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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2403.15001 |
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| _version_ | 1866913900296929280 |
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| author | Wu, Mawei |
| author_facet | Wu, Mawei |
| contents | Let $\mathcal{C}$ be a small category. In this paper, we mainly study the category of modules $\mathfrak{M}\mbox{od-}\mathfrak{R}$ on ringed sites $(\mathbf{C},\mathfrak{R})$. We firstly reprove the Theorem A of the paper (M. Wu and F. Xu. Skew category algebras and modules on ringed finite sites. J. A. 631, 2023), then we characterize $\mathfrak{M}\mbox{od-}\mathfrak{R}$ in terms of the torsion modules on $Gr(\mathfrak{R})$, where $Gr(\mathfrak{R})$ is the linear Grothendieck construction of $\mathfrak{R}$. Finally, we investigate the hereditary torsion pairs, TTF triples and Abelian recollements of $\mathfrak{M}\mbox{od-}\mathfrak{R}$. When $\mathcal{C}$ is finite, the complete classifications of all these are given respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15001 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torsion pairs in categories of modules on ringed finite sites Wu, Mawei Representation Theory 13D30, 16S90, 18A25, 18F20, 18E05 Let $\mathcal{C}$ be a small category. In this paper, we mainly study the category of modules $\mathfrak{M}\mbox{od-}\mathfrak{R}$ on ringed sites $(\mathbf{C},\mathfrak{R})$. We firstly reprove the Theorem A of the paper (M. Wu and F. Xu. Skew category algebras and modules on ringed finite sites. J. A. 631, 2023), then we characterize $\mathfrak{M}\mbox{od-}\mathfrak{R}$ in terms of the torsion modules on $Gr(\mathfrak{R})$, where $Gr(\mathfrak{R})$ is the linear Grothendieck construction of $\mathfrak{R}$. Finally, we investigate the hereditary torsion pairs, TTF triples and Abelian recollements of $\mathfrak{M}\mbox{od-}\mathfrak{R}$. When $\mathcal{C}$ is finite, the complete classifications of all these are given respectively. |
| title | Torsion pairs in categories of modules on ringed finite sites |
| topic | Representation Theory 13D30, 16S90, 18A25, 18F20, 18E05 |
| url | https://arxiv.org/abs/2403.15001 |