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

H. Charalambous·A. Tchernev
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Abstract

Let $Q=k[x_1,..., x_n]$ 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 $M$.

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H. Charalambous, A. Tchernev. 2002-07-03. Free resolutions fo rmultigraded modules: a generalization of Taylor's construction. https://arxiv.org/abs/math/0207040

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