DERIVED CATEGORIES WITH APPLICATIONS TO REPRESENTATIONS OF ALGEBRAS

JUN-ICHI MIYACHI

Contents

1. Categories and Functors 1 2. Additive Categories and Abelian Categories 5 3. Krull-Schmidt Categories 11 4. Triangulated Categories 14 5. Frobenius Categories 19 6. Homotopy Categories 24 7. Quotient Categories 29 8. Quotient Categories of Triangulated Categories 33 9. Épaisse Subcategories 35 10. Derived Categories 40 11. Homotopy Limits 46 12. Derived Functors 52 12.1. Derived Functors 52 12.2. Way-out Functors 55 13. Double Complexes 56 14. Derived Functors of Bi-∂-functors 62 15. Bimodule Complexes 70 16. Tilting Complexes 74 17. Two-sided Tilting Complexes 78 17.1. The Case of Flat k-algebras 78 17.2. The Case of Projective k-algebras 81 17.3. The Case of Finite Dimensional k-algebras 84 18. Cotilting Bimodule Complexes 87 References 90 Index 91

1. Categories and Functors

Definition 1.1 (Category). We define a category C by the following data: 1. A class Ob C of elements called objects of C.

2. For a ordered pair (X, Y) of objects a set Hom_{C}(X, Y) of morphisms is given such that Hom_{C}(X, Y)∩Hom_{C}(X′, Y′) = ϕ for (X, Y) ≠ (X′, Y′) (an element f of Hom_{C}(X, Y) is called a morphism, and denote by f : X → Y).

Date: June 2000.

This is a seminar note of which I gave a lecture at Chiba University in June 2000.

3. For each triple (X, Y, Z) of objects of C a map

θ(X, Y, Z) : Hom_{C}(X, Y) × Hom_{C}(Y, Z) → Hom_{C}(X, Z)

(θ is called the composition map) is given.

4. The composition map θ is associative.

5. For each object X of C, there is a morphism 1_{X} : X → X such that for any g : Y → X, h : X → Z we have 1_{X}g = g, h1_{X} = h.

X ∈ Ob C (often X ∈ C) means that X is an object of C.

Example 1.2. The following often appear in this note.

1. 𝔖𝔢𝔱 is the category consisting of sets as objects and maps as morphisms.

2. 𝔄𝔟 is the category consisting of abelian groups as objects and group morphisms as morphisms.

3. For a ring A, Mod A is the category consisting of right A-modules as objects and A-homomorphisms as morphisms.

Definition 1.3 (Opposite Category). For a category C, the opposite category C^{op} of C is defined by

1. Ob C^{op} = Ob C.

(for X ∈ C, we denote by X^{op} ∈ C^{op} the same object)

2. For X^{op}, Y^{op} ∈ Ob C^{op},

Hom_{C^{op}}(X^{op}, Y^{op}) = Hom_{C}(Y, X).

(for f ∈ Hom_{C}(Y, X), we denote by f^{op} ∈ Hom_{C^{op}}(X^{op}, Y^{op}))

3. The composition map θ^{op} is defined by θ^{op}(f^{op}, g^{op}) = θ(g, f)^{op}.

Definition 1.4. Let f : X → Y be a morphism in a category C.

1. f is called a monomorphism if fu = fv implies u = v.

2. f is called an epimorphism if uf = vf implies u = v.

3. f is called a split monomorphism if there is g : Y → X such that gf = 1_{X}.

4. f is called a split epimorphism if there is g : Y → X such that fg = 1_{Y}.

5. f is called an isomorphism if there is g : Y → X such that gf = 1_{X} and fg = 1_{Y}.

We often write ↠ for an epimorphism, and ↣ for a monomorphism.

Definition 1.5 (Functor). For categories C and C′, a covariant functor (resp., contravariant functor) F : C → C′ consists of the following data:

1. A map F : Ob C → Ob C′.

2. For X, Y ∈ Ob C, a map

F_{X,Y} : Hom_{C}(X, Y) → Hom_{C′}(F X, F Y)

(resp., F_{X,Y} : Hom_{C}(X, Y) → Hom_{C′}(F Y, F X))

such that F (gf) = F (g)F (f) (resp., F (gf) = F (f)F (g)), F (1_{X}) = 1_{F (X)}. Here we write simply F (f) instead of F_{X,Y} (f).

Example 1.6. In a category C, for X ∈ C, we define the covariant (resp., contravariant) functor

\begin{matrix}h^{X} : C → 𝔖𝔢𝔱 \\ (resp., h_{X} : C → 𝔖𝔢𝔱) \end{matrix}

by h^{X}(Y) = Hom_{C}(X, Y) (resp., h_{X}(Y) = Hom_{C}(Y, X)).

Definition 1.7 (Functorial Morphism). For covariant (resp., contravariant) functors F, G : C → C′, a functorial morphism α : F → G consists of the following data:

1. For each X ∈ C, α_{X} : F X → GX in C′ is given.

2. For any morphism f : X → Y (resp., f : Y → X) in C, we have the following commutative diagram in C′

F X ⟶^{α_{X}} GX
F (f)↓  ↓^{G(f)}
F Y ⟶^{α_{Y}} GY.

In the case that a functorial morphism α is called a functorial isomorphism if α_{X} are isomorphisms for all X ∈ C.

We denote by Mor(F, G) the collection of all functorial morphisms from F to G.

Lemma 1.8 (Representable Functor). For a covariant (resp., contravariant) functor F : C → 𝔖𝔢𝔱, the following are equivalent for C ∈ C.

1. F is isomorphic to h^{C} (resp., h_{C}).

2. There exists c ∈ F (C) satisfying that for any X ∈ C and x ∈ F (X), there is a unique f ∈ Hom_{C}(C, X) (resp., f ∈ Hom_{C}(X, C)) such that x = F (f)(c).

A covariant (resp., contravariant) functor F : C → 𝔖𝔢𝔱 is called representable if there exists C ∈ C such that F is isomorphic to h^{C}. Similarly, a contravariant functor F′ : C → 𝔖𝔢𝔱 is called representable if there exists C ∈ C such that F is isomorphic to h_{C}.

Lemma 1.9 (Yoneda’s Lemma). For X ∈ C and a covariant (resp., contravariant) functor F : C → 𝔖𝔢𝔱, we have the bijection

θ_{−} : F X → Mor(h^{X}, F) (resp., θ_{−} : F X → Mor(h_{X}, F)),

where θ_{−} is defined by (θ_{x})_{Y} (f) = F (f)(x) for x ∈ F X, Y ∈ C, f ∈ h^{X}(Y) (resp., f ∈ h_{X}(Y)).

Corollary 1.10. For X, Y ∈ C, we have the bijection

h^{−} : Hom_{C}(Y, X) → Mor(h^{X}, h^{Y}) (resp., h_{−} : Hom_{C}(X, Y) → Mor(h_{X}, h_{Y})).

Definition 1.11. Let F : C → C′ be a functor.

1. F is called full if F_{X,Y} : Hom_{C}(X, Y) → Hom_{C}(F X, F Y) are surjective for all X, Y ∈ C.

2. F is called faithful if F_{X,Y} : Hom_{C}(X, Y) → Hom_{C}(F X, F Y) are injective for all X, Y ∈ C.

3. F is called dense if for any Y ∈ C′, there is X ∈ C such that Y is isomorphic to F X.

Definition 1.12 (Limit, Colimit). Let I, C be categories and X ∈ C. We denote by X_{I} : I → C the constant functor such that X_{I}(i) = X for all i ∈ I and X_{I}(f) = 1_{X} for all f ∈ Hom_{I}(i, j). For a functor F : I → C, an object X of C is called the colimit colim F (resp., the limit lim F) of F provided that for all Y ∈ C we have

Mor(F, Y_{I}) ≅ Hom_{C}(X, Y)

(resp., Mor(Y_{I}, F) ≅ Hom_{C}(Y, X)).

Definition 1.13 (Filtered Colimit). A small category I is called a filtered category provided that

1. For any i, j ∈ I, there exists k ∈ I and morphisms i → k, j → k in I.

2. For two morphisms f, g : i → j, there is a morphism h : j → k such that hf = hg.

For a covariant (resp., contravariant) functor F : I → C from a filtered category I to a category C, the filtered colimit \underrightarrow{lim} F (resp., the filtered limit \underleftarrow{lim} F) of F is the colimit colim F (resp., the limit lim F).

Definition 1.14 (Product, Coproduct). For a collection {X_{i}}_{i∈I} of objects indexed by a set I, X is called a coproduct ∐_{i∈I}X_{i} (resp., a product ∏_{i∈I}X_{i}) of {X_{i}}_{i∈I} provided that

1. There are a collection of morphisms {q_{i} : X_{i} → X}_{i∈I} (resp., {p_{i} : X → X_{i}}_{i∈I}).

2. For any Y ∈ C and {f_{i} : X_{i} → Y }_{i∈I} (resp., {p_{i} : Y → X_{i}}_{i∈I}), there exists a unique morphism f : X → Y (resp., f : Y → X) with f_{i} = fq_{i} (resp., f_{i} = p_{i}f) for all i.

If a coproduct ∐_{i∈I}X_{i} is also a product, then it is called a biproduct of {X_{i}}_{i∈I} and denoted by ⨁_{i∈I}X_{i}.

Definition 1.15 (Bifunctor). Let C_{1}, C_{2} and D be categories. The product category C_{1} × C_{2} is the category consisting of pairs (X_{1}, X_{2}) of objects X_{1} ∈ Ob C_{1} and X_{2} ∈ Ob C_{2} as objects, and pairs (f_{1}, f_{2}) of morphisms f_{1} in C_{1} and f_{2} in C_{2} as morphisms. A bifunctor is the functor F : C_{1} × C_{2} → D.

For bifunctors F, G : C_{1} × C_{2} → D, a bifunctorial morphism α : F → G is a functorial morphism of functors C_{1} × C_{2} → D.

Then a bifunctor F : C_{1} × C_{2} → D consists of the following data:

1. For X_{1} ∈ C_{1}, F (X_{1}, −) : C_{2} → D is a functor.

2. For X_{2} ∈ C_{2}, F (−, X_{2}) : C_{1} → D is a functor.

3. For a morphism f_{1} : X_{1} → Y_{1} in C_{1}, F (f_{1}, −) : F (X_{1}, −) → F (Y_{1}, −) is a functorial morphism. (or equivalently, for a morphism f_{2} : X_{2} → Y_{2} in C_{1}, F (−, f_{2}) : F (−, X_{2}) → F (−, X_{2}) is a functorial morphism.)

And a bifunctorial morphism α : F → G consists of the following data:

1. For each (X_{1}, X_{2}) ∈ C_{1} ×C_{2}, α_{(X_{1},X_{2})} : F (X_{1}, X_{2}) → G(X_{1}, X_{2}) in D is given.

2. For X_{1} ∈ C_{1}, α_{(X_{1},−)} : F (X_{1}, −) → G(X_{1}, −) is a functorial morphism.

3. For X_{2} ∈ C_{2}, α_{(−,X_{2})} : F (−, X_{2}) → G(−, X_{2}) is a functorial morphism. In the case that a bifunctorial morphism α is called a bifunctorial isomorphism if α_{(X_{1},X_{2})} are isomorphisms for all (X_{1}, X_{2}) ∈ C_{1} × C_{2}.

Definition 1.16 (Adjoint). Let F : C → C′, G : C′ → C be covariant functors. We say that F is a left adjoint of G (or G is a right adjoint of F) (denote by F ⊣ G) if there is a bifunctorial isomorphism t(−,?) : Hom_{C′}(F −,?) → Hom_{C}(−, G?). In this case, let σ_{X} = t(X, F X)(1_{F X}) and τ_{Y} = t(GY, Y)^{−1}(1_{GY}) for X ∈ C, Y ∈ C′. (σ : 1_{C} → GF and τ : F G → 1_{C′} are called the adjunction arrows.)

For contravariant functors F′ : C → C′, G′ : C′ → C, a pair (F′, G′) is called a right adjoint pair if there is a bifunctorial isomorphism t′(−,?) : Hom_{C′}(−, F?) → Hom_{C}(?, G′−). There are adjunction arrows σ : 1_{C} → G′F′ and τ : 1_{C′} → F′G′.

Remark 1.17. According to Corollary 1.10, it is easy to see that a right (resp., left) adjoint is uniquely determined up to isomorphism. And in the above, we have

Gτ∘σG = 1_{G} and τF ∘F σ = 1_{F} .

Theorem 1.18. For a covariant functor F : C → C′, the following hold.

1. F has a right adjoint if and only if h_{Y} ∘ F : C → 𝔖𝔢𝔱 is representable for any Y ∈ C′.

2. F has a left adjoint if and only if h^{X} ∘ F : C → 𝔖𝔢𝔱 is representable for any X ∈ C′.

Theorem 1.19. Let F : C → C′, G : C′ → C be covariant functors such that F ⊣ G. Then the following are equivalent.

1. G is fully faithful.

2. The adjunction arrow τ : F G → 1_{C′} is a functorial isomorphism.

Sketch.

\xymatrix{Hom_{C}(X, Y) \ar[d]_{G_{X,Y}} \ar[dr]^{Hom(τ_{X},Y)} \\ Hom_{C}(GX, GY) \ar[r]^{∼} & Hom_{C}(F GX, Y)}

Theorem 1.20 (Equivalence). For a functor F : C → C′, the following are equivalent.

1. F is fully faithful and dense.

2. There is a functor G : C′ → C such that GF ≅ 1_{C} and F G ≅ 1_{C′}. In this case, F is called an equivalence and we say that C and C′ are equivalent.

Theorem 1.21. Let F : C → C′ be a covariant functor and let G be a right adjoint of F. Then the following hold.

1. F preserves the colimit in C of any functor.

2. G preserves the limit in C′ of any functor.

2. Additive Categories and Abelian Categories

In a category C, an object U is called an initial object if for any X ∈ C Hom_{C}(U, X) has only one element, V is called a terminal object if for any X ∈ C Hom_{C}(X, V) has only one element, and O is called a null object if O is initial and terminal.

Definition 2.1 (Preadditive Category). A category C with a null object is called a preadditive category provided that Hom_{C}(X, Y) is an abelian group for any X, Y ∈ C, and that the composition map θ is bilinear.

Definition 2.2 (Additive functor). Let C, C′ be preadditive categories. A covariant (resp., contravariant) functor F : C → C′ between preadditive categories is called an additive functor provided that for X, Y ∈ C,

F_{X,Y} : Hom_{C}(X, Y) → Hom_{C′}(F X, F Y)
(resp., F_{X,Y} : Hom_{C}(X, Y) → Hom_{C′}(F Y, F X))

is a group morphism.

Proposition 2.3. Let {X_{i}}_{1≤i≤n} be a finite collection of objects of a preadditive category C. Then the following are equivalent.

1. A coproduct ∐_{i=1}^{n}X_{i} of {X_{i}}_{1≤i≤n} exists in C.

2. A product ∏_{i=1}^{n}X_{i} of {X_{i}}_{1≤i≤n} exists in C.

3. There exist an object X ∈ C and morphisms u_{i} : X_{i} → X, p_{i} : X → X_{i} (1 ≤ i ≤ n) such that

\begin{matrix}(a) & Σ_{i=1}^{n}u_{i}p_{i} = 1_{X}. \\ (b) & p_{i}u_{j} = \begin{cases}0 if i ≠ j \\ 1_{X_{i}} if i = j. \end{cases} \end{matrix}

Moreover, the above coproduct is naturally isomorphic to the above product.

Proposition 2.4. Let F : C → C′ be an additive functor between preadditive categories, {X_{i}}_{1≤i≤n} a finite collection of objects in C. If the coproduct ∐_{i=1}^{n}X_{i} exists in C, then the coproduct ∐_{i=1}^{n}F (X_{i}) exists in C′ and is canonically isomorphic to F (∐_{i=1}^{n}X_{i}).

Definition 2.5 (Additive Category). A preadditive category C is called an additive functor if C satisfies Proposition 2.3 for any finite collection of objects in C.

Example 2.6. Let C be an additive category. For M ∈ C, We define Add M (resp., add M) the full subcategory of C consisting of objects which are direct summands of coproducts (resp., finite coproducts) of copies of M. Then Add M (resp., add M) is an additive category.

Proposition 2.7 (Compact Object). For an object C of an additive category C, the following are equivalent.

1. For any morphism f : C → ∐_{i∈I} X_{i}, there exists a factorization

C ⟶^{f′} ∐_{j∈F} X_{j} ⟶^{μ_{F}} ∐_{i∈I} X_{i}

where F is a finite subset of I and μ_{F} is the canonical inclusion.

2. For any morphism f : C → ∐_{i∈I} X_{i}, there exists a finite subset F of I such that f = ∑_{j∈F} u_{j}p_{j}f where u_{j} are the structural morphisms and p_{j} are the canonical projections.

3. The functor h^{C} : C → 𝔄𝔟 preserves coproducts.

An object C ∈ C is called a compact object (often called a small object) of C if C satisfies the above conditions.

Exercise 2.8. Show that if a right A-module C is finitely generated, then C is a compact object in Mod A.

Corollary 2.9. Let C be a compact object of an additive category C, and B = End_{C}(C). The following hold.

1. For any object X ∈ C,we have isomorphisms

Hom_{C}(C^{(I)}, X) →^{∼} Hom_{B}(Hom_{C}(C, C^{(I)}), Hom_{C}(C, X))

→^{∼} Hom_{B}(Hom_{C}(C, C)^{(I)}, Hom_{C}(C, X))

if a coproduct C^{(I)} exists for a set I.

2. we have isomorphisms

Hom_{C}(C^{(I)}, C^{(J)}) →^{∼} Hom_{B}(Hom_{C}(C, C^{(I)}), Hom_{C}(C, C^{(J)}))

→^{∼} Hom_{B}(Hom_{C}(C, C)^{(I)}, Hom_{C}(C, C)^{(J)})

if coproducts C^{(I)}, C^{(J)} exist for sets I, J.

Proposition 2.10. Let C be an additive category, C′ a preadditive category and F : C → C′ a functor. If F preserves finite coproducts, then F is an additive functor.

Definition 2.11 (Special Morphisms). Let C be a preadditive category. For f : X → Y, g : X → Z and h : W → Y, we define the following.

1. (Cok f, cok f) = colim (X ⟶^{f} Y, X ⟶^{0} Y).

2. (Ker f, ker f) = lim (X ⟶^{f} Y, X ⟶^{0} Y).

3. (Im f, im f) = Ker(Y ⟶^{0} Cok f, cok f).

4. (Coim f, im f) = Cok(Ker f ⟶^{0} X, ker f).

5. PushOut(f, g) = colim (X ⟶^{f} Y, X ⟶^{g} Z).

6. PullBack(f, h) = lim (X ⟶^{f} Y, W ⟶^{h} Y).

Proposition 2.12. Let C be a preadditive category and let f : X → Y be a morphism in C such that there exist Ker f, Cok f, Coim f and Im f in C. Then there exists a unique morphism \overline{f} : Coim f → Im f such that we have the following commutative diagram

X ⟶^{f} Y
coim f↓  ↑^{im f}
Coim f ⟶^{\overline{f}} Im f

Definition 2.13 (Abelian Category). An additive category C is called an abelian category provided that

1. For any morphism f, there exist Ker f and Cok f in C.

2. For any morphism f, the above morphism \overline{f} is an isomorphism.

Definition 2.14. In an abelian category C, we consider the following sequence

… → X^{i−1} ⟶^{f^{i−1}} X^{i} ⟶^{f^{i}} X^{i+1} → ….

We say that the above sequence is exact at X^{i} if Ker f^{i} = Im f^{i−1}. If the above sequence is exact at each X^{i}, then we say that the above sequence is exact.

In the rest of this section, we deal with internal properties of an abelian category C.

Proposition 2.15 (Snake Lemma). Suppose that the following diagram is commutative

X ⟶^{f} Y ⟶^{g} Z ⟶ O
x↓  ↓^{y}  ↓^{z}
O ⟶ X′ ⟶^{f′} Y′ ⟶^{g′} Z′

where all rows are exact. Then we have the following induced exact sequence

Ker x → Ker y → Ker z → Cok x → Cok y → Cok z.

Moreover, f (resp., g′) is monic (resp., epic) if and only if so is Ker x → Ker y (resp., Cok y → Cok z).

Proposition 2.16 (Five Lemma). Suppose that the following diagram is commutative

X_{1} ⟶ X_{2} ⟶ X_{3} ⟶ X_{4} ⟶ X_{5}
↓^{f_{1}}  ↓^{f_{2}}  ↓^{f_{3}}  ↓^{f_{4}}  ↓^{f_{5}}
X_{1} ⟶ X_{2} ⟶ X_{3} ⟶ X_{4} ⟶ X_{5}

where all rows are exact. Then the following hold.

1. If f_{1} is epic, and f_{2}, f_{4} are monic, then f_{3} is monic.

2. If f_{5} is monic, and f_{2}, f_{4} are epic, then f_{3} is epic.

3. If f_{1} is epic, f_{5} is monic, and f_{2}, f_{4} are isomorphisms, then f_{3} is an isomorphism.

Proposition 2.17 (Pull Back 1). Suppose that the following diagram is commutative

\xymatrix{X \ar[d]^{(A)}_{x} \ar[r]^{f} & Y \ar[d]^{y} \\ X′ \ar[r]^{f′} & Y′}

Then the following hold.

1. The square (A) is pull back if and only if O → X \underrightarrow{[ \begin{matrix}f \\ x \end{matrix}]} Y ⊕X′ ⟶^{[ −y f′ ]} Y′ is exact.

2. The square (A) is push out if and only if X \underrightarrow{[ \begin{matrix}f \\ x \end{matrix}]} Y ⊕X′ ⟶^{[ −y f′ ]} Y′ → O is exact.

Proposition 2.18 (Pull Back 2). Suppose that the following diagram is commutative

\xymatrix{X \ar[d]^{(A)} \ar[r] & Y \ar[d]^{(B)} \ar[r] & Z \ar[d] \\ X′ \ar[r] & Y′ \ar[r] & Z′}

If the squares (A) and (B) are push out (resp., pull back), then so is the square (A) + (B).

Proposition 2.19 (Pull Back 3). Suppose that the following diagram is pull back (resp., push out)

\xymatrix{X \ar[d]_{x} \ar[r]^{f} & Y \ar[d]^{y} \\ X′ \ar[r]^{f′} & Y′}

Then the following hold.

1. If f′ (resp., f) is epic (resp., monic), then the above diagram is also push out (resp., pull back), and f (resp., f′) is also epic (resp., monic).

2. The induced morphism Ker f → Ker f′ is an isomorphism (resp., an epimorphism).

3. The induced morphism Cok f → Cok f′ is a monomorphism (resp., an isomorphism).

Proposition 2.20 (Exact Sequence 1). Suppose that the following diagram is commutative

O ⟶ X ⟶ Y ⟶ Z ⟶ O
↓  ↓  ∥
O ⟶ X′ ⟶ Y′ ⟶ Z ⟶ O

where all rows are exact. Then the square

\xymatrix{X \ar[d]^{EX} \ar[r] & Y \ar[d] \\ X′ \ar[r] & Y′}

is pull back and push out (this is called an exact square).

Proposition 2.21 (Exact Sequence 2). Suppose that the following diagram is commutative

O ⟶ X ⟶^{f} Y ⟶^{g} Z ⟶ O
↓^{x}  ↓^{y}  ∥
O ⟶ X′ ⟶^{f′} Y′ ⟶^{g′} Z ⟶ O

where all rows are exact. Then we have the following commutative diagram

O ⟶ X ⟶^{α} Y ⊕X′ ⟶^{β} Y′ ⟶ O
∥  ↓^{γ}  ↓^{g′}
O ⟶ X ⟶^{f} Y ⟶^{−g} Z ⟶ O

where α = [ \begin{matrix}f \\ x \end{matrix}], β = [^{−y} ^{f′}], γ = [^{1} ^{0} ], where all rows are exact.

Proposition 2.22 (Exact Sequence 3). Suppose that the following diagram is commutative

[figure: X_{1}; Y_{1}; Z_{1}; X_{2}; Y_{2}; Z_{2}; X_{3}; Y_{3}; Z_{3}; X_{4}; Y_{4}; Z_{4}]

where all rows are short exact sequences. If two of squares

\begin{matrix}X_{1} & ⟶ & X_{2} \\ ↓ & & ↓ \\ X_{3} & ⟶ & X_{4} \end{matrix} \begin{matrix}Y_{1} & ⟶ & Y_{2} \\ ↓ & & ↓ \\ Y_{3} & ⟶ & Y_{4} \end{matrix} \begin{matrix}Z_{1} & ⟶ & Z_{2} \\ ↓ & & ↓ \\ Z_{3} & ⟶ & Z_{4} \end{matrix}

are exact, then the rest is also exact.

Hint. Consider the following commutative diagram

O ⟶ X_{1} ⟶ Y_{1} ⟶ Z_{1} ⟶ O
↓  ↓  ↓
O ⟶ X_{1}⊕X_{2} ⟶ Y_{3}⊕Y_{2} ⟶ Z_{2}⊕Z_{1} ⟶ O
↓  ↓  ↓
O ⟶ X_{2} ⟶ Y_{4} ⟶ Z_{2} ⟶ O.

Exercise 2.23. The following hold.

1. For a sequence X ⟶^{f} Y ⟶^{g} Z, if for any M ∈ A,

0 → Hom_{A}(Z, M) ⟶^{Hom_{A}(g,M)} Hom_{A}(Y, M) ⟶^{Hom_{A}(f,M)} Hom_{A}(X, M) → 0

is exact, then

O → X ⟶^{f} Y ⟶^{g} Z → O

is split exact.

2. For a sequence X ⟶^{f} Y ⟶^{g} Z, if for any M ∈ A,

0 → Hom_{A}(M, X) ⟶^{Hom_{A}(M,f)} Hom_{A}(M, Y) ⟶^{Hom_{A}(M,g)} Hom_{A}(M, Z) → 0

is exact, then

O → X ⟶^{f} Y ⟶^{g} Z → O

is split exact.

Definition 2.24 (Abn Categories). We define the conditions of an abelian category C. (Ab3) We say that an abelian category C satisfies the condition Ab3 (resp., Ab3*)

if C has coproducts (resp., products) of objects indexed by arbitrary sets. (Ab4) We say that an abelian category C satisfies the condition Ab4 (resp., Ab4*)

provided that C satisfies the condition Ab3 (resp., Ab3*), and that the coproduct (resp., product) of monics (resp., epics) is monic (resp., epic).

(Ab5) We say that an abelian category C satisfies the condition Ab5 (resp., Ab5*) provided that C satisfies the condition Ab3 (resp., Ab3*), and that the filtered colimit (resp., filtered limit) of exact sequences is exact.

Proposition 2.25. The following hold.

1. In a category satisfying Ab3* and Ab5, any ∐ → ∏ is monic.

2. Ab5 ⇒ Ab4.

3. An abelian category C satisfies the condition Ab5 if and only if C satisfies the condition Ab3, and for a collection {X_{i}} of subobjects of an object X, we have

∑_{i}(X_{i} ∩ X′) = (∑_{i}X_{i}) ∩ X′

for any subobject X′ of X.

Example 2.26. For a ring A, Mod A satisfies the conditions Ab4*, Ab5.

3. Krull-Schmidt Categories

Let R be a ring with unity and let J(R) be the Jacobson radical of R. We call R a semiperfect ring if (i) R/ J(R) is a semi-simple Artinian ring, and (ii) any idempotent of R/ J(R) can be lifted to an idempotent of R.

Lemma 3.1 (Semiperfect Rings 1). The following hold.

1. A ring R is semiperfect if and only if R has a complete set of orthogonal primitive idempotents e_{i} (1 ≤ i ≤ n) such that each e_{i}Re_{i} is a local ring.

2. A ring R is semiperfect if and only if every finitely generated R-module has a projective cover.

Lemma 3.2 (Semiperfect Rings 2). Let R be a semiperfect ring and let e_{i} (1 ≤ i ≤ n) be a complete set of orthogonal primitive idempotents.

1. If f_{i} (1 ≤ i ≤ m) is another complete set of orthogonal primitive idempotents, then m = n and there is a permutation π such that Rf_{i} ≅ Re_{π(i)} for all i.

2. If f is an idempotent of R, then there are a permutation π and an integer t (1 ≤ t ≤ n) such that Rf ≅ ⨁_{i=1}^{t}Re_{π(i)} and R(1 − f) ≅ ⨁_{i=t+1}^{n}Re_{π(i)}.

3. If I is a two-sided ideal of R, then R/I is also semiperfect.

Proposition 3.3. Let C be an additive category, and let X ∈ C, B = End_{C}(X). If X′ is a direct summand of a finite coproduct of copies of X, we have

Hom_{C}(X′, Y) →^{∼} Hom_{B}(Hom_{C}(X, X′), Hom_{C}(X, Y)) (f ↦ Hom_{C}(X, f))

for all Y ∈ C.

Proof. There are q_{i} : X′ → X and p_{i} : X → X′ (1 ≤ i ≤ n) such that ∑_{i=1}^{n}p_{i}q_{i} = 1_{X′}. Let ϕ ∈ Hom_{B}(Hom_{C}(X, X′), Hom_{C}(X, Y)), for any g ∈ Hom_{C}(X, X′), we have

ϕ(g) = ϕ (∑_{i=1}^{n}p_{i}q_{i}g)
= ∑_{i=1}^{n}ϕ(p_{i})q_{i}g
= Hom_{C} (X, ∑_{i=1}^{n}ϕ(p_{i})q_{i}) (g).

Then Hom_{C}(X, −) is surjective. Let f ∈ Hom_{C}(X′, Y) such that Hom_{C}(X, f) = 0. Then fp_{i} = 0 for all i, and hence f = f∑_{i=1}^{n}p_{i}q_{i} = ∑_{i=1}^{n}fp_{i}q_{i} = 0.

Definition 3.4. Let C be an additive category. An object X of C is called indecomposable if X ≅ X_{1} ⊕ X_{2} implies X_{1} = O or X_{2} = O.

Definition 3.5 (Pre-Krull-Schmidt Category). An additive category C is called a pre-Krull-Schmidt category provided that End_{C}(X) is a semiperfect ring for each X ∈ C.

Proposition 3.6. Let C be a pre-Krull-Schmidt category. For any X ∈ C, there are indecomposable objects X_{i} (1 ≤ i ≤ n) such that

X ≅ ⨁_{i=1}^{n}X_{i}.

Proof. Given X ∈ C, since End_{C}(X) is a semiperfect ring, there is a natural number n_{X} such that End_{C}(X) has a complete set of orthogonal primitive idempotents e_{i} (1 ≤ i ≤ n_{X}). If X is not indecomposable, then we have a decomposition X = X_{1}⊕X_{2} with X_{i} ≠ O (i = 1, 2). By Lemma 3.2,2, Proposition 3.3, we have n_{X_{i}} < n_{X} (i = 1, 2). We get the statement by induction on n_{X}.

Proposition 3.7. Let C be a pre-Krull-Schmidt category. Then the following are equivalent.

1. For any object X ∈ C, X is indecomposable if and only if End_{C}(X) is a local ring.

2. For any object X ∈ C, for any e^{2} = e ∈ End_{C}(X) there exist Y ∈ C and q : Y → X, p : X → Y such that qp = e and pq = 1_{Y} (i.e. any idempotent of End_{C}(X) splits).

Proof. 1 ⇒ 2. For any object X ∈ C, by Proposition 3.6, we have X ≅ ⨁_{i=1}^{n}X_{i}, where X_{i} are indecomposable objects (1 ≤ i ≤ n). Then the compositions of natural morphisms X → X_{i} → X form a complete set of orthogonal primitive idempotents of End_{C}(X). By Lemma 3.2, 2, we get the statement 2.

2 ⇒ 1. Since End_{C}(X) is semiperfect, it is trivial.

Definition 3.8 (Krull-Schmidt Category). We call a pre-Krull-Schmidt category C a Krull-Schmidt category if C satisfies the equivalent conditions of Proposition 3.7.

Theorem 3.9 (Krull-Schmidt Theorem). Let C be a Krull-Schmidt category. For any X ∈ C, X is isomorphic to ⨁_{i=1}^{n}X_{i}, where X_{i} are indecomposable objects. Moreover, this decomposition is unique up to isomorphism (this is called a K-S decomposition).

Proof. By Propositions 3.6, 3.7, X ∈ C has a K-S decomposition ⨁_{i=1}^{n}X_{i}. Lemma 3.2 and Proposition 3.3 imply uniqueness of this decomposition.

Example 3.10. We denote by mod A the category of finitely presented right Amodules. Let R be a commutative complete local ring, A a finite R-algebra. Then mod A is a Krull-Schmidt category.

Definition 3.11 (Stable Category). Let C be an additive category, I an additive full subcategory of C. For X, Y ∈ C, let I(X, Y) be the subgroup of Hom_{C}(X, Y) generated by morphisms which factor through some object of I. We define the category \underline{C}_{I} as follows.

1. Ob \underline{C}_{I} = Ob C.

2. Hom_{\underline{C}_{I}}(X, Y) = Hom_{C}(X, Y)/I(X, Y). This category is called the stable category of C by I.

Remark 3.12. For X ∈ C, If X is a direct summand of some object of I, then X ≅ O in \underline{C}_{I}.

Theorem 3.13. Let C be an additive category, I an additive full subcategory of C. If C is a Krull-Schmidt category, then so is \underline{C}_{I}.

Proof. By Proposition 2.3, \underline{C}_{I} is clearly an additive category. For an indecomposable object X ∈ \underline{C}_{I}, we have a K-S-decomposition X = ⨁_{i=1}^{n}X_{i} in C. Let e_{i} : X ⟶^{p_{i}} X_{i} ⟶^{q_{i}} X be the canonical morphism in C (1 ≤ i ≤ n), and \underline{e}_{i} be the image of e_{i} in ∈ \underline{C}_{I}. If the number of e_{i} such that \underline{e}_{i} ≠ 0 is greater than 1, then by Lemma 3.2 this contradicts indecomposability of X. Thus we may assume that \underline{e}_{1} ≠ 0 and \underline{e}_{i} = 0 for i ≥ 2, and then q_{i}p_{i} factors through V_{i} ∈ I for i ≥ 2. Then X_{i} is a direct summand of V_{i}. Therefore, by Lemma 3.2,3, End_{\underline{C}_{I}}(X) ≅ End_{\underline{C}_{I}}(X_{1}) is a local ring. We complete the proof by Proposition 3.7.

Example 3.14. Let R be a commutative complete local ring, A a finite R-algebra, and proj A the full subcategory of mod A consisting of finitely generated projective right A-modules. Then the stable category \underline{mod}A of mod A by proj A is a Krull-Schmidt category.

4. Triangulated Categories

Throughout this section, unless otherwise stated, functors are covariant functors.

Definition 4.1. A triangulated category C is an additive category together with (1) an auto-equivalence T : C →^{∼} C, called the translation, and (2) a collection T of sextuples (X, Y, Z, u, v, w), called triangle (distinguished triangle) . These data are subject to the following four axioms: (TR1) (1) Every sextuple (X, Y, Z, u, v, w) which is isomorphic to a triangle is a

triangle.

(2) Every morphism u : X → Y is embedded in a triangle (X, Y, Z, u, v, w).

(3) The triangle (X, X, O, 1_{X}, 0, 0) is a triangle for all X ∈ C. (TR2) A triangle (X, Y, Z, u, v, w) is a triangle if and only if (Y, Z, T X, v, w, −T u) is

a triangle. (TR3) For any triangles (X, Y, Z, u, v, w), (X′, Y′, Z′, u′, v′, w′) and morphisms f :

X → X′, g : Y → Y′ with gu = u′f, there exists h : Z → Z′ such that (f, g, h) is a homomorphism of triangles.

(TR4) (Octahedral axiom) For any two consecutive morphisms u : X → Y and v : Y → Z, if we embed u, vu and v in triangles (X, Y, Z′, u, i, i′), (X, Z, Y′, vu, k, k′) and (Y, Z, X′, v, j, j′), respectively, then there exist morphisms f : Z′ → Y′, g : Y′ → X′ such that the following diagram commute

X ⟶^{u} Y ⟶^{i} Z′ ⟶^{i′} T X
∥  ↓^{v}  ↓^{f}  ∥
X ⟶^{vu} Z ⟶^{k} Y′ ⟶^{k′} T X
↓^{j}  ↓^{g}  ↓^{T u}
X′  X′ ⟶^{j′} T Y
↓^{j′}  ↓^{(T i)j′}
T Y ⟶^{T i} T Z′

and the third column is a triangle. Sometimes, we write X[i] for T^{i}(X).

Definition 4.2 (∂-functor). Let C, C′ be triangulated categories. An additive functor F : C → C′ is called ∂-functor (sometimes exact functor) provided that there is a functorial isomorphism α : F T_{C} →^{∼} T_{C′}F such that (F X, F Y, F Z, F (u), F (v), α_{X}F (w)) is a triangle in C′ whenever (X, Y, Z, u, v, w)is a triangle in C. Moreover, if a ∂- functor F is an equivalence, then we say that C is triangle equivalent to C′, and

denote by C ≅^{t} C′

For (F, α), (G, β) : C → C′ ∂-functors, a functorial morphism ϕ : F → G is called a ∂-functorial morphism if (T_{C′}ϕ)α = βϕT_{C}.

We denote by ∂(C, C′) the collection of all ∂-functors from C to C′, and denote by ∂ Mor(F, G) the collection of ∂-functorial morphisms from F to G.

Definition 4.3. Given a triangulated category C with a translation T_{C}, we define the opposite triangulated category C^{op} the following

1. T_{C^{op}}(X^{op}) = T_{C}^{−1}(X).

2. X^{op} → Y^{op} → Z^{op} → T_{C^{op}}X^{op} is a distinguished triangle if T_{C}^{−1}X → Z → Y → X is a distinguished triangle in C.

Definition 4.4. A covariant additive functor H : C → C′ from a triangulated category to an abelian category is called a covariant cohomological functor, if whenever (X, Y, Z, u, v, w) is a triangle in C, the long sequence

… → H(T^{i}(X)) ⟶^{H(T^{i}(u))} H(T^{i}(Y)) ⟶^{H(T^{i}(v))} H(T^{i}(Z)) ⟶^{H(T^{i}(w))} H(T^{i+1}(X)) → …

is exact. If H is a cohomological functor, then we often write H^{i}(X) for H(T^{i}(X)), i ∈ ℤ. One defines a contravariant cohomological functor by reversing the arrows.

In this section, we deal with internal properties of a triangulated category C.

Proposition 4.5. The following hold.

1. If (X, Y, Z, u, v, w) is a triangle, then vu = 0, wv = 0 and T (u)w = 0.

2. For any W ∈ C, Hom_{C}(W, −) : C → 𝔄𝔟 (resp., Hom_{C}(−, W) : C → 𝔄𝔟) is a covariant (resp., contravariant) cohomological functor.

3. For any homomorphism of triangles (f, g, h) : (X, Y, Z, u, v, w)

→ (X′, Y′, Z′, u′, v′, w′), if two of f, g and h are isomorphisms, then the rest is also an isomorphism.

Proof. 1. According to (TR2), it suffices to show vu = 0. By (TR2) and (TR3) we have a commutative diagram

X ⟶^{1_{X}} X ⟶ O ⟶ T X
∥  ↓^{u}  ↓  ∥
X ⟶^{u} Y ⟶^{v} Z ⟶^{w} T X.

2. Let (X, Y, Z, u, v, w) be a triangle. Then, since by 1, vu = 0, we have Hom_{C}(W, v) ∘ Hom_{C}(W, u) = 0. Conversely, let g ∈ Hom_{C}(W, Y) such that Hom_{C}(W, v)(g) = vg = 0. Then by (TR3) there exists f ∈ Hom_{C}(W, Y) which makes the following diagram commutes

W ⟶^{1_{W}} W ⟶ O ⟶ T W
f↓  ↓^{g}  ↓  ↓^{T f}
X ⟶^{u} Y ⟶^{v} Z ⟶^{w} T X.

Thus g = Hom_{C}(W, u)(f) and the sequence Hom_{C}(W, X) → Hom_{C}(W, Y) → Hom_{C}(W, Z) is exact. It follows by (TR2) that Hom_{C}(W, −) is a cohomological functor.

3. According to (TR2), it is enough to deal with the case that f, g are isomorphisms. By 2 we have a commutative diagram with exact rows

Hom_{C}(T Y′, −) → Hom_{C}(T X′, −) →Hom_{C}(Z′, −) → Hom_{C}(Y′, −) →Hom_{C}(X′, −)
↓ Hom_{C}(T g,−) ↓ Hom_{C}(T f,−) ↓ Hom_{C}(h,−) ↓ Hom_{C}(g,−) ↓ Hom_{C}(f,−)
Hom_{C}(T Y, −) → Hom_{C}(T X, −) → Hom_{C}(Z, −) → Hom_{C}(Y, −) → Hom_{C}(X, −).

Thus, since by 5 lemma Hom_{C}(h, −) is an isomorphism, it follows by Yoneda lemma that h is an isomorphism.

Proposition 4.6. Let F : C → C′ be a ∂-functor between triangulated categories. If G : C′ → C is a right (resp., left) adjoint of F, then G is also a ∂-functor.

Proof. For X ∈ C′, we have a functorial isomorphisms

Hom_{C}(−, GT X) ≅ Hom_{C′}(F −, T X)
≅ Hom_{C′}(T^{−1}F −, X)
≅ Hom_{C′}(F T^{−1}−, X)
≅ Hom_{C}(T^{−1}−, GX)
≅ Hom_{C}(−, T GX).

Then we have a functorial isomorphism β : GT_{C′} →^{∼} T_{C}G. For a triangle (X, Y, Z, u, v, w) of C′, let (GX, GY, Z′, Gu, v′, w′) be a triangle of C. Since there is a morphism of triangles

\xymatrix{F GX \ar[d] \ar[r] & F GY \ar[d] \ar[r] & F Z′ \ar[d] \ar[r] & F T GX \ar[d] \ar[r]^{αG} & T F GX \ar[dl] \\ X \ar[r] & Y \ar[r] & Z \ar[r] & T X.}

Then we have a commutative diagram

[figure: GX; GY; Z′; T GX; β^{−1}; GX; GY; GZ; GT X; β; T GX.]

Since Hom_{C}(M, G−) ≅ Hom_{C′}(F M, −), (GX, GY, GZ, Gu, Gv, Gw) induces a long exact sequence. We apply Hom_{C}(M, −) to the above diagram, then by 5 lemma, we have an isomorphism from (GX, GY, Z′, Gu, v′, w′) to (GX, GY, GZ, Gu, Gv, β_{X}Gw).

Proposition 4.7. The following hold.

1. If ∐_{i∈I}X_{i} (resp., ∏_{i∈I}X_{i}) exists in C for {X_{i}}_{i∈I}, then there is an isomorphism

\begin{matrix}α : ∐_{i∈I}T X_{i} →^{∼} T ∐_{i∈I}X_{i} \\ (resp., β : ∏_{i∈I}T X_{i} →^{∼} T ∏_{i∈I}X_{i}). \end{matrix}

2. For a collection of triangles (X_{i}, Y_{i}, Z_{i}, u_{i}, v_{i}, w_{i}) (i ∈ I), if ∐_{i∈I}X_{i}, ∐_{i∈I}Y_{i}, ∐_{i∈I}Y_{i} (resp., ∏_{i∈I}X_{i}, ∏_{i∈I}Y_{i}, ∏_{i∈I}Y_{i}) exist in C, then

\begin{matrix}(∐_{i∈I}X_{i}, ∐_{i∈I}Y_{i}, ∐_{i∈I}Z_{i}, ∐_{i∈I}u_{i}, ∐_{i∈I}v_{i}, α∐_{i∈I}w_{i}) \\ (resp., (∏_{i∈I}X_{i}, ∏_{i∈I}Y_{i}, ∏_{i∈I}Z_{i}, ∏_{i∈I}u_{i}, ∏_{i∈I}v_{i}, β∏_{i∈I}w_{i})) \end{matrix}

is a triangle.

Proof. 1. We have isomorphisms

Hom_{C}(T ∐_{i∈I}X_{i}, −) ≅ Hom_{C}(∐_{i∈I}X_{i}, T^{−1}−)
≅ ∏_{i∈I} Hom_{C}(X_{i}, T^{−1}−)
≅ ∏_{i∈I} Hom_{C}(T X_{i}, −)
≅ Hom_{C}(∐_{i∈I}T X_{i}, −).

2. Let (∐_{i∈I}X_{i}, ∐_{i∈I}Y_{i}, Z′, ∐_{i∈I}u_{i}, v, w) be a triangle. Then we have a commutative diagram

∐_{i∈I}X_{i} \underrightarrow{∐_{i∈I}u_{i}} ∐_{i∈I}Y_{i} \underrightarrow{∐_{i∈I}v_{i}} ∐_{i∈I}Z_{i} \underrightarrow{α∐_{i∈I}v_{i}} T ∐_{i∈I}X_{i}
∥  ∥  ↓  ∥
∐_{i∈I}X_{i} \underrightarrow{∐_{i∈I}u_{i}} ∐_{i∈I}Y_{i} ⟶^{v} Z′  ⟶^{w} T ∐_{i∈I}X_{i}.

Applying Hom_{C}(M, −) to the above, by 5 lemma, we complete the proof.

Proposition 4.8. The following hold.

1. A triangle (X, Y, Z, u, v, 0) is isomorphic to (X, Z⊕X, Z, [ \begin{matrix}0 \\ 1 \end{matrix} ] , [^{1} ^{0} ] , 0).

2. For a morphism of triangles

X ⟶^{u} Y ⟶^{v} Z ⟶^{w} T X
∥  ^{f}↓  ↓^{g}  ∥
X′ ⟶^{u′} Y′ ⟶^{v′} Z′ ⟶^{w′} T X

there exists g′ : Z → Z′ such that

Y \underrightarrow{[ \begin{matrix}v \\ f \end{matrix}]} Z⊕Y′ ⟶^{[ −g′ v′ ]} Z′ ⟶^{(T u)w′} T Y

is a triangle.

Proof. 1. Since Hom_{C}(Z, Z) ⟶^{0} Hom_{C}(Z, T X), by Proposition 4.5, there is s : Z → Y such that vs = 1_{Z}. Then we have a commutative diagram

X ⟶^{μ} Z⊕X ⟶^{π} Z ⟶^{0} T X
∥  ^{α}↓  ∥  ∥
X ⟶^{u} Y ⟶^{v} Z ⟶^{0} T X

where μ = [ \begin{matrix}0 \\ 1 \end{matrix} ], π = [^{1} ^{0} ], α = [^{s u}].

2. Since T^{−1}Z′ ⟶^{−T^{−1}w′} X ⟶^{u′} Y′ ⟶^{v′} Z′ is triangle, we have a commutative diagram

T^{−1}Z′ ⟶^{−T^{−1}w′} X ⟶^{u′} Y′ ⟶^{v′} Z′
∥  ↓^{u}  ↓^{x}  ∥
T^{−1}Z′ ⟶^{−uT^{−1}w′} Y ⟶^{α} M ⟶^{β} Z′
↓^{v}  ↓^{y}  ↓^{−w′}
Z  Z ⟶^{w} T X
↓^{w}  ↓^{0}
T X ⟶^{T u′} T Y′

By 1, Y′ ⟶^{x} M ⟶^{y} Z ⟶^{0} T Y′ is isomorphic to Y′ ⟶^{μ} Z⊕Y′ ⟶^{π} Z ⟶^{0} T Y′. Then we

have a commutative diagram

T^{−1}Z′ ⟶^{−T^{−1}w′} X ⟶^{u′} Y′ ⟶^{v′} Z′
∥  ↓^{u}  ↓^{μ}  ∥
T^{−1}Z′ ⟶^{−uT^{−1}w′} Y ⟶^{α} Z⊕Y′ ⟶^{β} Z′
↓^{v}  ↓^{π}  ↓^{−w′}
Z  Z ⟶^{w} T X
↓^{w}  ↓^{0}
T X ⟶^{T u′} T Y′

where μ = [ \begin{matrix}0 \\ 1 \end{matrix} ], π = [^{1} ^{0} ], α = [ \begin{matrix}v \\ f′ \end{matrix} ], β = [^{−g′′} ^{v′}], and v′f′ = g′′v, f′u = u′,

w = w′g′. Since (f′ − f)u = 0, there is h : Z → Y′ such that f′ = f + hv. Hence we have a commutative diagram

Y ⟶^{α′} Z⊕Y′ ⟶^{β′} Z ⟶^{(T u)w′} T Y
∥  ^{ϕ}↓  ∥  ∥
Y ⟶^{α} Z⊕Y′ ⟶^{β} Z ⟶^{(T u)w′} T Y

where α′ = [ \begin{matrix}v \\ f \end{matrix}], β′ = [^{−g′} ^{v′}], ϕ = [ \begin{matrix}1 & 0 \\ h & 1 \end{matrix} ].

Proposition 4.9 (9 Lemma). Any commutative diagram in C

X′ ⟶^{u′} Y′
x′↓  ↓^{y′}
X ⟶^{u} Y

can be embedded in a diagram

\xymatrix{X′ \ar[d]^{x′} \ar[r]^{u′} & Y′ \ar[d]^{y′} \ar[r]^{v′} & Z′ \ar[d]^{z′} \ar[r]^{w′} & T X′ \ar[d]^{T x′} \\ X \ar[d]^{x} \ar[r]^{u} & Y \ar[d]^{y} \ar[r]^{v} & Z \ar[d]^{z} \ar[r]^{w} & T X \ar[d]^{T x} \\ X′′ \ar[d]^{x′′} \ar[r]^{u′′} & Y′′ \ar[d]^{y′′} \ar[r]^{v′′} & Z′′ \ar[d]^{−}^{z′′} \ar[r]^{w′′} & T X′′ \ar[d]^{−T x′′} \\ T X′ \ar[r]^{T u′} & T Y′ \ar[r]^{T v′} & T Z′ \ar[r]^{−T w′} & T^{2}X′} .

which is commutative without the right and bottom corner, - anti-commutative, where all rows and columns are triangles.

Proof. According to (TR4), we have three commutative diagrams

[figure: X′; u′; Y′; y′; v′; Z′; γ; w′; T X′; X′; y′u′; Y; y; α; A; δ; β; T X′; T u′; Y′′; y′′; Y′′; (T v′)y′′; y′′; T Y′; T Y′; T v′; T Z′]

[figure: X′; x′; X; u; x; X′′; ε; x′′; T X′; X′; ux′; Y; v; α; A; η; β; T X′; T x′; Z; w; Z; (T x)w; w; T X; T X; T x; T X′′]

[figure: X′′; ε; A; δ; η; Z; z; (T x)w; T X′′; X′′; u′′; Y′′; (T v′)y′′; v′′; Z′′; z′′; w′′; T X; T ε; T Z′; −T γ; T Z′; −T z′′; −T γ; T A; T A; T η; T Z]

In particular, we have u′′ = δε, v = ηα, y = δα, z′ = ηγ, zη = v′′δ and (T x′′)w′′ = (T β)(T ε)w′′ = −(T β)(T γ)z′′ = −(T w′)z′. Then it is easy to get the diagram.

5. Frobenius Categories

Definition 5.1 (Exact Category). Let C be an additive category which is embedded as a full subcategory of an abelian category A, and suppose that C is closed under extensions in A. Let S be a collection of exact sequences in A

O → X ⟶^{u} Y ⟶^{v} Z → O.

u is called an admissible monomorphism, and v is called an admissible epimorphism.

A pair (C, S) is called an exact category in the sense of Quillen provided that (EX1) Any split sequence of which all terms are in C is in S. (EX2) The composition of admissible monomorphisms (resp., epimorphisms) is also

an admissible monomorphism (resp., epimorphism).

