Explicit Betti Numbers for Skeletons of Chordal Clique Complexes and Their Alexander Duals

Abstract

Let ${\Delta }_{\mathbf{r}}={\Delta }_{\mathbf{r}}(n_1,\dots ,n_e)$ be the clique complex of a chordal graph with maximal cliques $S_1,\dots ,S_e$ in a leaf order, where $n_m=|S_m|$, $r_m=|S_{m+1}\cap (S_1\cup \cdots \cup S_m)|$, $\mathbf{r}=(r_1,\dots ,r_{e-1})$, and $N_{\mathbf{r}}$ is the number of vertices. We determine the graded Betti numbers ${\beta }_{i,j}(\mathbb{K}[({\Delta }_{\mathbf{r}}{)}_{(k)}])$ of every skeleton and derive explicit formulas for its regularity, projective dimension, depth, multiplicity, Cohen-Macaulayness, and initially Cohen-Macaulayness. We also compute its extremal Betti numbers and describe its integral homology and homotopy type. For the Alexander dual ${\Delta }_{\mathbf{r}}^{\vee }$, we construct an explicit minimal multigraded resolution whose shifts are determined by $N_{\mathbf{r}}-n_m$ and $N_{\mathbf{r}}-r_m$, and obtain its graded and multigraded Betti numbers, canonical module, Cohen-Macaulay type, $a$-invariant, and Gorenstein criterion. We further show that every proper skeleton $({\Delta }_{\mathbf{r}}^{\vee }{)}_{(k)}$ is Cohen-Macaulay and level, determine its Betti numbers and type, and characterize its Gorenstein cases. Finally, we show that the graded Betti table of $\mathbb{K}[{\Delta }_{\mathbf{r}}^{\vee }]$ determines the multisets $n_m$ and $r_m$ up to a common shift.

25 pages. Revised version with strengthened results