scholarly journals Tight closure of powers of ideals and tight hilbert polynomials

2019 ◽  
Vol 169 (2) ◽  
pp. 335-355
Author(s):  
KRITI GOEL ◽  
J. K. VERMA ◽  
VIVEK MUKUNDAN
Keyword(s):  
Local Ring ◽  
Hilbert Function ◽  
Simplicial Complexes ◽  
Primary Ideal ◽  
Hilbert Polynomial ◽  
Prime Characteristic ◽  
Tight Closure ◽  
Powers Of Ideals ◽  
Hilbert Polynomials

AbstractLet (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.

Reduction numbers and Hilbert polynomials of ideals in higher dimensional CohenMacaulay local rings

1992 ◽  
Vol 111 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Yinghwa Wu
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Primary Ideal ◽  
Local Rings ◽  
Hilbert Polynomials ◽  
Higher Dimensional ◽  
Reduction Number ◽  
Minimal Reduction ◽  
System Of Parameters

Throughout, (R, m) will denote a d-dimensional CohenMacaulay (CM for short) local ring having an infinite residue field and I an m-primary ideal in R. Recall that an ideal J I is said to be a reduction of I if Ir+1 = JIr for some r 0, and a reduction J of I is called a minimal reduction of I if J is generated by a system of parameters. The concepts of reduction and minimal reduction were first introduced by Northcott and Rees12. If J is a reduction of I, define the reduction number of I with respect to J, denoted by rj(I), to be min {r 0 Ir+1 = JIr}. The reduction number of I is defined as r(I) = min {rj(I)J is a minimal reduction of I}. The reduction number r(I) is said to be independent if r(I) = rj(I) for every minimal reduction J of I.

A note on reductions of ideals with an application to the generalized Hilbert function

1954 ◽  
Vol 50 (3) ◽  
pp. 353-359 ◽  
Author(s):  
D. G. Northcott ◽  
D. Rees
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Hilbert Function ◽  
Residue Field ◽  
Primary Ideal ◽  
System Of Parameters ◽  
A Minor ◽  
The Ideal

1. Throughout this note Q will denote a local ring, m will denote its maximal ideal, q will denote a primary ideal belonging to m and k will denote the residue field Q/m. It will not be assumed that k is infinite, but we shall suppose that Q and k both have the same characteristic. Now let υ1, υ2 …,υd be a system of parameters contained in q, so that d = dim Q; then according to the definition given in (2) the ideal (υl υ2,…, υd) is a reduction of q if (υ1 υ2, …, υd) qm = qm+1 for at least one value of m. The use of the concept lies in the fact that such a reduction is, in a certain sense, a very good approximation to q itself; but the notion does, however, suffer from a minor disadvantage in that, if k is finite, q need not have any reductions. In §3 we shall generalize the notion of a reduction in such a way that we overcome this difficulty, and in such a way that the results concerning reductions obtained in (2) acquire some useful extensions.

scholarly journals Hilbert polynomials of the algebras of $SL_ 2$-invariants

2018 ◽  
Vol 10 (2) ◽  
pp. 303-312
Author(s):  
N.B. Ilash
Keyword(s):  
Quadratic Forms ◽  
Hypergeometric Series ◽  
Hilbert Function ◽  
Dimensional Space ◽  
Hilbert Polynomial ◽  
Binary Forms ◽  
Hilbert Polynomials ◽  
Joint Invariants ◽  
Narayana Numbers ◽  
Classical Invariant

We consider one of the fundamental problems of classical invariant theory, the research of Hilbert polynomials for an algebra of invariants of Lie group $SL_2$. Form of the Hilbert polynomials gives us important information about the structure of the algebra. Besides, the coefficients and the degree of the Hilbert polynomial play an important role in algebraic geometry. It is well known that the Hilbert function of the algebra $SL_n$-invariants is quasi-polynomial. The Cayley-Sylvester formula for calculation of values of the Hilbert function for algebra of covariants of binary $d$-form $\mathcal{C}_{d}= \mathbb{C}[V_d\oplus \mathbb{C}^2]_{SL_2}$ (here $V_d$ is the $d+1$-dimensional space of binary forms of degree $d$) was obtained by Sylvester. Then it was generalized to the algebra of joint invariants for $n$ binary forms. But the Cayley-Sylvester formula is not expressed in terms of polynomials.In our article we consider the problem of computing the Hilbert polynomials for the algebras of joint invariants and joint covariants of $n$ linear forms and $n$ quadratic forms. We express the Hilbert polynomials $\mathcal{H} \mathcal{I}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i, \mathcal{H}(\mathcal{C}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i,$ $\mathcal{H}(\mathcal{I}^{(n)}_2,i)=\dim(\mathcal{I}^{(n)}_2)_i, \mathcal{H}(\mathcal{C}^{(n)}_2,i)=\dim(\mathcal{C}^{(n)}_2)_i$ of those algebras in terms of quasi-polynomial. We also present them in the form of Narayana numbers and generalized hypergeometric series.

