Ratliff–Rush filtration, regularity and depth of higher associated graded modules: Part I

Puthenpurakal, Tj

doi:10.1016/j.jpaa.2005.12.004

Cited by 33 publications

(6 citation statements)

References 23 publications

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“…It turns out that we need some extra assumptions in order to get a complete description of this case. Nevertheless, the theorem extends results of Elias-Valla (see [20]), Guerrieri-Rossi (see [36]), Itoh (see [55]), Sally (see [88]) and Puthenpurakal (see [68]).…”

Section: Already Introduced In (216)supporting

confidence: 82%

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Hilbert Functions of Filtered Modules

Rossi,

Valla

2007

Preprint

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Section: Already Introduced In (216)supporting

confidence: 82%

“…Notice that inequality 1. was already proved by Itoh in [56,12]. Assertion 2. of Corollary 3.2.4 was proved by Puthenpurakal in [68].…”

Section: Bounds For E 2 (M)mentioning

confidence: 84%

Hilbert Functions of Filtered Modules

Rossi,

Valla

2007

Preprint

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“…Filter-regular elements in the case when M is finitely generated have a lot of applications in the study of blow-up algebras [26]. An early version of quasi-finite modules is in [18,Proposition 4•7]. However it was somewhat unexpected to us that the existence of filter-regular elements for quasi-finite modules have implications in question ( * ) above.…”

Section: Introductionmentioning

confidence: 99%

Hilbert polynomials and powers of ideals

Herzog¹,

Puthenpurakal²,

Verma³

2008

Math. Proc. Camb. Phil. Soc.

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Abstract. The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal I in the polynomial ring S = K[x1, . . . , xn] and a finitely generated graded S-module, the Hilbert coefficients ei(M/I k M ) are polynomial functions. Given two families of graded ideals (I k ) k≥0 and (J k ) k≥0 with J k ⊂ I k for all k with the property that J k J ℓ ⊂ J k+ℓ and I k I ℓ ⊂ I k+ℓ for all k and ℓ, and such that the algebras A = L k≥0 J k and B = L k≥0 I k are finitely generated, we show the function k →0 (I k /J k ) is of quasipolynomial type, say given by the polynomials P0, . . . , Pg−1.we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that lim k→∞ ℓ(Γm (S/I k ))/k n ∈ Q. We also study analogous statements in the local case.

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“…It is of some interest to find the degrees of the polynomials ε i M,I (x) and t A I,i (M, x). For instance in [16,Theorem 18] it was proved that if M is a maximal Cohen-Macaulay A-module and I = m then deg t A m,1 (M, x) < d − 1 if and only if M is free. In [10,Theorem I] this result was generalized to arbitrary finitely generated modules with projective dimension at least 1 over Cohen-Macaulay local rings.…”

Section: Introductionmentioning

confidence: 99%

Bass and Betti Numbers of $A/I^n.$

Kadu¹,

Puthenpurakal²

2019

Preprint

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Let (A, m, k) be a Gorenstein local ring of dimension d ≥ 1. Let I be an ideal of A with ht(I) ≥ d − 1. We prove that the numerical functionis given by a polynomial of degree d − 1 in the case when i ≥ d + 1 and curv(I n ) > 1 for all n ≥ 1. We prove a similar result for the numerical functionunder the assumption that A is a Cohen-Macaulay local ring. We note that there are many examples of ideals satisfying the condition curv(I n ) > 1, for all n ≥ 1. We also consider more general functions n → ℓ(Tor A i (M, A/In) for a filtration {In} of ideals in A. We prove similar results in the case when M is a maximal Cohen-Macaulay A-module and {In = I n } is the integral closure filtration, I an m-primary ideal in A.

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Ratliff–Rush filtration, regularity and depth of higher associated graded modules: Part I

Cited by 33 publications

References 23 publications

Hilbert Functions of Filtered Modules

Hilbert Functions of Filtered Modules

Hilbert polynomials and powers of ideals

Bass and Betti Numbers of $A/I^n.$

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