Groupoid graded rings and their categories of graded modules
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917133046251520 |
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| author | Antony, Caio del Río, Ángel |
| author_facet | Antony, Caio del Río, Ángel |
| contents | 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 Năstăsescu, we characterize when $gr-(R,G,X)$ is equivalent to the category of modules over a unital ring. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_07722 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Groupoid graded rings and their categories of graded modules Antony, Caio del Río, Ángel Rings and Algebras 16W50, 20L05, 18A40 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 Năstăsescu, we characterize when $gr-(R,G,X)$ is equivalent to the category of modules over a unital ring. |
| title | Groupoid graded rings and their categories of graded modules |
| topic | Rings and Algebras 16W50, 20L05, 18A40 |
| url | https://arxiv.org/abs/2512.07722 |