Graded analogues of one-parameter subgroups and applications to the cohomology of $G{L}_{m|n(r)}$

Abstract

We introduce a family ${\mathbb{M}}_{r;f,\eta }$ of infinitesimal supergroup schemes, which we call multiparameter supergroups, that generalize the infinitesimal Frobenius kernels ${\mathbb{G}}_{a(r)}$ of the additive group scheme ${\mathbb{G}}_a$. Then, following the approach of Suslin, Friedlander, and Bendel, we use functor cohomology to define characteristic extension classes for the general linear supergroup $G{L}_{m|n}$, and we calculate how these classes restrict along homomorphisms $\rho :{\mathbb{M}}_{r;f,\eta }\to G{L}_{m|n}.$ Finally, we apply our calculations to describe (up to a finite surjective morphism) the spectrum of the cohomology ring of the $r$-th Frobenius kernel $G{L}_{m|n(r)}$ of the general linear supergroup $G{L}_{m|n}$.

Corrected some algebra misidentifications in Proposition 3.1.4 and in subsequent results. Other minor corrections and changes to improve readability

References in corpus (5)

• Cohomological finite-generation for finite supergroup schemes
• On support varieties for Lie superalgebras and finite supergroup schemes
• Support varieties for rational representations
• On the Existence of Mock Injective Modules for Algebraic Groups
• On the cohomological spectrum and support varieties for infinitesimal unipotent supergroup schemes

Cited by in corpus (5)

• On support varieties for Lie superalgebras and finite supergroup schemes
• 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
• Superized Troesch complexes and cohomology for strict polynomial superfunctors