Modules with Cosupport and Injective Functors

Abstract

Several authors have studied the filtered colimit closure $\varinjlim\mathcal{B}$ of a class $\mathcal{B}$ of finitely presented modules. Lenzing called $\varinjlim\mathcal{B}$ the category of modules with support in $\mathcal{B}$, and proved that it is equivalent to the category of flat objects in the functor category $(\mathcal{B}^\mathrm{op},\mathsf{Ab})$. In this paper, we study the category $({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}$ of modules with cosupport in $\mathcal{B}$. We show that $({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}$ is equivalent to the category of injective objects in $(\mathcal{B},\mathsf{Ab})$, and thus recover a classical result by Jensen-Lenzing on pure injective modules. Works of Angeleri-Hügel, Enochs, Krause, Rada, and Saorín make it easy to discuss covering and enveloping properties of $({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}$, and furthermore we compare the naturally associated notions of $\mathcal{B}$-coherence and $\mathcal{B}$-noetherianness. Finally, we prove a number of stability results for $\varinjlim\mathcal{B}$ and $({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}$. Our applications include a generalization of a result by Gruson-Jensen and Enochs on pure injective envelopes of flat modules.