Diagonal is representable then any morphism is representable

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP

up vote
3
down vote

favorite

Ariyan Javanpeykar said here in comments that,

If the diagonal is representable, then isn’t any morphism $Srightarrow mathcal{X}$ with $S$ a scheme representable?

I could not find the statement (Thanks to my bad searching skills). I would like to prove this and use this to deduce something else.

A stack $mathcal{X}$ over a scheme $T$ is a stack over category “schemes over $T$” i.e., we have a functor $mathcal{X}rightarrow Sch/T$. We can talk about $2$-fiber product here which I denote by $mathcal{X}times_Tmathcal{X}$.

Consider diagonal $Delta:mathcal{X}rightarrow mathcal{X}times_{T}mathcal{X}$. This is a morphism of stacks. (In stacks project, they simply write $mathcal{X}rightarrow mathcal{X}times mathcal{X}$. It is somewhat confusing. May be they have fixed notation somewhere but I think specifying $T$ is a good idea.)

We call a morphism of stacks $F:mathcal{M}rightarrow mathcal{N}$ to be representable if, given a morphism of stacks $G:Srightarrow mathcal{N}$, the product $mathcal{M}times_{mathcal{N}}S$ is a scheme.

Suppose $Delta$ is representable. Consider a map of stacks $F:Srightarrow mathcal{X}$. I want to see if $F$ is representable. For that, I take a morphism of stacks $G:Xrightarrow mathcal{X}$ and prove that $Stimes_{mathcal{X}}X$ is a scheme.

As $Delta:mathcal{X}rightarrow mathcal{X}times mathcal{X}times_ Tmathcal{X}$ is representable, to use the representability property of this map, I have to start with a scheme and a map from that scheme to $mathcal{X}times_T mathcal{X}$.

I have $F:Srightarrow mathcal{X}$ and $G:Xrightarrow mathcal{X}$. We can consider $(F,G):Stimes_TXrightarrow mathcal{X}_Tmathcal{X}$. As $Stimes_TX$ is a scheme, we can consider the map $(F,G):Stimes_TXrightarrow mathcal{X}times_Tmathcal{X}$.

As $Delta:mathcal{X}rightarrow mathcal{X}times_T mathcal{X}$
is representable, this means that $mathcal{X}times_{mathcal{X}times_Tmathcal{X}}(Stimes_TX)$ is a scheme. I did not prove explicitly, but I am almost sure that $mathcal{X}times_{mathcal{X}times_Tmathcal{X}}(Stimes_TX)$ is isomorphic to $Stimes_{mathcal{X}}T$ which is what I wanted to see.

Is this proof correct?

Any comments are welcome.

share|cite|improve this question

  • Please do let me know if any statement is not clear.
    – Praphulla Koushik
    Nov 29 at 11:11

  • It seems like a sentence cut off in the second paragraph: “…for any object $C$ of $mathcal{C}$…”
    – WSL
    Nov 29 at 11:20

  • 3

    @WSL : Does it look better now.? Thanks for pointing out..
    – Praphulla Koushik
    Nov 29 at 11:26

up vote
3
down vote

favorite

Ariyan Javanpeykar said here in comments that,

If the diagonal is representable, then isn’t any morphism $Srightarrow mathcal{X}$ with $S$ a scheme representable?

I could not find the statement (Thanks to my bad searching skills). I would like to prove this and use this to deduce something else.

A stack $mathcal{X}$ over a scheme $T$ is a stack over category “schemes over $T$” i.e., we have a functor $mathcal{X}rightarrow Sch/T$. We can talk about $2$-fiber product here which I denote by $mathcal{X}times_Tmathcal{X}$.

Consider diagonal $Delta:mathcal{X}rightarrow mathcal{X}times_{T}mathcal{X}$. This is a morphism of stacks. (In stacks project, they simply write $mathcal{X}rightarrow mathcal{X}times mathcal{X}$. It is somewhat confusing. May be they have fixed notation somewhere but I think specifying $T$ is a good idea.)

