Is the Hochschild chain complex $C_*(A, A)$ a $B_\infty$-module over the Hochschild cochain complex $C^*(A, A)$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T06:31:39Z http://mathoverflow.net/feeds/question/85258 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85258/is-the-hochschild-chain-complex-c-a-a-a-b-infty-module-over-the-hochschi Is the Hochschild chain complex $C_*(A, A)$ a $B_\infty$-module over the Hochschild cochain complex $C^*(A, A)$? motivique 2012-01-09T14:22:15Z 2012-01-11T07:49:52Z <p>Let $A$ be a dg-algebra, or more generally an $A_\infty$-algebra. Then it is well known that the Hochschild cochain complex $C^*(A, A)$ computing Hochschild cohomology is a $B_\infty$-algebra, see for example, the paper of Bernhard Keller "Derived invariance of higher structures on the Hochschild complex" available on his pageweb. </p> <p>I would like to know whether the Hochschild chain complex $C_*(A, A)$ (which computes Hochschild homology) is a $B_\infty$-module over the Hochschild cochain complex $C^*(A, A)$? </p> <p>Can someone give me the precise defintion, or a precise reference, of the action of $C^*(A,A)$ </p> <p>on $C_*(A, A)$ if the answer is Yes? </p> http://mathoverflow.net/questions/85258/is-the-hochschild-chain-complex-c-a-a-a-b-infty-module-over-the-hochschi/85261#85261 Answer by DamienC for Is the Hochschild chain complex $C_*(A, A)$ a $B_\infty$-module over the Hochschild cochain complex $C^*(A, A)$? DamienC 2012-01-09T15:13:57Z 2012-01-11T07:49:52Z <p>This is the subject of Section 2 in <a href="http://arxiv.org/abs/0805.3444" rel="nofollow">that paper</a> (sorry for self-promotion). Chains actually have two $B_\infty$-module structures (over cochains). Those two module structures are moreover compatible (see Theorem 2.4 of the above paper for a precise statement). </p>