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I have a few questions about the use of cohomology. Firstly,we use cohomology to measure the obstruction of a section from a global section, so what can we do about a global section? I got very confused when reading Hartshorne's book, and I lack of some certain motivations. Could someone introduce how cohomology of sheaves works and what's the essence of "a global section" (why we are interested in a global section)? Actually I was very interested in number theory,I know a bit about class field theory, but only in terms of ideals. I do not understand its adelic version, and how it finally goes into representation theory, can someone give me some useful remark, and a list of books (Of course from elementary to advanced level, actually I am poor at analysis,I need some remark about modern analysis, how the idea was raised, especially). Finally, while reading Hartshorne, which I found very hard to go on, I read Qing Liu's Arithmetic Curves, could someone give me some remark about why we are interested in fiber curves, and what kind of problem can be solved with it? It talks about models of a curve, and treats with reductions of a curve, what is that?

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    Dear Yoshinobu, you are asking too many questions simultaneously and each of them is too general. Try to be more specific and you will see that users of this site will answer in a very helpful and competent way.2012-03-23
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    those are many questions in one, tightly packed paragraph. anyway, for a manifold to be orientable is the same as having a global non-vanishing section of top differential forms. as far as cohomology goes, which I've always used as just a tool, you can realise singular cohomology as sheaf cohomology (with coefficients in the constant sheaf) and you recover a nice bunch of invariants of spaces. for cohomology of coherent sheaves it's a similar story: you have a nice bunch of invariants of varieties. AFANTG: I know nothing about number theory :(2012-03-23
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    As for the motivation for global sections, cohomology etc. look for the Riemann Roch Theorem in the literature.2012-03-23
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    @Yoshinobu: The motivations behinds sheaves and sections and cohomology are fundamentally geometric. The idea is that we have some local data, and we want to know how we can pass from coherent collections of local data to global data. For example, on a manifold, any closed differential form always has a local antiderivative – but sometimes we want a global antiderivative. The de Rham cohomology groups measure precisely how close this is to being true. I'm sure similar examples exist in algebraic number theory.2012-03-23
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    I did some copyediting on this question. (In particular I changed "Liu Qing" to "Qing Liu". I acknowledge that putting the surname first is a longstanding practice in Asian cultures, but his name is appearing here in the context of authorship of a book, and the book gives his name as "Qing Liu".)2012-03-23
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    Like everyone else said, WAY too many questions in one post. At least use paragraphs or something. Also, to answer your question about why we would care about global sections, showing that a rank $k$ vector bundle is trivial is equivalent to showing that it admits $k$ linearly independent global sections, which is equivalent to finding a global section of an associated Stiefel bundle.2012-03-23

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