Cohomological finite-generation for finite supergroup schemes

Abstract

In this paper we compute extension groups in the category of strict polynomial superfunctors and thereby exhibit certain "universal extension classes" for the general linear supergroup. Some of these classes restrict to the universal extension classes for the general linear group exhibited by Friedlander and Suslin, while others arise from purely super phenomena. We then use these extension classes to show that the cohomology ring of a finite supergroup scheme---equivalently of a finite-dimensional cocommutative Hopf superalgebra---over a field is a finitely-generated algebra. Implications for the rational cohomology of the general linear supergroup are also discussed.

Corrected minor typos and added additional details to some proofs

References in corpus (2)

• A new proof of the Mullineux conjecture
• Crystal structures arising from representations of $GL(m|n)$

Cited by in corpus (11)

• On support varieties for Lie superalgebras and finite supergroup schemes
• Cohomology for Drinfeld doubles of some infinitesimal group schemes
• Graded analogues of one-parameter subgroups and applications to the cohomology of $G{L}_{m|n(r)}$
• On the structure of graded commutative exponential funtors
• Support schemes for infinitesimal unipotent supergroups
• On the cohomological spectrum and support varieties for infinitesimal unipotent supergroup schemes
• Connectedness of cup products for polynomial representations of $G{L}_n$ and applications
• Cohomology of twisted polynomial superfunctors through the twisting spectral sequence
• The Steinberg Tensor Product Theorem for General Linear Group Schemes in the Verlinde Category
• Representation Theory of General Linear Supergroups in Characteristic 2
• Superized Troesch complexes and cohomology for strict polynomial superfunctors