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Article overview
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Cohen--Macaulaynees for symbolic power ideals of edge ideals | Giancarlo Rinaldo
; Naoki Terai
; Ken-ichi Yoshida
; | Date: |
9 Mar 2012 | 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]. | Source: | arXiv, 1203.1967 | Services: | Forum | Review | PDF | Favorites |
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