Noetherian spectral space comes from noetherian ring?
Let $X$ be a spectral space (en.wikipedia.org/wiki/Spectral_space), i.e. a space of the form $textrm{Spec}(A)$ for some commutative ring $A$. If $X$ is noetherian, does there also exist a noetherian ring $B$ such that $X=textrm{Spec}(B)$?
ag.algebraic-geometry ac.commutative-algebra gn.general-topology
add a comment |
Let $X$ be a spectral space (en.wikipedia.org/wiki/Spectral_space), i.e. a space of the form $textrm{Spec}(A)$ for some commutative ring $A$. If $X$ is noetherian, does there also exist a noetherian ring $B$ such that $X=textrm{Spec}(B)$?
ag.algebraic-geometry ac.commutative-algebra gn.general-topology
1
$mathrm{Spec}$ is an (anti-)equivalence from commutative rings to affine scheme, so two rings are isomorphic iff their Spec's are. So, if such a noetherian $B$ exists, your $A$ was already isomorphic to it.
– Qfwfq
Nov 10 at 18:19
1
Oh yes, you're totally right, it's the underlying top space of the Spec not the scheme
– Qfwfq
Nov 10 at 18:21
7
I wonder if Hochster's thesis addresses this? Off the top of my head, I don't know how to make the following Noetherian topological space ${p,q,r}$, with open sets ${ {p,q,r}, {p,q}, {p} }$, as the spectrum of a Noetherian ring (it's the spectrum of the non-discrete valuation ring associated to $mathbb{Z} times mathbb{Z}$ with the lex order).
– Karl Schwede
Nov 10 at 18:31
1
@KarlSchwede - You may want to take look at my comment below.
– Pierre-Yves Gaillard
Nov 10 at 20:03
add a comment |
Let $X$ be a spectral space (en.wikipedia.org/wiki/Spectral_space), i.e. a space of the form $textrm{Spec}(A)$ for some commutative ring $A$. If $X$ is noetherian, does there also exist a noetherian ring $B$ such that $X=textrm{Spec}(B)$?
ag.algebraic-geometry ac.commutative-algebra gn.general-topology
Let $X$ be a spectral space (en.wikipedia.org/wiki/Spectral_space), i.e. a space of the form $textrm{Spec}(A)$ for some commutative ring $A$. If $X$ is noetherian, does there also exist a noetherian ring $B$ such that $X=textrm{Spec}(B)$?
ag.algebraic-geometry ac.commutative-algebra gn.general-topology
ag.algebraic-geometry ac.commutative-algebra gn.general-topology
edited Nov 10 at 18:17
მამუკა ჯიბლაძე
7,846243110
7,846243110
asked Nov 10 at 18:05
Hans
817512
817512
1
$mathrm{Spec}$ is an (anti-)equivalence from commutative rings to affine scheme, so two rings are isomorphic iff their Spec's are. So, if such a noetherian $B$ exists, your $A$ was already isomorphic to it.
– Qfwfq
Nov 10 at 18:19
1
Oh yes, you're totally right, it's the underlying top space of the Spec not the scheme
– Qfwfq
Nov 10 at 18:21
7
I wonder if Hochster's thesis addresses this? Off the top of my head, I don't know how to make the following Noetherian topological space ${p,q,r}$, with open sets ${ {p,q,r}, {p,q}, {p} }$, as the spectrum of a Noetherian ring (it's the spectrum of the non-discrete valuation ring associated to $mathbb{Z} times mathbb{Z}$ with the lex order).
– Karl Schwede
Nov 10 at 18:31
1
@KarlSchwede - You may want to take look at my comment below.
– Pierre-Yves Gaillard
Nov 10 at 20:03
add a comment |
1
$mathrm{Spec}$ is an (anti-)equivalence from commutative rings to affine scheme, so two rings are isomorphic iff their Spec's are. So, if such a noetherian $B$ exists, your $A$ was already isomorphic to it.
