Rank varieties and $\pi$-points for elementary supergroup schemes

TL;DR

This paper introduces a support theory for elementary supergroup schemes over fields of characteristic p ≥ 3, generalizing existing concepts of π-points and classifying localising subcategories in their stable module categories.

Contribution

It defines a new notion of π-points for elementary supergroup schemes using graded algebra maps, extending prior theories for finite group schemes and elementary abelian groups.

Findings

Classified parity change invariant localising subcategories.

Developed a generalized support theory for supergroup schemes.

Connected support theory with classification of localising subcategories.

Abstract

We develop a support theory for elementary supergroup schemes, over a field of positive characteristic $p\ge 3$, starting with a definition of a $\pi$-point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and $\pi$-points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra $k[t,\tau]/(t^p-\tau^2)$, where $t$ has even degree and $\tau$ has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme.