# faithfully flat

Let $A$ be a commutative ring. Then $M$ if *faithfully flat* if for any $A$-modules $P,Q$, and $R$, we have

$0\to P\to Q\to R\to 0$ |

is exact if and only if the $M$-tensored sequence

$0\to M{\otimes}_{A}P\to M{\otimes}_{A}Q\to M{\otimes}_{A}R\to 0$ |

is exact. (Note that the “if and only if” clause makes this stronger than the definition of flatness).

Equivalently, an $A$-module $M$ is faithfully flat iff $M$ is flat and the functor^{} $-{\otimes}_{A}M$ is a faithful functor^{} (and hence the name).

Title | faithfully flat |
---|---|

Canonical name | FaithfullyFlat |

Date of creation | 2013-03-22 14:35:55 |

Last modified on | 2013-03-22 14:35:55 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 5 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 16D40 |