Hi guys, I'm studying Cech cohomology of sheaves and I've the following doubt: If you have a coherent sheaf $\mathcal{F}$ in a compact complex variety $X$ then the cohomology groups are finite dimensional vector spaces (that's ok).

But it's true that we can reduce ("contract" or something) the analysis of the whole space to the support of the sheaf (to study there the cohomology)?

For example, suppose that $\mbox{Supp}(\mathcal{F})=\{p\}$ is a single point. What we can say about the cohomology of $X$ and the cohomology of $\{p\}$? It's true that the stalk $\mathcal{F}_p$ is a finite dimensional vector space?

Thank you very much!

`$H^0(X,\mathcal F)=\mathcal F_p$`

is finite dimensional, $H^i(X,\mathcal F)=0$ if $i>0$. $\endgroup$