Modules with cosupport and injective functors

TL;DR

This paper explores the categories of modules with support and cosupport in a class B, establishing their equivalences to categories of flat and injective objects, and applies these findings to module theory and pure injective envelopes.

Contribution

It introduces the category of modules with cosupport in B, showing its equivalence to injective objects in (B,Ab), and generalizes results on pure injective envelopes of flat modules.

Findings

(Mod-R)^B is equivalent to the category of injective objects in (B,Ab).

Provides stability results for lim(B) and (Mod-R)^B.

Generalizes results on pure injective envelopes of flat modules.

Abstract

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