Acyclicity of complexes of flat modules

Let $R$ be a noetherian commutative ring, and $\mathbb F: ...\rightarrow F_2\rightarrow F_1\rightarrow F_0\rightarrow 0$ a complex of flat $R$-modules. We prove that if $κ(\mathfrak p)\otimes_R\mathbb F$ is acyclic for every $\mathfrak p\in\Spec R$, 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 $κ(\mathfrak p)\otimes_R \mathbb F$ is exact for every $\mathfrak p\in\Spec R$, 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).

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