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I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from algebraic geometry. I don't want a too rigorous approach, made of a lot of definition and propositions but instead I would like to find an introduction which gives the main ideas and many examples. Thank you

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Could you expand a bit on your background? How much homological algebra do you know already? I think feeling comfortable with what's in one of Hartshorne's chapters is the bare minimum for such an undertaking to be realistic without too much pain. If you've mastered that, then reading Chapters 1, 2 and 10 of Weibel might be a quite easy place to start (beware the many typos!). Then there's the book of Freitag-Kiehl, (Étale cohomology and the Weil conjectures) containing an exposition specifically adapted to the needs of arithmetic geometry. – t.b. May 16 '11 at 12:12

Gelfand and Manin's "Methods of homological algebra" explains the subject quite nicely, though it takes them some pages to develop the theory (and there are many typos, at least in the first edition). Personally, I also use the first three chapters of Huybrecht's "Fourier-Mukai transforms in algebraic geometry", which is a bit more condensed but has a nice overview of the situation in the algebro-geometric setting. For more advanced material you could have a look at the notes on

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You'll find Bernhard Keller's notes for a short course on the subject in his web page. His exposition is characteristically lucid and clear. His focus is representation theory, so they may not match your interests, though.

I also like a lots the to-the-pointness approach taken by Dieter Happel in his book about triangulated categories.

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I like "Sheaves in Topology" by Dimca a lot.

I also second the suggestion of Gelfand/Manin. They actually have two books for some reason, "Homological Algebra" and "Methods of Homological Algebra", which are quite similar but have slightly different focus/applications. Both of them are worthwhile, and I think either one of them could be useful to you.

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As already said in a previous answer, Fourier-Mukai transforms in Algebraic Geometry by Huybrechts is a good reference. The first time I studied derived categories (and triangulated categories) I used the first part of Residues and Duality by R. Hartshorne.

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If you interested in better understanding the triangulated category structure of the derived category, I suggest you to read chapter 1 of Neeman's "Triangulated Categories". You can find much results explained in a very precise way. A good exercise would be to compare the definitions and results that appear in the above textbook with those in the already suggested Gelfand/Manin book. In this second textbook a very explicit approach for the derived and homotopic categories is taken, and you can surely detect typos by rewriting all computations that are relevant for your studies.

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