Monomial Ideals Research Articles

Computational methods for t-spread monomial ideals

Let $K$ be a field and $S=K[x_1,\ldots,x_n]$ a standard polynomial ring over $K$. In this paper, we give new combinatorial algorithms to compute the smallest $t$-spread lexicographic set and the smallest $t$-spread strongly stable set containing a given set of $t$-spread monomials of $S$. Some technical tools allowing to compute the cardinality of $t$-spread strongly stable sets avoiding their construction are also presented. Such functions are also implemented in a \emph{Macaulay2} package, \texttt{TSpreadIdeals}, to ease the computation of well-known results about algebraic invariants for $t$-spread ideals.

Algebraic implications of neighborhood hypergraphs and their transversal hypergraphs

In this paper, we unfold balanced and totally balanced neighborhood hypergraphs to discover new classes of normally torsion-free monomial ideals. As a consequence, we establish that the closed neighborhood ideals and the dominating ideals of strongly chordal graphs are normally torsion-free. We discuss the stable sets of associated primes of the dominating ideals of cycles and characterize all the cycles with normally torsion-free dominating ideals.

The signature of a monomial ideal

<p>The irreducible decomposition of a monomial ideal has played an important role in combinatorial commutative algebra, with applications beyond pure mathematics, such as biology. Given a monomial ideal $ I $ of a polynomial ring $ S = k[{\bf x}] $ over a field $ k $ and variables $ {\bf x} = \{x_1, \ldots, x_n\} $, its incidence matrix, is the matrix whose rows are indexed by the variables $ {\bf x} $ and whose columns are indexed by its minimal generators. The main contribution of this paper is the introduction of a novel invariant of a monomial ideal $ I $, termed its signature, which could be thought of as a type of canonical form of its incidence matrix, and the proof that two monomial ideals with the same signature have essentially the same irreducible decomposition.</p>

Minimal lattice points in the Newton polyhedron and application to normal ideals

Let a1,...,an be positive integers and let ∆=NP(a1,...,an) be the Newton polyhedron associated to these integers, that is, the convex hull in Rn of the axial points that haveaiin the xi-axis. We give some characterization of the minimal elements of ∆, and then use this characterization to give an alternative simpler proof of a main result of [7] on the normality of monomial ideals.

On Regularity and Projective Dimension of Invariant Chains of Monomial Ideals

On Regularity and Projective Dimension of Invariant Chains of Monomial Ideals

Mapping cones of monomial ideals over exterior algebras

Let K be a field, V a finite dimensional K-vector space and E the exterior algebra of V. We analyze iterated mapping cone over E. If I is a monomial ideal of E with linear quotients, we show that the mapping cone construction yields a minimal graded free resolution F of I via the Cartan complex. Moreover, we provide an explicit description of the differentials in F when the ideal I has a regular decomposition function. Finally, we get a formula for the graded Betti numbers of a new class of monomial ideals including the class of strongly stable ideals.

Integral Closure of Powers of Generalized Edge Ideals

This article studies a new class of monomial ideals associated with a simple graph 𝐺, called generalized edge ideal, denoted by 𝐼𝑔(𝐺). Assuming that all the vertices 𝑥 have an exponent greater than 1 in 𝐼𝑔(𝐺), we completely characterize the graph 𝐺 for which 𝐼𝑔(𝐺) is integrally closed, and show that this is equivalent to 𝐼𝑔(𝐺) being normal i.e., all integral powers of 𝐼𝑔(𝐺) are integrally clased. We also give a necessary and sufficient condition for when 𝐺 is the star-shaped graph. Finally, we give a necessary and sufficient condition when the generalized edge ideal of a complete graph is integrally closed.

The Eliahou-Kervaire resolution over a skew polynomial ring

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings.

Face Posets of Tropical Polyhedra and Monomial Ideals

We exhibit several posets arising from commutative algebra, order theory, tropical convexity as potential face posets of tropical polyhedra, and we clarify their inclusion relations. We focus on monomial tropical polyhedra, and deduce how their geometry reflects properties of monomial ideals. Their vertex-facet lattice is homotopy equivalent to a sphere and encodes the Betti numbers of an associated monomial ideal.

Normally torsion-free edge ideals of weighted oriented graphs

Let be the edge ideal of a weighted oriented graph D, let G be the underlying graph of D, and let be the n-th symbolic power of I defined using the minimal primes of I. We prove that if and only if the following conditions hold: (i) every vertex of D with weight greater than 1 is a sink and (ii) G has no triangles. Using a result of Mandal and Pradhan and the classification of normally torsion-free edge ideals of graphs, we prove that for all if and only if the following conditions hold: (a) every vertex of D with weight greater than 1 is a sink and (b) G is bipartite. If I has no embedded primes, conditions (a) and (b) classify when I is normally torsion-free. Using polyhedral geometry and integral closure, we give necessary conditions for the equality of ordinary and symbolic powers of monomial ideals with a minimal irreducible decomposition. Then, we classify when the dual of the edge ideal of a weighted oriented graph is normally torsion-free.

