Finite generation of some cohomology rings via twisted tensor product and Anick resolutions

TL;DR

This paper proves that certain cohomology rings of pointed Hopf algebras of dimension p^3 over a field of characteristic p>2 are finitely generated, using advanced algebraic resolutions and spectral sequences.

Contribution

It introduces new methods combining twisted tensor product resolutions and Anick resolutions to establish finite generation of these cohomology rings.

Findings

Cohomology rings of specific pointed Hopf algebras are finitely generated.

Application of May spectral sequences to these algebraic structures.

Extension of classification results for finite dimensional Hopf algebras in positive characteristic.

Abstract

Over a field of prime characteristic $p>2$, we prove that the cohomology rings of some pointed Hopf algebras of dimension $p^3$ are finitely generated. These are Hopf algebras arising in the ongoing classification of finite dimensional Hopf algebras in positive characteristic, and include bosonizations of Nichols algebras of Jordan type in a general setting as well as their liftings when $p=3$. Our techniques are applications of twisted tensor product resolutions and Anick resolutions in combination with May spectral sequences.