The decomposition of Hochschild cohomology and Gerstenhaber operations
type: Item
instance of: scholarly article
title: The decomposition of Hochschild cohomology and Gerstenhaber operations
zbMATH Open document ID: 0851.16004
DOI: 10.1016/0022-4049(94)00136-7
author: Nantel Bergeron
author: H. Lewis Wolfgang
published in: Journal of Pure and Applied Algebra
publication date: 1996-11-24T00:00:00Z
review text: If \(A\) is a commutative unital algebra over a field of characteristic zero and \(M\) is a symmetric \(A\)-bimodule, Gerstenhaber and Schack have shown that there are Hodge-type decompositions of Hochschild (co)homology: \[ \begin{aligned} H(A,M)&=H_{1,n-1} (A,M) \oplus H_{2,n-2}(A,M) \oplus \dots \oplus H_{n,0} (A,M);\\ H^n(A,M)&=H^{1,n-1}(A,M) \oplus H^{2,n-2} (A,M) \oplus \dots \oplus H^{n,0}(A,M).\end{aligned} \] In the above decompositions, the first summands \(H_{1,n-1} (A,M)\) and \(H^{1,n-1} (A,M)\) are known to be Harrison (co)homology while the last summands \(H_{n,0}(A,M)\) and \(H^{n,0}(A,M)\) can be described in terms of differential forms and skew multiderivations respectively. The authors study the intermediate components of the above decompositions in terms of free Lie algebras. They describe all the components by generalizing to \(k\)-shuffles. The Hochschild cohomology \(H^*(A,A)\) is endowed with graded commutative cup product and graded Lie bracket. The authors answer in the negative the question of Gerstenhaber and Schack that whether these products are graded with respect to the decomposition i.e., do we have \(H^{(j)} \cup H^{(k)} \subseteq H^{j+k}\) and \([H^{(j)}, H^{(k)}] \subseteq H^{(j+k-1)}\), where \(H^{(k)}=\bigoplus_n H^{k,n-k} (A,A)\)? A counter example is given in Section 8. However, they prove that the products are filtered with respect \(H^{(\leq k)}=\bigoplus_{j \leq k} H^{(j)}\), that is, \(H^{(j)} \cup H^{(k)} \subseteq H^{(\leq j+k)}\) and \([H^{(j)}, H^{(k)}] \subseteq H^{(\leq j+k-1)}\).
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 856971
zbMATH Keywords: Lie products
zbMATH Keywords: symmetric bimodules
zbMATH Keywords: Hodge-type decompositions of Hochschild (co)homology
zbMATH Keywords: Harrison (co)homology
zbMATH Keywords: free Lie algebras
zbMATH Keywords: \(k\)-shuffles
zbMATH Keywords: graded commutative cup products
zbMATH Keywords: graded Lie brackets
reviewed by: Veereshwar A. Hiremath
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)00136-7
OpenAlex ID: W2080785703
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description: scientific article; zbMATH DE number 856971