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

Question

I want to know exactly how derived functor cohomology and Cech cohomology can fail to be the same.

I started worrying about this from Dinakar Muthiah's answer to an MO question, and Brian Conrad's comments 1 2 to another MO question. Let $\mathcal{F}$ be a sheaf of abelian groups on a space $X$. (Here I want to be a little vague about what a "space" means. I'm thinking of either a scheme or a topological space.) Then Čech cohomology of $X$ with respect to a cover $U \to X$ can be defined as cohomology of the complex

$$ \mathcal{F}(U) \to \mathcal{F}(U^{[2]}) \to \mathcal{F}(U^{[3]}) \to \dotsb $$

where $U^{[ n ]} = U \times_X U \times_X \dotsb \times_X U$. The total Čech cohomology of $X$, $\smash{\check H}^{ * }(X, \mathcal{F}) $, is then given by taking the colimit over all covers $U$ of $X$. Now if the following condition is satisfied:

Condition 1: For sufficiently many covers $U$, the sheaf $\mathcal{F}\rvert_{U^{[ n ]}}$ is an acyclic sheaf for each $n$

then this cohomology will agree with the derived functor version of sheaf cohomology. We have,

$$\smash{\check H}^{ * }(X, \mathcal{F}) \cong H^*(X; \mathcal{F}).$$

I am told, however, that even if $\mathcal{F}\rvert_U$ is acyclic this doesn't imply that it is acyclic on the intersections. It is still okay if this condition fails for some covers as long as it is satisfied for enough covers. However I am also told that there are spaces for which there is no cover satisfying condition 1.

Instead you can replace your covers by hypercovers. Basically this is an augmented simplicial object $$V_\bullet \to X$$ which you use instead of the simplicial object $U^{[ \bullet +1 ]} \to X$. There are some conditions which a simplicial object must satisfy in order to be a hypercover, but I don't want to get into it here. You can then define cohomology with respect to a hypercover analogously to Čech cohomology with respect to a cover, and then take a colimit. This seems to always reproduce derived functor sheaf cohomology.

So my question is when is this really necessary?

Question 1: What is the easiest example of a scheme and a sheaf of abelian groups (specifically representable ones such as $\mathbb{G}_m$) for which Čech cohomology of that sheaf and derived functor cohomology disagree?

Question 2: What is the easiest example of a (Hausdorff) topological space and a reasonable sheaf for which Čech cohomology and derived functor cohomology disagree?

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I also want to be a little flexible about what a "cover" is supposed to be. I definitely want to allow interesting Grothendieck topologies, and would be interested in knowing if passing to a different Grothendieck topology changes the answer. It changes both the notion of sheaf and the notion of Čech cohomology, so I don't really know what to expect.

Also, I edited question 1 slightly from the original version, which just asked about quasi-coherent sheaves. Brian Conrad kindly pointed out to me that for any quasi-coherent sheaf the Čech cohomology and the sheaf cohomology will agree (at least with reasonable assumptions on our scheme, like quasi-compact quasi-separated?) and that the really interesting case is for more general sheaves of groups.

Answer 1

Q1: A very simple example is given in Grothendieck's Tohoku paper "Sur quelques points d'algebre homologique", sec. 3.8. Edit: The space is the plane, and the sheaf is constructed by using a union of two irreducible curves intersecting at two points.

Q2: Cech cohomology and derived functor cohomology coincide on a Hausdorff paracompact space (the proof is given in Godement's "Topologie algébrique et théorie des faisceaux"). I don't know of an example on a non paracompact space where they differ.

Answer 2

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.