Static modules and stable Clifford theory

Abstract

The object of this paper is to express an important result of Dade [2, Theorem 7.41, on equivalence of categories, from classical stable Clifford theory to one in pure ring theory. In fact, in [2], utilising the properties of fully group-graded rings and modules, Dade described an extended version of Cline’s stable Clifford theory [ I]. In the sequel we will work in the category of right d-modules, where d is an arbitrary ring. In our work no group action is involved and the rings and modules are free from any group-grading. Our work is based on a type of module, called a “static module related to a fixed d-module JY” or “d-static d-module” (Definition 2.1). If .d and S# are rings and p: .02 -+ 3 is an identity-preserving ring homomorphism, then we will be interested in those B-modules which are A-static as &‘-modules under the restriction of p. A source of interest is the following example: Let Y be a group, X a normal subgroup of 9, and F any field. Let J$! be a g-stable S[S]-module. Then the 9[?3]-modules of interest here are exactly those which are .X-static as F[X]-modules. Our first main theorem, Theorem 4.9, below shows that the category of all such modules is equivalent to a certain category of d-modules, where In many cases, the latter category is much easier to deal with. Our second main theorem, Theorem 5.5, below shows that the category whose objects are those F[ZJ]-modules which weakly divide & as B[X]- modules is equivalent to the category whose objects are those &-modules which are projective of finite type as %modules, where 9 = End,,c.X ,(.M).