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?