Can you give two closed immersions $(f,f^sharp),(g,g^sharp):(X,mathcal O_X)to (Y,mathcal O_Y)$ such that $f=g$ but $f^sharpneq g^sharp$? [closed]

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Can you give two closed immersions of schemes $(f,f^sharp),(g,g^sharp):(X,mathcal O_X)to (Y,mathcal O_Y)$ such that $f=g$ but $f^sharpneq g^sharp$?

algebraic-geometry schemes

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asked Jul 30 at 5:32

Born to be proud

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closed as unclear what you're asking by Leucippus, José Carlos Santos, max_zorn, Jose Arnaldo Bebita Dris, Claude Leibovici Aug 2 at 8:52

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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Can you give two closed immersions of schemes $(f,f^sharp),(g,g^sharp):(X,mathcal O_X)to (Y,mathcal O_Y)$ such that $f=g$ but $f^sharpneq g^sharp$?

algebraic-geometry schemes

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asked Jul 30 at 5:32

Born to be proud

41429

closed as unclear what you're asking by Leucippus, José Carlos Santos, max_zorn, Jose Arnaldo Bebita Dris, Claude Leibovici Aug 2 at 8:52

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Can you give two closed immersions of schemes $(f,f^sharp),(g,g^sharp):(X,mathcal O_X)to (Y,mathcal O_Y)$ such that $f=g$ but $f^sharpneq g^sharp$?

algebraic-geometry schemes

share|cite|improve this question

asked Jul 30 at 5:32

Born to be proud

41429

Can you give two closed immersions of schemes $(f,f^sharp),(g,g^sharp):(X,mathcal O_X)to (Y,mathcal O_Y)$ such that $f=g$ but $f^sharpneq g^sharp$?

algebraic-geometry schemes

share|cite|improve this question

asked Jul 30 at 5:32

Born to be proud

41429

share|cite|improve this question

share|cite|improve this question

asked Jul 30 at 5:32

Born to be proud

41429

asked Jul 30 at 5:32

Born to be proud

41429

asked Jul 30 at 5:32

Born to be proud

41429

41429

closed as unclear what you're asking by Leucippus, José Carlos Santos, max_zorn, Jose Arnaldo Bebita Dris, Claude Leibovici Aug 2 at 8:52

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

closed as unclear what you're asking by Leucippus, José Carlos Santos, max_zorn, Jose Arnaldo Bebita Dris, Claude Leibovici Aug 2 at 8:52

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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1 Answer
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Let $k$ be a field, $Y=mathbbA^2_k= mathrmSpeck[x,y]$ and $X= mathrmSpeck[z]/(z^2)$. Consider the morphisms defined by

$$k[x,y] rightarrow k[z]/(z^2)$$

with

$$f^sharpcolon x mapsto z, y mapsto 0$$

and

$$g^sharp colon x mapsto 0, y mapsto z.$$

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answered Jul 30 at 12:10

Louis

2,28011228

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So this give two different closed subschemes of $Y$, doesn't it?
  – Born to be proud
  Jul 30 at 16:47
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  @Borntobeproud Yes, one with fuzz in the $x$ direction, and the other in the $y$ direction.
  – Fan Zheng
  Jul 31 at 18:37
  

  
  
  
  

add a comment | 

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
2
down vote

Let $k$ be a field, $Y=mathbbA^2_k= mathrmSpeck[x,y]$ and $X= mathrmSpeck[z]/(z^2)$. Consider the morphisms defined by

$$k[x,y] rightarrow k[z]/(z^2)$$

with

$$f^sharpcolon x mapsto z, y mapsto 0$$

and

$$g^sharp colon x mapsto 0, y mapsto z.$$

share|cite|improve this answer

answered Jul 30 at 12:10

Louis

2,28011228

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So this give two different closed subschemes of $Y$, doesn't it?
  – Born to be proud
  Jul 30 at 16:47
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  @Borntobeproud Yes, one with fuzz in the $x$ direction, and the other in the $y$ direction.
  – Fan Zheng
  Jul 31 at 18:37
  

  
  
  
  

add a comment | 

up vote
2
down vote

Let $k$ be a field, $Y=mathbbA^2_k= mathrmSpeck[x,y]$ and $X= mathrmSpeck[z]/(z^2)$. Consider the morphisms defined by

$$k[x,y] rightarrow k[z]/(z^2)$$

with

$$f^sharpcolon x mapsto z, y mapsto 0$$

and

$$g^sharp colon x mapsto 0, y mapsto z.$$

share|cite|improve this answer

answered Jul 30 at 12:10

Louis

2,28011228

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So this give two different closed subschemes of $Y$, doesn't it?
  – Born to be proud
  Jul 30 at 16:47
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  @Borntobeproud Yes, one with fuzz in the $x$ direction, and the other in the $y$ direction.
  – Fan Zheng
  Jul 31 at 18:37
  

  
  
  
  

add a comment | 

up vote
2
down vote

up vote
2
down vote

Let $k$ be a field, $Y=mathbbA^2_k= mathrmSpeck[x,y]$ and $X= mathrmSpeck[z]/(z^2)$. Consider the morphisms defined by

$$k[x,y] rightarrow k[z]/(z^2)$$

with

$$f^sharpcolon x mapsto z, y mapsto 0$$

and

$$g^sharp colon x mapsto 0, y mapsto z.$$

share|cite|improve this answer

answered Jul 30 at 12:10

Louis

2,28011228

Let $k$ be a field, $Y=mathbbA^2_k= mathrmSpeck[x,y]$ and $X= mathrmSpeck[z]/(z^2)$. Consider the morphisms defined by

$$k[x,y] rightarrow k[z]/(z^2)$$

with

$$f^sharpcolon x mapsto z, y mapsto 0$$

and

$$g^sharp colon x mapsto 0, y mapsto z.$$

share|cite|improve this answer

answered Jul 30 at 12:10

Louis

2,28011228

share|cite|improve this answer

share|cite|improve this answer

answered Jul 30 at 12:10

Louis

2,28011228

answered Jul 30 at 12:10

Louis

2,28011228

answered Jul 30 at 12:10

Louis

2,28011228

2,28011228

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So this give two different closed subschemes of $Y$, doesn't it?
  – Born to be proud
  Jul 30 at 16:47
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  @Borntobeproud Yes, one with fuzz in the $x$ direction, and the other in the $y$ direction.
  – Fan Zheng
  Jul 31 at 18:37
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So this give two different closed subschemes of $Y$, doesn't it?
  – Born to be proud
  Jul 30 at 16:47
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  @Borntobeproud Yes, one with fuzz in the $x$ direction, and the other in the $y$ direction.
  – Fan Zheng
  Jul 31 at 18:37
  

  
  
  
  

So this give two different closed subschemes of $Y$, doesn't it?
– Born to be proud
Jul 30 at 16:47

So this give two different closed subschemes of $Y$, doesn't it?
– Born to be proud
Jul 30 at 16:47

2

2

@Borntobeproud Yes, one with fuzz in the $x$ direction, and the other in the $y$ direction.
– Fan Zheng
Jul 31 at 18:37

@Borntobeproud Yes, one with fuzz in the $x$ direction, and the other in the $y$ direction.
– Fan Zheng
Jul 31 at 18:37

add a comment |