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.

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.

Family of quotients of some special rings

2018 ◽  
Vol 17 (12) ◽  
pp. 1850233 ◽  
Author(s):  
Maryam Salimi
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Proper Ideal ◽  
Rees Algebra ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
Local Cohomology Modules ◽  
Cohomology Modules ◽  
The Ideal

Let [Formula: see text] be a commutative Noetherian ring and let [Formula: see text] be a proper ideal of [Formula: see text]. We study some properties of a family of rings [Formula: see text] that are obtained as quotients of the Rees algebra associated with the ring [Formula: see text] and the ideal [Formula: see text]. We deal with the strongly cotorsion property of local cohomology modules of [Formula: see text], when [Formula: see text] is a local ring. Also, we investigate generically Cohen–Macaulay, generically Gorenstein, and generically quasi-Gorenstein properties of [Formula: see text]. Finally, we show that [Formula: see text] is approximately Cohen–Macaulay if and only if [Formula: see text] is approximately Cohen–Macaulay, provided some special conditions.

scholarly journals Self-linked curve singularities

1990 ◽  
Vol 120 ◽  
pp. 129-153 ◽  
Author(s):  
Jürgen Herzog ◽  
Bernd Ulrich
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Complete Intersection ◽  
Three Dimensional ◽  
Regular Local Ring ◽  
Rees Algebra ◽  
Curve Singularities ◽  
Noetherian Property

Let S be a three-dimensional regular local ring and let I be a prime ideal in S of height two. This paper is motivated by the question of when I is a set-theoretic complete intersection and when the symbolic Rees algebra S(I) = ⊕n≥0I(n)tn is Noetherian. The connection between the two problems is given by a result of Cowsik which says that the Noetherian property of S(I) implies that I is a set-theoretic complete intersection ([1]).

On complete d-sequences and the defining ideals of Rees algebras

1989 ◽  
Vol 106 (3) ◽  
pp. 445-458 ◽  
Author(s):  
Sam Huckaba
Keyword(s):  
Local Ring ◽  
Homomorphic Image ◽  
Polynomial Ring ◽  
Rees Algebra ◽  
Homogeneous Ideal ◽  
Rees Algebras ◽  
Generating Sets ◽  
Degree Bounds ◽  
Relation Type ◽  
Reduction Number

AbstractIf R is a Noetherian local ring and I = (x1, …, xn)R is an ideal of R then the Rees algebra R[It] can be represented as a homomorphic image of the polynomial ring R[Z1, …, Zn]. The kernel is a homogeneous ideal, and the smallest of the degree bounds among all generating sets, called the relation type of I, is independent of the representation. We derive formulae connecting the relation type of I with the reduction number of I when the analytic spread of I exceeds height(I) by one. In the process we define complete d-sequences with respect to I and use them to help achieve our results. In addition some results on the behaviour of the relation type modulo an element are proved, and examples where the relation type is explicitly computed are presented.

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.

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

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

Minimal Generators of the Defining Ideal of the Rees Algebra Associated with a Rational Plane Parametrization with μ = 2

2014 ◽  
Vol 66 (6) ◽  
pp. 1225-1249 ◽  
Author(s):  
Teresa Cortadellas Benítez ◽  
Carlos D'Andrea
Keyword(s):  
Projective Plane ◽  
Rational Curve ◽  
Rees Algebra ◽  
Homogeneous Polynomials ◽  
Rational Plane ◽  
Minimal Generators ◽  
The Ideal

AbstractWe exhibit a set of minimal generators of the defining ideal of the Rees Algebra associated with the ideal of three bivariate homogeneous polynomials parametrizing a proper rational curve in projective plane, having a minimal syzygy of degree 2.

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.

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