Skip to main content
MathOverflow

Return to Answer

corrected the formulae again; fixed a typo
Source Link
algori
algori
  • 23.5k
  • 3
  • 100
  • 152

Consider $X = \mathbb{P}^{1}$, $U =\mathbb{A}^{1}$ and j the inclusion of a affine chart in the projective line. The complement of the affine chart is a point P, let i be the inclusion of this point in the projective line. Consider the sheaf $\mathcal{F} = \mathcal{O}(-2)$ on the projective line. One has $dim H^{1}( \mathbb{P}^{1}, \mathcal{F} ) =1$. One has an exact sequence :

$0 -> j_{!} ( \mathcal{F}_{U} ) -> \mathcal{F} i_{*} ( \mathcal{F}_{P} ) -> 0$$0 \to j_{!} ( \mathcal{F}_{U} ) \to \mathcal{F} \to i_{*} ( \mathcal{F}_{P} ) \to 0$

$i_{*} ( \mathcal{F}|_{P} ) $ is a skyscraper sheaf over P so its $H^{1}$ is 0.

$H^{1}$ of $\mathcal{F}$ is not 0. By long exact sequence in cohomology, we have $H^{1}$ of $j_{!}( \mathcal{F}|_{U} )$ is not 0.

But on the other hand $\mathcal{F}|_{U}$ is a trivial sheaf over a affine scheme so its $H^{1}$ is 0.

This example seems to show that the expected relation is not true.

Consider $X = \mathbb{P}^{1}$, $U =\mathbb{A}^{1}$ and j the inclusion of a affine chart in the projective line. The complement of the affine chart is a point P, let i be the inclusion of this point in the projective line. Consider the sheaf $\mathcal{F} = \mathcal{O}(-2)$ on the projective line. One has $dim H^{1}( \mathbb{P}^{1}, \mathcal{F} ) =1$. One has an exact sequence :

$0 -> j_{!} ( \mathcal{F}_{U} ) -> \mathcal{F} i_{*} ( \mathcal{F}_{P} ) -> 0$

$i_{*} ( \mathcal{F}|_{P} ) $ is a skyscraper sheaf over P so its $H^{1}$ is 0.

$H^{1}$ of $\mathcal{F}$ is not 0. By long exact sequence in cohomology, we have $H^{1}$ of $j_{!}( \mathcal{F}|_{U} )$ is not 0.

But on the other hand $\mathcal{F}|_{U}$ is a trivial sheaf over a affine scheme so its $H^{1}$ is 0.

This example seems to show that the expected relation is not true.

Consider $X = \mathbb{P}^{1}$, $U =\mathbb{A}^{1}$ and j the inclusion of a affine chart in the projective line. The complement of the affine chart is a point P, let i be the inclusion of this point in the projective line. Consider the sheaf $\mathcal{F} = \mathcal{O}(-2)$ on the projective line. One has $dim H^{1}( \mathbb{P}^{1}, \mathcal{F} ) =1$. One has an exact sequence :

$0 \to j_{!} ( \mathcal{F}_{U} ) \to \mathcal{F} \to i_{*} ( \mathcal{F}_{P} ) \to 0$

$i_{*} ( \mathcal{F}|_{P} ) $ is a skyscraper sheaf over P so its $H^{1}$ is 0.

$H^{1}$ of $\mathcal{F}$ is not 0. By long exact sequence in cohomology, we have $H^{1}$ of $j_{!}( \mathcal{F}|_{U} )$ is not 0.

But on the other hand $\mathcal{F}|_{U}$ is a trivial sheaf over a affine scheme so its $H^{1}$ is 0.

This example seems to show that the expected relation is not true.

deleted 2 characters in body; deleted 38 characters in body
Source Link
user25309
user25309
  • 6.9k
  • 31
  • 53

Consider $X = \mathbb{P}^{1}$, $U =\mathbb{A}^{1}$ and j the inclusion of a affine chart in the projective line. The complement of the affine chart is a point P, let i be the inclusion of this point in the projective line. Consider the sheaf $\mathcal{F} = \mathcal{O}(-2)$ on the projective line. One has $dim H^{1}( \mathbb{P}^{1}, \mathcal{F} ) =1$. One has an exact sequence :

$0 \longrightarrow j_{!} (\mathcal{F}_{U}) \longrightarrow \mathcal{F} i_{*} (\mathcal{F}_{P}) \longrightarrow 0$$0 -> j_{!} ( \mathcal{F}_{U} ) -> \mathcal{F} i_{*} ( \mathcal{F}_{P} ) -> 0$

$i_{*} (\mathcal{F}_{P}) $$i_{*} ( \mathcal{F}|_{P} ) $ is a skyscraper sheaf over P so its $H^{1}$ is 0.   

$H^{1}$ of $\mathcal{F}$ is not 0. By long exact sequence in cohomology, we have $H^{1}$ of $j_{!}(\mathcal{F}|_{U})$$j_{!}( \mathcal{F}|_{U} )$ is not 0.

But on the other hand $\mathcal{F}|_{U}$ is a trivial sheaf over a affine scheme so its $H^{1}$ is 0.

This example seems to show that the expected relation is not true.

Consider $X = \mathbb{P}^{1}$, $U =\mathbb{A}^{1}$ and j the inclusion of a affine chart in the projective line. The complement of the affine chart is a point P, let i be the inclusion of this point in the projective line. Consider the sheaf $\mathcal{F} = \mathcal{O}(-2)$ on the projective line. One has $dim H^{1}( \mathbb{P}^{1}, \mathcal{F} ) =1$. One has an exact sequence :

$0 \longrightarrow j_{!} (\mathcal{F}_{U}) \longrightarrow \mathcal{F} i_{*} (\mathcal{F}_{P}) \longrightarrow 0$

$i_{*} (\mathcal{F}_{P}) $ is a skyscraper sheaf over P so its $H^{1}$ is 0.  $H^{1}$ of $\mathcal{F}$ is not 0. By long exact sequence in cohomology, we have $H^{1}$ of $j_{!}(\mathcal{F}|_{U})$ is not 0.

