Cohen--Macaulaynees for symbolic power ideals of edge ideals

Abstract

Let $S=K[x_1,...,x_n]$ be a polynomial ring over a field $K$. Let $I(G)\subseteq S$ denote the edge ideal of a graph $G$. We show that the $\ell$th symbolic power $I(G{)}^{(\ell )}$ is a Cohen-Macaulay ideal (i.e., $S/I(G{)}^{(\ell )}$ is Cohen-Macaulay) for some integer $\ell \ge 3$ if and only if $G$ is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers $I(G{)}^{(\ell )}$ are Cohen-Macaulay ideals. Similarly, we characterize graphs $G$ for which $S/I(G{)}^{(\ell )}$ has (FLC). As an application, we show that an edge ideal $I(G)$ is complete intersection provided that $S/I(G{)}^{\ell }$ is Cohen-Macaulay for some integer $\ell \ge 3$. This strengthens the main theorem in [Effective Cowsik-Nori theorem for edge ideals by M.Crupi, G.Rinaldo, N.Terai, and K.Yoshida, Comm. Alg. 38 (2010), 3347-3357].

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Cited by in corpus (1)

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