On the cohomological spectrum and support varieties for infinitesimal unipotent supergroup schemes

TL;DR

This paper establishes a homeomorphism between the cohomological spectrum of certain infinitesimal supergroup schemes and a variety of supergroup homomorphisms, providing new insights into their support varieties and extending these results to more general cases.

Contribution

It introduces a natural homeomorphism between the cohomological spectrum and a variety of supergroup homomorphisms for infinitesimal elementary supergroup schemes, and extends this to broader classes.

Findings

Homeomorphism between |G| and al N_r(G) for elementary supergroup schemes.

Identification of support varieties for finite-dimensional modules when r=1.

Extension of the homeomorphism to arbitrary infinitesimal unipotent supergroup schemes.

Abstract

We show that if $G$ is an infinitesimal elementary supergroup scheme of height $\leq r$, then the cohomological spectrum $|G|$ of $G$ is naturally homeomorphic to the variety $\mathcal{N}_r(G)$ of supergroup homomorphisms $\rho: \mathbb{M}_r \rightarrow G$ from a certain (non-algebraic) affine supergroup scheme $\mathbb{M}_r$ into $G$. In the case $r=1$, we further identify the cohomological support variety of a finite-dimensional $G$-supermodule $M$ as a subset of $\mathcal{N}_1(G)$. We then discuss how our methods, when combined with recently-announced results by Benson, Iyengar, Krause, and Pevtsova, can be applied to extend the homeomorphism $\mathcal{N}_r(G) \cong |G|$ to arbitrary infinitesimal unipotent supergroup schemes.