# Detecting nilpotence and projectivity over finite unipotent supergroup schemes

## TL;DR

This paper establishes criteria for detecting nilpotence and projectivity in the cohomology and modules of finite unipotent supergroup schemes, extending classical results to the supergroup context with applications to Steenrod algebra subalgebras.

## Contribution

It proves that nilpotence and projectivity can be detected via restrictions to elementary sub-supergroup schemes, generalizing known group scheme results to supergroups.

## Key findings

- Nilpotence in cohomology is detected by restrictions to elementary supergroups.
- Projectivity of modules is characterized by restrictions to elementary supergroups.
- Application to detection theorems for Steenrod algebra subalgebras.

## Abstract

This work concerns the representation theory and cohomology of a finite unipotent supergroup scheme $G$ over a perfect field $k$ of positive characteristic $p\ge 3$. It is proved that an element $x$ in the cohomology of $G$ is nilpotent if and only if for every extension field $K$ of $k$ and every elementary sub-supergroup scheme $E\subseteq G_K$, the restriction of $x_K$ to $E$ is nilpotent. It is also shown that a $kG$-module $M$ is projective if and only if for every extension field $K$ of $k$ and every elementary sub-supergroup scheme $E\subseteq G_K$, the restriction of $M_K$ to $E$ is projective. The statements are motivated by, and are analogues of, similar results for finite groups and finite group schemes, but the structure of elementary supergroups schemes necessary for detection is more complicated than in either of these cases. One application is a detection theorem for the nilpotence of cohomology, and projectivity of modules, over finite dimensional Hopf subalgebras of the Steenrod algebra.