In commutative ring, flat is equivalent to locally free
$begingroup$
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
1,0741725
$endgroup$
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
1,0741725
$endgroup$
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
1,0741725
$endgroup$
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
1,0741725
asked Jan 17 '18 at 8:08
chí trung châuchí trung châu
1,0741725
asked Jan 17 '18 at 8:08
chí trung châuchí trung châu
1,0741725
asked Jan 17 '18 at 8:08
chí trung châuchí trung châu
1,0741725
asked Jan 17 '18 at 8:08
chí trung châuchí trung châu
1,0741725
1,0741725
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 |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2608804%2fin-commutative-ring-flat-is-equivalent-to-locally-free%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2608804%2fin-commutative-ring-flat-is-equivalent-to-locally-free%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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