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 .