# Spectral sequences in Hypercohomology of sheaves

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?

-
As (almost) always, one of the two spectral sequences of the bicomplex degenerates and is used to identify the target of the other spectral sequence. –  Fernando Muro Apr 15 '12 at 22:28
$C^\bullet (F^q)(X)$ is usually not exact. It was formed by taking a resolution of the sheaf and then applying the global section functor to the whole thing. –  Matt Apr 15 '12 at 22:30
Thanks, I'd forgotten to apply the global section functor, I didn't see it, I've just posted the question: mathoverflow.net/questions/95524, for when the sheaves are acyclic –  Samuel Mf Apr 29 '12 at 22:12
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).
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9580793380737305, "perplexity": 208.0433525805157}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394009871907/warc/CC-MAIN-20140305085751-00053-ip-10-183-142-35.ec2.internal.warc.gz"}