2021/12/20 by Friedlander, Eric M.
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2112.10382
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 ↦ Π(\mathbb G)M to \mathbb G-modules of the π-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 ↦ \mathbb P\mathfrak C(\mathbb G)M of supports for an algebraic group of exponential type, a theory which uses 1-parameter subgroups \mathbb Ga → \mathbb G. We extend our support theory to bounded complexes of \mathbb G-modules, C^\bullet ↦ Π(\mathbb G)C^\bullet. We introduce the tensor triangulated category StMod(\mathbb G), the Verdier quotient of the bounded derived category Db(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 ↦ Π(\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 Π(\mathbb G) and the classification of (r)-complete, localizing subcategories of StMod(\mathbb G) in terms of StMod(\mathbb G)-realizable subsets of Π(\mathbb G).