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.
Algebraic Geometry – Derived Categories of Coherent Sheaves: Suggested References
ag.algebraic-geometryct.category-theorymirror-symmetryreference-requestsheaf-theory
Related Solutions
As always, the source you use may be related to what your goals are. To give some perspective, recall there are several ways to define sheaf cohomology, and Serre and Hartshorne feature different methods. Serre used Cech cohomology, and there the important long exact sequence property does not always hold. He was able to prove however that it does hold for "coherent" sheaves. One big advantage of Cech theory is its easier computability in specific cases, such as on projective space. Hartshorne presents first Grothendieck's theory of derived functor cohomology, but then proves it agrees with Cech cohomology before using that theory to compute the cohomology of coherent sheaves on projective space. But if you want to learn the derived functor theory you must choose Hartshorne over Serre.
The distinction made above between schemes and varieties is also relevant. Serre teaches Cech cohomology on varieties,and Hartshorne presents derived functor cohomology on schemes. If you are only interested in varieties, or prefer learning cohomology in the easier setting of varieties, then you may prefer Serre's FAC. Another good source is the book Algebraic Varieties by George Kempf, where the derived functor theory is presented on varieties and used for basic computations, including coherent cohomology of projective space and even the full Riemann Roch theorem, before being linked with Cech theory. So if you want to learn to make computations with the abstract derived functor theory you might prefer George's treatment, although some details are missing there, and some misprints exist.
Finally there are slight differences in Serre's and Hartshorne's results which can be relevant in some settings. E.g. in Beauville's book on surfaces, he uses Cech theory to relate rank two vector bundles on curves with ruled surfaces. To prove that all ruled surfaces arise from vector bundles he then uses Serre's result that Cech H^2 vanishes on a curve with coefficients in any sheaf coherent or not. (He also gives a second argument.) But this vanishing theorem for Cech cohomology does not follow from Hartshorne's treatment, since he proves vanishing for derived functor cohomology but does not relate derived functor and Cech cohomology on non coherent sheaves above degree one.
There is a sentence in Hartshorne, at the end of chapter III, section 2, page 212 in the 1977 edition, which says that Serre proved vanishing for "coherent sheaves on algebraic curves and projective algebraic varieties", whereas the correct statement would be that he proved it "for curves, and for coherent sheaves on projective algebraic varieties". Since Robin is very careful, one wonders whether some well intentioned copy editor did not change this sentence's meaning unwittingly to make it flow better.
One can see some initial answers in Huybrechts' book "Fourier-Mukai Transforms in Algebraic Geometry"
([Huybrechts], Prop 4.1) If two smooth projective varieties are derived equivalent, then they have the same dimension.
([Huybrechts], Prop 3.10) If $X$ is Noetherian, then $X$ is connected if and only if $D^b(Coh X)$ is indecomposable.
I heard about the first part of the following argument as a folklore, someone should correct me if I am wrong:
(EDIT: The following is WRONG. However, there is some interesting discussion in the comments to this answer.)
- The compact objects in $D^b(Coh X)$ are precisely those quasi-isomorphic to bounded complex of locally free sheaves, so $X$ is regular if and only if every object in $D^b(Coh X)$ is compact.
Best Answer
Kapustin-Orlov'a survey of derived categories of coherent sheaves is pretty good,
but more slow/elementary exposition starting with fundamentals of derived categories is in an earlier survey of Orlov
There are also Orlov's handwritten slides in djvu from a 5-lecture course in Bonn
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
...and also his Handbook of Algebra entry on derived categories: pdf