On the Lefschetz property for quotients by monomial ideals containing squares of variables

Let Δ be an (abstract) simplicial complex on n vertices. One can define the Artinian monomial algebra A ( Δ ) = k [ x 1 , … , x n ] / 〈 x 1 2 , … , x n 2 , I Δ 〉 , where k is a field of characteristic 0 and I Δ is the Stanley-Reisner ideal associated to Δ . In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of A ( Δ ) in terms of the simplicial complex Δ . We are able to completely analyze when the WLP holds in degree 1, complementing work by Migliore, Nagel and Schenck on the WLP for quotients by quadratic monomials. We give a complete characterization of all 2-dimensional pseudomanifolds Δ such that A ( Δ ) satisfies the WLP. We also construct Artinian Gorenstein algebras that fail the WLP by combining our results and the standard technique of Nagata idealization.

Generalized weights of codes over rings and invariants of monomial ideals

Generalized weights of codes over rings and invariants of monomial ideals

On constructions related to the generalized Taylor complex

In this paper, we extend constructions and results for the Taylor complex to the generalized Taylor complex constructed by Herzog. We construct an explicit DG-algebra structure on the generalized Taylor complex and extend a result of Katthän on quotients of the Taylor complex by DG-ideals. We introduce a generalization of the Scarf complex for families of monomial ideals, and show that this complex is always a subcomplex of the minimal free resolution of the sum of these ideals. We also give an example of an ideal where the generalized Scarf complex strictly contains the standard Scarf complex. Moreover, we introduce the notion of quasitransverse monomial ideals, and prove a list of results relating to Golodness, Koszul homology, and other homological properties for such ideals.

A Taylor resolution over complete intersections

The Taylor resolution is a fundamental object in the study of free resolutions over the polynomial ring, due to its explicit formula, cellular/combinatorial structure, and applicability to any and all monomial ideals. This paper generalizes the Taylor resolution to complete intersection rings via the Eisenbud–Shamash construction.

Differentially fixed ideals in affine semigroup rings

We give a complete characterization of when monomial ideals are fixed by differential operators of affine semigroup rings over [Formula: see text]. Perhaps surprisingly, every monomial ideal is fixed by an infinite set of homogeneous differential operators and is in fact determined by them. This opens up a new tool for studying monomial ideals. We explore applications of this to (mixed) multiplier ideals and other variants as well as give examples of detecting ideal membership in integrally closed powers and symbolic powers of squarefree monomial ideals.

Normality and associated primes of closed neighborhood ideals and dominating ideals

In this paper, we first give some sufficient criteria for normality of monomial ideals. As applications, we show that closed neighborhood ideals of complete bipartite graphs are normal, and hence satisfy the (strong) persistence property. We also prove that dominating ideals of complete bipartite graphs are nearly normally torsion-free. In addition, we show that dominating ideals of [Formula: see text]-wheel graphs, under certain condition, are normal.

Toric rings arising from vertex cover ideals

We extend the sortability concept to monomial ideals which are not necessarily generated in one degree and as an application we obtain normal Cohen-Macaulay toric rings attached to vertex cover ideals of graphs. Moreover, we consider a construction on a graph called a clique multi-whiskering which always produces vertex cover ideals with componentwise linear powers.

Combinatorial Bounds of the Regularity of Elimination Ideals

Elimination ideals are monomial ideals associated to simple graphs, not necessarily square–free, was introduced by Anwar and Khalid. These ideals are Borel type. In this paper, we obtain sharp combinatorial upper bounds of the Castelnuovo–Mumford regularity of elimination ideals corresponding to certain family of graphs.

Ombinatorial Bounds of the Regularity of Elimination Ideals

Elimination ideals are monomial ideals associated to simple graphs, not necessarily square--free, was introduced by Anwar and Khalid. These ideals are Borel type. In this paper, we obtain sharp combinatorial upper bounds of the Castelnuovo--Mumford regularity of elimination ideals corresponding to certain family of graphs.

Linear resolutions and quasi-linearity of monomial ideals

We introduce the notion of quasi-linearity and prove it is necessary for a monomial ideal to have a linear resolution and clarify all the quasi-linear monomial ideals generated in degree $2$. We also introduce the notion of a strongly linear monomial over a monomial ideal and prove that if $\mathbf {u}$ is a monomial strongly linear over $I$ then $I$ has a linear resolution (respectively is quasi-linear) if and only if $I+\mathbf {u}\mathfrak {p}$ has a linear resolution (respectively is quasi-linear). Here $\mathfrak {p}$ is any monomial prime ideal.