Modules with cosupport and injective functors

    5 Citations (Scopus)

    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 (Bop, Ab) . In this paper, we study the category (Mod-RB) of modules with cosupport in B. We show that (Mod-RB) 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-Hügel, Enochs, Krause, Rada, and Saorín 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-RB). Our applications include a generalization of a result by Gruson-Jensen and Enochs on pure injective envelopes of flat modules.

    Original languageEnglish
    JournalAlgebras and Representation Theory
    Volume13
    Issue number5
    Pages (from-to)543-560
    Number of pages18
    ISSN1386-923X
    DOIs
    Publication statusPublished - Oct 2010

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