Related papers: Rigid monomial ideals

Given $\Sigma\subset\mathbb K[x_1,\ldots,x_k]$, any finite collection of linear forms, some possibly proportional, and any $1\leq a\leq |\Sigma|$, it has been conjectured that $I_a(\Sigma)$, the ideal generated by all $a$-fold products of…

For any finite poset $P$ we have the poset of isotone maps $\text{Hom}(P,\mathbb{N})$, also called $P^{op}$-partitions. To any poset ideal ${\mathcal J}$ in $\text{Hom}(P,\mathbb{N})$, finite or infinite, we associate monomial ideals: the…

The shedding vertices of simplicial complexes are studied from an algebraic point of view. Based on this perspective, we introduce the class of ass-decomposable monomial ideals which is a generalization of the class of Stanley-Reisner…

The main result of this paper is that all antichains are finite in the poset of monomial ideals in a polynomial ring, ordered by inclusion. We present several corollaries of this result, both simpler proofs of results already in the…

Let $S_n$ be a polynomial ring with $n$ variables over a field and $\{I_n\}_{n \geq 1}$ a chain of ideals such that each $I_n$ is a monomial ideal of $S_n$ fixed by permutations of the variables. In this paper, we present a way to determine…

In 2017, Cooper et al. proposed a conjecture providing a lower bound for the Waldschmidt constant of monomial ideals. We confirm this conjecture for some classes of monomial ideals. Recently, M\'endez, Pinto, and Villarreal formulated a…

In this paper we study the normality of monomial ideals using linear programming and graph theory. We give normality criteria for monomial ideals, for ideals generated by monomials of degree two, and for edge ideals of graphs and clutters…

This paper focuses on asymptotic properties of random monomial ideals through a statistical viewpoint. It extends the study of redundancy in monomial ideals by analyzing the poset density of the LCM-lattice. We explore how this density…

Let R=k[x_1,...,x_n] be a polynomial ring over a field k. Let J={j_1,...,j_t} be a subset of [n]={1,...,n}, and let m_J denote the ideal (x_{j_1},...,x_{j_t}) of R. Given subsets J_1,...,J_s of [n] and positive integers a_1,...,a_s, we…

Let $X$ be an $(m\times n)$-matrix of indeterminates, and let $J$ be the ideal generated by a set $\mathcal{S}$ of maximal minors of $X$. We construct the linear strand of the resolution of $J$. This linear strand is determined by the…

Let $K$ be a field, $S$ a polynomial ring and $E$ an exterior algebra over $K$, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in $S$ and $E$ when passing to their generic…

Let $A$ be a semigroup whose only invertible element is 0. For an $A$-homogeneous ideal we discuss the notions of simple $i$-syzygies and simple minimal free resolutions of $R/I$. When $I$ is a lattice ideal, the simple 0-syzygies of $R/I$…

We show that the regularity of monomial ideals whose associated prime ideals are totally ordered by inclusion is linearly bounded.

In this paper we show that the Stanley depth, as well as the usual depth, are essentially determined by the lcm-lattice. More precisely, we show that for quotients $I/J$ of monomial ideals $J\subset I$, both invariants behave monotonic with…

We survey some recent results on the minimal graded free resolution of a square-free monomial ideal. The theme uniting these results is the point-of-view that the generators of a monomial ideal correspond to the maximal faces (the facets)…

Let $K$ be a field, $V$ a $K$-vector space with basis $e_1,\ldots,e_n$, and $E$ the exterior algebra of $V$. To a given monomial ideal $I\subsetneq E$ we associate a special monomial ideal $J$ with generators in the same degrees as those of…

Let $I$ be a perfect ideal of height 3 in a Gorenstein local ring $R$. Let $\mathbb{F}$ be the minimal free resolution of $I$. A sequence of linear maps, which generalize the multiplicative structure of $\mathbb{F}$, can be defined using…

A sparse generic matrix is a matrix whose entries are distinct variables and zeros. Such matrices were studied by Giusti and Merle who computed some invariants of their ideals of maximal minors. In this paper we extend these results by…

In this paper we study monomial ideals attached to posets, introduce generalized Hibi rings and investigate their algebraic and homological properties. The main tools to study these objects are Groebner basis theory, the concept of…

A celebrated theorem of Fr\"oberg gives a complete combinatorial classification of quadratic square-free monomial ideals with a linear resolution. A generalization of this theorem to higher degree square-free monomial ideals is an active…