On the Stanley depth and size of monomial ideals

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,...,x_n]$ be the polynomial ring in $n$ variables over the field $\mathbb{K}$. For every monomial ideal $I\subset S$, We provide a recursive formula to determine a lower bound for the Stanley depth of $S/I$. We use this formula to prove the inequality ${\rm sdepth}(S/I)\geq {\rm size}(I)$ for a particular class of monomial ideals

Intersecting faces of a simplicial complex via algebraic shifting

A family $\mathcal{A}$ of sets is {\it $t$-intersecting} if the cardinality of the intersection of every pair of sets in $\mathcal{A}$ is at least $t$, and is an {\it $r$-family} if every set in $\mathcal{A}$ has cardinality $r$. A well-known theorem of Erd\H{o}s, Ko, and Rado bounds the cardinality of a $t$-intersecting $r$-family of subsets of an $n$-element set, or equivalently of $(r-1)$-dimensional faces of a simplex with $n$ vertices. As a generalization of the Erd\H{o}s-Ko-Rado theorem, Borg presented a conjecture concerning the size of a $t$-intersecting $r$-family of faces of an arbitrary simplicial complex. He proved his conjecture for shifted complexes. In this paper we give a new proof for this result based on work of Woodroofe. Using algebraic shifting we verify Borg's conjecture in the case of sequentially Cohen-Macaulay $i$-near-cones for $t=i$.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1001.0313 by other autho

On the Stanley depth of squarefree monomial ideals

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over the field $\mathbb{K}$. Suppose that $\mathcal{C}$ is a chordal clutter with $n$ vertices and assume that the minimum edge cardinality of $\mathcal{C}$ is at least $d$. It is shown that $S/I(c_d(\mathcal{C}))$ satisfies Stanley's conjecture, where $I(c_d(\mathcal{C}))$ is the edge ideal of the $d$-complement of $\mathcal{C}$. This, in particular shows that $S/I$ satisfies Stanley's conjecture, where $I$ is a quadratic monomial ideal with linear resolution. We also define the notion of Schmitt--Vogel number of a monomial ideal $I$, denoted by ${\rm sv}(I)$ and prove that for every squarefree monomial ideal $I$, the inequalities ${\rm sdepth}(I)\geq n-{\rm sv}(I)+1$ and ${\rm sdepth}(S/I)\geq n-{\rm sv}(I)$ hold

On the hh-vector of (SrS_r) simplicial complexes

We give a negative answer to a question proposed in [3], regarding the $h$-vector of ($S_r$) simplicial complexes.Comment: To appear in J. Commut. Algebr

On the Stanley depth of powers of edge ideals

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 bipartite connected components. We prove that for every positive integer $k$, the inequalities ${\rm sdepth}(I^k/I^{k+1})\geq p$ and ${\rm sdepth}(S/I^k)\geq p$ hold. As a consequence, we conclude that $S/I^k$ satisfies the Stanley's inequality for every integer $k\geq n-1$. Also, it follows that $I^k/I^{k+1}$ satisfies the Stanley's inequality for every integer $k\gg 0$. Furthermore, we prove that if (i) $G$ is a non-bipartite graph, or (ii) at least one of the connected components of $G$ is a tree with at least one edge, then $I^k$ satisfies the Stanley's inequality for every integer $k\geq n-1$. Moreover, we verify a conjecture of the author in special cases

Saturation of Generalized Partially Hyperbolic Attractors

We prove the saturation of a generalized partially hyperbolic attractor of a $C^2$ map. As a consequence, we show that any generalized partially hyperbolic horseshoe-like attractor of a $C^1$-generic diffeomorphism has zero volume. In contrast, by modification of Poincar\'e cross section of the geometric model, we build a $C^1$-diffeomorphism with a partially hyperbolic horseshoe-like attractor of positive volume

