Free resolutions fo rmultigraded modules: a generalization of Taylor's construction

Abstract

Let $Q=k[x_1,...,x_n]$ be a polynomial ring over a field $k$ with the standard $N^n$-grading. Let $\varphi$ 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 $T_{\bullet }(\varphi )$ of finite free $N^n$-graded $Q$-modules that generalizes Taylor's resolution. This complex provides a free resolution for the cokernel $M$ of $\varphi$ when $\varphi$ satisfies certain rank criteria. We also introduce the Scarf complex of $\varphi$, and a notion of ``generic'' morphism. Our main result is that the Scarf complex of $\varphi$ is a minimal free resolution of $M$ when $\varphi$ is minimal and generic. Finally, we introduce the LCM-lattice for $\varphi$ and establish its significance in determining the minimal resolution of $M$.

LaTeX, 15 pages