Reduction number

Let (S, n) be a 2-dimensional regular local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Let I * be the leading ideal of I in the associated graded ring gr n (S), and set R = S/I and m = n/I. In [GHK2], we prove that if µ G (I *) = n, then I * contains a homogeneous system {ξ i } 1≤i≤n of generators such that deg ξ i + 2 ≤ deg ξ i+1 for 2 ≤ i ≤ n−1, and htG(ξ1, ξ2, • • • , ξn−1) = 1, and we describe precisely the Hilbert series H(gr m (R), λ) in terms of the degrees c i of the ξ i and the integers d i , where di is the degree of Di = GCD(ξ1,. .. , ξi). To the complete intersection ideal I = (f, g)S we associate a positive integer n with 2 ≤ n ≤ c 1 + 1, an ascending sequence of positive integers (c 1 , c 2 ,. .. , c n), and a descending sequence of integers (d1 = c1, d2,. .. , dn = 0) such that ci+1 − ci > di−1 − di > 0 for each i with 2 ≤ i ≤ n − 1. We establish here that this necessary condition is also sufficient for there to exist a complete intersection ideal I = (f, g) whose leading ideal has these invariants. We give several examples to illustrate our theorems.

Let (S, n) be a Noetherian local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Assume that the associated graded ring gr n (S) of S with respect to n is a UFD. We examine generators of the leading form ideal I * of I in gr n (S) and prove that I * is a perfect ideal of gr n (S), if I * is 3-generated. Thus, in this case, letting R = S/I and m = n /I, if gr n (S) is Cohen-Macaulay, then gr m (R) = gr n (S)/I * is Cohen-Macaulay. As an application, we prove that if (R, m) is a one-dimensional Gorenstein local ring of embedding dimension 3, then gr m (R) is Cohen-Macaulay if the reduction number of m is at most 4.

For a regular ideal having a principal reduction in a Noetherian ring we consider the structural numbers that arise from taking the Ratliff-Rush closure of the ideal and its powers. In particular, we analyze the interconnections among these numbers and the relation type and reduction number of the ideal. We prove that certain inequalites hold in general among these invariants, while for ideals contained in the conductor of the integral closure of the ring we obtain sharper results that do not hold in general. We provide applications to the one-dimensional local setting and present a sequence of examples in this context.

Let (A, m) be a Cohen-Macaulay local ring of dimension d ≥ 1 and I an ideal in A. Let M be a finitely generated maximal Cohen-MacaulayAmodule. Let I be a locally complete intersection ideal with ht M (I) = d − 1, l M (I) = d and reduction number at most one. We prove that the polynomial n → ℓ(Tor A 1 (M, A/I n+1)) either has degree d − 1 or F I (M) is a free F (I)−module.

In this article, we define a class of binomial ideals associated to a simplicial complex. This class of ideals appears in the presentation of fiber cones of codimension 2 lattice ideals \cite{hm}, and in the work of Barile and Morales \cite{bm2}, \cite{bm3}, \cite{bm4}. We compute the reduction number of Binomial extensions of Simplicial ideals. This extends all the previous results

Let R be a local Cohen-Macaulay ring, let I be an R-ideal, and let G be the associated graded ring of I. We give an estimate for the depth of G when G is not necessarily Cohen-Macaulay. We assume that I is either equimultiple, or has analytic deviation one, but we do not have any restriction on the reduction number. We also give a general estimate for the depth of G involving the first r + powers of I, where r denotes the Castelnuovo regularity of G and denotes the analytic spread of I.

Let R be a local Cohen-Macaulay ring, let I be an R-ideal, and let G be the associated graded ring of I. We give an estimate for the depth of G when G is not necessarily Cohen-Macaulay. We assume that I is either equimultiple, or has analytic deviation one, but we do not have any restriction on the reduction number. We also give a general estimate for the depth of G involving the first r + powers of I, where r denotes the Castelnuovo regularity of G and denotes the analytic spread of I.

