On the Facet Ideal of an Expanded Simplicial Complex

Abstract

For a simplicial complex ∆, the affect of the expansion functor on combinatorial properties of ∆ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papers. In this paper, we consider the facet ideal I(∆) and its Alexander dual which we denote by J∆ to see how the expansion functor alter the algebraic properties of these ideals. It is shown that for any expansion ∆ α the ideals J∆ and J∆α have the same total Betti numbers and their Cohen-Macaulayness are equivalent, which implies that the regularities of the ideals I(∆) and I(∆ α ) are equal. Moreover, the projective dimensions of I(∆) and I(∆ α ) are compared. In the sequel for a graph G, some properties that are equivalent in G and its expansions are presented and for a Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable) graph G, we give some conditions for adding or removing a vertex from G, so that the remaining graph is still Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable).