Stable Clifford theory for divisorially graded rings

Abstract

Dade [D1, Theorem 7.4] obtained an important result on the equivalence of categories, extending the classical stable Clifford theory. He used the theory of strongly graded rings. Recently, this work has been generalized to arbitrary graded rings, see E. Dade [D2], [D3], J.L. Gomez Pardo and C. Nǎstǎsescu [GN ], C. Nǎstǎsescu and F. Van Oystaeyen [NVO2]. In the classical case the stable Clifford theory relates isomorphism classes of simple modules on a strongly graded ring R which are direct sums of a fixed simple Re-module, where Re is the component of degree e, with the isomorphism classes of simple modules on a crossed product. The aim of this paper is extend the foregoing result to C-cocritical modules, where C is a localizing subcategory, on divisorially graded rings. We start with a relative version of Clifford theory using the simple objects of the quotient category. We investigated the situation of the so-called divisorially graded rings introduced by F. Van Oystaeyen in the commutative case and then generalized by many other author to more general situations (see the monograph [LVVO] and the references quoted there). We will work in the categories of R-Mod and R − gr, thus we prefer use the a general Grothendieck category and the concept of static objects in this kind of category to establish our basic results. The paper is organized as follows. After a Section of preliminaries, we introduce the notion of static objects in quotient categories in the next section. If we have adjoints functors between two Grothendieck categories A and B and a localizing subcategory C of A, then we show that under certain conditions it is possible to obtain an equivalence between the category of static objects inA/C and the category of co-static objects of some quotient category of B. In the last Section, we apply