Related papers: Arithmetical rank of strings and cycles

Given a square-free monomial ideal $I$, satisfying certain hypotheses, in a polynomial ring $R$ over a field $\mathbb{K}$, we compute the projective dimension of $I$. Specifically, we focus on the cases where the 1-skeleton of an associated…

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 show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex or a path, the arithmetical rank equals the projective dimension.

We present a closed formula and a simple algorithmic procedure to compute the projective dimension of square-free monomial ideals associated to string or cycle hypergraphs. As an application, among these ideals we characterize all the…

We prove a sufficient and a necessary condition for a square-free monomial ideal $J$ associated to a (dual) hypergraph to have projective dimension equal to the minimal number of generators of $J$ minus 2. We also provide an effective…

Given a square-free monomial ideal $I$ in a polynomial ring $R$ over a field $\mathbb{K}$, one can associate it with its LCM-lattice and its hypergraph. In this short note, we establish the connection between the LCM-lattice and the…

We consider the edge ideals of large classes of graphs with whiskers and for these ideals we prove that the arithmetical rank is equal to the big height. Then we extend these results to other classes of squarefree monomial ideals, generated…

Let $I\subset S=K[x_1,...,x_n]$ be a lexsegment edge ideal or the Alexander dual of such an ideal. In both cases it turns out that the arithmetical rank of $I$ is equal to the projective dimension of $S/I.$

Let $G$ be a connected and simple graph on the vertex set $[n]$. To the graph $G$ one can associate the generalized binomial edge ideal $J_{m}(G)$ in the polynomial ring $R=K[x_{ij}: i \in [m], j \in [n]]$. We provide a lower bound for the…

We give an affirmative answer to a question due to J. He and A. Van Tuyl, proving that the arithmetical rank of a special monomial ideal equals to the projective dimension of corresponding quotient module.

We prove that, for the edge ideal of a graph whose cycles are pairwise vertex-disjoint, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes.

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.…

Let $\H$ be a unimodular hypergraph over the vertex set $[n]$ and let $J(\H)$ be the cover ideal of $\H$ in the polynomial ring $R=K[x_1,\ldots,x_n]$. We show that $\reg J(\H)^s$ is a linear function in $s$ for all $s\geqslant r\left\lceil…

Let $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\mathrm{deg}\ x_i = 1$ and $I \subset S$ a homogeneous ideal of $S$ with $\dim S/I = d$. The Hilbert series of $S/I$ is of the form…

We show that for the edge ideals of a certain class of forests, the arithmetical rank equals the projective dimension.

Let $I$ be a monomial squarefree ideal of a polynomial ring $S$ over a field $K$ such that the sum of every three different of its minimal prime ideals is the maximal ideal of $S$, or more general a constant ideal. We associate to $I$ a…

Let $I_{G} \subset K[x_{1},...,x_{m}]$ be the toric ideal associated to a finite graph $G$. In this paper we study the binomial arithmetical rank and the $G$-homogeneous arithmetical rank of $I_G$ in 2 cases: $G$ is bipartite, $I_G$ is…

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…

We give a new proof of Hilbert's Syzygy Theorem for monomial ideals. In addition, we prove the following. If S=k[x_1,...,x_n] is a polynomial ring over a field, M is a squarefree monomial ideal in S, and each minimal generator of M has…

We show that a monomial ideal $I$ has projective dimension $\leq$ 1 if and only if the minimal free resolution of $S/I$ is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the…