Monomial Ideals Research Articles

Behaviour of the Normalized Depth Function

Let $I\subset S=K[x_1,\dots,x_n]$ be a squarefree monomial ideal, $K$ a field. The $k$th squarefree power $I^{[k]}$ of $I$ is the monomial ideal of $S$ generated by all squarefree monomials belonging to $I^k$. The biggest integer $k$ such that $I^{[k]}\ne(0)$ is called the monomial grade of $I$ and it is denoted by $\nu(I)$. Let $d_k$ be the minimum degree of the monomials belonging to $I^{[k]}$. Then, $\text{depth}(S/I^{[k]})\ge d_k-1$ for all $1\le k\le\nu(I)$. The normalized depth function of $I$ is defined as $g_{I}(k)=\text{depth}(S/I^{[k]})-(d_k-1)$, $1\le k\le\nu(I)$. It is expected that $g_I(k)$ is a non-increasing function for any $I$. In this article we study the behaviour of $g_{I}(k)$ under various operations on monomial ideals. Our main result characterizes all cochordal graphs $G$ such that for the edge ideal $I(G)$ of $G$ we have $g_{I(G)}(1)=0$. They are precisely all cochordal graphs $G$ whose complementary graph $G^c$ is connected and has a cut vertex. As a far-reaching application, for given integers $1\le s<m$ we construct a graph $G$ such that $\nu(I(G))=m$ and $g_{I(G)}(k)=0$ if and only if $k=s+1,\dots,m$. Finally, we show that any non-increasing function of non-negative integers is the normalized depth function of some squarefree monomial ideal.

Some counterexamples for (strong) persistence property and (nearly) normally torsion-freeness

In this article, we present some rare counterexamples, which are related to (strong) persistence property and (nearly) normally torsion-freeness of monomial ideals. They may be useful for researchers in this field to construct the other counterexamples refuting some conjectures.

Profunctors Between Posets and Alexander Duality

We consider profunctors between posets and introduce their graph and ascent. The profunctors text {Pro}(P,Q) form themselves a poset, and we consider a partition mathcal {I}sqcup mathcal {F} of this into a down-set mathcal {I} and up-set mathcal {F}, called a cut. To elements of mathcal {F} we associate their graphs, and to elements of mathcal {I} we associate their ascents. Our basic results is that this, suitably refined, preserves being a cut: We get a cut in the Boolean lattice of subsets of the underlying set of Q times P. Cuts in finite Booleans lattices correspond precisely to finite simplicial complexes. We apply this in commutative algebra where these give classes of Alexander dual square-free monomial ideals giving the full and natural generalized setting of isotonian ideals and letterplace ideals for posets. We study text {Pro}({mathbb N}, {mathbb N}). Such profunctors identify as order preserving maps f: {mathbb N}rightarrow {mathbb N}cup {infty }. For our applications when P and Q are infinite, we also introduce a topology on text {Pro}(P,Q), in particular on profunctors text {Pro}({mathbb N},{mathbb N}).

The saturation number of monomial ideals

Let be the polynomial ring over a field and be the homogeneous maximal ideal of S. For an ideal , let be the minimum number k for which . 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.

Multiplicities of jumping numbers

We study multiplicities of jumping numbers of multiplier ideals in a smooth variety of arbitrary dimension. We prove that the multiplicity function is a quasi-polynomial, hence proving that the Poincar\'e series is a rational function. We further study when the various components of the quasi-polynomial have the highest possible degree and relate it to jumping numbers contributed by Rees valuations. Finally, we study the special case of monomial ideals.

On the strong persistence property and normality of cover ideals of theta graphs

In this paper, we show that the cover ideals of theta graphs are normal square-free monomial ideals, and also satisfy the strong persistence property. Moreover, we re-prove a well-known result which is related to the cover ideals of bipartite graphs. We finally terminate this paper by stating some theorems which introduce some classes of normal monomial ideals.

Green-Lazarsfeld index of square-free monomial ideals and their powers

Let K be a field and I be a square-free monomial ideal in the polynomial ring K[x1,…,xn]. The Green-Lazarsfeld index, index(I), counts the number of steps to reach to a syzygy minimally generated by a nonlinear form in a graded minimal free resolution of I. In this paper, we study this invariant for I and its powers from a combinatorial point of view. We characterize all square-free monomial ideals I generated in degree 3 such that index(I)>1. Utilizing this result, we also characterize all square-free monomial ideals generated in degree 3 such that index(I)>1 and index(I2)=1. In case n≤5, it is shown that index(Ik)>1 for all k if I is any square-free monomial ideal with index(I)>1.

Regularity of symbolic powers of square-free monomial ideals

Regularity of symbolic powers of square-free monomial ideals

Computing with jets

We introduce a Macaulay2 package for working with jet schemes. The main method constructs jets of ideals, polynomial rings and their quotients, ring homomorphisms, affine varieties, and (hyper)graphs. The package also includes additional methods to compute principal components and radicals of jets of monomial ideals.