(EX3) Given the following commutative diagram in A

O ⟶ X ⟶ Y ⟶ Z ⟶ O
↓  ↓  ∥
O ⟶ X′ ⟶ Y′ ⟶ Z ⟶ O

where all rows are exact, if the top row is in S and X′ ∈ C, then the bottom row is in S.

(EX4) Given the following commutative diagram in A

O ⟶ X ⟶ Y′ ⟶ Z′ ⟶ O
∥  ↓  ↓
O ⟶ X ⟶ Y ⟶ Z ⟶ O

where all rows are exact, if the bottom row is in S and Z′ ∈ C, then the top row is in S.

An object X in C is called S-projective (resp., S-injective) if for any admissible epimorphisms (resp., monomorphisms) v : Y → Z, Hom_{C}(X, v) (resp., Hom_{C}(v, X)) is surjective.

Definition 5.2 (Frobenius Category). An exact category (C, S) is called a Frobenius category if (C, S) has enough S-projectives and enough S-injectives and if S-projectives coincide with S-injectives.

Let Q be the full subcategory of C consisting of S-projective objects. A stable category \underline{C} is the category \underline{C}_{Q}.

Proposition 5.3. In a Frobenius category (C, S), we consider the following commutative diagram

O ⟶ X ⟶ I ⟶ X_{1} ⟶ O
f↓  ↓  ↓^{f′}
O ⟶ X′ ⟶ I′ ⟶ X′_{1} ⟶ O

where I, I′ are S-injective, with all rows in S. Then the image \underline{f}′ is uniquely determined by f in \underline{C}.

Remark 5.4. For all X ∈ C we choose the elements O → X ⟶^{μ_{X}} I(X) ⟶^{π_{X}} T X → O in S, with I(X) being S-injective. According to Proposition 5.3, an object T X is uniquely determined up to isomorphism in \underline{C} independently of choice of the above sequence, but \underline{f}′ is depend on their choice. Then we can understand the induced functor T : \underline{C} → \underline{C} only if we know O → X ⟶^{μ_{X}} I(X) ⟶^{π_{X}} T X → O in S for all X ∈ C.

Proposition 5.5. T is an auto-equivalence of \underline{C}.

Definition 5.6 (Triangle). In a Frobenius category (C, S), let u : X → Y be an morphism in C. By taking M(u) = PushOut(u, μ_{X}), we have the following commutative diagram in C

O ⟶ X ⟶^{μ_{X}} I(X) ⟶^{π_{X}} T X ⟶ O
u↓  ↓^{x}  ∥
O ⟶ Y ⟶^{v} M(u) ⟶^{w} T X ⟶ O

with all rows in S. Then in \underline{C} the sequence

X ⟶^{\underline{u}} Y ⟶^{\underline{v}} M(u) ⟶^{\underline{w}} T X

is called a standard triangle. Let T be a collection of sextuples which are isomorphic to standard triangles in \underline{C}.

Lemma 5.7. In a Frobenius category (C, S), let w : M(u) → T X be the morphism in Definition 5.6. Consider the following commutative diagram in C

O ⟶ X ⟶^{−u′} Y′ ⟶^{v′} M(u) ⟶ O
∥  _{x′}↓  ↓^{w}
O ⟶ X ⟶^{μ} I(X) ⟶^{π} T X ⟶ O

with all rows in S, then the sextuple X ⟶^{\underline{u′}} Y′ ⟶^{\underline{v′}} M(u) ⟶^{\underline{w}} T X is isomorphic to the above triangle X ⟶^{\underline{u}} Y ⟶^{\underline{v}} M(u) ⟶^{\underline{w}} T X in \underline{C}.

Proof. By Proposition 2.21, we have the following commutative diagram

O ⟶ X ⟶^{β} I(X) ⊕ Y ⟶^{γ} M(u) ⟶ O
∥  ↓^{δ}  ↓^{w}
O ⟶ X ⟶^{μ} I(X) ⟶^{−π} T X ⟶ O

where β = [ \begin{matrix}μ \\ u \end{matrix}], γ = [^{−x v}], δ = [^{1} ^{0} ], with all rows in S. Since the right and bottom rectangle in the previous diagram is pull back, there exists η : Y′ → I(X) ⊕ Y such that we have the following commutative diagram in \underline{C}

X ⟶^{\underline{u′}}  Y′  ⟶^{\underline{v′}} M(u) ⟶^{\underline{w}} T X
∥  ↓\underline{η}  ∥  ∥
X ⟶^{\underline{β}} I(X) ⊕ Y ⟶^{\underline{γ}} M(u) ⟶^{\underline{w}} T X
∥  ↓\underline{ε}  ∥  ∥
X ⟶^{\underline{u}}  Y  ⟶^{\underline{v}} M(u) ⟶^{\underline{w}} T X

where ε = [^{0} ^{1} ], all vertical arrows are isomorphisms in \underline{C}.

Proposition 5.8. In a Frobenius category (C, S), the image of any element O → X ⟶^{u} Y ⟶^{v} Z → O of S can be embedded in a triangle X ⟶^{\underline{u}} Y ⟶^{\underline{v}} Z ⟶^{\underline{w}} T X in \underline{C}.

Proof. Since I(X) is S-injective and O → X ⟶^{−u} Y ⟶^{v} Z → O ∈ S, we have a commutative diagram

O ⟶ X ⟶^{−u} Y ⟶^{v} Z ⟶ O
∥  ↓  ↓^{w}
O ⟶ X ⟶^{μ} I(X) ⟶^{π} T X ⟶ O.

By Lemma 5.7, we get the statement.

Theorem 5.9. Let (C, S) be a Frobenius category. Then (\underline{C}, T ) is a triangulated category.

Proof. we show that (C, S) satisfies the axioms of a triangulated category. (TR1) It is trivial. (TR2) Let (X, Y, Z, \underline{u}, \underline{v}, \underline{w}) be a standard triangle. Then we have a commutative

diagram

O ⟶ X ⟶^{μ_{X}} I(X) ⟶^{π_{X}} T X ⟶ O
↓^{u}  ↓^{x}  ∥
O ⟶ Y ⟶^{v}  Z  ⟶^{w} T X ⟶ O
↓^{μ_{Y}}  ↓^{α}  ∥
O ⟶ I(Y) ⟶^{β} I(Y)⊕T X ⟶^{γ} T X ⟶ O
↓^{π_{Y}}  ↓^{δ}
T Y  T Y

where α = [ \begin{matrix}s \\ w \end{matrix}], β = [ \begin{matrix}1 \\ 0 \end{matrix} ], γ = [^{0} ^{1} ], δ = [^{π_{Y}} ^{u′}], π_{Y} s + u′w = 0, and with all rows in S. Since sxμ_{X} = [^{1} ^{0} ] αxμ_{X} = [^{1} ^{0} ] βμ_{Y} u = μ_{Y} u, we have a commutative diagram

O ⟶ X ⟶^{μ_{X}} I(X) ⟶^{π_{X}} T X ⟶ O
↓^{u}  ↓^{sx}  ↓^{−u′}
O ⟶ Y ⟶^{μ_{Y}} I(Y) ⟶^{π_{Y}} T Y ⟶ O.

Then we have \underline{u}′ = −T \underline{u}, and we have a commutative diagram in \underline{C}

Y ⟶^{v} Z ⟶^{w} T X  ⟶^{u′} T Y
∥  ∥  ↓^{ϵ}  ∥
Y ⟶^{v} Z ⟶^{α} I(Y)⊕T X ⟶^{δ} T Y

where ε = [ \begin{matrix}0 \\ 1 \end{matrix} ] is an isomorphism in \underline{C}. Hence a sextuple (Y, Z, T X, \underline{v}, \underline{w}, −T \underline{u}) is a triangle . The reverse implication is similar by Lemma 5.7.

(TR3) Let (X_{i}, Y_{i}, Z_{i}, \underline{u}_{i}, \underline{v}_{i}, \underline{w}_{i}), be standard triangles (i = 1, 2), and

O ⟶ X_{i} ⟶^{μ_{X_{i}}} I(X_{i}) ⟶^{π_{X_{i}}} T X_{i} ⟶ O
u_{i}↓  ↓^{x_{i}}  ∥
O ⟶ Y_{i} ⟶^{v_{i}} Z_{i} ⟶^{w_{i}} T X_{i} ⟶ O

commutative diagrams with all rows in S. Let f : X_{1} → X_{2}, g : Y_{1} → Y_{2} be morphisms satisfying \underline{u}_{2}\underline{f} = \underline{g}\underline{u}_{1} in \underline{C}. Since I(X_{1}) is S-injective, there exists t : I(X_{1}) → Y_{2} such that gu_{1} − u_{2}f = tμ_{X_{1}}. Since I(X_{2}) is S-injective, we have a commutative diagram

X_{1} ⟶^{μ_{X_{1}}} I(X_{1})
f↓  ↓^{r}
X_{2} ⟶^{μ_{X_{2}}} I(X_{2}).

Then we have equations

\begin{aligned}x_{2}rμ_{X_{1}} & = x_{2}μ_{X_{2}}f \\ & = v_{2}u_{2}f \end{aligned}

\begin{aligned}v_{2}gu_{1} & = v_{2}u_{2}f + v_{2}tμ_{X_{1}} \\ & = (x_{2}r + v_{2}t)μ_{X_{1}} \end{aligned}

Since

X_{1} ⟶^{μ_{X_{1}}} I(X_{1})
u_{1}↓  ↓^{x_{1}}
Y_{1} ⟶^{v_{1}} Z_{1}

is push out, there exists h : M(u_{1}) → M(u_{2}) such that v_{2}g = hv_{1} and x_{2}r + v_{2}t = hx_{1}. Then there is f′ : T X_{1} → T X_{2} such that f′w_{1} = w_{2}h. Since

f′π_{X_{1}} = f′w_{1}x_{1} = w_{2}hx_{1} = w_{2}(x_{2}r + v_{2}t) = w_{2}x_{2}r = π_{X_{2}}r

we have \underline{f}′ = T \underline{f}, and hence a morphism (\underline{f}, \underline{g}, \underline{h}) from (X_{1}, Y_{1}, Z_{1}, \underline{u}_{1}, \underline{v}_{1}, \underline{w}_{1}) to (X_{2}, Y_{2}, Z_{2}, \underline{u}_{2}, \underline{v}_{2}, \underline{w}_{2}).

(TR4) Let (X, Y, Z′, \underline{u}, \underline{i}, \underline{i}′), (X, Z, Y′, \underline{vu}, \underline{k}, \underline{k}′) and (Y, Z, X′, \underline{v}, \underline{j}, \underline{j}′) be triangles in \underline{C}. We have a commutative diagram in C

[figure: O; X; EX; u; μ_{X}; I(X); x; π_{X}; T X; O; O; Y; EX; v; i; Z′; f; i′; T X; O; O; Z; k; Y′; k′; T X; O]

where all rows are in S. Since i is an admissible monomorphism, μ′_{Y} = μ_{Z′}i is also an admissible monomorphism and there is an admissible epimorphism π′_{Y} such that π_{Z′} = (T′i)π′_{Y}. Then we have a commutative diagram in C

[figure: X; EX; u; μ_{X}; I(X); x; π_{X}; T X; T′u; Y; EX; v; i; Z′; EX; f; μ_{Z′}; I(Z′); z′; _{π′_{Y}}; T′Y; T′i; T Z′; Z; k; Y′; g_{1}; X′′; j′_{1}; T′Y; T′i; T Z′]

Therefore, we have a triangle in \underline{C}

Z′ ⟶^{\underline{f}} Y′ ⟶^{\underline{g_{1}}} X′ ⟶^{\underline{(T′i)j′_{1}}} T Z′.

Since j′_{1}g_{1}fx = (T′u)π_{X} = (T′u)k′fx and j′_{1}g_{1}k = 0 = (T′u)k′k in C, and

X ⟶^{u} Y ⟶^{v} Z
μ_{X} ↓  ↓^{k}
I(X) ⟶^{x} Z′ ⟶^{f} Y′

is push out, we have j′_{1}g_{1} = (T′u)k′. Since I(Y) and I(Z′) are S-injective, there exist α : T Y → T′Y and β : X′ → X′′ such that

[figure: Y; I(Y); T Y; Z; X′; T Y; Y; I(Z′); T′Y; Z; X′′; T′Y]

is commutative in C, where \underline{α} is an isomorphism in \underline{C}. Then by Proposition 2.22, \underline{β} is an isomorphism in \underline{C}, and T \underline{i} = (\underline{T′i})\underline{α}, T \underline{u} = \underline{α}^{−1}\underline{T′u}. Let \underline{g} = \underline{β}^{−1}\underline{g_{1}}, \underline{j} = \underline{β}^{−1}\underline{g_{1}}\underline{k}, \underline{j}′ = \underline{α}^{−1}\underline{j′_{1}}\underline{β}, then we have the octahedral diagram of Definition 4.1.

Example 5.10. Let A be a self-injective algebra over a field k, and O → Ω → A⊗_{k}A ⟶^{μ} A → O an exact sequence, where μ is the multiplication map. Then mod A is a Frobenius category, its stable category \underline{mod}A is a triangulated category with a translation functor Hom_{A}(Ω, −).

6. Homotopy Categories

Throughout this section, A is an abelian category and B is an additive subcategory of A which is closed under isomorphisms.

Definition 6.1 (Complex). Let B be an additive category. A complex (cochain complex) is a collection X^{▪} = (X^{n}, d_{X}^{n} : X^{n} → X_{X}^{n+1})_{n∈ℤ} of objects and morphisms of B such that d_{X}^{n+1}d_{X}^{n} = 0. A complex X^{▪} = (X^{n}, d_{X}^{n} : X^{n} → X_{X}^{n+1})_{n∈ℤ} is called bounded below (resp., bounded above, bounded) if X^{n} = O for sufficiently small (resp., large, large and small) n.

A complex X^{▪} = (X^{n}, d_{X}^{n}) is called a stalk complex if there exists an integer n_{0} such that X^{i} = O if i ≠ n_{0}. We identify objects of B with a stalk complexes of degree 0.

A morphism f of complexes X^{▪} to Y^{▪} is a collection of morphisms f^{n} : X^{n} → Y^{n} which commute with the maps of complexes

f^{n+1}d_{X}^{n} = d_{Y}^{n} f^{n}.

We denote by C(B) (resp., C^{+}(B), C^{−}(B), C^{b}(B)) the category of complexes (resp., bounded below complexes, bounded above complexes, bounded complexes) of B. An auto-equivalence T : C(B) → C(B) is called translation if (T X^{▪})^{n} = X^{n+1} and (T d_{X})^{n} = −d_{X}^{n+1} for any complex X^{▪} = (X^{n}, d_{X}^{n}).

Proposition 6.2. The following hold.

1. C^{∗}(B) is an additive category, where ∗ = nothing, +, −, b. Moreover, if B has products (resp., coproducts), then C(B) has also products (resp., coproducts).

2. C^{∗}(A) is an abelian category, where ∗ = nothing, +, −, b. Moreover, if A satisfies the condition Ab3* (resp., Ab3), then C(A) also satisfies the condition Ab3* (resp., Ab3).

Definition 6.3. For u ∈ Hom_{C(B)}(X^{▪}, Y^{▪}), the mapping cone of u is a complex M^{▪}(u) with

\begin{aligned}M^{n}(u) & = X^{n+1}⊕Y^{n}, \\ d_{M^{▪}(u)}^{n} & = [ \begin{matrix}−d_{X}^{n+1} & 0 \\ u^{n+1} & d_{X}^{n} \end{matrix} ] : X^{n+1}⊕Y^{n} → X^{n+2}⊕Y^{n+1}. \end{aligned}

Moreover, for 1_{X} ∈ Hom_{C(B)}(X^{▪}, X^{▪}), let I^{▪}(X^{▪}) = M^{▪}(1_{X}).

Definition 6.4. Let S_{C^{∗}(B)} be the collection of exact sequences O → X^{▪} ⟶^{f} Y^{▪} ⟶^{g} Z^{▪} → O of complexes of C^{∗}(B) such that

O → X^{n} ⟶^{f^{n}} Y^{n} ⟶^{g^{n}} Z^{n} → O

are split exact for all n ∈ ℤ, where ∗ = nothing, +, −, b. In this case, we call f (resp., g) a term-split monomorphism (resp., a term-split epimorphism).

Proposition 6.5. For u ∈ Hom_{C(B)}(X^{▪}, Y^{▪}), we have an exact sequence in C(A)

O → Y^{▪} ⟶^{μ_{u}} M^{▪}(u) ⟶^{π_{u}} T X^{▪} → O

where μ_{u} = [ \begin{matrix}0 \\ 1 \end{matrix} ], π_{u} = [^{1} ^{0} ]. Moreover, the above sequence belongs to S_{C(B)}.

Lemma 6.6. For X^{▪} ∈ C(B), we have

I^{▪}(X^{▪}) ≅ (X^{n+1}⊕X^{n}, [ \begin{matrix}0 & 0 \\ 1_{X^{n+1}} & 0 \end{matrix} ] : X^{n+1}⊕X^{n} → X^{n+2}⊕X^{n+1}).

Proof.

X^{n+1}⊕X^{n} ⟶^{d_{M^{▪}(1_{X})}^{n}} X^{n+2}⊕X^{n+1}
α^{n}↓  ↓_{α^{n+1}}
X^{n+1}⊕X^{n} ⟶^{δ^{n}} X^{n+2}⊕X^{n+1}

where α^{n} = [ \begin{matrix}1_{X^{n+1}} & d_{X}^{n} \\ 0 & 1_{X^{n}} \end{matrix} ] and δ^{n} = [ \begin{matrix}0 & 0 \\ 1_{X^{n+1}} & 0 \end{matrix} ].

Lemma 6.7. The category (C(B), S_{C(B)}) is an exact category.

Proposition 6.8. The category (C^{∗}(B), S_{C^{∗}(B)}) is a Frobenius category, where ∗ = nothing, +, −, b.

Proof. Let X^{▪} ∈ C(B), then by Lemma 6.6 we have

I^{▪}(X^{▪}) ≅ ⨁_{n∈ℤ}I^{▪}(X^{n})[−n].

where X^{n} is a stalk complex of degree 0 (Note that the above biproduct exists by Exercise 6.18). It is easy to see that I^{▪}(X^{n})[−n] is S_{C(B)}-projective and S_{C(B)}-injective, and then I^{▪}(X^{▪}) is S_{C(B)}-projective and S_{C(B)}-injective. For any S_{C(B)}-injective complex Y^{▪}, by Proposition 6.5, Y^{▪} is a direct summand of I^{▪}(Y^{▪}), and hence Y^{▪} is S_{C(B)}-projective. Similarly, any S_{C(B)}-projective complex is S_{C(B)}-injective. According to Proposition 6.5, it is easy to see that C(B) has enough S_{C(B)}-injectives and enough S_{C(B)}-projectives.

Definition 6.9 (Homotopy Category). A homotopy category K^{∗}(B) of B is the stable category of (C^{∗}(B), S_{C^{∗}(B)}) by the full subcategory I_{C^{∗}(B)} of S_{C^{∗}(B)}-injective objects, where ∗ = nothing, +, −, b.

Remark 6.10. For u ∈ Hom_{C(B)}(X^{▪}, Y^{▪}), we have a commutative diagram

O ⟶ X^{▪} ⟶^{μ_{X}} I^{▪}(X^{▪}) ⟶^{π_{X}} T X^{▪} ⟶ O
u↓  ↓^{x}  ∥
O ⟶ Y^{▪} ⟶^{μ_{u}} M^{▪}(u) ⟶^{π_{u}} T X^{▪} ⟶ O

where x = [ \begin{matrix}1 & 0 \\ 0 & u \end{matrix}], with all rows in S_{C(B)}. By the proof of Proposition 6.8, the definition of M^{▪}(u) coincides with the one of Definition 5.6.

Proposition 6.11. A category K^{∗}(B) is a triangulated category, where ∗ = nothing, +, −, b.

Proposition 6.12. If an exact sequence O → X^{▪} ⟶^{u} Y^{▪} ⟶^{v} Z^{▪} → O in C^{∗}(B) belongs to S_{C^{∗}(B)}, then it can be embedded in a triangle X^{▪} ⟶^{\underline{u}} Y^{▪} ⟶^{\underline{v}} Z^{▪} ⟶^{\underline{w}} T X^{▪} in K^{∗}(B), where ∗ = nothing, +, −, b.

Proof. By Proposition 5.8.

Definition 6.13 (Homotopy Relation). Two morphisms f, g ∈ Hom_{C(B)}(X^{▪}, Y^{▪}) is said to be homotopic (denote by f ≃_{h} g) if there is a collection of morphisms

h = (h^{n}), h^{n} : X^{n} → Y^{n−1} such that

f^{n} − g^{n} = d_{Y}^{n−1}h^{n} + h^{n+1}d_{X}^{n}

for all n ∈ ℤ. For X^{▪}, Y^{▪} ∈ C^{∗}(B), Htp_{C(B)}(X^{▪}, Y^{▪}) is the subgroup of Hom_{C(B)}(X^{▪}, Y^{▪}) consisting of morphisms which are homotopic to 0.

Proposition 6.14. For a morphism f ∈ Hom_{C(B)}(X^{▪}, Y^{▪}) (∗ = nothing, +, −, b), the following are equivalent.

1. f ∈ Htp_{C(B)}(X^{▪}, Y^{▪}).

2. f factors through X^{▪} ⟶^{μ_{X}} I^{▪}(X^{▪}).

3. f factors through I^{▪}(T^{−1}Y^{▪}) ⟶^{π_{T^{−1}Y}} Y^{▪}.

4. f ∈ I_{C(B)}(X^{▪}, Y^{▪}). In particular, Htp_{C(B)}(X^{▪}, Y^{▪}) = I_{C(B)}(X^{▪}, Y^{▪}).

Proof. 1 ⇔ 2. Let h = (h^{n}) be a homotopy morphism f ≃_{h} 0, and let ϕ = (ϕ^{n}) :

X^{▪} → Y^{▪}, ϕ^{n} = [^{h^{n+1}} ^{f^{n}}]. Then for all n ∈ ℤ we have

ϕ^{n}d_{M^{▪}(1_{X})}^{n} = [^{h^{n+1}} ^{f^{n}}] [ \begin{matrix}−d_{X}^{n} & 0 \\ 1_{X^{n}} & d_{X}^{n−1} \end{matrix} ]
= [−h^{n+1}d_{X}^{n} + f^{n} f^{n}d_{X}^{n−1}]
= [d_{Y}^{n} h^{n} d_{Y}^{n−1}f^{n−1}]
= d_{Y}^{n−1}ϕ^{n−1},

\begin{aligned}ϕ^{n}μ_{X}^{n} & = [^{h^{n+1}} ^{f^{n}}] [ \begin{matrix}0 \\ 1_{X^{n}} \end{matrix} ] \\ & = f^{n}. \end{aligned}

Conversely, let ϕ = (ϕ^{n}) : X^{▪} → Y^{▪} be a morphism such that f = ϕμ_{X^{▪}}. By the same calculation in the above, there is a homotopy morphism h = (h^{n}), h^{n} : X^{n} → Y^{n+1} such that

f^{n} = d_{Y}^{n−1}h^{n} + h^{n+1}d_{X}^{n}

for all n ∈ ℤ. Thus f ∈ Htp_{C(B)}(X^{▪}, Y^{▪}).

2 ⇔ 4. Since I^{▪}(X^{▪}) is I_{C(B)}-injective, 2 ⇒ 4 is trivial. Conversely, since μ_{X^{▪}} : X^{▪} → I^{▪}(X^{▪}) is an admissible monomorphism, it is also trivial.

3 ⇔ 4. The same as 2 ⇔ 4.

Corollary 6.15. The canonical functor C(B) → K(B) preserves products and coproducts. In particular, we have

1. If B has products (resp., coproducts), then so has K(B).

2. If A satisfies the condition Ab3* (resp., Ab3), then K(A) has products (resp., coproducts) .

Proof. By Proposition 6.14, we have exact sequences for X^{▪}, Y^{▪} ∈ C(B)

Hom_{C(B)}(I^{▪}(X^{▪}), Y^{▪}) → Hom_{C(B)}(X^{▪}, Y^{▪}) → Hom_{K(B)}(X^{▪}, Y^{▪}) → 0,
Hom_{C(B)}(X^{▪}, I^{▪}(T^{−1}Y^{▪})) → Hom_{C(B)}(X^{▪}, Y^{▪}) → Hom_{K(B)}(X^{▪}, Y^{▪}) → 0.

Then it is easy.

Corollary 6.16. Let B′ be another additive category, and F : C(B) → C(B′) an additive functor. If F satisfies the conditions

(a) there exists an functorial isomorphism α : F T_{C(B)} → T_{C(B′)}F,

(b) for any morphism u : X^{▪} → Y^{▪} in C(B), we have a commutative diagram

F X^{▪} ⟶^{F u} F Y^{▪} ⟶^{F μ_{u}} F M^{▪}(u) ⟶^{α_{X}F π_{u}} T_{C(B′)}F X^{▪}
∥  ∥  ^{s}↓  ∥
F X^{▪} ⟶^{F u} F Y^{▪} ⟶^{μ_{F u}} M^{▪}(F u) ⟶^{π_{F u}} T_{C(B′)}F X^{▪},

then F induces a ∂-functor F′ : K(B) → K(B′).

Remark 6.17. By the proof of Proposition 6.14, X^{▪} belongs to I_{C(B)} if and only if X^{▪} is a direct summand of I^{▪}(X^{▪}). Hence it is easy to see that any object of I_{C(B)} is isomorphic to I^{▪}(Z^{▪}) for some Z^{▪} ∈ C(B).

Exercise 6.18 (Biproduct). Let n ≥ 0, and let X_{i}^{▪} : … → X_{i}^{n−1} → X_{i}^{n} be complexes of C^{−}(B) indexed by i ∈ ℕ. Prove the following.

1. ∐_{i∈ℕ}T^{i}X_{i}^{▪} ≅ ∏_{i∈ℕ}T^{i}X_{i}^{▪} in C(B). Thus ⨁_{i∈ℕ}T^{i}X_{i}^{▪} exists in C(B).

