So recently I asked here about interesting results from cohomology that would be suitable for a small first semester independent project in algebraic topology.
Čech cohomology was suggested there which seems interesting to me. I have seen the following theorem in a few different places:
Let $X$ be a manifold of dimension $n$. Then if $\mathcal{F}$ is a coherent sheaf (in the language of manifolds I think this corresponds to a vector bundle) then \begin{equation} H^{m} \left( X, \mathcal{F} \right) = 0 \quad \text{for } m > n. \end{equation}
Now this seems to be a manifold equivalent of a vanishing theorem of Grothendieck found on page 208 of Hartshorne.
Could someone give me a reference to the first theorem I mentioned about the vanishing of Čech cohomology for manifolds? I haven't been able to find this in a text despite Googling, and I wonder if it had a particular name?
Secondly, is my claim that they are actually the same theorem in different contexts a correct interpretation or not?
The key ingredient you need is the notion of covering dimension. The point is then that each $n$-dimensional manifold has covering dimension $n$ and each (say, paracompact Hausdorff) space $X$ of covering dimension $n$ satisfies $H^k(X,{\mathcal F})=0$ for $k>n$ and any sheaf ${\mathcal F}$ of abelian groups, simply because one can arrange for a sequence of coverings with empty $n+2$-fold intersections. See for instance this wikipeda article for the basic definitions and references.