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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 on cohomological descent.

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up vote 20 down vote accepted

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.

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I do not think that sheaves of abelian groups need to be locally acyclic. Let me say what I mean in an example. Take $X=\mathbb{C}^{2}$ with the classical (metric) topology. Let $\mathcal{F} = \mathbb{Z}_{D}$ where $D=\mathbb{C}^{\times} \times \{0\}$.

Then for any arbitrarily small polydisk $U$ containing $(0,0)$ we have $H^{1}(U,\mathbb{Z}_{D}) = H^{1}(U \cap D, \mathbb{Z})$

is not vanishing. It seems in this example that $\underline{H}^{1}(\mathcal{F})_{(0,0)}$ does not vanish.

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Oh, I could have just used $\mathbb{C}^{\times}$ in $\mathbb{C}$ –  Oren Ben-Bassat Mar 28 '11 at 10:11
    
Just to clarify, the assumptions of locally affine and quasi coherence are crucial. The simple example of the push forward of the constant sheaf $\mathbb{Z}$ under the inclusion of $\mathbb{C}^{\times}$ into $\mathbb{C}$ gives a sheaf of abelian groups which is not locally acyclic. –  Oren Ben-Bassat Nov 1 '11 at 17:33
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