Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen--Macaulay vertex-weighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen--Macaulay and satisfy Alexander dualityComment: Special volume dedicated to Professor Antonio Campillo, Springer, to appea

Cohen-Macaulay Weighted Oriented Chordal and Simplicial Graphs

Herzog, Hibi, and Zheng classified the Cohen-Macaulay edge ideals of chordal graphs. In this paper, we classify Cohen-Macaulay edge ideals of (vertex) weighted oriented chordal and simplicial graphs, a more general class of monomial ideals. In particular, we show that the Cohen-Macaulay property of these ideals is equivalent to the unmixed one and hence, independent of the underlying field.Comment: 7 pages, 1 figur

Matching powers of monomial ideals and edge ideals of weighted oriented graphs

We introduce the concept of matching powers of monomial ideals. Let $I$ be a monomial ideal of $S=K[x_1,\dots,x_n]$, with $K$ a field. The $k$th matching power of $I$ is the monomial ideal $I^{[k]}$ generated by the products $u_1\cdots u_k$ where $u_1,\dots,u_k$ is a monomial regular sequence contained in $I$. This concept naturally generalizes that of squarefree powers of squarefree monomial ideals. We study depth and regularity functions of matching powers of monomial ideals and edge ideals of weighted oriented graphs. We show that the last nonvanishing power of a quadratic monomial ideal is always polymatroidal and thus has a linear resolution. When $I$ is a non-quadratic edge ideal of a weighted oriented forest, we characterize when $I^{[k]}$ has a linear resolution

Normally torsion-free edge ideals of weighted oriented graphs

Let $I=I(D)$ be the edge ideal of a weighted oriented graph $D$, let $G$ be the underlying graph of $D$, and let $I^{(n)}$ be the $n$-th symbolic power of $I$ defined using the minimal primes of $I$. We prove that $I^2=I^{(2)}$ if and only if (i) every vertex of $D$ with weight greater than $1$ is a sink and (ii) $G$ has no triangles. As a consequence, using a result of Mandal and Pradhan, and the classification of normally torsion-free edge ideals of graphs, it follows that $I^n=I^{(n)}$ for all $n\geq 1$ if and only if (a) every vertex of $D$ with weight greater than $1$ is a sink and (b) $G$ is bipartite. If $I$ has no embedded primes, conditions (a) and (b) classify when $I$ is normally torsion-free. Using polyhedral geometry and integral closure, we give necessary conditions for the equality of ordinary and symbolic powers of monomial ideals with a minimal irreducible decomposition. Then, we classify when the Alexander dual of the edge ideal of a weighted oriented graph is normally torsion-free.Comment: We added some result

Regularity of symbolic and ordinary powers of weighted oriented graphs and their upper bounds

In this paper, we compare the regularities of symbolic and ordinary powers of edge ideals of weighted oriented graphs. For a weighted oriented graph $D$, we give a lower bound for \reg(I(D)^{(k)}), if $V^+$ are sinks. If $D$ has an induced directed path $(x_i,x_j),(x_j,x_r) \in E(D)$ of length $2$ with $w(x_j)\geq 2$, then we show that \reg(I(D)^{(k)})\leq \reg(I(D)^k) for all $k\geq 2$. In particular, if $D$ is bipartite, then the above inequality holds for all $k\geq 2$. For any weighted oriented graph $D$, if $V^+$ are sink vertices, then we show that \reg(I(D)^{(k)}) \leq \reg(I(D)^k) with $k=2,3$. We further study when these regularities are equal. As a consequence, we give sharp linear upper bounds for regularity of symbolic powers of certain classes of weighted oriented graphs. Furthermore, we compare the regularity of symbolic powers of weighted oriented graphs $D$ and $D'$, where $D'$ is obtained from $D$ by adding a pendent. We show linear upper bounds for regularity of symbolic and ordinary powers of complete graph $K_n$ and $K_n'$.Comment: This is an updated version of arXiv preprint arXiv:2308.04705. Comments are welcom