Acyclicity of complexes of flat modules

Abstract

Let $R$ be a noetherian commutative ring, and

\mathbb{F} : \cdots \rightarrow F_{2} \rightarrow F_{1} \rightarrow F_{0} \rightarrow 0

a complex of flat $R$-modules. We prove that if $\kappa(\mathfrak{p}) \otimes_{R} \mathbb{F}$ is acyclic for every $\mathfrak{p} \in \operatorname{Spec} R$, then $\mathbb{F}$ is acyclic, and $H_{0}(\mathbb{F})$ is $R$-flat. It follows that if $\mathbb{F}$ is a (possibly unbounded) complex of flat $R$-modules and $\kappa(\mathfrak{p}) \otimes_{R} \mathbb{F}$ is exact for every $\mathfrak{p} \in \operatorname{Spec} R$, then $\mathbb{G} \otimes_{R}^{\bullet} \mathbb{F}$ is exact for every $R$-complex $\mathbb{G}$. If, moreover, $\mathbb{F}$ is a complex of projective $R$-modules, then it is null-homotopic (follows from Neeman's theorem).