2. ∐_{i∈ℕ}T^{i}X_{i}^{▪} ≅ ∏_{i∈ℕ}T^{i}X_{i}^{▪} in K(B). Thus ⨁_{i∈ℕ}T^{i}X_{i}^{▪} exists in K(B).

Let n ≥ 0, and let Y_{i}^{▪} : Y_{i}^{0} → Y_{i}^{1} → … → Y_{i}^{n} be complexes of C^{b}(B) indexed by i ∈ ℤ. Prove the following.

1. ∐_{i∈ℤ}T^{i}Y_{i}^{▪} ≅ ∏_{i∈ℤ}T^{i}Y_{i}^{▪} in C(B). Thus ⨁_{i∈ℤ}T^{i}Y_{i}^{▪} exists in C(B).

2. ∐_{i∈ℤ}T^{i}Y_{i}^{▪} ≅ ∏_{i∈I}T^{i}Y_{i}^{▪} in K(B). Thus ⨁_{i∈ℤ}T^{i}Y_{i}^{▪} exists in K(B).

Proposition 6.19. Let R be a commutative complete local ring, A a finite Ralgebra, and B is a Krull-Schmidt subcategory of mod A. Then K^{b}(B) is also a Krull-Schmidt category.

Proof. Let X^{▪} ∈ C^{b}(B). Since we may assume that X^{▪} = X^{0} → X^{1} → … → X^{n}, End_{C^{b}(B)}(X^{▪}) is a subring of ∏_{i=0}^{n} End_{A}(X^{i}), and hence End_{C^{b}(B)}(X^{▪}) is semiperfect. It is clear that any idempotent of End_{C^{b}(B)}(X^{▪}) splits. Therefore C^{b}(B) is a Krull-Schmidt category. According to Theorem 3.13, we complete the proof.

Definition 6.20. Let A be an abelian category. For a complex X^{▪} = (X^{n}, d_{X}^{n} : X^{n} → X_{X}^{n+1})_{n∈ℤ} of A, we define an objects of A for all n ∈ ℤ

Z^{n}(X^{▪}) = Ker d_{X}^{n}
B^{n}(X^{▪}) = Im d_{X}^{n−1}
C^{n}(X^{▪}) = Cok d_{X}^{n−1}
H^{n}(X^{▪}) = Z^{n}(X^{▪})/ B^{n}(X^{▪})
this is called nth cohomology,

and define complexes

Z^{▪}(X^{▪}) = (Z^{n}(X^{▪}), 0)_{n∈ℤ}
B^{▪}(X^{▪}) = (B^{n}(X^{▪}), 0)_{n∈ℤ}
C^{▪}(X^{▪}) = (C^{n}(X^{▪}), 0)_{n∈ℤ}
H^{▪}(X^{▪}) = (H^{n}(X^{▪}), 0)_{n∈ℤ}.

A complex X^{▪} = (X^{n}, d_{X}^{n}) is called an acyclic complex if H^{n}(X^{▪}) = 0 for all n ∈ ℤ.

Remark 6.21. Since we have a commutative diagram

O ⟶ B^{n}(X^{▪}) ⟶ X^{n} ⟶ C^{n}(X^{▪}) ⟶ O
↓  ∥  ↓
O ⟶ Z^{n}(X^{▪}) ⟶ X^{n} ⟶ B^{n+1}(X^{▪}) ⟶ O,

where all rows are exact, by snake lemma, we have a short exact sequence

O → H^{n}(X^{▪}) → C^{n}(X^{▪}) → B^{n+1}(X^{▪}) → O.

Exercise 6.22. Let A be an abelian category, and P a projective object of A. For X^{▪} ∈ C(A), show that

Hom_{K(A)}(P, X^{▪}[i]) ≅ Hom_{A}(P, H^{i}(X^{▪}))

for all i.

Proposition 6.23. Let A be an abelian category, and let O → X^{▪} → Y^{▪} → Z^{▪} → O be exact in C(A). Then we have the induced long exact sequence

… → H^{n}(X^{▪}) → H^{n}(Y^{▪}) → H^{n}(Z^{▪}) → H^{n+1}(X^{▪}) → …

Proof. According to snake lemma, we have a commutative diagram

C^{n}(X^{▪}) ⟶ C^{n}(Y^{▪}) ⟶ C^{n}(Z^{▪}) ⟶ O
↓  ↓  ↓
O ⟶ Z^{n+1}(X^{▪}) ⟶ Z^{n+1}(Y^{▪}) ⟶ Z^{n+1}(Z^{▪})

where all rows are exact. Then we get the exact sequence

H^{n}(X^{▪}) → H^{n}(Y^{▪}) → H^{n}(Z^{▪}) → H^{n+1}(X^{▪}) → H^{n+1}(Y^{▪}) → H^{n+1}(Z^{▪}),

by snake lemma.

Remark 6.24. Let u be a morphism of Hom_{C(A)}(X^{▪}, Y^{▪}). According to Proposition 6.5, we have an exact sequence in C(A)

O → Y^{▪} ⟶^{μ_{u}} M^{▪}(u) ⟶^{π_{u}} T X^{▪} → O.

Then we have the induced long exact sequence

… → H^{n}(X^{▪}) ⟶^{δ^{n}} H^{n}(Y^{▪}) → H^{n}(M^{▪}(u)) → H^{n+1}(X^{▪}) ⟶^{δ^{n+1}} ….

Moreover, it is not hard to see that δ^{n} = H^{n}(u) for all n ∈ ℤ (cf. Proposition 10.4).

Lemma 6.25. For n ∈ ℤ, The functor H^{n} : C(A) → A factors through K(A).

Proof. According to Remark 6.17, all objects of I_{C(B)} are acyclic. Then by Proposition 6.14, it is trivial.

Proposition 6.26. If X^{▪} ⟶^{\underline{u}} Y^{▪} ⟶^{\underline{v}} Z^{▪} ⟶^{\underline{w}} T X^{▪} is a triangle in K(A), then we have the induced long exact sequence

… → H^{n}(X^{▪}) ⟶^{H^{n}(\underline{u})} H^{n}(Y^{▪}) ⟶^{H^{n}(\underline{v})} H^{n}(Z^{▪}) ⟶^{H^{n}(\underline{w})} H^{n+1}(X^{▪}) → ….

Proof. According to Remark 6.10, for a representative u we have a commutative diagram

O ⟶ X^{▪} ⟶ I^{▪}(X^{▪}) ⟶ T X^{▪} ⟶ O
u↓  ↓  ∥
O ⟶ Y^{▪} ⟶^{v} M^{▪}(u) ⟶^{w} T X^{▪} ⟶ O

where all rows belong to S_{C(A)}, with v = μ_{u}, w = π_{u}. By Proposition 6.23, we have the induced long exact sequence

… → H^{n}(X^{▪}) → H^{n}(Y^{▪}) → H^{n}(M^{▪}(u)) → H^{n+1}(X^{▪}) → ….

By Remark 6.24, Proposition 6.25, we get the statement.

7. Quotient Categories

Definition 7.1 (Multiplicative System). A multiplicative system in a category C is a collection S of morphisms in C which satisfies the following axioms: (FR1) (1) 1_{X} ∈ S for every X ∈ C.

(2) For s, t ∈ S, if st is defined, then st ∈ S.

(FR2) (1) Every diagram in C

\begin{matrix}X & ⟶^{s} & Y \\ f & ↓ \\ X′ \end{matrix}

with s ∈ S, can be completed to a commutative square

X ⟶^{s} Y
f↓  ↓^{g}
X′ ⟶^{t} Y′

with s, t ∈ S.

(2) Every diagram in C

Y
↓^{g}
X′ ⟶^{t} Y′

with t ∈ S, can be completed to a commutative square

X ⟶^{s} Y
f↓  ↓^{g}
X′ ⟶^{t} Y′

with s, t ∈ S. (FR3) For f, g ∈ Hom_{C}(X, Y) the following are equivalent.

(1) There exists s ∈ S such that sf = sg.

(2) There exists t ∈ S such that ft = gt.

Throughout this section, S is a multiplicative system of a category C.

Definition 7.2 (Saturated Multiplicative System). A multiplicative system S in a category C is called saturated if it satisfies the following axiom: (FR0) For a morphism s in C, if there exist f, g such that sf, gs ∈ S, then s ∈ S.

Definition 7.3. For a morphism f : X → Y, we set source(f) = X and sink(f) = Y.

For a multiplicative system S, each X ∈ C, S^{X} is a category defined by

1. Ob(S^{X}) = {s ∈ S | source(s) = X},

2. Hom_{S}^{X} (s, s′) = {f ∈ Hom_{C}(sink(s), sink(s′))|s′ = fs} for s, s′ ∈ Ob(S^{X}), and S_{X} is a category defined by

1. Ob(S_{X}) = {t ∈ S | sink(t) = X},

2. Hom_{S_{X}} (t, t′) = {f ∈ Hom_{C}(source(t), source(t′))|t = t′f} for t, t′ ∈ Ob(S_{X}).

Lemma 7.4. For any X ∈ C, S^{X} satisfies the following axioms: (L1) For any f_{1} ∈ Hom_{S}^{X} (s, s_{1}′), f_{2} ∈ Hom_{S}^{X} (s, s_{2}′), there exist s′′ ∈ S^{X} and

g_{1} ∈ Hom_{S}^{X} (s_{1}′, s′′), g_{2} ∈ Hom_{S}^{X} (s_{2}′, s′′) such that g_{1}f_{1} = g_{2}f_{2}. (L2) For any f_{1}, f_{2} ∈ Hom_{S}^{X} (s, s′), there exist s′′ ∈ S^{X} and g ∈ Hom_{S}^{X} (s′, s′′)

such that gf_{1} = gf_{2}. (L3) S^{X} is connected.

Definition 7.5. For X, Y ∈ C, we define a covariant functor

h^{X} ∘ sink : S^{Y} → 𝔖𝔢𝔱

where h^{X} ∘ sink(s) = Hom_{C}(X, sink(s)) for s ∈ S^{Y}, and a contravariant functor

h_{Y} ∘ source : S_{X} → 𝔖𝔢𝔱

where h_{Y} ∘ source(t) = Hom_{C}(source(t), Y) for t ∈ S_{X}.

Lemma 7.6. Let X, Y ∈ C. Define a relation , on the collection

{(f, s)|s ∈ S^{Y} , f ∈ Hom_{C}(X, sink(s))}

as follows: (f_{1}, s_{1}) ∼ (f_{2}, s_{2}) if and only if there exist h_{1} ∈ Hom_{S}^{Y} (s_{1}, s′), h_{2} ∈ Hom_{S}^{Y} (s_{2}, s′) such that (h_{1}f_{1}, s′) = (h_{2}f_{2}, s′). Then ∼ is an equivalence relation and we have

colim_{S^{Y}} h^{X} ∘ sink = {(f, s)|s ∈ S^{Y} , f ∈ Hom_{C}(X, sink(s))}/ ∼ .

(Write the equivalence class [(f, s)] for (f, s), where f ∈ Hom_{C}(X, sink(s)), s ∈ S^{Y}.)

Remark 7.7 (Set-Theoretic Remark). In the above, we dealt with S^{Y} as a small category (i.e. Ob(S^{Y}) is a set). In general, we don’t know the existence of the above colimit. But the above colimit exists if there is a small subcategory S’^{Y} of S^{Y} satisfying the following (this category is called a cofinal subcategory):

For any Y ∈ C, S^{Y} satisfies the following axiom: (Co) For any s ∈ S^{Y}, there exists a morphism f : s → s′ with s′ ∈ S’^{Y}.

Then colim_{S’^{Y}} h^{X} ∘ sink exists, and we have

colim_{S^{Y}} h^{X} ∘ sink = colim_{S’^{Y}} h^{X} ∘ sink .

Lemma 7.8. For any X, Y, Z ∈ C we have a well-defined mapping

colim_{S^{Y}} h^{X} ∘ sink ×colim_{S^{Z}} h^{Y} ∘ sink → colim_{S^{Z}} h^{X} ∘ sink

which is defined as follows: with each pair ([(f, s)], [(g, t)]), since by (FR2) there exist s′ ∈ S with source(s′) = sink(t), g′ ∈ Hom_{C}(sink(s), sink(s′)) such that g′s = s′g, we associate the equivalence class [(g′f, s′t)].

Sketch.

\xymatrix{X \ar[dr]^{f} & Y \ar[d]^{s} \ar[dr]^{g} & Z \ar[d]^{t} \\  & Y′ \ar[dr]^{g′} & Z′ \ar[d]^{s′} \\  &  & Z′′}

Definition 7.9 (Quotient Category). We define a category S^{−1} C, called the quotient category of C, as follows:

1. Ob(S^{−1} C) = Ob(C).

2. For X, Y ∈ Ob(C), the morphism set is given by

Hom_{S^{−1} C}(X, Y) = colim_{S^{Y}} h^{X} ∘ sink .

3. For X, Y, Z ∈ Ob(S^{−1} C), the law of composition is given by

Hom_{S^{−1} C}(X, Y) × Hom_{S^{−1} C}(Y, Z) → Hom_{S^{−1} C}(X, Z),
(([(f, s)], [(g, t)]) ↦ [(g′f, s′t)]),

where [(g′, s′)] ∈ Hom_{S^{−1} C}(sink(s), sink(t)) with g′s = s′g.

4. The identity of X ∈ Ob(S^{−1} C) is given by the equivalence class [(1_{X}, 1_{X})].

Definition 7.10 (Quotient Functor). We have a functor Q : C → S^{−1} C, called the quotient functor, such that

Q(X) = X for X ∈ C, Q(f) = [(f, 1_{Y})] for f ∈ Hom_{C}(X, Y).

Proposition 7.11 (Basic Properties). The following hold.

1. Q : C → S^{−1} C sends null objects to null objects.

2. For f, g ∈ Hom_{C}(X, Y) the following are equivalent.

(1) Q(f) = Q(g).

(2) There exists s ∈ S^{Y} such that sf = sg.

(3) There exists t ∈ S_{X} such that ft = gt.

3. The following hold.

(1) Q(s) is an isomorphism for all s ∈ S.

(2) For any X, Y ∈ S^{−1} C we have

Hom_{S^{−1} C}(X, Y) = {Q(s)^{−1}Q(f)|s ∈ S^{Y} , f ∈ Hom_{C}(X, sink(s))}
= {Q(g)Q(t)^{−1}|t ∈ S_{X}, g ∈ Hom_{C}(source(t), Y)}.

4. For f ∈ Hom_{C}(X, Y) the following are equivalent.

(1) Q(f) is an isomorphism.

(2) There exist morphisms g, h in C with gf, fh ∈ S.

5. Assume S is saturated. Then for any f ∈ Hom_{C}(X, Y) the following hold. (1) Q(f) is an isomorphism if and only if f ∈ S.

(2) If there exists s ∈ S^{Y} with sf ∈ S, then f ∈ S.

(3) If there exists t ∈ S_{X} with ft ∈ S, then f ∈ S.

6. For any X, Y ∈ C we have a bijection

ζ = ζ_{X,Y} : colim_{S^{Y}} h^{X} ∘ sink → colim_{S_{X}} h_{Y} ∘ source

which associates with each [(f, s)] the equivalence class of (t, g) with ft = sg, and its inverse

η = η_{X,Y} : colim_{S_{X}} h_{Y} ∘ source → colim_{S^{Y}} h^{X} ∘ sink

which associates with each [(t, g)] the equivalence class of (f, s) with ft = sg.

Proposition 7.12 (Uniqueness of Quotient). Let C′ be another category and F : C → C′ a functor such that F (s) is an isomorphism for all s ∈ S. Then there exists a unique functor \overline{F} : S^{−1} C → C′ such that F = \overline{F} Q.

\xymatrix{C \ar[d]_{Q} \ar[dr]^{F} \\ S^{−1} C \ar@{-->}[r]_{\overline{F}} & C′}

Sketch. Since every object of S^{−1} C is of the form QX for X ∈ C, we can define \overline{F} : S^{−1} C → C′ as follows. Let \overline{F} (QX) = F (X) for QX ∈ S^{−1} C and \overline{F} ([(f, s)]) = (F s)^{−1}F f for [(f, s)] ∈ Hom_{S^{−1} C}(QX, QY). Then we have F = \overline{F} Q and the required property.

Proposition 7.13. Let C′ be another category and F, G : S^{−1} C → C′ functors. Then we have a bijective correspondence

Mor(F, G) →^{∼} Mor(F Q, GQ), (ζ ↦ ζ_{Q}).

Proof. By the proof of the above lemma, it is easy.

Corollary 7.14. Let S̃ = {f|Q(f) is an isomorphism in S^{−1} C}. Then S̃ is a saturated multiplicative system, and S̃^{−1} C is equivalent to S^{−1} C.

Remark 7.15. Considering S^{−1} C, by Corollary 7.14, we may assume S is a saturated multiplicative system.

Proposition 7.16. Let D be a full subcategory of C. Assume S ∩D is a multiplicative system in D and one of the following conditions is satisfied:

(a) For any s ∈ S^{Y} with Y ∈ D, there exists f ∈ Hom_{C}(sink(s), Y′) with Y′ ∈ D such that fs ∈ S.

(b) For any t ∈ S_{X} with X ∈ D, there exists g ∈ Hom_{C}(X′, source(t)) with X′ ∈ D such that tg ∈ S.

Then the canonical functor (S ∩D)^{−1}D → S^{−1} C is fully faithful, so that (S ∩D)^{−1}D can be considered as a full subcategory of S^{−1} C.

Proof. By the same reason of Remark 7.7.

Proposition 7.17. If C is an additive category, then S^{−1} C is an additive category, and Q : C → S^{−1} C is additive functor.

Proof. It is easy to see that S^{−1} C satisfies the condition of Definition 2.1. For X = ∐_{i=1}^{n}X_{i}, by Proposition 2.3 we have F X = ∐_{i=1}^{n}F X_{i} in S^{−1} C. Therefore, S^{−1} C is an additive category and Q is an additive category by Definition 2.5, Proposition 2.10.

8. Quotient Categories of Triangulated Categories

Definition 8.1 (Compatible with Triangle). A multiplicative system S in a triangulated category C is said to be compatible with the triangulation if it satisfies the following axioms: (FR4) For a morphism u in C, u ∈ S if and only if T u ∈ S. (FR5) For triangles (X, Y, Z, u, v, w), (X′, Y′, Z′, u′, v′, w′) and morphisms f : X →

X′, g : Y → Y′ in S with gu = u′f, there exists h : Z → Z′ in S such that (f, g, h) is a homomorphism of triangles.

Throughout this section, C is a triangulated category, we assume that S is a saturated multiplicative system of C which is compatible with the triangulation, and Q : C′ → S^{−1} C is a quotient functor.

Lemma 8.2. There exists a unique auto-functor T_{S^{−1} C} : S^{−1} C → S^{−1} C such that QT_{C} = T_{S^{−1} C}Q.

We simply write T for T_{S^{−1} C}.

Lemma 8.3. Let (X, Y, Z, u, v, w), (X′, Y′, Z′, u′, v′, w′) be triangles in C, and let α ∈ Hom_{S^{−1} C}(QX, QX′), β ∈ Hom_{S^{−1} C}(QY, QY′) such that (Qu′)α = βQ(u). Then there exists γ ∈ Hom_{S^{−1} C}(QZ, QZ′) such that

QX ⟶^{Qu} QY ⟶^{Qv} QZ ⟶^{Qw} T QX
α↓  ↓^{β}  ↓^{γ}  ↓^{T α}
QX′ ⟶^{Qu′} QY′ ⟶^{Qv′} QZ′ ⟶^{Qw′} T QX′

is commutative.

Proof. There are f : X → X′_{1}, s : X′ → X′_{1}, g : Y → Y′, t : Y′ → Y′_{1} such that α = (Qs)^{−1}Qf, β = (Qt)^{−1}Qg and s, t ∈ S. Then there are u′_{1} : X′_{1} → Y′_{2}, s_{1} : Y′ → Y′_{2} such that u′_{1}s = s_{1}u′, s_{1} ∈ S. Since (Qu′)α = βQ(u), there are s_{2} : Y′_{2} → Y′_{3}, t_{1} : Y′_{1} → Y′_{3} such that s_{2}u′_{1}f = t_{1}gu, s_{2}s_{1} = t_{1}t and s_{2}s_{1} ∈ S.

\xymatrix{X \ar[dr]^{f} & X′ \ar[d]^{s} \ar[dr]^{u′} &  & X \ar[r]^{u} & Y \ar[dr]^{g} & Y′ \ar[d]^{t} \\  & X′_{1} \ar[dr]^{u′_{1}} & Y′ \ar[d]^{s_{1}} &  &  & Y_{1} \ar[d]^{t_{1}} \\  &  & Y′_{2} \ar[d]^{s_{2}} &  &  & Y′_{3} \\  &  & Y′_{3}}

Therefore there are h : Z → Z′_{1}, s_{3} : Z′ → Z′_{1} such that we have a commutative diagram

X ⟶^{u} Y ⟶^{v} Z ⟶^{w} T X
f↓  ↓^{t_{1}g}  ↓^{h}  ↓^{T f}
X_{1} ⟶^{s_{2}u′_{1}} Y′_{2} ⟶^{v′} Z′_{1} ⟶^{w′′} T X′_{1}
s↑  ↑^{s_{2}s_{1}}  ↑^{s_{3}}  ↑^{T s}
X′ ⟶^{u′} Y′ ⟶^{v′} Z′ ⟶^{w′} T X.

with s_{3} ∈ S. Since (Qs_{2}s_{1})^{−1}(Qt_{1})Qg = (Qt_{1}t)^{−1}(Qt_{1})Qg = (Qt)^{−1}Qg = β, let γ = (Qs_{3})^{−1}h, then γ satisfies the statement.

Definition 8.4 (Triangulation). A sextuple (QX′, QY′, QZ′, λ, μ, ν) in S^{−1} C is called a triangle if there exists a triangle (X, Y, Z, u, v, w) in C such that (QX′, QY′, QZ′, λ, μ, ν) is isomorphic to (QX, QY, QZ, Qu, Qv, Qw).

Theorem 8.5. S^{−1} C is a triangulated category and Q : C → S^{−1} C is a ∂-functor.

Proof. Since for any morphism α : QX → QY in S^{−1} C there are f : X → Y_{1}, s : Y → Y_{1}, t : X_{1} → Y, g : X_{1} → Y such that we have commutative diagram

X  X ⟵^{Qt} X_{1}
Qf↓  ↓^{α}  ↓^{Qg}
Y_{1} ⟵^{Qs} Y  Y

with s, t ∈ S, it is easy.

Proposition 8.6. Let A be an abelian category and H : C → A a cohomological functor such that H(s) are isomorphisms for all s ∈ S. Then there exists a unique cohomological functor \overline{H} : S^{−1} C → A such that H = \overline{H}Q.

Proposition 8.7. Let D be another triangulated category and F = (F, θ) : C → D a ∂-functor such that F s are isomorphisms for all s ∈ S. Then there exists a unique ∂-functor \overline{F} = (\overline{F} , \overline{θ}) : S^{−1} C → D such that F = \overline{F} Q and θ = \overline{θ}Q.

\xymatrix{C \ar[d]_{Q} \ar[dr]^{F} \\ S^{−1} C \ar@{-->}[r]_{\overline{F}} & C′}

Proposition 8.8. Let D be another triangulated category and F = (F, θ), G = (G, η) : S^{−1} C → D ∂-functors. Then we have a bijective correspondence

∂ Mor(F, G) →^{∼} ∂ Mor(F Q, GQ), (ζ ↦ ζ_{Q}).

Proof. By Proposition 7.13, it remains to check the following commutativity. For X ∈ C, ψ ∈ ∂ Mor(F Q, GQ), we have

F T_{S^{−1} C}QX  F QT_{C}X ⟶^{θQ} T_{D}F QX
ψT_{C} ↓  ↓^{T_{D}ψ}
GT_{S^{−1} C}QX  GQT_{C}X ⟶_{ηQ} T_{D}GQX.

9. Épaisse Subcategories

Definition 9.1 (Épaisse Subcategory). An épaisse subcategory U of a triangulated category C is a triangulated full subcategory of C such that if u ∈ Hom_{C}(X, Y) factors through (an object of) U and is embedded in a triangle (X, Y, Z, u, v, w) in C with Z ∈ U, then X, Y ∈ U.

Proposition 9.2. For a triangulated full subcategory U of C, the following are equivalent.

1. U is an épaisse subcategory of C.

2. U is closed under direct summands.

Proof. 1 ⇒ 2. Let X, Y ∈ C such that X⊕Y ∈ U. By Proposition 4.8, we have a triangle (T^{−1}Y, X, X⊕Y, 0, μ, π). Since 0 : T^{−1}Y → X factors through O ∈ U, T^{−1}Y, X ∈ U, and hence Y, X ∈ U.

2 ⇒ 1. Let (X, Y, Z, u, v, w) be a triangle such that Z ∈ U and u factors through Y′ ∈ U, then we have a morphism of triangles

X ⟶^{u′} Y′ ⟶ Z′ ⟶ T X
∥  ↓_{u′′}  ↓  ∥
X ⟶^{u} Y ⟶ Z ⟶ T X.

By Proposition 4.8, we have a triangle (Y′, Z′⊕Y, Z, ∗, ∗, ∗). We have Z′⊕Y ∈ U, and then Y ∈ U. Therefore Y, Z ∈ U implies X ∈ U.

Definition 9.3. For an épaisse subcategory U of a triangulated category C, we denote by Φ(U) the collection of morphisms u in U. such that M(u) ∈ U.

Lemma 9.4. Let U be an épaisse subcategory of a triangulated category C. For a morphism f in C, the following are equivalent.

1. f factors through some object of U.

2. There exists s ∈ Φ(U) such that sf = 0.

3. There exists t ∈ Φ(U) such that ft = 0.

Proof. 1 ⇔ 2. If f factors through U ∈ U, then we have a commutative diagram

X  X
u↓  ↓^{f}
U ⟶^{x} Y ⟶^{s} Z ⟶^{z} T U.

where the bottom row is a triangle, and with s ∈ Φ(U). We have sf = 0. Conversely, if sf = 0 with s ∈ Φ(U), then there exists u such that we also have the above commutative diagram. Therefore f = xu.

