On the cohomology of finite tensor categories

TL;DR

This paper investigates the conjecture that finite tensor categories have finitely generated cohomology, establishing an equivalence with the finitely generated Hochschild cohomology of certain endomorphism algebras.

Contribution

It proves that the finite generation of cohomology in finite tensor categories is equivalent to the finite generation of Hochschild cohomology for their projective generator endomorphism algebras.

Findings

Finite tensor categories have finitely generated cohomology if and only if their projective generator endomorphism algebras have finitely generated Hochschild cohomology.

The paper establishes an equivalence between two conjectures in the theory of tensor categories.

Abstract

It has been conjectured that finite tensor categories have finitely generated cohomology. We show that this is equivalent to finitely generated Hochschild cohomology for the endomorphism algebras of the projective generators.