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Hochschild Cohomology

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Hochschild Cohomology Hochschild Cohomology Sarah Witherspoon Introduction At the end of the nineteenth century, Poincar´e created invariants to distinguish different topological spaces and their features, foreshadowing the homology and cohomol- ogy groups that would appear later. Towards the middle of the twentieth century, these notions were imported from topology to algebra: The subject of group homology and cohomology was founded by Eilenberg and Mac Lane, and the subject of Hochschild homology 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, category theory, functional analysis, topology, and Gerhard Hochschild, 2003. beyond. There are strong connections to cyclic homology 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 ring 퐴 that is also a vector space over a theoretical and computational techniques for better under- field 푘 such as ℝ or ℂ. We require the multiplication map 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 center 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 cup product, 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 function 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).
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