Affine duality and cofiniteness

Hartshorne, Robin

doi:10.1007/bf01404554

Cited by 263 publications

(218 citation statements)

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“…Hartshorne [7] defined that an R-module N is I-cofinte, if Supp(N ) ⊆ V (I) and Ext i R (R/I, N ) is a finitely generated R-module for all i. …”

Section: The Resultsmentioning

confidence: 99%

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Results of Certain Local Cohomology Modules

Mafi¹,

Talemi²

2014

Bulletin of the Korean Mathematical Society

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“…Hartshorne [7] defined that an R-module N is I-cofinte, if Supp(N ) ⊆ V (I) and Ext i R (R/I, N ) is a finitely generated R-module for all i. …”

Section: The Resultsmentioning

confidence: 99%

“…In [6], Grothendieck conjectured that the module Hom R (R/I, H i I (M )) is finitely generated for all i ≥ 0. Hartshorne provided a counter-example to this conjecture in [7]. However, this conjecture is true in many cases see for example [1], [3], [4], [11] and [12].…”

Section: Introductionmentioning

confidence: 99%

Results of Certain Local Cohomology Modules

Mafi¹,

Talemi²

2014

Bulletin of the Korean Mathematical Society

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“…Hartshorne [10] defined an R-module M to be I-cofinite if Supp M ⊆ V (I) and Ext j R (R/I, M ) is finitely generated for all j and asked: For which rings R and ideals I are the modules H i I (M ) I-cofinite for all i and all finitely generated modules M ?…”

Section: Introductionmentioning

confidence: 99%

Finiteness Properties of Extension Functors of Cofinite Modules

Irani

Bahmanpour

2013

Bulletin of the Korean Mathematical Society

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Abstract. Let R be a commutative Noetherian ring, I an ideal of R and T be a non-zero I-cofinite R-module with dim(T ) ≤ 1. In this paper, for any finitely generated R-module N with support in V (I), we show that the R-modules Ext i R (T, N ) are finitely generated for all integers i ≥ 0. This immediately implies that if I has dimension one (i.e., dim R/I = 1), then Ext i R (H j I (M ), N ) is finitely generated for all integers i, j ≥ 0, and all finitely generated R-modules M and N , with Supp(N ) ⊆ V (I).

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“…Cofiniteness of modules and local cohomology modules have been studied by many authors, c.f. [DeM], [Ha2], [M1], [M2], and [M3]. In section 2, the main aim is to give a generalization of Brodmann and Lashgari's result by showing that the R-module…”

Section: Introductionmentioning

confidence: 99%

“…Also, Hartshorne gave an example that the conjecture is not true for modules in place of the ring, cf. [Ha2].…”

Section: Introductionmentioning

confidence: 99%

Associated primes and cofiniteness of local cohomology modules

Dibaei

Yassemi

2005

manuscripta math.

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Let a be an ideal of Noetherian ring R and let M be an R-module such that Ext i R (R/a, M ) is finite R-module for every i. If s is the first integer such that the local cohomology module H s a (M ) is non a-cofinite, then we show that Hom R (R/a, H s a (M )) is finite. Specially, the set of associated primes of H s a (M ) is finite. Next assume (R, m) is a local Noetherian ring and M is a finite module. We study the last integer n such that the local cohomology module H n a (M ) is not m-cofinite and show that n just depends on the support of M . IntroductionLet R be a Noetherian ring, a an ideal, and M an R-module. An important problem in commutative algebra is determining when the set of associated primes of the ith local cohomology

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Affine duality and cofiniteness

Cited by 263 publications

References 5 publications

Results of Certain Local Cohomology Modules

Results of Certain Local Cohomology Modules

Finiteness Properties of Extension Functors of Cofinite Modules

Associated primes and cofiniteness of local cohomology modules

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