Related papers: A note on Stanley's conjecture for monomial ideals

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…

Given a tree T on n vertices, there is an associated ideal I of a polynomial ring in n variables over a field, generated by all paths of a fixed length of T. We show that such an ideal always satisfies the Konig property and classify all…

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…

We compute the Stanley depth for the quotient ring of a square free Veronese ideal and we give some bounds for the Stanley depth of a square free Veronese ideal. In particular, it follows that both satisfy the Stanley's conjecture.

Among other results, we prove that if $I$ is a monomial ideal of $S=K[x_1,\ldots,x_n]$, where $K$ is a field, and $a\geq b-1\geq0$ are integers such that $a+b\leq\mathrm{proj~dim}(S/I)$, then $$t_{a+b}\leq…

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 $M$.…

We introduce the concept of Stanley decompositions in the localized polynomial ring $S_f$ where $f$ is a product of variables, and we show that the Stanley depth does not decrease upon localization. Furthermore it is shown that for monomial…

In this paper, we introduce the concept of $k$-clean monomial ideals as an extension of clean monomial ideals and present some homological and combinatorial properties of them. Using the hierarchal structure of $k$-clean ideals, we show…

Let $R = K[X_1, ..., X_n]$ be a polynomial ring over some field $K$. In this paper, we prove that the $k$-th syzygy module of the residue class field $K$ of $R$ has Stanley depth $n-1$ for $\lfloor n/2 \rfloor \leq k < n$, as it had been…

In [7] we obtained a formula for the Hilbert depth of squarefree Veronese ideals in a standard graded polynomial ring by relating it to the Hilbert depth of powers of the irrelevant maximal ideal. In this paper, we prove that these two…

The Stanley's Conjecture on Cohen-Macaulay multigraded modules is studied especially in dimension 2. In codimension 2 similar results were obtained by Herzog, Soleyman-Jahan and Yassemi. As a consequence of our results Stanley's Conjecture…

Let $I$ be a squarefree monomial ideal of $S=K[x_1,\ldots,x_n]$. We prove that if $\operatorname{hdepth}(S/I)\leq 8$ or $n\leq 10$ then $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)-1$.

We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $I\subset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for…

Let $I$ be a monomial ideal in a polynomial ring $S=K[x_1,\ldots,x_n]$ over a field $K$ with $n=2$ or $3$, and let $\overline{I}$ be its integral closure. We will show that $\text{reg} (\overline{I}) \le \text{reg} (I)$. Furthermore, if $I$…

We extend a result of Minh and Trung to get criteria for $\depth I=\depth\sqrt{I}$ where $I$ is an unmixed monomial ideal of the polynomial ring $S=K[x_1,..., x_n]$. As an application we characterize all the pure simplicial complexes…

We construct monomial ideals with the property that their depth function has any given number of strict local maxima.

We prove that a monomial ideal $I$ generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also…

We define nice partitions of the multicomplex associated to a Stanley ideal. As the main result we show that if the monomial ideal $I$ is a CM Stanley ideal, then $I^p$ is a Stanley ideal as well, where $I^p$ is the polarization of $I$.

Let $I$ be a monomial ideal of $S=K[x_1,\ldots,x_n]$. We show that the following are equivalent: (i) $I$ is principal, (ii) $\operatorname{hdepth}(I)=n$, (iii) $\operatorname{hdepth}(S/I)=n-1$. Assuming that $I$ is squarefree, we prove that…

Let $I$ be a squarefree monomial ideal of $S=K[x_1,\ldots,x_n]$. We prove that if $\operatorname{hdepth}(S/I)\leq 6$ of $n\leq 9$ then $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)$, giving a positive answer to a problem putted…