Depth, Stanley depth and regularity of ideals associated to graphs

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 prove that ${\rm sdepth}(J)\geq n-\nu_{o}(G)$ and ${\rm sdepth}(S/J)\geq n-\nu_{o}(G)-1$, where $\nu_{o}(G)$ is the ordered matching number of $G$. We also prove the inequalities ${\rm sdepth}(J^k)\geq {\rm depth}(J^k)$ and ${\rm sdepth}(S/J^k)\geq {\rm depth}(S/J^k)$, for every integer $k\gg 0$, when $G$ is a bipartite graph. Moreover, we provide an elementary proof for the known inequality ${\rm reg}(S/I)\leq \nu_{o}(G)$

Stability of depth and Stanley depth of symbolic powers of squarefree monomial ideals

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Assume that $I\subset S$ is a squarefree monomial ideal. For every integer $k\geq 1$, we denote the $k$-th symbolic power of $I$ by $I^{(k)}$. Recently, Monta\~no and N\'u\~nez-Betancourt \cite{mn} proved that for every pair of integers $m, k\geq 1$,$${\rm depth}(S/I^{(m)})\leq {\rm depth}(S/I^{(\lceil\frac{m}{k}\rceil)}).$$We provide an alternative proof for this inequality. Moreover, we reprove the known results that the sequence $\{{\rm depth}(S/I^{(k)})\}_{k=1}^{\infty}$ is convergent and$$\min_k{\rm depth}(S/I^{(k)})=\lim_{k\rightarrow \infty}{\rm depth}(S/I^{(k)})=n-\ell_s(I),$$where $\ell_s(I)$ denotes the symbolic analytic spread of $I$. We also determine an upper bound for the index of depth stability of symbolic powers of $I$. Next, we consider the Stanley depth of symbolic powers and prove that the sequences $\{{\rm sdepth}(S/I^{(k)})\}_{k=1}^{\infty}$ and $\{{\rm sdepth}(I^{(k)})\}_{k=1}^{\infty}$ are convergent and the limit of each sequence is equal to its minimum. Furthermore, we determine an upper bound for the indices of sdepth stability of symbolic powers

Regularity of symbolic powers of cover ideals of graphs

Let $G$ be a graph which belongs to either of the following classes: (i) bipartite graphs, (ii) unmixed graphs, or (iii) claw--free graphs. Assume that $J(G)$ is the cover ideal $G$ and $J(G)^{(k)}$ is its $k$-th symbolic power. We prove that$$k{\rm deg}(J(G))\leq {\rm reg}(J(G)^{(k)})\leq (k-1){\rm deg}(J(G))+|V(G)|-1.$$We also determine families of graphs for which the above inequalities are equality

Stanley depth of the integral closure of monomial ideals

Let $I$ be a monomial ideal in the polynomial ring $S=\mathbb{K}[x_1,...,x_n]$. We study the Stanley depth of the integral closure $\bar{I}$ of $I$. We prove that for every integer $k\geq 1$, the inequalities ${\rm sdepth} (S/\bar{I^k}) \leq {\rm sdepth} (S/\bar{I})$ and ${\rm sdepth} (\bar{I^k}) \leq {\rm sdepth} (\bar{I})$ hold. We also prove that for every monomial ideal $I\subset S$ there exist integers $k_1,k_2\geq 1$, such that for every $s\geq 1$, the inequalities ${\rm sdepth} (S/I^{sk_1}) \leq {\rm sdepth} (S/\bar{I})$ and ${\rm sdepth} (I^{sk_2}) \leq {\rm sdepth} (\bar{I})$ hold. In particular, $\min_k \{{\rm sdepth} (S/I^k)\} \leq {\rm sdepth} (S/\bar{I})$ and $\min_k \{{\rm sdepth} (I^k)\} \leq {\rm sdepth} (\bar{I})$. We conjecture that for every integrally closed monomial ideal $I$, the inequalities ${\rm sdepth}(S/I)\geq n-\ell(I)$ and ${\rm sdepth} (I)\geq n-\ell(I)+1$ hold, where $\ell(I)$ is the analytic spread of $I$. Assuming the conjecture is true, it follows together with the Burch's inequality that Stanley's conjecture holds for $I^k$ and $S/I^k$ for $k\gg 0$, provided that $I$ is a normal ideal