Vertex-source pairs and the Clifford theory of finite group crossed product algebras

Abstract

Let G be a finite group and let A be a G-crossed product -algebra where is a commutative neotherian local complete ring such that the field is of characteristic p. We study the Clifford Theory of A over the 1-component A 1 of A with emphasis on the vertex-source pairs of indecomposable A-modules. We present the following application of our results (Section 5): Let be a field of characteristic p that is “big enough”, let so that , let that is covered by b and let T be the G-stabilizer of e. Thus e is a central idempotent of AT . As is known, there is a twisted group algebra C of T over k such that the categories C-mod and Ab-mod are equivalent. Let X be an indecomposable (resp. simple) C-module. Then from a vertex-source pair of X, we give a formula for a vertex-source pair of a corresponding indecomposable (resp. simple) Ab-module .