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.

  • 5
    $\begingroup$ 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 Apr 8 '13 at 9:42
  • 2
    $\begingroup$ 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 Apr 8 '13 at 13:42

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.

  • $\begingroup$ Note that is is now Section 20.25. It's best to reference the tag, which is 01FP. $\endgroup$ – David Corwin Sep 10 '20 at 2:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.