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Contravariant functors on the category of finitely presented modules

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Abstract

If R is an associative ring with identity, a theory of minimal flat resolutions is developed in the category ((R-mod)op, Ab) of contravariant functors G: (R-mod)op → Ab from the category R-mod of finitely presented left R-modules to the category Ab of abelian groups. For a left R-module M, it is shown that the flat contravariant functor (−, M) is cotorsion if and only if M is pure-injective. This is applied to characterize when a flat resolution of an object F in ((R-mod)op, Ab) is minimal, and is used to construct a minimal flat resolution of F, given a projective presentation.

It is shown that the injective objects of ((R-mod)op, Ab) are precisely those of the form Ext1(−, M), where M is pure-injective, and if m: M → PE(M) is the pure-injective envelope of M, then Ext1(−, m): Ext1(−, M) → Ext1(−, PE(M)) is an injective envelope of Ext1(−, M) in ((R-mod)op, Ab). M ↦ Ext1(−, M) yields an explicit equivalence between the subcategory of injective objects of ((R-mod)op, Ab) and the category of pure-injective left R-modules, modulo morphisms that factor through an injective. The characterization of minimal flat resolutions is also used to describe the relationship between the minimal flat resolution in ((R-mod)op, Ab) of a functor F on the stable category and its minimal injective copresentation in ((R-mod)op, Ab).

A final application is a description of the contravariant Gabriel spectrum of R, the set of indecomposable injective objects of the functor category ((R-mod)op, Ab). The points are in bijective correspondence with the set of pure-injective indecomposable left R-modules, which correspond to the points of the covariant Gabriel spectrum of R. It is proved that both Gabriel spectra of R may be partitioned into an open and a closed set such that this canonical bijection restricts to a homeomorphism on each.

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Authors and Affiliations

  1. The Ohio State University (Lima), Lima, OH, 45804, USA

    Ivo Herzog

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  1. Ivo Herzog
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The author is partially supported by NSF Grant DMS05-01207.

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Herzog, I. Contravariant functors on the category of finitely presented modules. Isr. J. Math. 167, 347–410 (2008). https://doi.org/10.1007/s11856-008-1052-8

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  • DOI: https://doi.org/10.1007/s11856-008-1052-8

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