Spectral sequences in Hypercohomology of sheaves

Question

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?

Answer 1

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).