Matlis reflexive modules on complete rings

Jara, Pascual; Santos, Eva

doi:10.1016/0022-4049(95)00012-7

Journal of Pure and Applied Algebra

1996

DOI: 10.1016/0022-4049(95)00012-7

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Matlis reflexive modules on complete rings

Pascual Jara

Eva Santos

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1999

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“…For example, dropping the condition that R be complete or weakening local to semilocal ( [1], [2], [6], [8], [9]). …”

mentioning

confidence: 99%

Generalized Matlis duality

Belshoff

Enochs

Rozas

1999

Proc. Amer. Math. Soc.

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Abstract. Let R be a commutative noetherian ring and let E be the minimal injective cogenerator of the category of R-modules. A module M is said to be reflexive with respect to E if the natural evaluation map from M to Hom R (Hom R (M, E), E) is an isomorphism. We give a classification of modules which are reflexive with respect to E. A module M is reflexive with respect to E if and only if M has a finitely generated submodule S such that M/S is artinian and R/ ann(M ) is a complete semi-local ring.Matlis and Gabriel in [7] and [5] considered modules over a complete local ring R. They showed that if the dual of an R-module is taken with respect to E R (k) (the injective envelope of the residue field k of R), then finitely generated and artinian modules are reflexive.Various authors have considered related questions. For example, dropping the condition that R be complete or weakening local to semilocalIn this paper we let R be any commutative noetherian ring and let E be the minimal injective cogenerator of the category of R-modules. We give a classification of modules which are reflexive with respect to E. The result is that a module M is reflexive with respect to E if and only if M has a finitely generated submodule S such that M/S is artinian and if R/I is a complete semilocal ring where I = ann(M ).We denote by Ω the maximal spectrum of R, and we let E = m∈Ω E R (R/m) be the minimal injective cogenerator in the category of R-modules.∨∨ is an isomorphism we say that M is (Matlis) reflexive. We note that for any M , the map M → M ∨∨ is an injection. From this it is easy to conclude that ann(M ) = ann(M ∨ ). If S ⊂ R is a multiplicative set and the canonical mapIf R is a local ring we letR denote its completion. If M is finitely generated we note thatR ⊗ R M ∼ =M (the completion of M ). We writeR ⊗ R M ∼ =M = M to mean that M →R ⊗ R M ∼ =M is an isomorphism.We note that if m ∈ Ω and M is a finitely generated R-module M , then Hom R (M, E(R/m)) = 0 if and only if ann(M ) ⊂ m.

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“…For example, dropping the condition that R be complete or weakening local to semilocal ( [1], [2], [6], [8], [9]). …”

mentioning

confidence: 99%

Generalized Matlis duality

Belshoff

Enochs

Rozas

1999

Proc. Amer. Math. Soc.

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Matlis reflexive modules on complete rings

Cited by 1 publication

References 9 publications

Generalized Matlis duality

Generalized Matlis duality

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