Betti numbers of path ideals of cycles and line

Details Betti numbers of path ideals of cycles and line Betti numbers of path ideals of cycles and line

2011-10-30

arxiv.org/abs/1110.6653v1

Mathematics

Commutative Algebra

Scientific paper

Let $K$ be a field and $G$ be a finite ordered simple graph with vertex set $V = \{x_1, ..., x_n\}$ and edge set. We define $I_t (G)$ to be the ideal of $K [x_1, ..., x_n]$ generated by the monomials of the form $x_{i_1}x_{i_2} ... x_{i_t}$ where $x_{i_1}, x_{i_2}, ..., x_{i_t}$ is a path of $G$. Path ideals were introduced by Conca and De Negri in 1999. Later in 2009 He and Van Tuyl studied the sequential Cohen-Macaulayness of $I_t (G)$ where $G$ is a rooted tree, and most recently the graded Betti numbers of $I_t (G)$ where $G$ is a rooted tree were investigated by Bouchat, Tai Ha and O'keefe. In this paper we give a formula to compute all the graded Betti numbers of $I_t (G)$ where G is a cycle and a line. We also give a formula to compute the projective dimension and Castelnuovo-Mumford regularity of $I_t (G)$ where G is a cycle and a line.

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