Some change of ring theorems for matlis reflexive modules

Belshoff, Richard

doi:10.1080/00927879408825040

Cited by 9 publications

(8 citation statements)

References 6 publications

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“…Corollary 3.2 implies that Tor R i (M/N, M ′ ) mini-max, so Tor R i (M, M ′ ) is mini-max by Lemma 1.25(c). Parts (b) and (c) now follow from Lemmas 1.20 and 1.21.A special case of the next result is contained in[3, Theorem 3].…”

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confidence: 84%

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Homology of artinian and Matlis reflexive modules, I

Kubik

Leamer

Sather-Wagstaff

2011

Journal of Pure and Applied Algebra

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a b s t r a c tLet R be a commutative local noetherian ring, and let L and L ′ be R-modules. We investigate the properties of the functors Tor R i (L, −) and Ext i R (L, −). For instance, we show the following: (a) if L and L ′ are artinian, then Tor R i (L, L ′ ) is artinian, and Ext i R (L, L ′ ) is noetherian over the completion  R; (b) if L is artinian and L ′ is Matlis reflexive, then Ext i R (L, L ′ ), Ext i R (L ′ , L), and Tor R i (L, L ′ ) are Matlis reflexive.Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.

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confidence: 84%

“…The next result contains Theorem 4 from the introduction. A special case of it can be found in[3, Theorem 3].…”

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confidence: 98%

Homology of artinian and Matlis reflexive modules, I

Kubik

Leamer

Sather-Wagstaff

2011

Journal of Pure and Applied Algebra

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“…It is easy to see thatR is a Matlis re exive R-module. [2,Theorem 3]. On the other hand, by Theorem 2.3, M ⊗ R E(R=m) is strongly cotorsion then using Lemma 2.1, Hom R (M;R) is strongly torsion free.…”

Section: Strongly Torsion Free and Copure At Modulesmentioning

confidence: 95%

“…We recall that an R-module M is copure at (copure injective) if Tor R 1 (E; M )=0 (Ext 1 R (E; M )=0) for any injective R-module E. Also M is strongly copure at (strongly copure injective) if Tor R i (E; M )=0 (Ext i R (E; M )=0) for all i ¿ 1. From [2], over a local ring (R; m) an R-module M is said to be Matlis re exive if M ∼ = Hom R (Hom R (M; E(R=m)); E(R=m)).…”

Section: Introductionmentioning

confidence: 99%

Strongly torsion free, copure flat and Matlis reflexive modules

Sazeedeh

2004

Journal of Pure and Applied Algebra

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In this paper we show that if R is a ring of Krull dimension d and M is a strongly copure at (respectively, strongly copure injective) module, then E ⊗R M (respectively, HomR(E; M )) is strongly cotorsion (respectively, strongly torsion free) for any injective module E. We obtain a new characterization for copure at modules. We prove that M is copure at if and only if Ext 1 R (M; F) = 0 for all at cotorsion modules F. Moreover, if (R; m) is a Cohen-Macaulay local ring and M is strongly copure at, then HomR(M; F) is strongly torsion free. Let (R; m) be a Cohen-Macaulay local ring of Krull dimension d and M be a Matlis re exive strongly torsion free R-module. We show thatM is a maximal Cohen-MacaulayR-module. Also, if R is as above and M a Matlis re exive strongly copure injective R-module, then HomR(E(R=m); M ) is either zero or a maximal Cohen-MacaulayR-module.

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

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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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Some change of ring theorems for matlis reflexive modules

Cited by 9 publications

References 6 publications

Homology of artinian and Matlis reflexive modules, I

Homology of artinian and Matlis reflexive modules, I

Strongly torsion free, copure flat and Matlis reflexive modules

Generalized Matlis duality

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