  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.
