Groupoid graded rings and their categories of graded modules

TL;DR

This paper explores the structure of categories of graded modules over groupoid-graded rings with local units, establishing functorial relationships, equivalences, and connections to module categories over rings.

Contribution

It characterizes functors and equivalences between categories of graded modules over groupoid-graded rings, extending Morita theory and relating these categories to modules over rings with local units.

Findings

Categories are equivalent to modules over rings with local units.

Characterization of when functors are equivalences.

Generalization of Morita theory for groupoid-graded rings.

Abstract

Let $G$ be a groupoid acting on a set $X$ and let $R$ be a $G$-graded ring with graded local units. We study the main properties of the category $gr-(R,G,X)$ of $X$-graded $R$-modules and adjoint functors between categories of this kind. We characterize the latter in terms of tensor-like and hom-like functors. As an application we obtain a characterization of equivalences between such categories in the spirit of Morita theory. Then we introduce restriction and induction functors between categories of type $gr-(R,G,X)$ and show that many functors between such categories can be realized naturally as a restriction or induction functor. This includes the forgetful functor, the functor associating a $G$ graded $R$-module with the $H$-graded module formed by the sum of the homogeneous components of degree in subgroupoid $H$, and the one associating it with the collapsing of homogeneous components in cosets modulo $H$ for $H$ a wide subgrupoid. We characterize when a restriction or induction functor is an equivalence of categories. Finally, we prove that $gr-(R,G,X)$ is always equivalent to the category of modules over a ring with local units and, generalizing a result of Menini and N\u{a}st\u{a}sescu, we characterize when $gr-(R,G,X)$ is equivalent to the category of modules over a unital ring.