Support Varieties and stable categories for algebraic groups

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 $\Pi(\mathbb G)$.