Note on analytic spread and asymptotic sequences

1983 ◽  
Vol 93 (1) ◽  
pp. 49-55 ◽  
Author(s):  
L. J. Ratliff
Keyword(s):  
Local Ring ◽  
Prime Divisor ◽  
New Approaches ◽  
Analytic Spread ◽  
Minimal Prime ◽  
Equality Holding

Since the foundational paper (10) by Northcott and Rees in 1954 there have been quite a few papers concerning reductions of ideals and the analytic spread of an ideal. One particular line of investigation concerning the analytic spread l(I) of an ideal I in a local ring (R, M) was begun in 1972 by Burch in (5), where it was shown that l(I) ≤ altitude R – min (grade R/In; n ≥ 1). This result was sharpened in 1980–81 by Brodmann in three papers, (2, 3, 4). Therein he showed that the sets {grade R/In; n ≥ 1} and {grade In−1/In; n ≥ 1} stabilize for all large n, and calling the stable values t and t*, respectively, it holds that t ≤ t* and l(I) ≤ altitude R – t* when I is not nilpotent. He then gave a case (involving R being quasi-unmixed) when equality holds. In 1981 in (20) Rees used two new approaches to Burch's inequality, and he proved two nice results which may both be stated as: l(I) ≤ altitude R – s(I) with equality holding when R is quasi-unmixed; here, s(I) = min {height P; P is a minimal prime divisor of (M, u) R[tI, u]}– 1 (in the first theorem), and s(I) is the length of a maximal asymptotic sequence over I (in the second theorem).

Form rings and projective equivalence

1986 ◽  
Vol 99 (3) ◽  
pp. 447-456 ◽  
Author(s):  
Daniel Katz ◽  
L. J. Ratliff
Keyword(s):  
Local Ring ◽  
Prime Divisor ◽  
Noetherian Ring ◽  
Noetherian Rings ◽  
Local Rings ◽  
Prime Divisors ◽  
Research Problems ◽  
Form Ring ◽  
Divisor Of Zero ◽  
Minimal Prime

If I and J are ideals in a Noetherian ring R, then I and J are projectively equivalent in case (Ii)a = (Jj)a for some positive integers i, j (where Ka denotes the integral closure in R of the ideal K) and the form ring F(R, I) of R with respect to I is the graded ring R/I ⊕ I/I2 ⊕ I2/I3 ⊕ …. These two concepts have played an important role in many research problems in commutative algebra, so they have been deeply studied and many of their properties have been discovered. In a recent paper [13] they were combined to show that a semi-local ring R is unmixed if and only if for every ideal J in R there exists a projectively equivalent ideal J in R such that every prime divisor of zero in F(R, J) has the same depth. It seems to us that results similar to this are interesting and potentially quite useful, so in this paper we prove several additional such theorems. Namely, it is shown that all ideals in all local rings have a projectively equivalent ideal whose form ring is fairly nice. Also, a characterization similar to the just mentioned result in [13] is given for the class of local rings whose completions have no embedded prime divisors of zero, and several analogous new characterizations are given for locally unmixed Noetherian rings. In particular, it is shown that if I is an ideal in an unmixed local ring R such that height(I) = l(I) (where l(I) denotes the analytic spread of I), then there exists a projectively equivalent ideal J in R such that Ass (F(R, J)) has exactly m elements, all minimal, where m is the number of minimal prime divisors of I (so if I is open, then F(R, J) has exactly one prime divisor of zero and is a locally unmixed Noetherian ring).

scholarly journals Some Studies on Semi-local Rings

1951 ◽  
Vol 3 ◽  
pp. 23-30 ◽  
Author(s):  
Masayoshi Nagata
Keyword(s):  
Local Ring ◽  
Prime Divisor ◽  
Chain Condition ◽  
Local Rings ◽  
Proposition 8 ◽  
Minimal Prime

The concept of semi-local rings was introduced by C. Chevalley [1], which the writer has generalized in a recent paper [7] by removing the chain condition. The present paper aims mainly at the study of completions of semi-local rings. First in § 1 we investigate semi-local rings which are subdirect sums of semi-local rings, and we see in § 2 that a Noetherian semi-local ringRis complete if (and only if)R/pis complete for every minimal prime divisorpof zero ideal, together with some other properties. Further we consider in § 3 subrings of the completion of a semi-local ring. § 4 gives some supplementary remarks to [7], Chapter II, Proposition 8.

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 equimultiplicity

1982 ◽  
Vol 91 (2) ◽  
pp. 207-213 ◽  
Author(s):  
M. Herrmann ◽  
U. Orbanz
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Analytic Spread

This note consists of some investigations about the condition ht(A) = l(A) where A is an ideal in a local ring and l(A) is the analytic spread of A (9).In (4) we proved the following: If R is a local ring and P a prime ideal such that R/P is regular then (under some technical assumptions) ht(P) = l(P) is equivalent to the equimultiplicity e(R) = e(RP). Also for a general ideal A (which need not be prime), the condition ht(A) = l(A) can be translated into an equality of certain multiplicities (see Theorem 0).

scholarly journals Monochromatic and Zero-Sum Sets of Nondecreasing Modified Diameter

10.37236/1054 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
David Grynkiewicz ◽  
Rasheed Sabar
Keyword(s):  
Positive Integer ◽  
Cyclic Group ◽  
Prime Divisor ◽  
Zero Sum ◽  
Equality Holding

