A constructive approach to minimal free resolutions of path ideals of trees

Year 2017, Volume: 4 Issue: 1, 23 - 35, 11.01.2017

Rachelle R. Bouchat Tricia Muldoon Brown

https://doi.org/10.13069/jacodesmath.63088

Abstract

For a rooted tree $\Gamma ,$ we consider path ideals of $\Gamma$, which are ideals that are generated by all directed paths of a fixed length in $\Gamma$. In this paper, we provide a combinatorial description of the minimal free resolution of these path ideals. In particular, we provide a class of subforests of $\Gamma$ that are in one-to-one correspondence with the multi-graded Betti numbers of the path ideal as well as providing a method for determining the projective dimension and the Castelnuovo-Mumford regularity of a given path ideal.

Keywords

Betti numbers , Path ideals , Rooted trees , Monomial ideals

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