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Support varieties and stable categories for algebraic groups

Part of: Linear algebraic groups and related topics Representation theory of groups

Published online by Cambridge University Press:  27 March 2023

Eric M. Friedlander
Eric M. Friedlander*
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA [email protected]
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Abstract

We consider rational representations of a connected linear algebraic group $\mathbb {G}$ over a field $k$ of positive characteristic $p > 0$. We introduce a natural extension $M \mapsto \Pi (\mathbb {G})_M$ to $\mathbb {G}$-modules of the $\pi$-point support theory for modules $M$ for a finite group scheme $G$ and show that this theory is essentially equivalent to the more ‘intrinsic’ and ‘explicit’ theory $M \mapsto \mathbb {P}\mathfrak{C}(\mathbb {G})_M$ of supports for an algebraic group of exponential type, a theory which uses $1$-parameter subgroups $\mathbb {G}_a \to \mathbb {G}$. We extend our support theory to bounded complexes of $\mathbb {G}$-modules, $C^\bullet \mapsto \Pi (\mathbb {G})_{C^\bullet }$. We introduce the tensor triangulated category $\mathit {StMod}(\mathbb {G})$, the Verdier quotient of the bounded derived category $D^b(\mathit {Mod}(\mathbb {G}))$ by the thick subcategory of mock injective modules. Our support theory satisfies all the ‘standard properties’ for a theory of supports for $\mathit {StMod}(\mathbb {G})$. As an application, we employ $C^\bullet \mapsto \Pi (\mathbb {G})_{C^\bullet }$ to establish the classification of $(r)$-complete, thick tensor ideals of $\mathit {stmod}(\mathbb {G})$ in terms of locally $\mathit {stmod}(\mathbb {G})$-realizable subsets of $\Pi (\mathbb {G})$ and the classification of $(r)$-complete, localizing subcategories of $\mathit {StMod}(\mathbb {G})$ in terms of locally $\mathit {StMod}(\mathbb {G})$-realizable subsets of $\Pi (\mathbb {G})$.

Keywords

support varietiesstable categoriesalgebraic groups

MSC classification

Primary: 20G05: Representation theory 20C20: Modular representations and characters
Secondary: 20G10: Cohomology theory
Type
Research Article
Information
Compositio Mathematica , Volume 159 , Issue 4 , April 2023 , pp. 746 - 779
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

Dedicated to the memory of Brian Parshall

The author was partially supported by the Simons Foundation.

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