Let X be a topological space, let $\mathcal{U} = \{U\_i\}$ be a cover of X, and let $\mathcal{F}$ be a sheaf of abelian groups on X. If X is separated, each $U\_i$ is affine, and $\mathcal{F}$ is quasi-coherent, then Cech cohomology computes derived functor cohomology; in general one only gets a spectral sequence
$$
H^p(\mathcal{U},\underline{H}^q(\mathcal{F})) \Rightarrow H^{p+q}(X,\mathcal{F})
$$
where $\underline{H}^q(\mathcal{F})$ is the presheaf $U \mapsto H^q(U,\mathcal{F}|\_U)$.



> 
> **Question**: For q > 0, $\underline{H}^q(\mathcal{F})$ sheafifies to 0.
> 
> 
> 


For a quasi-coherent sheaf $\mathcal{F}$ this is clear because cohomology vanishes on affines. Is this really true in general? Brian Conrad states this in the introduction to his [notes](http://math.stanford.edu/~conrad/papers/hypercover.pdf) on cohomological descent.



Cech to derived spectral sequence and sheafification
Yes, this is true in general.


It suffices to show the stalks vanish. Pick $x \in X$ and take an injective resolution $0 \to {\cal F} \to I^0 \to \cdots$. For any open $U$ containing $x$, we get a chain complex


$$0 \to I^0(U) \to I^1(U) \to \cdots$$


whose cohomology groups are $H^p(U,{\cal F}|\_U)$.


Taking direct limits of these sections gives the chain complex


$$0 \to I^0\_x \to I^1\_x \to \cdots$$


of stalks, which has zero cohomology in positive degrees because the original complex was a resolution. However, direct limits are exact and so we find


$$0 = {\rm colim}\_{x \in U} H^p(U,{\cal F}|\_U) = {\underline H}^p({\cal F})\_x$$


as desired.


Generally, cohomology tells you the obstructions to patching local solutions into global solutions, and this says that locally those obstructions vanish.


