On the vanishing of local cohomology modules

Huneke, Craig; Lyubeznik, Gennady

doi:10.1007/bf01233420

Cited by 107 publications

(76 citation statements)

References 18 publications

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“…The cohomological dimension has been studied by several authors; see, for example, Faltings [7], Hartshorne [9] and Huneke−Lyubeznik [11]. In particular in [7] and [11], several upper bounds for cohomological dimension were obtained. The main aim of this article is to establish lower bounds for cohomological dimension of finitely generated modules over a local ring.…”

Section: Introductionmentioning

confidence: 99%

“…For an R-module M , the cohomological dimension of M with respect to a is defined as cd(a, M) := max{i ∈ Z : H i a (M ) = 0}. The cohomological dimension has been studied by several authors; see, for example, Faltings [7], Hartshorne [9] and Huneke−Lyubeznik [11]. In particular in [7] and [11], several upper bounds for cohomological dimension were obtained.…”

Section: Introductionmentioning

confidence: 99%

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Cohomological dimension of certain algebraic varieties

Divaani-Aazar

Naghipour

Tousi

2002

Proc. Amer. Math. Soc.

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Abstract. Let a be an ideal of a commutative Noetherian ring R. For finitely generated R-modules M and N with Supp N ⊆ Supp M , it is shown that cd(a, N) ≤ cd(a, M). Let N be a finitely generated module over a local ring (R, m) such that MinRN = AsshRN . Using the above result and the notion of connectedness dimension, it is proved that cd(a, N) ≥ dim N − c(N/aN ) − 1. Here c(N ) denotes the connectedness dimension of the topological space Supp N . Finally, as a consequence of this inequality, two previously known generalizations of Faltings' connectedness theorem are improved.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Cohomological dimension of certain algebraic varieties

Divaani-Aazar

Naghipour

Tousi

2002

Proc. Amer. Math. Soc.

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“…For an extensive list of references and some open questions we recommend to consult the very nice survey article [17].…”

Section: The Master Planmentioning

confidence: 99%

D-modules and Cohomology of Varieties

Walther

2002

Computations in Algebraic Geometry With Macaulay 2

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In this chapter we introduce the reader to some ideas from the world of differential operators. We show how to use these concepts in conjunction with Macaulay 2 to obtain new information about polynomials and their algebraic varieties.Gröbner bases over polynomial rings have been used for many years in computational algebra, and the other chapters in this book bear witness to this fact. In the mid-eighties some important steps were made in the theory of Gröbner bases in non-commutative rings, notably in rings of differential operators. This chapter is about some of the applications of this theory to problems in commutative algebra and algebraic geometry.Our interest in rings of differential operators and D-modules stems from the fact that some very interesting objects in algebraic geometry and commutative algebra have a finite module structure over an appropriate ring of differential operators. The prime example is the ring of regular functions on the complement of an affine hypersurface. A more general object is theČech complex associated to a set of polynomials, and its cohomology, the local cohomology modules of the variety defined by the vanishing of the polynomials. More advanced topics are restriction functors and de Rham cohomology.With these goals in mind, we shall study applications of Gröbner bases theory in the simplest ring of differential operators, the Weyl algebra, and develop algorithms that compute various invariants associated to a polynomial f . These include the Bernstein-Sato polynomial b f (s), the set of differential operators J(f s ) which annihilate the germ of the function f s (where s is a new variable), and the ring of regular functions on the complement of the variety of f .For a family f 1 , . . . , f r of polynomials we study the associatedČech complex as a complex in the category of modules over the Weyl algebra. The algorithms are illustrated with examples. We also give an indication what other invariants associated to polynomials or varieties are known to be computable at this point and list some open problems in the area.

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“…This means that in R we can use the second vanishing theorem, due to Ogus, Hartshorne-Speiser and Huneke-Lyubeznik (see [3], Theorem 1. Ja (R) so that λ 2,2 (R/I) = a−1+1.…”

Section: Pure Dimension Twomentioning

confidence: 99%

On the Lyubeznik numbers of a local ring

Walther

2000

Proc. Amer. Math. Soc.

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Abstract. We collect some information about the invariants λ p,i (A) of a commutative local ring A containing a field introduced by G. Lyubeznik in 1993 (Finiteness properties of local cohomology modules, Invent. Math. 113, 41-55). We treat the cases dim(A) equal to zero, one and two, thereby answering in the negative a question raised in Lyubeznik's paper. In fact, we will show that λ p,i (A) has in the two-dimensional case a topological interpretation.

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On the vanishing of local cohomology modules

Cited by 107 publications

References 18 publications

Cohomological dimension of certain algebraic varieties

Cohomological dimension of certain algebraic varieties

D-modules and Cohomology of Varieties

On the Lyubeznik numbers of a local ring

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