A formula for the core of an ideal

Polini, Claudia; Ulrich, Bernd

doi:10.1007/s00208-004-0560-z

Cited by 35 publications

(60 citation statements)

References 20 publications

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“…As a contrast to (15), we start by providing an example of a Cohen-Macaulay domain where we use Theorem 2.12 to compute explicitly the core as well as all minimal reductions for one of its (non-basic) maximal ideals. Moreover, this example validates a conjecture by Corso, Polini, and Ulrich [13, Conjecture 5.1] which was mentioned (and studied) later in [33,47,52]. The conjecture sustains that if R is a Cohen-Macaulay ring, I is an ideal of R of analytic spread ≥ 1 (subject to some additional assumptions), and J is a minimal reduction of I with reduction number r, then core(I) = (J r+1 : I r ).…”

Section: Minimal Reductionssupporting

confidence: 87%

“…Recently, in a series of papers [12,13,47], Corso, Polini and Ulrich gave explicit descriptions for the core of certain ideals in Cohen-Macaulay local rings, extending the results of [31]. In 1997, Mohan [43] investigated the core of a module over a twodimensional regular local ring and was inspired by the original work of Huneke and Swanson.…”

Section: Introductionmentioning

confidence: 95%

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Core of ideals in integral domains

Kabbaj

Mimouni

2016

Journal of Algebra

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Section: Minimal Reductionssupporting

confidence: 87%

Section: Introductionmentioning

confidence: 95%

Core of ideals in integral domains

Kabbaj

Mimouni

2016

Journal of Algebra

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“…Hence there is significant difficulty in computing this ideal. The question of finding explicit formulas that compute the core has been addressed in the work of Corso, Huneke, Hyry, Polini, Smith, Swanson, Trung, Ulrich and Vitulli [2,3,[8][9][10][11]15,16]. Moreover, Hyry and Smith have discovered a connection with a conjecture by Kawamata on the nonvanishing of sections of line bundles [11].…”

Section: Introductionmentioning

confidence: 98%

Computing the core of ideals in arbitrary characteristic

Fouli

2008

Journal of Algebra

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Let R be a local Gorenstein ring with infinite residue field of arbitrary characteristic. Let I be an R-ideal with g = ht I > 0, analytic spread , and let J be a minimal reduction of I . We further assume that I satisfies G and depth R/I j dim R/I − j + 1 for 1 j − g. The question we are interested in is whether core(I ) = J n+1 : b∈I (J, b) n for n 0. In the case of analytic spread Polini and Ulrich show that this is true with even weaker assumptions [C. Polini, B. Ulrich, A formula for the core of an ideal, Math. Ann. 331 (2005) 487-503, Theorem 3.4]. We give a negative answer to this question for higher analytic spreads and suggest a formula for the core of such ideals.

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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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A formula for the core of an ideal

Cited by 35 publications

References 20 publications

Core of ideals in integral domains

Core of ideals in integral domains

Computing the core of ideals in arbitrary characteristic

Core versus graded core, and global sections of line bundles

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