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 $\mathfrak{M}od-\mathfrak{R}$ 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 $\mathfrak{M}od-\mathfrak{R}$ 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 $\mathfrak{M}od-\mathfrak{R}$. When $\mathcal{C}$ is finite, the complete classifications of all these are given respectively.