Torsion Pairs in Categories of Modules on Ringed Finite Sites

Let C be a small category. In this paper, we mainly study the category of modules Mod-R on ringed sites (C,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 Mod-R in terms of the torsion presheaves on Gr(R), where Gr(R) is the linear Grothendieck construction of R. Finally, we investigate the hereditary torsion pairs, TTF triples and Abelian recollements of Mod-R. When C is finite, the complete classifications of all these are given respectively.