Monomial Ideals Research Articles

When quotient of a polynomial or power series ring by a monomial ideal is half-factorial

Suppose that K is a field, S = K [ x 1 , … , x n ] or S = K [ [ x 1 , … , x n ] ] . We present a characterization of those monomial ideals I of S, for which S/I is a half-factorial ring. This characterization is related to the structure of the base field K and also depends on the combinatorics of a graph constructed from the monomial ideal I. We also show that if R [ x ] or R [ [ x ] ] is half-factorial for an arbitrary commutative ring R, then R is an integral domain.

Newton–Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals

Newton–Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals

Colon Structure of Associated Primes of Monomial Ideals

Colon Structure of Associated Primes of Monomial Ideals

Factorization in the monoid of integrally closed ideals

Given a Noetherian ring A, the collection of integrally closed ideals in A which contain a nonzerodivisor forms a cancellative monoid under the operation I * J = IJ ¯ , the integral closure of the product. The monoid is torsion-free and atomic. Restricting to monomial ideals in a polynomial ring, there is a surjective homomorphism from the Integral Polytope Group onto the Grothendieck group of integrally closed monomial ideals under translation invariance of their Newton Polyhedra. The Integral Polytope Group, the Grothendieck group of polytopes with integer vertices under Minkowski addition and translation invariance, has an explicit basis, allowing for explicit factoring in the monoid.

Products and powers of principal symmetric ideals

Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.

N-Absorbing number for some monomial ideals

We compute the absorbing number of the edge ideals of special graphs such as complete graphs, path graphs, and cycle graphs. We also compute the absorbing number for other monomial ideals. In special cases, we show that the length of a prime filtration of a monomial ideal [Formula: see text] is an upper bound for its absorbing number.

The weak Lefschetz property and mixed multiplicities of monomial ideals

The weak Lefschetz property and mixed multiplicities of monomial ideals

On Posets, Monomial Ideals, Gorenstein Ideals and their Combinatorics

On Posets, Monomial Ideals, Gorenstein Ideals and their Combinatorics

On the Normally Torsion-Freeness of Square-Free Monomial Ideals

On the Normally Torsion-Freeness of Square-Free Monomial Ideals

Simon conjecture and the $$\text{ v }$$-number of monomial ideals

Simon conjecture and the $$\text{ v }$$-number of monomial ideals

Symbolic defect of monomial ideals

Given a monomial ideal I, we study two functions that quantify ways to measure the difference between symbolic powers and usual powers of I. In many cases we determine the asymptotic growth rate of these two functions. We also perform explicit computations by using the symbolic polyhedron.

Hadamard Product of Monomial Ideals and the Hadamard Package

In this paper, we generalize and study the concept of Hadamard product of ideals of projective varieties to the case of monomial ideals. We have a research direction similar to the one of the join of monomial ideals contained in a paper of Sturmfels and Sullivant. In the second part of the paper, we give a brief tutorial on the Hadamard.m2 package of Macaulay2.

Cohen–Macaulayness of Vertex Splittable Monomial Ideals

In this paper, we give a new criterion for the Cohen–Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an inductive way of checking the Cohen–Macaulay property. As a result, we obtain characterizations for Gorenstein, level and pseudo-Gorenstein vertex splittable ideals. Furthermore, we provide new and simpler combinatorial proofs of known Cohen–Macaulay criteria for several families of monomial ideals, such as (vector-spread) strongly stable ideals and (componentwise) polymatroidals. Finally, we characterize the family of bi-Cohen–Macaulay graphs by the novel criterion for the Cohen–Macaulayness of vertex splittable ideals.

Specialization of integral closure of ideals by general elements

In this paper, we prove a result similar to results of Itoh [J. Algebra 150 (1992), pp. 101–117] and Hong-Ulrich [J. Lond. Math. Soc. (2) 90 (2014), pp. 861–878], proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form.

Generalized splittings of monomial ideals

Eliahou and Kervaire defined splittable monomial ideals and provided a relationship between the Betti numbers of the more complicated ideal in terms of the less complicated pieces. We extend the concept of splittable monomial ideals showing that an ideal which was not splittable according to the original definition is splittable in this more general definition. Further, we provide a generalized version of the result concerning the relationship between the Betti numbers.

Forcing the weak Lefschetz property for equigenerated monomial ideals

We classify the minimal number of generators of artinian equigenerated monomial ideals I I such that k [ x 1 , … , x n ] / I \Bbbk [x_1,\ldots ,x_n]/I is forced to have the weak Lefschetz property.

Finitely generated saturated multi-Rees algebras

We study the question of finite generation of saturated multi-Rees algebras and investigate the asymptotic behaviour of related length functions. In the setup of excellent local domains, we show that the saturated multi-Rees algebra of a finite collection of ideals is finitely generated when the analytic spread is not maximal and the associated length function eventually agrees with a polynomial. Similar results are obtained when we restrict to two-dimensional local UFDs with no restrictions on the analytic spread. We further prove that the saturated multi-Rees algebra of finitely many monomial ideals in a polynomial ring modulo an irreducible monomial ideal, is always finitely generated. In this case, the corresponding length function is shown to exhibit piecewise quasi-polynomial behaviour. We also produce multi-ideal versions of a theorem of Amao.

Barile–Macchia resolutions

We construct cellular resolutions for monomial ideals via discrete Morse theory. In particular, we develop an algorithm to create homogeneous acyclic matchings and we call the cellular resolutions induced from these matchings Barile–Macchia resolutions. These resolutions are minimal for edge ideals of weighted oriented forests and (most) cycles. As a result, we provide recursive formulas for graded Betti numbers and projective dimension. Furthermore, we compare Barile–Macchia resolutions to those created by Batzies and Welker and some well-known simplicial resolutions. Under certain assumptions, whenever the above resolutions are minimal, so are Barile–Macchia resolutions.

Factorization properties of quotients of polynomial and power series rings by monomial ideals

Suppose that K is a field, S = K [ x 1 , … , x n ] or S = K [ [ x 1 , … , x n ] ] and I is a monomial ideal of S. We study factorization properties of the ring R = S / I . In particular, we present conditions equivalent to R being présimplifiable, a bounded factorization ring, a finite factorization ring or a unique factorization ring.

Equality of Ordinary and Symbolic Powers of Some Classes of Monomial Ideals

Equality of Ordinary and Symbolic Powers of Some Classes of Monomial Ideals