Castelnuovo Regularity and Graded Rings Associated to an Ideal

Johnston, Bernard; Katz, Daniel

doi:10.2307/2160792

Cited by 17 publications

(18 citation statements)

References 2 publications

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“…These indices acquire a sharp relief when the normalization n≥0 I n t n is Cohen-Macaulay (Theorem 2.5). This result, whose proof follows ipsis literis the characterization of Cohen-Macaulayness in the Rees algebras of I-adic filtrations ( [1], [6], [10]), has various consequences. It is partly used to motivate the treatment in Section 3 of the Sally module of the normalization algebra as a vehicle to study the number of generators and their degrees.…”

Section: Introductionmentioning

confidence: 93%

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Indices of normalization of ideals

Polini

Ulrich

Vasconcelos

et al. 2019

Journal of Pure and Applied Algebra

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We derive numerical estimates controlling the intertwined properties of the normalization of an ideal and of the computational complexity of general processes for its construction. In [8], this goal was carried out for equimultiple ideals via the examination of Hilbert functions. Here we add to this picture, in an important case, how certain Hilbert functions provide a description of the locations of the generators of the normalization of ideals of dimension zero. We also present a rare instance of normalization of a class of homogeneous ideals by a single colon operation.

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Section: Introductionmentioning

confidence: 93%

“…Expectably, normalization indices are easier to obtain when the normalization of the ideal is Cohen-Macaulay. The following is directly derived from the known characterizations of Cohen-Macaulayness of Rees algebras of ideals in terms of associated graded rings and reduction numbers ([1],[6],[10]). Theorem 2.5.…”

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confidence: 99%

Indices of normalization of ideals

Polini

Ulrich

Vasconcelos

et al. 2019

Journal of Pure and Applied Algebra

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“…[33,18,27,30,1,21,28]), so that one can construct various classes of ideals I for which S is locally Cohen-Macaulay on X. We list here only the most interesting applications of Theorem 2.4.…”

Section: 52(1) and 262]) From This It Follows That [Hmentioning

confidence: 99%

Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

Harbourne

Trung

2005

Trans. Amer. Math. Soc.

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Abstract. This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a surprising complete answer to a question poised by Cutkosky and Herzog. Let Y be the blow-up of a projective scheme X = Proj R along the ideal sheaf of I ⊂ R. IntroductionLet X be a projective scheme over a field k. An arithmetic Macaulayfication of X is a proper birational morphism π : Y → X such that Y has an arithmetically Cohen-Macaulay embedding, i.e. there exists a Cohen-Macaulay standard graded k-algebra A such that Y ∼ = Proj A. Inspired by the problem of resolution of singularities, it was asked when X has an arithmetic Macaulayfication. The local version of this problem (arithmetic Macaulayfication of local rings) has been extensively studied in the literature and recently solved by Kawasaki [22]. An important aspect of the global problem is to determine, given a proper birational morphism Y → X, if Y has an arithmetically Cohen-Macaulay embedding, and if it does, which embeddings of Y are arithmetically Cohen-Macaulay.Let R be a standard graded k-algebra and let I ⊂ R be a homogeneous ideal such that X = Proj R and Y is the blow-up of X along the ideal sheaf of I. It was observed by Cutkosky and Herzog [9] • Y is equidimensional and Cohen-Macaulay,In the first part of this paper, we study The invariant e 0 is a projective version of the a * -invariant, which is the largest non-vanishing degree of the graded local cohomology modules [29,32]. The invariant ε comes from the asymptotic linearity of the CastelnuovoMumford regularity of powers of ideals ([31, 10, 23, 34]). We will see that the bounds c > d(I)e + ε and e > e 0 are the best possible (Theorem 2.3 and Example 2.5). The existence of linear bounds on c and e is not hard to prove. The novelty we claim here is that an explicit description for the best possible bounds is obtained. Moreover, if the Rees algebra R[It] is locally Cohen-Macaulay on X, then e 0 = 0 and we can replace the second condition by the weaker condition thatThese results strengthen and unify all previously-known results on the Cohen-Macaulayness of k[(I e ) c ] which were obtained by different methods.In the second part of this paper, we investigate the more difficult question of when Y is an arithmetically Cohen-Macaulay blow-up of X; that is, when there exists a standard graded k-algebra R and an ideal J ⊂ R, such that X = Proj R, Y is the blow-up of X along the ideal sheaf of J, and R[Jt] is a Cohen-Macaulay ring. Given R and I, we will concentrate on ideals J ⊆ I which are generated by the elements of (I The paper is organized as follows. In Section 1, we introduce the notion of a projective a * -invariant which governs how sheaf cohomology behaves through blowing up morphisms. The material in this section is interesting on its own right. In Section 2, we study the Cohen-Macaulayness of rings of the form k[(I e ) c ] which correspond to projective embeddings of Y . The last section of the paper deals with the problem of when Y is an arithmetically Cohen-Macaulay blow-up ...

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“…Indeed, Proj(gr I (R)) is the exceptional fiber of the blow-up of Spec(R) along the subvariety V (I). Strong efforts have been given in the last thirty years to detect conditions on R and I which guarantee that gr I (R) has sufficiently high depth (more precisely, gr I (R) being Cohen-Macaulay or almost Cohen-Macaulay), due to the reason that high depth of the associated graded ring forces the vanishing of its cohomology groups and thereby allows one to compute, or bound, relevant numerical invariants such as the Castelnuovo-Mumford regularity or the number and degrees of the defining equations of the blow-up (see for instance, [15] and [14]).…”

Section: Introductionmentioning

confidence: 99%

Generalized stretched ideals and Sally's conjecture

Mantero

Xie

2016

Journal of Pure and Applied Algebra

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We introduce the concept of j-stretched ideals in a Noetherian local ring. This notion generalizes to arbitrary ideals the classical notion of stretched m-primary ideals of Sally and Rossi-Valla, as well as the concept of ideals of minimal and almost minimal j-multiplicity introduced by Polini-Xie. One of our main theorems states that, for a j-stretched ideal, the associated graded ring is Cohen-Macaulay if and only if two classical invariants of the ideal, the reduction number and the index of nilpotency, are equal. Our second main theorem, presenting numerical conditions which ensure the almost Cohen-Macaulayness of the associated graded ring of a j-stretched ideal, provides a generalized version of Sally's conjecture. This work, which also holds for modules, unifies the approaches of Rossi-Valla and Polini-Xie and generalizes simultaneously results on the Cohen-Macaulayness or almost Cohen-Macaulayness of the associated graded module by several authors, including

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Castelnuovo Regularity and Graded Rings Associated to an Ideal

Abstract: Abstract. We compare the Castelnuovo regularity defined with respect to different homogeneous ideals in a graded ring and use the result we obtain to prove a generalized Goto-Shimoda theorem for ideals of positive height in a Cohen-Macaulay local ring.

Cited by 17 publications

References 2 publications

Indices of normalization of ideals

Indices of normalization of ideals

Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

Generalized stretched ideals and Sally's conjecture

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