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Since now we know that R in your question refers to real line equipped with standard topology, sheaf cohomology will always have H^i(F) = 0 for i>1 — depending on how you define sheaf cohomology this …

I think Fourier-Mukai transform is related to the Fourier transforms you described through the space A \times \check A which is symplectic and somehow relevant, though I don't know the details. The r …

Yes, there is a big class of modules that have an intuition different from the abstract algebra, namely the ones that come from an algebraic geometry. If $R$ is a (say, Noetherian) commutative ring, t …

Ah, great question! I'm not a big expert, but one thing it obviously does is constructing the sheaf M from the locally free bundles (locally free = projective). For example, consider a skyscraper she …

One thing I understand is that vanishing cycles are more than just about singularities — there's a derived version that is more interesting. I'd like to get an answer myself. This is also important fo …

The book of Kashiwara and Schapira is quite focused and technical. I won't recommend it as an introduction to sheaves, since the abstract language of sheaves and homological algebra is most useful whe …