1 ⇔3. Similarly.

Proposition 9.5. Let U be an épaisse subcategory of a triangulated category C. Then Φ(U) is a saturated multiplicative system which is compatible with the triangulation.

Proof. We use the following diagram to check the axioms of a multiplicative system.

X ⟶^{u} Y ⟶^{i} Z′ ⟶^{i′} T X
∥  ↓^{v}  ↓^{z}  ∥
X ⟶^{w} Z ⟶^{k} Y′ ⟶^{k′} T X
↓^{j}  ↓^{y}  ↓^{T u}
X′  X′ ⟶^{j′} T Y
↓^{j′}  ↓^{x}
T Y ⟶^{T i} T Z′

Diagram A

(FR0) Let v : X → Y, u : Y → Z, r : Z → U be morphisms such that ru, uv ∈ Φ(U). Then we have a commutative diagram

Y ⟶^{u} Z ⟶^{j} X′ ⟶^{j′} T Y
∥  ↓^{r}  ↓^{l}  ∥
Y ⟶^{ru} U ⟶^{p} V ⟶^{q} T Y

with V ∈ U. Since x = (T i)j′ = (T i)ql, x factors through V ∈ U. Since uv ∈ Φ(U), we have Diagram A with Y′ ∈ U. Therefore, X′, Z′ ∈ U, and hence u ∈ Φ(U).

(FR1) (1) Since O ∈ U, it is trivial. (2) If u, v ∈ Φ(U), then we have Diagram A with Z′, X′ ∈ U. Then Y′ ∈ U, and hence vu ∈ Φ(U).

(FR2) (1) Given v ∈∈ Φ(U), i, we have Diagram A with X′ ∈ U. Then z ∈ Φ(U).

(2) Given z ∈ Φ(U), k, we have Diagram A with X′ ∈ U. Then v ∈ Φ(U). (FR3) By Lemma 9.4. (FR4) It is trivial. (FR5) By Proposition 4.9, it is easy.

Theorem 9.6. For a saturated multiplicative system S of a triangulated category C which is compatible with the triangulation, let Ψ(S) be the full subcategory of C consisting of objects X such that QX = O, where Q : C → S^{−1} C is the canonical quotient. Then Ψ(S) is an épaisse subcategory of C.

Hence, Φ and Ψ induce one to one correspondence between épaisse subcategories and saturated multiplicative systems which is compatible with the triangulation.

Proof. Let (QX, QY, QZ, Qu, Qv, Qw) be the image of a triangle (X, Y, Z, u, v, w) of C. If two objects of QX, QY, QZ are O, then the rest one is clearly O. Then it is easy to see that Ψ(S) is a triangulated full subcategory of C. If Z ∈ Ψ(S) and u factors through some object of Ψ(S), then QZ = O and Qu factors through O. Therefore, QX = QY = O and hence X, Y ∈ Ψ(S).

Definition 9.7. For an épaisse subcategory U of a triangulated category C, we denote by C/U the quotient category Φ(U)^{−1}C. In this case, we say that 0 → U → C → C/U → 0 is an exact sequence of triangulated categories.

Proposition 9.8. Let C be a triangulated category, U an épaisse subcategory of C, and Q : C → C/U the canonical quotient. For M ∈ C, the following are equivalent.

1. For every f : X → Y ∈ Φ(U), Hom_{C}(f, M) : Hom_{C}(Y, M) → Hom_{C}(X, M) is bijective.

2. Hom_{C}(U, M) = 0.

3. For every X ∈ C, Q_{X,M} : Hom_{C}(X, M) → Hom_{C/U}(QX, QM) is bijective.

Proof. 1 ⇒ 2. For every object U ∈ U, O → U ⟶^{1} U → O is a triangle. Then 0 = Hom_{C}(O, M) ≅ Hom_{C}(U, M).

2 ⇒ 3. Every morphism of Hom_{C/U}(QX, QM) is represented by a diagram

\xymatrix{X′ \ar[d]^{s} \ar[dr]^{f} \\ X & M}

where s is contained in a triangle U → X′ ⟶^{s} X → T U with U ∈ U. Then there exists f′ : X → M in C such that f = f′s, because Hom_{C}(U, M) = 0. Hence Q_{X,M} is surjective. Let U → X′ → X → T U be a triangle with U ∈ U. If a morphism g : X → M satisfies gs = 0, then there exist u : T U → M such that g = ut. Therefore g = 0, because u ∈ Hom_{C}(U, M) = 0. Hence Q_{X,M} is injective.

3 ⇒ 1. Let f : X → Y be a morphism in Φ(U). Then we have the following commutative diagram

Hom_{C}(Y, M)  ⟶^{Hom(f,M)}  Hom_{C}(X, M)
Q_{Y,M} ↓  ↓^{Q_{X,M}}
Hom_{C/U}(QY, QM) ⟶^{Hom(Qf,QM)} Hom_{C/U}(QX, QM)

According to 3, Q_{X,M} and Q_{Y,M} are bijective. Since QU = 0, Hom(Qf, QM) is bijective. Hence Hom(f, M) is bijective.

Definition 9.9 (U-Local Object). An object M is called U-local (resp., U-colocal) if it satisfies the equivalent conditions (resp., the dual conditions ) of Proposition 9.8 . Let 0 → U → C ⟶^{Q} C/U → 0 be an exact sequence of triangulated categories.

The right (resp., left ) adjoint of Q is called a section functor. If there exists a section functor S, then {C/U; Q, S} is called a localization (resp., colocalization) of C, and 0 → U → C ⟶^{Q} C/U → 0 is called localization (resp., colocalization) exact.

Lemma 9.10. Let {C/U; Q, S} be a localization of C. For every object V ∈ C/U, SV is U-local.

Proof. For every f : X → Y ∈ Φ(U), we have a commutative diagram

Hom_{C}(Y, SV) ⟶^{Hom(f,SV)} Hom_{C}(X, SV)
↓^{≀}  ↓^{≀}
Hom_{C/U}(QY, V) ⟶^{Hom(Qf,V)} Hom_{C/U}(QX, V)

Therefore Hom(f, SV) is an isomorphism. By Proposition 9.8, SV is U-local.

Proposition 9.11. Let {C/U; Q, S} be a localization of C, and τ : QS → 1_{C/U} and σ : 1_{C} → SQ adjunction arrows. Then the following hold.

1. τ is an isomorphism (i.e. S is fully faithful ).

2. For every object X ∈ C, the triangle U → X ⟶^{σ_{X}} SQX → T U satisfies that U is in U.

Proof. 1. For every X ∈ C and Y ∈ C/U, we have a commutative diagram

Hom_{C}(X, SY)  Hom_{C}(X, SY)
Q_{X,SY} ↓  ↓^{≀}
Hom_{C/U}(QX, QSY) ⟶^{Hom(QX,τ_{Y} )} Hom_{C/U}(QX, Y).

By Proposition 9.8 and Lemma 9.10, Q_{X,SY} is an isomorphism. Then Hom(QX, τ_{Y}) is an isomorphism. For any Z ∈ C/U, there exists X ∈ C such that Z ≅ QX. Hence τ is an isomorphism.

2. It suffices to show that for any X ∈ C, Qσ_{X} is an isomorphism. By the property of adjunction arrows, we have QX ⟶^{Qσ_{X}} QSQX ⟶^{τ_{QX}} QX = 1_{QX}, and hence Qσ_{X} is an isomorphism.

Corollary 9.12. Let M ∈ C. Then M is U-local if and only if M ≅ SQM.

Proposition 9.13. Let C and C′ be triangulated categories, F : C → C′ a ∂- functor which has a fully faithful right adjoint S : C′ → C. Then F induces an equivalence between C/ Ker F and C′.

Proof. By the universal property of Q : C → C/ Ker F, we have the following commutative diagram

\xymatrix{C \ar[d]_{Q} \ar[dr]^{F} \\ C/ Ker F \ar[r]_{F′} & C′}

If f : X → Y is a morphism in C, then F f is an isomorphism if and only if Qf is an isomorphism. For every object M ∈ C, F M → F SF M is an isomorphism, and then QM → QSF M is an isomorphism. Therefore Q → QSF is an isomorphism.

By the universal property of Q and QSF = QSF′Q, we have 1_{C/ Ker F} ≅ QSF′. Since, F′QS = F S ≅ 1_{C′}, F′ is an equivalence.

Definition 9.14 (stable t-structure). For full subcategories U and V of C, (U, V) is called a stable t-structure in C provided that

1. U and V are stable for translations.

2. Hom_{C}(U, V) = 0.

3. For every X ∈ C, there exists a triangle U → X → V → T U with U ∈ U and V ∈ V.

Proposition 9.15. Let C be a triangulated category, (U, V) a stable t-structure in C. Then the following hold.

1. For X ∈ C, Hom_{C}(X, V) = 0 if and only if X is isomorphic to an object of U.

2. For Y ∈ C, Hom_{C}(U, Y) = 0 if and only if Y is isomorphic to an object of V.

3. Let Ũ be the full subcategory of C consisting objects which are isomorphic to objects of U. Then Ũ is an épaisse subcategory of C.

4. Let Ṽ be the full subcategory of C consisting objects which are isomorphic to objects of V. Then Ṽ is an épaisse subcategory of C.

Proof. 1. For X ∈ C, we have a triangle

U_{X} ⟶^{τ_{X}} X ⟶^{σ_{X}} V_{X} → T U_{X}.

If Hom_{C}(X, V) = 0, then σ_{X} = 0 and U_{X} ≅ X⊕T^{−1}V_{X}. Therefore, T^{−1}V_{X} = O and X ≅ U_{X}, because of Hom_{C}(U_{X}, T^{−1}V_{X}) = 0.

2. Similarly.

3, 4. By 1, 2, it is trivial.

Proposition 9.16. Let C be a triangulated category. If {V; Q, S} is a localization of C, then S is fully faithful, (U, SV) is a stable t-structure, where U = Ker Q. Conversely, if (U, V) is a stable t-structure in C, then the canonical inclusion S : V → C has a left adjoint Q such that {V; Q, S} is a localization.

Proof. Let {V; Q, S} be a localization of C. Then, by

\begin{aligned}Hom_{C}(U, SV) & ≅ Hom_{C}(QU, V) \\ & = 0 \end{aligned}

and Proposition 9.13, it is clear that S is fully faithful and (U, SV) is a stable t-structure. Conversely, let (U, V) be a stable t-structure in C. For X ∈ C, let U_{X} → X → V_{X} → T U_{X} be triangle such that U_{X} ∈ U and V_{X} ∈ V. Since Hom_{C}(U, V) = 0 and U and V are stable under translations, for any V ∈ V, we have an isomorphism

Hom_{C}(V_{X}, V) ≅ Hom_{C}(X, V).

According to Theorem 1.18, S : V → C has a left adjoint Q such that {V; Q, S} is a localization.

Remark 9.17. Similarly, if (U, V) is a stable t-structure in C, then there is a functor Q : C → U such that {U; Q, S′} is a colocalization of C, where S′ : U → C is the canonical embedding. Conversely, if {U; Q, S′} is a colocalization of C, then (S′U, Ker Q) is a stable t-structure in C.

10. Derived Categories

Throughout this section, A is an abelian category.

Definition 10.1. For X^{▪}, Y^{▪} ∈ K^{∗}(A), a morphism u ∈ Hom_{C(A)}(X^{▪}, Y^{▪}) is called a quasi-isomorphism if H^{n}(u) are isomorphisms for all n ∈ ℤ, where ∗ = nothing, +, −, b.

K^{∗,ϕ}(A) is a full subcategory of K^{∗}(A) consisting of complexes of which all homologies are O, where ∗ = nothing, +, −, b.

Proof. It is easy to see that K^{∗,ϕ}(A) is an épaisse subcategory of K^{∗}(A). By Proposition 6.26, it is easy.

Definition 10.2 (Derived Category). The derived category D^{∗}(A) of an abelian category A is K^{∗}(A)/ K^{∗,ϕ}(A), where ∗ = nothing, +, −, b.

Remark 10.3. For two morphisms f, g : X^{▪} → Y^{▪} in C(A), f = g ⇒ f ≃_{h} g ⇒

f ≅ g in D(A) ⇒ H^{n}(f) ≅ H^{n}(g) for all n. The converse implications do not hold.

Proposition 10.4. If O → X^{▪} ⟶^{u} Y^{▪} ⟶^{v} Z^{▪} → 0 is a exact sequence in C(A), then it can be embedded in a triangle in D(A)

QX^{▪} ⟶^{Qu} Y^{▪} ⟶^{Qv} QZ^{▪} ⟶^{Qw} T QX^{▪}.

Proof. According to Remark 6.10, we have a commutative diagram in C(A)

O  O
↓  ↓
O ⟶ X^{▪} ⟶ I^{▪}(X^{▪}) ⟶ T X^{▪} ⟶ O
u↓  ↓^{x}  ∥
O ⟶ Y^{▪} ⟶^{v′} M^{▪}(u) ⟶^{w} T X^{▪} ⟶ O
v↓  ↓^{s}
Z^{▪}  Z^{▪}
↓  ↓
O  O

where all rows and columns are exact. Then X^{▪} ⟶^{u} Y^{▪} ⟶^{v′} Z^{▪} ⟶^{w} T X^{▪} is a triangle in K(A). Since I^{▪}(X^{▪}) ∈ K^{ϕ}(A), by Proposition 6.23, s is a quasi-morphism, and hence we have a commutative diagram in D(A)

QX^{▪} ⟶^{Qu} QY^{▪} ⟶^{Qv′} Q M^{▪}(u) ⟶^{Qw} T QX^{▪}
∥  ∥  ↓^{Qs}  ∥
QX^{▪} ⟶^{Qu} QY^{▪} ⟶^{Qv} QZ ⟶^{Qw(Qs)^{−1}} T QX^{▪}.

Definition 10.5. A full subcategory A′ of A is called a thick abelian full subcategory if A′ is an abelian exact full subcategory which is closed under extensions.

For ∗ = nothing, +, −, b, we denote by K_{A′}^{∗}(A) the full subcategory of K^{∗}(A) consisting of complexes X^{▪} ∈ K(A) with H^{n}(X^{▪}) ∈ A′ for all n ∈ ℤ.

Moreover, we set D_{A′}^{∗}(A) = K_{A′}^{∗}(A)/ K^{∗,ϕ}(A), where ∗ = nothing, +, −, b.

Definition 10.6 (Truncations). For a complex X^{▪} = (X^{i}, d^{i}), we define the following truncations:

σ_{>n}X^{▪} : … → 0 → Im d^{n} → X^{n+1} → X^{n+2} → …,
σ_{≤n}X^{▪} : … → X^{n−2} → X^{n−1} → Ker d^{n} → 0 → …,
σ′_{≥n}X^{▪} : … → 0 → Cok d^{n−1} → X^{n+1} → X^{n+2} → …,
σ′_{<n}X^{▪} : … → X^{n−2} → X^{n−1} → Im d^{n−1} → 0 → …,
τ_{≥n}X^{▪} : … → 0 → X^{n} → X^{n+1} → X^{n+2} → …,
τ_{≤n}X^{▪} : … → X^{n−2} → X^{n−1} → X^{n} → 0 → ….

Then we have exact sequences in C(A)

O → σ_{≤n}(X^{▪}) → X^{▪} → σ_{>n}(X^{▪}) → O
O → σ′_{<n}(X^{▪}) → X^{▪} → σ′_{≥n}(X^{▪}) → O
O → τ_{≥n}(X^{▪}) → X^{▪} → τ_{≤n+1}(X^{▪}) → O

Then it is easy to see that

H^{i}(σ_{>n}X^{▪}) = \begin{cases}O & if i ≤ n \\ H^{i}(X^{▪}) & if i > n \end{cases}

H^{i}(σ′_{≥n}X^{▪}) = \begin{cases}O & if i < n \\ H^{i}(X^{▪}) & if i ≥ n \end{cases}

H^{i}(σ_{≤n}X^{▪}) = \begin{cases}H^{i}(X^{▪}) & if i ≤ n \\ O & if i > n \end{cases}

H^{i}(σ′_{<n}X^{▪}) = \begin{cases}H^{i}(X^{▪}) & if i < n \\ O & if i ≥ n \end{cases}

Proposition 10.7. The following hold.

1. The canonical functor D^{∗}(A) → D(A) is fully faithful, where ∗ = +, −.

2. The canonical functor D^{b}(A) → D^{∗}(A) is fully faithful, where ∗ = +, −.

3. The canonical functor D_{A′}^{∗}(A) → D^{∗}(A) is fully faithful, where ∗ = +, −, b.

Proof. According to Definitions 10.2, 10.5, it suffices to check the condition of Proposition 7.16. Let X^{▪} ∈ K^{−}(A), Y^{▪} ∈ K(A) and a quasi-isomorphism X^{▪} → Y^{▪} in K(A). Then we may assume that there is n such that H^{i}(Y^{▪}) = 0 for all i > n. Then the morphism Y^{▪} → σ_{≤n}(Y^{▪}) is a quasi-isomorphism, and σ_{≤n}(Y^{▪}) ∈ K^{−}(A). For the other cases, similarly.

Let Inj A (resp., Proj A) be the full subcategory of A consisting of injective (resp., projective) objects.

Lemma 10.8. For X^{▪} ∈ K(A) and I^{▪} ∈ K^{+}(Inj A) (resp., P^{▪} ∈ K^{−}(Proj A)), if X^{▪} is acyclic, then we have

Hom_{K(A)}(X^{▪}, I^{▪}) = 0.
(resp., Hom_{K(A)}(P^{▪}, X^{▪}) = 0)

Corollary 10.9. The following hold.

1. If A has enough injectives, then we have an isomorphism

Hom_{K(A)}(X^{▪}, I^{▪}) ≅ Hom_{D(A)}(X^{▪}, I^{▪})

for X^{▪} ∈ K(A), I^{▪} ∈ K^{+}(Inj A).

2. If A has enough projectives, then we have an isomorphism

Hom_{K(A)}(P^{▪}, Y^{▪}) ≅ Hom_{D(A)}(P^{▪}, Y^{▪})

for P^{▪} ∈ K^{−}(Proj A), Y^{▪} ∈ K(A).

Proof. By Lemma 10.8 and Proposition 9.8, and their dual.

Lemma 10.10. The following hold.

1. Let L be a collection of objects of A such that every object X ∈ A is a image of an epimorphism from some object of L. Then for any X^{▪} ∈ K^{−}(A), there exists P^{▪} ∈ K^{−}(L) and a morphism f : P^{▪} → X^{▪} in K(A) such that f is a quasi-isomorphism.

2. Let A′ be a thick abelian full subcategory of A such that every object X ∈ A′ is a image of an epimorphism from some object of (Proj A) ∩ A′. Then for any X^{▪} ∈ K_{A′}^{−}(A), there exists P^{▪} ∈ K^{−}((Proj A) ∩ A′) and a morphism f : P^{▪} → X^{▪} in K(A) such that f is a quasi-isomorphism.

Proof. 1. Given an object X ∈ K^{−}(A), we may assume that X^{i} = O for all i > 0. By the backward induction on n, we construct a complex P^{▪} ∈ K^{−}(Proj A). as follows. Let Z′^{n} and Z^{n} be Z^{n}(P^{▪}) and Z^{n}(X^{▪}), respectively. Assume we have a commutative diagram

[figure: Z′^{n}; P^{n}; Z^{n}; X^{n}]

We take a pull back M^{n} of X^{n−1} → Z^{n} ← Z′^{n}, and take an epimorphism from P^{n−1} → M^{n}. Then by Proposition 2.19, H^{n}(P^{▪}) ≅ H^{n}(X^{▪}) and the induced morphism Z^{n}(P^{▪}) → Z^{n}(X^{▪}) is epic.

[figure: Z′^{n−1}; P^{n−1}; M^{n−1}; P B; Z′^{n}; P^{n}; Z^{n−1}; X^{n−1}; X^{n−1}; Z^{n}; X^{n}]

2. Given an object X ∈ K_{A′}^{−}(A), we may assume that X^{i} = O for all i > 0. By the backward induction on n, we construct a complex P^{▪} ∈ K^{−}((Proj A) ∩ A′). as follows. Let B′^{n} and B^{n} (resp., C′^{n} and C^{n}) be Z^{n}(P^{▪}) and Z^{n}(X^{▪}) (resp., C^{n}(P^{▪}) and C^{n}(X^{▪}), respectively. Assume we have a commutative diagram

\xymatrix{P^{n} \ar[d] \ar@{->>}[r] & C′^{n} \ar[d] \\ X^{n} \ar@{->>}[r] & C^{n}}

where P^{n}, C′^{n} ∈ A′. Then B′^{n} ∈ A′. We take a pull back C′^{n−1} of C^{n−1} → B^{n} ← B′^{n}. Then by Proposition 2.19, H^{n−1}(P^{▪}) ≅ H^{n−1}(X^{▪}). Since A′ is closed under extensions, C′^{n−1} ∈ A′. Therefore we can take an epimorphism from P^{n−1} → C′^{n−2}, with P^{n−1} ∈ (Proj A) ∩ A′. Since P^{n−1} is projective, we have a morphism P^{n−1} → X^{n−1}, and we have a commutative diagram

[figure: P^{n−1}; C′^{n−1}; P B; B′^{n−1}; P^{n}; C′^{n}; X^{n−1}; C^{n−1}; B^{n−1}; X^{n}; C^{n}]

Proposition 10.11. The following hold.

1. If A has enough projectives, then

K^{−}(Proj A) ≅^{t} D^{−}(A).

2. If A has enough injectives, then

K^{+}(Inj A) ≅^{t} D^{+}(A).

3. Let A′ be a thick abelian full subcategories of A such that A′ has enough A-projectives in A′. Then we have

D^{∗}(A′) ≅^{t} D_{A′}^{∗}(A)

where ∗ = −, b.

4. Let A′ be a thick abelian full subcategories of A such that A′ has enough A-injectives in A′. Then we have

D^{∗}(A′) ≅^{t} D_{A′}^{∗}(A)

where ∗ = +, b.

Proof. 1. By Lemmas 10.8, 10.10, (K^{−}(Proj A), K^{−,ϕ}(A)) is a stable t-structure in K^{−}(A). According to Proposition 9.16, Remark 9.17, we get the statement.

2. Similarly.

3. Since we have the canonical full embedding K^{−}(A′) → K^{−}(A), it suffices to check the condition of Proposition 7.16. Let X^{▪} ∈ K^{−}(A′), Y^{▪} ∈ K^{−}(A), and Y^{▪} → X^{▪} a quasi-isomorphism in K^{−}(A). Since all homologies of Y^{▪} are in A′, by Lemma 10.10, we have X′^{▪} → Y^{▪} is a quasi-isomorphism, with X′^{▪} ∈ D^{−}(A′).

4. Similarly.

Definition 10.12. In the case of A having enough projectives (resp., injectives), we denote by K^{−,b}(Proj A) (resp., K^{+,b}(Inj A)) the triangulated full subcategory of K^{−}(Proj A) (resp., K^{+}(Inj A)) consisting of complexes of which homologies are bounded.

Corollary 10.13. The following hold.

1. If A has enough projectives, then

K^{−,b}(Proj A) ≅^{t} D^{b}(A).

2. If A has enough injectives, then

K^{+,b}(Inj A) ≅^{t} D^{b}(A).

Example 10.14. For a coherent ring A, let mod A be the full subcategory of Mod A consisting of right coherent A-modules. Then mod A is an thick abelian full subcategory of Mod A. Therefore, we have

D^{∗}(mod A) ≅^{t} D_{mod A}^{∗}(Mod A)

where ∗ = −, b. We often write D_{c}^{∗}(Mod A) for D_{mod A}^{∗}(Mod A).

Definition 10.15 (Yoneda Ext). For X, Y ∈ A and n ∈ ℕ, let Exact_{A}^{n}(X, Y) be the set of exact sequences in A of the form

Σ : O → Y → X_{n−1} → … → X_{0} → X → O.

For Σ_{1}, Σ_{2} ∈ Exact_{A}^{n}(X, Y), we write Σ_{1} → Σ_{2} if we have a commutative diagram

Σ_{1} :O ⟶ Y ⟶ X_{n−1} ⟶ … ⟶ X_{0} ⟶ X ⟶ O
↓  ∥  ↓  …  ↓  ∥
Σ_{2} :O ⟶ Y ⟶ X′_{n−1} ⟶ … ⟶ X′_{0} ⟶ X ⟶ O.

And, we define Σ_{1} ∼ Σ_{m} if there are Σ_{i} (2 ≤ i ≤ m − 1) such that Σ_{i} ⇄ Σ_{i+1} (1 ≤ i ≤ m − 1), where ⇄ means → or ←. Then ∼ is a equivalent relation on Exact_{A}^{n}(X, Y). We denote Exact_{A}^{n}(X, Y)/ ∼ by Ext_{A}^{n}(X, Y).

Proposition 10.16. For X, Y ∈ A and n ∈ ℕ, We have a bifunctorial isomorphism

Ext_{A}^{n}(X, Y) →^{∼} Hom_{D(A)}(X, T^{n}Y).

Proof. Let Σ ∈ Exact_{A}^{n}(X, Y) has the form

Σ : O → Y → X_{n−1} → … → X_{0} → X → O.

Then we have a commutative diagram

Y [n − 1] :O ⟶ Y ⟶ O
↓  ↓  ↓  ↓
M^{▪} : O ⟶ X_{n−1} ⟶ X_{n−2} ⟶ … ⟶ X_{1} ⟶ X_{0} ⟶ O
↓  ↓  ↓  ↓
X :  O ⟶ X ⟶ O

Therefore, we have a triangle Y [n − 1] → M^{▪} → X ⟶^{ϕ(Σ)} Y [n]. It is easy to see that ϕ_{X,Y} : Ext_{A}^{n}(X, Y) → Hom_{D(A)}(X, Y [n]) is a bifunctorial isomorphism (left to the reader).

