In commutative ring, flat is equivalent to locally free
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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
commutative-algebra
asked Jan 17 '18 at 8:08
chí trung châuchí trung châu
1,0801725
$endgroup$
add a comment |
$begingroup$
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
commutative-algebra
asked Jan 17 '18 at 8:08
chí trung châuchí trung châu
1,0801725
$endgroup$
1
$begingroup$
I also don't know how to remove the Noetherian hypothesis, so I suspect the first statement is false with no Noetherian hypotheses.
$endgroup$
– Qiaochu Yuan
Jan 17 '18 at 9:11
$begingroup$
See mathoverflow.net/questions/33522/flatness-and-local-freeness .
$endgroup$
– darij grinberg
Jan 13 at 13:14
add a comment |
$begingroup$
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
commutative-algebra
asked Jan 17 '18 at 8:08
chí trung châuchí trung châu
1,0801725
$endgroup$
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
commutative-algebra
commutative-algebra
asked Jan 17 '18 at 8:08
chí trung châuchí trung châu
1,0801725
asked Jan 17 '18 at 8:08
chí trung châuchí trung châu
1,0801725
asked Jan 17 '18 at 8:08
chí trung châuchí trung châu
1,0801725
asked Jan 17 '18 at 8:08
chí trung châuchí trung châu
1,0801725
asked Jan 17 '18 at 8:08
chí trung châuchí trung châu
1,0801725
1,0801725
1
$begingroup$
I also don't know how to remove the Noetherian hypothesis, so I suspect the first statement is false with no Noetherian hypotheses.
$endgroup$
– Qiaochu Yuan
Jan 17 '18 at 9:11
$begingroup$
See mathoverflow.net/questions/33522/flatness-and-local-freeness .
$endgroup$
– darij grinberg
Jan 13 at 13:14
add a comment |
1
$begingroup$
I also don't know how to remove the Noetherian hypothesis, so I suspect the first statement is false with no Noetherian hypotheses.
$endgroup$
– Qiaochu Yuan
Jan 17 '18 at 9:11
$begingroup$
See mathoverflow.net/questions/33522/flatness-and-local-freeness .
$endgroup$
– darij grinberg
Jan 13 at 13:14
1
1
$begingroup$
I also don't know how to remove the Noetherian hypothesis, so I suspect the first statement is false with no Noetherian hypotheses.
$endgroup$
– Qiaochu Yuan
Jan 17 '18 at 9:11
$begingroup$
I also don't know how to remove the Noetherian hypothesis, so I suspect the first statement is false with no Noetherian hypotheses.
$endgroup$
– Qiaochu Yuan
Jan 17 '18 at 9:11
$begingroup$
See mathoverflow.net/questions/33522/flatness-and-local-freeness .
$endgroup$
– darij grinberg
Jan 13 at 13:14
$begingroup$
See mathoverflow.net/questions/33522/flatness-and-local-freeness .
$endgroup$
– darij grinberg
Jan 13 at 13:14
add a comment |
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$begingroup$
I also don't know how to remove the Noetherian hypothesis, so I suspect the first statement is false with no Noetherian hypotheses.
$endgroup$
– Qiaochu Yuan
Jan 17 '18 at 9:11
$begingroup$
See mathoverflow.net/questions/33522/flatness-and-local-freeness .
$endgroup$
– darij grinberg
Jan 13 at 13:14