New approaches to finite generation of cohomology rings

Van C. Nguyen; Xingting Wang; Sarah Witherspoon

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arXiv · 1911.04552

New approaches to finite generation of cohomology rings

Van C. Nguyen·Xingting Wang·Sarah Witherspoon

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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 $A$.

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Van C. Nguyen, Xingting Wang, Sarah Witherspoon. 2019-11-11. New approaches to finite generation of cohomology rings. https://arxiv.org/abs/1911.04552

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