# Diagonal is representable then any morphism is representable

Clash Royale CLAN TAG#URR8PPP

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?

• 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

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

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?

• 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

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

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?

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?

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

edited Nov 29 at 12:31

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

• @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

• @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

@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

active

oldest

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.

• 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

active

oldest

active

oldest

active

oldest

active

oldest

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.

• 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

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.

• 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

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.

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.

Leo Alonso

5,19122136

5,19122136

• 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

• 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

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

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