Object-unital groupoid graded modules

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 $\grmod$.