5
$\begingroup$

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.

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 $C^*(A, A)$?

Can someone give me the precise defintion, or a precise reference, of the action of $C^*(A,A)$

on $C_*(A, A)$ if the answer is Yes?

$\endgroup$
6
$\begingroup$

This is the subject of Section 2 in that paper (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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.