Stanley-Reisner ideals of higher independence complexes of chordal graphs

For $t\ge 2$, the $t$-independence complex ${\mathrm{I}\mathrm{n}\mathrm{d}}_t(G)$ of a graph $G$ is the collection of all $A\subseteq V(G)$ such that each connected component of the induced subgraph $G[A]$ has at most $t-1$ vertices. The topology of ${\mathrm{I}\mathrm{n}\mathrm{d}}_t(G)$ is intimately related to the combinatorial property of $G$. In this article, we consider the Stanley-Reisner ideal $J_t(G)$ of ${\mathrm{I}\mathrm{n}\mathrm{d}}_t(G)$ and focus on its algebraic properties. We prove that for a chordal graph $G$ and for all $t$ $$\mathrm{r}\mathrm{e}\mathrm{g}(R\mathrm{/}{J}_t(G))=(t-1){\nu }_t(G)\text{ and }\mathrm{p}\mathrm{d}(R\mathrm{/}{J}_t(G))=\mathrm{b}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}(J_t(G)),$$ where ${\nu }_t(G)$ denotes the induced matching number of the corresponding hypergraph of $J_t(G)$, and $\mathrm{r}\mathrm{e}\mathrm{g}$, $\mathrm{p}\mathrm{d}$ and $\mathrm{b}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}$ stand for the regularity, projective dimension, and big height, respectively. As a consequence of the above results, we combinatorially characterize when the Stanley-Reisner ideal of the $t$-independence complex of a chordal graph has a linear resolution as well as when it satisfies the Cohen-Macaulay property. The above formulas and their consequences can be seen as a nice generalization of the classical results corresponding to the edge ideals of chordal graphs.