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I would like to learn about derived functors. Which reference do you advise ?

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The answers show that various approaches might be good, but to help can you more easily can you tell us what is the area in which you need them. On that and on your background mathematical interests the most appropriate answer for you will depend... a lot. –  Tim Porter Feb 4 '10 at 15:01
    
You are so right. –  Wanderer Feb 4 '10 at 15:08
    
I don't really need them. But I feel like I need to know more about cohomology. I know a little bit of singular/simplicial/de Rham/Cech cohomologies. If often hear (in seminars) of the R^i f, and I would like to know who they are. Thanks for all for the answers. I was thinking of the Hartshorne, I know the Schapira-Kashiwara and I like it for it preciseness. I was more looking for some typed lectures than for a book. Thanks again. –  user2330 Feb 4 '10 at 15:42
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I have to agree strongly with Ame's answer, in part. Weibel is a great place to go for the formalism. Once you have a little bit of the formalism, though, where to go depends on interests. To really see a lot of the power of derived functors, Hartshorne chapter 3 has some good theorems and exercises using them (though he does bounce back and forth with Cech cohomology...) and Huybrecht's "Fourier Mukai Transforms in Algebraic Geometry" book is very clear (at least to me) in the first few chapters, where he discusses derived categories (Note: I do disagree with Harry, this is a SECOND step, not a first for most) and makes extensive use of them, as he's interested in talking about derived categories, and they're the most natural thing in the universe there.

Also, the following papers might help:

Fourier Mukai Transforms and Applications to String Theory talks about how FM transforms (which are compositions of derived functors) help in string theory

Derived categories for the working mathematician This is a wonderful paper, very clear about what the derived category is, and might help in conjunction with the books above.

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I have to second the recommendation of Huybrecht's book regarding derived categories. –  Lars Feb 4 '10 at 19:47
    
That paper "FM and applications" looks awesome, thanks for pointing it out. –  Kevin H. Lin Feb 5 '10 at 4:22
    
I printed it out, but since then, I've drifted a bit away from FM stuff proper and into more classical Hodge theory stuff, but yeah, it looked good to me, and I figured it might be helpful. –  Charles Siegel Feb 5 '10 at 4:51
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The classic book "An introduction to homological algebra" by Weibel is an excellent reference, although it is not an easy read. I would certainly not recommend the book by Kashiwara and Schapira. Rotman is decent, but not as clear as Weibel, and it is full of typos.

When learning about derived functors, also try to see how they are used in an "applied context" (algebraic geometry/algebraic topology/any other applications which might interest you...).

I don't think going straight to derived categories (as Harry Gindi suggests) is a good idea.

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I was going to recommend Weibel's book too. This is where I first read about derived functors as a grad student, and I found it relatively easy going. At least it was easy to pick up the gist and some absolute basics of what was going on, which is about my level of command today. I would recommend that a person learn about derived functors (and spectral sequences) before they embark on a study of derived categories. I knew a very bright second-year undergraduate -- ten years ago -- who took a reading course in the latter without knowing the former, and it didn't work out so well. –  Pete L. Clark Feb 4 '10 at 14:55
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The biggest difficulty with Weibel for me has always been tha the does things homologically, but everything else I've ever seen with hte stuff is cohomological. From Delta Functors to spectral sequences. –  Charles Siegel Feb 4 '10 at 15:01
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Kashiwara and Schapira is totally readable without having any prior knowledge. I don't see what's so wrong with it. To be honest though, I haven't read the later chapters. –  Harry Gindi Feb 4 '10 at 15:09
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I agree that the book doesn't require any prior knowledge. But that doesn't mean that it is (pedagogically) well written. I have always had the feeling that they make things more complicated than they actually are, and more dense than they should be. –  Wanderer Feb 4 '10 at 15:16
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The homological vs cohomological issue stems from a cultural divide between algebraic topologists and algebraic geometers, both of whom have several good reasons for using one convention over the other. If you can come up with a better solution to this than upper-vs-lower indexing conventions then many people would appreciate it. –  Tyler Lawson Feb 4 '10 at 16:10
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It's perhaps in the "second step" category of references, but the book by Gelfand and Manin is a classic reference for derived categories. (Later editions have fewer typos I think).

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Rotman - "Introduction to homological algebra"

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There are some nice lecture notes by Jon Woolf on derived categories, derived functors and triangulated categories.

It's only three lectures'-worth, and the emphasis is more on derived and triangulated categories than derived functors. Still, maybe you'll find them useful. The lectures were for an audience of category theorists, but Woolf is a mostly-algebraic geometer and they finish with a brief description of some advanced applications to geometry.

Oh, and I like Weibel too.

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Peter Hilton and Urs Stammbach's Introduction to Homological Algebra is a great, more or less slow paced introduction to the classical theory. MacLane's Homology also makes a great introduction, provided you have a little background to appreciate its examples.

IMO it is not a good idea to start with derived categories. In most cases that approach results in the quite wrong idea that homological algebra is an absolutely abstract and obstruse subject with no contact with 'concrete math'.

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If you already know a cohomology theory I'd go straight to derived categories: it's what I did and, although it wasn't easy, it was sure worth it. Richard Thomas's notes suggested by Charles Siegel are excellent, as a sort of supplement I'd recommend Behrang Noohi's notes http://www.mth.kcl.ac.uk/~noohi/papers/Tehran.pdf

As far as books go, I definitely suggest reading the last chapter of Aluffi's great book Algebra Chapter 0. It doesn't reach derived categories, but it gives you every tool you need to approach them later.

After those I'd say the first chapter of Kashiwara-Schapira's Sheaves on Manifolds is very good, as it is fast and concise. Gelfand-Manin is also worth checking out, although full of mistakes. But if you're really brave I'd read Kashiwara-Schapira's Categories and Sheaves: it's the best (but only if you need all the gory details).

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Aluffi's book is very good indeed. It is not very known yet, but I'm convinced that this will change soon enough :) –  Wanderer Feb 4 '10 at 23:41
    
I couldn't agree more. I'm hoping for a chapter 1... –  babubba Feb 4 '10 at 23:47
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