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29 March 2024
 
  » arxiv » 1203.1967

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Cohen--Macaulaynees for symbolic power ideals of edge ideals
Giancarlo Rinaldo ; Naoki Terai ; Ken-ichi Yoshida ;
Date 9 Mar 2012
AbstractLet $S = K[x_1,..., x_n]$ 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].
Source arXiv, 1203.1967
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