Derived Categories, Spring 2011

Professor A.J. de Jong, Columbia university, Department of Mathematics.

Organizational: The talks will be 2x45 minutes with a short break. Time and place: Fridays 10:30 AM in Room 312.

Introductory lecture topics (more than one topic per lecture):

1. Gabriel-Zisman localization.
2. A bit on triangulated categories.
3. Localizing triangulated categories.
4. Homotopy category of complexes in an additive category as a triangulated category.
5. Construction of the derived category of an abelian category as in Verdier's thesis.
6. Construction of the filtered derived category of an abelian category.
7. Bounded below derived category of an abelian category with enough injectives.
8. Deligne's general discussion of derived functors in Exposee XVII in SGA4.
9. Spaltenstein's paper on unbounded resolutions.

More advanced lecture topics:

1. Neeman's paper on Grothendieck duality.
2. Derived category modules over a ring determines the ring.
3. Beilinson's decomposition theorem for D^b_{coh}(P^n).
4. Bondal-Orlov: D^b_{coh}(X) determines X in some case.
5. Orlov's computation of Aut(D^b(A)) with A an abelian variety.
6. Orlov's description of the triangulated category associated to a hypersurface singularity in terms of matrix factorisations.
7. Bondal-van den Bergh: D^b_{coh}(X) is saturated when X is a smooth proper variety.

Please email if you have more ideas for talks. Please email if you are interested in giving one of the talks. Here are some references:

• Jean-Louis Verdier, Des categories derivees des categories abeliennes, Asterisque No. 239 (1996)
• Pierre Deligne, Cohomologie a support propres, Exposee XVII, SGA4.
• Beilinson, Bernstein, Deligne, Faisceaux pervers, Asterisque, 100.
• Bondal, van den Bergh, Generators and representability of functors in commutative and noncommutative geometry Mosc. Math. J. 3 (2003), no. 1, 1-36, 258.
• Bondal, Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math. 125 (2001), no. 3, 327-344.
• Orlov, Derived categories of coherent sheaves on abelian varieties and equivalences between them, Izv. Math. 66 (2002), no. 3, 569-594.
• Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, Progr. Math., 270, Birkhäuser Boston, Inc., Boston, MA, 2009.
• Beilinson, The derived category of coherent sheaves on P^n, Selecta Math. Soviet. 3 (1983/84), no. 3, 233-237.
• Thomason, The classification of triangulated subcategories, Compositio Math. 105 (1997), no. 1, 1-27.
• Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121-154.
• Neeman, Triangulated categories, Annals of Mathematics Studies, 148.
• Neeman, The Grothendieck duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205-236.
• Theo Bühler, Exact categories, Expo. Math. 28 (2010), no. 1, 1-69.
• Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456.
• Amnon Yekutieli, Dualizing complexes, Morita equivalence and the derived Picard group of a ring, J. London Math. Soc. (2) 60 (1999), no. 3, 723-746.