Contents -- 1 Triangulated categories -- 1.1 Additive categories and functors -- 1.2 Triangulated categories and exact functors -- 1.3 Equivalences of triangulated categories -- 1.4 Exceptional sequences and orthogonal decompositions -- 2 Derived categories: a quick tour -- 2.1 Derived category of an abelian category -- 2.2 Derived functors -- 2.3 Spectral sequences -- 3 Derived categories of coherent sheaves -- 3.1 Basic structure -- 3.2 Spanning classes in the derived category -- 3.3 Derived functors in algebraic geometry -- 3.4 Grothendieck-Verdier duality -- 4 Derived category and canonical bundle - I -- 4.1 Ample (anti-)canonical bundle -- 4.2 Autoequivalences for ample (anti-)canonical bundle -- 4.3 Ample sequences in derived categories -- 5 Fourier-Mukai transforms -- 5.1 What it is and Orlov's result -- 5.2 Passage to cohomology -- 6 Derived category and canonical bundle - II -- 6.1 Kodaira dimension under derived equivalence -- 6.2 Geometrical aspects of the Fourier-Mukai kernel -- 6.3 Nefness under derived equivalence -- 6.4 Derived equivalence versus birationality -- 6.5 Recap: Kodaira dimension, canonical ring, etc. -- 7 Equivalence criteria for Fourier-Mukai transforms -- 7.1 Fully faithful -- 7.2 Equivalences -- 7.3 Canonical quotients -- 8 Spherical and exceptional objects -- 8.1 Autoequivalences induced by spherical objects -- 8.2 Braid group actions -- 8.3 Beilinson spectral sequence -- 8.4 They go together -- 9 Abelian varieties -- 9.1 Basic definitions and facts -- 9.2 The Poincaré bundle as a Fourier-Mukai kernel -- 9.3 Sl[Sub(2)]-action -- 9.4 Derived equivalences of abelian varieties -- 9.5 Autoequivalences of abelian varieties -- 10 K3 surfaces -- 10.1 Recap: K3 surfaces -- 10.2 Derived equivalence of K3 surfaces -- 10.3 Recap: Moduli spaces of sheaves -- 11 Flips and flops.

11.1 Preparations: Closed embeddings and blow-ups -- 11.2 Derived categories under blow-up -- 11.3 The standard flip -- 11.4 The Mukai flop -- 12 Derived categories of surfaces -- 12.1 Recap: Enriques classification of algebraic surfaces -- 12.2 Minimal surfaces with kod = -∞, 2 -- 12.3 Surfaces with torsion canonical bundle -- 12.4 Properly elliptic surfaces -- 13 Where to go from here -- 13.1 McKay correspondence for derived categories -- 13.2 Homological mirror symmetry -- 13.3 D-branes and stability conditions -- 13.4 Twisted derived categories -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- O -- P -- Q -- R -- S -- T -- Y.