# faithfully flat

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

 $\displaystyle 0\rightarrow P\rightarrow Q\rightarrow R\rightarrow 0$

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

 $\displaystyle 0\rightarrow M\otimes_{A}P\rightarrow M\otimes_{A}Q\rightarrow M% \otimes_{A}R\rightarrow 0$

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 FaithfullyFlat 2013-03-22 14:35:55 2013-03-22 14:35:55 mathcam (2727) mathcam (2727) 5 mathcam (2727) Definition msc 16D40