School
“Introductory School Derived Categories”

Problem sheet for the exercise class on Thursday

• Tom Bridgeland (Sheffield): Introduction to Derived Categories and Stability Conditions
  

  
  Lecture Notes
  
  

  Lecture 1: Recap on derived and triangulated categories. Recall of basic properties of the derived category, including triangulated structure and truncation functors. Derived functors and example of dualizing functor for sheaves on a surface.

  Lecture 2: Basic properties of derived categories of smooth projective varieties (Ext-finiteness, Serre duality, Riemann-Roch). Gabriel's Theorem. The Bondal-Orlov Theorem.

  Lecture 3: Picture of derived category of P^1. Equivalence with derived category of Kronecker quiver. Derived category of an elliptic curve and its symmetry group.

  Lecture 4: Derived categories of surfaces. Slope and Gieseker stability. Mukai-Orlov derived Torelli theorem for K3 surfaces.

  Lecture 5: T-stuctures, torsion pairs and tilting. Example of threefold flop as a moduli space of two-step complexes.

  Lecture 6: Stability conditions. Definition and alternative characterisation in terms of t-structures. Main deformation result. Construction of examples on surfaces.



• Alexander Kuznetsov(Steklov Institute): Semiorthogonal decomposition of derived categories
  

  
  Lecture Notes (scanned, courtesy of M. Akhtar)
  
  

  Lecture 1: Semiorthogonal decompositions. Exceptional collections. Examples from directed quivers.

  Lecture 2: Mutations. Braid group action. Serre functor. Dual collections. Resolution of the diagonal.

  Lecture 3: The gluing functor and bimodule. Gluing of triangulated categories. Behaviour of invariants.

  Lecture 4: Beilinson's collection. Kapranov's collections. Other homogeneous spaces.

  Lecture 5: Projectivization of a vector bundle. Severi-Brauer varieties. The blowup formula. Flips and flops.

  Lecture 6: Fibrations in quadrics. Intersection of quadrics. Examples from homological projective duality.



• Timothy Logvinenko (Cardiff): Introduction to DG-categories
  
  

  Lecture Notes (Lectures 1-3)
  
  

  Lecture 1: DG-categories. DG-modules. Properties of DG-modules: h-projective, perfect, acyclic.

  Lecture 2: Derived category of a DG-category. Bimodules, duals and adjunctions.

  Lecture 3: Homotopy and Morita homotopy categories of small DG-categories. Quasi-functors.

  Lecture 4: Twisted complexes. Pre-triangulated and Kc-triangulated categories. Ordinary and Morita DG-enhancements.

  Lecture 5: Applications to algebraic geometry.

  
  

  Possible extra lecture(s): Model theory. A-infinity categories.