Alright, here I go again, don't know if I'm missing something here but let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I want to compute the cohomology of this complex. The hypercohomology of this complex is the cohomology of the complex


tot$(C^\bullet(F^\bullet)(X))$


of global sections right? So fine, but


$C^\bullet(F^q)$


is an exact sequence.


Doesn't that make the spectral sequence


$''E^{p,q}\_2 = H^P(H^q(X,F)) = 0 $ degenerate at 2 and hence the Hypercohomology of $F^\bullet$ 0?



Spectral sequences in Hypercohomology of sheaves
You seem to be concluding that the hypercohomology of **any** cochain complex $F^\bullet$ must vanish (except, perhaps, in degree zero)?


To see where you've gone wrong, start with your favorite sheaf ${\cal O}$, and let $F^\bullet$ be an injective resolution of ${\cal O}$. Then (pretty much directly from the definition) the hypercohomology of $F^\bullet$ coincides with the cohomology of ${\cal O}$. So as long as ${\cal O}$ has any nonvanishing higher cohomology, $F^\bullet$ is a counterexample to your conclusion.


Now apply your argument to $F^\bullet$ and see where it goes wrong. (Hint: Keep Matt's comment in mind).