Normality Criteria for Monomial Ideals

Normality Criteria for Monomial Ideals

Applications of Bar Code to Involutive Divisions and a “Greedy” Algorithm for Complete Sets

Given a finite set of terms U in n variables, we describe an algorithm which finds – if it exists – an ordering on the variables such that U is a complete set according to Janet involutive division. The algorithm, based on Bar Codes for monomial ideals, is able to adjust the variables ordering with a sort of backtracking technique, thus allowing to find the desired ordering without trying all the n! possible ones.

Vector-spread monomial ideals and Eliahou–Kervaire type resolutions

We introduce the class of vector-spread monomial ideals. This notion generalizes that of t-spread ideals introduced by Ene, Herzog and Qureshi. In particular, we focus on vector-spread strongly stable ideals, we compute their Koszul cycles and describe their minimal free resolution. As a consequence the graded Betti numbers and the Poincaré series are determined. Finally, we consider a generalization of algebraic shifting theory for such a class of ideals.

Neighborly partitions and the numerators of Rogers–Ramanujan identities

We prove two partition identities which are dual to the Rogers–Ramanujan identities. These identities are inspired by (and proved using) a correspondence between three kinds of objects: a new type of partitions (neighborly partitions), monomial ideals and some infinite graphs.

Regular sequences on squares of monomial ideals

We use initially regular sequences that consist of linear sums to explore the depth of $$R/I^2$$ , when I is a monomial ideal in a polynomial ring R. We give conditions under which these linear sums form regular or initially regular sequences on $$R/I^2$$ . We then obtain a criterion for when $${{\,\mathrm{depth}\,}}R/I^2>1$$ and a lower bound on $${{\,\mathrm{depth}\,}}R/I^2$$ .

Correction to: Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers [J. ALGEBRAIC COMBIN. 27 (2008), NO. 2, 215-245

Correction to: Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers [J. ALGEBRAIC COMBIN. 27 (2008), NO. 2, 215-245

Convex bodies and asymptotic invariants for powers of monomial ideals

Continuing a well established tradition of associating convex bodies to monomial ideals, we initiate a program to construct asymptotic Newton polyhedra from decompositions of monomial ideals. This is achieved by forming a graded family of ideals based on a given decomposition. We term these graded families powers since they generalize the notions of ordinary and symbolic powers. Asymptotic invariants for these graded families are expressed as solutions to linear optimization problems on the respective convex bodies. This allows to establish a lower bound on the Waldschmidt constant of a monomial ideal by means of a more easily computable invariant, which we introduce under the name of naive Waldschmidt constant.

On the resurgence and asymptotic resurgence of homogeneous ideals

Let \({\mathbb {K}}\) be a field and \(R = {\mathbb {K}}[x_1, \ldots , x_n]\). We obtain an improved upper bound for asymptotic resurgence of squarefree monomial ideals in R. We study the effect on the resurgence when sum, product and intersection of ideals are taken. We obtain sharp upper and lower bounds for the resurgence and asymptotic resurgence of cover ideals of finite simple graphs in terms of associated combinatorial invariants. We also explicitly compute the resurgence and asymptotic resurgence of cover ideals of several classes of graphs. We characterize a graph being bipartite in terms of the resurgence and asymptotic resurgence of edge and cover ideals. We also compute explicitly the resurgence and asymptotic resurgence of edge ideals of some classes of graphs.

When Is a Graded Free Complex Exact?

Minimal free resolutions of a finitely generated module over a polynomial ring S=k[x], with variables x={x1,…,xn} and a field k have been extensively studied. Almost all the results in the literature about minimal free resolutions give their Betti numbers, that is, the ranks of the free modules in the resolution at each degree. Several techniques have been developed to compute Betti numbers, making this a manageable problem in many cases. However, a description of the differentials in the resolution is rarely given, as this turns out to be a more difficult problem. The main purpose of this article is to give a criterion to check when a graded free complex of an S-module is exact. Unlike previous similar criteria, this one allows us to give a description of the differentials using the combinatorics of the S-module. The criterion is given in terms of the Betti numbers of the resolutions in each degree and the set of columns of the matrix representation of the differentials. In the last section, and with the aim of illustrating how to use the criterion, we apply it to one of the first better-understood cases, the edge ideal of the complete graph. However, this criterion can be used to give an explicit description of the differentials of a resolution of several monomial ideals such as the duplication of an ideal, the edge ideal of a cograph, etc.

Frobenius methods in combinatorics

We survey results produced from the interaction between methods in prime characteristic and combinatorial commutative algebra. We showcase results for edge ideals, toric varieties, Stanley-Reisner rings, and initial ideals that were proven via Frobenius. We also discuss results for monomial ideals obtained using Frobenius-like maps. Finally, we present results for F-pure rings that were inspired by work done for Stanley-Reisner rings.

Standard pairs for monomial ideals in semigroup rings

We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Standard pairs can be used as a data structure to encode such monomial ideals, providing an alternative to generating sets that is well suited to computing intersections, decompositions, and multiplicities. We give algorithms to compute standard pairs from generating sets and vice versa and make all of our results effective. We assume that the underlying semigroup ring is positively graded, but not necessarily normal. The lack of normality is at the root of most challenges, subtleties, and innovations in this work.