Monoidal invariance of the cohomological dimension of Hopf algebras: the finite case

TL;DR

This paper proves that the cohomological dimension of Hopf algebras remains invariant under monoidal equivalences of their comodule categories, extending previous results to the finite case without requiring bijective antipodes.

Contribution

It provides a direct proof of the invariance of global dimension for finite Hopf algebras under monoidal equivalences, removing the need for bijective antipodes.

Findings

Global dimensions of finite Hopf algebras are invariant under monoidal equivalences.

The proof does not rely on Gorenstein projective dimensions.

The assumption of bijective antipodes is unnecessary for the invariance.

Abstract

A consequence of the recent work of Ren and Zhu on Gorenstein projective dimensions of modules over Hopf algebras is that if $A$ and $B$ are Hopf algebras with bijective antipodes having equivalent linear tensor categories of comodules and both having finite global dimensions, then their global dimensions coincide. In this note we provide a direct proof of this result, without using Gorenstein projective dimensions, and we notice that the assumption on the bijectivity of the antipodes can be removed.