Stanley–Reisner ideals of higher independence complexes of chordal graphs

For [Formula: see text], the [Formula: see text]-independence complex [Formula: see text] of a graph [Formula: see text] is the collection of all [Formula: see text] such that each connected component of the induced subgraph [Formula: see text] has at most [Formula: see text] vertices. The topology of [Formula: see text] is intimately related to the combinatorial property of [Formula: see text]. In this paper, we consider the Stanley–Reisner ideal [Formula: see text] of [Formula: see text] and focus on its algebraic properties. We prove that for a chordal graph [Formula: see text] and for all [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the induced matching number of the corresponding hypergraph of [Formula: see text], and [Formula: see text], [Formula: see text] and [Formula: see text] 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 [Formula: see text]-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.