A little bit of extra functoriality for Ext and the computation of the Gerstenhaber bracket

TL;DR

This paper presents a new method to compute the Gerstenhaber bracket action of HH^1(A) on HH^*(A) using arbitrary projective resolutions, simplifying calculations without needing comparison maps.

Contribution

It introduces a novel approach to compute the Gerstenhaber bracket action directly from any projective resolution, bypassing the traditional bar resolution comparison.

Findings

Simplifies the computation of the Gerstenhaber bracket action

Eliminates the need for comparison maps between resolutions

Provides a more flexible method for Hochschild cohomology calculations

Abstract

We show that the action of the Lie algebra HH^1(A) of outer derivations of an associative algebra A on the Hochschild cohomology HH^*(A) of A given by the Gerstenhaber bracket can be computed in terms of an arbitrary projective resolution of A as an A-bimodule, without having recourse to comparison maps between the resolution and the bar resolution.