faithfully flat
FaithfullyFlat
2004-09-13 01:37:27
2005-03-18 22:10:59
Definition
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Let $A$ be a commutative ring. Then $M$ if \emph{faithfully flat} if for any $A$-modules $P, Q$, and $R$, we have
\begin{align*}
0\ra P\ra Q\ra R\ra 0
\end{align*}
is exact if and only if the $M$-tensored sequence
\begin{align*}
0\ra M\otimes_A P\ra M\otimes_A Q\ra M\otimes_A R\ra 0
\end{align*}
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).