Irreducible of finite Krull dimension implies quasi-compact? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)A closed point in the closure of any point in the closure of any point of an irreducible schemeWhen is an irreducible scheme quasi-compact?not locally of finite type implies not universally closed?If $X,Y$ are regular and of finite type over $S$, can $Xtimes _S Y$ be embedded into a regular $S$-scheme? Krull dimension in equivariant cohomologyKrull dimension and Morley rankQuasi-finite morphisms of stacksQuasi-compactness of irreducible separated scheme locally of finite typeWhen does the image of a morphism of schemes support scheme structure?Sufficient condition for quasi-compactness of scheme morphismSchemes monomorphing into affine scheme of dimension 1
Irreducible of finite Krull dimension implies quasi-compact?
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)A closed point in the closure of any point in the closure of any point of an irreducible schemeWhen is an irreducible scheme quasi-compact?not locally of finite type implies not universally closed?If $X,Y$ are regular and of finite type over $S$, can $Xtimes _S Y$ be embedded into a regular $S$-scheme? Krull dimension in equivariant cohomologyKrull dimension and Morley rankQuasi-finite morphisms of stacksQuasi-compactness of irreducible separated scheme locally of finite typeWhen does the image of a morphism of schemes support scheme structure?Sufficient condition for quasi-compactness of scheme morphismSchemes monomorphing into affine scheme of dimension 1
5
$begingroup$
Let $X$ be the underlying space of a scheme.
- If $X$ is irreducible of finite Krull dimension, is it necessarily
quasi-compact? - Is it necessarily Noetherian?
- What if we assume not
only that Krull dimension is finite but also that it is 1?
at.algebraic-topology gn.general-topology schemes
edited 3 hours ago
András Bátkai
3,84142342
asked 3 hours ago
schematic_ftmschematic_ftm
261
New contributor
schematic_ftm is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
5
$begingroup$
Let $X$ be the underlying space of a scheme.
- If $X$ is irreducible of finite Krull dimension, is it necessarily
quasi-compact? - Is it necessarily Noetherian?
- What if we assume not
only that Krull dimension is finite but also that it is 1?
at.algebraic-topology gn.general-topology schemes
edited 3 hours ago
András Bátkai
3,84142342
asked 3 hours ago
schematic_ftmschematic_ftm
261
New contributor
schematic_ftm is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
5
5
5
$begingroup$
Let $X$ be the underlying space of a scheme.
- If $X$ is irreducible of finite Krull dimension, is it necessarily
quasi-compact? - Is it necessarily Noetherian?
- What if we assume not
only that Krull dimension is finite but also that it is 1?
at.algebraic-topology gn.general-topology schemes
edited 3 hours ago
András Bátkai
3,84142342
asked 3 hours ago
schematic_ftmschematic_ftm
261
New contributor
schematic_ftm is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Let $X$ be the underlying space of a scheme.
- If $X$ is irreducible of finite Krull dimension, is it necessarily
quasi-compact? - Is it necessarily Noetherian?
- What if we assume not
only that Krull dimension is finite but also that it is 1?
at.algebraic-topology gn.general-topology schemes
at.algebraic-topology gn.general-topology schemes
edited 3 hours ago
András Bátkai
3,84142342
asked 3 hours ago
schematic_ftmschematic_ftm
261
New contributor
schematic_ftm is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 3 hours ago
András Bátkai
3,84142342
asked 3 hours ago
schematic_ftmschematic_ftm
261
New contributor
schematic_ftm is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 3 hours ago
András Bátkai
3,84142342
edited 3 hours ago
András Bátkai
3,84142342
edited 3 hours ago
András Bátkai
3,84142342
3,84142342
asked 3 hours ago
schematic_ftmschematic_ftm
261
New contributor
schematic_ftm is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 3 hours ago
schematic_ftmschematic_ftm
261
asked 3 hours ago
schematic_ftmschematic_ftm
261
261
New contributor
schematic_ftm is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
schematic_ftm is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
schematic_ftm is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
10
$begingroup$
The answer is no for all these questions. Take the line with infinite origins: the scheme obtained by gluing an infinite amount of copies of $mathbbA^1$ along the open subsets $mathbbG_m$. This has Krull dimension 1 (there are only closed points and the unique generic point) and it is irreducible (the only proper nonempty closed subsets are the closed points) but it is not quasi-compact (it contains an infinite discrete set), and so in particular not Noetherian.
answered 3 hours ago
Denis NardinDenis Nardin
9,16223565
$endgroup$
1
$begingroup$
Denis Nardin, I am not sure about the second paragraph. I thought nilpotent thickening never affects the underlying topological space. So while your nilpotent thickening gives a non-locally Noetherian scheme, does it give a non-locally Noetherian space (about which is the question)?
$endgroup$
– Stepan Banach
2 hours ago
1
$begingroup$
@StepanBanach Ah sorry, I was thinking of a non-Noetherian scheme. Let me correct it.
$endgroup$
– Denis Nardin
2 hours ago
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "504"
;
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
);
);
schematic_ftm is a new contributor. Be nice, and check out our Code of Conduct.
draft saved
draft discarded
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%2fmathoverflow.net%2fquestions%2f328305%2firreducible-of-finite-krull-dimension-implies-quasi-compact%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
10
$begingroup$
The answer is no for all these questions. Take the line with infinite origins: the scheme obtained by gluing an infinite amount of copies of $mathbbA^1$ along the open subsets $mathbbG_m$. This has Krull dimension 1 (there are only closed points and the unique generic point) and it is irreducible (the only proper nonempty closed subsets are the closed points) but it is not quasi-compact (it contains an infinite discrete set), and so in particular not Noetherian.
answered 3 hours ago
Denis NardinDenis Nardin
9,16223565
$endgroup$
1
$begingroup$
Denis Nardin, I am not sure about the second paragraph. I thought nilpotent thickening never affects the underlying topological space. So while your nilpotent thickening gives a non-locally Noetherian scheme, does it give a non-locally Noetherian space (about which is the question)?
