Equimultiple Ideals Research Articles

Indices of normalization of ideals

We derive numerical estimates controlling the intertwined properties of the normalization of an ideal and of the computational complexity of general processes for its construction. In [18], this goal was carried out for equimultiple ideals via the examination of Hilbert functions. Here we add to this picture, in an important case, how certain Hilbert functions provide a description of the locations of the generators of the normalization of ideals of dimension zero. We also present a rare instance of normalization of a class of homogeneous ideals by a single colon operation.

Reduction numbers of equimultiple ideals

Reduction numbers of equimultiple ideals

Degrees of multiplicity functions for equimultiple ideals

Let J be an equimultiple ideal of height a in a formally equidimensional local ring (R,m). If I is a proper ideal that contains J, we show that the degree of the multiplicity function fJ,I(n)=e(In/Jn) is at most a with equality if and only if J is not a reduction of I. As a consequence, we are able to define a unique filtration J⊆J[a]⊆…⊆J[1]⊆J‾ between the ideal J and its integral closure J‾ with J[k] being the largest ideal containing J such that deg⁡fJ,I(n)≤a−k−1. Further results consider the ideal J[1] and its relation to the S2-ification of the Rees algebra R[Jt].

The equations of Rees algebras of equimultiple ideals of deviation one

We describe the equations of the Rees algebra R ( I ) \mathbf {R}(I) of an equimultiple ideal I I of deviation one provided that I I has a reduction generated by a regular sequence x 1 , … , x s x_1,\ldots ,x_s such that the initial forms x 1 ∗ , … , x s − 1 ∗ x_1^*,\ldots ,x_{s-1}^* are a regular sequence in the associated graded ring. In particular, we prove that there is a single equation of maximum degree in a minimal generating set of the equations of R ( I ) \mathbf {R}(I) , which recovers some previous known results.

The multiplicity and the Cohen–Macaulayness of extended Rees algebras of equimultiple ideals

Let I be an equimultiple ideal of Noetherian local ring A. This paper gives some multiplicity formulas of the extended Rees algebras T = A [ It , t - 1 ] . In the case A generalized Cohen–Macaulay, we determine when T is Cohen–Macaulay and as an immediate consequence we obtain e.g., some criteria for the Cohen–Macaulayness of Rees algebra R ( I ) over a Cohen–Macaulay ring in terms of reduction numbers and ideals.

Normalization of ideals and Briançon–Skoda numbers

We establish bounds for the coefficiente1(I) of the Hilbert function of the integral closure filtration of equimultiple ideals. These values are shown to help control all algorithmic processes of normalization that make use of extensions satisfying the condition S2 of Serre.

On the mixed multiplicity and the multiplicity of blow-up rings of equimultiple ideals

Let (S, m) be a graded algebra of dimension d generated by finitely many elements of degree 1 over a field k and a homogeneous equimultiple ideal I of S with ht I= h>0. In this paper we will show that if a 1⩽ a 2⩽⋯⩽ a h is the degree sequence of a minimal homogeneous reduction of I, then the mixed multiplicity e( m [d−i],I [i])=a 1a 2⋯a ie(S) for all 0< i< h and the multiplicity of Rees algebra e(R(I))=[1+∑ i=1 h−1 a 1a 2⋯a i]e(S) .

On some properties of (fc)-sequences of ideals in local rings

The paper characterizes the length of maximal sequences satisfying conditions (i) and (ii) of (FC)-sequences, and proves some properties of (FC)-sequences, such as a bound on their lengths. As a consequence we get some results for mixed multiplicities and multiplicities of Rees rings of equimultiple ideals. We also prove that if I is an ideal of positive height and x 1 , x 2 ..., x p is an arbitrary maximal sequence in I satisfying conditions (i) and (ii) of (FC)-sequences, then (x 1 , x 2 ,..., x p ) is a reduction of I.

