The cup product on Hochschild cohomology via twisting cochains and applications to Koszul rings

Abstract

Given an acyclic twisting cochain $\pi :C\to A$, from a curved dg coalgebra $C$ to a dg algebra $A$, we show that the associated twisted hom complex ${\mathrm{Hom}}_k^{\pi }(C,A)$ has cohomology equal to the Hochschild cohomology of $A$, as a graded ring. As a corollary we find that the Hochschild cohomology of a Koszul algebra $A$, along with its cup product, is a subquotient of the tensor product algebra $A^{!}\otimes A$ of $A$ with its Koszul dual.

arXiv comment24 pages, presentation streamlined and all results extended to the dg setting. Changed title from The Cup Product on Hochschild Cohomology for Localizations of Filtered Koszul Algebras