But on the other hand $\mathcal{F}|_{U}$ is a trivial sheaf over a affine scheme so its $H^{1}$ is 0.

This example seems to show that the expected relation is not true.

Consider $X = \mathbb{P}^{1}$, $U =\mathbb{A}^{1}$ and j the inclusion of a affine chart in the projective line. The complement of the affine chart is a point P, let i be the inclusion of this point in the projective line. Consider the sheaf $\mathcal{F} = \mathcal{O}(-2)$ on the projective line. One has $dim H^{1}( \mathbb{P}^{1}, \mathcal{F} ) =1$. One has an exact sequence :

$0 -> j_{!} ( \mathcal{F}_{U} ) -> \mathcal{F} i_{*} ( \mathcal{F}_{P} ) -> 0$

$i_{*} ( \mathcal{F}|_{P} ) $ is a skyscraper sheaf over P so its $H^{1}$ is 0. 

$H^{1}$ of $\mathcal{F}$ is not 0. By long exact sequence in cohomology, we have $H^{1}$ of $j_{!}( \mathcal{F}|_{U} )$ is not 0.

But on the other hand $\mathcal{F}|_{U}$ is a trivial sheaf over a affine scheme so its $H^{1}$ is 0.

This example seems to show that the expected relation is not true.

fixed the formulae
Source Link
algori
algori
  • 23.5k
  • 3
  • 100
  • 152

Consider $X = \mathbb{P}^{1}$, $U =\mathbb{A}^{1}$ and j the inclusion of a affine chart in the projective line. The complement of the affine chart is a point P, let i be the inclusion of this point in the projective line. Consider the sheaf $\mathcal{F} = \mathcal{O}(-2)$ on the projective line. One has $dim H^{1}( \mathbb{P}^{1}, \mathcal{F} ) =1$. One has an exact sequence :

'$0 \longrightarrow j_{!} ( \mathcal{F}_{U} ) \longrightarrow \mathcal{F} i_{*} ( \mathcal{F}_{P} ) \longrightarrow 0$'$0 \longrightarrow j_{!} (\mathcal{F}_{U}) \longrightarrow \mathcal{F} i_{*} (\mathcal{F}_{P}) \longrightarrow 0$

$i_{*} ( \mathcal{F}_{P} ) $$i_{*} (\mathcal{F}_{P}) $ is a skyscraper sheaf over P so its $H^{1}$ is 0.

   $H^{1}$ of $\mathcal{F}$ is not 0. By long exact sequence in cohomology, we have $H^{1}$ of $j_{!}( \mathcal{F}|_{U} )$$j_{!}(\mathcal{F}|_{U})$ is not 0.

But on the other hand $\mathcal{F}|_{U}$ is a trivial sheaf over a affine scheme so its $H^{1}$ is 0.

This example seems to show that the expected relation is not true.

Consider $X = \mathbb{P}^{1}$, $U =\mathbb{A}^{1}$ and j the inclusion of a affine chart in the projective line. The complement of the affine chart is a point P, let i be the inclusion of this point in the projective line. Consider the sheaf $\mathcal{F} = \mathcal{O}(-2)$ on the projective line. One has $dim H^{1}( \mathbb{P}^{1}, \mathcal{F} ) =1$. One has an exact sequence :

'$0 \longrightarrow j_{!} ( \mathcal{F}_{U} ) \longrightarrow \mathcal{F} i_{*} ( \mathcal{F}_{P} ) \longrightarrow 0$'

$i_{*} ( \mathcal{F}_{P} ) $ is a skyscraper sheaf over P so its $H^{1}$ is 0.

 $H^{1}$ of $\mathcal{F}$ is not 0. By long exact sequence in cohomology, we have $H^{1}$ of $j_{!}( \mathcal{F}|_{U} )$ is not 0.

But on the other hand $\mathcal{F}|_{U}$ is a trivial sheaf over a affine scheme so its $H^{1}$ is 0.

This example seems to show that the expected relation is not true.

Consider $X = \mathbb{P}^{1}$, $U =\mathbb{A}^{1}$ and j the inclusion of a affine chart in the projective line. The complement of the affine chart is a point P, let i be the inclusion of this point in the projective line. Consider the sheaf $\mathcal{F} = \mathcal{O}(-2)$ on the projective line. One has $dim H^{1}( \mathbb{P}^{1}, \mathcal{F} ) =1$. One has an exact sequence :

$0 \longrightarrow j_{!} (\mathcal{F}_{U}) \longrightarrow \mathcal{F} i_{*} (\mathcal{F}_{P}) \longrightarrow 0$

$i_{*} (\mathcal{F}_{P}) $ is a skyscraper sheaf over P so its $H^{1}$ is 0.  $H^{1}$ of $\mathcal{F}$ is not 0. By long exact sequence in cohomology, we have $H^{1}$ of $j_{!}(\mathcal{F}|_{U})$ is not 0.

But on the other hand $\mathcal{F}|_{U}$ is a trivial sheaf over a affine scheme so its $H^{1}$ is 0.

This example seems to show that the expected relation is not true.

deleted 1 characters in body; added 2 characters in body; added 2 characters in body; added 10 characters in body
Source Link
user25309
user25309
  • 6.9k
  • 31
  • 53
improved formatting
Source Link
Charles Staats
Charles Staats
  • 7.3k
  • 5
  • 67
  • 86
Source Link
user25309
user25309
  • 6.9k
  • 31
  • 53