– Qfwfq
Nov 10 at 18:19
1
Oh yes, you're totally right, it's the underlying top space of the Spec not the scheme
– Qfwfq
Nov 10 at 18:21
7
I wonder if Hochster's thesis addresses this? Off the top of my head, I don't know how to make the following Noetherian topological space ${p,q,r}$, with open sets ${ {p,q,r}, {p,q}, {p} }$, as the spectrum of a Noetherian ring (it's the spectrum of the non-discrete valuation ring associated to $mathbb{Z} times mathbb{Z}$ with the lex order).
– Karl Schwede
Nov 10 at 18:31
1
@KarlSchwede - You may want to take look at my comment below.
– Pierre-Yves Gaillard
Nov 10 at 20:03
1
1
$mathrm{Spec}$ is an (anti-)equivalence from commutative rings to affine scheme, so two rings are isomorphic iff their Spec's are. So, if such a noetherian $B$ exists, your $A$ was already isomorphic to it.
– Qfwfq
Nov 10 at 18:19
$mathrm{Spec}$ is an (anti-)equivalence from commutative rings to affine scheme, so two rings are isomorphic iff their Spec's are. So, if such a noetherian $B$ exists, your $A$ was already isomorphic to it.
– Qfwfq
Nov 10 at 18:19
1
1
Oh yes, you're totally right, it's the underlying top space of the Spec not the scheme
– Qfwfq
Nov 10 at 18:21
Oh yes, you're totally right, it's the underlying top space of the Spec not the scheme
– Qfwfq
Nov 10 at 18:21
7
7
I wonder if Hochster's thesis addresses this? Off the top of my head, I don't know how to make the following Noetherian topological space ${p,q,r}$, with open sets ${ {p,q,r}, {p,q}, {p} }$, as the spectrum of a Noetherian ring (it's the spectrum of the non-discrete valuation ring associated to $mathbb{Z} times mathbb{Z}$ with the lex order).
– Karl Schwede
Nov 10 at 18:31
I wonder if Hochster's thesis addresses this? Off the top of my head, I don't know how to make the following Noetherian topological space ${p,q,r}$, with open sets ${ {p,q,r}, {p,q}, {p} }$, as the spectrum of a Noetherian ring (it's the spectrum of the non-discrete valuation ring associated to $mathbb{Z} times mathbb{Z}$ with the lex order).
– Karl Schwede
Nov 10 at 18:31
1
1
@KarlSchwede - You may want to take look at my comment below.
– Pierre-Yves Gaillard
Nov 10 at 20:03
@KarlSchwede - You may want to take look at my comment below.
– Pierre-Yves Gaillard
Nov 10 at 20:03
add a comment |
1 Answer
1
active
oldest
votes
Graph $N_5$ with poset order topology (i.e. poset $M={p,q,r}, P_2={p,q}, P_1={p}, Q={r}, N=phi$) is not Spec($A$) for Noetherian $A$ because if $a in Q-P_2$ then 1 = dim$(A/a)$ = dim$(A)-1$ = 2 by the principal ideal theorem.
6
See also top of p. 48 in Ring constructions on spectral spaces by Christopher Francis Tedd escholar.manchester.ac.uk/jrul/item/?pid=uk-ac-man-scw:307012 --- link to the PDF file: escholar.manchester.ac.uk/api/…
– Pierre-Yves Gaillard
Nov 10 at 19:57
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: "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
});
}
});
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%2f315031%2fnoetherian-spectral-space-comes-from-noetherian-ring%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
Graph $N_5$ with poset order topology (i.e. poset $M={p,q,r}, P_2={p,q}, P_1={p}, Q={r}, N=phi$) is not Spec($A$) for Noetherian $A$ because if $a in Q-P_2$ then 1 = dim$(A/a)$ = dim$(A)-1$ = 2 by the principal ideal theorem.
