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I am trying to understand why global sections of a sheaf are "important" or interesting objects of study. Perhaps I have too weak of a background to appreciate it (and that is certainly an acceptable answer), but the sense I've gotten is that sheaf cohomology is all about trying to understand $\Gamma(X, \mathcal{F})$, the global sections of $X$ over $\mathcal{F}$ (I think I said that right...?). I think I would be more motivated to study sheaf cohomology if I cared more about global sections.

I'm quite new to this area of study, but I have a few examples in mind that anyone could understand: the sheaf of continuous/differentiable/smooth functions on an open subset of $\mathbb{C}^n$. I've been told to think of sections like vector fields, but I'm having trouble even making that connection. Even if that was clear to me, it's not clear at all why I should care about them in more abstract scheme-theoretic settings, like the sheaf of regular functions on a affine/projective variety's open sets.

As for my background, I'm looking to understand these ideas because I hear they are rather useful in number theory (interpreting $Spec( \mathcal{O}_K )$ as a scheme leads to the notion of field extensions giving rise to ramified coverings).

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    $\begingroup$ I think this question is better suited to mathstack. The point is that global sections measure some invariants of the space (depending on the sheaf). $\endgroup$
    – user40276
    Commented Aug 20, 2015 at 7:00
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    $\begingroup$ To add to the previous comment: in algebraic geometry most of the things we care about are global sections (or zero-sets of global sections) of sheaves on a scheme. For example, the set of surfaces of degree 59 in $\mathbf P^{19}$ passing through your favourite 33 points can be (more or less) viewed as the set of global sections of a certain sheaf. More generally, linear systems of divisors, vector fields, differential forms, vanishing conditions along subvarieties, and so on can all be treated in this language. And all of this makes sense in significant generality, which allows one to... $\endgroup$
    – Lazzaro Campeotti
    Commented Aug 20, 2015 at 15:48
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    $\begingroup$ apply powerful geometric ideas in non-geometric situations (such as number theory). Maybe you are being led astray a bit by the example you have in mind: for example, on a manifold, a smooth function on an open set can always be extended to the whole space (by using bump functions), so the language of sheaves and global sections seems superfluous there. By contrast, in algebraic geometry there are no bump functions, and indeed projective varieties have no nonconstant regular functions. Global sections of (some) sheaves can be thought of as substitutes for regular functions. A good place to... $\endgroup$
    – Lazzaro Campeotti
    Commented Aug 20, 2015 at 15:51
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    $\begingroup$ start is to try to understand line bundles (=invertible sheaves) on projective space, and how their global sections yield hypersurfaces. (In my first comment, by the way, "surfaces" should be "hypersurfaces".) $\endgroup$
    – Lazzaro Campeotti
    Commented Aug 20, 2015 at 15:54
  • $\begingroup$ @potentiallydense Even constant sheaves on manifolds are important (they give the number of connected components). And locally constant sheaves gives the monodromy. So they're not superfluous. The fact that the sheaves are in general fine (so, for instance, the sheaves of differential forms are acyclic) makes things easier, but not superfluous. Furthermore almost all ideas of algebraic geometry still work in the differentiable setting (for instance, line bundles as cohomology classes as you cited, and, more generally, gerbes). $\endgroup$
    – user40276
    Commented Aug 20, 2015 at 18:59

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