Derived categories of coherent sheaves: suggested references? - MathOverflow

Derived categories of coherent sheaves: suggested references?

2010-10-17T06:35:49Z

I am interested in learning about the derived categories of coherent sheaves, the work of Bondal/Orlov and T. Bridgeland. Can someone suggest a reference for this, very introductory one with least prerequisites. As I was looking through the papers of Bridgeland, I realized that much of the theorems are stated for Projective varieties (not schemes), I've just started learning Scheme theory in my Algebraic Geometry course,my background in schemes is not very good but I am fine with Sheaves. It would be better if you suggest some reference where everything is developed in terms of Projective varieties.

Answer by ABayer for Derived categories of coherent sheaves: suggested references?

2010-10-17T13:52:44Z

A good introduction to derived categories of coherent sheaves are Caldaru's notes Derived categories of sheaves: a skimming.

As for Bridgeland's work, I would recommend reading his papers directly, using Huybrecht's book as a reference (as mentioned by Francesco above, "Fourier-Mukai transforms in algebraic geometry"). Specifically for Bridgeland stability conditions, I also have some short notes on my homepage, but again, you should also read his original papers.

Answer by Piotr Achinger for Derived categories of coherent sheaves: suggested references?

2010-10-17T20:13:26Z

Scheme theory won't be that important (as you said, most of the time you are concerned with smooth projective varieties over $\mathbb C$). However, you didn't mention about your experience in homological algebra. Huybrechts' book is a pleasure to read, but it can't serve as a textbook in homological algebra. I would recommend e.g. reading first first few chapters of Weibel's book (1, 2, 3, 5, 10). Gelfand-Manin "Methods of Homological Algebra" is also a good reference here.

Moreover, there is a short introduction by R. P. Thomas "Derived categories for the working mathematician".

Answer by Leo Alonso for Derived categories of coherent sheaves: suggested references?

2010-10-18T08:57:32Z

Let me suggest a perhaps longer program to master derived categories of coherent sheaves. In a comment you said "I don't know much Homological Algebra", so first you have to master the basics of abelian categories and ext and tor of modules. The first chapters of Weibel's "Homological Algebra" may be a useful reference.

This said, I agree with Polizzi's suggestion of Huybrechts' book. The only issue I would point out is that the book concentrates on the bounded derived category of coherent sheaves. It is enough for its application to the classification of non-singular varieties. But if you have varieties with singularities in mind then things may get more complicated. First there is a difference between perfect complexes and bounded complexes of coherent sheaves. Second one sometimes need to use infinite methods, therefore one is forced to consider quasi-coherent sheaves and unbounded derived categories. For this beautiful theory, a good introduction is (as I have suggested here before) the first chapters of Lipman's "Notes on Derived Functors and Grothendieck Duality", (in Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics, no. 1960, Springer, 2009).

This is not a shortest path, but in my experience getting into general methods is that they pay back later. Your mileage may vary.

Answer by Zoran Škoda for Derived categories of coherent sheaves: suggested references?

2010-10-18T12:24:54Z

Kapustin-Orlov'a survey of derived categories of coherent sheaves is pretty good,

A. N. Kapustin, D. O. Orlov, Lectures on mirror symmetry, derived categories, and D-branes, Uspehi Mat. Nauk 59 (2004), no. 5(359), 101--134; translation in Russian Math. Surveys 59 (2004), no. 5, 907--940, math.AG/0308173

but more slow/elementary exposition starting with fundamentals of derived categories is in an earlier survey of Orlov

D. O. Orlov, Derived categories of coherent sheaves and equivalences between them, Uspekhi Mat. Nauk, 2003, Vol. 58, issue 3(351), pp. 89–172, Russian pdf, English transl. in Russian Mathematical Surveys (2003),58(3):511, doi link, pdf at Orlov's webpage (not on arXiv!)

There are also Orlov's handwritten slides in djvu from a 5-lecture course in Bonn

djvu, but the link is temporary

For derived categories per se, apart from Gelfand-Manin methods book and Weibel's hoological algebra remember that a really good expositor is Bernhard Keller. E.g. his text

Bernhard Keller, Introduction to abelian and derived categories, pdf

...and also his Handbook of Algebra entry on derived categories: pdf