On the dimension of the non-Cohen–Macaulay locus of local rings admitting dualizing complexes

1991 ◽  
Vol 109 (3) ◽  
pp. 479-488 ◽  
Author(s):  
Nguyen Tu Cuong
Keyword(s):  
Local Ring ◽  
Prime Ideals ◽  
Local Rings ◽  
Large Power ◽  
Systems Of Parameters ◽  
The Difference ◽  
Image Position ◽  
Minimal Prime ◽  
Positive Integers ◽  
Dualizing Complexes

In this paper we mainly consider local rings admitting dualizing complexes. It is well-known that if a Noetherian local ring A admits a dualizing complex, then the non-Cohen–Macaulay (abbreviated CM) locus of A is closed in the Zariski topology (cf. [8, 10]). If the dimension of this locus is zero and A is equidimensional, i.e. the punctured spectrum of A is locally CM and dim(A/P) = dim (A) for all minimal prime ideals P ∈ Ass (A), then A is a generalized CM ring and its structure is well-understood (see [2, 12]). For instance, one of the characterizations of generalized CM rings is the conditions that for any parameter ideal q contained in a large power of the maximal ideal m of A, the difference between length and multiplicityis independent of the choice of q. However, if the dimension of the non-CM locus is larger than zero, little is known about how this dimension is related to the structure of the local ring A. The purpose of this paper is to show that if M is a finitely generated A-module, then there exist systems of parameters x = (x1, …, xd) (where d = dim M) such that the differenceis a polynomial in n1, …, nd for all positive integers n1, …, nd and the degree of IM(n1, …, nd;x) is independent of the choice of x. We shall also give various characterizations of this degree by using the notion of reducing systems of parameters of Auslander and Buchsbaum[l]. In particular, if the module M is equidimensional we shall show that the degree of IM(n1, …, nd;x) is equal to the dimension of the non-CM locus of M.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

The Poincaré Series Of Stretched Cohen-Macaulay Rings

1980 ◽  
Vol 32 (5) ◽  
pp. 1261-1265 ◽  
Author(s):  
Judith D. Sally
Keyword(s):  
Local Ring ◽  
Poincaré Series ◽  
Local Rings ◽  
Embedding Dimension ◽  
Loop Spaces ◽  
Gorenstein Rings ◽  
Poincare Series ◽  
Image Position

There are relatively few classes of local rings (R, m) for which the question of the rationality of the Poincaré serieswhere k = R/m, has been settled. (For an example of a local ring with non-rational Poincaré series see the recent paper by D. Anick, “Construction of loop spaces and local rings whose Poincaré—Betti series are nonrational”, C. R. Acad. Sc. Paris 290 (1980), 729-732.) In this note, we compute the Poincaré series of a certain family of local Cohen-Macaulay rings and obtain, as a corollary, the rationality of the Poincaré series of d-dimensional local Gorenstein rings (R, m) of embedding dimension at least e + d – 3, where e is the multiplicity of R. It follows that local Gorenstein rings of multiplicity at most five have rational Poincaré series.

scholarly journals On the least degree of polynomials bounding above the differences between lengths and multiplicities of certain systems of parameters in local rings

1992 ◽  
Vol 125 ◽  
pp. 105-114 ◽  
Author(s):  
Nguyen Tu Cuong
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Local Rings ◽  
Finitely Generated ◽  
Parameter Ideal ◽  
Systems Of Parameters ◽  
The Difference

Let A be a commutative local Noetherian ring with the maximal ideal m and M a finitely generated A-module, d = dim M. It is well-known that the difference between the length and the multiplicity of a parameter ideal q of Mgives a lot of informations on the structure of the module M. For instance, M is a Cohen-Macaulay (CM for short) module if and only if IM(q) = 0 for some parameter ideal q or M is Buchsbaum module (see [S-V]) if and only if IM(q) is a constant for all parameter ideals q of M.

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 ON A NEW INVARIANT OF FINITELY GENERATED MODULES OVER LOCAL RINGS

2010 ◽  
Vol 09 (06) ◽  
pp. 959-976 ◽  
Author(s):  
NGUYEN TU CUONG ◽  
DOAN TRUNG CUONG ◽  
HOANG LE TRUONG
Keyword(s):  
Local Ring ◽  
Negative Integer ◽  
Local Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Polynomial Type ◽  
Systems Of Parameters ◽  
Finitely Generated Modules

Let M be a finitely generated module on a local ring R and [Formula: see text] a filtration of submodules of M such that do < d1 < ⋯ < dt = d, where di = dim Mi. This paper is concerned with a non-negative integer [Formula: see text] which is defined as the least degree of all polynomials in n1, …, nd bounding above the function [Formula: see text] We prove that [Formula: see text] is independent of the choice of good systems of parameters [Formula: see text]. When [Formula: see text] is the dimension filtration of M, we can use the polynomial type of Mi/Mi-1 and the dimension of the non-sequentially Cohen–Macaulay locus of M to compute [Formula: see text], and also to study the behavior of it under local flat homomorphisms.

