Acyclicity of complexes of flat modules

Abstract

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).