A constructive approach to minimal free resolutions of path ideals of trees
Keywords:
Betti numbers, Path ideals, Rooted trees,, Monomial idealsAbstract
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.
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