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Agnieszka Bodzenta June 12, 2019 HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY AGNIESZKA BODZENTA Contents 1. Categories, functors, natural transformations 2 1.1. Direct product, coproduct, fiber and cofiber product 4 1.2. Adjoint functors 5 1.3. Limits and colimits 5 1.4. Localisation in categories 5 2. Abelian categories 8 2.1. Additive and abelian categories 8 2.2. The category of modules over a quiver 9 2.3. Cohomology of a complex 9 2.4. Left and right exact functors 10 2.5. The category of sheaves 10 2.6. The long exact sequence of Ext-groups 11 2.7. Exact categories 13 2.8. Serre subcategory and quotient 14 3. Triangulated categories 16 3.1. Stable category of an exact category with enough injectives 16 3.2. Triangulated categories 22 3.3. Localization of triangulated categories 25 3.4. Derived category as a quotient by acyclic complexes 28 4. t-structures 30 4.1. The motivating example 30 4.2. Definition and first properties 34 4.3. Semi-orthogonal decompositions and recollements 40 4.4. Gluing of t-structures 42 4.5. Intermediate extension 43 5. Perverse sheaves 44 5.1. Derived functors 44 5.2. The six functors formalism 46 5.3. Recollement for a closed subset 50 1 2 AGNIESZKA BODZENTA 5.4. Perverse sheaves 52 5.5. Gluing of perverse sheaves 56 5.6. Perverse sheaves on hyperplane arrangements 59 6. Derived categories of coherent sheaves 60 6.1. Crash course on spectral sequences 60 6.2. Preliminaries 61 6.3. Hom and Hom 64 6.4. Serre duality 66 6.5. Derived functors in algebraic geometry 66 6.6. Grothendieck-Verdier duality 72 6.7. Spanning classes in the derived category 73 7. Full exceptional collections 79 7.1. Beilinson's result 80 7.2. Equivalence of categories 80 7.3. Braid group action 81 7.4. Glued t-structure 82 7.5. Full exceptional collections on P2 and Markov numbers 83 7.6. Full exceptional collections on homogeneous spaces and toric varieties 83 7.7. Derived category under blow-up and projective bundles 84 8. Modern approach 90 8.1. The stable category of spectra 90 8.2. DG categories and DG enhancements 91 8.3. Infinity categories 92 8.4. Stable infinity categories 94 8.5. Examples of stable infinity categories 96 8.6. Derived algebraic geometry 98 8.7. Why do we care about derived algebraic geometry 99 References 100 1. Categories, functors, natural transformations A category C is the data of a class of objects Ob(C) and a family of morphisms Mor(C). Every morphism has a source and a target, we denote by HomC(C1;C2) the collection of objects whose source is C1 and target is C2. We assume that there exists an associative HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 3 composition HomC(C2;C3) × HomC(C1;C2) ! HomC(C1;C3) and that every object has 0 00 the identity IdC 2 HomC(C; C) such that for any f 2 HomC(C; C ), g 2 HomC(C ;C), f ◦ IdC = f and IdC ◦g = g. A subcategory D ⊂ C is a category such that Ob(D) ⊂ Ob(C) and Mor(D) ⊂ Mor(C). A morphism f 2 HomC(C1;C2) is an isomorphism if there exists g 2 HomC(C2;C1) such that f ◦ g = IdC2 and g ◦ f = IdC1 . op The opposite category C has the same objects as C while HomCop (C1;C2) = HomC(C2;C1). Examples of categories include the categories of Sets, (pointed) topological spaces, (abelian) groups, the category ∆: Objects of ∆ are [n], for n = 0; 1;:::. Hom∆([m]; [n]) is the set of nonincreasing mappings from f0; : : : ; mg to f0; : : : ; ng. A (covariant) functor F : C!D is the data of a mapping ObC! ObD, C 7! F (C) and a mapping Mor(C) ! Mor(D), ' 7! F (') such that F ( ') = F ( )F (') and F (IdC ) = op IdF (C). A contravariant functor C!D is a functor C !D. A natural transformation η : F ! G of functors C!D is the data of ηC 2 HomD(F (C);G(C)), for any C 2 Ob(C). Morphisms ηC induce commutative diagrams ηC2 F (C2) / G(C2) O O F (') G(') ηC1 F (C1) / G(C1) for any ' 2 HomC(C1;C2). We say that a functor F : C!D is faithful if the map F : Mor(C) ! Mor(D) is injective. F is full if the map is surjective. It is essentially surjective if every object in D is isomorphic to F (C), for some C 2 Ob(C). A subcategory D ⊂ C is full if the embedding functor D!C is fully faithful. Functor F : C!D is an equivalence if there exists G: D!C and natural transformations η : IdC ! G ◦ F , ν : IdD ! F ◦ G such that ηC and νD are isomorphisms, for all C 2 Ob(C), D 2 Ob(D). Exercise 1.1. Show that a functor F : C!D is an equivalence if and only if it is fully faithful and essentially surjective. Examples of functors C op C • Any object C 2 C defines functors h : C! Set and hC : C ! Set via h (C1) = C HomC(C; C1), hC (C1) = HomC(C1;C). Functors h , hC are representable. 