Betti numbers of multigraded modules of generic type

Let R=k[x1,…,xm]R=k[x1,…,xm] be the polynomial ring over a field kk with the standard ZmZm-grading (multigrading), let L be a Noetherian multigraded R  -module, let βi,α(L)βi,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 βi,α(L)βi,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≥1i≥1 in which βi,α(L)βi,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.