PlanetMath: faithfully flat

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faithfully flat

(Definition)

Let be a commutative ring. Then if faithfully flat if for any -modules , and , we have

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

is exact if and only if the

-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 -module is faithfully flat iff is flat and the functor $-\otimes_A M$ is a faithful functor (and hence the name).

"faithfully flat" is owned by mathcam.

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Cross-references: faithful functor, functor, flat, iff, clause, sequence, commutative ring

This is version 2 of faithfully flat, born on 2004-09-13, modified 2005-03-18.
Object id is 6167, canonical name is FaithfullyFlat.
Accessed 1512 times total.

Classification:

16D40 (Associative rings and algebras :: Modules, bimodules and ideals :: Free, projective, and flat modules and ideals)

Pending Errata and Addenda

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