Object-unital groupoid graded modules

TL;DR

This paper studies modules over object-unital groupoid graded rings, focusing on how module properties relate between graded modules and their underlying ungraded modules, extending classical module theory results.

Contribution

It analyzes the relationship between properties of graded modules and their underlying modules over object-unital groupoid graded rings, extending classical module theory to this context.

Findings

Characterization of when properties like projective or injective are preserved under the forgetful functor.

Structural properties of the category of graded modules over such rings.

Extension of classical module theory results to the graded setting.

Abstract

In a previous article (see \cite{CNP}), we introduced and analyzed ring-theoretic properties of object unital $\mathcal{G}$-graded rings $R$, where $\mathcal{G}$ is a groupoid. In the present article, we analyze the category $\grmod$ of unitary $\G$-graded modules over such rings. Following ideas developed earlier by one of the authors in \cite{lundstrom2004}, we analyze the forgetful functor $U \colon \grmod \to \rmod$ and aim to determine properties $\mathcal{P}$ for which the following implications are valid for modules $M$ in $\grmod$: $M$ is $\mathcal{P}$ $\Rightarrow$ $U(M)$ is $\mathcal{P}$; $U(M)$ is $\mathcal{P}$ $\Rightarrow$ $M$ is $\mathcal{P}$. Here we treat the cases when $\mathcal{P}$ is any of the properties: direct summand, projective, injective, free, simple and semisimple. Moreover, graded versions of results concerning classical module theory are established, as well as some structural properties related to the category $\grmod$.