Remark 10.17. Assume A has enough injectives. For X, Y ∈ A, let

O → Y → I^{0} → I^{1} → …

be an injective resolution, that is, Y → I^{▪} is a quasi-isomorphism. Then by Corollary 10.9, it is easy to see that

Ext_{A}^{n}(X, Y) ≅ Hom_{D(A)}(X, T^{n}Y)
≅ Hom_{D(A)}(X, T^{n}I^{▪})
≅ Hom_{K(A)}(X, T^{n}I^{▪})
≅ H^{n}(Hom_{A}(X, I^{▪})).

The last term is Ext_{A}^{n}(X, Y) in the sense of standard homological algebra.

Definition 10.18. Let A be an abelian category with enough injectives. A complex X^{▪} ∈ K^{∗}(A) is said to have finite injective dimension if there is n ∈ ℤ such that Hom_{D^{∗}(A)}(M, X^{▪}[i]) = 0 for all M ∈ A and i > n, where ∗ = nothing, +, −, b.

We denote by K^{∗}(A)_{fid} the full subcategory of K^{∗}(A) consisting of X^{▪} ∈ K^{∗}(A) which have finite injective dimension.

Moreover, for a thick abelian full subcategory A′ of A, we denote by K_{A′}^{∗}(A)_{fid} the full subcategory of K_{A′}^{∗}(A) consisting of X^{▪} ∈ K_{A′}^{∗}(A) which have finite injective dimension.

Proposition 10.19. Let A be an abelian category with enough injectives. Then K^{∗}(A)_{fid} and K_{A′}^{∗}(A)_{fid} are quotientizing subcategories of K^{∗}(A).

Proof. For u : X^{▪} → Y^{▪} in K^{∗}(A)_{fid}, let X^{▪} ⟶^{u} Y^{▪} → M^{▪}(u) → X^{▪}[1] be a triangle. For M ∈ A, by applying Hom_{K^{∗}(A)}(M, −) to the triangle, we have M^{▪}(u) ∈ K^{∗}(A)_{fid}. Therefore, K^{∗}(A)_{fid} is a triangulated full subcategory of K^{∗}(A). By Proposition 9.2, K^{∗,ϕ}(A)_{fid} = K^{ϕ}(A) ∩ K^{∗}(A)_{fid} is an épaisse subcategory of K^{∗}(A). According to Proposition 7.16, K^{∗}(A)_{fid} is a quotientizing subcategory of K^{∗}(A). In the case of K_{A′}^{∗}(A)_{fid}, similarly.

Definition 10.20. For ∗ = nothing, +, −, b, D^{∗}(A)_{fid} = K^{∗}(A)_{fid}/ K^{∗,ϕ}(A)_{fid} and D_{A′}^{∗}(A)_{fid} = K_{A′}^{∗}(A)_{fid}/ K^{∗,ϕ}(A)_{fid}.

Proposition 10.21. Let A be an abelian category with enough injectives. Then the following are equivalent for X^{▪} ∈ K^{+}(A).

1. For any integer n_{1} ∈ ℤ there is n_{2} ∈ ℤ such that Hom_{D(A)}(Y^{▪}, X^{▪}[i]) = 0 for all i > n_{2} and all complexes Y^{▪} ∈ K^{+}(A) with H^{j}(Y^{▪}) = O for j < n_{1}.

2. X^{▪} ∈ K^{+}(A)_{fid}.

3. There exists I^{▪} ∈ K^{b}(Inj A) such that X^{▪} ≅ I^{▪} in D^{+}(A).

Proof. 1 ⇒ 2. It is trivial.

2 ⇒ 3. Let n be an integer such that Hom_{D(A)}(M, X^{▪}[i]) = 0 for all i > n. We take I^{▪} ∈ K^{+}(Inj A) which has a quasi-isomorphism X^{▪} → I^{▪} in K^{+}(A). For i > n, we have isomorphisms

Hom_{D(A)}(Z^{i}(I^{▪})[−i], X^{▪}) ≅ Hom_{K(A)}(Z^{i}(I^{▪}), I^{▪}[i])
= 0.

This means that the canonical morphisms I^{i−1} → Z^{i}(I^{▪}) is split epic, and B^{i}(I^{▪}) = Z^{i}(I^{▪}). Then σ_{≤n}I^{▪} → I^{▪} is an isomorphism in K(A) and σ_{≤n}I^{▪} ∈ K^{b}(Inj A).

3 ⇒ 1. By Corollary 10.9, it is easy.

Corollary 10.22. Let A be an abelian category with enough injectives. Then we have a triangle equivalent

K^{b}(Inj A) ≅^{t} D^{+}(A)_{fid}.

Proposition 10.23. Let B be an additive full subcategory of an abelian category A which is closed under direct summands, and K̃^{b}(B) the triangulated full subcategory of K^{−}(B) consisting of objects which are isomorphic to an object of K^{b}(B) in K^{−}(B). Then K̃^{b}(B) is an épaisse subcategory of K^{−}(B).

46 JUN-ICHI MIYACHI

Proof. Let X^{▪} ∈ K^{b}(B), and Y^{▪} is direct summand of X^{▪} in K^{−}(B). Then Y^{▪} is a direct summand of X^{▪}⊕I^{▪}(Y^{▪}) in C^{−}(B). Then we have a split exact sequence in C^{−}(B)

O → Y^{▪} → X^{▪}⊕I^{▪}(Y^{▪}) → Z^{▪} → O.

Since X^{▪} ∈ K^{b}(B), there is n ∈ ℤ such that we have a split exact sequence in C^{−}(B)

O → τ_{≤n}Y^{▪} → τ_{≤n}I^{▪}(Y^{▪}) → τ_{≤n}Z^{▪} → O.

Since H^{i}(τ_{≤n}I^{▪}(Y^{▪})) = O for i ≠ n and B^{i}(τ_{≤n}I^{▪}(Y^{▪})) ∈ B, H^{i}(τ_{≤n}Y^{▪}) = O for i ≠ n and B^{i}(τ_{≤n}Y^{▪}) ∈ B. Hence Y^{▪} ≅ σ_{>n−2}Y^{▪} in K^{−}(B) with σ_{>n−2}Y^{▪} ∈ K^{b}(B).

11. Homotopy Limits

Throughout this section C is a triangulated category with arbitrary coproducts. Definition 11.1. A triangulated full subcategory L of C is called localizing if (L1) Every direct summand of an object in L is in L.

(L2) Every coproduct of objects in L is in L. Lemma 11.2. Let L of C be a localizing subcategory, then C/L has arbitrary coproducts, and the quotient C → C/L preserves coproducts. Proof. Let {X_{I}}_{i∈I} be a collection of objects of C. It suffices to show

1. Any collection of morphisms X_{i} ⟶^{f_{i}} Y in C/L can be lifted to a morphism

∐_{i}X_{i} ⟶^{f} Y in C/L.

2. a morphism ∐_{i}X_{i} ⟶^{f} Y in C/L such that all X_{i} ⟶^{q_{i}} ∐_{i}X_{i} ⟶^{f} Y = 0 in C/L, then f = 0.

1. A morphism X_{i} ⟶^{f_{i}} Y in C/L is a diagram in C

\xymatrix{Xi^{′} \ar[d]^{s_{i}} \ar[dr]^{f′_{i}} \\ X_{i} & Y}

where X_{i} → X_{i} → Xi^{′′} → T Xi^{′} is a triangle, with X′′_{i} ∈ L. Thus we get a diagram

\xymatrix{∐_{i}Xi^{′} \ar[d]^{∐s_{i}} \ar[dr]^{∑f′_{i}} \\ ∐_{i}X_{i} & Y.}

Since ∐_{i}X_{i} → ∐_{i}X_{i} → ∐_{i}Xi^{′′} → T ∐_{i}Xi^{′} is a triangle, we have a morphism

∐_{i}X_{i} ⟶^{f} Y in C/L.

2. Given a morphism ∐_{i}X_{i} ⟶^{f} Y in C/L, it corresponds to a diagram

\xymatrix{∐_{i}X_{i} \ar[dr]^{f′} & Y \ar[d]^{t} \\  & Y}

with t ∈ Φ(L). If the composite X_{i} → ∐_{i}X_{i} → Y = 0 in C/L, then we have a diagram in C

\xymatrix{X_{i} \ar[r]^{q_{i}} & ∐_{i}X_{i} \ar[dr]^{f′} & Y \ar[d]^{t} \\  &  & Y′}

which corresponds to 0 in C/L. Then by Proposition 7.11 and Lemma 9.4, every

X_{i} ⟶^{q_{i}} ∐_{i}X_{i} ⟶^{f′} Y′ factors through Z_{i} ∈ L. Thus f factorizes as

\xymatrix{∐_{i}X_{i} \ar[r] & ∐_{i}Z_{i} \ar[dr] & Y \ar[d]^{t} \\  &  & Y′.}

Since L is localizing, ∐_{i}Z_{i} ∈ L and f = 0.

Corollary 11.3. Let A be an abelian category satisfying the condition Ab4. Then D(A) has arbitrary coproducts.

Proof. According to Corollary 6.15, K(A) has arbitrary coproducts. For a collection of quasi-isomorphisms X_{i}^{▪} ⟶^{f_{i}} Y_{i}^{▪} (i ∈ I), H^{n}(∐_{i}f_{i}) ≅ ∐_{i} H^{n}(f_{i}) is isomorphic for all n ∈ ℤ. Thus K^{ϕ}(A) is localizing, and Lemma 11.2 can be applied.

Definition 11.4. For a sequence {X_{i} → X_{i+1}}_{i∈ℕ} (resp., {X_{i+1} → X_{i}}_{i∈ℕ}) of morphisms in C, the homotopy colimit (resp., limit) of the sequence is the third (resp., second) term of the triangle

∐_{i}X_{i} ⟶^{1− shift} ∐_{i}X_{i} → hlim_{⟶} X_{i} → T ∐_{i}X_{i}

(resp., T^{−1}∏_{i}X_{i} → hlim_{⟵} X_{i} → ∏_{i}X_{i} ⟶^{1− shift} ∐_{i}X_{i})

where the above shift morphism is the coproduct (resp., product) of X_{i} ⟶^{f_{i}} X_{i+1}

(resp., X_{i+1} ⟶^{f_{i}} X_{i}) (i ∈ ℕ).

Exercise 11.5. In the category 𝔄𝔟, prove the following.

1. For a sequence of morphisms {X_{i+1} ⟶^{f_{i}} X_{i}}_{i∈ℕ}, if there is n ∈ ℕ such that f_{i} are epimorphisms for all i ≥ n, then ∏_{i}X_{i} ⟶^{1− shift} ∏_{i}X_{i} is epic.

2. For a sequence of morphisms {X_{i} ⟶^{f_{i}} X_{i+1}}_{i∈ℕ}, ∐_{i}X_{i} ⟶^{1− shift} ∐_{i}X_{i} is monic.

Lemma 11.6. The following hold.

1. Assume A satisfies the condition Ab3. For a sequence of morphisms {X_{i} ⟶^{f_{i}} X_{i+1}}_{i∈ℕ}, if there is n ∈ ℕ such that f_{i} are split monomorphisms for all i ≥ n, then we have a split exact sequence

O → ∐_{i}X_{i} ⟶^{1− shift} ∐_{i}X_{i} → \underrightarrow{lim} X_{i} → O.

2. Assume A satisfies the condition Ab3*. For a sequence of morphisms {X_{i+1} ⟶^{f_{i}} X_{i}}_{i∈ℕ}, if there is n ∈ ℕ such that f_{i} are split epimorphisms for all i ≥ n, then we have a split exact sequence

O → \underleftarrow{lim} X_{i} → ∏_{i}X_{i} ⟶^{1− shift} ∏_{i}X_{i} → O.

Proof. 1, For any M ∈ A, we have a commutative diagram

Hom_{A}(∐_{i}X_{i}, M) ⟶^{1− shift} Hom_{A}(∐_{i}X_{i}, M)
↓^{≀}  ↓^{≀}
∏_{i} Hom_{A}(X_{i}, M) ⟶^{1− shift} ∏_{i} Hom_{A}(X_{i}, M).

By Exercise 11.5, the bottom horizontal morphism is epic.

2. Similarly.

Proposition 11.7. The following hold.

1. Assume A satisfies the condition Ab3. and X_{i}^{▪} → X_{i+1}^{▪} a sequence of complexes in C(A) satisfying that for each j ∈ ℤ there is n ∈ ℕ such that X_{i}^{j} → X_{i+1}^{j} are split monomorphisms for all i ≥ n. Then we have an exact sequence in C(A)

O → ∐_{i}X_{i}^{▪} ⟶^{1− shift} ∐_{i}X_{i}^{▪} → \underrightarrow{lim} X_{i}^{▪} → O

which belongs to S_{C(A)}. In particular, \underrightarrow{lim} X_{i}^{▪} ≅ hlim_{⟶} X_{i}^{▪} in K(A).

2. Assume A satisfies the condition Ab3*. and X_{i+1}^{▪} → X_{i}^{▪} a sequence of complexes in C(A) satisfying that for each j ∈ ℤ there is n ∈ ℕ such that X_{i}^{j} → X_{i+1}^{j} are split epimorphisms for all i ≥ n. Then we have an exact sequence in C(A)

O → \underleftarrow{lim} X_{i}^{▪} → ∏_{i}X_{i}^{▪} ⟶^{1− shift} ∏_{i}X_{i}^{▪} → O

which belongs to S_{C(A)}. In particular, \underleftarrow{lim} X_{i}^{▪} ≅ hlim_{⟵} X_{i}^{▪} in K(A).

Proof. 1. By Lemma 11.6, for any j we have a split exact sequence

O → ∐_{i}X_{i}^{j} ⟶^{1− shift} ∐_{i}X_{i}^{j} → \underrightarrow{lim} X_{i}^{j} → O.

Then we have an exact sequence in C(A)

O → ∐_{i}X_{i}^{▪} ⟶^{1− shift} ∐_{i}X_{i}^{▪} → \underrightarrow{lim} X_{i}^{▪} → O

which belongs to S_{C(A)}. The last assertion follows by Proposition 6.12. 2. Similarly.

Remark 11.8. The above \underrightarrow{lim} X_{i}^{▪} and \underleftarrow{lim} X_{i}^{▪} are the filtered colimit and the filtered limit in C(A), but are not the filtered colimit and the filtered limit in K(A) (see Lemma 16.17).

Remark 11.9. 1. If A satisfies the condition Ab5, then for a sequence {X_{i}^{▪} → X_{i+1}^{▪}}_{i∈ℕ} of morphisms in D(A), we have exact sequences

O → ∐_{i} H^{n}(X_{i}^{▪}) → ∐_{i} H^{n}(X_{i}^{▪}) → H^{n}(hlim_{⟶} X_{i}^{▪}) → O

for all n ∈ ℕ.

2. If A satisfies the condition Ab5 and {X_{i}^{▪} → X_{i+1}^{▪}}_{i∈ℕ} a sequence of morphisms in C(A), then by Proposition 2.25, 3 we have an exact sequence in C(A)

O → ∐_{i}X_{i}^{▪} → ∐_{i}X_{i}^{▪} → \underrightarrow{lim} X_{i}^{▪} → O

and we have a quasi-isomorphism

hlim_{⟶} X_{i}^{▪} → \underrightarrow{lim} X_{i}^{▪}.

3. Assume A satisfies the condition Ab4*, and let {X_{i+1}^{▪} → X_{i}^{▪}}_{i∈ℕ} be a sequence of morphisms in D(A) satisfying that for any n ∈ ℤ there is k ∈ ℕ such that H^{n}(X_{i+1}^{▪}) ≅ H^{n}(X_{i}^{▪}) for all i > k. Then we have exact sequences

O → H^{n}(hlim_{⟵} X_{i}^{▪}) → ∏_{i} H^{n}(X_{i}^{▪}) → ∏_{i} H^{n}(X_{i}^{▪}) → O

for all n ∈ ℕ.

Proposition 11.10. For an abelian category A, the following hold.

1. If A satisfies the condition Ab4 with enough projectives, then every object of K(A) is quasi-isomorphic to a complex P^{▪} of projectives with Hom_{K(A)}(P^{▪}, K^{ϕ}(A)) = 0.

2. If A satisfies the condition Ab4* with enough injectives, then every object of K(A) is quasi-isomorphic to a complex P^{▪} of injectives with Hom_{K(A)}(K^{ϕ}(A), I^{▪}) = 0.

Proof. 1. For a complex X^{▪} ∈ K(A), we have morphisms of complexes σ_{≤i}X^{▪} → σ_{≤i+1}X^{▪} → X^{▪}. According to Lemma 10.10, there is P_{i}^{▪} ∈ K^{−}(Proj A) which has a quasi-isomorphism P_{i}^{▪} → σ_{≤i}X^{▪}, and we have a commutative diagram in K(A)

P_{i}^{▪} ⟶ P_{i+1}^{▪}
↓  ↓
σ_{≤i}X^{▪} ⟶ σ_{≤i+1}X^{▪}

By Remark 11.9, we have a quasi-isomorphism

hlim_{⟶} P_{i}^{▪} → hlim_{⟶} σ_{≤i}X^{▪}.

By Proposition 11.7, hlim_{⟶} σ_{≤i}X^{▪} ≅ \underrightarrow{lim} σ_{≤i}X^{▪} = X^{▪} in K(A). By the construction, hlim_{⟶} P_{i}^{▪} is a complex of projectives. Since we have an exact sequence

∏_{i} Hom_{K(A)}(T P_{i}^{▪}, K^{ϕ}(A)) → Hom_{K(A)}(P^{▪}, K^{ϕ}(A)) → ∏_{i} Hom_{K(A)}(T P_{i}^{▪}, K^{ϕ}(A)),

we have Hom_{K(A)}(P^{▪}, K^{ϕ}(A)) = 0 by Lemma 10.8.

2. Similarly.

Definition 11.11. 1. In the case that A satisfies the condition Ab4 with enough projectives, we define the triangulated full subcategory K^{s}(Proj A) of K(A) consisting of complexes P^{▪} of projectives such that Hom_{K(A)}(P^{▪}, K^{ϕ}(A)) = 0. Then (K^{s}(Proj A), K^{ϕ}(A)) is a stable t-structure in K(A).

2. In the case that A satisfies the condition Ab4* with enough injectives, we define the triangulated full subcategory K^{s}(Inj A) of K(A) consisting of complexes I^{▪} of injectives such that Hom_{K(A)}(K^{ϕ}(A), I^{▪}) = 0. Then (K^{ϕ}(A), K^{s}(Inj A)) is a stable t-structure in K(A).

They are often called K-projective complexes (resp., K-injective complexes).

Proposition 11.12. The following hold.

1. If A satisfies the condition Ab4 with enough projectives, then we have an isomorphism

Hom_{K(A)}(P^{▪}, Y^{▪}) ≅ Hom_{D(A)}(P^{▪}, Y^{▪})

for P^{▪} ∈ K^{s}(Proj A), Y^{▪} ∈ K(A).

2. If A satisfies the condition Ab4* with enough injectives, then we have an isomorphism

Hom_{K(A)}(X^{▪}, I^{▪}) ≅ Hom_{D(A)}(X^{▪}, I^{▪})

for X^{▪} ∈ K(A), I^{▪} ∈ K^{s}(Inj A).

Theorem 11.13. The following hold.

1. If A satisfies the condition Ab4 with enough projectives, then we have a triangle equivalence

K^{s}(Proj A) ≅^{t} D(A).

2. If A satisfies the condition Ab4* with enough injectives, then we have a triangle equivalence

K^{s}(Inj A) ≅^{t} D(A).

Proof. 1. By Proposition 11.10, (K^{s}(Proj A), K^{ϕ}(A)) is a stable t-structure in K(A). According to Proposition 9.16, Remark 9.17, we get the statement.

2. Similarly.

Remark 11.14 (Set-Theoretic Remark 2). Conversely, in the case that A satisfies the condition Ab4 with enough projectives, we can define D^{−}(A) = K^{−}(Proj A), and D(A) = K^{s}(Proj A). Then we can bypass Remark 7.7.

Similarly, in the case that A satisfies the condition Ab4* with enough injectives, we can define D^{+}(A) = K^{+}(Inj A), and D(A) = K^{s}(Inj A).

Proposition 11.15. The following hold.

1. Let C be a triangulated category with coproducts. For a sequence {X_{i} ⟶^{f_{i}} X_{i+1}}_{i∈ℕ}, if there is n ∈ ℕ such that f_{i} are split monomorphisms for all i ≥ n, then the structural morphism ∐_{i}X_{i} → hlim_{⟶} X_{i} is a split epimorphism.

2. Let C be a triangulated category with products. For a sequence {X_{i+1} ⟶^{f_{i}} X_{i}}_{i∈ℕ}, if there is n ∈ ℕ such that f_{i} are split epimorphisms for all i ≥ n, then the structural morphism hlim_{⟵} X_{i} → ∏_{i}X_{i} is a split monomorphism.

Proof. 1. For any M ∈ C and n ∈ ℕ, we have a commutative diagram

Hom_{C}(∐_{i}T^{n}X_{i}, M) ⟶^{1− shift} Hom_{C}(∐_{i}T^{n}X_{i}, M)
↓^{≀}  ↓^{≀}
∏_{i} Hom_{C}(T^{n}X_{i}, M) ⟶^{1− shift} ∏_{i} Hom_{C}(T^{n}X_{i}, M).

Then by Exercise 11.5.2, Hom_{C}(∐_{i}T^{n}X_{i}, M) ⟶^{1− shift} Hom_{C}(∐_{i}T^{n}X_{i}, M) is epic, and then T^{n}∐_{i}X_{i} ⟶^{1− shift} T^{n}∐_{i}X_{i} is split monic. Therefore, hlim_{⟶} X_{i} → T ∐_{i}X_{i} = 0, and hence ∐_{i}X_{i} → hlim_{⟶} X_{i} is split epic.

2. Similarly.

Lemma 11.16. Let C be a triangulated category with coproducts. For a sequence {X_{i} ⟶^{f_{i}} X_{i+1}}_{i∈ℕ} where X_{i} = X and f_{i} = 1_{X} for all i, we have an isomorphism in C

hlim_{⟶} X_{i} ≅ X.

Proof. Let ∐_{i}X_{i} ⟶^{p} ∐_{i}X_{i} ⟶^{q} hlim_{⟶} X_{i} ⟶^{r} ∐_{i}T X_{i} be a triangle, and α = ∑_{i}(1_{X})_{i} : ∐_{i}X_{i} → X. By easy calculation, the following hold.

(a) αp = 0.

(b) If a morphism ϕ : ∐_{i}X_{i} → Y satisfies ϕp = 0, then there is a unique f : X → Y such that ϕ = fα.

By the property of hlim_{⟶} and the above, there exist h : X → hlim_{⟶} X_{i} and k : hlim_{⟶} X_{i} → X such that α = kq and q = hα. Since q = hkq and α = khα, we have hk = 1 and kh = 1 by (b) and Proposition 11.15.

Proposition 11.17. Let C be a triangulated category with coproducts. Let e : X → X be a morphism in C such that e^{2} = e. Then e splits in C.

Proof. We consider three sequences

(A) X ⟶^{e} X ⟶^{e} . . ..

(B) X ⟶^{1−e} X ⟶^{1−e} . . ..

(C) X⊕X \underrightarrow{[ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ]} X⊕X \underrightarrow{[ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ]} . . ..

Then we have an isomorphism α = [ \begin{matrix}e & 1−e \\ 1−e & e \end{matrix} ] : (A)⊕(B) → (C) of sequences. Thus hlim_{⟶} (C) ≅ Y ⊕ Z in C, where Y = hlim_{⟶} (A), Z = hlim_{⟶} (B). For a sequence (C), we have a commutative diagram

∐_{i}(X ⊕ X)_{i}  ⟶^{1−shift}  ∐_{i}(X ⊕ X)_{i} ⟶ hlim_{⟶} (C)
↓^{≀}  ↓^{≀}
(∐_{i}X_{i}) ⊕ (∐_{i}X_{i}) ⟶^{1⊕(1−shift)} (∐_{i}X_{i}) ⊕ (∐_{i}X_{i}).

By Lemma 11.16, we have hlim_{⟶} (C) ≅ X in C. On the other hand, we have a commutative diagram

(∐_{i}X_{i}) ⊕ (∐_{i}X_{i}) (1−shift)⊕(1−shift) ⟶ (∐_{i}X_{i}) ⊕ (∐_{i}X_{i}) ⟶^{β} Y ⊕ Z

↓^{α}  ↓^{α}  ↓^{γ}
∐_{i}(X ⊕ X)_{i}  ⟶^{1−shift}  ∐_{i}(X ⊕ X)_{i}  \underrightarrow{∑_{i}(0 1)_{i}} X

where β = [ \begin{matrix}a & 0 \\ 0 & b \end{matrix}] and γ = [^{ε η}] are isomorphisms. According to Proposition 11.15, (0 1) and β = [ \begin{matrix}a & 0 \\ 0 & b \end{matrix}] are epic, then a and b are epic. Since (1 − e)εb = (1 − e)(0 1) [ \begin{matrix}1−e \\ e \end{matrix}] = 0, we have (1 − e)ε = 0. Similarly, we have eη = 0. Hence we have a morphism δ : X → Y such that εδ = e and δε = 1_{Y}.

Corollary 11.18. Let L be a triangulated full subcategory L of C. If L satisfies the condition (L2) every coproduct of objects in L is in L, then L is a localizing subcategory of C.

Corollary 11.19. Let R be a commutative complete local ring, A a finite Ralgebra. Then D^{b}(mod A) is a Krull-Schmidt category.

