Mixing
Question about the proof of rigidity lemma (Mumford, GIT, Proposition 6.1)
$begingroup$
Let $p:X to S$ be flat, $S$ is connected and $H^{0}(X_{s},mathscr{O}_{X_{s}})=kappa(s)$, $forall s in S$. ($X_s$ is the fibre at $s$).
In the first case, we assume $epsilon:S to X$ is a section and $S$ consists of one point. Mumford claims that $p_*mathscr{O}_X=mathscr{O}_S$.
(1) What does $kappa(s)$ mean here? The residue field?
(2) If it means the residue field, then $p_*mathscr{O}_X(S)=mathscr{O}_X(X)=kappa(s)=mathscr{O}_S(S)$. This implies that $S=mathrm{Spec}(kappa(s))$. How to prove that $kappa(s)=mathscr{O}_S(S)$? What if $S$ is the spectrum of a ring? By Atiyah-Macdonald, Chapter 8, there do exist a ring with unique prime ideal that is not a field.
algebraic-geometry
edited Jan 13 at 9:42
user26857
39.3k124183
asked Oct 1 '16 at 7:20
Y. LiY. Li
436
$endgroup$
migrated from mathoverflow.net Oct 1 '16 at 10:01
This question came from our site for professional mathematicians.
add a comment |
$begingroup$
Let $p:X to S$ be flat, $S$ is connected and $H^{0}(X_{s},mathscr{O}_{X_{s}})=kappa(s)$, $forall s in S$. ($X_s$ is the fibre at $s$).
In the first case, we assume $epsilon:S to X$ is a section and $S$ consists of one point. Mumford claims that $p_*mathscr{O}_X=mathscr{O}_S$.
(1) What does $kappa(s)$ mean here? The residue field?
(2) If it means the residue field, then $p_*mathscr{O}_X(S)=mathscr{O}_X(X)=kappa(s)=mathscr{O}_S(S)$. This implies that $S=mathrm{Spec}(kappa(s))$. How to prove that $kappa(s)=mathscr{O}_S(S)$? What if $S$ is the spectrum of a ring? By Atiyah-Macdonald, Chapter 8, there do exist a ring with unique prime ideal that is not a field.
algebraic-geometry
edited Jan 13 at 9:42
user26857
39.3k124183
asked Oct 1 '16 at 7:20
Y. LiY. Li
436
$endgroup$
migrated from mathoverflow.net Oct 1 '16 at 10:01
This question came from our site for professional mathematicians.
$begingroup$
Why do you write that $O_X(X)=k(s)$? This is not true in general in the setup you describe.
$endgroup$
– potentially dense
Oct 1 '16 at 7:38
$begingroup$
Hint: apply Nakayama's lemma to the homomorphism $mathcal{O}_Srightarrow p_*mathcal{O}_X$. The question would have been more appropriate on MSE.
$endgroup$
– abx
Oct 1 '16 at 8:17
add a comment |
$begingroup$
Let $p:X to S$ be flat, $S$ is connected and $H^{0}(X_{s},mathscr{O}_{X_{s}})=kappa(s)$, $forall s in S$. ($X_s$ is the fibre at $s$).
In the first case, we assume $epsilon:S to X$ is a section and $S$ consists of one point. Mumford claims that $p_*mathscr{O}_X=mathscr{O}_S$.
(1) What does $kappa(s)$ mean here? The residue field?
(2) If it means the residue field, then $p_*mathscr{O}_X(S)=mathscr{O}_X(X)=kappa(s)=mathscr{O}_S(S)$. This implies that $S=mathrm{Spec}(kappa(s))$. How to prove that $kappa(s)=mathscr{O}_S(S)$? What if $S$ is the spectrum of a ring? By Atiyah-Macdonald, Chapter 8, there do exist a ring with unique prime ideal that is not a field.
algebraic-geometry
edited Jan 13 at 9:42
user26857
39.3k124183
asked Oct 1 '16 at 7:20
Y. LiY. Li
436
$endgroup$
Let $p:X to S$ be flat, $S$ is connected and $H^{0}(X_{s},mathscr{O}_{X_{s}})=kappa(s)$, $forall s in S$. ($X_s$ is the fibre at $s$).
