Lexsegment Ideals Research Articles

Linear resolutions of t-spread lexsegment ideals via Betti splittings

Let [Formula: see text] be a polynomial ring in [Formula: see text] variables with coefficients over a field [Formula: see text]. A [Formula: see text]-spread lexsegment ideal [Formula: see text] of [Formula: see text] is a monomial ideal generated by a [Formula: see text]-spread lexsegment set. We determine all [Formula: see text]-spread lexsegment ideals with linear resolution by means of Betti splittings. As applications we provide formulas for the Betti numbers of such a class of ideals and furthermore we characterize all incompletely [Formula: see text]-spread lexsegment ideals with linear quotients.

Shifting operations and completely t–spread lexsegment ideals

In this article we introduce the concepts of arbitrary t–spread lexsegments and of arbitrary t–spread lexsegment ideals with t a positive integer. These concepts are a natural generalization of arbitrary lexsegments and arbitrary lexsegment ideals. An ideal generated by an arbitrary t–spread lexsegment is called completely t–spread lexsegment if it is equal to the intersection of an initial t–spread lexsegment ideal and of a final t–spread lexsegment ideal. We study the class of arbitrary t–spread lexsegment ideals. In particular, we characterize all completely t–spread lexsegment ideals. Moreover, we classify all completely t–spread lexsegment ideals with a linear resolution.

Classification of Cohen–Macaulay t-spread lexsegment ideals via simplicial complexes

We study the minimal primary decomposition of completely t-spread lexsegment ideals via simplicial complexes. We determine some algebraic invariants of such a class of t-spread ideals. Hence, we classify all t-spread lexsegment ideals which are Cohen–Macaulay.

Eakin-Sathaye-Type Theorems for Joint Reductions and Good Filtrations of Ideals

Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for $\mathbb N^{s}$ -graded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for $\mathbb N$ -graded filtrations. Several examples related to lex-segment ideals, contracted ideals in 2-dimensional regular local rings and the filtration of integral and tight closures of powers of ideals in hypersurface rings are constructed to show effectiveness of these bounds.

Lexsegment Ideals and Their h-Polynomials

Let S = K[x1, . . . , xn] denote the polynomial ring in n variables over a field K with each deg xi = 1 and I ⊂ S a homogeneous ideal of S with dim S/I = d. The Hilbert series of S/I is of the form hS/I(λ)/(1 − λ)d, where hS/I(λ) = h0 + h1λ + h2λ2 + ⋯ + hsλs with hs≠ 0 is the h-polynomial of S/I. Given arbitrary integers r ≥ 1 and s ≥ 1, a lexsegment ideal I of S = K[x1,…,xn], where n ≤ max{r,s} + 2, satisfying reg(S/I) = r and deg hS/I(λ) = s will be constructed.

Algebraic Properties of Universal Squarefree Lexsegment Ideals

Let K be a field and let A=K[X1,…,Xn] be the polynomial ring in X1,…,Xn with coefficients in K. In this paper we study the universal squarefree lexsegment ideals, and put our attention on their combinatorics computing some invariants. Moreover, we study the link between such a special class of squarefree lexsegment ideals and the so-called s-sequences.

On the Relation Type of Fiber Cone

In this article, we study the relation type of the fiber cone of certain special classes of ideals in Noetherian local rings. We show that in any Noetherian local ring, if deviation of I is 1, and depth $(G(\mathcal F_{L})) \geq \ell -1$ , then the relation types of $\mathcal {R}(I)$ and F L (I) are equal. We also prove that for lexsegment ideals in K[x,y], where K is a field, the relation types of the fiber cone and the Rees algebra are equal.

Normally Torsion-free Lexsegment Ideals

In this paper we characterize all the lexsegment ideals which are normally torsion-free. This will provide a large class of normally torsion-free monomial ideals which are not square-free. Our characterization is given in terms of the ends of the lexsegment. We also prove that, for lexsegment ideals, the property being normally torsion-free is equivalent to the property of the depth function being constant.

Lexsegment Ideals Are Sequentially Cohen-Macaulay

The associated primes of an arbitrary lexsegment ideal I ⊆ S=K[x1,…,xn] are determined. As application it is shown that S/I is a pretty clean module, therefore S/I is sequentially Cohen-Macaulay and satisfies Stanley's conjecture.

Classes of Sequentially Cohen-Macaulay Squarefree Monomial Ideals

We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen-Macaulay. As an application, we give a complete characterization of the completely squarefree lexsegment ideals which are sequentially Cohen-Macaulay and we also derive formulas for some homological invariants of this class of ideals.

Algorithms for strongly stable ideals

Strongly stable monomial ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers among saturated ideals with a given Hilbert polynomial, in this note we present three algorithms to produce all strongly stable ideals with certain prescribed properties: the saturated strongly stable ideals with a given Hilbert polynomial, the almost lexsegment ideals with a given Hilbert polynomial, and the saturated strongly stable ideals with a given Hilbert function. We also establish results for estimating the complexity of our algorithms.