Let $m$ be a positive integer whose smallest prime divisor is denoted by $p$, and let ${\Bbb Z}_m$ denote the cyclic group of residues modulo $m$. For a set $B=\{x_1,x_2,\ldots,x_m\}$ of $m$ integers satisfying $x_1 < x_2 < \cdots < x_m$, and an integer $j$ satisfying $2\leq j \leq m$, define $g_j(B)=x_j-x_1$. Furthermore, define $f_j(m,2)$ (define $f_j(m,{\Bbb Z}_m)$) to be the least integer $N$ such that for every coloring $\Delta : \{1,\ldots,N\}\rightarrow \{0,1\}$ (every coloring $\Delta : \{1,\ldots,N\}\rightarrow {\Bbb Z}_m$), there exist two $m$-sets $B_1,B_2\subset \{1,\ldots,N\}$ satisfying: (i) $\max(B_1) < \min(B_2)$, (ii) $g_j(B_1)\leq g_j(B_2)$, and (iii) $|\Delta (B_i)|=1$ for $i=1,2$ (and (iii) $\sum_{x\in B_i}\Delta (x)=0$ for $i=1,2$). We prove that $f_j(m,2)\leq 5m-3$ for all $j$, with equality holding for $j=m$, and that $f_j(m,{\Bbb Z}_m)\leq 8m+{m\over p}-6$. Moreover, we show that $f_j(m,2)\ge 4m-2+(j-1)k$, where $k=\left\lfloor\left(-1+\sqrt{{8m-9+j\over j-1}}\right){/2}\right\rfloor$, and, if $m$ is prime or $j\geq{m\over p}+p-1$, that $f_j(m,{\Bbb Z}_m)\leq 6m-4$. We conclude by showing $f_{m-1}(m,2)=f_{m-1}(m,{\Bbb Z}_m)$ for $m\geq 9$.

scholarly journals The asymptotic nature of the analytic spread

1979 ◽  
Vol 86 (1) ◽  
pp. 35-39 ◽  
Author(s):  
M. Brodmann
Keyword(s):  
Local Ring ◽  
Result Section ◽  
Easy Consequence ◽  
Noetherian Local Ring ◽  
Section 8 ◽  
Asymptotic Nature ◽  
Analytic Spread ◽  
Image Position ◽  
Almost All ◽  
Local Nature

In (3), corollary, p. 373) Burch gives the following inequality for the analytic spread l(I) of an ideal I of a noetherian local ring (R, m):In this paper we shall improve this by showing that the number min depth (R/In) may be replaced by the asymptotic value of depth (R/In) for large n (which exists) (see Section (2)). By its definition (see (6), def. 3)) the analytic spread is of asymptotic nature, i.e. depends on the modules In/mIn = Un only for large n. We shall prove a stronger result, Section (4), which also shows the asymptotic nature of l(I). This result might be interesting for itself, particularly as it is not of local nature. Once Section (4) is proved and once we know that depth (R/In) is asymptotically constant (which turns out to be an easy consequence of (1), (1)), our improved inequality is easily established: Indeed, replacing R by R/xR where x is regular with respect to almost all modules (R/In), we perform a change which affects only finitely many of the modules Un (see Section (8)).

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) &lt; 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 On the vanishing and the positivity of intersection multiplicities over local rings with small non complete intersection loci

1994 ◽  
Vol 136 ◽  
pp. 133-155 ◽  
Author(s):  
Kazuhiko Kurano
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Minimal Prime Ideal ◽  
Local Rings ◽  
Regular Local Ring ◽  
Commutative Noetherian Ring ◽  
Intersection Multiplicities ◽  
Minimal Prime

Throughout this paperAis a commutative Noetherian ring of dimensiondwith the maximal ideal m and we assume that there exists a regular local ringSsuch thatAis a homomorphic image ofS, i.e.,A=S/Ifor some idealIofS. Furthermore we assume thatAis equi-dimensional, i.e., dimA= dimA/for any minimal prime idealofA. We put.

On a question of D. Rees on classical integral closure and integral closure relative to an Artinian module

2019 ◽  
Vol 19 (02) ◽  
pp. 2050033
Author(s):  
V. H. Jorge Pérez ◽  
L. C. Merighe
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Local Cohomology ◽  
Integral Closure ◽  
Minimal Prime Ideal ◽  
Local Cohomology Module ◽  
Artinian Module ◽  
Classical Integral ◽  
Minimal Prime ◽  
The Relationship

Let [Formula: see text] be a commutative Noetherian complete local ring and [Formula: see text] and [Formula: see text] ideals of [Formula: see text]. Motivated by a question of Rees, we study the relationship between [Formula: see text], the classical Northcott–Rees integral closure of [Formula: see text], and [Formula: see text], the integral closure of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text] (also called here ST-closure of [Formula: see text] on [Formula: see text]), in order to study a relation between [Formula: see text], the multiplicity of [Formula: see text], and [Formula: see text], the multiplicity of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text]. We conclude [Formula: see text] when every minimal prime ideal of [Formula: see text] belongs to the set of attached primes of [Formula: see text]. As an application, we show what happens when [Formula: see text] is a generalized local cohomology module.

Codimension and analytic spread

1972 ◽  
Vol 72 (3) ◽  
pp. 369-373 ◽  
Author(s):  
Lindsay Burch
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Upper Bound ◽  
Proper Ideal ◽  
Analytic Spread

In this paper, I shall establish the sufficiency of certain conditions on an ideal A of a local ring Q, and on a set {g1 …,gk} of elements of Q generating a proper ideal G, for the ideals A and G to be analytically disjoint. Hence I shall establish an upper bound for the analytic spread of A.The maximal ideal of Q will be denoted throughout by M, and it will be assumed that the field Q/M is infinite.

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