Betti numbers of thick trees

Abstract

A tree could be defined as follows. An edge is a tree. If T_k − 1 = ∪_i = 1^k − 1e_i is a tree with k − 1 edges e_i, and e_k an edge, then T_k = T_k − 1 ∪ e_k is a tree if T_k − 1 ∩ e_k is a point. We generalize this construction: A simplex S₁ of dimension  ≥ 1 is a thick tree. If G_k − 1 = ∪_i = 1^k − 1S_i is a thick tree, where S_i are simplices of dimension  ≥ 1, and S_k a new simplex of dimension  ≥ 1, then G_k − 1 ∪ S_k is a thick tree if G_k − 1 ∩ S_k is a point. All homological properties of Stanley-Reisner rings of thick trees are well known. We determine the Hilbert series and Betti numbers for Stanley-Reisner rings of skeletons of thick trees. From this one can read of projective dimension, regularity, and judge when they are Cohen-Macaulay.