Tight closure of powers of ideals and tight hilbert polynomials

Goel, Kriti; Mukundan, Vivek; Verma, J. K.

doi:10.1017/s0305004119000215

Cited by 13 publications

(10 citation statements)

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“…The purpose of this section is to find the tight Hilbert polynomial in certain diagonal hypersurface rings. As evidenced in [7], this will enable us to detect F -rationality of such rings. x 1 , x 2 , .…”

Section: The Tight Hilbert Polynomial For Diagonal Hypersurfacesmentioning

confidence: 91%

“…The objective of this paper is to determine the tight Hilbert polynomial of ideals generated by homogeneous system of parameters in diagonal hypersurface rings in prime characteristic. The tight Hilbert polynomial was introduced by K. Goel, V. Mukundan and J. K. Verma in [7]. Let (R, m) be a d-dimensional analytically unramified local ring of prime characteristic p > 0 and I be an m-primary ideal.…”

Section: Introductionmentioning

confidence: 99%

“…Recall that R is called an F -rational local ring if the ideals of the principal class, that is, ideals J generated by height J elements, are tightly closed. It is proved [7] that in a Cohen-Macaulay local ring R, e * 1 (I) = 0 if and only if R is F -rational. This indicates that the other tight Hilbert coefficients might play an important role in detecting other F -singularities.…”

Section: Introductionmentioning

confidence: 99%

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Tight closure of powers of parameter ideals in hypersurface rings and their tight Hilbert polynomials

Dubey¹,

Mukundan²,

Verma³

2021

Preprint

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In this paper we find the tight closure of powers of parameter ideals of certain diagonal hypersurface rings. In many cases the associated graded ring with respect to tight closure filtration turns out to be Cohen-Macaulay. This helps us find the tight Hilbert polynomial in these diagonal hypersurfaces. We determine the tight Hilbert polynomial in the following cases: (1) F -pure diagonal hypersurfaces where number of variables is equal to the degree of defining equation, (2) diagonal hypersurface rings where characteristic of the ring is one less than the degree of defining equation and (3) quartic diagonal hypersurface in four variables.

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Section: The Tight Hilbert Polynomial For Diagonal Hypersurfacesmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Tight closure of powers of parameter ideals in hypersurface rings and their tight Hilbert polynomials

Dubey¹,

Mukundan²,

Verma³

2021

Preprint

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“…If R is an analytically unramified local ring and I is an ideal of R, then Rees [19] proved that the integral closure filtration F = {I n } is an I-good filtration. Using [5], under the same conditions, the tight closure filtration T = {(I n ) * } is an I-good filtration.…”

Section: Multi-graded Filtrations Of Idealsmentioning

confidence: 99%

Eakin-Sathaye-Type Theorems for Joint Reductions and Good Filtrations of Ideals

2019

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Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for N sgraded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for N-graded filtrations. Several examples related to lex-segment ideals, contracted ideals in 2-dimensional regular local rings and the filtration of integral and tight closures of powers of ideals in hypersurface rings are constructed to show effectiveness of these bounds. Dedicated to Le Tuan Hoa on the occasion of his sixtieth birthday 1. Introduction The objective of this paper is to prove Eakin-Sathaye type theorems [4] for joint reductions and good filtrations of ideals. Recall that an ideal J contained in an ideal I in a commutative ring R is called a reduction of I if there is a non-negative integer n such that JI n = I n+1 . The concept of reduction of an ideal was introduced by Northcott and Rees [15]. It has become an important tool in many investigations in commutative algebra and algebraic geometry such as Hilbert-Samuel functions [18], blow-up algebras [6] singularities of hypersurfaces [23], number of defining equations of algebraic varieties [13] and many others. Research in this paper is motivated by the following result of Paul Eakin and Avinash Sathaye [4]. Let µ(I) denote the minimum number of generators of an ideal I in a local ring. Theorem 1.1. Let I an ideal of a local ring R with infinite residue field. If µ(I n ) < n+r r for some positive integers n and r then there is a reduction J of I generated by r elements such that JI n−1 = I n . The case of n = 2, r = 1 was proved by J. D. Sally [22]. The EST (Eakin-Sathaye Theorem) has been revisited by G. Caviglia [2], N. V. Trung [24] and Liam O'Carroll [16]. Caviglia used Green's hyperplane section theorem to give a new proof of the EST. The EST was generalised by Liam O'Carroll for complete reductions [16]. The versions of the EST proved in this paper follow the approach used by O'Carroll. In order to state his result we recall necessary definitions Date: June 21, 2018. Both of the first author and the second author are supported by UGC fellowship, Govt. of India.

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“…The other coefficients e * i (I) ∈ Z are called the tight Hilbert coefficients of I. The tight Hilbert polynomial was introduced in [4] where it was proved that an analytically unramified Cohen-Macaulay local ring R having prime characteristic is F-rational if and only if e * 1 (Q) = 0 for some ideal Q generated by a system of parameters of R. This paper is motivated by the following question of Craig Huneke Question 1.2. Is it true that an unmixed Noetherian local ring R is F-rational if and only if for some ideal Q of R generated by a system of parameters, e * 1 (Q) = 0?…”

Section: Introductionmentioning

confidence: 99%

Tight Hilbert Polynomial and F-rational local rings

Dubey¹,

Quy²,

Verma³

2021

Preprint

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Let (R, m) be a Noetherian local ring of prime characteristic p and Q be an m-primary parameter ideal. We give criteria for F-rationality of R using the tight Hilbert function. We obtain a lower bound for the tight Hilbert function of Q for equidimensional excellent local rings that generalises a result of Goto and Nakamura.We show that if dim R = 2, the Hochster-Huneke graph of R is connected and this lower bound is achieved then R is F-rational. Craig Huneke asked if the F -rationality of unmixed local rings may be characterized by the vanishing of e * 1 (Q). We construct examples to show that without additional conditions, this is not possible. Let R be an excellent, reduced, equidimensional Noetherian local ring and Q be generated by parameter test elements. We find formulas for e * 1 (Q), e * 2 (Q), . . . , e * d (Q) in terms of Hilbert coefficients of Q, lengths of local cohomology modules of R, and the length of the tight closure of the zero submodule of H d m (R). Using these we prove: R is F-rational ⇐⇒ e * 1 (Q) = e 1 (Q) ⇐⇒ depth R ≥ 2 and e * 1 (Q) = 0.

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Tight closure of powers of ideals and tight hilbert polynomials

Cited by 13 publications

References 13 publications

Tight closure of powers of parameter ideals in hypersurface rings and their tight Hilbert polynomials

Tight closure of powers of parameter ideals in hypersurface rings and their tight Hilbert polynomials

Eakin-Sathaye-Type Theorems for Joint Reductions and Good Filtrations of Ideals

Tight Hilbert Polynomial and F-rational local rings

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