# Graded analogues of one-parameter subgroups and applications to the cohomology of $GL_{m|n(r)}$

## TL;DR

This paper introduces multiparameter supergroups to study the cohomology of the general linear supergroup's Frobenius kernels, providing new tools and calculations for understanding their cohomological structure.

## Contribution

It defines a new family of infinitesimal supergroup schemes and applies functor cohomology to analyze characteristic classes and the cohomology ring of Frobenius kernels of supergroups.

## Key findings

- Defined multiparameter supergroups generalizing Frobenius kernels
- Calculated restriction of characteristic classes along supergroup homomorphisms
- Described the spectrum of the cohomology ring of $GL_{m|n(r)}$

## Abstract

We introduce a family $\mathbb{M}_{r;f,\eta}$ of infinitesimal supergroup schemes, which we call multiparameter supergroups, that generalize the infinitesimal Frobenius kernels $\mathbb{G}_{a(r)}$ of the additive group scheme $\mathbb{G}_{a}$. Then, following the approach of Suslin, Friedlander, and Bendel, we use functor cohomology to define characteristic extension classes for the general linear supergroup $GL_{m|n}$, and we calculate how these classes restrict along homomorphisms $\rho: \mathbb{M}_{r;f,\eta} \rightarrow GL_{m|n}.$ 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 $GL_{m|n(r)}$ of the general linear supergroup $GL_{m|n}$.