Generalized Stalk FunctorDoes Hom commute with stalks for locally free sheaves?Is invertibility a stalk-local property?Definition of stalk as a colimit of sheavesCan we define MaxSpec as a locally-ringed space?Is the zero set of a global section closed?Stalk of tensor product sheaf is tensor product of stalks via adjunctions and abstract nonsenseReducedness and colimitSchemes and locally ringed spacesStalk of the direct image of a locally free sheafStalk of a generic point of the special fiber
What do you call someone who likes to pick fights?
Cycles on the torus
Is "cogitate" used appropriately in "I cogitate that success relies on hard work"?
Is it a Cyclops number? "Nobody" knows!
How can a demon take control of a human body during REM sleep?
How exactly does an Ethernet collision happen in the cable, since nodes use different circuits for Tx and Rx?
How would an energy-based "projectile" blow up a spaceship?
Do Paladin Auras of Differing Oaths Stack?
I am the person who abides by rules, but breaks the rules. Who am I?
How to install round brake pads
How do spaceships determine each other's mass in space?
Professor forcing me to attend a conference, I can't afford even with 50% funding
Having the player face themselves after the mid-game
(Codewars) Linked Lists-Sorted Insert
Is there stress on two letters on the word стоят
What should I do when a paper is published similar to my PhD thesis without citation?
How can I portion out frozen cookie dough?
Yet another question on sums of the reciprocals of the primes
Why restrict private health insurance?
School performs periodic password audits. Is my password compromised?
The (Easy) Road to Code
Was it really inappropriate to write a pull request for the company I interviewed with?
Why aren't there more Gauls like Obelix?
Giving a career talk in my old university, how prominently should I tell students my salary?
Generalized Stalk Functor
Does Hom commute with stalks for locally free sheaves?Is invertibility a stalk-local property?Definition of stalk as a colimit of sheavesCan we define MaxSpec as a locally-ringed space?Is the zero set of a global section closed?Stalk of tensor product sheaf is tensor product of stalks via adjunctions and abstract nonsenseReducedness and colimitSchemes and locally ringed spacesStalk of the direct image of a locally free sheafStalk of a generic point of the special fiber
$begingroup$
Let $X$ be a locally ringed space. If $X$ has a generic point, then $textcolimit_U text dense in X mathcalO_X (U)$ is the stalk of $X$ at $U$. What is this in general? That is, if $X$ does not have a generic point, then what is this? Is this the product of the stalks of the irreducible components? Also, could I have a reference for this?
Edit: maybe it would be fair to call it the ring of rational functions.
algebraic-geometry
asked 2 days ago
Dean YoungDean Young
1,747721
$endgroup$
add a comment |
$begingroup$
Let $X$ be a locally ringed space. If $X$ has a generic point, then $textcolimit_U text dense in X mathcalO_X (U)$ is the stalk of $X$ at $U$. What is this in general? That is, if $X$ does not have a generic point, then what is this? Is this the product of the stalks of the irreducible components? Also, could I have a reference for this?
Edit: maybe it would be fair to call it the ring of rational functions.
algebraic-geometry
asked 2 days ago
Dean YoungDean Young
1,747721
$endgroup$
$begingroup$
I don't have references for this. But this colimit is not what you claim. For example, on $mathbbR$ with its euclidean topology and $mathcalO_X$ the sheaf of locally constant function with value in $mathbbZ$, then this colimit is the set of continuous functions $mathbbRto Z$ which are only defined on a dense open subset. This isn't the same thing as the set of all totally defined but not necessarily continuous functions $mathbbRto Z$.
$endgroup$
– Roland
yesterday
$begingroup$
If you leave this as an answer I will accept it. Do you know what the class of rings occuring as $textcolimit_U subset X text dense mathcalO_X (U)$ for some $X$ is?
$endgroup$
– Dean Young
yesterday
$begingroup$
Ok I will write an answer with some more examples. But I don't understand your comment, any ring works (for example on a one point topological, or an irreducible space).
