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faithfully flat
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(Definition)
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Let $A$ be a commutative ring. Then $M$ if faithfully flat if for any $A$ -modules $P, Q$ , and $R$ , we have
is exact if and only if the $M$ -tensored sequence
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).
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"faithfully flat" is owned by mathcam.
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Cross-references: faithful functor, functor, flat, iff, stronger, 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 2164 times total.
Classification:
| AMS MSC: | 16D40 (Associative rings and algebras :: Modules, bimodules and ideals :: Free, projective, and flat modules and ideals) |
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Pending Errata and Addenda
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