The decomposition of Hochschild cohomology and Gerstenhaber operations
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 \([H^{(j)}, H^{(k)}] \subseteq H^{(\leq j+k-1)}\).
Hodge-type decompositions of Hochschild (co)homology
scientific article; zbMATH DE number 856971
scientific article