$endgroup$
– Stepan Banach
2 hours ago
1
$begingroup$
@StepanBanach Ah sorry, I was thinking of a non-Noetherian scheme. Let me correct it.
$endgroup$
– Denis Nardin
2 hours ago
add a comment |
10
$begingroup$
The answer is no for all these questions. Take the line with infinite origins: the scheme obtained by gluing an infinite amount of copies of $mathbbA^1$ along the open subsets $mathbbG_m$. This has Krull dimension 1 (there are only closed points and the unique generic point) and it is irreducible (the only proper nonempty closed subsets are the closed points) but it is not quasi-compact (it contains an infinite discrete set), and so in particular not Noetherian.
answered 3 hours ago
Denis NardinDenis Nardin
9,16223565
$endgroup$
1
$begingroup$
Denis Nardin, I am not sure about the second paragraph. I thought nilpotent thickening never affects the underlying topological space. So while your nilpotent thickening gives a non-locally Noetherian scheme, does it give a non-locally Noetherian space (about which is the question)?
$endgroup$
– Stepan Banach
2 hours ago
1
$begingroup$
@StepanBanach Ah sorry, I was thinking of a non-Noetherian scheme. Let me correct it.
$endgroup$
– Denis Nardin
2 hours ago
add a comment |
10
10
10
$begingroup$
The answer is no for all these questions. Take the line with infinite origins: the scheme obtained by gluing an infinite amount of copies of $mathbbA^1$ along the open subsets $mathbbG_m$. This has Krull dimension 1 (there are only closed points and the unique generic point) and it is irreducible (the only proper nonempty closed subsets are the closed points) but it is not quasi-compact (it contains an infinite discrete set), and so in particular not Noetherian.
answered 3 hours ago
Denis NardinDenis Nardin
9,16223565
$endgroup$
The answer is no for all these questions. Take the line with infinite origins: the scheme obtained by gluing an infinite amount of copies of $mathbbA^1$ along the open subsets $mathbbG_m$. This has Krull dimension 1 (there are only closed points and the unique generic point) and it is irreducible (the only proper nonempty closed subsets are the closed points) but it is not quasi-compact (it contains an infinite discrete set), and so in particular not Noetherian.
answered 3 hours ago
Denis NardinDenis Nardin
9,16223565
edited 2 hours ago
answered 3 hours ago
Denis NardinDenis Nardin
9,16223565
answered 3 hours ago
Denis NardinDenis Nardin
9,16223565
answered 3 hours ago
Denis NardinDenis Nardin
9,16223565
9,16223565
1
$begingroup$
Denis Nardin, I am not sure about the second paragraph. I thought nilpotent thickening never affects the underlying topological space. So while your nilpotent thickening gives a non-locally Noetherian scheme, does it give a non-locally Noetherian space (about which is the question)?
$endgroup$
– Stepan Banach
2 hours ago
1
$begingroup$
@StepanBanach Ah sorry, I was thinking of a non-Noetherian scheme. Let me correct it.
$endgroup$
– Denis Nardin
2 hours ago
add a comment |
1
$begingroup$
Denis Nardin, I am not sure about the second paragraph. I thought nilpotent thickening never affects the underlying topological space. So while your nilpotent thickening gives a non-locally Noetherian scheme, does it give a non-locally Noetherian space (about which is the question)?
$endgroup$
– Stepan Banach
2 hours ago
1
$begingroup$
@StepanBanach Ah sorry, I was thinking of a non-Noetherian scheme. Let me correct it.
$endgroup$
– Denis Nardin
2 hours ago
1
1
$begingroup$
Denis Nardin, I am not sure about the second paragraph. I thought nilpotent thickening never affects the underlying topological space. So while your nilpotent thickening gives a non-locally Noetherian scheme, does it give a non-locally Noetherian space (about which is the question)?
$endgroup$
– Stepan Banach
2 hours ago
$begingroup$
Denis Nardin, I am not sure about the second paragraph. I thought nilpotent thickening never affects the underlying topological space. So while your nilpotent thickening gives a non-locally Noetherian scheme, does it give a non-locally Noetherian space (about which is the question)?
$endgroup$
– Stepan Banach
2 hours ago
1
1
$begingroup$
@StepanBanach Ah sorry, I was thinking of a non-Noetherian scheme. Let me correct it.
$endgroup$
– Denis Nardin
2 hours ago
$begingroup$
@StepanBanach Ah sorry, I was thinking of a non-Noetherian scheme. Let me correct it.
$endgroup$
– Denis Nardin
2 hours ago
add a comment |
schematic_ftm is a new contributor. Be nice, and check out our Code of Conduct.
draft saved
draft discarded
schematic_ftm is a new contributor. Be nice, and check out our Code of Conduct.
schematic_ftm is a new contributor. Be nice, and check out our Code of Conduct.
schematic_ftm is a new contributor. Be nice, and check out our Code of Conduct.
Thanks for contributing an answer to MathOverflow!
- 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.
draft saved
draft discarded
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%2fmathoverflow.net%2fquestions%2f328305%2firreducible-of-finite-krull-dimension-implies-quasi-compact%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