Support Varieties and stable cat3egories for algebraic groups

We develop a theory of supports M 7→ V (G)M for arbitrary Gmodules M for a linear algebraic group G of exponential type over a field k of positive characteristic p > 0. Our construction is an refinement and a generalization of an earlier approach by the author. We extend our support theory to apply to bounded complexes of G-modules, C• 7→ V (G)C• . We introduce the tensor triangulated category StMod(G), a Verdier quotient of the derived category D(Mod(G)), and we view StMod(G) as the stable module category of G-modules. Our support theory is well defined on StMod(G), satisfies the “standard properties” for a theory of supports, and extends the support theory for the Frobenius kernels G(r) ⊂ G as considered by Suslin, Bendel, and the author. We also consider the stable module category of finite dimensional G-modules, stmod(G). As an application of our support theory, we employ our support varieties to establish the classification of (r)-complete, thick tensor ideals of stmod(G) by locally G-finte subsets of V (G) and the classification of (r)-complete, localizing tensor ideals of the stable module category StMod(G) by G-realizable subsets of V (G).

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