$endgroup$
– Roland
yesterday
$begingroup$
If the space $X$ is irreducible then won't the ring $textcolimit_U subset X text dense mathcalO_X (U)$ be forced to be local? Maybe you could include a brief explanation of how to get a space $X$ with $textcolimit_U subset X text dense mathcalO_X (U)$ a ring of your choice.
$endgroup$
– Dean Young
yesterday
add a comment |
$begingroup$
Let $X$ be a locally ringed space. If $X$ has a generic point, then $textcolimit_U text dense in X mathcalO_X (U)$ is the stalk of $X$ at $U$. What is this in general? That is, if $X$ does not have a generic point, then what is this? Is this the product of the stalks of the irreducible components? Also, could I have a reference for this?
Edit: maybe it would be fair to call it the ring of rational functions.
algebraic-geometry
asked 2 days ago
Dean YoungDean Young
1,747721
$endgroup$
Let $X$ be a locally ringed space. If $X$ has a generic point, then $textcolimit_U text dense in X mathcalO_X (U)$ is the stalk of $X$ at $U$. What is this in general? That is, if $X$ does not have a generic point, then what is this? Is this the product of the stalks of the irreducible components? Also, could I have a reference for this?
Edit: maybe it would be fair to call it the ring of rational functions.
algebraic-geometry
algebraic-geometry
asked 2 days ago
Dean YoungDean Young
1,747721
asked 2 days ago
Dean YoungDean Young
1,747721
edited yesterday
Dean Young
asked 2 days ago
Dean YoungDean Young
1,747721
asked 2 days ago
Dean YoungDean Young
1,747721
asked 2 days ago
Dean YoungDean Young
1,747721
1,747721
$begingroup$
I don't have references for this. But this colimit is not what you claim. For example, on $mathbbR$ with its euclidean topology and $mathcalO_X$ the sheaf of locally constant function with value in $mathbbZ$, then this colimit is the set of continuous functions $mathbbRto Z$ which are only defined on a dense open subset. This isn't the same thing as the set of all totally defined but not necessarily continuous functions $mathbbRto Z$.
$endgroup$
– Roland
yesterday
$begingroup$
If you leave this as an answer I will accept it. Do you know what the class of rings occuring as $textcolimit_U subset X text dense mathcalO_X (U)$ for some $X$ is?
$endgroup$
– Dean Young
yesterday
$begingroup$
Ok I will write an answer with some more examples. But I don't understand your comment, any ring works (for example on a one point topological, or an irreducible space).
$endgroup$
– Roland
yesterday
$begingroup$
If the space $X$ is irreducible then won't the ring $textcolimit_U subset X text dense mathcalO_X (U)$ be forced to be local? Maybe you could include a brief explanation of how to get a space $X$ with $textcolimit_U subset X text dense mathcalO_X (U)$ a ring of your choice.
$endgroup$
– Dean Young
yesterday
add a comment |
$begingroup$
I don't have references for this. But this colimit is not what you claim. For example, on $mathbbR$ with its euclidean topology and $mathcalO_X$ the sheaf of locally constant function with value in $mathbbZ$, then this colimit is the set of continuous functions $mathbbRto Z$ which are only defined on a dense open subset. This isn't the same thing as the set of all totally defined but not necessarily continuous functions $mathbbRto Z$.
$endgroup$
– Roland
yesterday
$begingroup$
If you leave this as an answer I will accept it. Do you know what the class of rings occuring as $textcolimit_U subset X text dense mathcalO_X (U)$ for some $X$ is?
$endgroup$
– Dean Young
yesterday
$begingroup$
Ok I will write an answer with some more examples. But I don't understand your comment, any ring works (for example on a one point topological, or an irreducible space).
$endgroup$
– Roland
yesterday
$begingroup$
If the space $X$ is irreducible then won't the ring $textcolimit_U subset X text dense mathcalO_X (U)$ be forced to be local? Maybe you could include a brief explanation of how to get a space $X$ with $textcolimit_U subset X text dense mathcalO_X (U)$ a ring of your choice.
