### Abstract

Let R be a noetherian commutative ring, and F : · · · → F_{2} → F_{1} → F_{0} → 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 H_{0}(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}⊗•_{R}F{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).

Original language | English |
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Pages (from-to) | 111-118 |

Number of pages | 8 |

Journal | Nagoya Mathematical Journal |

Volume | 192 |

Publication status | Published - Dec 1 2008 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Hashimoto, M. (2008). Acyclicity of complexes of flat modules.

*Nagoya Mathematical Journal*,*192*, 111-118.