Acyclicity of complexes of flat modules

Let R be a noetherian commutative ring, and F : · · · → F₂ → F₁ → F₀ → 0 a complex of flat R-modules. We prove that if κ(p{fraktur}) ⊗_R F{double-struck} is acyclic for every p{fraktur} ∈ Spec R, then F{double-struck} is acyclic, and H₀(F{double-struck}) is R-flat. It follows that if F{double-struck} is a (possibly unbounded) complex of flat R-modules and κ(p{fraktur}) ⊗_R F{double-struck} is exact for every p{fraktur} ∈ Spec R, then G{double-struck}⊗•_RF{double-struck} is exact for every R-complex G{double-struck}. If, moreover, F{double-struck} is a complex of projective R-modules, then it is null-homotopic (follows from Neeman's theorem).