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

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.

Keywords

Koszul algebra, Yoneda algebra, homological algebra, dg algebra, $A_{\infty}$-algebra

2010 Mathematics Subject Classification

16E40, 16E45, 16S37, 16W50, 18G55

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This work was also partially supported by UBACYT 20020130200169BA, UBACYT 20020130100533BA, PIP-CONICET 2012-2014 11220110100870, MathAmSud-GR2HOPF, PICT 2011-1510 and PICT 2012-1186.

Received 7 April 2016

Received revised 26 June 2017

Published 17 January 2018