On Nagata's Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (I)

2018 ◽  
Vol 85 (3-4) ◽  
pp. 356
Author(s):  
Paula Kemp ◽  
Louis J. Ratliff, Jr. ◽  
Kishor Shah
Keyword(s):  
Integral Closure ◽  
Prime Ideals ◽  
Local Rings ◽  
Finite Sets ◽  
Maximal Ideals ◽  
Valuation Rings ◽  
Canonical Bijection ◽  
Minimal Prime

<p>It is shown that, for all local rings (R,M), there is a canonical bijection between the set <em>DO(R)</em> of depth one minimal prime ideals ω in the completion <em><sup>^</sup>R</em> of <em>R</em> and the set <em>HO(R/Z)</em> of height one maximal ideals <em>̅M'</em> in the integral closure <em>(R/Z)'</em> of <em>R/Z</em>, where <em>Z </em>:<em>= Rad(R)</em>. Moreover, for the finite sets <strong>D</strong> := {<em>V*/V* </em>:<em>= (<sup>^</sup>R/ω)'</em>, ω ∈ DO(R)} and H := {<em>V/V := (R/Z)'<sub><em>̅M'</em></sub>, <em>̅M'</em> ∈ HO(R/Z)</em>}:</p><p>(a) The elements in <strong>D</strong> and <strong>H</strong> are discrete Noetherian valuation rings.</p><p>(b) <strong>D</strong> = {<em><sup>^</sup>V</em> ∈ <strong>H</strong>}.</p>

scholarly journals A Study of the Length Function of Generalized Fractions of Modules

2016 ◽  
Vol 60 (3) ◽  
pp. 721-737 ◽  
Author(s):  
Marcel Morales ◽  
Pham Hung Quy
Keyword(s):  
Local Ring ◽  
Length Function ◽  
Explicit Calculation ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Image Position ◽  
System Of Parameters ◽  
Positive Integers

AbstractLet be a Noetherian local ring and let M be a finitely generated R-module of dimension d. Let be a system of parameters of M and let be a d-tuple of positive integers. In this paper we study the length of generalized fractions M(1/(x1, … , xd, 1)), which was introduced by Sharp and Hamieh. First, we study the growth of the functionThen we give an explicit calculation for the function in the case in which M admits a certain Macaulay extension. Most previous results on this topic are improved in a clearly understandable way.

scholarly journals On non-commutative regular local rings

1976 ◽  
Vol 17 (2) ◽  
pp. 98-102 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Artinian Ring ◽  
Nilpotent Groups ◽  
Global Dimension ◽  
Prime Ideals ◽  
Homological Dimension ◽  
Local Rings ◽  
Universal Enveloping Algebras ◽  
Regular Local Rings

Let R be a ring (with identity). We shall call R a local ring if R is aright noetherian ring such that the Jacobson radical M is a maximal ideal (and so is the only maximal ideal), and R/M is a simple artinian ring. A local ring R with maximal ideal M is called regular if there exists a chainof ideals Mi of such that Mi–1/Mi is generated by a central regular element of R/Mi (1 ≦ i ≦ n). For such a ring R, Walker [6, Theorem 2. 7] proved that R is prime and n is the right global dimension of R, the Krull dimension of R, the homological dimension of theR-module R/M and the supremum of the lengths of chains of prime ideals of R. Such regular local rings will be called n-dimensional. The aim of this note is to give examples of regular local rings. These arise as localizations of universal enveloping algebras of nilpotent Lie algebras over fields and localizations of group algebras of certain finitely generated finite-by-nilpotent groups.

scholarly journals Noetherian Rings—Dimension and Chain Conditions

2005 ◽  
Vol 4 (3) ◽  
Author(s):  
Abhishek Banerjee
Keyword(s):  
Sufficient Conditions ◽  
Artinian Ring ◽  
Small Dimension ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Local Rings ◽  
Direct Products ◽  
Finite Set ◽  
Minimal Prime ◽  
Necessary And Sufficient

In this paper we look at the properties of modules and prime ideals in finite dimensional noetherian rings. This paper is divided into four sections. The first section deals with noetherian one-dimensional rings. Section Two deals with what we define a “zero minimum rings” and explores necessary and sufficient conditions for the property to hold. In Section Three, we come to the minimal prime ideals of a noetherian ring. In particular, we express noetherian rings with certain properties as finite direct products of noetherian rings with a unique minimal prime ideal, as an analogue to the expression of an artinian ring as a finite direct product of artinian local rings. Besides, we also consider the set of ideals I in R such that M ≠ I M for a given module M and show that a maximal element among these is prime. In Section Four, we deal with dimensions of prime ideals, Krull’s Small Dimension Theorem and generalize it (and its converse) to the case of a finite set of prime ideals. Towards the end of the paper, we also consider the sets of linear dependencies that might hold between the generators of an ideal and consider the ideals generated by the coefficients in such linear relations.

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