Properties of cofinite modules and applications to local cohomology

Melkersson, Leif

doi:10.1017/s0305004198003041

Cited by 64 publications

(20 citation statements)

References 7 publications

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“…These results are proved in [8] and [3], respectively. Melkersson [15] characterized those Artinian modules which are a-cofinite. For a survey of recent developments on cofiniteness properties of local cohomology, see Melkersson's interesting article [16].…”

Section: Introductionmentioning

confidence: 99%

Some results on local cohomology modules

Mafi

2006

Arch. Math.

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Let R be a commutative Noetherian ring, a an ideal of R, and let M be a finitely generated R-module. For a non-negative integer t, we prove that H t a (M) is a-cofinite whenever H t a (M) is Artinian and H i a (M) is a-cofinite for all i < t. This result, in particular, characterizes the a-cofiniteness property of local cohomology modules of certain regular local rings. Also, we show that for a local ring. This result, in conjunction with the first one, yields some interesting consequences. Finally, we extend Grothendieck's non-vanishing Theorem to a-cofinite modules.1. Introduction. Throughout this paper, we assume that R is a commutative Noetherian ring, a an ideal of R, and that M is an R-module. Let t be a non-negative integer. Grothendieck [4] introduced the local cohomology modules H t a (M) of M with respect to a. He proved their basic properties. For example, for a finitely generated module M, he proved that H t m (M) is Artinian for all t whenever R is local with maximal ideal m. In particular, it is shown that Hom R (R/m, H t m (M)) is finitely generated. Later Grothendieck asked in [5] whether a similar statement is valid if m is replaced by an arbitrary ideal. Hartshorne gave a counterexample in [6], where he also defined that an R-module M (not necessarily finitely generated) is a-cofinite, if Supp R (M) V (a) and Ext t R (R/a, M) is a finitely generated R-module for all t. He also asked when the local cohomology modules are a-cofinite. In this regard, the best known result is that when either a is principal or R is local and dim R/a = 1, then the modules H t a (M) are a-cofinite. These results are proved in [8] and [3], respectively. Melkersson [15] characterized those Artinian modules which are a-cofinite. For a survey of recent developments on cofiniteness properties of local cohomology, see Melkersson's interesting article [16]. One of the aim of this note is to show that, for a finitely generated module M, the module H t a (M) is a-cofinite whenever the modules H i a (M) are a-cofinite for all i < t and H t a (M) is Artinian. This result, in particular, characterizes the a-cofiniteness property of local cohomology modules of certain regular local rings (see Remark 2.3(ii)). Next, we assume that R is local with maximal ideal m.

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Section: Introductionmentioning

confidence: 99%

Some results on local cohomology modules

Mafi

2006

Arch. Math.

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“…In [16] we showed that an artinian module M with support in V(a) is a-cofinite if and only if 0 : M a has finite length. We extend this result to the class of minimax modules in 4.3.…”

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confidence: 99%

Modules cofinite with respect to an ideal

2005

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“…is a-cofinite and artinian for all i < n − 1, by [11,Corollary 1.7]. Therefore by the induction hypothesis there exists y 2 , .…”

Section: Resultsmentioning

confidence: 78%

Cofiniteness and coassociated primes of local cohomology modules

Aghapournahr¹,

Melkersson²

2009

MATH. SCAND.

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Let R be a noetherian ring, a an ideal of R such that dim R/a = 1 and M a finite R-module. We will study cofiniteness and some other properties of the local cohomology modules H i a (M ). For an arbitrary ideal a and an R-module M (not necessarily finite), we will characterize a-cofinite artinian local cohomology modules. Certain sets of coassociated primes of top local cohomology modules over local rings are characterized.2000 Mathematics Subject Classification. 13D45, 13D07.

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Properties of cofinite modules and applications to local cohomology

Cited by 64 publications

References 7 publications

Some results on local cohomology modules

Some results on local cohomology modules

Modules cofinite with respect to an ideal

Cofiniteness and coassociated primes of local cohomology modules

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