Actions of Picard Groups on Graded Rings

Abstract

Throughout this paper, G denotes a group with identity e. If R is a G-graded ring, we write R = ⊕g∈GRg and we refer to Re as the coefficient ring. We denote the category of graded (left) R-modules (i.e., those left R-modules M with a G-grading M = ⊕g∈GMg such that RhMg ⊆ Mhg) by R-gr. The aim of this paper is to present a method to reduce the study of one strongly graded ring R to the study of another strongly graded ring R′ that is more tractable. This reduction process has two aspects: reducing the coefficient ring or reducing the grading. Since we apply this method in our study of module-theoretic properties such as semisimplicity for graded rings, we require that this reduction process preserve the category of modules R-mod and the category of graded modules R-gr. This leads us to the notion of graded equivalence. In this paper, a Morita context is the usual tuple (A,B,P,Q, τ, μ) where we assume τ and μ are isomorphisms. A G-graded context is a Morita context (R,R′, P,Q, τ, μ), where R and R′ are G-graded rings, RPR′ and R′QR are graded bimodules (i.e., they have a grading which makes them into both graded R-modules and graded R′-modules on the corresponding side) and τ and μ are graded bimodule homomorphisms. See [14] for details about graded modules and homomorphisms. Given a functor F : R-gr → R′-gr, we say that F is a graded equivalence of categories if any of the following equivalent conditions hold (see for example [8]):