Componentwise linearity of ideals arising from graphs

Details Componentwise linearity of ideals arising from graphs Componentwise linearity of ideals arising from graphs

2009-11-10

arxiv.org/abs/0911.2026v1

Matematiche (Catania) 63 (2008), no. 2, 185--189 (2009)

Mathematics

Commutative Algebra

5 pages

Scientific paper

Let $G$ be a simple undirected graph on $n$ vertices. Francisco and Van Tuyl have shown that if $G$ is chordal, then $\bigcap_{\{x_i,x_j\}\in E_G} < x_i,x_j>$ is componentwise linear. A natural question that arises is for which $t_{ij}>1$ the ideal $\bigcap_{\{x_i,x_j\}\in E_G}< x_i, x_j>^{t_{ij}}$ is componentwise linear, if $G$ is chordal. In this report we show that $\bigcap_{\{x_i,x_j\}\in E_G} < x_i, x_j>^{t}$ is componentwise linear for all $n\geq 3$ and positive $t$, if $G$ is a complete graph. We give also an example where $G$ is chordal, but the intersection ideal is not componentwise linear for any $t>1$.

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