Cohomological finite-generation for finite supergroup schemes

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.