Mixing
Pushout on open immersions
$begingroup$
I'm currently stuck on this exercise.
I am asked to show that if $f_i:X→Y_i$ with $i=1,2$ are open immersions, then a pushout exists.
Recall that a pushout of $(f_1,f_2)$
is a triple $(Z,g_1,g_2)$ such that $Z$ is a scheme and $g_i$'s are the morphisms of schemes such that for all schemes $U$, the canonical map $g_1×g_2:hom(Z,U)→hom(Y_1,U)×_{hom(X,U)} hom(Y_2,U)$ is bijective.
I really don't know where to start to contruct the pushout, can somebody help me? Also a hint would be really nice. Thank you in advance.
algebraic-geometry
asked Jan 15 at 15:04
NoemiZNoemiZ
212
$endgroup$
add a comment |
$begingroup$
I'm currently stuck on this exercise.
I am asked to show that if $f_i:X→Y_i$ with $i=1,2$ are open immersions, then a pushout exists.
Recall that a pushout of $(f_1,f_2)$
is a triple $(Z,g_1,g_2)$ such that $Z$ is a scheme and $g_i$'s are the morphisms of schemes such that for all schemes $U$, the canonical map $g_1×g_2:hom(Z,U)→hom(Y_1,U)×_{hom(X,U)} hom(Y_2,U)$ is bijective.
I really don't know where to start to contruct the pushout, can somebody help me? Also a hint would be really nice. Thank you in advance.
algebraic-geometry
asked Jan 15 at 15:04
NoemiZNoemiZ
212
$endgroup$
2
$begingroup$
The pushout along open immersions is the gluing of $Y_1$ and $Y_2$ along their common open subscheme, $X$.
$endgroup$
– jgon
Jan 15 at 15:21
add a comment |
$begingroup$
I'm currently stuck on this exercise.
I am asked to show that if $f_i:X→Y_i$ with $i=1,2$ are open immersions, then a pushout exists.
Recall that a pushout of $(f_1,f_2)$
is a triple $(Z,g_1,g_2)$ such that $Z$ is a scheme and $g_i$'s are the morphisms of schemes such that for all schemes $U$, the canonical map $g_1×g_2:hom(Z,U)→hom(Y_1,U)×_{hom(X,U)} hom(Y_2,U)$ is bijective.
I really don't know where to start to contruct the pushout, can somebody help me? Also a hint would be really nice. Thank you in advance.
algebraic-geometry
asked Jan 15 at 15:04
NoemiZNoemiZ
212
$endgroup$
I'm currently stuck on this exercise.
I am asked to show that if $f_i:X→Y_i$ with $i=1,2$ are open immersions, then a pushout exists.
Recall that a pushout of $(f_1,f_2)$
is a triple $(Z,g_1,g_2)$ such that $Z$ is a scheme and $g_i$'s are the morphisms of schemes such that for all schemes $U$, the canonical map $g_1×g_2:hom(Z,U)→hom(Y_1,U)×_{hom(X,U)} hom(Y_2,U)$ is bijective.
I really don't know where to start to contruct the pushout, can somebody help me? Also a hint would be really nice. Thank you in advance.
algebraic-geometry
algebraic-geometry
asked Jan 15 at 15:04
NoemiZNoemiZ
212
asked Jan 15 at 15:04
NoemiZNoemiZ
212
asked Jan 15 at 15:04
NoemiZNoemiZ
212
asked Jan 15 at 15:04
NoemiZNoemiZ
212
asked Jan 15 at 15:04
NoemiZNoemiZ
212
212
2
$begingroup$
The pushout along open immersions is the gluing of $Y_1$ and $Y_2$ along their common open subscheme, $X$.
$endgroup$
– jgon
Jan 15 at 15:21
add a comment |
2
$begingroup$
The pushout along open immersions is the gluing of $Y_1$ and $Y_2$ along their common open subscheme, $X$.
$endgroup$
– jgon
Jan 15 at 15:21
2
2
$begingroup$
The pushout along open immersions is the gluing of $Y_1$ and $Y_2$ along their common open subscheme, $X$.
$endgroup$
– jgon
Jan 15 at 15:21
$begingroup$
The pushout along open immersions is the gluing of $Y_1$ and $Y_2$ along their common open subscheme, $X$.
$endgroup$
– jgon
Jan 15 at 15:21
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%2f3074543%2fpushout-on-open-immersions%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%2f3074543%2fpushout-on-open-immersions%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
2
$begingroup$
The pushout along open immersions is the gluing of $Y_1$ and $Y_2$ along their common open subscheme, $X$.
$endgroup$
– jgon
Jan 15 at 15:21