On the finite generation of the cohomology of abelian extensions of Hopf algebras

TL;DR

This paper proves that a class of finite-dimensional Hopf algebras called quasi-split satisfy the finite generation of their cohomology, confirming a conjecture for these algebraic structures and specific pointed Hopf algebras.

Contribution

It demonstrates that all quasi-split finite-dimensional Hopf algebras satisfy the finite generation cohomology conjecture, extending the conjecture's validity to new classes of Hopf algebras.

Findings

Quasi-split Hopf algebras satisfy the finite generation cohomology conjecture.

The result applies to a family of pointed Hopf algebras in odd characteristic.

Confirms the conjecture for specific algebraic structures introduced by Angiono, Heckenberger, and the first author.

Abstract

A finite-dimensional Hopf algebra is called quasi-split if it is Morita equivalent to a split abelian extension of Hopf algebras. Combining results of Schauenburg and Negron, it is shown that every quasi-split finite-dimensional Hopf algebra satisfies the finite generation cohomology conjecture of Etingof and Ostrik. This is applied to a family of pointed Hopf algebras in odd characteristic introduced by Angiono, Heckenberger and the first author, proving that they satisfy the aforementioned conjecture.