On the monoidal invariance of the cohomological dimension of Hopf algebras

TL;DR

This paper investigates whether the global dimension of Hopf algebras remains invariant under monoidal equivalences of their comodule categories, providing new positive results under various conditions and exploring related cohomological dimensions.

Contribution

It introduces new conditions under which the global dimension is a monoidal invariant and compares it with the Gerstenhaber-Schack cohomological dimension in cosemisimple cases.

Findings

Global dimension is invariant under monoidal equivalences for certain classes of Hopf algebras.

Equality between global dimension and Gerstenhaber-Schack cohomological dimension in cosemisimple cases when the latter is finite.

Introduction of the concept of twisted separable functor as a key tool.

Abstract

We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber-Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new concept of twisted separable functor.