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# Homology, Homotopy and Applications

## Volume 20 (2018)

### Number 1

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

Pages: 117 – 139

DOI: https://dx.doi.org/10.4310/HHA.2018.v20.n1.a8

#### Author

#### Abstract

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

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