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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.
add a comment 
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.

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
add a comment 
3
down vote
favorite
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.
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.

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
add a comment 

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
– Praphulla Koushik
Nov 29 at 11:11
– Praphulla Koushik
Nov 29 at 11:11
– WSL
Nov 29 at 11:20
– WSL
Nov 29 at 11:20
– Praphulla Koushik
Nov 29 at 11:26
– Praphulla Koushik
Nov 29 at 11:26
add a comment 
1 Answer
1
active
oldest
votes
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 1isomorphic to a scheme ($delta$ is representable) and this 1isomorphism 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.

1Thanks 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 
@praphullakoushik 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
add a comment 
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 1isomorphic to a scheme ($delta$ is representable) and this 1isomorphism 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.

1Thanks 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 
@praphullakoushik 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
add a comment 
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 1isomorphic to a scheme ($delta$ is representable) and this 1isomorphism 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.

1Thanks 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 
@praphullakoushik 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
add a comment 
4
down vote
accepted
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 1isomorphic to a scheme ($delta$ is representable) and this 1isomorphism 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.
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 1isomorphic to a scheme ($delta$ is representable) and this 1isomorphism 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.

1Thanks 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 
@praphullakoushik 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
add a comment 

1Thanks 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 
@praphullakoushik 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
– Praphulla Koushik
Nov 29 at 11:28
– Praphulla Koushik
Nov 29 at 11:28
– Praphulla Koushik
Nov 29 at 15:11
– Praphulla Koushik
Nov 29 at 15:11
– Leo Alonso
Nov 29 at 17:08
– Leo Alonso
Nov 29 at 17:08
– Praphulla Koushik
Nov 29 at 17:18
– Praphulla Koushik
Nov 29 at 17:18
add a comment 
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– Praphulla Koushik
Nov 29 at 11:11
– WSL
Nov 29 at 11:20
– Praphulla Koushik
Nov 29 at 11:26