In commutative ring, flat is equivalent to locally free

Asked Jan 17 '18 at 08:08

Active Jan 13 '19 at 09:39

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In wikipedia https://en.m.wikipedia.org/wiki/Flat_module , particularly Case of commutative rings, they say that

"In a commutative ring, a finitely generated module is flat if and only if it is locally free, i.e. $M_P$ is free for all prime ideals"

In Atiyah anf MacDonald's commutative algebra, they proved that

"In a commutative ring, a finitely generated module is flat if and only if it is locally flat, i.e. $M_P$ is flat for all prime ideals"

So does it mean "$M_P$ is free iff $M_P$ is flat"? How? $R_P$ is only local while we need it to be also Noetherian for the statement to be true?

Thank you for your help

asked Jan 17 '18 at 08:08

T C

1

I also don't know how to remove the Noetherian hypothesis, so I suspect the first statement is false with no Noetherian hypotheses. – Qiaochu Yuan Jan 17 '18 at 09:11
See https://mathoverflow.net/questions/33522/flatness-and-local-freeness . – darij grinberg Jan 13 '19 at 13:14

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