On the category of semi-graded modules
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918523313324032 |
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| author | Reyes, Armando |
| author_facet | Reyes, Armando |
| contents | Lezama \cite{LezamaLatorre2017} introduced the notion of semi-graded ring with the aim of generalizing $\mathbb{Z}$-graded rings and several families of noncommutative rings of polynomial type non-$\mathbb{N}$-graded such as the skew Poincaré-Birkhoff-Witt extensions defined by him \cite{GallegoLezama2010}. In a series of papers, \cite{Lezama2020, Lezama2021, LezamaGomez2019, LezamaLatorre2017}, he studied problems of non-commutative projective algebraic geometry generalizing the original ideas of Artin et al. \cite{Artin1992, ArtinSchelter1987, ArtinTateVandenBergh2007, ArtinTateVandenBergh1991, ArtinZhang1994} on $\mathbb{N}$-graded rings, in the categorical context of the category $\mathsf{SGR}-R$ of left semi-graded modules over a semi-graded ring $R$. In this note we prove that $\mathsf{SGR}-R$ possesses a canonical set of free generators via shifted twists, which endows the category with a {\em Grothendieck structure} and guarantees the existence of enough injective and projective objects. This categorical robustness allows us to formulate a semi-graded analogue of Baer's criterion for injectivity and to establish a first approach to the dual theory of projective resolutions using shifted twists. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_26374 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the category of semi-graded modules Reyes, Armando Category Theory Algebraic Geometry Quantum Algebra Rings and Algebras 14A22, 16S38, 16S80, 16U20, 16W60 Lezama \cite{LezamaLatorre2017} introduced the notion of semi-graded ring with the aim of generalizing $\mathbb{Z}$-graded rings and several families of noncommutative rings of polynomial type non-$\mathbb{N}$-graded such as the skew Poincaré-Birkhoff-Witt extensions defined by him \cite{GallegoLezama2010}. In a series of papers, \cite{Lezama2020, Lezama2021, LezamaGomez2019, LezamaLatorre2017}, he studied problems of non-commutative projective algebraic geometry generalizing the original ideas of Artin et al. \cite{Artin1992, ArtinSchelter1987, ArtinTateVandenBergh2007, ArtinTateVandenBergh1991, ArtinZhang1994} on $\mathbb{N}$-graded rings, in the categorical context of the category $\mathsf{SGR}-R$ of left semi-graded modules over a semi-graded ring $R$. In this note we prove that $\mathsf{SGR}-R$ possesses a canonical set of free generators via shifted twists, which endows the category with a {\em Grothendieck structure} and guarantees the existence of enough injective and projective objects. This categorical robustness allows us to formulate a semi-graded analogue of Baer's criterion for injectivity and to establish a first approach to the dual theory of projective resolutions using shifted twists. |
| title | On the category of semi-graded modules |
| topic | Category Theory Algebraic Geometry Quantum Algebra Rings and Algebras 14A22, 16S38, 16S80, 16U20, 16W60 |
| url | https://arxiv.org/abs/2605.26374 |