Support varieties and stable categories for algebraic groups

We consider rational representations of a connected linear algebraic group G over a field k of positive characteristic p > 0. We introduce a natural extension M ->(G)(M) to 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 -> P(sic)(G)(M) of supports for an algebraic group of exponential type, a theory which uses 1-parameter subgroups G(a) -> G. We extend our support theory to bounded complexes of G -modules, C-center dot ->Pi(G)(C)(center dot.) We introduce the tensor triangulated category StMod(G), the Verdier quotient of the bounded derived category D-b(Mod(G)) by the thick subcategory of mock injective modules. Our support theory satisfies all the standard properties' for a theory of supports for StMod(G). As an application, we employ C-center dot ->Pi(G)(C)(center dot.) to establish the classification of (r)-complete, thick tensor ideals of stmod(G) in terms of locally stmod(G)-realizable subsets of Pi(G) and the classification of (r)-complete, localizing subcategories of StMod(G) in terms of locally StMod(G)-realizable subsets of Pi(G).