Related papers: Ideals with componentwise linear powers

To a vector configuration one can associate a polynomial ideal generated by powers of linear forms, known as a power ideal, which exhibits many combinatorial features of the matroid underlying the configuration. In this note we observe that…

We investigate products J of ideals of "row initial" minors in the polynomial ring K[X] defined by a generic m-by-n matrix. Such ideals are shown to be generated by a certain set of standard bitableaux that we call superstandard. These…

We consider powers of lexsegment ideals with a linear resolution (equivalently, with linear quotients) which are not completely lexsegment ideals. We give a complete description of their minimal graded free resolution.

An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring $\text{Int}(D)=\{f\in K[x]\mid f(D)\subseteq D\}$,…

Let I=I(D) be the edge ideal of a weighted oriented graph D. We determine the irredundant irreducible decomposition of I. Also, we characterize the associated primes and the unmixed property of I. Furthermore, we give a combinatorial…

In this article we prove that every toric ideal associated with a gap-free graph $G$ has a squarefree lexicographic initial ideal. Moreover, in the particular case when the complementary graph of $G$ is chordal (i.e. when the edge ideal of…

Let $R$ be a commutative ring with unity. The co-maximal ideal graph of $R$, denoted by $\Gamma(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and…

We develop a functorial framework for the ideal theory of commutative semirings using coherent frames and spectral spaces. Two central constructions-the radical ideal functor and the $k$-radical ideal functor-are shown to yield coherent…

Let I be an ideal of height two in R=k[x_0,x_1] generated by forms of the same degree, and let K be the ideal of defining equations of the Rees algebra of I. Suppose that the second largest column degree in the syzygy matrix of I is e. We…

Let $\Phi(x,y)\in\mathbb{C}[x,y]$ be a symmetric polynomial of partial degree $d$. The graph $G(\Phi)$ is defined by taking $\mathbb{C}$ as set of vertices and the points of $\mathbb{V}(\Phi(x,y))$ as edges. We study the following problem:…

Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G) \subset S$ the edge ideal of a finite graph $G$ on $n$ vertices. Given a vector $\mathfrak{c}\in\mathbb{N}^n$ and an integer $q\geq 1$, we…

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 $A=k[x_1, \dots ,x_n]$.…

Assume that $G$ is a graph with edge ideal $I(G)$. For every integer $s\geq 1$, we denote the squarefree part of the $s$-th symbolic power of $I(G)$ by $I(G)^{\{s\}}$. We determine an upper bound for the regularity of $I(G)^{\{s\}}$ when…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over the field $K$, and let $I\subset S$ be a graded ideal. It is shown that for $k \gg0$ the postulation number of $I^k$ is bounded by a linear function of $k$, and it is a linear function…

Let $A$ be a commutative algebra equipped with an action of a group $G$. The so-called $G$-primes of $A$ are the equivariant analogs of prime ideals, and of central importance in equivariant commutative algebra. When $G$ is an infinite…

Let $G$ be a graph with $n$ vertices and let $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. Assume that $I(G)$ and $J(G)$ denote the edge ideal and the cover ideal of $G$, respectively. We…

Let I be the ideal of minors of a 2 by n matrix of linear forms with the expected codimension. In this paper we prove that the Rees algebra of I and its special fiber ring are Cohen-Macaulay and Koszul; in particular, they are quadratic…

In this article we associate to every lattice ideal $I_{L,\rho}\subset K[x_1,..., x_m]$ a cone $\sigma $ and a graph $G_{\sigma}$ with vertices the minimal generators of the Stanley-Reisner ideal of $\sigma $. To every polynomial $F$ we…

We relate the annihilators of graded components of the canonical module of a graded Cohen-Macaulay ring to colon ideals of powers of the homogeneous maximal ideal. In particular, we connect them to the core of the maximal ideal. An…

Let K be a field, m and n positive integers, and X = {x_1,...,x_n}, and Y = {y_1,..., y_m} sets of independent variables over K. Let A be the polynomial ring K[X] localized at (X). We prove that every prime ideal P in A^ = K[[X]] that is…