arXiv subjects
Publications and source records attributed to Enrico Sbarra.
In this survey paper we first present the main properties of sequentially Cohen-Macaulay modules. Some basic examples are provided to help the reader with quickly getting acquainted with this topic. We then discuss two generalizations of the notion of sequential Cohen-Macaulayness which are inspired by a theorem of Jürgen Herzog and the third author.
Inspired by a very recent work of A. Ðurić, S. Jevđenić and N. Stopar, we introduce a new definition of zero-divisor graphs attached to rings, that includes all of the classical definitions already known in the literature. We provide an interpretation of such graphs as images of a functor, that we call zero-divisor functor and which is associated with a family of special equivalence relations fixed beforehand. We thus recover and generalize many known results for zero-divisor graphs and provide a framework which might be useful for further investigations on this topic.
We survey most of the known results concerning the Eisenbud-Green-Harris Conjecture. Our presentation includes new proofs of several theorems, as well as a unified treatment of many results which are otherwise scattered in the literature. We include a final section with some applications, and examples.
Jet schemes and arc spaces received quite a lot of attention by researchers after their introduction, due to J. Nash, and established their importance as an object of study in M. Kontsevich's motivic integration theory. Several results point out that jet schemes carry a rich amount of geometrical information about the original object they stem from, whereas, from an algebraic point of view, little is know about them. In this paper we study some algebraic properties of jet schemes ideals of pfaffian varieties and we determine under which conditions the corresponding jet scheme varieties are irreducible.
The relationships between the invariants and the homological properties of $I$, ${\rm Gin}(I)$ and $I^{\rm lex}$ have been studied extensively over the past decades. A result of A. Conca, J. Herzog and T. Hibi points out some rigid behaviours of their Betti numbers. In this work we establish a local cohomology counterpart of their theorem. To this end, we make use of properties of sequentially Cohen-Macaulay modules and we study a generalization of such concept by introducing what we call partially sequentially Cohen-Macaulay modules, which might be of interest by themselves.
Given a homogeneous ideal I of a polynomial ring A=K[X_1,...,X_n] and a monomial order, we construct a new monomial ideal of A associated with I. We call it the zero-generic initial ideal of I with respect to the order and denote it with gin_0(I), or with Gin_0(I) whenever the order is the reverse-lexicographic order. When the characteristic of K is zero, a zero-generic initial ideal is the usual generic initial ideal. We show that Gin_0(I) is endowed with many interesting properties, some of which are easily seen, e.g., it is a strongly stable monomial ideal in any characteristic, and has the same Hilbert series as I; some other properties are less obvious: it shares with I the same Castelnuovo-Mumford regularity, projective dimension and extremal Betti numbers. Quite surprisingly, gin_0(I) also satisfies Green's Crystallization Principle, which is known to fail in positive characteristic. Thus, zero-generic initial ideals can be used as formal analogues of generic initial ideals computed in characteristic 0. We also prove the analogue for local cohomology of Pardue's Conjecture for Betti numbers: we show that the Hilbert functions of local cohomology modules of a quotient ring of a polynomial ring modulo a weakly stable ideal are independent of the characteristic of the coefficients field.
We study the set of circuits of a homogeneous ideal and that of its truncations, and introduce the notion of generic circuits set. We show how this is a well-defined invariant that can be used, in the case of initial ideals with respect to weights, as a counterpart of the (usual) generic initial ideal with respect to monomial orders. As an application we recover the existence of the generic fan introduced by Römer and Schmitz for studying generic tropical varieties. We also consider general initial ideals with respect to weights and show, in analogy to the fact that generic initial ideals are Borel-fixed, that these are fixed under the action of certain Borel subgroups of the general linear group.
We study, by means of embeddings of Hilbert functions, a class of rings which we call Shakin rings, i.e. quotients K[X_1,...,X_n]/a of a polynomial ring over a field K by ideals a=L+P which are the sum of a piecewise lex-segment ideal L, as defined by Shakin, and a pure powers ideal P. Our main results extend Abedelfatah's recent work on the Eisenbud-Green-Harris conjecture, Shakin's generalization of Macaulay and Bigatti-Hulett-Pardue theorems on Betti numbers and, when char(K)=0, Mermin-Murai theorem on the Lex-Plus-Power inequality, from monomial regular sequences to a larger class of ideals. We also prove an extremality property of embeddings induced by distractions in terms of Hilbert functions of local cohomology modules.
We prove an inequality between Hilbert functions of local cohomology modules supported in the homogeneous maximal ideal of standard graded algebras over a field, within the framework of embeddings of posets of Hilbert functions. As a main application, we prove an analogue for local cohomology of Evans' Lex-Plus-Power Conjecture for Betti numbers. This results implies some cases of the classical Lex-Plus-Power Conjecture, namely an inequality between extremal Betti numbers. In particular, for the classes of ideals for which the Eisenbud-Green-Harris Conjecture is currently known, the projective dimension and the Castelnuovo-Mumford regularity of a graded ideal do not decrease by passing to the corresponding Lex-Plus-Power ideal.
We characterise the class of one-cogenerated Pfaffian ideals whose natural generators form a Gröbner basis with respect to any anti-diagonal term-order. We describe their initial ideals as well as the associated simplicial complexes, which turn out to be shellable and thus Cohen-Macaulay. We also provide a formula for computing their multiplicity.
We study bounds for the Castelnuovo-Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular our aim is to give a positive answer to a question posed by Bayer and Mumford, by showing that the known upper bound in characteristic zero holds true also in positive characteristic. We first analyze Giusti's proof, which provides the result in characteristic 0, giving some insight on the combinatorial properties needed in that context. For the general case we provide a new argument which employs Bayer and Stillman criterion for detecting regularity.
We characterize the class of ideals of a polynomial ring such that the hilbert series of their graded local cohomology modules is maximal.
The main result of the paper states that for a graded ideal I in a polynomial ring R over a field of characteristic 0, the Hilbert functions of the local cohomology modules of R/I and of R/Gin(I) coincide if and only if R/I is sequentially Cohen-Macaulay.