Betti categories of graded modules and applications to monomial ideals and toric rings

Abstract

We introduce the notion of Betti category for graded modules over suitably graded polynomial rings, and more generally for modules over certain small categories. Our categorical approach allows us to treat simultaneously many important cases, such as monomial ideals and toric rings. We prove that in these cases the Betti category is a finite combinatorial object that completely determines the structure of the minimal free resolution. For monomial ideals, the Betti category is the same as the Betti poset that we studied in a previous article. We describe in detail and with examples how the theory applies to the toric case, and provide an analog for toric rings of the lcm-lattice for monomial ideals.