2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64455Let $Q=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ with the standard $N^n$-grading. Let $ϕ$ 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 $T_\bullet(ϕ)$ of finite free $N^n$-graded $Q$-modules that generalizes Taylor's resolution. This complex provides a free resolution for the cokernel $M$ of $ϕ$ when $ϕ$ satisfies certain rank criteria. We also introduce the Scarf complex of $ϕ$, and a notion of ``generic'' morphism. Our main result is that the Scarf complex of $ϕ$ is a minimal free resolution of $M$ when $ϕ$ is minimal and generic. Finally, we introduce the LCM-lattice for $ϕ$ and establish its significance in determining the minimal resolution of $M$.LaTeX, 15 pagesCommutative Algebra13D02; 13C05text