Algebraic properties of the path ideal of a tree

Details Algebraic properties of the path ideal of a tree Algebraic properties of the path ideal of a tree

2009-02-05

arxiv.org/abs/0902.0902v3

Mathematics

Commutative Algebra

Added some additional hypotheses to old Theorem 3.6; now Lemma 3.6 and Theorem 3.7

Scientific paper

The path ideal (of length t >=2) of a graph G is the monomial ideal, denoted I_t(G), whose generators correspond to the directed paths of length t in G. We study some of the algebraic properties of I_t(G) when G is a tree. We first show that I_t(G) is the facet ideal of a simplicial tree. As a consequence, the quotient ring R/I_t(G) is always sequentially Cohen-Macaulay, and the Betti numbers of R/I_t(G) do not depend upon the characteristic of the field. We study the case of the line graph in greater detail at the end of the paper.

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