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Example Wanted: When Does Cech Cohomology Fail to be the same as Derived Functor Cohomology?

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

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The question linked to by Steven provides an answer to the question as stated: take $X$ to be the (Zariski) topological space corresponding to the affine plane, and $\cF$ to be the sheaf described. However, it looks like there is still no answer to Q2 of the linked question: is there a (non-paracompact) Hausdorff space $X$... –  Sam Gunningham Feb 21 '13 at 1:24
    
There you go, Sam. –  David Roberts Feb 21 '13 at 2:23
    
Maybe this should not be deleted because David's answer is not covered by the answer of the other question. –  Benjamin Steinberg Mar 23 '13 at 1:09
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marked as duplicate by Martin Brandenburg, Ryan Budney, Eric Wofsey, Chris Gerig, Dan Petersen Feb 21 '13 at 3:38

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1 Answer

In the paper Pathologies in cohomology of non-paracompact Hausdorff spaces, Stefan Schröer constructs a Hausdorff space which is not paracompact, and for which sheaf cohomology with values in the sheaf of germs of $S^1$-valued functions does not agree with the Čech cohomology (for example - the same is true for other sheaves).

The space is constructed by taking the countably infinite join of disks $D^2$, with the CW-structure consisting of two 0-cells, two 1-cells and a single 2-cell, using one of the 0-cells as a basepoint. Then he takes a coarser topology, whereby the open sets are open sets from the CW-topology, but only those that either don't contain the basepoint, or contain all but finitely many of the closed disks. With this topology the space is not paracompact, but is $\sigma$-compact, Lindelöf, metacompact.... and contractible! Sheaf cohomology is non-trivial however.

An interesting point to note is that this is not a k-space, and the k-ification of this space is the original CW-complex.

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