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arXiv · 1712.05434

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

Christopher M. Drupieski·Jonathan R. Kujawa
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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.

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Christopher M. Drupieski, Jonathan R. Kujawa. 2017-12-14. On the cohomological spectrum and support varieties for infinitesimal unipotent supergroup schemes. https://doi.org/10.1007/978-3-030-11521-0

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