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Let $X$ be a reasonable topological space. If $\mathcal{F}$ is a sheaf of abelian groups then Cech cohomology gives us a method to compute the cohomology groups $H^p(X, \mathcal{F})$ - the main input being the sections $\mathcal{F}(U)$ for various open sets $U \subset X$.

I would like to have a similar procedure to compute hypercohomology of a finite complex $C^\cdot$ of abelian sheaves on $X$ (or coherent sheaves on a scheme). Is this possible, and is there a reference you can recommend? I couldn't find this in the Stacks project, which incidently explains how to compute cohomology of a complex (not hypercohomology) via a Cech argument.

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    $\begingroup$ I can't think of a reference, but I have a vague recollection that this is how you do it: since the Cech complex of a sheaf is functorial, if we have a complex of sheaves $\mathcal{F}^{\cdot}$ we can take the Cech complex of each term to get a double complex $C^{\cdot,\cdot}$. Now just take the associated simple complex, $\mathrm{Tot}(C^{\cdot,\cdot})$ - in suitably nice situations, this computes hypercohomology of $\mathcal{F}^{\cdot}$. $\endgroup$
    – ChrisLazda
    Commented Apr 8, 2013 at 9:42
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    $\begingroup$ Indeed regarding "sufficiently nice": it suffices that for each $n$, the complex $C^{n, \bullet}$ computes the cohomology of $\scr{F}^n$ (consider the spectral sequence taking "vertical cohomology" first). $\endgroup$
    – Tom Bachmann
    Commented Apr 8, 2013 at 13:42

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There is a nice treatment of it in chapter 1 of Brylisnki's Loop Spaces, Characteristic Classes and Geometric Quantization. In the Stacks projects, look for section 19.19.

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  • $\begingroup$ Note that is is now Section 20.25. It's best to reference the tag, which is 01FP. $\endgroup$
    – David Corwin
    Commented Sep 10, 2020 at 2:07

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