derived9 - Derived categories. Winter 2008/09 Igor V....

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Unformatted text preview: Derived categories. Winter 2008/09 Igor V. Dolgachev May 5, 2009 ii Contents 1 Derived categories 1 1.1 Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Derived categories . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4 Spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2 Derived McKay correspondence 47 2.1 Derived category of coherent sheaves . . . . . . . . . . . . . . . . 47 2.2 Fourier-Mukai Transform . . . . . . . . . . . . . . . . . . . . . . 59 2.3 Equivariant derived categories . . . . . . . . . . . . . . . . . . . . 75 2.4 The Bridgeland-King-Reid Theorem . . . . . . . . . . . . . . . . 86 2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3 Reconstruction Theorems 105 3.1 Bondal-Orlov Theorem . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2 Spherical objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.3 Semi-orthogonal decomposition . . . . . . . . . . . . . . . . . . . 121 3.4 Tilting objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 iii iv CONTENTS Lecture 1 Derived categories 1.1 Abelian categories We assume that the reader is familiar with the concepts of categories and func- tors. We will assume that all categories are small , i.e. the class of objects Ob( C ) in a category C is a set. A small category can be defined by two sets Mor( C ) and Ob( C ) together with two maps s, t : Mor( C ) Ob( C ) defined by the source and the target of a morphism. There is a section e : Ob( C ) Mor( C ) for both maps defined by the identity morphism. We identify Ob( C ) with its image under e . The composition of morphisms is a map c : Mor( C ) s,t Mor( C ) Mor( C ). There are obvious properties of the maps ( s, t, e, c ) expressing the axioms of associativity and the identity of a category. For any A, B Ob( C ) we denote by Mor C ( A, B ) the subset s- 1 ( A ) t- 1 ( B ) and we denote by id A the element e ( A ) Mor C ( A, A ). A functor from a category C defined by (Ob( C ) , Mor( C ) , s, t, c, e ) to a cate- gory C defined by (Ob( C ) , Mor( C ) , s , t , c , e ) is a map of sets F : Mor( C ) Mor( C ) which is compatible with the maps ( s, t, c, e ) and ( s , t , c , e ) in the obvious way. In particular, it defines a map Ob( C ) Ob( C ) which we also denote by F . For any category C we denote by C op the dual category , i.e. the category (Mor( C ) , Ob( C ) , s , t , c, e ), where s = t, t = s . A contravariant functor from C to C is a functor from C op to C . For any two categories C and D we denote by D C (or by Funct( C , D )) the category of functors from C to D . Its set of objects are functors F : C D . Its set of morphisms with source F 1...
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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derived9 - Derived categories. Winter 2008/09 Igor V....

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