# Representations and cohomology of a family of finite supergroup schemes

## TL;DR

This paper investigates the cohomology and representation theory of a specific family of finite supergroup schemes, establishing key relations, detection properties, and explicit cohomology ring computations for particular cases.

## Contribution

It introduces new cohomological relations for these supergroup schemes and fully determines their cohomology rings in minimal cases, advancing understanding of their structure.

## Key findings

- Cohomology mod nilpotents is detected on proper sub-supergroup schemes.
- Explicit cohomology rings computed for smallest cases.
- Established a relation in the cohomology ring of the family.

## Abstract

We examine the cohomology and representation theory of a family of finite supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes (\mathbb G_{a(r)}\times (\mathbb Z/p)^s)$. In particular, we show that a certain relation holds in the cohomology ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-super\-group schemes. This special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes, in a separate work of the authors joint with Iyengar and Krause. We also completely determine the cohomology ring in the smallest cases, namely $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb G_{a(1)}$ and $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb Z/p$. The computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes.