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Iversen, in "Cohomology of Sheaves," proves a number of theorems about the cohomological dimension of a locally compact space. In particular, if $\mathcal{F}$ is any sheaf on $\mathbb{R}^1$, it is proved that the compactly supported cohomology of $\mathcal{F}$ vanishes in degree greater than one.

The argument is quite slick: namely, let $\mathcal{F}$ be a sheaf, and let $\alpha \in H^2_c(X, \mathcal{F})$ a non-vanishing cohomology class. By generalities about direct limits and sheaf cohomology, Iversen argues that there is a minimal closed set $Z \subset \mathbb{R}^1$ such that $H^2(Z, \mathcal{F}) $ is not zero---namely, by Zorn it suffices to show that if $\alpha$ restricts to nonzero a cohomology class on a descending chain of closed subsets, then it restricts to a nonzero class on the intersection. (This is the fact alluded to about direct limits.) Now if $Z$ is this minimal set, the Mayer-Vietoris sequence(!) shows that $\alpha$ cannot map to zero in both $H^2_c(Z \cap [t, \infty))$ and $H^2_c(Z \cap (-\infty, t])$ (or it would be zero because the cohomology of a point is trivial). But this is a contradiction if $t$ is chosen appropriately.

Is this true in ordinary sheaf cohomology? The Mayer-Vietoris sequence is certainly still fine, but without the compactness the claim about cohomology and direct limits should fail. (E.g. in degree zero. For example, take $\bigoplus (i_n)_*(\mathbb{Z})$ where $i_n$ is the inclusion of $\{n\}$, so that global sections over $\mathbb{R}$ is an infinite product $\prod \mathbb{Z}$, clearly much bigger than the colimit (which is an infinite direct sum).) Nonetheless it would be intuitively very appealing if $\mathbb{R}$ had no sheaf cohomology in dimensions greater than one.

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Sheaf cohomology of a paracompact space can be computed using Čech cohomology. Now, the topological dimension of $\mathbb R$ is a one, so there is a cofinal family of open coverings of $\mathbb R$ for which the Čech complex is trivial above dimension $1$. That $\mathbb R$ has cohomological dimension one follows (well, one has to check that the dimension is not actually zero, of course)

An enjoyable reference for all this is Godement's book Théorie de faisceaux.

Later: One can also use Godement's theorem 4.14.2 which states that if $X$ is a metrisable space with $\operatorname{cd}X\leq n$, then every subspace $Y\subseteq X$ has $\operatorname{cd}Y\leq n$.

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Thanks! (and extra characters) –  Akhil Mathew Jun 8 '11 at 13:14

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