푥푦 = 푦푥 − 1 of the Weyl algebra that distinguishes it from Towards Structural Understanding the relation 푥푦 = 푦푥 of the polynomial ring. Conversely, given a Hochschild 2-cocycle, in order for it to be associ- Due to the many appearances of Hochschild cohomology ated to a deformation, other cocycles defined via the Ger- in mathematics, it is important to understand Hochschild stenhaber bracket [,] must be coboundaries. This con- cohomology spaces well. One approach is to understand 푛 nection to Hochschild cohomology can be profitable. For first the precursor spaces 퐶 (퐴) of multilinear functions 푛 example, the classical Poincar´e–Birkhoff–Witttheorem for from 퐴 to 퐴, with the goal of understanding better their 푛 Lie algebras in this language states that the universal en- subspaces 푍 (퐴) of Hochschild 푛-cocycles and the quo- 푛 푛 푛 veloping algebra of a Lie algebra is a deformation of a poly- tients 푍 (퐴)/퐵 (퐴) = HH (퐴). 푛 nomial ring, and one proof passes through Hochschild co- Identify the vector space 퐶 (퐴) with that of all linear ⊗푛 homology. A notable special case of algebraic deforma- functions from 퐴 = 퐴⊗푘 ⋯⊗푘 퐴 (푛 tensor factors) to 퐴. tion is deformation quantization, that is, deformation of Stasheff [8] considered some additional structure on the a commutative algebra for which the Hochschild 2-cocycle graded vector space is a Poisson bracket. 푇(퐴) = 퐴⊗푛, ⨁ Representation theory. Some techniques in representa- 푛≥0 tion theory rely on the graded commutative property of the Hochschild under cup product that we namely that of a coalgebra. This structure is given by saw earlier. This property opens the door to commutative deconcatenating tensor products in all possible ways and algebraic and algebraic geometric techniques. Specifically, summing. Stasheff showed that Hochschild cohomology each 퐴- 푀 is associated to a module for the graded comes from the subspace of all coderivations on this coal- commutative Hochschild cohomology ring HH∗(퐴), and gebra. He also initiated the study of higher multiplications this module itself is a cohomology ring H∗(푀) defined by and comultiplications on 푀. Under some finiteness conditions, this yields an as- 퐶∗(퐴) = 퐶푛(퐴) ⨁ signment of an algebraic variety, called a support variety, to 푛≥0 each 퐴-module 푀, that is, the of the annihila- tor in HH∗(퐴) of H∗(푀). This assignment grants access to and on other spaces, thus founding the subject of 퐴∞- algebras and related structures. Deligne [1] conjectured geometric information encoding representation-theoretic ∗ properties. For example, in some settings, the dimension further structure on 퐶 (퐴), that it is an algebra over an op- of the support variety of 푀 is equal to a measure of com- erad of little discs, one version of which can be viewed as plexity of 푀. This theory is both parallel to and more existence of a particular infinity structure. The conjecture was proven in different ways by different groups of mathe- maticians around the turn of the millenium; see [6] for a

784 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6 ∗ discussion and references. Operads underlie other impor- to be a composition with 푑푛, that is, 푑푛(훿) = 훿 ∘ 푑푛, for all ∗ ∗ tant properties of 퐶 (퐴) and thus of Hochschild cohomol- 푛 > 1, and define 푑0 to be the zero map. It turns out that ogy. See [6, 9] and references therein for an introduction there is a vector space isomorphism for each 푛, to operads. 