Separable equivalences, finitely generated cohomology and finite tensor categories

TL;DR

This paper proves that finitely generated cohomology remains invariant under separable equivalences for all algebras, confirming a conjecture for finite symmetric tensor categories and deriving related homological invariants.

Contribution

It establishes the invariance of finitely generated cohomology under separable equivalences and proves the finite generation conjecture for finite symmetric tensor categories in characteristic zero.

Findings

Cohomology invariance under separable equivalences

Finite generation of cohomology for symmetric tensor categories

Determination of representation and Rouquier dimensions

Abstract

We show that finitely generated cohomology is invariant under separable equivalences for all algebras. As a result, we obtain a proof of the finite generation of cohomology for finite symmetric tensor categories in characteristic zero, as conjectured by Etingof and Ostrik. Moreover, for such categories we also determine the representation dimension and the Rouquier dimension of the stable category. Finally, we recover a number of results on the cohomology of stably equivalent and singularly equivalent algebras.