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 On Fiber Cones of 𝑚-Primary Ideals

2007 ◽  
Vol 59 (1) ◽  
pp. 109-126 ◽  
Author(s):  
A. V. Jayanthan ◽  
Tony J. Puthenpurakal ◽  
J. K. Verma
Keyword(s):  
Local Ring ◽  
Primary Ideal ◽  
Reduction Number ◽  
Fiber Cone ◽  
Minimal Reduction ◽  
Primary Ideals

AbstractTwo formulas for the multiplicity of the fiber cone of an 𝑚-primary ideal of a d-dimensional Cohen–Macaulay local ring (R, 𝑚) are derived in terms of the mixed multiplicity ed–1(𝑚|I), the multiplicity e(I), and superficial elements. As a consequence, the Cohen–Macaulay property of F(I) when I has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of I and lengths of certain ideals. We also characterize the Cohen–Macaulay and Gorenstein properties of fiber cones of 𝑚–primary ideals with a d–generated minimal reduction J satisfying ℓ(I2/JI) = 1 or ℓ(I𝑚/J𝑚) = 1.

scholarly journals VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY

2018 ◽  
Vol 61 (03) ◽  
pp. 705-725
Author(s):  
DIPANKAR GHOSH ◽  
TONY J. PUTHENPURAKAL
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Regular Sequence ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Syzygy Modules ◽  
Tor Modules ◽  
Necessary And Sufficient ◽  
Minimal Multiplicity

AbstractLet R be a d-dimensional Cohen–Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f), where f := f1,. . .,fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if ExtSi(M, N) = 0 for some (d + c + 1) consecutive values of i ⩾ 2, then ExtSi(M, N) = 0 for all i ⩾ 1. Moreover, if this holds true, then either projdimR(M) or injdimR(N) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.

scholarly journals Linearity defect of the residue field of short local rings

2016 ◽  
Vol 16 (09) ◽  
pp. 1750163
Author(s):  
Rasoul Ahangari Maleki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Residue Field ◽  
Positive Answer ◽  
Local Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Linear Resolution

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text] and residue field [Formula: see text]. The linearity defect of a finitely generated [Formula: see text]-module [Formula: see text], which is denoted [Formula: see text], is a numerical measure of how far [Formula: see text] is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether [Formula: see text] implies [Formula: see text], in the case when [Formula: see text].

scholarly journals CONTRACTED, -FULL AND RELATED CLASSES OF IDEALS IN LOCAL RINGS

2013 ◽  
Vol 55 (3) ◽  
pp. 669-675
Author(s):  
DAVID E. RUSH
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Local Rings

AbstractThe class of $\mathfrak{m}$-full and four related classes of ideals in a local ring (R, $\mathfrak{m}$) are extended by replacing $\mathfrak{m}$ with other ideals and the resulting classes of ideals are compared. It is shown that contracted ideals are $\mathfrak{m}$-full in a local ring with infinite residue field.

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.

On Special Fiber Rings of Modules

2019 ◽  
Vol 72 (1) ◽  
pp. 225-242
Author(s):  
Cleto B. Miranda-Neto
Keyword(s):  
Local Ring ◽  
Arbitrary Dimension ◽  
Residue Field ◽  
Independent Interest ◽  
General Situation ◽  
Finitely Generated ◽  
Fiber Ring ◽  
Noetherian Local Ring ◽  
Reduction Number

AbstractWe prove results concerning the multiplicity as well as the Cohen–Macaulay and Gorenstein properties of the special fiber ring $\mathscr{F}(E)$ of a finitely generated $R$-module $E\subsetneq R^{e}$ over a Noetherian local ring $R$ with infinite residue field. Assuming that $R$ is Cohen–Macaulay of dimension 1 and that $E$ has finite colength in $R^{e}$, our main result establishes an asymptotic length formula for the multiplicity of $\mathscr{F}(E)$, which, in addition to being of independent interest, allows us to derive a Cohen–Macaulayness criterion and to detect a curious relation to the Buchsbaum–Rim multiplicity of $E$ in this setting. Further, we provide a Gorensteinness characterization for $\mathscr{F}(E)$ in the more general situation where $R$ is Cohen–Macaulay of arbitrary dimension and $E$ is not necessarily of finite colength, and we notice a constraint in terms of the second analytic deviation of the module $E$ if its reduction number is at least three.

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.

Reductions of ideals in local rings

1954 ◽  
Vol 50 (2) ◽  
pp. 145-158 ◽  
Author(s):  
D. G. Northcott ◽  
D. Rees
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Reduction Process ◽  
Local Rings ◽  
Central Concept ◽  
Separate Paper ◽  
Associative Law ◽  
System Of Parameters ◽  
Algebraic Multiplicities ◽  
Special Case

This paper contains some contributions to the analytic theory of ideals. The central concept is that of a reduction which is defined as follows: if and are ideals and ⊆ , then is called a reduction of if n = n+1 for all large values of n. The usefulness of the concept depends mainly on two facts. First, it defines a relationship between two ideals which is preserved under homomorphisms and ring extensions; secondly, what we may term the reduction process gets rid of superfluous elements of an ideal without disturbing the algebraic multiplicities associated with it. For example, the process when applied to a primary ideal belonging to the maximal ideal of a local ring gives rise to a system of parameters having the same multiplicity; but the methods work almost equally well for an arbitrary ideal and bring to light some interesting facts which are rather obscured in the special case. The concept seems to be suitable for a variety of applications. The present paper contains one instance which is a generalized form of the associative law for multiplicities (see § 8), and the authors hope to give other illustrations in a separate paper.

scholarly journals Homological Invariants of Local Rings

1963 ◽  
Vol 22 ◽  
pp. 219-227 ◽  
Author(s):  
Hiroshi Uehara
Keyword(s):  
Local Ring ◽  
Betti Numbers ◽  
Residue Field ◽  
Unit Element ◽  
Minimal System ◽  
Local Rings ◽  
Homology Algebra ◽  
Homological Invariants ◽  
Minimal System Of Generators ◽  
The Relationship

In this paper R is a commutative noetherian local ring with unit element 1 and M is its maximal ideal. Let K be the residue field R/M and let {t1,t2,…, tn) be a minimal system of generators for M. By a complex R<T1. . ., Tp> we mean an R-algebra* obtained by the adjunction of the variables T1. . ., Tp of degree 1 which kill t1,…, tp. The main purpose of this paper is, among other things, to construct an R-algebra resolution of the field K, so that we can investigate the relationship between the homology algebra H (R < T1,…, Tn>) and the homological invariants of R such as the algebra TorR(K, K) and the Betti numbers Bp = dimk TorR(K, K) of the local ring R. The relationship was initially studied by Serre [5].

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