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.
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19$\begingroup$ "Fourier-Mukai transforms in algebraic geometry" by D. Huybrechts $\endgroup$– Francesco PolizziCommented Oct 17, 2010 at 7:38
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2$\begingroup$ If you want to learn about derived categories in the modern way, take a look at M. Hovey's Model Categories chapters 1 and 2 (don't worry, they're pretty short and not too bad a read, although you might want to skip the parts on the small object argument). After that, there are two ways you can go: The computationally more useful but substantially less natural approach of working with triangulated categories, or by working with simplicial localizations and resolutions. $\endgroup$– Harry GindiCommented Oct 17, 2010 at 13:39
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1$\begingroup$ Hi,Verma,you might want to take Rosenberg's course now, it is of great help $\endgroup$– Shizhuo ZhangCommented Oct 17, 2010 at 16:58
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11$\begingroup$ I disagree with Harry's implicit statement that there is a unique "modern way" to approach derived categories. Also, I was under the impression that Hovey's book does not say much about coherent sheaves on projective varieties. $\endgroup$– S. Carnahan ♦Commented Oct 18, 2010 at 0:33
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3$\begingroup$ @Scott: Certainly, but I think it's been evident from the beginning that triangulated categories were not the canonical "right" notion. $\endgroup$– Harry GindiCommented Oct 18, 2010 at 1:13
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4 Answers
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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 homological 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
answered Oct 18, 2010 at 12:24
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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.
answered Oct 17, 2010 at 13:52
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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.
answered Oct 18, 2010 at 8:57
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$\begingroup$ Thanks Leo, time is not a constraint for me since I am a first year grad student. Of course I want to learn stuff quickly, but better understanding is my priority. $\endgroup$– J VermaCommented Oct 18, 2010 at 19:03
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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".
answered Oct 17, 2010 at 20:13
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$\begingroup$ I don't know much Homological Algebra, could you please tell me what topics to look at. I'll highly appreciate if you could outline a roadmap to Bridgeland Stability starting from Projective varieties.Thanks. $\endgroup$– J VermaCommented Oct 17, 2010 at 21:06