I have been learning about the cohomology of sheaves, and I have the following questions.
Suppose we have a sheaf $\mathscr{F}\in Sh_X$ where $X$ is some topological space. Now, for any open $U\subset X$, we have a map $\mathscr{F}\to j_*\mathscr{F}|_U$ (where $j:U\hookrightarrow X$ is the inclusion) induced by the restriction maps. Now $H^0(X,j_*\mathscr{F}|_U) = H^0(U,\mathscr{F}|_U)$. As a result, we have a map of cohomologies $H^i(X, \mathscr{F})\to H^i(U,\mathscr{F} |_U)$. Now, as far as I can see, it is easy to conclude that we have a presheaf $\scr{G}$, given by \begin{align*} U\mapsto H^i(U,\mathscr{F}|_U) \end{align*} for each $i\ge 0$.
Now, what I am unsure about are the following points:
- Is $\scr{G}$ a sheaf? I can't think of any way to prove that it is.
- If $\scr{G}$ is not a sheaf, what is its sheafification? Are there are any well behaved cases where the sheafification is actually something nice?
I would be very grateful is someone helps me out. Thank you.