Nagoya Mathematical Journal

Acyclicity of complexes of flat modules

Mitsuyasu Hashimoto

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Let $R$ be a noetherian commutative ring, and

\mathbb{F} : \cdots \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 $\mathfrak{p} \in \operatorname{Spec} R$, 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 $\mathfrak{p} \in \operatorname{Spec} R$, 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).

Article information

Nagoya Math. J., Volume 192 (2008), 111-118.

First available in Project Euclid: 22 December 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13C11: Injective and flat modules and ideals
Secondary: 13C10: Projective and free modules and ideals [See also 19A13]


Hashimoto, Mitsuyasu. Acyclicity of complexes of flat modules. Nagoya Math. J. 192 (2008), 111--118.

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