Proof. For a complex Y^{▪} ∈ D^{b}(mod A), we may assume that H^{m}(Y^{▪}) ≠ 0, H^{n}(Y^{▪}) ≠ 0, and H^{i}(Y^{▪}) = 0 for i < m, n < i for some m < n. We define TL(Y^{▪}) = n−m. For complexes X^{▪}, Y^{▪} ∈ D^{b}(mod A), by induction on the lexicographic order (TL(X^{▪}), TL(Y^{▪})), we show that Hom_{D^{b}(mod A)}(X^{▪}, Y^{▪}) is a finitely generated R-module. Let Y_{1}^{▪} = σ_{≤n−1}(Y^{▪}) and Y_{2}^{▪} = σ_{>n−1}(Y^{▪}), then we have a triangle in D^{b}(mod A)

Y_{1}^{▪} → Y^{▪} → Y_{2}^{▪} → T Y_{1}^{▪}.

Then we have an exact sequence

Hom_{D^{b}(mod A)}(X^{▪}, Y_{1}^{▪}) → Hom_{D^{b}(mod A)}(X^{▪}, Y^{▪}) → Hom_{D^{b}(mod A)}(X^{▪}, Y_{2}^{▪}).

By the assumption, Hom_{D^{b}(mod A)}(X^{▪}, Y_{1}^{▪}) and Hom_{D^{b}(mod A)}(X^{▪}, Y_{2}^{▪}) are finitely generated R-modules. Then Hom_{D^{b}(mod A)}(X^{▪}, Y^{▪}) is a finitely generated R-module. Similarly, for a triangle σ_{≤n−1}(X^{▪}) → X^{▪} → σ_{>n−1}(X^{▪}) → T σ_{≤n−1}(X^{▪}), we have the same result. In particular, End_{D^{b}(mod A)}(X^{▪}) is a semiperfect ring. For an idempotent e ∈ End_{D^{b}(mod A)}(X^{▪}), by Example 10.14, we may consider an idempotent in D_{c}^{b}(Mod A) ⊂ D^{b}(Mod A). By Proposition 11.17, there exist a complex Y^{▪} ∈ D^{b}(Mod A) and morphisms p : X^{▪} → Y^{▪}, q : Y^{▪} → X^{▪} such that qp = e and pq = 1_{Y^{▪}}. Since every H^{i}(Y^{▪}) is a direct summand of H^{i}(X^{▪}), Y^{▪} ∈ D_{c}^{b}(Mod A). According to Proposition 3.7, we complete the proof.

12. Derived Functors

Throughout this section, A, B and C are abelian categories.

12.1. Derived Functors.

Definition 12.1. A triangulated full subcategory K^{∗}(A) of is called a quotientizing subcategory (often called localizing subcategory) if the canonical functor

K^{∗}(A)/ K^{∗,ϕ}(A) → D(A)

is fully faithful, where K^{∗,ϕ}(A) = K^{ϕ}(A) ∩ K^{∗}(A). If K^{∗}(A) is a quotientizing subcategory of K(A), we denote by D^{∗}(A) the quotient category K^{∗}(A)/ K^{∗,ϕ}(A) and by Q_{A}^{∗} : K^{∗}(A) → D^{∗}(A) the canonical quotient functor.

Definition 12.2 (Right Derived Functor). Let K^{∗}(A) be a quotientizing subcategory of K(A) and F : K^{∗}(A) → K(B) a ∂-functor. The right derived functor of F is a ∂-functor

R^{∗}F : D^{∗}(A) → D(B)

together with a functorial morphism of ∂-functors

ξ ∈ ∂ Mor(Q_{B}F, R^{∗}F Q_{A}^{∗})

with the following property: For G ∈ ∂(D^{∗}(A), D(B)) and ζ ∈ ∂ Mor(Q_{B}F, GQ_{A}^{∗}), there exists a unique morphism η ∈ ∂ Mor(R^{∗}F, G) such that

ζ = (ηQ_{A}^{∗})ξ.

In other words, we can simply write the above using functor categories. For triangulated categories C, C′, the ∂-functor category ∂(C, C′) is the category (?) consisting of ∂-functors from C to C′ as objects and ∂-functorial morphisms as morphisms. Then we have

∂ Mor(Q_{B}F, −Q_{A}^{∗}) ≅ ∂ Mor(R^{∗}F, −)

as functors from ∂(D^{∗}(A), D(B)) to 𝔖𝔢𝔱 (See Lemma 1.8).

Definition 12.3. Let K^{∗}(A) be a quotientizing subcategory of K^{∗}(A) and F : K^{∗}(A) → K(B) a ∂-functor. When F has a right derived functor R^{∗}F : D^{∗}(A) → D(B), we define R^{i} F = H^{i} R^{∗}F : D^{∗}(A) → B (i ∈ ℤ).

Proposition 12.4. Let K^{∗}(A) be a quotientizing subcategory of K(A) and F : K^{∗}(A) → K(B) a ∂-functor. Assume F has a right derived functor R^{∗}F : D^{∗}(A) → D(B). Then for any exact sequence O → X^{▪} → Y^{▪} → Z^{▪} → O in C(A) we have a long exact sequence

… → R^{i} F (X^{▪}) → R^{i} F (Y^{▪}) → R^{i} F (Z^{▪}) → R^{i+1} F (X^{▪}) → ….

Proof. By Proposition 10.4, it is easy.

Theorem 12.5 (Existence Theorem). Let K^{∗}(A) be a quotientizing subcategory of K(A) and F : K^{∗}(A) → K(B) a ∂-functor. Assume there exists a triangulated full subcategory L of K^{∗}(A) such that

(a) for any X^{▪} ∈ K^{∗}(A) there is a quasi-isomorphism X^{▪} → I^{▪} with I^{▪} ∈ L,

(b) Q_{B}F (L^{ϕ}) = {O}, where L^{ϕ} = K^{ϕ}(A) ∩ L. Then there exists the right derived functor (R^{∗}F, ξ) such that ξ_{I} : Q_{B}F I^{▪} → R^{∗}F I^{▪} is a quasi-isomorphism for I^{▪} ∈ L.

Proof. Let E : L → K^{∗}(A) be the embedding functor, then by the assumption (a) and Proposition 7.16 the canonical functor \overline{E} : L/L^{ϕ} → D^{∗}(A) is an equivalence. Let J : D^{∗}(A) → L/L^{ϕ} be a quasi-inverse of \overline{E}. By the assumption (b) and Proposition 8.7 there is a ∂-functor \overline{F} : L/L^{ϕ} → D(B) such that Q_{B}F E = \overline{F} Q_{L}, where Q_{L} : L → L/L^{ϕ} is the canonical quotient. Put R^{∗}F = \overline{F} J. Since Q_{A}^{∗}E = \overline{E}Q_{L}, we have

∂ Mor(Q_{B}F E, GQ_{A}^{∗}E) ≅ ∂ Mor(\overline{F} Q_{L}, G\overline{E}Q_{L})
≅ ∂ Mor(\overline{F} J\overline{E}, G\overline{E})
≅ ∂ Mor(\overline{F} J, G).

It remains to show that

∂ Mor(Q_{B}F, GQ_{A}^{∗}) ≅ ∂ Mor(Q_{B}F E, GQ_{A}^{∗}E) (ϕ ↦ ϕE).

Let ϕ ∈ ∂ Mor(Q_{B}F, GQ_{A}^{∗}) with ϕE = 0. For any X^{▪} ∈ K^{∗}(A) there exists I^{▪} ∈ L which has a quasi-isomorphism s : X^{▪} → I^{▪}. Then ϕ_{X} = (GQ_{A}^{∗}s)^{−1}ϕ_{I}Q_{B}F s = 0, and hence ϕ = 0. Given ψ ∈ ∂ Mor(Q_{B}F E, GQ_{A}^{∗}E), for any X^{▪} ∈ K^{∗}(A), let ϕ_{X} = (GQ_{A}^{∗}s)^{−1}ψ_{I}Q_{B}F s for some quasi-isomorphism s : X^{▪} → I^{▪}, with I^{▪} ∈ L. For another quasi-isomorphism s′ : X^{▪} → I′^{▪}, by the assumption (a), we have a commutative diagram

X^{▪} ⟶^{s} I^{▪}
s′↓  ↓_{t′}
I′^{▪} ⟶^{t} I′′^{▪}

where all morphisms are quasi-isomorphisms and I′′^{▪} ∈ L. Then we have

(GQ_{A}^{∗}s)^{−1}ψ_{I}Q_{B}F s = (GQ_{A}^{∗}t′s)^{−1}ψ_{I′′}Q_{B}F t′s
= (GQ_{A}^{∗}ts′)^{−1}ψ_{I′′}Q_{B}F ts′
= (GQ_{A}^{∗}s′)^{−1}ψ_{I′}Q_{B}F s′.

It is not hard to see that ϕ ∈ ∂ Mor(Q_{B}F, GQ_{A}^{∗}). The last assertion is easy to check.

Corollary 12.6. Assume that there exists an additive subcategory I of A such that (a) every X ∈ A has a monomorphism to an object in I.

(b) for an exact sequence O → X → Y → Z → O in A with X ∈ I, Y ∈ I if and only if Z ∈ I,

(c) for an exact sequence O → X → Y → Z → O in A with X, Y, Z ∈ I, then O → F X → F Y → F Z → O is exact in B.

For any additive functor F : A → B, R^{+}F : D^{+}(A) → D(B) exists.

Proof. Let L = K^{+}(I), then it is easy to see that L satisfies the conditions of Theorem 12.5.

Proposition 12.7. Let K^{∗∗}(A) ⊂ K^{∗}(A) be quotientizing subcategories of K(A) and F : K^{∗}(A) → K(B) a ∂-functor. Assume K^{∗}(A) has a triangulated full subcategory L such that

(a) for any X^{▪} ∈ K^{∗}(A), there exists a quasi-isomorphism X^{▪} → I^{▪} with I^{▪} ∈ L,

(b) for any X^{▪} ∈ K^{∗∗}(A), there exists a quasi-isomorphism X^{▪} → I^{▪} with I^{▪} ∈ L ∩ K^{∗∗}(A), and

(c) Q_{B}F (L^{ϕ}) = {O}, where L^{ϕ} = K^{ϕ}(A) ∩ L. Then both F and F |_{K^{∗∗}(A)} have the right derived functors (R^{∗}F, ξ) and (R^{∗∗}(F |_{K^{∗∗}(A)}), ζ), respectively, and the canonical ∂-functorial morphism

ϕ : R^{∗∗}(F |_{K^{∗∗}(A)}) → R^{∗}F |_{K^{∗∗}(A)}

is an isomorphism.

Proof. By Theorem 12.5 both F and F |_{K^{∗∗}(A)} have the right derived functors (R^{∗}F, ξ) and (R^{∗∗}(F |_{K^{∗∗}(A)}), ζ), respectively, and we have a unique ∂-functorial morphism

ϕ : R^{∗∗}(F |_{K^{∗∗}(A)}) → R^{∗}F |_{K^{∗∗}(A)}

such that ξ|_{K^{∗∗}(A)} = (ϕQ_{A}^{∗∗})ζ. For any I^{▪} ∈ L ∩ K^{∗∗}(A), by Theorem 12.5 both ξ_{I} and ζ_{I} are isomorphisms, so that ϕ_{QI} is an isomorphism. Thus, since by assumption (b), the canonical functor Q : L ∩ K^{∗∗}(A) → D^{∗∗}(A) is dense, ϕ is an isomorphism.

Proposition 12.8. Let K^{∗}(A) be a quotientizing subcategory of K(A) and F : K^{∗}(A) → K(B) a ∂-functor. Let K^{†}(B) be a quotientizing subcategory of K(B) and G : K^{†}(B) → K(C) a ∂-functor. Assume

(a) K^{∗}(A) has a triangulated full subcategory L for which the assumptions 1, 2 of Theorem 12.5 are satisfied,

(b) K^{†}(B) has a triangulated full subcategory M for which the assumptions 1, 2 of Theorem 12.5 are satisfied, and

(c) F (K^{∗}(A)) ⊂ K^{†}(B) and F (L) ⊂ M.

Then F, G and GF have the right derived functors (R^{∗}F, ξ), (R^{†}G, ζ) and (R^{∗}(GF), η) with R^{∗}F (D^{∗}(A)) ⊂ C^{†}(B), and the canonical homomorphism

ϕ : R^{∗}(GF) → R^{†}G ∘ R^{∗}F

is an isomorphism.

Proof. By Theorem 12.5 F and G have the right derived functors (R^{∗}F, ξ) and (R^{†}G, ζ), respectively. Let X^{▪} ∈ L be acyclic. Then, since Q(F (X^{▪})) = 0, F (X^{▪}) is acyclic and Q(G(F (X^{▪}))) = 0. Thus, again by Theorem 12.5 GF has a right derived functor (R^{∗}(GF), η). Also, for any X^{▪} ∈ D^{∗}(A), since we have a quasiisomorphism X^{▪} → I^{▪} with I^{▪} ∈ L, R^{∗}F (X^{▪}) ≅ R^{∗}F (Q(I^{▪})) ≅ Q(F (I^{▪})) ∈ D^{†}(B). Thus by Theorem 12.5 we have a unique homomorphism of ∂-functors

ϕ : R^{∗}(GF) → R^{†}G∘R^{∗}F

such that (R^{†}Gξ)(ζF) = (ϕQ)η. Let I^{▪} ∈ L. Then ξ_{I}, ζ_{F I} and η_{I} are isomorphisms, so that ϕ_{QI} is an isomorphism. Thus ϕ is an isomorphism, because Q : L → D^{∗}(A) is dense

12.2. Way-out Functors.

Definition 12.9 (Way-out Functor). Let A, B be abelian categories, A′ a thick abelian full subcategory of A. Let K^{∗}(A) be a quotientizing subcategory of K(A). A ∂-functor F : D_{A′}^{∗}(A) → D(B) is called way-out right (resp., way-out left) provided that for any n_{1} ∈ ℤ there exists n_{2} ∈ ℤ such that if X^{▪} ∈ D^{∗}(A) is a complex with H^{i}(X^{▪}) = O for all i < n_{2} (resp., i > n_{2}), then H^{i}(F X^{▪}) = O for all i < n_{1} (resp., i > n_{1}).

Moreover, F is called way-out in both directions if F is way-out right and way-out left.

Lemma 12.10. For a complex X^{▪} ∈ C(A), we have triangles in D(A) 1.

τ_{≥n−1}X^{▪} → τ_{≥n}X^{▪} → X^{n}[−n] → τ_{≥n−1}X^{▪}[1].

2.

H^{n}(X^{▪})[−n] → σ_{>n−1}X^{▪} → σ_{>n}X^{▪} → H^{n}(X^{▪})[1 − n].

Proof. By 10.6, we have an exact sequence

O → Y^{▪} → σ_{>n−1}X^{▪} → σ_{>n}X^{▪} → O,

where Y^{▪} : Y^{n−1} → Y^{n} = Im d_{X}^{n−1} → Ker d_{X}^{n}. Then it is easy to see Y^{▪} ≅ H^{n}(X^{▪})[−n].

Proposition 12.11. Let A, B be abelian categories, A′ a thick abelian full subcategory of A. Let F^{∗}, G^{∗} : D_{A′}^{∗}(A) → D(B) be ∂-functors, and η^{∗} ∈ ∂ Mor(F^{∗}, G^{∗}), where ∗ = nothing, +, −, b. Then the following hold.

1. If η^{b}(X) are isomorphisms for all X ∈ A′, then η^{b} is an isomorphism.

2. Assume that F and G are way-out right. If η^{+}(X) are isomorphisms for all X ∈ A′, then η^{+} is an isomorphism.

3. Assume that F and G are way-out in both directions. If η(X) are isomorphisms for all X ∈ A′, then η is an isomorphism.

4. Let I be a collection of objects of A′ such that every X ∈ A′ admits a monomorphism to an object of I. Assume that F and G are way-out right. If η^{∗}(I) are isomorphisms for all I ∈ I, then η^{∗}(X) are isomorphisms for all X ∈ A′.

Proof. 1. Let X^{▪} ∈ D^{b}(A). For n ≫ 0, σ_{>n}X^{▪} = O, and then η(σ_{>n}X^{▪}) is an isomorphism. By Lemma 12.10, η(σ_{>n−1}X^{▪}) is an isomorphism. Since X^{▪} ∈ D^{b}(A), we get the statement.

2. Let X^{▪} ∈ D^{+}(A). For any n ∈ ℤ, we show that H^{n}(η(X^{▪})) is an isomorphism. Put n_{1} = n + 1. Then there exists n_{2} ∈ ℤ such that if Y^{▪} ∈ D^{∗}(A) is a complex with H^{i}(Y^{▪}) = O for all i < n_{2}, then H^{i}(F Y^{▪}) = O and H^{i}(GY^{▪}) = O for all i < n_{1}. Since H^{i}(σ_{>n_{2}}X^{▪}) = O for i < n_{1}, we have

\begin{aligned}H^{n}(F σ_{>n_{2}}X^{▪}) & = H^{n−1}(F σ_{>n_{2}}X^{▪}) = O, \\ H^{n}(Gσ_{>n_{2}}X^{▪}) & = H^{n−1}(Gσ_{>n_{2}}X^{▪}) = O. \end{aligned}

Considering a triangle σ_{≤n_{2}}X^{▪} → X^{▪} → σ_{>n_{2}}X^{▪} → σ_{≤n_{2}}X^{▪}[1], we have a commutative diagram

H^{n}(F σ_{≤n_{2}}X^{▪}) ⟶^{∼} H^{n}(F X^{▪})
≀↓  ↓
H^{n}(Gσ_{≤n_{2}}X^{▪}) ⟶^{∼} H^{n}(GX^{▪}).

where all horizontal arrows are isomorphisms. By 1, the left vertical arrow are an isomorphism, and hence η(H^{n}(X^{▪})) is an isomorphism.

3. As in 2, for any X^{▪} ∈ D(A), η(σ_{>0}X^{▪}) is an isomorphism. Considering σ_{≤0}X^{▪} → X^{▪} → σ_{>0}X^{▪} → σ_{≤0}X^{▪}[1], η(X^{▪}) is an isomorphism.

4. For X ∈ A′, by the dual of Lemma 10.10, there is a resolution I^{▪} with each I^{i} ∈ I. By replacing σ by τ in 1 and 2, we have the statement.

Proposition 12.12. Let A, B be abelian categories, A′, B′ thick abelian full subcategories of A, B, respectively. Let F^{∗} : D_{A′}^{∗}(A) → D(B) be a ∂-functor, where ∗ = nothing, +, −, b. Then the following hold.

1. If F^{b}(X) ∈ D_{B′}(B) for all X ∈ A′, then F^{b}(X) ∈ D_{B′}(B) for all X^{▪} ∈ D_{A′}^{b}(A).

2. Assume that F and G are way-out right. If F^{b}(X) ∈ D_{B′}(B) for all X ∈ A′, then F^{b}(X) ∈ D_{B′}(B) for all X^{▪} ∈ D_{A′}^{+}(A).

3. Assume that F and G are way-out in both directions. If F^{b}(X) ∈ D_{B′}(B) for all X ∈ A′, then F^{b}(X) ∈ D_{B′}(B) for all X^{▪} ∈ D_{A′}(A).

4. Let I be a collection of objects of A′ such that every X ∈ A′ admits a monomorphism to an object of I. Assume that F and G are way-out right. If F (I) ∈ D_{B′}(B) for all I ∈ I, then F (X) ∈ D_{B′}(B) for all X ∈ A′.

Proof. The same as the proof of Proposition 12.12 (left to the reader).

13. Double Complexes

Throughout this section, A is an abelian category.

Definition 13.1 (Double Complex). A double complex C^{▪▪} is a bigraded object (C^{p,q})_{p,q∈ℤ} of A together with d_{I}^{p,q} : C^{p,q} → C^{p+1,q} and d_{II}^{p,q} : C^{p,q} → C^{p,q+1} such that

\begin{aligned}C^{▪q} & = (C^{p,q}, d_{I}^{p,q} : C^{p,q} → C^{p+1,q}) \\ C^{p▪} & = (C^{p,q}, d_{II}^{p,q} : C^{p,q} → C^{p,q+1}) \end{aligned}

are complexes satisfying d_{I}^{p,q+1}d_{II}^{p,q} + d_{II}^{p+1,q}d_{I}^{p,q} = 0.

A morphism f of double complexes X^{▪▪} to Y^{▪▪} is a collection of morphisms f^{p,q} : X^{p,q} → Y^{p,q} such that f^{▪q} : X^{▪q} → Y^{▪q} and f^{p▪} : X^{p▪} → Y^{p▪} are morphisms of complexes for all p, q ∈ ℤ.

We denote by C^{2}(A) the categories of double complexes of A. Auto-equivalences T_{I}, T_{II} : C^{2}(A) → C^{2}(A) are called the translations if (T_{I}X^{▪▪})^{p,q} = X^{p+1,q} and (T_{I}d_{# X})^{p,q} = −d_{# X}^{p+1,q} and (T_{II}X^{▪▪})^{p,q} = X^{p,q+1} and (T_{II}d_{# X})^{p,q} = −d_{# X}^{p,q+1} for any complex X^{▪▪} = (X^{p,q}, d_{I X}^{p,q}, d_{II X}^{p,q}), where # = I, II.

Moreover, an r-tuple complex C^{▪^{r}} is an r-tuple graded object (C^{p})_{p∈ℤ^{r}} of A together with d_{i}^{p} : C^{p} → C^{p+e_{i}} (1 ≤ i ≤ r) such that

\begin{aligned}d_{i}^{2} & = 0 (1 ≤ i ≤ r), \\ d_{i}d_{j} + d_{j}d_{i} & = 0 for all i, j, \end{aligned}

where e_{i} = (0, …, 0, 1, 0, …, 0)_{ith component}.

Proposition 13.2. Let C_{I}(A) (resp., C_{II}(A)) be the full subcategory of C^{2}(A) consisting of complexes X^{▪▪} such that X^{p,q} = O for all q ≠ 0 (resp., p ≠ 0). Then we have C(A) ≅ C_{I}(A) ≅ C_{II}(A) and C^{2}(A) ≅ C(C_{I}(A)) ≅ C(C_{II}(A)). In particular, C^{2}(A) is an abelian category.

Proof. We define a functor F_{I} : C^{2}(A) → C(C_{I}(A)) as follows. For any double complex X^{▪▪} = (X^{p,q}, d_{I X}^{p,q}, d_{II X}^{p,q}), F_{I}(X^{p,q}, d_{I X}^{p,q}, d_{II X}^{p,q}) = (X^{▪,q}, d_{II X}^{▪,q}) where X^{▪,q} = (X^{p,q}, (−1)^{q}d_{I X}^{p,q}). For a morphism f : X^{▪▪} → Y^{▪▪}, F_{I}(f)^{p,q} = f^{p,q}. Then it is easy to see that F_{I} is an equivalence.

By the above, we can deal with C^{2}(A) as C(C_{I}(A)) or C(C_{II}(A)).

Definition 13.3 (Truncations). For a double complex X^{▪▪} = (X^{p,q}, d_{I}^{p,q}, d_{II}^{p,q}), we define the following truncations:

(σ_{>n}^{I}X^{▪▪})^{p,q} = \begin{cases}O if p < n \\ Im d_{I}^{p,q} if p = n \\ X^{p,q} if p > n \end{cases} (σ_{≤n}^{I}X^{▪▪})^{p,q} = \begin{cases}X^{p,q} if p < n \\ Ker d_{I}^{p,q} if p = n \\ O if p > n \end{cases}

(σ_{>n}^{II}X^{▪▪})^{p,q} = \begin{cases}O if q < n \\ Im d_{II}^{p,q} if q = n \\ X^{p,q} if q > n \end{cases} (σ_{≤n}^{II}X^{▪▪})^{p,q} = \begin{cases}X^{p,q} if q < n \\ Ker d_{II}^{p,q} if q = n \\ O if q > n \end{cases}

(τ_{≤n}^{I}X^{▪▪})^{p,q} = \begin{cases}X^{p,q} if p ≤ n \\ O if p > n \end{cases}  (τ_{≥n}^{I}X^{▪▪})^{p,q} = \begin{cases}O if p < n \\ X^{p,q} if p ≥ n \end{cases}

(τ_{≤n}^{II}X^{▪▪})^{p,q} = \begin{cases}X^{p,q} if q ≤ n \\ O if q > n \end{cases}  (τ_{≥n}^{II}X^{▪▪})^{p,q} = \begin{cases}O if q < n \\ X^{p,q} if q ≥ n \end{cases}

Lemma 13.4. For a double complex X^{▪▪} = (X^{p,q}, d_{I}, d_{II}), the following hold.

1. We have exact sequences in C^{2}(A)

O → σ_{≤n}^{#}X^{▪▪} → X^{▪▪} → σ_{>n}^{#}X^{▪▪} → O

for all n ∈ ℤ with # = I, II.

2. We have exact sequences in C^{2}(A)

O → τ_{≥n}^{#}X^{▪▪} → X^{▪▪} → τ_{≤n−1}^{#}X^{▪▪} → O

for all n ∈ ℤ with # = I, II.

Definition 13.5 (Total Complexes). For a double complex X^{▪▪} = (X^{p,q}, d_{I}^{p,q}, d_{II}^{p,q}), we define the total complexes

Tot C^{▪▪} = (X^{n}, d^{n}), where X^{n} = ∐_{p+q=n}C^{p,q}, d^{n} = ∐_{p+q=n}d_{I}^{p,q} + d_{II}^{p,q}

Tot^{∧}C^{▪▪} = (Y^{n}, d^{n}), where Y^{n} = ∏_{p+q=n}C^{p,q}, d^{n} = ∏_{p+q=n}d_{I}^{p,q} + d_{II}^{p,q}.

Moreover, for an r-tuple complex X^{▪^{r}} = (X^{p}, d_{i}^{p}) (1 ≤ i ≤ r), we define the total complexes

Tot C^{▪^{r}} = (X^{n}, d^{n}), where X^{n} = ∐_{|p|=n}C^{p}, d^{n} = ∐_{|p|=n} ∑_{i=1}^{r} d_{i}^{p}

Tot^{∧}C^{▪^{r}} = (Y^{n}, d^{n}), where Y^{n} = ∏_{|p|=n}C^{p}, d^{n} = ∏_{|p|=n} ∑_{i=1}^{r} d_{i}^{p},

where |p| = |(p_{1}, …, p_{r})| = p_{1} + … + p_{r}.

