New approaches to finite generation of cohomology rings

TL;DR

This paper explores new methods to establish the finite generation of cohomology rings in representation theory, using algebraic, spectral sequence, and deformation techniques across different algebraic structures.

Contribution

It introduces approaches to verify finite generation conditions for cohomology rings, including module algebra conditions, spectral sequence collapse, and deformation from finite fields.

Findings

Finite generation condition can be checked via invariant subrings in module algebras.

Spectral sequences collapse for certain algebra classes imply finite generation.

Deformation from finite fields preserves finite generation conditions.

Abstract

In support variety theory, representations of a finite dimensional (Hopf) algebra $A$ can be studied geometrically by associating any representation of $A$ to an algebraic variety using the cohomology ring of $A$. An essential assumption in this theory is the finite generation condition for the cohomology ring of $A$ and that for the corresponding modules. In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra $A$, we show that the finite generation condition on $A$-modules can be replaced by a condition on any affine commutative $A$-module algebra $R$ under the assumption that $R$ is integral over its invariant subring $R^A$. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if $A$ is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra $A'$ over a finite field, where $A$ can be viewed as a deformation of $A'$, and prove that if the finite generation condition holds for $A'$, then the same condition holds for $A$.