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This https://arxiv.org/abs/2403.15001 has been replaced.
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Torsion pairs in categories of modules on ringed finite sites
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 $…