Finiteness properties of local cohomology modules (an application ofD-modules to commutative algebra)

Lyubeznik, Gennady

doi:10.1007/bf01244301

Cited by 275 publications

(311 citation statements)

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“…In the isolated singular case, statement (1) was already pointed out in [Lyu93] to follow from a result of Ogus [Ogu73,Theorem 2.3]. Observing the proof in [GLS98] we first note that part (3) is independent of the characteristic whereas the other parts distinctively use characteristic zero.…”

Section: Introductionmentioning

confidence: 92%

“…Let (R, m) be a regular local ring of dimension n and let A = R/I be a quotient of R. In where the multiplicity e( ) can be described as follows: The main results of [Lyu93,HS93] state that the module H a m (H n−i I (R)) is injective. As it is supported at the maximal ideal it is isomorphic to a finite direct sum of e copies of the injective hull E R/m ∼ = H n m (R) of the residue field of R. This integer e is the multiplicity.…”

Section: Introductionmentioning

confidence: 99%

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Local cohomology multiplicities in terms of étale cohomology

Blickle¹,

Bondu²

2005

Ann. inst. Fourier

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In this paper we give an interpretation of the invariants λa,i(A) introduced by Lyubeznik in [Lyu93] for a reasonably general class of singularities. In positive characteristic it is the newly introduced class of close to F -rational varieties and the invariants are described in terms of étale cohomology with Z/pZ-coefficients. This result presents the first application of Emerton and Kisin's Riemann-Hilbert type correspondence to local algebra. In fact our proof works in characteristic zero as well so that we obtain generalizations of results on these invariants which were previously obtained for isolated singularities by analytic techniques.

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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Local cohomology multiplicities in terms of étale cohomology

Blickle¹,

Bondu²

2005

Ann. inst. Fourier

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“…One of the important and hard problems in commutative algebra is determining the annihilator of the local cohomology module H i I (M ). This problem has been studied by several authors; see for example [1], [2], [9], [10], [11], [15], [16] and [18]. Recall that, for an R-module M , the cohomological dimension of M with respect to I is defined as cd(I, M ) := max{i ∈ Z : H i I (M ) = 0}.…”

Section: Introductionmentioning

confidence: 99%

Exactness of Ideal Transforms and Annihilators of Top Local Cohomology Modules

Bahmanpour¹

2015

Journal of the Korean Mathematical Society

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Abstract. Let (R, m) be a commutative Noetherian local domain, M a non-zero finitely generated R-module of dimension n > 0 and I be an ideal of R. In this paper it is shown that if x 1 , . . . , xt (1 ≤ t ≤ n) be a subset of a system of parameters for M , then the R-module H t (x 1 ,...,xt) (R) is faithful, i.e., Ann H t (x 1 ,...,xt) (R) = 0. Also, it is shown that, if H i I (R) = 0 for all i > dim R − dim R/I, then the R-module H dim R−dim R/I I (R) is faithful. These results provide some partially affirmative answers to the Lynch's conjecture in [10]. Moreover, for an ideal I of an arbitrary Noetherian ring R, we calculate the annihilator of the top local cohomology module H 1 I (M ), when H i I (M ) = 0 for all integers i > 1. Also, for such ideals we show that the finitely generated R-algebra D I (R) is a flat Ralgebra.

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“…Sharp [13] when the field k has positive characteristic and G. Lyubeznik [18] in the characteristic zero case (see also [20] for a characteristic-free approach). In particular, they proved that Bass numbers of these local cohomology modules are finite.…”

Section: Introductionmentioning

confidence: 99%