The structure of the core of ideals

Abstract. D. Rees and J. Sally defined the core of an R-ideal I as the intersection of all (minimal) reductions of I. However, it is not easy to give an explicit characterization of it in terms of data attached to the ideal. Until recently, the only case in which a closed formula was known is the one of integrally closed ideals in a two-dimensional regular local ring, due to C. Huneke and I. Swanson. The main result of this paper explicitly describes the core of a broad class of ideals with good residual properties in an arbitrary local Cohen-Macaulay ring. We also find sharp bounds on the number of minimal reductions that one needs to intersect to get the core.

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“…We begin by reviewing some facts and results from [6] and [9]. Let R be a Noetherian ring, I an R-ideal of height g, and s an integer.…”

Section: Residually S 2 Idealsmentioning

confidence: 99%

“…The spirit of [9] was close to the one of [25]. In this paper we shift interest instead: Our main goal is to give an explicit formula for core(I) in the spirit of [17], by which we mean a formula that only involves operations inside the ring R itself.…”

Section: Introductionmentioning

confidence: 99%

Core and residual intersections of ideals

Corso

Polini

Ulrich

2002

Trans. Amer. Math. Soc.

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“…Because H i (X, O X (K X + mD)) = 0 for all m ≥ n − i and all i > 0, a standard argument 6 shows that the map of global sections…”

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confidence: 99%

“…6 The standard argument is this: break the complex into several short exact complexes of sheaves. Then look at the corresponding long exact complexes of cohomology, beginning with the 0-th cohomology of the short exact sequence arising from the right-most part of the complex.…”

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confidence: 99%

On a non-vanishing conjecture of Kawamata and the core of an ideal

Hyry¹,

Smith²

2003

ajm

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Abstract. We show, under suitable hypothesis which are sharp in a certain sense, that the core of an m-primary ideal in a regular local ring of dimension d is equal to the adjoint (or multiplier) ideal of its d-th power. This generalizes the fundamental formula for the core of an integrally closed ideal in a two dimensional regular local ring due to Huneke and Swanson. We also find a generalization of this result to singular (non-regular) settings, which we show to be intimately related to the problem of finding non-zero sections of ample line bundles on projective varieties. In particular, we show that a graded analog of our formula for core would imply a remarkable conjecture of Kawamata predicting that every adjoint ample line bundle on a smooth variety admits a non-zero section.

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“…The core first arose in the work of Rees and Sally [25] because of its connection with Briançon-Skoda theorems, and has recently been the subject of active investigation in commutative algebra; see [11,2,3,16,24,13].…”

Section: Introductionmentioning

confidence: 99%

Core versus graded core, and global sections of line bundles

Hyry

Smith

2003

Trans. Amer. Math. Soc.

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Abstract. We find formulas for the graded core of certain m-primary ideals in a graded ring. In particular, if S is the section ring of an ample line bundle on a Cohen-Macaulay complex projective variety, we show that under a suitable hypothesis, the core and graded core of the ideal of S generated by all elements of degrees at least N (for some, equivalently every, large N ) are equal if and only if the line bundle admits a non-zero global section. We also prove a formula for the graded core of the powers of the unique homogeneous maximal ideal in a standard graded Cohen-Macaulay ring of arbitrary characteristic. Several open problems are posed whose solutions would lead to progress on a non-vanishing conjecture of Kawamata.

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The structure of the core of ideals

Cited by 38 publications

References 13 publications

Core and residual intersections of ideals

Core and residual intersections of ideals

On a non-vanishing conjecture of Kawamata and the core of an ideal

Core versus graded core, and global sections of line bundles

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