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

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.