scholarly journals RELATIVE HILBERT CO-EFFICIENTS

2017 ◽  
Vol 59 (3) ◽  
pp. 729-741
Author(s):  
AMIR MAFI ◽  
TONY J. PUTHENPURAKAL ◽  
RAKESH B. T. REDDY ◽  
HERO SAREMI
Keyword(s):  
Local Ring ◽  
Hilbert Coefficient ◽  
Primary Ideals

AbstractLet (A,${\mathfrak{m}$) be a Cohen–Macaulay local ring of dimensiondand letI⊆Jbe two${\mathfrak{m}$-primary ideals withIa reduction ofJ. Fori= 0,. . .,d, leteiJ(A) (eiI(A)) be theith Hilbert coefficient ofJ(I), respectively. We call the numberci(I,J) =eiJ(A) −eiI(A) theith relative Hilbert coefficient ofJwith respect toI. IfGI(A) is Cohen–Macaulay, thenci(I,J) satisfy various constraints. We also show that vanishing of someci(I,J) has strong implications on depthGJn(A) forn≫ 0.

scholarly journals Notes on irreducible ideals

1985 ◽  
Vol 31 (3) ◽  
pp. 321-324
Author(s):  
David J. Smith
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Primary Ideal ◽  
Regular Local Ring ◽  
Finite Intersection ◽  
Irreducible Components ◽  
Primary Ideals

Every ideal of a Noetherian ring may be represented as a finite intersection of primary ideals. Each primary ideal may be decomposed as an irredundant intersection of irreducible ideals. It is shown that in the case that Q is an M-primary ideal of a local ring (R, M) satisfying the condition that Q: M = Q + Ms−1 where s is the index of Q, then all irreducible components of Q have index s. (Q is “index-unmixed”.) This condition is shown to hold in the case that Q is a power of the maximal ideal of a regular local ring, and also in other cases as illustrated by examples.

scholarly journals Equimultiple coefficient ideals

2017 ◽  
Vol 121 (1) ◽  
pp. 5 ◽  
Author(s):  
P. H. Lima ◽  
V. H. Jorge Pérez
Keyword(s):  
Local Ring ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Analytic Spread ◽  
Minimal Prime ◽  
Primary Ideals

Let $(R,\mathfrak {m})$ be a quasi-unmixed local ring and $I$ an equimultiple ideal of $R$ of analytic spread $s$. In this paper, we introduce the equimultiple coefficient ideals. Fix $k\in \{1,\dots ,s\}$. The largest ideal $L$ containing $I$ such that $e_{i}(I_{\mathfrak{p} })=e_{i}(L_{\mathfrak{p} })$ for each $i \in \{1,\dots ,k\}$ and each minimal prime $\mathfrak{p} $ of $I$ is called the $k$-th equimultiple coefficient ideal denoted by $I_{k}$. It is a generalization of the coefficient ideals introduced by Shah for the case of $\mathfrak {m}$-primary ideals. We also see applications of these ideals. For instance, we show that the associated graded ring $G_{I}(R)$ satisfies the $S_{1}$ condition if and only if $I^{n}=(I^{n})_{1}$ for all $n$.

On coefficient ideals

2018 ◽  
Vol 167 (02) ◽  
pp. 285-294 ◽  
Author(s):  
R. CALLEJAS-BEDREGAL ◽  
V. H. JORGE PÉREZ ◽  
M. DUARTE FERRARI
Keyword(s):  
Local Ring ◽  
Structure Theorem ◽  
Noetherian Local Ring ◽  
Analytic Spread ◽  
A Chain ◽  
Primary Ideals

AbstractLet (R, 𝔪) be a Noetherian local ring and I an arbitrary ideal of R with analytic spread s. In [3] the authors proved the existence of a chain of ideals I ⊆ I[s] ⊆ ⋅⋅⋅ ⊆ I[1] such that deg(PI[k]/I) < s − k. In this article we obtain a structure theorem for this ideals which is similar to that of K. Shah in [10] for 𝔪-primary ideals.

