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

TL;DR

This paper establishes a new method to compute Hochschild cohomology of Koszul algebras using twisting cochains, revealing it as a subquotient of the tensor product of the algebra with its Koszul dual.

Contribution

It introduces a novel approach linking twisted hom complexes to Hochschild cohomology via acyclic twisting cochains, with applications to Koszul rings.

Findings

Hochschild cohomology of A is isomorphic to the cohomology of a twisted hom complex.

Hochschild cohomology of Koszul algebra A is a subquotient of A^! ⊗ A.

The method provides a new perspective on the algebraic structure of Hochschild cohomology.

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 $\mathrm{Hom}^\pi_k(C,A)$ 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.