Monomial ideals of minimal depth

Abstract Let S be a polynomial algebra over a field. We study classes of monomial ideals (as for example lexsegment ideals) of S having minimal depth. In particular, Stanley's conjecture holds for these ideals. Also we show that if I is a monomial ideal with Ass(S/I) = {P1, P2, ..., Ps} and Pi ⊄ ∑s1=j≠i Pj for all i ∊ [s], then Stanley’s conjecture holds for S/I.

ON THE MINIMAL GRADED FREE RESOLUTION OF POWERS OF LEXSEGMENT IDEALS

We consider powers of lexsegment ideals with a linear resolution (equivalently, with linear quotients) which are not completely lexsegment ideals. We give a complete description of their minimal graded free resolution.

Pure and Cohen–Macaulay Simplicial Complexes Associated with Squarefree Lexsegment Ideals

We classify the unmixed squarefree lexsegment ideals and determine those which are Cohen–Macaulay.

Minimal graded resolutions of reverse lexsegment ideals

Let k be a field, and let S = k[x1, …, xn] be the polynomial ring in x1, …, xn with coefficients in the field k. We study the minimal graded free S-resolutions of reverse lexsegment ideals of S. We discuss the extremal Betti numbers of initial reverse lexsegment ideals of S. Moreover, we analyze all reverse lexsegment ideals with linear resolution.

Non-linear behaviour of Castelnuovo–Mumford regularity

It is shown that the Castelnuovo–Mumford regularity of lex-segment ideals of ordinary powers as well as of Frobenius powers of a homogeneous polynomial ideal is a function of polynomial type of huge degree.

Powers of lexsegment ideals with linear resolution

All powers of lexsegment ideals with linear resolution (equivalently, with linear quotients) have linear quotients with respect to suitable orders of the minimal monomial generators. For a large subclass of lexsegment ideals, the corresponding Rees algebra has a quadratic Grobner basis, thus it is Koszul. We also find other classes of monomial ideals with linear quotients whose powers have linear quotients too.

Properties of lexsegment ideals

We show that any lexsegment ideal with linear resolution has linear quotients with respect to a suitable ordering of its minimal monomial generators. For completely lexsegment ideals with linear resolution we show that the decomposition function is regular. For arbitrary lexsegment ideals we compute the depth and the dimension. As application we characterize the Cohen--Macaulay lexsegment ideals.

A characterization of stable and Borel ideals

In this paper a combinatorial characterization of stable, lex-segment and Borel ideals in P(n) := k[X1, ..., Xn] (assuming chark = 0) is given. The last ones are special monomial ideals, widely studied after the fundamental result of A. Galligo [7] (chark = 0) and D. Bayer–M. Stillman [1] (any characteristic) stating that for every homogeneous ideal a ⊆ P(n) and generic change of coordinates the initial ideal does not vary and is actually fixed by the Borel subgroup of triangular matrices in Gl(n,k). Such a monomial ideal is called generic initial ideal of a and denoted gin (a). Also the first ones are monomial ideals (including the Borel ones) studied after S. Eliahou–M. Kervaire [6] that gave a handy way of computing their minimal free resolutions. We want to characterize stable, Borel and lex-segment ideals a ⊆ P(n) by means of the associated order ideal N (a) ⊆ N, corresponding to the terms outside a. We note first that N (a) reflects the geometric structure of the reduced union of linear varieties X(a) ⊆ P associated to a ⊆ P(n), via the well-known lifting-construction due to Hartshorne (see [13]). To each monomial ideal a ⊆ P(n) we associate a numerical character, denoted e(a) and called e-vector of a, consisting of a sequence of symbols ∞ and positive integers intercalated by finite groups of /’s. We notice that symbols ∞ do not occur if a is 0-dimensional. The numerical character e(a) gives in a very compact way the ‘geometric configuration’ of the order ideal N (a) ⊆ N and a canonical system of generators of a that in the stable case is precisely the minimal system G(a) of monomial generators of a (see Thm. 3.6). Namely, assuming a ⊆ P(n) stable, let G(a) = {m1, . . . , mr} and let m1 <L m2 <L · · · <L mr , where <L is the lexicographic ordering, then the entries of e(a) belonging to N are ordinately

Piecewise lexsegment ideals in exterior algebras

The problem of describing the Hilbert functions of homogeneous ideals of an exterior algebra over a field containing a fixed monomial ideal is considered. For this purpose the notion of a piecewise lexsegment ideal in an exterior algebra is introduced generalizing the notion of a lexsegment ideal. It is proved that if is a piecewise lexsegment ideal, then it is possible to describe the Hilbert functions of the homogeneous ideals containing in a way similar to that suggested by Kruskal and Katona for the situation . Moreover, a generalization of the extremal properties of lexsegment ideals is obtained (the inequality for the Betti numbers).