We call a morphism of stacks $F:mathcal{M}rightarrow mathcal{N}$ to be representable if, given a morphism of stacks $G:Srightarrow mathcal{N}$, the product $mathcal{M}times_{mathcal{N}}S$ is a scheme.

Suppose $Delta$ is representable. Consider a map of stacks $F:Srightarrow mathcal{X}$. I want to see if $F$ is representable. For that, I take a morphism of stacks $G:Xrightarrow mathcal{X}$ and prove that $Stimes_{mathcal{X}}X$ is a scheme.

As $Delta:mathcal{X}rightarrow mathcal{X}times mathcal{X}times_ Tmathcal{X}$ is representable, to use the representability property of this map, I have to start with a scheme and a map from that scheme to $mathcal{X}times_T mathcal{X}$.

I have $F:Srightarrow mathcal{X}$ and $G:Xrightarrow mathcal{X}$. We can consider $(F,G):Stimes_TXrightarrow mathcal{X}_Tmathcal{X}$. As $Stimes_TX$ is a scheme, we can consider the map $(F,G):Stimes_TXrightarrow mathcal{X}times_Tmathcal{X}$.

As $Delta:mathcal{X}rightarrow mathcal{X}times_T mathcal{X}$
is representable, this means that $mathcal{X}times_{mathcal{X}times_Tmathcal{X}}(Stimes_TX)$ is a scheme. I did not prove explicitly, but I am almost sure that $mathcal{X}times_{mathcal{X}times_Tmathcal{X}}(Stimes_TX)$ is isomorphic to $Stimes_{mathcal{X}}T$ which is what I wanted to see.

Is this proof correct?

Any comments are welcome.

share|cite|improve this question

  • Please do let me know if any statement is not clear.
    – Praphulla Koushik
    Nov 29 at 11:11

  • It seems like a sentence cut off in the second paragraph: “…for any object $C$ of $mathcal{C}$…”
    – WSL
    Nov 29 at 11:20

  • 3

    @WSL : Does it look better now.? Thanks for pointing out..
    – Praphulla Koushik
    Nov 29 at 11:26

up vote
3
down vote

favorite

up vote
3
down vote

favorite

Ariyan Javanpeykar said here in comments that,

If the diagonal is representable, then isn’t any morphism $Srightarrow mathcal{X}$ with $S$ a scheme representable?

I could not find the statement (Thanks to my bad searching skills). I would like to prove this and use this to deduce something else.

A stack $mathcal{X}$ over a scheme $T$ is a stack over category “schemes over $T$” i.e., we have a functor $mathcal{X}rightarrow Sch/T$. We can talk about $2$-fiber product here which I denote by $mathcal{X}times_Tmathcal{X}$.

Consider diagonal $Delta:mathcal{X}rightarrow mathcal{X}times_{T}mathcal{X}$. This is a morphism of stacks. (In stacks project, they simply write $mathcal{X}rightarrow mathcal{X}times mathcal{X}$. It is somewhat confusing. May be they have fixed notation somewhere but I think specifying $T$ is a good idea.)

We call a morphism of stacks $F:mathcal{M}rightarrow mathcal{N}$ to be representable if, given a morphism of stacks $G:Srightarrow mathcal{N}$, the product $mathcal{M}times_{mathcal{N}}S$ is a scheme.

Suppose $Delta$ is representable. Consider a map of stacks $F:Srightarrow mathcal{X}$. I want to see if $F$ is representable. For that, I take a morphism of stacks $G:Xrightarrow mathcal{X}$ and prove that $Stimes_{mathcal{X}}X$ is a scheme.

As $Delta:mathcal{X}rightarrow mathcal{X}times mathcal{X}times_ Tmathcal{X}$ is representable, to use the representability property of this map, I have to start with a scheme and a map from that scheme to $mathcal{X}times_T mathcal{X}$.

I have $F:Srightarrow mathcal{X}$ and $G:Xrightarrow mathcal{X}$. We can consider $(F,G):Stimes_TXrightarrow mathcal{X}_Tmathcal{X}$. As $Stimes_TX$ is a scheme, we can consider the map $(F,G):Stimes_TXrightarrow mathcal{X}times_Tmathcal{X}$.

