New approaches to finite generation of cohomology rings

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 RA. 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.