Reduction numbers of equimultiple ideals

Let (A, m) be a local ring and Z and an ideal of A. In this paper, all local rings are assumed to have infinite residue fields. An ideal .Z 5 Z is called a reduction of Z if Zflfl = Z”J for some nonnegative integer n. A reduction J is called a minimal reduction if it does not properly contain a reduction of I. These notions were introduced by Northcott and Rees [16]. They proved that minimal reductions of Z do always exist, and every minimal reduction of Z is minimally generated by E(Z):= dim 0, k 0 P/ml” which is called the anQ~y~~c spread of 1. If J is a minimal reduction of I, we define the reduction number of I with respect to J, denoted by r,(Z), to be the last nonnegative integer n such that I”+’ = Z”J. The reduction number of Z is defined by r{Z) = min{r.,(Z); J is a minimal reduction of Z 3. Reduction numbers have been proven to be very useful in studying the Cohen-Macaulay (abbr. C-M} and Gorenstein property of the associated graded ring G(Z) = @,I>OZn/Zn+l and the Rees algebra A[Zt] = a,, z 0 Z”t” of I. It was initiated by Sally, Goto and Shimoda, and is intensively studied now by various authors (see, e.g., [l] for references). Therefore it is of great interest to study properties of reduction numbers: see 14, 10,12-14, 19,231. The main aim of this paper is to give upper bounds for the reduction numbers of equimultiple ideals. Recall that ht Z 1, easy

Links of Prime Ideals and Their Rees Algebras

In a previous paper we exhibited the somewhat surprising property that most direct links of prime ideals in Gorenstein rings are equimultiple ideals with reduction number 1. This led to the construction of large families of Cohen-Macaulay Rees algebras. The first goal of this paper is to extend this result to arbitrary Cohen-Macaulay rings. The means of the proof are changed since one cannot depend so heavily on linkage theory. We then study the structure of the Rees algebra of these links; more specifically, we describe their canonical module in sufficient detail to be able to characterize self-linked prime ideals. In the last section multiplicity estimates for classes of such ideals are established.

Cohen-Macaulay and Gorenstein property of Rees algebras of non-singular equimultiple prime ideals

We give criteria for the Cohen-Macaulay and Gorenstein property of Rees algebras of height 2 non-singular equimultiple prime ideals in terms of explicite representations of the associated graded rings. As consequences, we show that in general, the Cohen-Macaulay resp. Gorenstein property of such Rees algebras does not imply the Cohen-Macaulay resp. Gorenstein property of the base ring and that these properties depend upon the characteristic.

Rees algebras of non-singular equimultiple prime ideals

In a recent paper [HI2] the first named author and S. Ikeda have investigated Gorenstein properties under blowing up of height two equimultiple ideals from the arithmetical point of view. The problem is to what extent does the Gorenstein property of the Rees algebra R(a) = of an equimultiple ideal a of a local ring (A, m) describe the structure of A and a. One result is that if A is a generalized Cohen-Mac aulay ring (i.e. a ring of finite local cohomology) with dim (A) ⊇ 4 and if there is an height two equimultiple prime ideal p such that R(p) is Gorenstein, then A is Gorenstein and p is generated by a regular sequence (see [HI2], Theorem 2.6). That has led to the question whether this result is still valid for dim (∧) = 3. ∧ partial answer was given in [HI2], Proposition 2.10, where “equimultiplicity” was replaced by the stronger condition “p/p2 is flat over A/p”

On the Gorenstein property of Rees Algebras

Let (A,M) be a noetherian local ring. For certain equimultiple ideals I in A we try to relate the Gorenstein property of the Rees algebra⊕n and of A itself. In particular we n≥0 treat the case of equimultiple prime ideals of height two and the case I=M. The results underscore a natural conjecture, s. Thm. 2.6. and 3.2.

Graded Cohen-Macaulay rings associated to equimultiple ideals

Graded Cohen-Macaulay rings associated to equimultiple ideals