Graded Morita Theory for Infinite Groups

Abstract

Having shown that not all graded rings are graded equivalent (via classical Morita theory) to a skew group ring, we extend classical Morita theory, which is based on rings with identity, to a generalized graded Morita theory for rings with local units. This enables us to give necessary and sufficient conditions for graded Morita equivalence between two rings graded by a group. We show that the strongly graded property is a graded Morita invariant and we show that a graded ring is graded Morita equivalent to a skew group ring (namely, (R#G)∗G) if and only if it is strongly graded. These results extend the classical Cohen-Montgomery duality theory to rings graded by infinite groups. The fundamental tools include rings with local units and the results of Boisen.