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I was looking for a good book where to learn something about sheaf theory. I'm really a beginner with homological algebra :I know what homology is and what are derived functors but I do kot really know anything about derived categories etc. Is there a way I can start to understand something about sheaf with my level of knowledge and where?

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    $\begingroup$ I'm not sure sheaves have much to do with homological algebra or derived categories. You can do homological / derived things to them, certainly, but if you want to learn about sheaves, it's not a book on homological algebra / derived categories you want. I learnt about sheaves and schemes from Vakil's online "algebraic geometry" notes: math.stanford.edu/~vakil/216blog . $\endgroup$ – Billy Apr 1 '18 at 21:32
  • $\begingroup$ Thank you. I said that becaus all the book I found about sheaves started with an higly cathegorical/homological approach, so I thought the two things were strictly linked $\endgroup$ – Tommaso Scognamiglio Apr 1 '18 at 21:36
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Very often, one use sheaf theory to prove theorems, so they are usually introduced in algebraic geometry textbooks. So book specifically devoted to sheaf theory are usually a bit technical.

I would say the best reference to understand sheaf cohomology is Forster, Lectures on Riemann surfaces. It introduces in a very down-to-earth way the first Čech cohomology group and deduce all the classical theorems of algebraic geometry (Riemann-Roch, Abel-Jacobi theorem, and other) using sheaf cohomology as a main tool. It's only for curves but it's already good.

Classical references are Serre's Faisceaux algébriques cohérents, and the book by Godement.

More modern references are Sheaves on manifolds by Kashiwara and Shapira (using analytic methods), and Sheaves in topology by Dimca (more oriented toward singular algebraic varieties and perverse sheaves).

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