Related papers: Stanley depth of monomial ideals

In analogy to the skeletons of a simplicial complex and their Stanley--Reisner ideals we introduce the skeletons of an arbitrary monomial ideal $I\subset S=K[x_1,...,x_n]$. This allows us to compute the depth of $S/I$ in terms of its…

We prove that if $I$ is a monomial ideal with linear quotients in a ring of polynomials $S$ in $n$ indeterminates and $\operatorname{depth}(S/I)=n-2$, then $\operatorname{sdepth}(S/I)=n-2$ and, if $I$ is squarefree,…

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

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

Let $S$ be a ring of polynomials in finitely many variables over a field. In this paper we give lower bounds for depth and Stanley depth of modules of the type $S/I^t$ for $t\geq1$, where $I$ is the edge ideal of some caterpillar and…

The aim of this paper is to study the Stanley depth of symbolic powers of a squarefree monomial ideal. We prove that for every squarefree monomial ideal $I$ and every pair of integers $k, s\geq 1$, the inequalities ${\rm sdepth}…

In this paper we study the behavior of the size of a monomial ideal under polarization and under generic deformations. As an application, we extend a result relating the size and the Stanley depth of a squarefree monomial ideal obtained by…

We prove that the edge ideals of line and cyclic graphs and their quotient rings satisfy the Stanley conjecture. We compute the Stanley depth for the quotient ring of the edge ideal associated to a cycle graph of length $n$, given a precise…

Let $I$ be the edge ideal of a clutter $\mathcal{C}$ in a polynomial ring $S$. In this paper, we present estimations of the Stanley depth of $I$ as well as the Stanley regularity of $S/I$, in terms of combinatorial data from the clutter…

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 $S=K[x_1,\ldots,x_n]$ be the ring of polynomials over a field $K$. Given two monomial ideals $0\subset I\subsetneq J \subset S$, we present a new method to compute the Hilbert depth of $J/I$. As an application, we show that if $u\in S$…

Let $S=K[x_1,...,x_n]$ be a polynomial ring in $n$ variables over the field $K$. For integers $1\leq t< n$ consider the ideal $I=(x_1,...,x_t)\cap(x_{t+1}, ...,x_n)$ in $S$. In this paper we bound from above the Stanley depth of the ideal…

Let $K$ be a infinite field, $S=K[x_1,\ldots,x_n]$ and $0\subset I\subsetneq J\subset S$ two squarefree monomial ideals. In a previous paper we proved a new formula for the Hilbert depth of $J/I$. In this paper, we illustrate how one can…

We study the family of depolarizations of a squarefree monomial ideal $I$, i.e. all monomial ideals whose polarization is $I$. We describe a method to find all depolarizations of $I$ and study some of the properties they share and some they…

We give upper bounds for the Stanley depth of edge ideals of certain k-partite clutters. In particular, we generalize a result of Ishaq about the Stanley depth of the edge ideal of a complete bipartite graph. A result of Pournaki, Seyed…

We show that all monomial ideals in the polynomial ring in at most 3 variables are pretty clean and that an arbitrary monomial ideal $I$ is pretty clean if and only if its polarization $I^p$ is clean. This yields a new characterization of…

Let $K$ be a field and $S=K[x_1,...,x_n]$. In 1982, Stanley defined what is now called the Stanley depth of an $S$-module $M$, denoted $\sdepth(M)$, and conjectured that $\depth(M) \le \sdepth(M)$ for all finitely generated $S$-modules $M$.…

Let $I\subset S$ be a graded ideal of a standard graded polynomial ring $S$ with coefficients in a field $K$, and let $\text{v}(I)$ be the $\text{v}$-number of $I$. In previous work, we showed that for any graded ideal $I\subset S$…

Let $K$ be a field and $I$ a monomial ideal of the polynomial ring $S=K[x_1,..., x_n]$ generated by monomials $u_1,u_2,..., u_t$. We show that $S/I$ is pretty clean if either: 1) $u_1,u_2,..., u_t$ is a filter-regular sequence, 2)…

We give sharp bounds for the Stanley depth of a special class of ideals of Borel type.