$endgroup$
– Dean Young
yesterday
$begingroup$
I don't have references for this. But this colimit is not what you claim. For example, on $mathbbR$ with its euclidean topology and $mathcalO_X$ the sheaf of locally constant function with value in $mathbbZ$, then this colimit is the set of continuous functions $mathbbRto Z$ which are only defined on a dense open subset. This isn't the same thing as the set of all totally defined but not necessarily continuous functions $mathbbRto Z$.
$endgroup$
– Roland
yesterday
$begingroup$
I don't have references for this. But this colimit is not what you claim. For example, on $mathbbR$ with its euclidean topology and $mathcalO_X$ the sheaf of locally constant function with value in $mathbbZ$, then this colimit is the set of continuous functions $mathbbRto Z$ which are only defined on a dense open subset. This isn't the same thing as the set of all totally defined but not necessarily continuous functions $mathbbRto Z$.
$endgroup$
– Roland
yesterday
$begingroup$
If you leave this as an answer I will accept it. Do you know what the class of rings occuring as $textcolimit_U subset X text dense mathcalO_X (U)$ for some $X$ is?
$endgroup$
– Dean Young
yesterday
$begingroup$
If you leave this as an answer I will accept it. Do you know what the class of rings occuring as $textcolimit_U subset X text dense mathcalO_X (U)$ for some $X$ is?
$endgroup$
– Dean Young
yesterday
$begingroup$
Ok I will write an answer with some more examples. But I don't understand your comment, any ring works (for example on a one point topological, or an irreducible space).
$endgroup$
– Roland
yesterday
$begingroup$
Ok I will write an answer with some more examples. But I don't understand your comment, any ring works (for example on a one point topological, or an irreducible space).
$endgroup$
– Roland
yesterday
$begingroup$
If the space $X$ is irreducible then won't the ring $textcolimit_U subset X text dense mathcalO_X (U)$ be forced to be local? Maybe you could include a brief explanation of how to get a space $X$ with $textcolimit_U subset X text dense mathcalO_X (U)$ a ring of your choice.
$endgroup$
– Dean Young
yesterday
$begingroup$
If the space $X$ is irreducible then won't the ring $textcolimit_U subset X text dense mathcalO_X (U)$ be forced to be local? Maybe you could include a brief explanation of how to get a space $X$ with $textcolimit_U subset X text dense mathcalO_X (U)$ a ring of your choice.
$endgroup$
– Dean Young
yesterday
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
As I said in the comment, the colimit is not the product of all stalks. (In fact, it does not work if $X$ has a generic point). Here are some examples of this colimits :
Let $X=mathbbR$ with its euclidean topology and $mathcalO_X=underlinemathbb Z$ be the constant sheaf. Then $undersetU text dense in XoperatornamecolimmathcalO_X(U)$ is the set of continuous function with value in $mathbbZ$ which are only define on a dense open subset. This isn't the same thing as the set of all totally defined but not necessarily continuous functions $mathbbRto Z$.
Now the previous example is not a locally ringed space (the colimit indeed makes sense for any space and any sheaf, not necessarily a locally ringed space and its structure sheaf). But if you prefer this case, consider again $X=mathbbR$ with its euclidean topology and $mathcalO_X=mathcalC^0$ the sheaf of continuous function. Then $undersetU text dense in XoperatornamecolimmathcalO_X(U)$ is the set of continuous function defined on an dense open subset. Again this is different from the set of all not necessarily continuous function $mathbbRto Z$.
