Normally torsion-free edge ideals of weighted oriented graphs

Abstract

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.

Figures (1)

• Figure 1: The partition $\{L_i(C)\}_{i=1}^3$ of $C$.

Theorems & Definitions (38)

• Remark 2.1
• Lemma 2.2
• Theorem 2.3
• Corollary 2.4
• Proposition 2.5
• Lemma 2.6
• proof
• Lemma 2.7
• Lemma 2.8
• proof