Acyclicity of complexes of flat modules

TL;DR

This paper proves that for complexes of flat modules over a noetherian ring, acyclicity can be checked locally at prime ideals, and explores implications for tensor products and projective complexes.

Contribution

It establishes a local-global principle for acyclicity of complexes of flat modules over noetherian rings, extending known results to unbounded complexes.

Findings

Acyclicity of flat complexes can be verified locally at prime ideals.

If tensoring with residue fields yields acyclicity, then the complex is globally acyclic.

Complexes of projective modules are null-homotopic under certain conditions.

Abstract

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 $\kappa(\mathfrak p)\otimes_R\mathbb F$ 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 $\kappa(\mathfrak p)\otimes_R \mathbb F$ 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).