# What is a good reference (preferably thorough) for the Derived Category of a scheme/orbifold/stack?

I've sort of circled around the idea of derived categories a few times, read a few introductory papers ("Derived Categories for the working mathematician", e.g.), and feel now that this is something that I need to actually learn.

So what is a good reference textbook-type resource for this material? Expository papers are great to read, but they only go so far in terms of building up a working understanding of the material.

Ideally this reference would contain some information about derived categories of orbifolds and some discussion of the relation to moduli problems, cotangent complices, etc. Bonus points if it has many exercises to work through.

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There is the first chapter of the book "Sheaves on manifolds" by Kashiwara-Shapira. –  Damian Rössler May 10 '13 at 16:44
The Stacks Project is great! –  Dylan Wilson May 10 '13 at 16:45
@Simon, I apologize in advance if my suggestion is innapropriate, but if you're not familiar with derived categories I would first go for derived categories of rings, then for derived categories of Grothendieck abelian categories, and finally I'd consider the particular cases you're interested in. –  Fernando Muro May 10 '13 at 18:16
For derived categories I would second the recommendation of Damian Rössler. For derived categories of sheaves on varieties I would recommend the survey of Orlov (Russian Mathematical Surveys, 2003); though it is not a textbook it is still quite thorough (about 80 pages). For schemes and stacks the only thorough reference I'm aware of is the Stacks project. –  Adeel May 11 '13 at 7:47
Unfortunately there is not the derived category $D(X)$ of a scheme or stack $X$, although this notation is used quite often. But it can refer to the bounded, bounded below, bounded above derived category of the abelian category of Zariski, étale, lisse-étale etc. quasi-coherent or coherent sheaves on $X$ ... –  Martin Brandenburg Jun 17 '13 at 15:31

I'll just mention some sources that are not already in the other question indicated by Franz's answer, in case you need to become more comfortable with the derived machinery itself. (The book by Huybrechts on Fourier-Mukai Transforms in the other answer will probably be the best, depending on your interests, and there is also "Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics", by Bartocci, Bruzzo, and Ruipérez that contains additional information and good references).

So here are the other references that might help with derived categories themselves:

I think most of the texts or papers treating derived categories applied to algebraic geometry are a bit too terse when it comes to derived categories themselves. There is a nice book called "Interactions Between Homotopy Theory and Algebra", which is from a summer school that was held in Chicago, and in this book there are two nice lectures by Henning Krause on derivated categories. His second lecture actually consists of about 40 exercises, which might help you with understanding the machinery of derived categories without having to worry about how they work in algebraic geometry yet. These two chapters are also available on the arXiv (Lecture, Exercises), though I recommend the whole book too.

Yet another book is the edited volume "Handbook of Tilting Theory" (LMS 332), which is a series of longer (most introductory papers) on tilting theory, so you'll see plenty of applications to module categories and representation theory, but there is also a chapter on the Fourier-Mukai transform there.

In the book "Derived Equivalences for Group Rings" (König, Zimmerman, et al.), there are several chapters that include introductions to aspects of derived categories including unbounded derived categories and many examples, which might also be useful.

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Perhaps this question (and related answers) might help you : A down-to-earth introduction to the uses of derived categories

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