As $Delta:mathcal{X}rightarrow mathcal{X}times_T mathcal{X}$
is representable, this means that $mathcal{X}times_{mathcal{X}times_Tmathcal{X}}(Stimes_TX)$ is a scheme. I did not prove explicitly, but I am almost sure that $mathcal{X}times_{mathcal{X}times_Tmathcal{X}}(Stimes_TX)$ is isomorphic to $Stimes_{mathcal{X}}T$ which is what I wanted to see.

Is this proof correct?

Any comments are welcome.

share|cite|improve this question

Ariyan Javanpeykar said here in comments that,

If the diagonal is representable, then isn’t any morphism $Srightarrow mathcal{X}$ with $S$ a scheme representable?

I could not find the statement (Thanks to my bad searching skills). I would like to prove this and use this to deduce something else.

A stack $mathcal{X}$ over a scheme $T$ is a stack over category “schemes over $T$” i.e., we have a functor $mathcal{X}rightarrow Sch/T$. We can talk about $2$-fiber product here which I denote by $mathcal{X}times_Tmathcal{X}$.

Consider diagonal $Delta:mathcal{X}rightarrow mathcal{X}times_{T}mathcal{X}$. This is a morphism of stacks. (In stacks project, they simply write $mathcal{X}rightarrow mathcal{X}times mathcal{X}$. It is somewhat confusing. May be they have fixed notation somewhere but I think specifying $T$ is a good idea.)

We call a morphism of stacks $F:mathcal{M}rightarrow mathcal{N}$ to be representable if, given a morphism of stacks $G:Srightarrow mathcal{N}$, the product $mathcal{M}times_{mathcal{N}}S$ is a scheme.

Suppose $Delta$ is representable. Consider a map of stacks $F:Srightarrow mathcal{X}$. I want to see if $F$ is representable. For that, I take a morphism of stacks $G:Xrightarrow mathcal{X}$ and prove that $Stimes_{mathcal{X}}X$ is a scheme.

As $Delta:mathcal{X}rightarrow mathcal{X}times mathcal{X}times_ Tmathcal{X}$ is representable, to use the representability property of this map, I have to start with a scheme and a map from that scheme to $mathcal{X}times_T mathcal{X}$.

I have $F:Srightarrow mathcal{X}$ and $G:Xrightarrow mathcal{X}$. We can consider $(F,G):Stimes_TXrightarrow mathcal{X}_Tmathcal{X}$. As $Stimes_TX$ is a scheme, we can consider the map $(F,G):Stimes_TXrightarrow mathcal{X}times_Tmathcal{X}$.

As $Delta:mathcal{X}rightarrow mathcal{X}times_T mathcal{X}$
is representable, this means that $mathcal{X}times_{mathcal{X}times_Tmathcal{X}}(Stimes_TX)$ is a scheme. I did not prove explicitly, but I am almost sure that $mathcal{X}times_{mathcal{X}times_Tmathcal{X}}(Stimes_TX)$ is isomorphic to $Stimes_{mathcal{X}}T$ which is what I wanted to see.

Is this proof correct?

Any comments are welcome.

ag.algebraic-geometry dg.differential-geometry ct.category-theory schemes stacks

share|cite|improve this question

share|cite|improve this question

share|cite|improve this question

share|cite|improve this question

edited Nov 29 at 12:31

asked Nov 29 at 10:41

Praphulla Koushik

6451421

6451421

  • Please do let me know if any statement is not clear.
    – Praphulla Koushik
    Nov 29 at 11:11

  • It seems like a sentence cut off in the second paragraph: “…for any object $C$ of $mathcal{C}$…”
    – WSL
    Nov 29 at 11:20

  • 3

    @WSL : Does it look better now.? Thanks for pointing out..
    – Praphulla Koushik
    Nov 29 at 11:26

  • Please do let me know if any statement is not clear.
    – Praphulla Koushik
    Nov 29 at 11:11

  • It seems like a sentence cut off in the second paragraph: “…for any object $C$ of $mathcal{C}$…”
    – WSL
    Nov 29 at 11:20

