多重分次模的自由分解：Taylor构造的推广
Free resolutions fo rmultigraded modules: a generalization of Taylor's construction
摘要：
令$Q=k[x_1,..., x_n]$是域$k$上的具有标准$N^n$分级的多项式环。设$ϕ$是有限自由$N^n$ -分次$Q$ -模的态射.我们把最初出现在单项理想的上下文中的几个概念和结构翻译到这个环境中。首先，利用Buchsbaum-Rim复形的一个修正，构造了有限自由$N^n$ -分次$Q$ -模的一个典型复形$T_\bullet(ϕ)$，推广了Taylor分解.当$ϕ$满足一定的秩准则时，该复合体为$ϕ$的协核$M$提供了一个自由分解。我们还介绍了$ϕ$的Scarf复形和"泛型“态射的概念。我们的主要结果是：当$ϕ$是极小的且一般的时，$ϕ$的Scarf复形是$M$的极小自由分解.最后，我们引入了$ϕ$的LCM-格，并建立了它在确定$M$的最小分辨率方面的意义。
英文摘要：
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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