Related papers: Simplicial ideals, 2-linear ideals and arithmetica…

We study initial algebras of determinantal rings, defined by minors of generic matrices, with respect to their classical generic point. This approach leads to very short proofs for the structural properties of determinantal rings. Moreover,…

Our main theorems provide a single geometric setting in which polynomial representatives for Schubert classes in the integral cohomology ring of the flag manifold are determined uniquely, and have positive coefficients for geometric…

In this short note we prove that the projective dimension of a sequentially Cohen-Macaulay square-free monomial ideal is equal to the maximal height of its minimal primes (also known as the big height), or equivalently, the maximal…

Let $I$ be a monomial ideal of the polynomial ring $S=K[x_1,...,x_4]$ over a field $K$. Then $S/I$ is sequentially Cohen-Macaulay if and only if $S/I$ is pretty clean. In particular, if $S/I$ is sequentially Cohen-Macaulay then $I$ is a…

Let $m_{12}$, $m_{13}$, ..., $m_{n-1,n}$ be the slopes of the $\binom{n}{2}$ lines connecting $n$ points in general position in the plane. The ideal $I_n$ of all algebraic relations among the $m_{ij}$ defines a configuration space called…

The main purpose of this paper is to introduce and study minimal and maximal ideals defined on ideal topological spaces. Also, we define and investigate the concepts of ideal quotient and annihilator of any subfamily of $2^X$, where $2^X$…

Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s)^{\ell} \oplus k(-2s+1)$, where $s \geq3$ and…

Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$ and $I\subset S$ be a squarefree monomial ideal generated in degree $n-2$. Motivated by the remarkable behavior of the powers of $I$ when $I$ admits a linear resolution, as…

Let $A$ be a commutative Noetherian local ring with maximal ideal $\mathfrak{m}$, and let $I$ be an ideal. The fiber cone is then an image of the polynomial ring over the residue field $A/\mathfrak{m}$. The kernel of this map is called the…

Consider a height two ideal, $I$, which is minimally generated by $m$ homogeneous forms of degree $d$ in the polynomial ring $R=k[x,y]$. Suppose that one column in the homogeneous presenting matrix $\f$ of $I$ has entries of degree $n$ and…

In the present paper, we aim to classify monomial ideals whose all matching powers are Cohen-Macaulay. We especially focus our attention on edge ideals. The Cohen-Macaulayness of the last matching power of an edge ideal is characterized,…

We consider path ideals associated to special classes of posets such as tree posets and cycles. We express their property of being sequentially Cohen-Macaulay in terms of the underlying poset. Moreover, monomial ideals, which arise from the…

In a recent work, Fouli and Lin generalized a Villarreal's result and showed that if each connected components of the line graph of a squarefree monomial ideal contains at most a unique odd cycle, then this ideal is of linear type. In this…

In this paper, we introduce and study families of squarefree monomial ideals called clique ideals and independence ideals that can be associated to a finite graph. A family of clique ideals with linear resolutions has been characterized.…

Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions derived from simplicial complexes. For a generic monomial ideal, the associated primes satisfy a saturated…

The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible…

We consider ideals in the ring $\mathbb{Z}_2[x_1,\ldots, x_n]$ that contain the polynomials $x_i^2 - x_i$ for $i = 1, \ldots, n$ and give various results related to the one-to-one correspondence between these ideals and the subsets of…

We study the class of squarefree principal vector-spread Borel ideals. We compute the minimal primary decomposition of these ideals and thereby we prove that they are sequentially Cohen-Macaulay. As the final conclusion of our results, we…

We recall a numerical criteria for Cohen--Macaulayness related to system of parameters, and introduce monomial ideals of K\"onig type which include the edge ideals of K\"onig graphs. We show that a monomial ideal is of K\"onig type if and…

We study several properties of multihomogeneous prime ideals. We show that the multigraded generic initial ideal of a prime has very special properties, for instance, its radical is Cohen-Macaulay. We develop a comprehensive study of…