  • 3

    @WSL : Does it look better now.? Thanks for pointing out..
    – Praphulla Koushik
    Nov 29 at 11:26

Please do let me know if any statement is not clear.
– Praphulla Koushik
Nov 29 at 11:11

Please do let me know if any statement is not clear.
– Praphulla Koushik
Nov 29 at 11:11

It seems like a sentence cut off in the second paragraph: “…for any object $C$ of $mathcal{C}$…”
– WSL
Nov 29 at 11:20

It seems like a sentence cut off in the second paragraph: “…for any object $C$ of $mathcal{C}$…”
– WSL
Nov 29 at 11:20

3

3

@WSL : Does it look better now.? Thanks for pointing out..
– Praphulla Koushik
Nov 29 at 11:26

@WSL : Does it look better now.? Thanks for pointing out..
– Praphulla Koushik
Nov 29 at 11:26

1 Answer
1

active

oldest

votes

up vote
4
down vote

accepted

I guess it is correct (and may be rendered in a simpler way). Ideed, let $delta:mathcal{X}rightarrow mathcal{X}times_{T}mathcal{X}$ be the diagonal map. If it is representable then every morphism $u : S → mathcal{X}$ is representable. For $v : V → mathcal{X}$ another morphism with $V$ a scheme, we have that
$$
S times_{mathcal{X}} V cong mathcal{X} times_{mathcal{X}timesmathcal{X}} (S times_T V)
$$

is 1-isomorphic to a scheme ($delta$ is representable) and this 1-isomorphism turns out to be an isomorphism because $S times_{mathcal{X}} V$ is in fact a category fibered in sets, therefore corresponds to a scheme.

For your second question, I don’t see any special use of the category of schemes, the displayed isomorphism is a basic categorical fact.

share|cite|improve this answer

  • 1

    Thanks for the clarification. 🙂
    – Praphulla Koushik
    Nov 29 at 11:28

  • You can suggest some reference if are free.. Not specifically for this question but for general notion of stacks.
    – Praphulla Koushik
    Nov 29 at 15:11

  • @praphulla-koushik I guess a very nice entry point is Vistoli “Notes on Grothendieck topologies, fibered categories and descent theory” (homepage.sns.it/vistoli/descent.pdf) for stacks in general. The book “Algebraic Spaces and Stacks” by Martin Olsson is a nice modern introduction to algebraic stacks.
    – Leo Alonso
    Nov 29 at 17:08

  • I have seen Vistoli’s notes… What ever I know (not much), I learned from his notes.. I see it every now and then to get some idea… I have not seen Martin Olsson’s book. I will have a look at that.. 🙂
    – Praphulla Koushik
    Nov 29 at 17:18

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’,
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
});

}
});

draft saved
draft discarded

StackExchange.ready(
function () {
StackExchange.openid.initPostLogin(‘.new-post-login’, ‘https%3a%2f%2fmathoverflow.net%2fquestions%2f316484%2fdiagonal-is-representable-then-any-morphism-is-representable%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

up vote
4
down vote

accepted

I guess it is correct (and may be rendered in a simpler way). Ideed, let $delta:mathcal{X}rightarrow mathcal{X}times_{T}mathcal{X}$ be the diagonal map. If it is representable then every morphism $u : S → mathcal{X}$ is representable. For $v : V → mathcal{X}$ another morphism with $V$ a scheme, we have that
$$
S times_{mathcal{X}} V cong mathcal{X} times_{mathcal{X}timesmathcal{X}} (S times_T V)
$$

is 1-isomorphic to a scheme ($delta$ is representable) and this 1-isomorphism turns out to be an isomorphism because $S times_{mathcal{X}} V$ is in fact a category fibered in sets, therefore corresponds to a scheme.

