Lifting idempotents and Clifford theory

Abstract

Let N be a normal subgroup of a finite group G and let R be a noetherian complete local commutative ring. Clifford theory deals with the relationship between RG-modules and RN-modules, using induction from N to G or restriction from G to N. Since Clifford's 1937 paper [1], the theory is well understood for irreducible representations (see also [2, w For an indecomposable RNmodule W, several authors have proved a going-up theorem describing how Ind~ W decomposes (see [2, w One purpose of this paper is to prove (in Section 2) a going-down theorem for indecomposable modules (analogous to Clifford's theorem), based on a refinement of the lifting idempotents theorem, presented in Section 1. The going-up and going-down theorems are actually equivalent in the sense that each can be dcrivcd as a corollary to the other one. One main assumption is necessary for the going-down theorem: the RG-module we start from must be projective relative to H. The whole procedure is presented in the more general context of Clifford systems. The paper concludes in Section 3 with another application of the lifting idempotents theorem, concerning the behaviour of indecomposable modules under ground ring extensions.