Modules with cosupport and injective functors

Henrik Granau Holm

doi:10.1007/s10468-009-9136-7

Modules with cosupport and injective functors

5 Citationer (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 (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-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-R^B). Our applications include a generalization of a result by Gruson-Jensen and Enochs on pure injective envelopes of flat modules.

Originalsprog	Engelsk
Tidsskrift	Algebras and Representation Theory
Vol/bind	13
Udgave nummer	5
Sider (fra-til)	543-560
Antal sider	18
ISSN	1386-923X
DOI	https://doi.org/10.1007/s10468-009-9136-7
Status	Udgivet - okt. 2010

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10.1007/s10468-009-9136-7

Citationsformater

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AB - 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.

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DO - 10.1007/s10468-009-9136-7

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JO - Algebras and Representation Theory

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ER -

Modules with cosupport and injective functors

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