Derived category and so on

Question

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

Answer 1

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 http://therisingsea.org.

Answer 2

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.

Answer 3

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.