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.

scholarly journals Upper bounds of Hilbert coefficients and Hilbert functions

2008 ◽  
Vol 145 (1) ◽  
pp. 87-94 ◽  
Author(s):  
JUAN ELIAS
Keyword(s):  
Finite Number ◽  
Local Ring ◽  
Maximal Ideal ◽  
Upper Bound ◽  
Upper Bounds ◽  
Primary Ideal ◽  
Hilbert Functions ◽  
Hilbert Coefficients ◽  
Hilbert Coefficient

AbstractLet (R, m) be a d-dimensional Cohen–Macaulay local ring. In this paper we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a m-primary ideal I ⊂ R that improves all known upper bounds unless for a finite number of cases, see Remark 2.3. We also provide new upper bounds of the Hilbert functions of I extending the known bounds for the maximal ideal.

On powers of ideals generated by R-sequences in a Noetherian local ring

1973 ◽  
Vol 74 (3) ◽  
pp. 441-444 ◽  
Author(s):  
A. Caruth
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Primary Decomposition ◽  
Finite Intersection ◽  
Noetherian Local Ring ◽  
Powers Of Ideals ◽  
Primary Ideals ◽  
Unique Maximal Ideal

In a Noetherian commutative ring with identity, every ideal can be expressed (not necessarily uniquely) as a finite intersection of primary ideals (called a primary decomposition). This note is concerned with powers of ideals generated by subsets of an R-sequence in a local ring R (i.e. a Noetherian commutative ring R with identity possessing a unique maximal ideal m) and with a decomposition of such ideals.

The second Hilbert coefficient of modules with almost maximal depth

2019 ◽  
Vol 18 (12) ◽  
pp. 1950240
Author(s):  
Van Duc Trung
Keyword(s):  
Lower Bound ◽  
Local Ring ◽  
Upper Bound ◽  
Primary Ideal ◽  
Finitely Generated ◽  
Dimension Formula ◽  
Maximal Depth ◽  
Graded Module ◽  
Hilbert Coefficient

Let [Formula: see text] be a good [Formula: see text]-filtration of a finitely generated [Formula: see text]-module [Formula: see text] of dimension [Formula: see text], where [Formula: see text] is a local ring and [Formula: see text] is an [Formula: see text]-primary ideal of [Formula: see text]. In the case of depth [Formula: see text], we give an upper bound for the second Hilbert coefficient [Formula: see text] generalizing the results by Huckaba–Marley, and Rossi–Valla proved that [Formula: see text] is Cohen–Macaulay. We also give a condition for the equality, which relates to the depth of the associated graded module [Formula: see text]. A lower bound on [Formula: see text] is proved generalizing a result by Rees and Narita.

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 On reduction exponents of ideals with Gorenstein formring

1995 ◽  
Vol 38 (3) ◽  
pp. 449-463 ◽  
Author(s):  
M. Herrmann ◽  
C. Huneke ◽  
J. Ribbe
Keyword(s):  
Local Ring ◽  
General Point ◽  
Regular Local Ring ◽  
Rees Algebra ◽  
Reduction Number ◽  
Primary Ideals ◽  
The Ideal

This paper studies questions connected with when the Rees algebra of an ideal or the formring of an ideal is Gorenstein. The main results are for ideals of small analytic deviation, and for m-primary ideals of a regular local ring (R, m). The general point proved is that the Gorenstein property forces (and is sometimes equivalent to) lowering the reduction number of the ideal by one from the value predicted if one only assumes the Rees algebra or formring is Cohen–Macaulay.

On Generalized Regular Local Ring

2015 ◽  
Vol 3 (1) ◽  
pp. 145-152
Author(s):  
Zubayda Ibraheem ◽  
Naeema Shereef
Keyword(s):  
Local Ring ◽  
Regular Local Ring
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