Let $X$ be any irreducible space (for example a point) and $mathcalO_X=underlineR$ be a constant sheaf with value in some ring $R$. Then for any $U$, $mathcalO_X(U)=R$. It follows that $undersetU text dense in XoperatornamecolimmathcalO_X(U)=R$. In particular, any ring can happen as this colimit (if one require that $X$ is a locally ringed space, then indeed $R$ must be local). Of course, this is different from the product of all stalks if $X$ has more than one point.
answered yesterday
RolandRoland
7,40411015
$endgroup$
add a comment |
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%2f3139383%2fgeneralized-stalk-functor%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
As I said in the comment, the colimit is not the product of all stalks. (In fact, it does not work if $X$ has a generic point). Here are some examples of this colimits :
Let $X=mathbbR$ with its euclidean topology and $mathcalO_X=underlinemathbb Z$ be the constant sheaf. Then $undersetU text dense in XoperatornamecolimmathcalO_X(U)$ is the set of continuous function with value in $mathbbZ$ which are only define on a dense open subset. This isn't the same thing as the set of all totally defined but not necessarily continuous functions $mathbbRto Z$.
Now the previous example is not a locally ringed space (the colimit indeed makes sense for any space and any sheaf, not necessarily a locally ringed space and its structure sheaf). But if you prefer this case, consider again $X=mathbbR$ with its euclidean topology and $mathcalO_X=mathcalC^0$ the sheaf of continuous function. Then $undersetU text dense in XoperatornamecolimmathcalO_X(U)$ is the set of continuous function defined on an dense open subset. Again this is different from the set of all not necessarily continuous function $mathbbRto Z$.
Let $X$ be any irreducible space (for example a point) and $mathcalO_X=underlineR$ be a constant sheaf with value in some ring $R$. Then for any $U$, $mathcalO_X(U)=R$. It follows that $undersetU text dense in XoperatornamecolimmathcalO_X(U)=R$. In particular, any ring can happen as this colimit (if one require that $X$ is a locally ringed space, then indeed $R$ must be local). Of course, this is different from the product of all stalks if $X$ has more than one point.
answered yesterday
RolandRoland
7,40411015
$endgroup$
add a comment |
$begingroup$
As I said in the comment, the colimit is not the product of all stalks. (In fact, it does not work if $X$ has a generic point). Here are some examples of this colimits :
Let $X=mathbbR$ with its euclidean topology and $mathcalO_X=underlinemathbb Z$ be the constant sheaf. Then $undersetU text dense in XoperatornamecolimmathcalO_X(U)$ is the set of continuous function with value in $mathbbZ$ which are only define on a dense open subset. This isn't the same thing as the set of all totally defined but not necessarily continuous functions $mathbbRto Z$.
Now the previous example is not a locally ringed space (the colimit indeed makes sense for any space and any sheaf, not necessarily a locally ringed space and its structure sheaf). But if you prefer this case, consider again $X=mathbbR$ with its euclidean topology and $mathcalO_X=mathcalC^0$ the sheaf of continuous function. Then $undersetU text dense in XoperatornamecolimmathcalO_X(U)$ is the set of continuous function defined on an dense open subset. Again this is different from the set of all not necessarily continuous function $mathbbRto Z$.
Let $X$ be any irreducible space (for example a point) and $mathcalO_X=underlineR$ be a constant sheaf with value in some ring $R$. Then for any $U$, $mathcalO_X(U)=R$. It follows that $undersetU text dense in XoperatornamecolimmathcalO_X(U)=R$. In particular, any ring can happen as this colimit (if one require that $X$ is a locally ringed space, then indeed $R$ must be local). Of course, this is different from the product of all stalks if $X$ has more than one point.
answered yesterday
RolandRoland
7,40411015
$endgroup$
add a comment |
$begingroup$
As I said in the comment, the colimit is not the product of all stalks. (In fact, it does not work if $X$ has a generic point). Here are some examples of this colimits :
Let $X=mathbbR$ with its euclidean topology and $mathcalO_X=underlinemathbb Z$ be the constant sheaf. Then $undersetU text dense in XoperatornamecolimmathcalO_X(U)$ is the set of continuous function with value in $mathbbZ$ which are only define on a dense open subset. This isn't the same thing as the set of all totally defined but not necessarily continuous functions $mathbbRto Z$.
