Suppose $X$ is a closed submanifold of $Y$ and the inclusion map is $i:X\to Y$. Let $\mathscr{F}$ be a sheaf of abelian groups over $X$. Prove that for any $k\geq 0$, we have $$ H^k(Y,i_*\mathscr{F})\cong H^k(X,\mathscr{F}). $$
I am learning some basis of sheaf theory for the first time from the appendix of Complex Geometry- An Introduction by Daniel Huybrechts. It is obvious to see $i_*\mathscr{F}(V)=\mathscr{F}(i^{-1}V)$ makes $i_*\mathscr{F}$ a presheaf and it is also easy to verify the two additional sheaf condition, so the Čech homology makes sense. I just went through the construction of Čech cohomology (using open covering first and than take the direct limit) and wonder after such complex construction, how to identify the two cohomology groups. I would like to find a detailed reference or answer about the problem.
Appreciate any help!
This follows from taking a flasque resolution, which I believe is how sheaf cohomology is defined in that appendix.
Indeed if $0 \to \mathscr{F} \to \mathscr{G}^\bullet$ is a flasque resolution of $\mathscr{F}$, then $0 \to i_* \mathscr{F} \to i_*\mathscr{G}^\bullet$ is a flasque resolution of $i_*\mathscr{F}$, and applying $\Gamma(X, -)$ gives the same complex in either case.
We should justify that $i_*$ is exact when $i_Y: Y \hookrightarrow X$ is the inclusion of a closed submanifold. This can be checked on the stalks, given that $$(i_*\mathscr{F})_P = \begin{cases} \mathscr{F}_P & P \in Y\\ 0 & P \notin Y \end{cases}$$ is the extensions by zero.
As for a reference, this is lemma III.2.10 in Hartshorne, and section III.2 does this in detail, defining cohomology as the right derived functors of $\Gamma(X, -)$.