For your second question, I don’t see any special use of the category of schemes, the displayed isomorphism is a basic categorical fact.

share|cite|improve this answer

  • 1

    Thanks for the clarification. 🙂
    – Praphulla Koushik
    Nov 29 at 11:28

  • You can suggest some reference if are free.. Not specifically for this question but for general notion of stacks.
    – Praphulla Koushik
    Nov 29 at 15:11

  • @praphulla-koushik I guess a very nice entry point is Vistoli “Notes on Grothendieck topologies, fibered categories and descent theory” (homepage.sns.it/vistoli/descent.pdf) for stacks in general. The book “Algebraic Spaces and Stacks” by Martin Olsson is a nice modern introduction to algebraic stacks.
    – Leo Alonso
    Nov 29 at 17:08

  • I have seen Vistoli’s notes… What ever I know (not much), I learned from his notes.. I see it every now and then to get some idea… I have not seen Martin Olsson’s book. I will have a look at that.. 🙂
    – Praphulla Koushik
    Nov 29 at 17:18

up vote
4
down vote

accepted

I guess it is correct (and may be rendered in a simpler way). Ideed, let $delta:mathcal{X}rightarrow mathcal{X}times_{T}mathcal{X}$ be the diagonal map. If it is representable then every morphism $u : S → mathcal{X}$ is representable. For $v : V → mathcal{X}$ another morphism with $V$ a scheme, we have that
$$
S times_{mathcal{X}} V cong mathcal{X} times_{mathcal{X}timesmathcal{X}} (S times_T V)
$$

is 1-isomorphic to a scheme ($delta$ is representable) and this 1-isomorphism turns out to be an isomorphism because $S times_{mathcal{X}} V$ is in fact a category fibered in sets, therefore corresponds to a scheme.

For your second question, I don’t see any special use of the category of schemes, the displayed isomorphism is a basic categorical fact.

share|cite|improve this answer

  • 1

    Thanks for the clarification. 🙂
    – Praphulla Koushik
    Nov 29 at 11:28

  • You can suggest some reference if are free.. Not specifically for this question but for general notion of stacks.
    – Praphulla Koushik
    Nov 29 at 15:11

  • @praphulla-koushik I guess a very nice entry point is Vistoli “Notes on Grothendieck topologies, fibered categories and descent theory” (homepage.sns.it/vistoli/descent.pdf) for stacks in general. The book “Algebraic Spaces and Stacks” by Martin Olsson is a nice modern introduction to algebraic stacks.
    – Leo Alonso
    Nov 29 at 17:08

  • I have seen Vistoli’s notes… What ever I know (not much), I learned from his notes.. I see it every now and then to get some idea… I have not seen Martin Olsson’s book. I will have a look at that.. 🙂
    – Praphulla Koushik
    Nov 29 at 17:18

up vote
4
down vote

accepted

up vote
4
down vote

accepted

I guess it is correct (and may be rendered in a simpler way). Ideed, let $delta:mathcal{X}rightarrow mathcal{X}times_{T}mathcal{X}$ be the diagonal map. If it is representable then every morphism $u : S → mathcal{X}$ is representable. For $v : V → mathcal{X}$ another morphism with $V$ a scheme, we have that
$$
S times_{mathcal{X}} V cong mathcal{X} times_{mathcal{X}timesmathcal{X}} (S times_T V)
$$

is 1-isomorphic to a scheme ($delta$ is representable) and this 1-isomorphism turns out to be an isomorphism because $S times_{mathcal{X}} V$ is in fact a category fibered in sets, therefore corresponds to a scheme.

For your second question, I don’t see any special use of the category of schemes, the displayed isomorphism is a basic categorical fact.

share|cite|improve this answer

I guess it is correct (and may be rendered in a simpler way). Ideed, let $delta:mathcal{X}rightarrow mathcal{X}times_{T}mathcal{X}$ be the diagonal map. If it is representable then every morphism $u : S → mathcal{X}$ is representable. For $v : V → mathcal{X}$ another morphism with $V$ a scheme, we have that
$$
S times_{mathcal{X}} V cong mathcal{X} times_{mathcal{X}timesmathcal{X}} (S times_T V)
$$

is 1-isomorphic to a scheme ($delta$ is representable) and this 1-isomorphism turns out to be an isomorphism because $S times_{mathcal{X}} V$ is in fact a category fibered in sets, therefore corresponds to a scheme.

