Does the Čech cohomology always give rise to a long exact sequences given a short exact sequence of sheaves?


Clearly that cannot occur for sheaves on a paracomact (perhaps also Hausdorff, I'm not sure about that) space, but the argument one uses there relies on the topological assumptions in a crucial way and cannot be generalized to an arbitrary space, not in a straight-forward manner anyway.


Thank you for shedding any light on this issue!



Does the Čech cohomology always yield long exact sequences from short ones?
It should be no. Sorry, I don't have the time to write anything detailed, so I'll just leave assembly instructions. Take a look at my answer here:
 [equivalence of Grothendieck-style versus Cech-style sheaf cohomology](https://mathoverflow.net/questions/4214/equivalence-of-grothendieck-style-versus-cech-style-sheaf-cohomology/18674#18674)
describing an example due to Grothendieck where Čech differs from derived functor cohomology.
If you write out what you get from the sequence 
$$0\to K\to \mathbb{Z}\_X\to \mathbb{Z}\_Y\to 0$$
described there, the Čech sequence should fail to be exact at $\check{H}^2(K)$.




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Here are some details. $X$ is the affine plane over a field with the Zariski topology.
$Y\subset X$ is the union of two irreducible curves $Y\_i$ meeting at two points. If you are not familiar with algebraic geometry, the key point is that the sheaves $\mathbb{Z}\_X$, $\mathbb{Z}\_{Y\_i}$ are flasque hence acylic. From the long exact sequence for sheaf cohomology (= derived functors of global sections), we have an exact sequence
$$ H^1(X,\mathbb{Z}\_X)=0\to H^1(Y,\mathbb{Z}\_Y)\to H^2(X, K)\to H^2(X,\mathbb{Z}\_X)=0$$ 
Using Mayer-Vietor, the second, and therefore third group, is just $\mathbb{Z}$.
Now consider the sequence (I'm not saying exact sequence),
$$ \check{H}^1(X,\mathbb{Z}\_X)\to \check{H}^1(Y,\mathbb{Z}\_Y)\to \check{H}^2(X, K)\to $$
It is known that $\check{H}^1$ coincides with $H^1$, but that $\check{H}^2(K)=0$ [cf Grothendieck "Sur quelques points...", sect 3.8]. So exactness of the second sequence fails.


