F-modules: applications to local cohomology and D-modules in characteristic p&gt;0.

Lyubeznik, Gennady

doi:10.1515/crll.1997.491.65

Cited by 218 publications

(335 citation statements)

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“…One of the key results of the theory (which was essentially proved by Lyubeznik in [Lyu97]) is that the category of locally finitely generated unit R[F ]-modules is abelian, and that every such M has finite length in that category [Lyu97, Theorem 3.2].…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

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: Annales De L'institut Fouriermentioning

confidence: 99%

Local cohomology multiplicities in terms of étale cohomology

Blickle¹,

Bondu²

2005

Ann. inst. Fourier

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“…Since T is Dedekind we have V is a regular ring of characteristic p > 0. So Ass V H i JV (V ) is finite by [7] or [9]. By the independent theorem we have H i J (V ) ∼ = H i JV (V ).…”

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confidence: 93%

“…The Lyubeznik conjecture has affirmative answers in several cases: for regular rings of prime characteristic (cf. [7,9]); for regular local and affine rings of characteristic zero (cf. [8]); for unramified regular local rings of mixed characteristic (cf.…”

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confidence: 99%

Some results on local cohomology of polynomial and formal power series rings: The one dimensional case

Quy

2016

Journal of Algebra

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Abstract. In this paper, we prove several results on the finiteness of local cohomology of polynomial and formal power series rings. In particular, we give a partial affirmative answer for a question of L. Núñez-Betancourt in [J. Algebra 399 (2014), 770-781]. IntroductionThe motivation of this paper is the following conjecture of G. Lyubeznik: If R is a regular ring, then each local cohomology module H i I (R) has finitely many associated prime ideals. The Lyubeznik conjecture has affirmative answers in several cases: for regular rings of prime characteristic (cf. [7,9]); for regular local and affine rings of characteristic zero (cf. [8]); for unramified regular local rings of mixed characteristic (cf. [11,13]) and for smooth Z-algebras (cf. [2]). The method of the proof of these results is considering the module structure of local cohomology over non-commutative rings, D-modules (resp. F -modules). The finiteness of these module structures (for example, finite length) yields the finiteness of Ass S H i I (R). Motivated by the above finiteness results, M. Hochster raised the following related question (cf. [14, Question 1.1]): Question 1. Let (R, m, k) be a local ring and S a flat extension of R with regular closed fiber. Then is Ass S H 0 mS (H i I (S)) = V (mS) ∩ Ass S H i I (S) finite for every ideal I ⊂ S and for every integer i ≥ 0? Suppose S is a flat extension of R with regular fibers. It is worth to note that if Question 1 has an affirmative answer, then the finiteness conditions of Ass S H i I (S) and Ass R H i I (S) are equivalent. In [14], L. Núñez-Betancourt gave a positive answer for Question 1 when S is either R[x 1 , ..., x n ] or R[[x 1 , ..., x n ]] and dim R/(I ∩ R) ≤ 1. In that paper, he introduced the notion of Σ-finite D-modules. It should be noted that Σ-finite D-modules maybe not have finite length but they have finitely many associated primes. Núñez-Betancourt asked the following question (cf. [14, Question 5.1] ]]. In Section 3 we modify the definition of Σ-finite D-modules for rings that not necessarily local rings. We prove that H j J (S) is Σ-finite for every ideal J ⊆ S satisfying dim R/(J ∩ R) = 0 (cf. Proposition 3.7). Applying this result we give a positive answer for Question 2 when dim R/(J ∩ R) ≤ 1 (cf. Theorem 3.8). Moreover, a finiteness result of associated primes of local cohomology is given (cf. Corollary 3.9).In Section 4 we consider the following problem.Question 3. Suppose that dim R = 1 and S is either The next interesting case of Lyubeznik's conjecture is seem to be the case S = R[x 1 , ..., x n ] with R is a Dedekind domain (containing the field of rational numbers). This is a special case of Question 3. In this section we will give a partial affirmative answer of Question 3 in the case R contains a field of positive characteristic (cf. Proposition 4.4). It should be noted that H. Dao and the author showed that local cohomology of Stanley-Reisner rings over a field of positive characteristic have only finitely many associated primes, see [4] for a more general result...

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“…Local cohomology modules have a natural structure over the ring of k-linear differential operators D R|k (see [18], [19] …”

Section: Introductionmentioning

confidence: 99%