Coherent Sheaf supported in a point - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T18:20:55Z http://mathoverflow.net/feeds/question/122806 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/122806/coherent-sheaf-supported-in-a-point Coherent Sheaf supported in a point Peter 2013-02-24T16:10:12Z 2013-02-24T16:10:12Z <p>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). </p> <p>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)?</p> <p>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?</p> <p>Thank you very much!</p>