Free resolutions for multigraded modules: a generalization of Taylor's construction

Let Q = k[x(1),...,x(n)] be a polynomial ring over a field k with the standard N-n-grading. Let phi be a morphism of finite free N-n-graded Q-modules. We translate to this setting several notions and constructions that appear originally in the context of monomial ideals. First, using a modification of the Buchsbaum-Rim complex, we construct a canonical complex T-.(phi) of finite free N-n-graded Q-modules that generalizes Taylor's resolution. This complex provides a free resolution for the cokernel M of phi when phi satisfies certain rank criteria. We also introduce the Scarf complex of phi, and a notion of "generic" morphism. Our main result is that the Scarf complex of phi is a minimal free resolution of M when phi is minimal and generic. Finally, we introduce the LCM-lattice for phi and establish its significance in determining the minimal resolution of M.