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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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    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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