show/hide this revision's text 6 corrected the formulae again; fixed a typo

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 :

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

$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.
show/hide this revision's text 5 deleted 2 characters in body; deleted 38 characters in body

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 :

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

$i_{*} ( \mathcal{F}_{P}) \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.
show/hide this revision's text 4 fixed the formulae

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

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

$i_{*} (\mathcal{F}_{P} ) \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.
show/hide this revision's text 3 deleted 1 characters in body; added 2 characters in body; added 2 characters in body; added 10 characters in body
show/hide this revision's text 2 improved formatting
show/hide this revision's text 1