푛 ∗ ∗ For many questions involving Hochschild cohomology, HH (퐴) ≅ ker(푑푛+1)/im(푑푛), ∗ it can be helpful first to refine the vector space 퐶 (퐴) itself, where HH푛(퐴) is the Hochschild cohomology space of since this space is quite large and can conceal important in- 퐴 in degree 푛, as defined earlier. The latter space above, formation. We look next at such an alternative approach. ∗ ∗ 푛 ker(푑푛+1)/im(푑푛), is also denoted Ext퐴푒 (퐴, 퐴), and is inde- Resolutions pendent of the choice of resolution (2) as a consequence of existence of comparison maps between resolutions. If We wish to place the Hochschild cohomology vector space ∗ we choose the bar resolution as described above, we return HH (퐴) into a bigger picture by giving an equivalent def- to the definition of Hochschild cohomology we gave ear- inition via an arbitrary resolution next. In fact, results lier. If we take any other choice of free resolution of 퐴, we and examples mentioned earlier depend heavily on this obtain a new (equivalent) definition of the Hochschild co- approach. homology spaces HH푛(퐴). The resulting zoo of choices of The algebra 퐴 acts on itself by left and right multipli- definitions of HH푛(퐴) vastly expands our toolbox for han- cation, making it into an 퐴-. Equivalently, it is 푒 표푝 dling it. a left module over the enveloping algebra 퐴 = 퐴 ⊗푘 퐴 표푝 In comparison, the dual notion of Hochschild homol- where 퐴 is the opposite algebra (that is, 퐴 with the op- 퐴푒 ogy HH (퐴) is given by Tor (퐴, 퐴), and generally for any posite multiplication, 푎 ⋅ 푏 = 푏푎 for all 푎 and 푏 in 퐴). 푛 푛 표푝 퐴-bimodule 푀 and 푛 ≥ 0, The left 퐴푒-module structure on 퐴 is given by 푛 푛 HH (퐴, 푀) = Ext퐴푒 (퐴, 푀), (푎 ⊗푘 푏) ⋅ 푐 = 푎푐푏 퐴푒 HH푛(퐴, 푀) = Tor푛 (푀, 퐴) for all 푎 and 푐 in 퐴 and 푏 in 퐴표푝. Considering 퐴 as an 퐴푒-module in this way, we ask for a sequence of free 퐴푒- are the Hochschild cohomology and homology vector modules 퐹푛 (that is, each 퐹푛 must be a of copies spaces with coefficients in 푀. These spaces also have of 퐴푒), further structure and uses that we will not discuss here. See [5, 12, 13] for more information. 푑3 / 푑2 / 푑1 / 휀 / / ⋯ 퐹 퐹 퐹 퐴 0 (2) 2 1 0 Cup Product Revisited 푒 with 퐴 -module homomorphisms 휀 ∶ 퐹0 → 퐴 and 푑푛 ∶ Cup products can be found on an arbitrary resolution by 퐹푛 → 퐹푛−1 for all 푛 ≥ 1. We require im(푑1) = ker(휀) and using an algorithm analogous to a slide rule. In relation ∗ im(푑푛) = ker(푑푛−1) for all 푛 > 1; that is, the sequence (2) to the sequence (2), let 훿 be an element of ker(푑푚+1) ′ ∗ is exact. Put another way, we say that the sequence (2), or and let 훿 be an element of ker(푑푛+1), representing ele- 푚 푛 rather the collection of bimodules 퐹푛 and maps 푑푛, is a ments of HH (퐴) and HH (퐴), respectively. That is, 훿 푒 1 푒 free resolution of the 퐴 -module 퐴. An example is the bar is an 퐴 -module homomorphism from 퐹푚 to 퐴 for which ′ 푒 resolution of 퐴 for which 퐹푛 = 퐴 ⊗푘 ⋯ ⊗푘 퐴 (푛 + 2 factors), 훿∘푑푚+1 = 0 and 훿 is an 퐴 -module homomorphism from ′ 휀(푎 ⊗푘 푏) = 푎푏, and 퐹푛 to 퐴 for which 훿 ∘ 푑푛+1 = 0. Consider two copies of the sequence (2), one above the other: 푑푛(푎0 ⊗푘 ⋯ ⊗푘 푎푛+1) / / / / / 푛 ⋯ 퐹푛 ⋯ 퐹1 퐹0 퐴 푖 = ∑(−1) 푎0 ⊗푘 ⋯ ⊗푘 푎푖푎푖+1 ⊗푘 ⋯ ⊗푘 푎푛+1 푖=0 / / / / / for all 푎0, … , 푎푛+1 in 퐴. A resemblance to simplices and ⋯ 퐹푛 ⋯ 퐹1 퐹0 퐴 boundary maps in the theory of recalls roots in . Slide the top copy 푛 places to the right so that 퐹푛 is now In general, consider the vector space of all 퐴푒-module above 퐹0, called shifting the resolution. By virtue of being free, the 퐴푒-module 퐹 may be mapped to 퐹 via an 퐴푒- homomorphisms from 퐹 to 퐴, denoted Hom 푒 (퐹 , 퐴) for 푛 0 푛 퐴 푛 ′ ′ ′ each 푛. Define the map module homomorphism 훿0 in such a way that 훿 = 휀 ∘ 훿0, as in the diagram below. Similarly, there are 퐴푒-module ∗ 푑 ∶ 푒 (퐹 , 퐴) → 푒 (퐹 , 퐴) ′ 푛 Hom퐴 푛−1 Hom퐴 푛 homomorphisms 훿푖 ∶ 퐹푛+푖 → 퐹푖 for 1 ≤ 푖 ≤ 푚 for which the diagram below commutes; that is, composing maps 1 푒 More generally, we may take each 퐹푛 to be a projective 퐴 -module for a projec- along any two paths with the same start and end produces ′ tive resolution. the same result. We call the collection of such maps 훿푖 a

JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 785 chain map: Specifically, write 퐹∗ for the truncation of sequence (2) in which we have deleted the rightmost nonzero term 퐴. Let 푑푚+푛 / 푑푛+2 / 푑푛+1 / 퐹푚+푛 ⋯ 퐹푛+1 퐹푛 > Δ be any chain map from 퐹∗ to the 퐹∗ ⊗퐴 퐹∗ > ∼ >>훿′ ′ ′ ′ > that extends the canonical isomorphism 퐴 ⟶ 퐴 ⊗퐴 퐴. 훿푚 훿1 훿0 >    > 푑 훿′ + (−1)푛훿′ 푑 푑 푑 푑 We require the map 푖+1 [푖+1] [푖] 푖+푛 to be equal 푚 / 2 / 1 / 휀 / ′ ′ 퐹푚 ⋯ 퐹1 퐹0 퐴 to the difference between (훿 ⊗퐴 1)Δ푖+푛 and (1 ⊗퐴 훿 )Δ푖+푛 for all 푖, as well as requiring 훿′ to satisfy an initial condi- Now define [0] tion. (Here we take the value of 훿′ to be 0 on all 퐹 with 훿 ⌣ 훿′ = 훿 ∘ 훿′ , 푗 푚 푗 ≠ 푛.) The collection of such maps 훿′ is called a 푒 [푖] an 퐴 -module homomorphism from 퐹푚+푛 to 퐴: lifting of 훿′ [11,13]. Now define

퐹푚+푛 ′ ′ (푚−1)(푛−1) ′ [훿, 훿 ] = 훿 ∘ 훿 − (−1) 훿 ∘ 훿[푛], ′ [푚] 훿푚  ′ 푒 훿∘훿푚 퐹푚 an 퐴 -module homomorphism from 퐹푚+푛−1 to 퐴. Then [훿, 훿′] represents the Gerstenhaber bracket in Hochschild 훿 푚+푛−1  cohomology HH (퐴). If we take the bar resolution as ′ 퐴 described earlier, one choice of maps 훿[푖] produces exactly the formula for the Gerstenhaber bracket that we had be- Then 훿 ⌣ 훿′ indeed represents the cup product in the 푚+푛 fore. Hochschild cohomology space HH (퐴). If we take the This alternate definition of Gerstenhaber bracket, onan bar resolution as described earlier, one choice of maps 훿′ 푖 arbitrary resolution 퐹 , is due to Volkov [11] in this gen- leads to precisely the formula for cup product that we had ∗ eral form. The notion of homotopy lifting is directly re- before. lated to Stasheff’s coderivations [8] in case 퐹 is the bar This alternate definition of cup product, on an arbitrary ∗ resolution. A method for finding homotopy liftings un- resolution, increases the versatility and therefore the use- der some additional conditions is in an earlier paper with fulness of this operation. Often we choose a resolution Negron [7]. In case 퐹 is the bar resolution, homotopy lift- tailored specifically to a given type of algebra, aiding both ∗ ings of 1-cocycles are related to earlier, more general results computation and theoretical understanding. The above of Su´arez–Alvarez´ [10]. All of these new techniques have method of defining cup products is constructive andis been used in further computational and theoretical work used in practice to compute them. There are yet other on Gerstenhaber brackets for some classes of algebras, and equivalent definitions of cup product, also useful, that we they promise to lead to yet further advances. do not discuss here. References Gerstenhaber Bracket Revisited [1] A. A. Voronov and M. Gerstenhaber, Higher operations and Gerstenhaber brackets may be found on an arbitrary res- the Hochschild complex, Funct. Anal. Appl. 29 (1995), no. 1, olution by a similar technique to that described above 1–5. MR1328534 for cup products. Again, in relation to the sequence (2), [2] M. Gerstenhaber, The cohomology structure of an associative ∗ ′ let 훿 be an element of ker(푑푚+1) and let 훿 be an ele- ring, Ann. Math. 78 (1963), no. 2, 267–288. MR161898 ∗ 푚 ment of ker(푑푛+1), representing elements of HH (퐴) and [3] G. Hochschild, On the cohomology groups of an associative HH푛(퐴), respectively. Again we line up two copies of the algebra, Ann. Math. 46 (1945), no. 2, 58–67. MR11076 sequence (2), but this time we slide the top copy 푛 − 1 [4] G. Hochschild, B. Kostant, and A. Rosenberg, Differential places to the right, so that 퐹 is above 퐹 . We are inter- forms on regular affine algebras, Trans. Amer. Math. Soc. 102 푛−1 0 (1962), 383–408. MR142598 퐴푒 훿′ ∶ 퐹 → 퐹 ested in -module homomorphisms [푖] 푖+푛−1 푖 as [5] J.