Mumford's reconstruction of topology via local ring and f.g field extension over algebraically closed field
$begingroup$
Let $K$ be a f.g. field extension over an algebraically closed field $k$. Let $(O_x,m_x)$ be a family of maximal local subrings of $K$.(Here maximal means there is no further local ring $(O,m)subset K$ s.t. $(O,m)supset (O_x,m_x)$.)
Then he mentioned topology can be recovered as follows "for each $fin K$, let $U_f$ be the set of local rings in $X$ containing $f$."
$textbf{Q1:}$ Is it true $(O_x,m_x)$ had better be DVR?(It is clear that DVR satisfies maximality condition but it is not clear the converse holds. DVR corresponds to smooth objects but there is no good faith that variety is smooth.) What is the condition to see there is no further local ring dominating over $(O_x,m_x)$?
$textbf{Q2:}$ Is what written in the book correct? I feel he means that $U_f$ should be the set of local rings in $X$ with $f$ not contained in maximal ideals.
$textbf{Q3:}$ Let $X$ be a prevariety. Then $X$ is a variety iff for all $x,yin X$ s.t. $xneq y$, there is no local ring $Osubset k(X)$ s.t. $O$ dominates both $O_x$ and $O_y$. The book is not going to prove converse. What is the reference for the converse statement?(It feels like some sort of point separation statement.)
Ref. Mumford Red book 2nd Edition(pg 40, Chpt 1, Sec 6. Local Criterion)
abstract-algebra algebraic-geometry commutative-algebra
asked Jan 3 at 21:45
user45765user45765
2,6882724
$endgroup$
add a comment |
$begingroup$
Let $K$ be a f.g. field extension over an algebraically closed field $k$. Let $(O_x,m_x)$ be a family of maximal local subrings of $K$.(Here maximal means there is no further local ring $(O,m)subset K$ s.t. $(O,m)supset (O_x,m_x)$.)
Then he mentioned topology can be recovered as follows "for each $fin K$, let $U_f$ be the set of local rings in $X$ containing $f$."
$textbf{Q1:}$ Is it true $(O_x,m_x)$ had better be DVR?(It is clear that DVR satisfies maximality condition but it is not clear the converse holds. DVR corresponds to smooth objects but there is no good faith that variety is smooth.) What is the condition to see there is no further local ring dominating over $(O_x,m_x)$?
$textbf{Q2:}$ Is what written in the book correct? I feel he means that $U_f$ should be the set of local rings in $X$ with $f$ not contained in maximal ideals.
$textbf{Q3:}$ Let $X$ be a prevariety. Then $X$ is a variety iff for all $x,yin X$ s.t. $xneq y$, there is no local ring $Osubset k(X)$ s.t. $O$ dominates both $O_x$ and $O_y$. The book is not going to prove converse. What is the reference for the converse statement?(It feels like some sort of point separation statement.)
Ref. Mumford Red book 2nd Edition(pg 40, Chpt 1, Sec 6. Local Criterion)
abstract-algebra algebraic-geometry commutative-algebra
asked Jan 3 at 21:45
user45765user45765
2,6882724
$endgroup$
add a comment |
$begingroup$
Let $K$ be a f.g. field extension over an algebraically closed field $k$. Let $(O_x,m_x)$ be a family of maximal local subrings of $K$.(Here maximal means there is no further local ring $(O,m)subset K$ s.t. $(O,m)supset (O_x,m_x)$.)
Then he mentioned topology can be recovered as follows "for each $fin K$, let $U_f$ be the set of local rings in $X$ containing $f$."
$textbf{Q1:}$ Is it true $(O_x,m_x)$ had better be DVR?(It is clear that DVR satisfies maximality condition but it is not clear the converse holds. DVR corresponds to smooth objects but there is no good faith that variety is smooth.) What is the condition to see there is no further local ring dominating over $(O_x,m_x)$?
$textbf{Q2:}$ Is what written in the book correct? I feel he means that $U_f$ should be the set of local rings in $X$ with $f$ not contained in maximal ideals.
$textbf{Q3:}$ Let $X$ be a prevariety. Then $X$ is a variety iff for all $x,yin X$ s.t. $xneq y$, there is no local ring $Osubset k(X)$ s.t. $O$ dominates both $O_x$ and $O_y$. The book is not going to prove converse. What is the reference for the converse statement?(It feels like some sort of point separation statement.)
Ref. Mumford Red book 2nd Edition(pg 40, Chpt 1, Sec 6. Local Criterion)
abstract-algebra algebraic-geometry commutative-algebra
asked Jan 3 at 21:45
user45765user45765
2,6882724
$endgroup$
Let $K$ be a f.g. field extension over an algebraically closed field $k$. Let $(O_x,m_x)$ be a family of maximal local subrings of $K$.(Here maximal means there is no further local ring $(O,m)subset K$ s.t. $(O,m)supset (O_x,m_x)$.)
Then he mentioned topology can be recovered as follows "for each $fin K$, let $U_f$ be the set of local rings in $X$ containing $f$."
$textbf{Q1:}$ Is it true $(O_x,m_x)$ had better be DVR?(It is clear that DVR satisfies maximality condition but it is not clear the converse holds. DVR corresponds to smooth objects but there is no good faith that variety is smooth.) What is the condition to see there is no further local ring dominating over $(O_x,m_x)$?
$textbf{Q2:}$ Is what written in the book correct? I feel he means that $U_f$ should be the set of local rings in $X$ with $f$ not contained in maximal ideals.
$textbf{Q3:}$ Let $X$ be a prevariety. Then $X$ is a variety iff for all $x,yin X$ s.t. $xneq y$, there is no local ring $Osubset k(X)$ s.t. $O$ dominates both $O_x$ and $O_y$. The book is not going to prove converse. What is the reference for the converse statement?(It feels like some sort of point separation statement.)
Ref. Mumford Red book 2nd Edition(pg 40, Chpt 1, Sec 6. Local Criterion)
abstract-algebra algebraic-geometry commutative-algebra
abstract-algebra algebraic-geometry commutative-algebra
asked Jan 3 at 21:45
user45765user45765
2,6882724
asked Jan 3 at 21:45
user45765user45765
2,6882724
asked Jan 3 at 21:45
user45765user45765
2,6882724
asked Jan 3 at 21:45
user45765user45765
2,6882724
asked Jan 3 at 21:45
user45765user45765
2,6882724
2,6882724
add a comment |
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%2f3061041%2fmumfords-reconstruction-of-topology-via-local-ring-and-f-g-field-extension-over%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%2f3061041%2fmumfords-reconstruction-of-topology-via-local-ring-and-f-g-field-extension-over%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
