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I'm studying sheaf cohomology, and I've seen that Čech and derived functor cohomologies agree, at least on paracompact Hausdorff topological spaces.

Is there a simple example of a topological space $X$ with a sheaf $\mathcal F$ such that these two cohomologies don't agree? (I don't have any knowledge of schemes, I want $X$ to be a topological space)

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A scheme is in particular a topological space, and their sheaf cohomology only depends on the topology. In fact, they are very simple as topological spaces. :p – Zhen Lin Feb 20 '13 at 22:14
More seriously, one can prove without any hypotheses at all that Čech cohomology (after taking the direct limit over all open covers) always computes the correct $H^1$, so any counterexample has to be in $H^2$ or higher. – Zhen Lin Feb 20 '13 at 22:22
So, is there such an example for $H^2$? – Henry Wensleydale Feb 20 '13 at 22:35
Answered on MO. – Zhen Lin Feb 21 '13 at 8:22

So as to answer this question, I will give a link to a paper:

Abstract: "We construct a non-paracompact Hausdorff space for which Cech cohomology does not coincide with sheaf cohomology. Moreover, the sheaf of continuous real-valued functions is neither soft nor acyclic, and our space admits non-numerable principal bundles."

The construction is quite long, and so I'll refer you to that paper.

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1… You're an active MO user. Didn't you see this was already answered there, with the same reference? – Pedro Tamaroff Jun 10 '14 at 3:38
@PedroTamaroff Forgive me; didn't see that. I made the answer CW. Again, I apologize for not seeing the answer at MO. – Sanath K. Devalapurkar Jun 10 '14 at 3:51

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