On the Facet Ideal of an Expanded Simplicial Complex

Abstract

For a simplicial complex $$\Delta $$ , the effect of the expansion functor on combinatorial properties of $$\Delta $$ and algebraic properties of its Stanley–Reisner ring has been studied in some previous papers. In this paper, we consider the facet ideal $$I(\Delta )$$ and its Alexander dual which we denote by $$J_{\Delta }$$ to see how the expansion functor alters the algebraic properties of these ideals. It is shown that for any expansion $$\Delta ^{\alpha }$$ the ideals $$J_{\Delta }$$ and $$J_{\Delta ^{\alpha }}$$ have the same total Betti numbers and their Cohen–Macaulayness is equivalent, which implies that the regularities of the ideals $$I(\Delta )$$ and $$I(\Delta ^{\alpha })$$ are equal. Moreover, the projective dimensions of $$I(\Delta )$$ and $$I(\Delta ^{\alpha })$$ 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 (respectively, 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 (respectively, sequentially Cohen–Macaulay and shellable).