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

Abstract

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

Fixed some algebra misidentifications, primarily in Sections 1.3 and 3.3. Simplified the proof of Proposition 3.3.6

References in corpus (3)

• Cohomological finite-generation for finite supergroup schemes
• On support varieties for Lie superalgebras and finite supergroup schemes
• Graded analogues of one-parameter subgroups and applications to the cohomology of $G{L}_{m|n(r)}$

Cited by in corpus (3)

• On support varieties for Lie superalgebras and finite supergroup schemes
• Graded analogues of one-parameter subgroups and applications to the cohomology of $G{L}_{m|n(r)}$
• Support schemes for infinitesimal unipotent supergroups