scholarly journals Hilbert-Samuel function and Grothendieck group

2000 ◽  
Vol 43 (1) ◽  
pp. 73-94
Author(s):  
Koji Nishida
Keyword(s):  
Local Ring ◽  
Grothendieck Group ◽  
Residue Field ◽  
Primary Ideal ◽  
Hilbert Polynomial ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Graded Module ◽  
Primary Ideals

AbstractLet (A, m) be a Noetherian local ring such that the residue field A/m is infinite. Let I be arbitrary ideal in A, and M a finitely generated A-module. We denote by ℓ(I, M) the Krull dimension of the graded module ⊕n≥0InM/mInM over the associated graded ring of I. Notice that ℓ(I, A) is just the analytic spread of I. In this paper, we define, for 0 ≤ i ≤ ℓ = ℓ(I, M), certain elements ei(I, M) in the Grothendieck group K0(A/I) that suitably generalize the notion of the coefficients of Hilbert polynomial for m-primary ideals. In particular, we show that the top term eℓ (I, M), which is denoted by eI(M), enjoys the same properties as the ordinary multiplicity of M with respect to an m-primary ideal.

The Hilbert function of two ideals

1957 ◽  
Vol 53 (3) ◽  
pp. 568-575 ◽  
Author(s):  
P. B. Bhattacharya
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Hilbert Function ◽  
Projective Dimension ◽  
Primary Ideal ◽  
Homogeneous Ideal ◽  
Artin Ring ◽  
Ring Of Polynomials ◽  
1 Samuel

It is well known that Hubert's function of a homogeneous ideal in the ring of polynomials K[x0, …, xm], where K is a field and x0, …, xm are independent indeterminates over K, is, for large values of r, a polynomial in r of degree equal to the projective dimension of (1). Samuel (4) and Northcott (2) have both shown that if the field K is replaced by an Artin ring A, is still a polynomial in r for large values of r. Applying this generalization Samuel (4) has shown that in a local ring Q the length of an ideal qρ, where q is a primary ideal belonging to the maximal ideal m of Q, is, for sufficiently large values of ρ, a polynomial in ρ whose degree is equal to the dimension of Q.

Frobenius closure and prime characteristic singularities

2020 ◽  
Author(s):  
◽  
Kyle Logan Maddox
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Sufficient Condition ◽  
Zariski Topology ◽  
Prime Characteristic ◽  
Joint Work ◽  
University Of Missouri ◽  
Mild Conditions ◽  
Flat Maps ◽  
The University

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] This dissertation outlines several results about prime characteristic singularities for which the nilpotent part under the induced Frobenius action on local cohomology is either finite colength or the entire module, collectively referred to here as nilpotent singularities. First, we establish a sufficient condition for the finiteness of the Frobenius test exponent for a local ring and apply it to conclude that nilpotent singularities have finite Frobenius test exponent. In joint work with Jennifer Kenkel, Thomas Polstra, and Austyn Simpson, we show that under mild conditions nilpotent singularities descend and ascend along faithfully flat maps. Consequently, we then prove that the loci of primes which are weakly F-nilpotent and F-nilpotent are open in the Zariski topology for rings which are either F-finite or essentially of fiiite type over an excellent local ring.

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

2021 ◽  
Vol 28 (01) ◽  
pp. 13-32
Author(s):  
Nguyen Tien Manh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Graded Modules ◽  
Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

scholarly journals On a generalization of test ideals

2004 ◽  
Vol 175 ◽  
pp. 59-74 ◽  
Author(s):  
Nobuo Hara ◽  
Shunsuke Takagi
Keyword(s):  
Prime Characteristic ◽  
Tight Closure ◽  
Test Ideal ◽  
Important Object ◽  
The Ideal

AbstractThe test ideal τ(R) of a ring R of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal τ(at) associated to a given ideal a with rational exponent t ≥ 0. We first prove a key lemma of this paper (Lemma 2.1), which gives a characterization of the ideal τ(at). As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal τ(R). Moreover, we prove an analogue of so-called Skoda’s theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the “modified Briançon-Skoda theorem.”

On local cohomology and Hilbert function of powers of ideals

2003 ◽  
Vol 112 (1) ◽  
pp. 77-92 ◽  
Author(s):  
L� Tu�n Hoa ◽  
Eero Hyry
Keyword(s):  
Hilbert Function ◽  
Local Cohomology ◽  
Powers Of Ideals

Hilbert's function in a semi-lattice

1959 ◽  
Vol 55 (3) ◽  
pp. 239-243
Author(s):  
A. Learner
Keyword(s):  
Local Ring ◽  
Polynomial Ring ◽  
Hilbert Function ◽  
Homogeneous Ideal ◽  
Samuel 1

Samuel (1) introduced a generalized Hilbert function, written Xq(r, a) and defined for arbitrary ideals a in a local ring Q with maximal ideai m. where q is m-primary.Northcott(2) proved that for a homogeneous ideal ã in a polynomial ring A[X1, …, Xn], where A = Q/q, this is equal to the ordinary Hilbert function χ(r, ã).

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