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.



Derived categories of coherent sheaves: suggested references?
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](https://arxiv.org/abs/math/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](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=629&volume=58&year=2003&issue=3&fpage=89&what=fullt&option_lang=eng), English transl. in Russian Mathematical Surveys (2003),58(3):511, [doi link](http://dx.doi.org/10.1070/RM2003v058n03ABEH000629), [pdf](http://www.mi.ras.ru/%7Eorlov/papers/Uspekhi2003.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](http://www.irb.hr/korisnici/zskoda/orlovMPIslides.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](http://www.math.jussieu.fr/%7Ekeller/publ/cam.pdf)


...and also his Handbook of Algebra entry on derived categories:
[pdf](http://www.math.jussieu.fr/%7Ekeller/publ/dcu.pdf)


