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For some time I've been trying to find an answer to the question "why do we care about or compute sheaf cohomology". As far as I can tell books like Hartshorne treat this as something we already want to compute and do not motivate it. Some posts I've seen talk about how cohomology measures obstructions for glueing local data into global data.

However, they do not explain how this works. For example if I have an affine cover of a scheme and some local data. How do the cohomology groups inform me if I can glue this data together to get global data. I don't even know how to phrase this problem in terms of cohomology.

If you could be as explicit as possible perhaps even giving some examples that would be greatly appreciated.

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    $\begingroup$ Sheaf cohomology is an essential tool in geometry and the more geometry you do the more you'll see it used. As an example, sheaf cohomology of the constant sheaf on a space gives fundamental invariants of the space which one can use to e.g. begin to classify certain types of spaces, or proves two spaces are not isomorphic, and so on. And sure it measures obstructions for glueing local data into global data -- even H^0 does this. Local sections of a sheaf on an affine cover are often non-trivial, but existence of global sections tells you if you can glue the local sections together. $\endgroup$
    – Kevin Buzzard
    Commented Jan 10, 2017 at 19:59
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    $\begingroup$ "Gluing of local data" is sheaves, not cohomology. The cohomology has meaning in many cases as a home for global obstructions. In particular, degree-1 cohomology and degree-2 cohomology often have very down-to-earth meaning as places where one builds obstructions to global honest construction problems. The case of infinitesimal flat deformation of a smooth scheme (done very concretely in SGA1) is an excellent example in the context of coherent cohomology. $\endgroup$
    – nfdc23
    Commented Jan 10, 2017 at 22:19
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    $\begingroup$ I'm not sure what stage you're at, but --- are you familiar with Cech cohomology? The definition of Cech cohomology quite clearly captures the idea of finding obstructions to glueing, and then there are theorems relating Cech cohomology to ordinary sheaf cohomology, so that under good circumstances they are identical. $\endgroup$
    – Steven Landsburg
    Commented Jan 10, 2017 at 23:30
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    $\begingroup$ @R.Leyland I'll bite and try to explain it: suppose you have a family of local sections $\{f_i\}_{i\in I}$ relative to some open cover $\{U_i\}_{i\in I}$. Then what it is the obstruction to gluing them? It is the collection $\{g_{ij}\}$ where $g_{ij}$ is a section on $U_i\cap U_j$ given by $f_i-f_j$. But this is a Čech cocycle! It turns out that you can find $\{h_i\}$ so that $\{f_i+h_i\}$ glue together iff this Čech cocycle is a coboundary. Often, using exact sequences of sheaves, you can ask for the $h_i$'s to live in some subsheaf which has trivial Čech cohomology. $\endgroup$
    – Denis Nardin
    Commented Jan 11, 2017 at 7:31
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    $\begingroup$ @DenisNardin I'm really curious; could you expand on this last bit: "Often, using exact sequences of sheaves, you can ask for the $h_i$'s to live in some subsheaf which has trivial Čech cohomology" ? $\endgroup$
    – ಠ_ಠ
    Commented Mar 17, 2017 at 0:35

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