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Derived Categories DERIVED CATEGORIES 05QI Contents 1. Introduction 2 2. Triangulated categories 2 3. The definition of a triangulated category 2 4. Elementary results on triangulated categories 5 5. Localization of triangulated categories 13 6. Quotients of triangulated categories 19 7. Adjoints for exact functors 24 8. The homotopy category 26 9. Cones and termwise split sequences 26 10. Distinguished triangles in the homotopy category 33 11. Derived categories 37 12. The canonical delta-functor 39 13. Filtered derived categories 42 14. Derived functors in general 45 15. Derived functors on derived categories 52 16. Higher derived functors 55 17. Triangulated subcategories of the derived category 59 18. Injective resolutions 62 19. Projective resolutions 67 20. Right derived functors and injective resolutions 69 21. Cartan-Eilenberg resolutions 71 22. Composition of right derived functors 72 23. Resolution functors 73 24. Functorial injective embeddings and resolution functors 75 25. Right derived functors via resolution functors 77 26. Filtered derived category and injective resolutions 77 27. Ext groups 85 28. K-groups 89 29. Unbounded complexes 90 30. Deriving adjoints 93 31. K-injective complexes 94 32. Bounded cohomological dimension 97 33. Derived colimits 99 34. Derived limits 103 35. Operations on full subcategories 104 36. Generators of triangulated categories 106 37. Compact objects 108 38. Brown representability 110 This is a chapter of the Stacks Project, version fac02ecd, compiled on Sep 14, 2021. 1 DERIVED CATEGORIES 2 39. Admissible subcategories 112 40. Postnikov systems 114 41. Essentially constant systems 119 42. Other chapters 121 References 122 1. Introduction 05QJ We first discuss triangulated categories and localization in triangulated categories. Next, we prove that the homotopy category of complexes in an additive category is a triangulated category. Once this is done we define the derived category of an abelian category as the localization of the homotopy category with respect to quasi-isomorphisms. A good reference is Verdier’s thesis [Ver96]. 2. Triangulated categories 0143 Triangulated categories are a convenient tool to describe the type of structure inherent in the derived category of an abelian category. Some references are [Ver96], [KS06], and [Nee01]. 3. The definition of a triangulated category 05QK In this section we collect most of the definitions concerning triangulated and pre- triangulated categories. 0144Definition 3.1. Let D be an additive category. Let [n]: D → D, E 7→ E[n] be a collection of additive functors indexed by n ∈ Z such that [n] ◦ [m] = [n + m] and [0] = id (equality as functors). In this situation we define a triangle to be a sextuple (X,Y,Z,f,g,h) where X, Y, Z ∈ Ob(D) and f : X → Y , g : Y → Z and h : Z → X[1] are morphisms of D.A morphism of triangles (X,Y,Z,f,g,h) → (X0,Y 0,Z0, f 0, g0, h0) is given by morphisms a : X → X0, b : Y → Y 0 and c : Z → Z0 of D such that b ◦ f = f 0 ◦ a, c ◦ g = g0 ◦ b and a[1] ◦ h = h0 ◦ c. A morphism of triangles is visualized by the following commutative diagram X / Y / Z / X[1] a b c a[1] X0 / Y 0 / Z0 / X0[1] Here is the definition of a triangulated category as given in Verdier’s thesis. 0145Definition 3.2. A triangulated category consists of a triple (D, {[n]}n∈Z, T ) where (1) D is an additive category, (2) [n]: D → D, E 7→ E[n] is a collection of additive functors indexed by n ∈ Z such that [n] ◦ [m] = [n + m] and [0] = id (equality as functors), and (3) T is a set of triangles called the distinguished triangles subject to the following conditions DERIVED CATEGORIES 3 TR1 Any triangle isomorphic to a distinguished triangle is a distinguished tri- angle. Any triangle of the form (X, X, 0, id, 0, 0) is distinguished. For any morphism f : X → Y of D there exists a distinguished triangle of the form (X,Y,Z,f,g,h). TR2 The triangle (X,Y,Z,f,g,h) is distinguished if and only if the triangle (Y, Z, X[1], g, h, −f[1]) is. TR3 Given a solid diagram f g h X / Y / Z / X[1] a b a[1] 0 0 f g h0 X0 / Y 0 / Z0 / X0[1] whose rows are distinguished triangles and which satisfies b ◦ f = f 0 ◦ a, there exists a morphism c : Z → Z0 such that (a, b, c) is a morphism of triangles. TR4 Given objects X, Y , Z of D, and morphisms f : X → Y , g : Y → Z, and distinguished triangles (X, Y, Q1, f, p1, d1), (X, Z, Q2, g ◦ f, p2, d2), and (Y, Z, Q3, g, p3, d3), there exist morphisms a : Q1 → Q2 and b : Q2 → Q3 such that (a) (Q1,Q2,Q3, a, b, p1[1] ◦ d3) is a distinguished triangle, (b) the triple (idX , g, a) is a morphism of triangles (X, Y, Q1, f, p1, d1) → (X, Z, Q2, g ◦ f, p2, d2), and (c) the triple (f, idZ , b) is a morphism of triangles (X, Z, Q2, g◦f, p2, d2) → (Y, Z, Q3, g, p3, d3). We will call (D, [], T ) a pre-triangulated category if TR1, TR2 and TR3 hold.1 The explanation of TR4 is that if you think of Q1 as Y/X, Q2 as Z/X and Q3 as Z/Y , then TR4(a) expresses the isomorphism (Z/X)/(Y/X) =∼ Z/Y and TR4(b) and TR4(c) express that we can compare the triangles X → Y → Q1 → X[1] etc with morphisms of triangles. For a more precise reformulation of this idea see the proof of Lemma 10.2. The sign in TR2 means that if (X,Y,Z,f,g,h) is a distinguished triangle then in the long sequence −h[−1] f g −f[1] −g[1] (3.2.1)05QL ... → Z[−1] −−−−→ X −→ Y −→ Z −→h X[1] −−−→ Y [1] −−−→ Z[1] → ... each four term sequence gives a distinguished triangle. As usual we abuse notation and we simply speak of a (pre-)triangulated category D without explicitly introducing notation for the additional data. The notion of a pre-triangulated category is useful in finding statements equivalent to TR4. We have the following definition of a triangulated functor. 014VDefinition 3.3. Let D, D0 be pre-triangulated categories. An exact functor, or a triangulated functor from D to D0 is a functor F : D → D0 together with given func- torial isomorphisms ξX : F (X[1]) → F (X)[1] such that for every distinguished tri- angle (X,Y,Z,f,g,h) of D the triangle (F (X),F (Y ),F (Z),F (f),F (g), ξX ◦ F (h)) is a distinguished triangle of D0. 1 We use [] as an abbreviation for the family {[n]}n∈Z. DERIVED CATEGORIES 4 An exact functor is additive, see Lemma 4.17. When we say two triangulated categories are equivalent we mean that they are equivalent in the 2-category of triangulated categories. A 2-morphism a :(F, ξ) → (F 0, ξ0) in this 2-category is simply a transformation of functors a : F → F 0 which is compatible with ξ and ξ0, i.e., F ◦ [1] / [1] ◦ F ξ a?1 1?a ξ0 F 0 ◦ [1] / [1] ◦ F 0 commutes. 05QMDefinition 3.4. Let (D, [], T ) be a pre-triangulated category. A pre-triangulated subcategory2 is a pair (D0, T 0) such that (1) D0 is an additive subcategory of D which is preserved under [1] and [−1], (2) T 0 ⊂ T is a subset such that for every (X,Y,Z,f,g,h) ∈ T 0 we have X, Y, Z ∈ Ob(D0) and f, g, h ∈ Arrows(D0), and (3) (D0, [], T 0) is a pre-triangulated category. If D is a triangulated category, then we say (D0, T 0) is a triangulated subcategory if it is a pre-triangulated subcategory and (D0, [], T 0) is a triangulated category. 0 In this situation the inclusion functor D → D is an exact functor with ξX : X[1] → X[1] given by the identity on X[1]. We will see in Lemma 4.1 that for a distinguished triangle (X,Y,Z,f,g,h) in a pre- triangulated category the composition g ◦ f : X → Z is zero. Thus the sequence (3.2.1) is a complex. A homological functor is one that turns this complex into a long exact sequence. 0147Definition 3.5. Let D be a pre-triangulated category. Let A be an abelian cate- gory. An additive functor H : D → A is called homological if for every distinguished triangle (X,Y,Z,f,g,h) the sequence H(X) → H(Y ) → H(Z) is exact in the abelian category A. An additive functor H : Dopp → A is called cohomological if the corresponding functor D → Aopp is homological. If H : D → A is a homological functor we often write Hn(X) = H(X[n]) so that H(X) = H0(X). Our discussion of TR2 above implies that a distinguished triangle (X,Y,Z,f,g,h) determines a long exact sequence (3.5.1) H(h[−1]) H(f) H(g) H(h) 0148 H−1(Z) / H0(X) / H0(Y ) / H0(Z) / H1(X) This will be called the long exact sequence associated to the distinguished triangle and the homological functor. As indicated we will not use any signs for the mor- phisms in the long exact sequence. This has the side effect that maps in the long exact sequence associated to the rotation (TR2) of a distinguished triangle differ from the maps in the sequence above by some signs.
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