The facet ideal of a simplicial complex

Abstract

To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By generalizing the notion of a tree from graphs to simplicial complexes, we show that ideals associated to trees satisfy sliding depth condition, and therefore have normal and Cohen-Macaulay Rees rings. We also discuss connections with the theory of Stanley-Reisner rings.