Resolutions of Square-free Monomial Ideals via Facet Ideals: a Survey

We survey some recent results on the minimal graded free resolution of a square-free monomial ideal. The theme uniting these results is the point-of-view that the generators of a monomial ideal correspond to the maximal faces (the facets) of a simplicial complex ∆. This correspondence gives us a new method, distinct from the Stanley-Reisner correspondence, to associate to a square-free monomial ideal a simplicial complex. In this context, the monomial ideal is called the facet ideal of ∆. Of particular interest is the case that all the facets have dimension one. Here, the simplicial complex is a simple graph G, and the facet ideal is usually called the edge ideal of G. Many people have been interested in understanding how the combinatorial data or structure of ∆ appears in or affects the minimal graded free resolution of the associated facet ideal. In the first part of this paper, we describe the current state-of-the-art with respect to this program by collecting together many of the relevant results. We sketch the main details of many of the proofs and provide pointers to the relevant literature for the remainder. In the second part we introduce some open questions which will hopefully inspire future research on this topic.