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.
Categories: math.RT math.GR