# Coherent Sheaf supported in a point

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!

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It is. $H^0(X,\mathcal F)=\mathcal F_p$ is finite dimensional, $H^i(X,\mathcal F)=0$ if $i>0$. – Serge Lvovski Feb 24 '13 at 16:35
And how I can proof that $H^0(X,\mathcal{F})=\mathcal{F}_p$? Thanks for answer! – Peter Feb 24 '13 at 16:39
I don't think this is so far from the definition. – S. Carnahan Feb 24 '13 at 17:46
For the proof that one can restrict to the support of the sheaf from the point of view of Cech cohomology : one can "sheafify" the construction of the Cech complex and it is clear that the sheaves in this complex are supported on the support of F. – user25309 Feb 24 '13 at 18:22