In the first case, we assume $epsilon:S to X$ is a section and $S$ consists of one point. Mumford claims that $p_*mathscr{O}_X=mathscr{O}_S$.
(1) What does $kappa(s)$ mean here? The residue field?
(2) If it means the residue field, then $p_*mathscr{O}_X(S)=mathscr{O}_X(X)=kappa(s)=mathscr{O}_S(S)$. This implies that $S=mathrm{Spec}(kappa(s))$. How to prove that $kappa(s)=mathscr{O}_S(S)$? What if $S$ is the spectrum of a ring? By Atiyah-Macdonald, Chapter 8, there do exist a ring with unique prime ideal that is not a field.
algebraic-geometry
algebraic-geometry
edited Jan 13 at 9:42
user26857
39.3k124183
asked Oct 1 '16 at 7:20
Y. LiY. Li
436
edited Jan 13 at 9:42
user26857
39.3k124183
asked Oct 1 '16 at 7:20
Y. LiY. Li
436
edited Jan 13 at 9:42
user26857
39.3k124183
edited Jan 13 at 9:42
user26857
39.3k124183
edited Jan 13 at 9:42
user26857
39.3k124183
39.3k124183
asked Oct 1 '16 at 7:20
Y. LiY. Li
436
asked Oct 1 '16 at 7:20
Y. LiY. Li
436
asked Oct 1 '16 at 7:20
Y. LiY. Li
436
436
migrated from mathoverflow.net Oct 1 '16 at 10:01
This question came from our site for professional mathematicians.
migrated from mathoverflow.net Oct 1 '16 at 10:01
This question came from our site for professional mathematicians.
$begingroup$
Why do you write that $O_X(X)=k(s)$? This is not true in general in the setup you describe.
$endgroup$
– potentially dense
Oct 1 '16 at 7:38
$begingroup$
Hint: apply Nakayama's lemma to the homomorphism $mathcal{O}_Srightarrow p_*mathcal{O}_X$. The question would have been more appropriate on MSE.
$endgroup$
– abx
Oct 1 '16 at 8:17
add a comment |
$begingroup$
Why do you write that $O_X(X)=k(s)$? This is not true in general in the setup you describe.
$endgroup$
– potentially dense
Oct 1 '16 at 7:38
$begingroup$
Hint: apply Nakayama's lemma to the homomorphism $mathcal{O}_Srightarrow p_*mathcal{O}_X$. The question would have been more appropriate on MSE.
$endgroup$
– abx
Oct 1 '16 at 8:17
$begingroup$
Why do you write that $O_X(X)=k(s)$? This is not true in general in the setup you describe.
$endgroup$
– potentially dense
Oct 1 '16 at 7:38
$begingroup$
Why do you write that $O_X(X)=k(s)$? This is not true in general in the setup you describe.
$endgroup$
– potentially dense
Oct 1 '16 at 7:38
$begingroup$
Hint: apply Nakayama's lemma to the homomorphism $mathcal{O}_Srightarrow p_*mathcal{O}_X$. The question would have been more appropriate on MSE.
$endgroup$
– abx
Oct 1 '16 at 8:17
$begingroup$
Hint: apply Nakayama's lemma to the homomorphism $mathcal{O}_Srightarrow p_*mathcal{O}_X$. The question would have been more appropriate on MSE.
$endgroup$
– abx
Oct 1 '16 at 8:17
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%2f1949066%2fquestion-about-the-proof-of-rigidity-lemma-mumford-git-proposition-6-1%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%2f1949066%2fquestion-about-the-proof-of-rigidity-lemma-mumford-git-proposition-6-1%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
$begingroup$
Why do you write that $O_X(X)=k(s)$? This is not true in general in the setup you describe.
$endgroup$
– potentially dense
Oct 1 '16 at 7:38
$begingroup$
Hint: apply Nakayama's lemma to the homomorphism $mathcal{O}_Srightarrow p_*mathcal{O}_X$. The question would have been more appropriate on MSE.
$endgroup$
– abx
Oct 1 '16 at 8:17