For your second question, I don’t see any special use of the category of schemes, the displayed isomorphism is a basic categorical fact.

share|cite|improve this answer

share|cite|improve this answer

share|cite|improve this answer

answered Nov 29 at 11:23

Leo Alonso

5,19122136

5,19122136

  • 1

    Thanks for the clarification. 🙂
    – Praphulla Koushik
    Nov 29 at 11:28

  • You can suggest some reference if are free.. Not specifically for this question but for general notion of stacks.
    – Praphulla Koushik
    Nov 29 at 15:11

  • @praphulla-koushik I guess a very nice entry point is Vistoli “Notes on Grothendieck topologies, fibered categories and descent theory” (homepage.sns.it/vistoli/descent.pdf) for stacks in general. The book “Algebraic Spaces and Stacks” by Martin Olsson is a nice modern introduction to algebraic stacks.
    – Leo Alonso
    Nov 29 at 17:08

  • I have seen Vistoli’s notes… What ever I know (not much), I learned from his notes.. I see it every now and then to get some idea… I have not seen Martin Olsson’s book. I will have a look at that.. 🙂
    – Praphulla Koushik
    Nov 29 at 17:18

  • 1

    Thanks for the clarification. 🙂
    – Praphulla Koushik
    Nov 29 at 11:28

  • You can suggest some reference if are free.. Not specifically for this question but for general notion of stacks.
    – Praphulla Koushik
    Nov 29 at 15:11

  • @praphulla-koushik I guess a very nice entry point is Vistoli “Notes on Grothendieck topologies, fibered categories and descent theory” (homepage.sns.it/vistoli/descent.pdf) for stacks in general. The book “Algebraic Spaces and Stacks” by Martin Olsson is a nice modern introduction to algebraic stacks.
    – Leo Alonso
    Nov 29 at 17:08

  • I have seen Vistoli’s notes… What ever I know (not much), I learned from his notes.. I see it every now and then to get some idea… I have not seen Martin Olsson’s book. I will have a look at that.. 🙂
    – Praphulla Koushik
    Nov 29 at 17:18

1

1

Thanks for the clarification. 🙂
– Praphulla Koushik
Nov 29 at 11:28

Thanks for the clarification. 🙂
– Praphulla Koushik
Nov 29 at 11:28

You can suggest some reference if are free.. Not specifically for this question but for general notion of stacks.
– Praphulla Koushik
Nov 29 at 15:11

You can suggest some reference if are free.. Not specifically for this question but for general notion of stacks.
– Praphulla Koushik
Nov 29 at 15:11

@praphulla-koushik I guess a very nice entry point is Vistoli “Notes on Grothendieck topologies, fibered categories and descent theory” (homepage.sns.it/vistoli/descent.pdf) for stacks in general. The book “Algebraic Spaces and Stacks” by Martin Olsson is a nice modern introduction to algebraic stacks.
– Leo Alonso
Nov 29 at 17:08

@praphulla-koushik I guess a very nice entry point is Vistoli “Notes on Grothendieck topologies, fibered categories and descent theory” (homepage.sns.it/vistoli/descent.pdf) for stacks in general. The book “Algebraic Spaces and Stacks” by Martin Olsson is a nice modern introduction to algebraic stacks.
– Leo Alonso
Nov 29 at 17:08

I have seen Vistoli’s notes… What ever I know (not much), I learned from his notes.. I see it every now and then to get some idea… I have not seen Martin Olsson’s book. I will have a look at that.. 🙂
– Praphulla Koushik
Nov 29 at 17:18

I have seen Vistoli’s notes… What ever I know (not much), I learned from his notes.. I see it every now and then to get some idea… I have not seen Martin Olsson’s book. I will have a look at that.. 🙂
– Praphulla Koushik
Nov 29 at 17:18

draft saved
draft discarded

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.

draft saved

draft discarded

StackExchange.ready(
function () {
StackExchange.openid.initPostLogin(‘.new-post-login’, ‘https%3a%2f%2fmathoverflow.net%2fquestions%2f316484%2fdiagonal-is-representable-then-any-morphism-is-representable%23new-answer’, ‘question_page’);
}
);

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

Related Post

Leave a Reply

Your email address will not be published. Required fields are marked *