On support varieties for Lie superalgebras and finite supergroup schemes

Abstract

We study the spectrum of the cohomology rings of cocommutative Hopf superalgebras, restricted and non-restricted Lie superalgebras, and finite supergroup schemes. We also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory. We completely characterize support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero, while for non-restricted Lie superalgebras we obtain results in positive characteristic that are strikingly similar to results of Duflo and Serganova in characteristic zero. Our computations for restricted Lie superalgebras and infinitesimal supergroup schemes provide natural generalizations of foundational results of Friedlander and Parshall and of Bendel, Friedlander, and Suslin in the classical setting.

32 pages, submitted

References in corpus (7)

• A new proof of the Mullineux conjecture
• Support varieties for selfinjective algebras
• On associated variety for Lie superalgebras
• Cohomological finite-generation for finite supergroup schemes
• Graded analogues of one-parameter subgroups and applications to the cohomology of $G{L}_{m|n(r)}$
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Cited by in corpus (5)

• Graded analogues of one-parameter subgroups and applications to the cohomology of $G{L}_{m|n(r)}$
• Support schemes for infinitesimal unipotent supergroups
• Support varieties and modules of finite projective dimension for modular Lie superalgebras (with an appendix on homological dimensions over Noether Algebras by Luchezar L. Avramov and Srikanth B. Iyengar)
• On the cohomological spectrum and support varieties for infinitesimal unipotent supergroup schemes
• Duflo-Serganova homology for exceptional modular Lie superalgebras with Cartan matrix