Graded analogues of one-parameter subgroups and applications to the cohomology of GLm|n(r)

Abstract

We introduce a family Mr;f,η of infinitesimal supergroup schemes, which we call multiparameter supergroups, that generalize the infinitesimal Frobenius kernels Ga(r) of the additive group scheme Ga. Then, following the approach of Suslin, Friedlander, and Bendel, we use functor cohomology to define characteristic extension classes for the general linear supergroup GLm|n, and we calculate how these classes restrict along homomorphisms ρ:Mr;f,η→GLm|n. Finally, we apply our calculations to describe (up to a finite surjective morphism) the spectrum of the cohomology ring of the r-th Frobenius kernel GLm|n(r) of the general linear supergroup GLm|n.