Betti numbers of multigraded modules of generic type

Let R=k[x₁,. .,x_m] be the polynomial ring over a field k with the standard Zm-grading (multigrading), let L be a Noetherian multigraded R-module, let β_i,α(L) the ith (multigraded) Betti number of L of multidegree α. We introduce the notion of a generic (relative to L) multidegree, and the notion of multigraded module of generic type. When the multidegree α is generic (relative to L) we provide a Hochster-type formula for β_i,α(L) as the dimension of the reduced homology of a certain simplicial complex associated with L. This allows us to show that there is precisely one homological degree i≥1 in which β_i,α(L) is non-zero and in this homological degree the Betti number is the β-invariant of a certain minor of a matroid associated to L. In particular, this provides a precise combinatorial description of all multigraded Betti numbers of L when it is a multigraded module of generic type.