In this article, we define a class of binomial ideals associated to a simplicial complex. This class of ideals appears in the presentation of fiber cones of codimension 2 lattice ideals \cite{hm}, and in the work of Barile and Morales \cite{bm2}, \cite{bm3}, \cite{bm4}. We compute the reduction number of Binomial extensions of Simplicial ideals. This extends all the previous results

This paper studies the core of an ideal in a Noetherian local or graded ring. By definition, the core of an ideal is the intersection of all reductions of the ideal. We provide computational formulae for the determination of the core of a graded ring, meaning the core of the unique homogeneous maximal ideal. We then apply the formulae to give answers to several questions raised by Corso, Polini and Ulrich. We are also able to answer in the positive a conjecture raised by these three authors concerning a closed formula for the core. We give a positive answer to their question in the case in which the ring is Cohen-Macaulay with a residue field of characteristic 0, and in the case the ideal is equimultiple.

In this article, we define a class of binomial ideals associated to a simplicial complex. This class of ideals appears in the presentation of fiber cones of codimension 2 lattice ideals \cite{hm}, and in the work of Barile and Morales \cite{bm2}, \cite{bm3}, \cite{bm4}. We compute the reduction number of Binomial extensions of Simplicial ideals. This extends all the previous results

Let R be a local Cohen-Macaulay ring, let I be an R-ideal, and let G be the associated graded ring of I. We give an estimate for the depth of G when G is not necessarily Cohen-Macaulay. We assume that I is either equimultiple, or has analytic deviation one, but we do not have any restriction on the reduction number. We also give a general estimate for the depth of G involving the first r + powers of I, where r denotes the Castelnuovo regularity of G and denotes the analytic spread of I.

We explicitly calculate the normal cones of all monomial primes which define the curves of the form (t L , t L+1 ,. .. , t L+n), where n ≤ 4. All of these normal cones are reduced and Cohen-Macaulay, and their reduction numbers are independent of the reduction. These monomial primes are new examples of integrally closed ideals for which the product with the maximal homogeneous ideal is also integrally closed. Substantial use was made of the computer algebra packages Maple and Macaulay2.

The main result of the paper confirms, for generic coordinates, a conjecture which states that r ( R / I ) ≤ r ( R / i n ( I ) ) r(R/I) \le r(R/in(I)) . Here I I is a homogeneous polynomial ideal in R R and r ( R / I ) r(R/I) and r ( R / i n ( I ) ) r(R/in(I)) are the reduction numbers.

Let G(I) be the associated graded ring of an ideal I in a Cohen-Macaulay local ring A. We give a sufficient condition for G(I) to be a Cohen-Macaulay ring. It is described in terms of the depths of A=I n for finitely many n, the reduction numbers of I Q for certain prime ideals Q and the Artin-Nagata property of I in the sense of Ulrich (Contemp. Math. vol. 159, pp. 373-400). The main theorem unifies several results which are already known in this aspect of the theory. Although the statement of the main theorem is rather complicated, we restate the conditions under special situations, so that they are practical and simple.

In this paper, we investigate and study the notion of left -biflatness of abstract Segal algebras, where  is a character on Banach algebra. Precisely, we give a necessary and sufficient condition for left -biflatness of abstract Segal algebras equipped with a left approximate identity. As an application, we show that if  is a Segal algebra on the locally group  and  is a character, then  is left -biflat if and only if  is amenable. Indeed, this is a generalization of [4, Theorem 3.4]. Moreover, we study the relationship between left -biflatness and inner -amenability and show that if the Banach algebra  is inner -amenable, then the notions of left -biflatness and left -amenability are equivalent.

The present paper compares properties of Ratliff-Rush closure of ‎‎‎an ‎ideal ‎with ‎its integral closure. Furthermore, ideals in which their associated graded ring has positive depth, are introduced as ideals for which all its powers are Ratliff-Rush ideals. While stating that each regular ideal is always a reduction of its associated Ratliff-Rush ideal, it expresses the command for calculating the Rutliff-Rush closure of an ideal by its reduction. This fact that Hilbert polynomial of an ideal has the same Hilbert polynomial its Ratliff-Rush closure, is from our other results. T‎he Ratliff-Rush closure of ideals is a good operation with respect to many properties, it carries information about associated primes of powers of ideals, about zerodivisors in the associated graded ring, preserves the Hilbert function of zero-dimensional ideals, etc.