Cohen–Macaulay properties of square-free monomial ideals

In this paper, we study simplicial complexes as higher-dimensional graphs in order to produce algebraic statements about their facet ideals. We introduce a large class of square-free monomial ideals with Cohen–Macaulay quotients, and a criterion for the Cohen–Macaulayness of facet ideals of simplicial trees. Along the way, we generalize several concepts from graph theory to simplicial complexes.