## Nagoya Mathematical Journal

### Acyclicity of complexes of flat modules

Mitsuyasu Hashimoto

#### Abstract

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

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

Dates
First available in Project Euclid: 22 December 2008

https://projecteuclid.org/euclid.nmj/1229955907

Mathematical Reviews number (MathSciNet)
MR2477613

Zentralblatt MATH identifier
1158.13003

#### Citation

Hashimoto, Mitsuyasu. Acyclicity of complexes of flat modules. Nagoya Math. J. 192 (2008), 111--118. https://projecteuclid.org/euclid.nmj/1229955907

#### References

• E. E. Enochs, Minimal pure injective resolutions of flat modules, J. Algebra, 105 (1987), 351--364.
• M. Hashimoto, Auslander-Buchweitz Approximations of Equivariant Modules, London Mathematical Society Lecture Note Series 282, Cambridge, 2000.
• A. Neeman, The homotopy category of flat modules, preprint.
• N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math., 65 (1988), 121--154.