Support Varieties and stable categories for algebraic groups

TL;DR

This paper develops a support variety theory for rational representations of algebraic groups over fields of positive characteristic, extending existing theories and classifying certain subcategories via support data.

Contribution

It introduces a new support theory for algebraic group modules, extending to complexes, and classifies tensor ideals and localizing subcategories using support varieties.

Findings

Support theory is equivalent to existing intrinsic support theories.

Supports satisfy standard properties for module categories.

Classification of tensor ideals and subcategories via support varieties.

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 $StMod(\mathbb G)$, the Verdier quotient of the bounded derived category $D^b(Mod(\mathbb G))$ by the thick subcategory of mock injective modules. Our support theory satisfies all the standard properties" for a theory of supports for $StMod(\mathbb G)$. As an application, we employ $C^\bullet \mapsto \Pi(\mathbb G)_{C^\bullet}$ to establish the classification of $(r)$-complete, thick tensor ideals of $stmod(\mathbb G)$ in terms of $stmod(\mathbb G)$-realizable subsets of $\Pi(\mathbb G)$ and the classification of $(r)$-complete, localizing subcategories of $StMod(\mathbb G)$ in terms of $StMod(\mathbb G)$-realizable subsets of $\Pi(\mathbb G)$.