Related papers: Squarefree monomial ideals with maximal depth

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…

Let $I\subset R=K[x_1, \ldots, x_n]$ be a square-free monomial ideal, $\mathfrak{q}$ be a prime monomial ideal in $R$, $h$ be a square-free monomial in $R$ with $\mathrm{supp}(h) \cap (\mathrm{supp}(\mathfrak{q}) \cup…

There is a natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction. Let H be a finite simple hypergraph, and let J = J(H) be its cover ideal in a polynomial ring R.…

The aim of this paper is to study the associated primes of powers of squarefree monomial ideals. Hypergraphs and squarefree monomial ideals are strongly connected. The cover ideal $J(H)$ of a hypergraph $H$ is the intersection of the primes…

An explicit construction is given of a minimal free resolution of the ideal generated by all squarefree monomials of a given degree. The construction relies upon and exhibits the natural action of the symmetric group on the syzygy modules.…

Let $I$ be a squarefree monomial ideal of a polynomial algebra over a field minimally generated by $f_1,...,f_r$ of degree $ d\geq 1$, and a set $E$ of monomials of degree $\geq d+1$. Let $J\subsetneq I$ be a squarefree monomial ideal…

Let $K$ be a field and $S=K[x_1,\ldots,x_n]$, the ring of polynomials in $n$ variables, over $K$. Using the fact that the Hilbert depth is an upper bound for the Stanley depth of a quotient of squarefree monomial ideals $0\subset…

We give a new combinatorial characterization of the big height of a squarefree monomial ideal leading to a new bound for the projective dimension of a monomial ideal.

Let $I\supsetneq J$ be two square free monomial ideals of a polynomial algebra over a field generated in degree $\geq 1$, resp. $\geq 2$ . Almost always when $I$ contains precisely one variable, the other generators having degrees $\geq 2$,…

We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of…

We introduce the concept of edgewise domination in clutters, and use it to provide an upper bound for the projective dimension of any squarefree monomial ideal. We then use a simple recursion to recover a formula for the projective…

We survey some recent results on the minimal graded free resolution of a square-free monomial ideal. The theme uniting these results is the point-of-view that the generators of a monomial ideal correspond to the maximal faces (the facets)…

Let $I$ be an arbitrary nonzero squarefree monomial ideal of dimension $d$ in a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_n]$. Let $\mu$ be the number of associated primes of $S/I$ of dimension $d$. We prove that the multiplicity of…

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 $G$ is a graph with edge ideal $I(G)$. We prove that the modules $S/\overline{I(G)^k}$ and…

Let $I$ be a square-free monomial ideal in a polynomial ring $R=K[x_1,\ldots, x_n]$ over a field $K$, $\mathfrak{m}=(x_1, \ldots, x_n)$ be the graded maximal ideal of $R$, and $\{u_1, \ldots, u_{\beta_1(I)}\}$ be a maximal independent set…

Let $I\subseteq S=K[x_1,\ldots,x_n]$ be a homogeneous ideal equipped with a monomial order $<$. We show that if $\operatorname{in}_<(I)$ is a square-free monomial ideal, then $S/I$ and $S/\operatorname{in}_<(I)$ have the same connectedness…

Let $K$ be a field and $S = K[x_1,\dots,x_n]$ be a polynomial ring over $K$. We discuss the behaviour of the extremal Betti numbers of the class of squarefree strongly stable ideals. More precisely, we give a numerical characterization of…

The aim of this paper is to characterize simplicial complexes which have standard graded vertex cover algebras. This property has several nice consequences for the squarefree monomial ideals defining these algebras. It turns out that such…

Let $I$ be a squarefree monomial ideal of a polynomial ring $S$. In this paper, we prove that the arithmetical rank of $I$ is equal to the projective dimension of $S/I$ when one of the following conditions is satisfied: (1) $\mu (I) \leq…

We define a new combinatorial object, which we call a labeled hypergraph, uniquely associated to any square-free monomial ideal. We prove several upper bounds on the regularity of a square-free monomial ideal in terms of simple…