  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
$T_\bullet(\phi)$ 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
$M$.