4 AGNIESZKA BODZENTA • Let ∆op Set be the category of simplicial sets. Its objects of ∆op Set are functors ∆op ! Set. Morphisms are natural transformations of functors. The geometric realisation is a functor j − j: ∆op Set ! Top. To a simplicial F1 set X = fXn = X([n])g it assigns jXj = n=0(∆n × Xn)=R where ∆n is the geometrical n-dimensional simplex n n+1 X ∆n = f(x0; : : : ; xn) 2 R j xi = 1; xi ≥ 0g i=0 and the equivalence relation R is defined as follows: (s; x) 2 ∆n × Xn is identified with (t; y) 2 ∆m × Xm if there exists f 2 Hom∆([m]; [n]) with y = X(f)x and F1 s = ∆f t. The topology on jXj is the weakest for which n=0 Xn × ∆n ! jXj is continuous. The map ∆f : ∆m ! ∆n is the unique linear mapping which sends vertex ei 2 ∆m to ef(i) 2 ∆n. • Another example of a functor is the Singular simplicial set Sing: Top ! ∆op Set. For a topological space Y , Sing(Y )(n) is the set of continuous maps ∆n ! Y For f 2 Hom∆([n]; [m]) the map Sing(Y )(f) maps ': ∆m ! Y to ' ◦ ∆f : ∆n ! Y . • A presheaf of sets is a functor (TopY )op ! Set. Here, Y is a topological space and TopY is the category whose objects are open subsets of Y and morphisms are inclusions U ! V . 1.1. Direct product, coproduct, fiber and cofiber product. Let X, Y be objects of a category C. The direct product X × Y is the object Z representing the functor C 7! hX (C) × hY (C) The direct sum X ⊕ Y is the object Z representing the functor C 7! hX (C) [ hY (C): Let S be an object of C. Define category CS whose objects are pairs (C; ') of objects of C and morphisms ': C ! S. Morphisms CS (C1;'1) ! (C2;'2) in CS are such f 2 HomC(C1;C2) that '2 ◦ f = '1. Let now X, Y be objects of CS, i.e. assume fixed ': X ! S, : Y ! S. The fiber product X ×S Y of X and Y over S is the direct product of (X; '), (Y; ) in CS considered as an object of C. Exercise 1.2. Write down the universal property of a fiber product. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 5 Given morphisms αX : S ! X, αY : S ! Y in a category C the cofiber product X tS Y of X and Y along S is an object Z 2 C together with βX : X ! Z, βY : Y ! Z satisfying the following universal property: given C in C and 'X : X ! C, 'Y : Y ! C such that 'X ◦αX = 'Y ◦αY , there exists unique ': Z ! C such that '◦βX = 'X and '◦βY = 'Y . Exercise 1.3. Present cofiber product as a coproduct in an appropriate category. 1.2. Adjoint functors. Let C, D be categories and F : C!D, G: D!C functors. Functor F is left adjoint to G, F a G if there exist natural transformations ": FG ! IdD, η : IdC ! GF , called the adjunction counit and unit, such that maps HomC(C; G(D)) ! HomD(F (C);D);' 7! "D ◦ F ('); HomD(F (C);D) ! HomC(C; G(D)); 7! G( ) ◦ ηC are inverse to each other. Exercise 1.4. Show that ": FG ! IdD, η : IdC ! GF yield F a G if and only if the F η ηG compositions F −! F GF −!"F F , G −! GF G −!G" G are the identity transformations. 1.3. Limits and colimits. Consider a category I and a functor F : I !C. A cone to F is an object N of C together with i : N ! F (i), for any i 2 I, such that for every α 2 HomI (i; j), F (α) ◦ i = j.A limit of F : I ! J is a cone (L; 'i) such that given any other cone (N; i) there exists a unique morphism u: N ! L such that 'i ◦ u = i. A cocone to F is an object W of C together with i : F (i) ! W , for any i 2 I, such that for every α 2 HomI (i; j), j ◦ F (α) = i.A colimit of F : I ! J is a cocone (T;'i) such that given any other cocone (W; i) there exists a unique morphism u: T ! W such that u ◦ 'i = i.
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