Monomial ideals of minimal depth

TURKISH JOURNAL OF MATHEMATICS, 2016

Let S = K[x1,. .. , xn] be a polynomial ring over a field K in n variables and I a squarefree monomial ideal of S with Schmitt-Vogel number sv(I). In this paper, we show that sdepth (I) ≥ max {1, n − 1 − ⌊ sv(I) 2 ⌋}, which improves the lower bound obtained by Herzog, Vladoiu, and Zheng. As some applications, we show that Stanley's conjecture holds for the edge ideals of some special n-cyclic graphs with a common edge.

In dimension two, we study complete monomial ideals combinatorially, their Rees algebras and develop effective means to find their defining equations.

Partially supported by NSF grant DMS-8301870. 159 160 J. HERZOG, W. V. VASCONCELOS AND R. VILLARREAL stein precisely when I is strongly Cohen-Macaulay ([9, (6.5)]).

Journal of Algebraic Combinatorics, 2011

Let K be a field and S = K[x 1 , . . . , x n ]. In 1982, Stanley defined what is now called the Stanley depth of an S-module M, denoted sdepth(M), and conjectured that depth(M) ≤ sdepth(M) for all finitely generated S-modules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when M = I/J with J ⊂ I being monomial S-ideals.

Illinois Journal of Mathematics, 2012

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 the lexsegment ideals the corresponding Rees algebra has a quadratic Gröbner basis, thus it is Koszul. We also find other classes of monomial ideals with linear quotients whose powers have linear quotients too.

In this paper we show that the depth and the Stanley depth of the factor of two monomial ideals is invariant under taking a so called canonical form. It follows easily that the Stanley Conjecture holds for the factor if and only if it holds for its canonical form. In particular, we construct an algorithm which simplifies the depth computation and using the canonical form we massively reduce the run time for the sdepth computation.

Arxiv preprint math/0702569, 2007

Let S = K[x1,...,xn] be a polynomial ring over a field K and I ⊂ S a monomial ideal. If S/I is Gorenstein of codimension three then a description of I is given in [1, Theorem 6.1] in terms of the minimal system of monomial generators. Here we are interested to describe monomial ideals I ...

Archiv der Mathematik

This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal I in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound µ(I 2 ) ≥ 9 for the number of minimal generators of I 2 . Recently, Gasanova constructed monomial ideals such that µ(I) > µ(I n ) for any positive integer n. In reference to them, we construct a certain class of monomial ideals such that µ(I) > µ(I 2 ) > • • • > µ(I n ) = (n + 1) 2 for any positive integer n, which provides one of the most unexpected behaviors of the function µ(I k ).

2002

In this paper we investigate the question of normality for special monomial ideals in a polynomial ring over a field. We first include some expository sections that give the basics on the integral closure of a ideal, the Rees algebra on an ideal, and some fundamental results on the integral closure of a monomial ideal.

Algebra Colloquium, 2014

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.

Let K be a field and S = K[x 1 , . . . , x n ] be the polynomial ring in n variables over K. Let G be a graph with n vertices. Assume that I = I(G) is the edge ideal of G and J = J(G) is its cover ideal. We prove that sdepth(J) ≥ n−ν o (G) and sdepth(S/J) ≥ n − ν o (G) − 1, where ν o (G) is the ordered matching number of G. We also prove the inequalities sdepth(J k ) ≥ depth(J k ) and sdepth(S/J k ) ≥ depth(S/J k ), for every integer k ≫ 0, when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg(S/I) ≤ ν o (G).

Bulletin of the Iranian Mathematical Society

We prove that a monomial ideal I generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also conjecture that the polymatroidal ideals can be characterized with linear quotients property with respect to the reverse lexicographical ordering of the minimal generators induced by every ordering of variables. We prove our conjecture in many special cases.

2013

In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for any edge E of the hypergraph, there exists an edge F in the 2-collage such that |E \ F | ≤ 1. The Castelnuovo-Mumford regularity of the edge ideal of a simple hypergraph is bounded above by a multiple of the minimum size of a 2-collage. We also give a recursive formula to compute the regularity of a vertex-decomposable hypergraph. Finally, we show that regularity in the graph case is bounded by a certain statistic based on maximal packings of nondegenerate star subgraphs.

Communications in Algebra

Let S = K[x1,. .. , xn] be the polynomial ring over a field K and m = (x1,. .. , xn) be the homogeneous maximal ideal of S. For an ideal I ⊂ S, let sat(I) be the minimum number k for which I : m k = I : m k+1. In this paper, we compute the saturation number of irreducible monomial ideals and their powers. We apply this result to find the saturation number of the ordinary powers and symbolic powers of some families of monomial ideals in terms of the saturation number of irreducible components appearing in an irreducible decomposition of these ideals. Moreover, we give an explicit formula for the saturation number of monomial ideals in two variables.

Fundamenta Mathematicae, 2004

We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the Hilbert-Samuel polynomial, and we compute bounds on the maximal order type.