-L. Loday, Cyclic Homology, Grundlehren der Mathema- in the diagram below: tischen Wissenschaften, vol. 301, Springer-Verlag, 1998. 퐹 / ⋯ / 퐹 / 퐹 MR1600246 푚+푛−1 푛 푛−1 [6] J. E. McClure and J. H. Smith, Operads and cosimpli- ′ ′ ′ 훿[푚] 훿[1] 훿[0] cial objects: an introduction, Axiomatic, enriched and mo-    / / / tivic , 133–171, NATO Sci. Ser. II Math. 퐹푚 ⋯ 퐹1 퐹0 Phys. Chem. 131, Kluwer Acad. Publ., Dordrecht, 2004. MR2061854 However instead of requiring that the diagram commute, ′ [7] C. Negron and S. Witherspoon, An alternate approach we require the homomorphisms 훿[푖] to comprise a homo- to the Lie bracket on Hochschild cohomology, Homology, topy between two maps defined in terms of 훿′ and a chain Homotopy and Applications 18 (2016), no. 1, 265–285. map from the sequence (2) to its tensor square over 퐴. MR3498646

786 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6 [8] J. Stasheff, The intrinsic bracket on the deformation complex FEATURED TITLE FROM of an associative algebra, J. Pure Appl. Algebra 89 (1993), 231–235. MR1239562 [9] J. Stasheff, What is an operad? Notices Amer. Math. Soc. 51 (2004), no. 6, 630–631. MR2064150 [10] M. Su´arez-Alvarez,´ A little bit of extra functoriality for Ext Theory of Semigroups and the computation of the Gerstenhaber bracket, J. Pure Appl. Algebra 221 (2017), no. 8, 1981–1998. MR3623179 and Applications Kalyan B. Sinha, Jawaharlal Nehru [11] Y. Volkov, Gerstenhaber bracket on the Hochschild cohomol- Centre for Advanced Scientific ogy via an arbitrary resolution, Proc. Edinburgh Math. Soc., Research, Bangladore, India, and Sachi 2019, 20 pp. MR3974969 Srivastava, University of Delhi South [12] C. A. Weibel, An Introduction to , Cam- Campus, New Delhi, India bridge University Press, 1994. MR1269324 This book combines the spirit of a text- book with that of a monograph on the [13] S. Witherspoon, Hochschild Cohomology for Algebras, topic of semigroups and their applications. Graduate Studies in Mathematics, 204, American Mathe- It is expected to have potential readers across a broad spectrum that matical Society, Providence, RI, 2019. MR3971234 includes operator theory, partial differential equations, harmonic analysis, probability and statistics, and classical and quantum mechanics. A reasonable amount of familiarity with real analysis, including the Lebesgue-integration theory, basic functional analysis, and bounded linear operators is assumed. However, any discourse on a theory of semigroups needs an introduction to unbounded linear operators, some elements of which have been included in the Appendix, along with the basic ideas of the Fourier transform and of Sobolev spaces. Chapters 4–6 contain advanced material not often found in textbooks but which have many interesting applications, such as the Feynman–Kac formula, the central limit theorem, and the construction of Markov semigroups. The exercis- es are given in the text as the topics are developed so that interested readers can solve these while learning that topic. Sarah Witherspoon Hindustan Book Agency; 2017; 180 pages; Hardcover; ISBN: 978-93-86279-63-7; List US$38; AMS members US$30.40; Order Credits code HIN/73 Opening image is courtesy of the AMS. Photo of Gerhard Hochschild is courtesy of George Bergman Titles published by the Hindustan Book Agency and the Mathematisches Forschungsinstitut Oberwolfach. (New Delhi, India) include studies in advanced mathematics, monographs, lecture notes, and/or Photo of Murray Gerstenhaber is courtesy of Ruth Gersten- conference proceedings on current topics of interest. haber. Photo of James Stasheff is courtesy of Felice Macera. Author photo is courtesy of Frank Sottile. Discover more books at bookstore.ams.org/hin.

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