Juan Elias

Rigid Hilbert functions

Abstract We prove that, given a local Cohen-Macaulay ring ( A,m ), suitable relations between the first two coefficients of the Hilbert polynomial determine the whole Hilbert function of A . The connection of these ideas with the Cohen-Macaulayness of the associated graded ring is also considered.

Number of generators of ideals

Let I be a homogeneous ideal of a polynomial ring over a field, v(I ) the number of elements of any minimal basis of I, e = e(I ) the multiplicity or degree of R/I, h = h(I ) the height or codimension of I, i = indeg ( I ) the initial degree of J , i.e. the minimal degree of non zero elements of I . This paper is mainly devoted to find bounds for v(I ) when I ranges over large classes of ideals. For instance we get bounds when I ranges over the set of perfect ideals with preassigned codimension and multiplicity and when I ranges over the set of perfect ideals with preassigned codimension, multiplicity and initial degree. Moreover all the bounds are sharp since they are attained by suitable ideals. Now let us make some historical remarks.

On the depth of the tangent cone and the growth of the Hilbert function

For a d-dimensional Cohen-Macaulay local ring (R, mn) we study the depth of the associated graded ring of R with respect to an rm-primary ideal I in terms of the Vallabrega-Valla conditions and the length of It+ /JIt, where J is a J minimal reduction of I and t > 1. As a corollary we generalize Sallys conjecture on the depth of the associated graded ring with respect to a maximal ideal to rm-primary ideals. We also study the growth of the Hilbert function.

On the coefficients of the Hilbert polynomial

Let (A, m) be Cohen-Macaulay local ring with maximal ideal m and dimension d. It is well known that for n > 0, the length of the A-module A/mn is given by iAAmn=eontd−1d−e1n+d−2d−1+⋯+(−1)ded. The integers paper an ei are called the Hilbert coefficients of A. In this paper an upper bound is given for e2 in terms of e0, e1 and the embedded codimension h of A. If d ≤ 2 and the bound is reached, A has a specified Hilbert function. Similarly, in the one-dimensional case, we study the extremal behaviour with respect to the known inequality e1≤e2−h2.

On the deep structure of the blowing-up of curve singularities

Let C be a germ of curve singularity embedded in ( k n , 0). It is well known that the blowing-up of C centred on its closed ring, Bl ( C ), is a finite union of curve singularities. If C is reduced we can iterate this process and, after a finite number of steps, we find only non-singular curves. This is the desingularization process. The main idea of this paper is to linearize the blowing-up of curve singularities Bl ( C ) → C . We perform this by studying the structure of [Oscr ] Bl ( C ) /[Oscr ] C as W -module, where W is a discrete valuation ring contained in [Oscr ] C . Since [Oscr ] Bl ( C ) /[Oscr ] C is a torsion W -module, its structure is determined by the invariant factors of [Oscr ] C in [Oscr ] Bl ( C ) . The set of invariant factors is called in this paper as the set of micro-invariants of C (see Definition 1·2). In the first section we relate the micro-invariants of C to the Hilbert function of C (Proposition 1·3), and we show how to compute them from the Hilbert function of some quotient of [Oscr ] C (see Proposition 1·4). The main result of this paper is Theorem 3·3 where we give upper bounds of the micro-invariants in terms of the regularity, multiplicity and embedding dimension. As a corollary we improve and we recover some results of [ 6 ]. These bounds can be established as a consequence of the study of the Hilbert function of a filtration of ideals g = { g [ r , i +1] } i [ges ] 0 of the tangent cone of [Oscr ] C (see Section 2). The main property of g is that the ideals g [ r , i +1] have initial degree bigger than the Castelnuovo–Mumford regularity of the tangent cone of [Oscr ] C . Section 4 is devoted to computation the micro-invariants of branches; we show how to compute them from the semigroup of values of C and Bl ( C ) (Proposition 4·3). The case of monomial curve singularities is especially studied; we end Section 4 with some explicit computations. In the last section we study some geometric properties of C that can be deduced from special values of the micro-invariants, and we specially study the relationship of the micro-invariants with the Hilbert function of [Oscr ] Bl ( C ) . We end the paper studying the natural equisingularity criteria that can be defined from the micro-invariants and its relationship with some of the known equisingularity criteria.

Poincaré Series and Deformations of Gorenstein Local Algebras

Let (A, 𝔪, K) be an Artinian Gorenstein local ring with K an algebraically closed field of characteristic 0. In the present article, we prove a structure theorem describing the Artinian Gorenstein local K-algebras satisfying 𝔪4 = 0. We use this result in order to prove that such a K-algebra has rational Poincaré series and it is smoothable in any embedding dimension, provided dim K 𝔪2/𝔪3 ≤ 4. We also prove that the generic Artinian Gorenstein local K-algebra with 𝔪4 = 0 has rational Poincaré series.

On the computation of the Ratliff¿Rush closure

Abstract Let R be a Cohen–Macaulay local ring with maximal ideal m . In this paper we present a procedure for computing the Ratliff–Rush closure of an m -primary ideal I ⊂ R .

A sharp bound for the minimal number of generators of perfect height two ideals

Let R a regular local ring of dimension N. In this paper we give a sharp bound for the minimal number of generators of perfect height two ideals I of R in terms of the multiplicity of R/I, and also some characterizations of the ideals which reach that bound. For a similar bound for the number of generators of height three ideals I of R such that R/I is Gorenstein see theorem 1 in [3].

The Hilbert function of a Cohen-Macaulay local algebra: Extremal gorenstein algebras

algebras, studied their resolutions, and extended Theorem 1 to the case A Cohen-Macaulay graded [F-L]. We extend the inequality to C.M. local algebras A that need not be graded. Here the associated graded algebra A* need not be Cohen-Macaulay; even if A* is CM., its socle type may be dif- ferent from that of A. Assume that I is an ideal defining a Gorenstein quotient A of

Upper bounds of Hilbert coefficients and Hilbert functions

Let ( R , m ) be a d -dimensional Cohen–Macaulay local ring. In this paper we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a m -primary ideal I ⊂ R that improves all known upper bounds unless for a finite number of cases, see Remark 2.3. We also provide new upper bounds of the Hilbert functions of I extending the known bounds for the maximal ideal.