arXiv:1206.6632 A Course on Derived Categories from arXiv Front: math.CT by Amnon Yekutieli These are notes for an advanced course given at Ben Gurion University in Spring 2012.
Caldararu in Homol alg folder (good!)
http://www.ncatlab.org/nlab/show/derived+category
On derived cats in terms of model cats: http://mathoverflow.net/questions/372/derived-categories-and-homotopy-categories
Krause lectures, in Homological algebra folder
Hartshorne LNM0020
Keller chapter in Handbook of Algebra vol 1, in Various folder under ALGEBRA
An appendix to Hartshorne: Residues and duality
Huybrechts: Fourier-Mukai transform in algebraic geometry (book)
Notes by Dolgachev, in Homological algebra folder
Gelfand-Manin: Algebra V
Gelfand and Manin: Methods book
Bondal-Orlov in Compositio
The derived category of R-modules is the homotopy category of the model category of chain complexes of R-modules.
Can define the derived category of an exact category: see Laumon in LNM 1016.
For brief intro, and longer treatment of equivariant derived cats, see Bernstein and Lunts, in Homol alg folder.
A nice note by May
Ask Bao for other refs. Bao said that automorphisms of the derived cat are just (3) the shifts () times the semidirect product of (2) automorphisms of the variety and (3) twists with (any???) line bundles)
Here are some notes by Murfet:
Memo: Let be a complex of abelian groups. Then is the set of homotopy classes of maps from to . Here I think the first complex is places in degree . (And when writing down a complex, the degree increases as one goes to the right - I think this means that the complex is cohomological.)
Can think of as the complexes of injectives in . For , think complexes of injectives, which have bounded cohomology.
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 33 [Li] J. Lipman, Notes on derived categories and derived functors, available at http://www.math.purdue.edu/lipman.
A list of references in an article by Orlov, related to the derived category of coherent sheaves on a variety.
J.-L. Verdier, Des categoryégories dérivées des categoryégories abéliennes, Astérisque 239, 1996.
A. Rosenberg, The spectrum of abelian categories and reconstructions of schemes, in Rings, Hopf Algebras, and Brauer groups, 257-274, Lectures Notes in Pure and Appl. Math. 197, Marcel Dekker, New York, 1998.
[Ke1] B. Keller, Introduction to abelian and derived categories, in Representations of reductive groups, edited by R. W. Carter and M. Geck, Cambridge University Press 1998, 41-62 (available on Keller’s webpage).
[Ke2] B. Keller, Derived categories and their uses, in Handbook of algebra, edited by M. Hazewinkel, Elsevier 1996 (available on Keller’s webpage).
The aim of this essay is to give an introduction to homological algebra, as it is applied in algebraic geometry and string theory (homological mirror symmetry), and/or in representation theory. As well as learning some abstract machinery, you will be gaining proficiency in either computing cohomology of coherent sheaves (geometry), or in representation theory (quivers, symplectic reflection algebras, finite groups). The category of representations of an algebra, or of coherent sheaves on an algebraic variety, is an abelian category; its derived category is built out of the category of chain complexes of such objects. Cutting your abelian category into pieces (for example, by partitioning your variety into an open subvariety and its closed complement) allows you to cut the derived category into pieces, but the derived category has ’more’ decompositions than the abelian. Moreover, different abelian categories may have the same derived category. There are two ways into this essay, for those with and without algebraic geometry. If you know some algebraic geometry: Begin with Fourier-Mukai transform, which is an involution on the derived category of an elliptic curve. Use this to describe all vector bundles on an elliptic curve. More generally, do this for abelian varieties. A nice consequence is the Torelli theorem. Then back up, and learn classical Koszul duality, the derived equivalence between modules for SV and ^V ; as well as Beilinson’s theorem describing all sheaves on Pn. ADHM correspondence, describing quiver moduli spaces. After this, try some explicit examples of flops by blowing up and down; this is due to Bondal- Orlov. You can continue in this way, and learn some of the derived category approach to 27 birational geometry. Some lovely classical computations can be studied in this language; see for example the papers of Kuznetsov. This is probably enough for an essay, but if you’re ambitious: Then read Bridgeland’s paper constructing smooth 3-fold flops as the moduli space of perverse point sheaves. Other nice examples of derived categories were inspired by physics; they are called Landau- Ginzburg models. These are an avatar of coherent sheaves on the space of vanishing cycles of a function f : X ! C, and were constructed by Orlov and Kontsevich. You may wish to play with these. At this point, you should read Bridgeland’s papers on stability conditions. He constructs a complex variety attached to an abelian or derived category, each point of which parameterizes a t-structure and a notion of stability condition. You should compute these spaces (or follow the computations in the literature!) for a line bundle over P2, at least. If you don’t know any algebraic geometry, this is still a good essay — derived equivalences are at the heart of representation theory — but the examples you should work with are representations of groups, or quivers, or Hecke algebras. One very rich source of examples are the ’cluster algebras’ of Fomin-Zelevinsky; Keller’s survey paper (listed below) is a fun introduction. But if you are interested in this essay, come talk to me and we’ll find a way into the subject that works best with your background. References For the basics of homological algebra, the textbooks of Weibel and Gelfand & Manin. Some survey papers: [1] Bridgeland, Spaces of stability conditions, arxiv/0611510 Bridgeland, Derived categories of coherent sheaves, arxiv/0602129 Bondal & Orlov, Derived categories of coherent sheaves, arxiv/0206295 [2] Keller, Cluster algebras, quiver representations and triangulated categories, arxiv/0807.1960
nLab page on Derived category