6
See also top of p. 48 in Ring constructions on spectral spaces by Christopher Francis Tedd escholar.manchester.ac.uk/jrul/item/?pid=uk-ac-man-scw:307012 --- link to the PDF file: escholar.manchester.ac.uk/api/…
– Pierre-Yves Gaillard
Nov 10 at 19:57
add a comment |
Graph $N_5$ with poset order topology (i.e. poset $M={p,q,r}, P_2={p,q}, P_1={p}, Q={r}, N=phi$) is not Spec($A$) for Noetherian $A$ because if $a in Q-P_2$ then 1 = dim$(A/a)$ = dim$(A)-1$ = 2 by the principal ideal theorem.
6
See also top of p. 48 in Ring constructions on spectral spaces by Christopher Francis Tedd escholar.manchester.ac.uk/jrul/item/?pid=uk-ac-man-scw:307012 --- link to the PDF file: escholar.manchester.ac.uk/api/…
– Pierre-Yves Gaillard
Nov 10 at 19:57
add a comment |
Graph $N_5$ with poset order topology (i.e. poset $M={p,q,r}, P_2={p,q}, P_1={p}, Q={r}, N=phi$) is not Spec($A$) for Noetherian $A$ because if $a in Q-P_2$ then 1 = dim$(A/a)$ = dim$(A)-1$ = 2 by the principal ideal theorem.
Graph $N_5$ with poset order topology (i.e. poset $M={p,q,r}, P_2={p,q}, P_1={p}, Q={r}, N=phi$) is not Spec($A$) for Noetherian $A$ because if $a in Q-P_2$ then 1 = dim$(A/a)$ = dim$(A)-1$ = 2 by the principal ideal theorem.
answered Nov 10 at 19:55
David Lampert
1,729169
1,729169
6
See also top of p. 48 in Ring constructions on spectral spaces by Christopher Francis Tedd escholar.manchester.ac.uk/jrul/item/?pid=uk-ac-man-scw:307012 --- link to the PDF file: escholar.manchester.ac.uk/api/…
– Pierre-Yves Gaillard
Nov 10 at 19:57
add a comment |
6
See also top of p. 48 in Ring constructions on spectral spaces by Christopher Francis Tedd escholar.manchester.ac.uk/jrul/item/?pid=uk-ac-man-scw:307012 --- link to the PDF file: escholar.manchester.ac.uk/api/…
– Pierre-Yves Gaillard
Nov 10 at 19:57
6
6
See also top of p. 48 in Ring constructions on spectral spaces by Christopher Francis Tedd escholar.manchester.ac.uk/jrul/item/?pid=uk-ac-man-scw:307012 --- link to the PDF file: escholar.manchester.ac.uk/api/…
– Pierre-Yves Gaillard
Nov 10 at 19:57
See also top of p. 48 in Ring constructions on spectral spaces by Christopher Francis Tedd escholar.manchester.ac.uk/jrul/item/?pid=uk-ac-man-scw:307012 --- link to the PDF file: escholar.manchester.ac.uk/api/…
– Pierre-Yves Gaillard
Nov 10 at 19:57
add a comment |
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2fmathoverflow.net%2fquestions%2f315031%2fnoetherian-spectral-space-comes-from-noetherian-ring%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
1
$mathrm{Spec}$ is an (anti-)equivalence from commutative rings to affine scheme, so two rings are isomorphic iff their Spec's are. So, if such a noetherian $B$ exists, your $A$ was already isomorphic to it.
– Qfwfq
Nov 10 at 18:19
1
Oh yes, you're totally right, it's the underlying top space of the Spec not the scheme
– Qfwfq
Nov 10 at 18:21
7
I wonder if Hochster's thesis addresses this? Off the top of my head, I don't know how to make the following Noetherian topological space ${p,q,r}$, with open sets ${ {p,q,r}, {p,q}, {p} }$, as the spectrum of a Noetherian ring (it's the spectrum of the non-discrete valuation ring associated to $mathbb{Z} times mathbb{Z}$ with the lex order).
– Karl Schwede
Nov 10 at 18:31
1
@KarlSchwede - You may want to take look at my comment below.
– Pierre-Yves Gaillard
Nov 10 at 20:03