Hochschild cohomology and group actions

TL;DR

This paper derives a formula for the Hochschild cohomology of categories with finite group actions, revealing a canonical splitting and providing tools for computing cohomology in fractional Calabi-Yau contexts.

Contribution

It establishes a Hochschild cohomology formula for invariant categories under finite group actions, assuming coprimality, and shows Serre functors act trivially on Hochschild cohomology.

Findings

Hochschild cohomology splits canonically with invariant subspace

Serre functors act trivially on Hochschild cohomology

Provides a method for computing Hochschild cohomology of fractional Calabi-Yau categories

Abstract

Given a finite group action on a (suitably enhanced) triangulated category linear over a field, we establish a formula for the Hochschild cohomology of the category of invariants, assuming the order of the group is coprime to the characteristic of the base field. The formula shows that the cohomology splits canonically with one summand given by the invariant subspace of the Hochschild cohomology of the original category. We also prove that Serre functors act trivially on Hochschild cohomology, and combine this with our formula to give a useful mechanism for computing the Hochschild cohomology of fractional Calabi-Yau categories.