Acyclicity of complexes of flat modules

Let $R$ be a noetherian commutative ring, and $$\mathbb{F}:\mathrm{.}\mathrm{.}\mathrm{.}\to F_2\to F_1\to F_0\to 0$$ a complex of flat $R$-modules. We prove that if $\kappa (\mathfrak{p}){\otimes }_R\mathbb{F}$ is acyclic for every $\mathfrak{p}\in \text{Spec}R$, 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 $\kappa(\mathfrak p)\otimes_R \mathbb F$isexactforevery$\mathfrak p\in\Spec R$,then$\mathbb G\otimes_R^\bullet\mathbb F$isexactforevery$R$-complex$\mathbb G$. If, moreover, $\mathbb{F}$ is a complex of projective $R$-modules, then it is null-homotopic (follows from Neeman’s theorem).