Using torsion theory to compute the algebraic structure of Hochschild (co)homology

Estanislao Herscovich

Homology, homotopy, and applications, Tome 20 (2018) no. 1, pp. 117-139.

Résumé

The aim of this article is to provide explicit formulas for the cup product on the Hochschild cohomology of any nonnegatively graded connected algebra $A$ and for the cap products on the Hochschild homology of $A$ with coefficients in any graded bimodule $M$ at the level of the complexes $\operatorname{Hom}_{A^{\mathrm{e}}}(P_{\bullet},A)$ and $M \otimes_{A^{\mathrm{e}}} P_{\bullet}$, resp., where $P_{\bullet}$ is a minimal projective resolution of the $A$-bimodule $A$, based on the $A_{\infty}$-algebra structure of $\mathcal{E}xt^{\bullet}_{A}(k,k)$. We remark that we do not (need to) construct any comparison map between $P_{\bullet}$ and the Hochschild resolution of $A$, or any lift $\Delta \colon P \rightarrow P \otimes_{A} P$ of the identity of $A$. The main tools we use come from torsion theory of $A_{\infty}$-algebras and of their $A_{\infty}$-bimodules.

Détail

DOI : 10.4310/HHA.2018.v20.n1.a8

Classification : 16E40, 16E45, 16S37, 16W50, 18G55
Keywords: Koszul algebra, Yoneda algebra, homological algebra, dg algebra, $A_{\infty}$-algebra