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Derived Categories. Winter 2008/09

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Derived Categories. Winter 2008/09 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 −1 −1 by MorC(A, B) the subset s (A) ∩ t (B) and we denote by idA the element e(A) ∈ MorC(A, A). A functor from a category C defined by (Ob(C), Mor(C), s, t, c, e) to a cate- gory C0 defined by (Ob(C0), Mor(C0), s0, t0, c0, e0) is a map of sets F : Mor(C) → Mor(C0) which is compatible with the maps (s, t, c, e) and (s0, t0, c0, e0) in the obvious way. In particular, it defines a map Ob(C) → Ob(C0) which we also denote by F . For any category C we denote by Cop the dual category, i.e. the category (Mor(C), Ob(C), s0, t0, c, e), where s0 = t, t0 = s.A contravariant functor from C to C0 is a functor from Cop to C. For any two categories C and D we denote by DC (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 F1 and target F2 are natural transformation of functors, i.e. maps Φ = (Φ1, Φ2): Mor(C) → Mor(D) × Mor(D) such that (s × t) ◦ Φ = (s × t) ◦ (F1,F2) and (t × s) ◦ Φ = (t × s) ◦ (F1,F2), and Φ2 ◦ F1 = F2 ◦ Φ1. It is also required that Φi(u ◦ v) = Φi(u) ◦ Φi(v) and Φi(idA) = idFi(A), for any A ∈ Ob(C). Let Sets be the category of small sets (i.e. subset of some set which we op can always enlarge if needed). We denote by Cb the category SetsC . A typical example is when we take C = Open(X) to be the category of open subsets of 1 2 LECTURE 1. DERIVED CATEGORIES a topological space X with inclusions as morphisms, a contravariant functor F : Open(X) → Sets is a presheaf on X. For this reason, the objects of Cb are called presheaves on C. Any A ∈ Ob(C) defines the presheaf φ◦ hA :(φ : X → Y ) → (MorC(Y, A) −→ MorC(X, A)) For any morphism u : A → A0 in C and any A ∈ Ob(C), composing on the left defines a map hA(A) → hA0 (A), and the set of such maps makes a morphism of functors hA → hA0 . This defines a functor 0 h : C → C, Mor (A, A ) → Mor (h , h 0 ) b C Cb A A called the Yoneda functor or the representation functor. According to the Yoneda lemma this functor is fully faithful, i.e. defines a bijection 0 Mor (A, A ) → Mor (h , h 0 ). C Cb A A Via the Yoneda functor, the category C becomes equivalent to a full subcategory 0 of the category Cb (a subcategory C is full if MorC0 (A, B) = MorC(A, B)). Also recall that a category C0 is equivalent to a category C if there exist functors F : C0 → C,G : C → C0 such that the compositions F ◦ G, G ◦ F are isomorphic (in the category of functors) to the identity functors. A presheaf F ∈ Cb is called representable if it is isomorphic to a functor of the form hS for some S ∈ C. We say that F and G are quasi-inverse functors. The object S is called the representing object of F . It is defined uniquely, up to isomorphism. Dually, one defines the functor hA : C → Sets whose value at an object X is A equal to MorC(A, X). A functor C → Sets isomorphic to a functor h is called corepresentable. Let S be a subcategory of the category Sets. A category C is called an S- category if for any X ∈ Ob(C) the presheaf hX takes values in S and for any (A → B) ∈ Mor(C), the map hX (A) → hX B) is a morphism in S. We will be interested in the case when S = Ab is the category of abelian groups. In this case an Ab-category is called a Z-category. It follows from the definition that the sets of morphisms is equipped with a structure of an abelian group, moreover, the left composition map MorC(A, B) → MorC(A, C) and the right composition map MorC(B, C) → MorC(A, C) are homomorphisms of groups. Another useful example is when A is the category of linear spaces over a field K. This allows to equip the sets of morphism with compatible structures of linear spaces. In this case a category is called K-linear. From now on, whenever we deal with a Z-category category we set HomC(A, B) := MorC(A, B). ab Let Cb be the category of abelian presheaves on C, i.e. the category of contravariant functors from C to Ab. If C is a Z-category, the Yoneda functor ab is the functor from C to Cb . For any abelian group A one defines the constant presheaf AC (or just A if no confusion arises) by 0 AC(S) = A, (S → S ) → idA. 1.1. ABELIAN CATEGORIES 3 ab In particular, we have the zero presheaf 0. For any F ∈ C , we have Hom ab (0,F ) = b Cb {0} and the zero represents the unique morphism 0 → F . A zero object of a Z-category is an object representing the functor 0. It may not exist, but when they exist all of them are isomorphic and can be identified with one object denoted by 0. The zero object 0 is characterized by the property that HomC(0, 0) = {0}. This immediately implies that 0 is the initial and the final object in C, i.e., for any X ∈ Ob(C) there exists a unique morphism 0 → X (resp. X → 0). It represents the zero element in HomC(0,X) (resp. in HomC(X, 0)). For any morphism u : F → G of abelian presheaves we can define the kernel ker(u) by ker(u)(A) = ker(F (A) → G(A)). It can be characterized by the following universal property: there is a morphism ker(u) → F such that the composition ker(u) → F → G is the zero morphism, and any morphism K → F with this property factors through the morphism ker(u) → F . We say that C has kernels if, for any X → Y the kernel of hX → hY ab in Cb is representable. By the Yoneda Lemma, the kernel ker(X → Y ) satisfies the universal property from above, and this can be taken as the equivalent definition. The definition of the cokernel coker(u) of a morphism X → Y in a pre- additive category is obtained by reversing the arrows. It comes with a unique morphism Y → coker(u) such that any morphism Y → K with the zero com- position F → G → K factors through G → coker(u) → K. In other words, coker(u) = ker(uop)op, where uop : G → F is a morphism in the dual category Cop and ker(uop)op is the morphism ker(u0) → G in Cop considered as a morphism G → ker(uop) in C. ab Note that Cb has cokernels defined by coker(F → G)(S) = coker(F (S) → G(S)). However, even when C = Ab where kernels and cokernels are the usual ones, coker(hA → hB) may not be representable. For example, if we take [n]: Z → Z the multiplication by n > 1 in Ab and u = h([n]) : hZ → hZ, then we get coker(u)(Z/nZ) = coker(hZ(Z/nZ) → hZ(Z/nZ)) = coker(0 → 0) = {0}, because Hom(Z/nZ, Z) = {0}. On the other hand, hcoker([n])(Z/nZ) = hZ/nZ(Z/nZ) = Hom(Z/nZ, Z/nZ) 6= {0}. It follows from the definition that there is a canonical morphism coker(ker(u) → s(u)) →α ker(t(u) → coker(u)), (1.1) 4 LECTURE 1. DERIVED CATEGORIES where u : s(u) → t(u) is a morphism in C (provided that all the objects in above exist). The morphism u factors as s(u) → coker(ker(u) → s(u)) → ker(t(u) → coker(u)) → t(u).
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