Co-Localization, Co-Support and Local Homology

Richardson, A. S.

doi:10.1216/rmjm/1181069391

Cited by 18 publications

(16 citation statements)

References 17 publications

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“…In [30], it is shown that, in case the ring R is Noetherian local, these definitions are equivalent. Now, let E be the injective hull of the direct sum of all simple R-modules and D R (−) be the functor Hom R (−, E), which is a natural generalization of Matlis duality functor to non-local rings (see [26]). Following [30], for a local ring R, we define a prime ideal p to be a coassociated prime of M if p is an associated prime of D R (M).…”

Section: Introductionmentioning

confidence: 99%

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On the finiteness properties of Matlis duals of local cohomology modules

Khashyarmansh

Khosh-Ahang

2008

Proc Math Sci

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Let R be a complete semi-local ring with respect to the topology defined by its Jacobson radical, a an ideal of R, and M a finitely generated R-module. Let D R (−) := Hom R (−, E), where E is the injective hull of the direct sum of all simple R-modules. If n is a positive integer such that Ext j R (R/a, D R (H t a (M))) is finitely generated for all t > n and all j 0, then we show that Hom R (R/a, D R (H n a (M))) is also finitely generated. Specially, the set of prime ideals in Coass R (H n a (M)) which contains a is finite.Next, assume that (R, m) is a complete local ring. We study the finiteness properties of D R (H r a (R)) where r is the least integer i such that H i a (R) is not Artinian.

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Section: Introductionmentioning

confidence: 99%

“…DEFINITION 2.2 (see [25] and [26]) For a commutative ring R, let R be the direct sum ⊕ m∈MaxSpec(R) R/m of all simple R-modules, E R be the injective hull of R , and D R (−) be the functor Hom R (−, E R ). To do this, we recall the definition of Matlis duality functor.…”

Section: Introductionmentioning

confidence: 99%

On the finiteness properties of Matlis duals of local cohomology modules

Khashyarmansh

Khosh-Ahang

2008

Proc Math Sci

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“…Remark 2.1 Following [9], for a multiplicatively closed subset S of the local ring (R, m), the co-localization of the R -module M relative to S is defined to be the…”

Section: Attached Prime Idealsmentioning

confidence: 99%

On the attached prime ideals of localcohomology modules defined by a pair of ideals

Habibi¹,

Jahangiri²,

Amoli³

2017

Turk J Math

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Let I and J be two ideals of a commutative Noetherian ring R and M be an R -module of dimension d . For each i ∈ N0 let H i I,J (−) denote the i -th right derived functor of ΓI,J (−) , where ΓI,J (M ) := {x ∈ M : I n x ⊆ Jx for n ≫ 1} . If R is a complete local ring and M is finite, then attached prime ideals of H d−1 I,J (M ) are computed by means of the concept of co-localization. Moreover, we illustrate the attached prime ideals of H t I,J (M ) on a nonlocal ring R , for t = dim M and t = cd (I, J, M ) , where cd (I, J, M ) is the last nonvanishing level of H i I,J (M ) .

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“…So, E is a direct summand of F 0 which implies that E is flat. Now, let E be the injective cogenerator of R, and let D R (−) be the functor Hom R (−, E), which is a natural generalization of the Matlis duality functor to non-local rings (see [9]). Following [14], for a local ring R, we define a prime ideal p to be a coassociated prime of M if p is an associated prime of D R (M ) and we set Coass(M ) = Ass(D R (M )).…”

Section: Proposition 21 Let F Be a Flat R-module And E Be An Injectmentioning

confidence: 99%