Stable functorial equivalence of blocks

Abstract

Let k be an algebraically closed field of characteristic p > 0, let R be a commutative ring and let F be an algebraically closed field of characteristic 0. We introduce the category F1 Rppk of stable diagonal p-permutation functors over R. We prove that the category F1 F ppk is semisimple and give a parametrization of its simple objects in terms of the simple diagonal p-permutation functors. We also introduce the notion of a stable functorial equivalence over R between blocks of finite groups. We prove that if G is a finite group and if b is a block idempotent of kG with an abelian defect group D and Frobenius inertial quotient E, then there exists a stable functorial equivalence over F between the pairs (G, b) and (D ⋊ E, 1)