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How to prove by universal properties that image sheaf is the sheafification of the image presheaf
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I'm working on an exercise from Vakil's notes.
Exercise 2.6.C Suppose $Phi:mathscrFtomathscrG$ is a morphism of sheaves of abelian groups, show that the image sheaf Im $Phi$ is the sheafification of the image presheaf.
There is a post discuss the same problem.
But I don't think the discussion given there is good enough to solve the problem.
Someone claimed in the comment that $ker(textcoker_prePhi)^sh = (ker_pretextcoker_prePhi)^sh$ but didn't give a proof.
Now I'm trying to come up with a proof just using the universal properties of $textCoker$, $textKer$ and sheafification.
But just using the categorial approach I found that there are some limitations that I couldn't pass.
So my question is: Is there any approach to the problem just using the universal properties?
Thank you very much for your help!
algebraic-geometry homological-algebra sheaf-theory
asked Jul 21 at 2:19
user263834
430213
add a comment |Â
up vote
1
down vote
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I'm working on an exercise from Vakil's notes.
Exercise 2.6.C Suppose $Phi:mathscrFtomathscrG$ is a morphism of sheaves of abelian groups, show that the image sheaf Im $Phi$ is the sheafification of the image presheaf.
There is a post discuss the same problem.
But I don't think the discussion given there is good enough to solve the problem.
Someone claimed in the comment that $ker(textcoker_prePhi)^sh = (ker_pretextcoker_prePhi)^sh$ but didn't give a proof.
Now I'm trying to come up with a proof just using the universal properties of $textCoker$, $textKer$ and sheafification.
But just using the categorial approach I found that there are some limitations that I couldn't pass.
So my question is: Is there any approach to the problem just using the universal properties?
Thank you very much for your help!
algebraic-geometry homological-algebra sheaf-theory
asked Jul 21 at 2:19
user263834
430213
2
The sheaf kernel of the sheafification of a morphism is the sheafification of the presheaf kernel of that morphsim because sheafification is exact. This should solve your problem.
– KReiser
Jul 21 at 2:58
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm working on an exercise from Vakil's notes.
Exercise 2.6.C Suppose $Phi:mathscrFtomathscrG$ is a morphism of sheaves of abelian groups, show that the image sheaf Im $Phi$ is the sheafification of the image presheaf.
There is a post discuss the same problem.
But I don't think the discussion given there is good enough to solve the problem.
Someone claimed in the comment that $ker(textcoker_prePhi)^sh = (ker_pretextcoker_prePhi)^sh$ but didn't give a proof.
Now I'm trying to come up with a proof just using the universal properties of $textCoker$, $textKer$ and sheafification.
But just using the categorial approach I found that there are some limitations that I couldn't pass.
So my question is: Is there any approach to the problem just using the universal properties?
Thank you very much for your help!
algebraic-geometry homological-algebra sheaf-theory
asked Jul 21 at 2:19
user263834
430213
I'm working on an exercise from Vakil's notes.
Exercise 2.6.C Suppose $Phi:mathscrFtomathscrG$ is a morphism of sheaves of abelian groups, show that the image sheaf Im $Phi$ is the sheafification of the image presheaf.
There is a post discuss the same problem.
But I don't think the discussion given there is good enough to solve the problem.
Someone claimed in the comment that $ker(textcoker_prePhi)^sh = (ker_pretextcoker_prePhi)^sh$ but didn't give a proof.
Now I'm trying to come up with a proof just using the universal properties of $textCoker$, $textKer$ and sheafification.
But just using the categorial approach I found that there are some limitations that I couldn't pass.
So my question is: Is there any approach to the problem just using the universal properties?
Thank you very much for your help!
algebraic-geometry homological-algebra sheaf-theory
asked Jul 21 at 2:19
user263834
430213
asked Jul 21 at 2:19
user263834
430213
asked Jul 21 at 2:19
user263834
430213
asked Jul 21 at 2:19
user263834
430213
430213
2
The sheaf kernel of the sheafification of a morphism is the sheafification of the presheaf kernel of that morphsim because sheafification is exact. This should solve your problem.
– KReiser
Jul 21 at 2:58
add a comment |Â
2
The sheaf kernel of the sheafification of a morphism is the sheafification of the presheaf kernel of that morphsim because sheafification is exact. This should solve your problem.
– KReiser
Jul 21 at 2:58
2
2
The sheaf kernel of the sheafification of a morphism is the sheafification of the presheaf kernel of that morphsim because sheafification is exact. This should solve your problem.
– KReiser
Jul 21 at 2:58
The sheaf kernel of the sheafification of a morphism is the sheafification of the presheaf kernel of that morphsim because sheafification is exact. This should solve your problem.
– KReiser
Jul 21 at 2:58
add a comment |Â
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2
The sheaf kernel of the sheafification of a morphism is the sheafification of the presheaf kernel of that morphsim because sheafification is exact. This should solve your problem.
– KReiser
Jul 21 at 2:58