Related papers: Chordality, $d$-collapsibility, and componentwise …

In this paper we investigate the class of rigid monomial ideals. We give a characterization of the minimal free resolutions of certain classes of these ideals. Specifically, we show that the ideals in a particular subclass of rigid monomial…

We show that the ideal generated by the $(n-2)$ minors of a general symmetric $n$ by $n$ matrix has an initial ideal that is the Stanley-Reisner ideal of the boundary complex of a simplicial polytope and has the same Betti numbers.

Associated to a simple undirected graph G is a simplicial complex whose faces correspond to the independent sets of G. We call a graph G shellable if this simplicial complex is a shellable simplicial complex in the non-pure sense of…

Using the recent results on square-free Gr\"obner degenerations by Conca and Varbaro, we proved that if a homogeneous ideal $I$ of a polynomial ring is such that its initial ideal $\mathrm{in}_<(I)$ is square-free and $\beta_0(I) =…

We introduce the concept of edgewise domination in clutters, and use it to provide an upper bound for the projective dimension of any squarefree monomial ideal. We then use a simple recursion to recover a formula for the projective…

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…

We introduce the notion of Betti category for graded modules over suitably graded polynomial rings, and more generally for modules over certain small categories. Our categorical approach allows us to treat simultaneously many important…

For a simplicial complex $\Delta$, the affect of the expansion functor on combinatorial properties of $\Delta$ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papers. In this paper, we consider the…

We prove a duality theorem for simplicial complexes arising from a combinatorial construction we define, which applies to the squarefree monomial complexes for Veronese ideals of projective spaces and weighted projective spaces. Our theorem…

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 study Stanley decompositions and show that Stanley's conjecture on Stanley decompositions implies his conjecture on partitionable Cohen-Macaulay simplicial complexes. We also prove these conjectures for all Cohen-Macaulay monomial ideals…

Let $\mathcal{C}$ be a clutter with a perfect matching $e_1,...,e_g$ of K\"onig type and let $\Delta_\mathcal{C}$ be the Stanley-Reisner complex of the edge ideal of $\mathcal{C}$. If all c-minors of $\mathcal{C}$ have a free vertex and…

For a given graph whose edges are labeled with general real numbers, we consider the set of functions from the vertex set into the Euclidean plane such that the distance between the images of neighbouring vertices is equal to the…

In the present paper, motivated by a conjecture of Jahan and Zheng, we prove that componentwise polymatroidal ideals have linear quotients. This solves positively a conjecture of Bandari and Herzog.

This paper introduces two new notions of graded linear resolution and graded linear quotients, which generalize the concepts of linear resolution property and linear quotient for modules over the polynomial ring $A=k[x_1, \dots ,x_n]$.…

Fr\"oberg's classical theorem about edge ideals with $2$-linear resolution can be regarded as a classification of graphs whose edge ideals have linearity defect zero. Extending his theorem, we classify all graphs whose edge ideals have…

In $2011$, Herzog, Hibi, and Ohsugi conjectured that if $J$ is the cover ideal of a chordal graph, then $J^s$ is componentwise linear for all $s \ge 1.$ In 2022, H\`a and Tuyl considered objects more general than chordal graphs and posed…

We say that a pure simplicial complex ${\mathbf K}$ of dimension $d$ satisfies the removal-collapsibility condition if ${\mathbf K}$ is either empty or ${\mathbf K}$ becomes collapsible after removing $\tilde \beta_d ({\mathbf K}; {\mathbb…

In categorical realizability, it is common to construct categories of assemblies and categories of modest sets from applicative structures. These categories have structures corresponding to the structures of applicative structures. In the…

We define a notion of ideal for objects in the category of abstract unitary Cuntz semigroups introduced in [3] and termed Cu$^\sim$. We show that the set of ideals of a Cu$^\sim$-semigroup has a complete lattice structure. In fact, we prove…