Maria Evelina Rossi

Hilbert functions of filtered modules

In this presentation we shall deal with some aspects of the theory of Hilbert functions of modules over local rings, and we intend to guide the reader along one of the possible routes through the last three decades of progress in this area of dynamic mathematical activity. Motivated by the ever increasing interest in this field, our goal is to gather together many new developments of this theory into one place, and to present them using a unifying approach which gives self-contained and easier proofs. In this text we shall discuss many results by different authors, following essentially the direction typified by the pioneering work of J. Sally. Our personal view of the subject is most visibly expressed by the presentation of Chapters 1 and 2 in which we discuss the use of the superficial elements and related devices. Basic techniques will be stressed with the aim of reproving recent results by using a more elementary approach. Over the past few years several papers have appeared which extend classical results on the theory of Hilbert functions to the case of filtered modules. The extension of the theory to the case of general filtrations on a module has one more important motivation. Namely, we have interesting applications to the study of graded algebras which are not associated to a filtration, in particular the Fiber cone and the Sally-module. We show here that each of these algebras fits into certain short exact sequences, together with algebras associated to filtrations. Hence one can study the Hilbert function and the depth of these algebras with the aid of the know-how we got in the case of a filtration.

Gröbner Flags and Gorenstein Algebras

The goal of this paper is to study the Koszul property and the property of having a Gröbner basis of quadrics for classical varieties and algebras as canonical curves, finite sets of points and Artinian Gorenstein algebras with socle in low degree. Our approach is based on the notion of Gröbner flags and Koszul filtrations. The main results are the existence of a Gröbner basis of quadrics for the ideal of the canonical curve whenever it is defined by quadrics, the existence of a Gröbner basis of quadrics for the defining ideal of s ≤ 2n points in general linear position in Pn, and the Koszul property of the ‘generic’ Artinian Gorenstein algebra of socle degree 3.

Isomorphism classes of short Gorenstein local rings via Macaulay’s inverse system

In this paper we study the isomorphism classes of Artinian Gorenstein local rings with socle degree three by means of Macaulay’s inverse system. We prove that their classification is equivalent to the projective classification of the hypersurfaces of P n of degree three. This is an unexpected result because it reduces the study of this class of local rings to the homogeneous case. The result has applications in problems concerning the punctual Hilbert scheme Hilbd(P n ) and

Castelnuovo-Mumford regularity and extended degree

Our main result shows that the Castelnuovo-Mumford regularity of the tangent cone of a local ring A is effectively bounded by the dimension and any extended degree of A. From this it follows that there are only a finite number of Hilbert-Samuel functions of local rings with given dimension and extended degree.

Koszul Algebras and Regularity

This is a survey paper on commutative Koszul algebras and Castelnuovo-Mumford regularity. Koszul algebras, originally introduced by Priddy, are graded K-algebras R whose residue field K has a linear free resolution as an R-module. The Castelnuovo-Mumford regularity is, after Krull dimension and multiplicity, perhaps the most important invariant of a finitely generated graded module M, as it controls the vanishing of both syzygies and the local cohomology modules of M.

Primary ideals with good associated graded ring

Abstract Let (A, M ) be a local Cohen–Macaulay ring of dimension d. Let I be an M -primary ideal and let J be the ideal generated by a maximal superficial sequence for I . Under these assumptions Valabrega and Valla (Nogoya Math. J. 72 (1978) 93–101) proved that the associated graded ring G of I is Cohen–Macaulay if and only if I n ∩J=JI n−1 for every integer n . In this paper we consider the class of the M -primary ideals I such that, for some positive integer k, we have I n ∩J=JI n−1 for n≤k and λ(I k+1 /JI k )≤1. In this case G need not be Cohen–Macaulay. In Theorem 2.2. we show that G is Cohen–Macaulay unless the ideals we are considering are of maximal Cohen–Macaulay type. One can use the ideas of Russi and Valla (Comm. Algebra 24(13) (1996) 4249–4261; Pure Appl. Algebra 122 (1997) 293–311) to prove that, for the ideals we consider, the depth of G is at least d−1 and that its h -vector has no negative components. We characterize the possible Hilbert function of G. Our approach gives proof of an extended version of a conjecture of Sally (proved in Russi and Valla Comm. Algebra 24(13) (1996) 4249–4261)) and independently in Wang (J. Algebra 190 (1997) 226–240) in the case I= M ). Several results proved in Huckaba (Comm. Algebra, to appear), Russi and Valla (Nogoya Math. J. 110 (1988) 81–111; Comm. Algebra 24(13) (1996) 4249–4261; J. Pure Appl. Algebra 122 (1997) 293–311) and Sally (J. Algebra 83 (1983) 325–333) are unified and generalized.

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.

Analytic Isomorphisms of compressed local algebras

In this paper we consider Artin local K-algebras with maximal length in the class of Artin algebras with given embedding dimension and socle type. They have been widely studied by several authors, among others by Iarrobino, Froberg and Laksov. If the local K-algebra is Gorenstein of socle degree 3, then the authors proved that it is canonically graded, i.e. analytically isomorphic to its associated graded ring, see (6). This unexpected result has been extended to compressed level K-algebras of socle degree 3 in (4). In this paper we end the investigation proving that extremal Artin Gorenstein local K-algebras of socle degree s � 4 are canonically graded, but the result does not extend to extremal Artin Gorenstein local rings of socle degree 5 or to compressed level local rings of socle degree 4 and type > 1. As a consequence we present results on Artin compressed local K-algebras having a specified socle type.

On the resolution of certain graded algebras

Let A = R/I be a graded algebra over the polynomial ring R = k[X 0 ,..., X n ]. Some properties of the numerical invariants in a minimal free resolution of A are discussed in the case A is a «Short Graded Algebra». When A is the homogeneous coordinate ring of a set of points in generic position in the projective space, several result are obtained on the line traced by some conjectures proposed by Green and Lazarsfeld in [GL] and Lorenzini in [L1]

Consecutive cancellations in Betti numbers of local rings

Let I be a homogeneous ideal in a polynomial ring P over a field. By Macaulays Theorem, there exists a lexicographic ideal L=Lex(I) with the same Hilbert function as I. Peeva has proved that the Betti numbers of P/I can be obtained from the graded Betti numbers of P/L by a suitable sequence of consecutive cancellations. We extend this result to any ideal I in a regular local ring (R,m) by passing through the associated graded ring. To this purpose it will be necessary to enlarge the list of the allowed cancellations. Taking advantage of Eliahou-Kervaires construction, several applications are presented. This connection between the graded perspective and the local one is a new viewpoint and we hope it will be useful for studying the numerical invariants of classes of local rings.