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