Now the previous example is not a locally ringed space (the colimit indeed makes sense for any space and any sheaf, not necessarily a locally ringed space and its structure sheaf). But if you prefer this case, consider again $X=mathbbR$ with its euclidean topology and $mathcalO_X=mathcalC^0$ the sheaf of continuous function. Then $undersetU text dense in XoperatornamecolimmathcalO_X(U)$ is the set of continuous function defined on an dense open subset. Again this is different from the set of all not necessarily continuous function $mathbbRto Z$.
Let $X$ be any irreducible space (for example a point) and $mathcalO_X=underlineR$ be a constant sheaf with value in some ring $R$. Then for any $U$, $mathcalO_X(U)=R$. It follows that $undersetU text dense in XoperatornamecolimmathcalO_X(U)=R$. In particular, any ring can happen as this colimit (if one require that $X$ is a locally ringed space, then indeed $R$ must be local). Of course, this is different from the product of all stalks if $X$ has more than one point.
answered yesterday
RolandRoland
7,40411015
$endgroup$
As I said in the comment, the colimit is not the product of all stalks. (In fact, it does not work if $X$ has a generic point). Here are some examples of this colimits :
Let $X=mathbbR$ with its euclidean topology and $mathcalO_X=underlinemathbb Z$ be the constant sheaf. Then $undersetU text dense in XoperatornamecolimmathcalO_X(U)$ is the set of continuous function with value in $mathbbZ$ which are only define on a dense open subset. This isn't the same thing as the set of all totally defined but not necessarily continuous functions $mathbbRto Z$.
Now the previous example is not a locally ringed space (the colimit indeed makes sense for any space and any sheaf, not necessarily a locally ringed space and its structure sheaf). But if you prefer this case, consider again $X=mathbbR$ with its euclidean topology and $mathcalO_X=mathcalC^0$ the sheaf of continuous function. Then $undersetU text dense in XoperatornamecolimmathcalO_X(U)$ is the set of continuous function defined on an dense open subset. Again this is different from the set of all not necessarily continuous function $mathbbRto Z$.
Let $X$ be any irreducible space (for example a point) and $mathcalO_X=underlineR$ be a constant sheaf with value in some ring $R$. Then for any $U$, $mathcalO_X(U)=R$. It follows that $undersetU text dense in XoperatornamecolimmathcalO_X(U)=R$. In particular, any ring can happen as this colimit (if one require that $X$ is a locally ringed space, then indeed $R$ must be local). Of course, this is different from the product of all stalks if $X$ has more than one point.
answered yesterday
RolandRoland
7,40411015
answered yesterday
RolandRoland
7,40411015
answered yesterday
RolandRoland
7,40411015
answered yesterday
RolandRoland
7,40411015
7,40411015
add a comment |
add a comment |
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%2f3139383%2fgeneralized-stalk-functor%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
$begingroup$
I don't have references for this. But this colimit is not what you claim. For example, on $mathbbR$ with its euclidean topology and $mathcalO_X$ the sheaf of locally constant function with value in $mathbbZ$, then this colimit is the set of continuous functions $mathbbRto Z$ which are only defined on a dense open subset. This isn't the same thing as the set of all totally defined but not necessarily continuous functions $mathbbRto Z$.
$endgroup$
– Roland
yesterday
$begingroup$
If you leave this as an answer I will accept it. Do you know what the class of rings occuring as $textcolimit_U subset X text dense mathcalO_X (U)$ for some $X$ is?
$endgroup$
– Dean Young
yesterday
$begingroup$
Ok I will write an answer with some more examples. But I don't understand your comment, any ring works (for example on a one point topological, or an irreducible space).
$endgroup$
– Roland
yesterday
$begingroup$
If the space $X$ is irreducible then won't the ring $textcolimit_U subset X text dense mathcalO_X (U)$ be forced to be local? Maybe you could include a brief explanation of how to get a space $X$ with $textcolimit_U subset X text dense mathcalO_X (U)$ a ring of your choice.
$endgroup$
– Dean Young
yesterday