Mathematical Research Letters
Volume 10 (2003)
Free resolutions for multigraded modules: a generalization of Taylor’s construction
Pages: 535 – 550
Let $Q=\Bbbk[x_1,\ldots, x_n]$ be a polynomial ring over a field $\Bbbk$ with the standard $\mathbb N^n$-grading. Let $\phi$ be a morphism of finite free $\mathbb N^n$-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 $\mathbb N^n$-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 $M$.