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

We show that if $G$ is an infinitesimal elementary supergroup scheme of height $\le r$, then the cohomological spectrum $\mathrm{\mid }G\mathrm{\mid }$ of $G$ is naturally homeomorphic to the variety ${\mathcal{N}}_r(G)$ of supergroup homomorphisms $\rho :{\mathbb{M}}_r\to G$ 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 ${\mathcal{N}}_r(G)\cong \mathrm{\mid }G\mathrm{\mid }$ to arbitrary infinitesimal unipotent supergroup schemes.