# On the facet ideal of an expanded simplicial complex

## TL;DR

This paper investigates how the expansion functor affects the algebraic properties of facet ideals of simplicial complexes and their duals, revealing invariances and conditions for preserving Cohen-Macaulayness.

## Contribution

It demonstrates that expansion preserves Betti numbers and Cohen-Macaulayness of facet ideals and explores conditions for maintaining Cohen-Macaulay properties in graph expansions.

## Key findings

- Betti numbers of dual ideals are preserved under expansion
- Cohen-Macaulayness is equivalent before and after expansion
- Conditions are provided for maintaining Cohen-Macaulay properties in graph modifications

## Abstract

For a simplicial complex $\Delta$, the affect 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 alter 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 are 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 (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).