Lemma 13.6. The following hold.

1. If A satisfies the condition Ab4, then the functor Tot : C^{2}(A) → C(A) preserves translations, exact sequences and coproducts.

2. If A satisfies the condition Ab4*, then the functor Tot^{∧} : C^{2}(A) → C(A) preserves translations, exact sequences and products.

Lemma 13.7. For a double complex X^{▪▪} = (X^{p,q}, d_{I}, d_{II}), the following hold.

1. If X^{p,q} = O for q < m, n < q for m ≤ n and X^{▪q} are acyclic complexes in C(A) for all q, then Tot X^{▪▪} is acyclic in C(A) .

2. If X^{p,q} = O for q < m, n < q for m ≤ n and X^{p▪} are acyclic complexes in C(A) for all p, then Tot X^{▪▪} is acyclic in C(A) .

3. Assume A satisfies the condition Ab4. If X^{p,q} = O for q > n and X^{▪q} are acyclic complexes in C(A) for all q, then Tot X^{▪▪} is acyclic in C(A) .

4. Assume A satisfies the condition Ab4*. If X^{p,q} = O for q < n and X^{▪q} are

acyclic complexes in C(A) for all q, then Tot^{∧}X^{▪▪} is acyclic in C(A) .

5. Assume A satisfies the condition Ab5. If X^{p,q} = O for q < n and X^{p▪} are acyclic complexes in C(A) for all p, then TotX^{▪▪} is acyclic in C(A) .

6. Assume A = Mod A for a ring A. If X^{p,q} = O for q > n and X^{p▪} are acyclic

complexes in C(A) for all p, then Tot^{∧}X^{▪▪} is acyclic in C(A) .

Proof. 1. Let n_{X} = n − m. By Lemma 13.4, we have an exact sequence in C^{2}(A)

O → τ_{≥n−1}^{II}X^{▪▪} → X^{▪▪} → τ_{≤n}^{II}X^{▪▪} → O.

Then by Lemma 13.6, we have the exact sequence in C(A)

O → Tot τ_{≥n−1}^{II}X^{▪▪} → Tot X^{▪▪} → X^{▪n}[−n] → O.

By the assumption of induction on n_{X}, Tot τ_{≥n−1}^{II}X^{▪▪} is acyclic. Then Tot X^{▪▪} is acyclic because X^{▪n}[−n] is acyclic.

2. Let n_{X} = n − m. By Lemma 13.4, we have an exact sequence in C^{2}(A)

O → σ_{≤n−1}^{II}X^{▪▪} → X^{▪▪} → σ_{>n−1}^{II}X^{▪▪} → O.

Then by Lemma 13.6, we have the exact sequence in C^{2}(A)

O → Tot σ_{≤n−1}^{II}X^{▪▪} → Tot X^{▪▪} → Tot σ_{>n−1}^{II}X^{▪n} → O.

By the assumption of induction on n_{X}, Tot σ_{≤n−1}^{II}X^{▪▪} is acyclic. It is easy to see that Tot σ_{>n−1}^{II}X^{▪n} ≅ M^{▪}(1_{X^{▪n}})[−n] is acyclic. Then Tot X^{▪▪} is acyclic.

3. By Lemma 13.4, we have the canonical morphisms in C^{2}(A)

τ_{≥−r}^{II}X^{▪▪} ⟶^{f_{r}} τ_{≥−(r+1)}^{II}X^{▪▪},

which are term-split monomorphisms for all p, q. Since Tot f_{r} : Tot τ_{≥−r}^{II}X^{▪▪} → Tot τ_{≥−(r+1)}^{II}X^{▪▪} is term-split monomorphisms in A, by Proposition 11.7, we have

hlim_{⟶} Tot τ_{≥−r}^{II}X^{▪▪} ≅ \underrightarrow{lim} Tot τ_{≥−r}^{II}X^{▪▪} ≅ Tot X^{▪▪}.

By 1, Tot τ_{≥−r}^{II}X^{▪▪} is acyclic for all r. Then hlim_{⟶} Tot τ_{≥−r}^{II}X^{▪▪} is acyclic, and hence so is Tot X^{▪▪}.

4. Dual of 3.

5. By Lemma 13.4, we have the canonical morphisms in C^{2}(A)

σ_{≤r}^{II}X^{▪▪} ⟶^{g_{r}} σ_{≤r+1}^{II}X^{▪▪}.

By Exercise 11.5, we have an exact sequence in C(A)

O → ∐_{r} Tot σ_{≤r}^{II}X^{▪▪} → ∐_{r} Tot σ_{≤r}^{II}X^{▪▪} → \underrightarrow{lim} Tot σ_{≤r}^{II}X^{▪▪} → O.

Then we have

hlim_{⟶}Totσ_{≤r}^{II}X^{▪▪} ≅ \underrightarrow{lim} Totσ_{≤r}^{II}X^{▪▪} ≅ TotX^{▪▪}.

By 2, Tot σ_{≤r}^{II}X^{▪▪} is acyclic for all r. Then hlim_{⟶} Tot σ_{≤r}^{II}X^{▪▪} is acyclic, and hence so is TotX^{▪▪}.

6. Dual of 5.

Definition 13.8. We define an embedding functor em^{I} : C(A) → C^{2}(A) as follows. For a complex X^{▪} ∈ C(A),

em^{I}(X^{▪})^{p,q} = \begin{cases}X^{p} if q = 0 \\ O othewise. \end{cases}

Definition 13.9 (Proper Exact). An exact sequence O → X^{▪} ⟶^{f} Y^{▪} ⟶^{g} Z^{▪} → O in C(A) is called proper exact if the induced sequence O → Z^{▪}(X^{▪}) → Z^{▪}(Y^{▪}) → Z^{▪}(Z^{▪}) → O is also exact. In this case, f (resp., g) is called a proper monomorphism (resp., a proper epimorphism).

A complex X^{▪} ∈ C(A) is called a proper projective complex (resp., a proper injective complex) if

X^{▪} ≅ ⨁_{n∈ℤ}P^{n}[−n]⊕⨁_{n∈ℤ} M^{▪}(1_{Q^{n}})[−n − 1]

where P^{n}, Q^{n} are projective (resp., injective) objects of A.

Lemma 13.10. For a complex Z^{▪} ∈ C(A), the following hold.

1. M^{▪} is proper projective if and only if for every proper epimorphism g : X^{▪} → Y^{▪} in C(A), Hom_{C(A)}(M^{▪}, g) is surjective.

2. M^{▪} is proper injective if and only if for every proper monomorphism f : X^{▪} → Y^{▪} in C(A), Hom_{C(A)}(f, M^{▪}) is surjective.

Proof. 1. If M^{▪} is proper projective, then it suffices to check the case that M^{▪} = P [−n] or M^{▪}(1_{P})[−n], where P is a projective object of A, n ∈ ℤ. For any proper epimorphism g : X^{▪} → Y^{▪}, we have commutative diagrams

Hom_{C(A)}(P [−n], X^{▪}) ⟶^{Hom_{C(A)}(P [−n],g)} Hom_{C(A)}(P [−n], Y^{▪})
↓^{≀}  ↓^{≀}
Hom_{A}(P, Z^{n}(X^{▪})) ⟶^{Hom_{A}(P,Z^{n}(g))} Hom_{A}(P, Z^{n}(Y^{▪})),
Hom_{C(A)}(M^{▪}(1_{P})[−n], X^{▪}) ⟶^{Hom_{C(A)}(M^{▪}(1_{P})[−n],g)} Hom_{C(A)}(M^{▪}(1_{P})[−n], Y^{▪})
↓^{≀}  ↓^{≀}
Hom_{A}(P, X^{n−1})  ⟶^{Hom_{A}(P,g^{n−1})}  Hom_{A}(P, Y^{n−1}).

Since P is projective, the bottom arrows of the above diagrams are surjective. Then the top arrows are also surjective. Conversely, let M^{▪} be a complex satisfying the surjective condition. For any epimorphism g : X → Y in A, we have a commutative diagram

Hom_{C(A)}(M^{▪}, X[−n]) ⟶^{Hom_{C(A)}(M^{▪},g[−n])} Hom_{C(A)}(M^{▪}, Y [−n])
↓^{≀}  ↓^{≀}
Hom_{A}(C^{n}(M^{▪}), X) ⟶^{Hom_{A}(C^{n}(M^{▪}),g)} Hom_{A}(C^{n}(M^{▪}), Y).

Since g[−n] : X[−n] → Y [−n] is proper epic, the top arrow is surjective. Then C^{n}(M^{▪}) are projective objects of A. It is easy to see that

M^{▪} ≅ ⨁_{n∈ℤ} M^{▪}(p^{n})[−n − 1],

where p^{n} : C^{n}(M^{▪}) → B^{n}(M^{▪}) are the canonical epimorphisms. We have a commutative diagram

Hom_{C(A)}(M^{▪}(p^{n}), M^{▪}(1_{X})) ⟶^{Hom_{C(A)}(M^{▪}(p^{n}),M^{▪}(g))} Hom_{C(A)}(M^{▪}(p^{n}), M^{▪}(1_{X}))
↓^{≀}  ↓^{≀}
Hom_{A}(B^{n}(M^{▪}), X)  ⟶^{Hom_{A}(B^{n}(M^{▪}),g)}  Hom_{A}(B^{n}(M^{▪}), Y).

Since M^{▪}(p^{n}) is proper projective, the top arrow is surjective, and then the bottom arrow is surjective. Therefore B^{n}(M^{▪}) is a projective object of A. Hence M^{▪} is proper projective.

2. Similarly.

Lemma 13.11 (Proper Resolutions). Assume that A has enough projectives (resp. injectives). Given a complex M^{▪} ∈ C(A), there are proper projective complexes (resp., proper injective complexes) X^{▪n} (n ≥ 0) which has a proper projective resolution (resp., a proper injective resolution) in C(A)

\begin{matrix}… → X^{▪−1} → X^{▪0} → M^{▪} → O \\ (resp., O → M^{▪} → X^{▪0} → X^{▪1} → …). \end{matrix}

Proof. For a complex M^{▪} ∈ C(A), we have exact sequences in C(A)

\begin{matrix}O → Z^{▪}(M^{▪}) → M^{▪} → B^{▪}(M^{▪})[1] → O \\ O → B^{▪}(M^{▪}) → Z^{▪}(M^{▪}) → H^{▪}(M^{▪}) → O. \end{matrix}

For each B^{n}(M^{▪}), there is a projective object Q^{n} which has a epimorphism Q^{n} → B^{n}(M^{▪}). Then we have an epimorphism ⨁_{n∈ℤ} M^{▪}(1_{Q^{n}})[−n] → B^{▪}(M^{▪})[1] and its lift ⨁_{n∈ℤ} M^{▪}(1_{Q^{n}})[−n] → M^{▪}. For each H^{n}(M^{▪}), there is a projective object P^{n} which has a epimorphism P^{n} → H^{n}(M^{▪}). Then we have a morphism ⨁_{n∈ℤ}P^{n}[−n] → B^{▪}(M^{▪}) and its lift ⨁_{n∈ℤ}P^{n}[−n] → Z^{▪}(M^{▪}) → M^{▪}. Then it is easy to see that ⨁_{n∈ℤ}P^{n}[−n] ⊕ ⨁_{n∈ℤ} M^{▪}(1_{Q^{n}})[−n] → M^{▪} is proper epimorphism. Then by induction we complete the proof.

The above proper resolution … → X^{▪−1} → X^{▪0} (resp., X^{▪0} → X^{▪1} → …) is called a proper projective (resp., injective) resolution of M^{▪} (they are often called Cartan-Eilenberg resolutions).

Proposition 13.12. The following hold.

1. Assume that Asatisfies the condition Ab4 with enough projectives. Given a complex M^{▪} ∈ C(A), let π : P^{▪▪} → M^{▪} be a proper projective resolution. Then

Tot π : Tot P^{▪▪} → M^{▪}

is a quasi-isomorphism in K(A), and Tot P^{▪▪} ∈ K^{s}(Proj A).

2. Assume that Asatisfies the condition Ab4* with enough injectives. Given a complex M^{▪} ∈ C(A), let μ : M^{▪} → I^{▪▪} be a proper injective resolution. Then

Tot^{∧}μ : M^{▪} → Tot^{∧}I^{▪▪}

is a quasi-isomorphism in K(A), and Tot^{∧}I^{▪▪} ∈ K^{s}(Inj A).

Proof. 1. We can consider π : P^{▪▪} → M^{▪} as π : P^{▪▪} → em^{I}M^{▪} in C^{2}(A). Then we have a commutative diagram

σ_{≤n}^{I}P^{▪▪} ⟶^{σ_{≤n}^{I}π} σ_{≤n}^{I}em^{I}M^{▪}
α_{n}↓  ↓^{β_{n}}
σ_{≤n+1}^{I}P^{▪▪} ⟶^{σ_{≤n}^{I}π} σ_{≤n+1}^{I}em^{I}M^{▪}

where α_{n}, β_{n} are term-split monomorphisms. Therefore we have a commutative diagram

Tot σ_{≤n}^{I}P^{▪▪} ⟶^{σ_{≤n}^{I}π} σ_{≤n}M^{▪}
Tot α_{n}↓  ↓^{β_{n}}
Tot σ_{≤n+1}^{I}P^{▪▪} ⟶^{σ_{≤n}^{I}π} σ_{≤n+1}M^{▪}

where Tot α_{n}, β_{n} are term-split monomorphisms, and all horizontal morphisms are

quasi-isomorphisms, because Tot(σ_{≤n}^{I}P^{▪▪} ⟶^{σ_{≤n}^{I}π} σ_{≤n}^{I}em^{I}M^{▪}) is acyclic by Lemma 13.7. According to Proposition 11.7, we have isomorphisms in D(A)

Tot P^{▪▪} = \underrightarrow{lim} Tot σ_{≤n}^{I}P^{▪▪}

≅ hlim_{⟶} Tot σ_{≤n}^{I}P^{▪▪}

≅ hlim_{⟶} σ_{≤n}M^{▪}

≅ \underrightarrow{lim} σ_{≤n}M^{▪}

= M^{▪}.

On the other hand, since Tot σ_{≤n}^{I}P^{▪▪} ∈ K^{−}(Proj A), Tot P^{▪▪} ≅ hlim_{⟶} Tot σ_{≤n}^{I}P^{▪▪} ∈ K^{s}(Proj A).

2. Similarly

Definition 13.13. For a morphism u : X^{▪▪} → Y^{▪▪} of double complexes, we define the complexes M_{I}^{▪▪}(u), M_{II}^{▪▪}(u) as follows.

M_{I}^{▪▪}(u)^{p,q} = X^{p+1,q} ⊕ Y^{p,q}
d_{I M_{I}^{▪▪}(u)}^{p,q} = [ \begin{matrix}−d_{I X}^{p+1,q} & 0 \\ u^{p+1,q} & d_{I Y}^{p,q} \end{matrix} ]
d_{II M_{I}^{▪▪}(u)}^{p,q} = [ \begin{matrix}−d_{II X}^{p+1,q} & 0 \\ 0 & d_{II Y}^{p,q} \end{matrix} ] ,
M_{II}^{▪▪}(u)^{p,q} = X^{p,q+1} ⊕ Y^{p,q}
d_{I M_{II}^{▪▪}(u)}^{p,q} = [ \begin{matrix}−d_{I X}^{p,q+1} & 0 \\ 0 & d_{I Y}^{p,q} \end{matrix} ]
d_{II M_{II}^{▪▪}(u)}^{p,q} = [ \begin{matrix}−d_{II X}^{p,q+1} & 0 \\ u^{p+1,q} & d_{II Y}^{p,q} \end{matrix} ] .

Proposition 13.14. For a morphism u : X^{▪▪} → Y^{▪▪} of double complexes, the following hold.

1. Tot M_{I}^{▪▪}(u) = M^{▪}(Tot u).

2. Tot M_{II}^{▪▪}(u) = M^{▪}(Tot u).

14. Derived Functors of Bi-∂-functors

Definition 14.1 (Bi-∂-functor). For triangulated categories C_{1}, C_{2} and D, a bi- ∂-functor (F, θ_{1}, θ_{2}) : C_{1} × C_{2} → D is a bifunctor F : C_{1} × C_{2} → D together with bifunctorial isomorphisms θ_{1} : F (T_{C_{1}}−,?) →^{∼} T_{D}F (−,?), θ_{2} : F (−, T_{C_{2}}?) →^{∼} T_{D}F (−,?) such that

(a) For each object X_{1} ∈ C_{1}, F (X_{1}, −) = (F (X_{1}, −), θ_{2}_{(X_{1},−)}) : C_{2} → D is a ∂-functor.

(b) For each object X_{2} ∈ C_{2}, F (−, X_{2}) = (F (−, X_{2}), θ_{1}_{(−,X_{2})}) : C_{1} → D is a ∂-functor.

For bi-∂-functors (F, θ_{1}, θ_{2}), (G, η_{1}, η_{2}) : C_{1} × C_{2} → D, a bifunctorial morphism ϕ : F → G is called a bi-∂-functorial morphism if (T_{D}ϕ)θ_{1} = η_{1}ϕ(T_{C_{1}} × 1_{C_{2}}), (T_{D}ϕ)θ_{2} = η_{2}ϕ(1_{C_{1}} × T_{C_{2}}).

We denote by ∂^{2}(C_{1} × C_{2}, D) the collection of all bi-∂-functors from C_{1} × C_{2} to D, and denote by ∂^{2} Mor(F, G) the collection of all bi-∂-functorial morphisms from F to G.

Proposition 14.2. Let C_{1}, C_{2} and C_{3} be triangulated categories, U_{i} épaisse subcategories of C_{i}, and Q_{i} : U_{i} → C_{i}/U_{i} the canonical quotients (i = 1, 2). Assume a bi-∂-functor F = (F, θ_{1}, θ_{2}) : C_{1} × C_{2} → C_{3} satisfies that

(a) F (U_{1}, C_{2}) = {O}.

(b) F (C_{1}, U_{2}) = {O}. Then there exists a unique bi-∂-functor \overline{F} = (\overline{F} , \overline{θ}_{1}, \overline{θ}_{2}) : C_{1}/U_{1} × C_{2}/U_{2} → C_{3} such that F = \overline{F} (Q_{1} × Q_{2}) and θ_{i} = \overline{θ}_{i}(Q_{1} × Q_{2}) (i = 1, 2).

\xymatrix{C_{1} × C_{2} \ar[d]_{Q_{1}×Q_{2}} \ar[dr]^{F} \\ C_{1}/U_{1} × C_{2}/U_{2} \ar@{-->}[r]_{\overline{F}} & C_{3}}

Sketch. We define the functor \overline{F} : C_{1}/U_{1} × C_{2}/U_{2} → C_{3} as follows. For X_{i} ∈ C_{i}, let \overline{F} (Q_{1}X_{1}, Q_{2}X_{2}) = F (X_{1}, X_{2}). For [(f_{i}, s_{i})] : X_{i} → Y_{i} in C_{i}/U_{i}, let \overline{F} ([(f_{1}, s_{1})], Q_{2}X_{2}) = F (s_{1}, X_{2})^{−1}F (f_{1}, X_{2}), \overline{F} (Q_{1}X_{1}, [(f_{2}, s_{2})]) = F (X_{1}, s_{2})^{−1}F (X_{1}, f_{2}). Let T_{i}, \overline{T}_{i} are translations of C_{i}, C_{i}/U_{i}, respectively (i = 1, 2, 3). Then we define

[figure: \overline{F} (\overline{T}_{1}Q_{1}X_{1}, Q_{2}X_{2}); \overline{θ}_{1}(Q_{1}X_{1},Q_{2}X_{2}); \overline{F} (Q_{1}T_{1}X_{1}, Q_{2}X_{2}); F (T_{1}X_{1}, X_{2}); θ_{1}(X_{1},X_{2}); T_{3}\overline{F} (Q_{1}X_{1}, Q_{2}X_{2}); T_{3}F (X_{1}, X_{2})]

[figure: \overline{F} (Q_{1}X_{1}, \overline{T}_{2}Q_{2}X_{2}); \overline{θ}_{2}(Q_{1}X_{1},Q_{2}X_{2}); \overline{F} (Q_{1}X_{1}, Q_{2}T_{2}X_{2}); F (X_{1}, T_{2}X_{2}); θ_{2}(X_{1},X_{2}); T_{3}\overline{F} (Q_{1}X_{1}, Q_{2}X_{2}); T_{3}F (X_{1}, X_{2})]

Then it is not hard to see that \overline{F} satisfies the assertions (left to the reader).

Proposition 14.3. Let C_{1}, C_{2} and C_{3} be triangulated categories, U_{i} épaisse subcategories of C_{i}, and Q_{i} : U_{i} → C_{i}/U_{i} the canonical quotients (i = 1, 2). For bi-∂-functors F = (F, θ_{1}, θ_{2}), G = (G, η_{1}, η_{2}) : C_{1}/U_{1} × C_{2}/U_{2} → C_{3}, we have a bijective correspondence

∂^{2} Mor(F, G) →^{∼} ∂^{2} Mor(F (Q_{1} × Q_{2}), G(Q_{1} × Q_{2})), (ζ ↦ ζ(Q_{1} × Q_{2})).

Definition 14.4 (Right Derived Functor). Let A_{i} be abelian categories, and let K^{∗_{i}}(A_{i}) be a quotientizing subcategory of K(A_{i}) and F : K^{∗_{1}}(A_{1}) × K^{∗_{2}}(A_{2}) → K(A_{3}) a bi-∂-functor (i = 1, 2, 3). The right derived functor of F is a bi-∂-functor

R^{∗_{1},∗_{2}}F : D^{∗_{1}}(A_{1}) × D^{∗_{2}}(A_{2}) → D(A_{3})

together with a functorial morphism of bi-∂-functors

ξ ∈ ∂^{2} Mor(Q_{A_{3}}F, R^{∗_{1},∗_{2}}F (Q_{A_{1}}^{∗_{1}} × Q_{A_{2}}^{∗_{2}}))

with the following property: For G ∈ ∂^{2}(D^{∗_{1}}(A_{1}) × D^{∗_{2}}(A_{2}), D(A_{3})) and ζ ∈ ∂^{2} Mor(Q_{A_{3}}F, G(Q_{A_{1}}^{∗_{1}} × Q_{A_{2}}^{∗_{2}})), there exists a unique morphism η ∈ ∂^{2} Mor(R^{∗_{1},∗_{2}}F, G) such that

ζ = (η(Q_{A_{1}}^{∗_{1}} × Q_{A_{2}}^{∗_{2}}))ξ.

In other words,

∂^{2} Mor(Q_{A_{3}}F, −(Q_{A_{1}}^{∗_{1}} × Q_{A_{2}}^{∗_{2}})) ≅ ∂^{2} Mor(R^{∗_{1},∗_{2}}F, −)

as functors from ∂^{2}(D^{∗_{1}}(A_{1}) × D^{∗_{2}}(A_{2}), D(A_{3})) to 𝔖𝔢𝔱 (See Lemma 1.8).

Theorem 14.5 (Existence Theorem). Let A_{i} be abelian categories (i = 1, 2, 3), and let K^{∗_{i}}(A_{i}) be a quotientizing subcategory of K(A_{i}) and F : K^{∗_{1}}(A_{1}) × K^{∗_{2}}(A_{2}) → K(A_{3}) a bi-∂-functor. Assume there exist triangulated full subcategories L_{i} of K^{∗_{i}}(A_{i}) (i = 1, 2) such that

(a) for any X_{i}^{▪} ∈ K^{∗_{i}}(A_{i}) there is a quasi-isomorphism X_{i}^{▪} → I_{i}^{▪} with I_{i}^{▪} ∈ L_{i},

(b) Q_{A_{3}}F (L_{1}^{ϕ}, L_{2}) = {O},

(c) Q_{A_{3}}F (L_{1}, L_{2}^{ϕ}) = {O},

where L_{i}^{ϕ} = K^{ϕ}(A_{i}) ∩ L_{i} (i = 1, 2). Then there exists the right derived functor (R^{∗_{1},∗_{2}}F, ξ) such that ξ_{(I_{1}^{▪},I_{2}^{▪})} : Q_{A_{3}}F (I_{1}^{▪}, I_{2}^{▪}) → R^{∗}F (I_{1}^{▪}, I_{2}^{▪}) is a quasi-isomorphism for (I_{1}^{▪}, I_{2}^{▪}) ∈ L_{1} × L_{2}.

Proof. Let Q_{i} : K^{∗_{i}}(A_{i}) → D^{∗_{i}}(A_{i}) be the canonical quotients, and let E_{i} : L_{i} → K^{∗_{i}}(A_{i}) be the embedding functors, then by the assumption 1 and Proposition 7.16 the canonical functor \overline{E}_{i} : L_{i}/L_{i}^{ϕ} → D^{∗_{i}}(A_{i}) are equivalences (i = 1, 2). Let J_{i} : D^{∗_{i}}(A_{i}) → L_{i}/L_{i}^{ϕ} be quasi-inverses of \overline{E}_{i} (i = 1, 2). By the assumption 2, 3 and Proposition 14.2 there is a bi-∂-functor

\overline{F} : L_{1}/L_{1}^{ϕ} × L_{2}/L_{2}^{ϕ} → D(A_{3})

such that Q_{3}F (E_{1} ×E_{2}) = \overline{F} (Q′_{1} ×Q′_{2}), where Q′_{i} : L_{i} → L_{i}/L_{i}^{ϕ} are the canonical quotients. Put R^{∗_{1},∗_{2}}F = \overline{F} (J_{1} × J_{2}). Since (Q_{1}E_{1} × Q_{2}E_{2}) = (\overline{E}_{1}Q′_{1} × \overline{E}_{2}Q′_{2}), we have

∂^{2} Mor(Q_{3}F (E_{1} × E_{2}), G(Q_{1}E_{1} × Q_{2}E_{2}))
≅ ∂^{2} Mor(\overline{F} (Q′_{1} × Q′_{2}), G(\overline{E}_{1}Q′_{1} × \overline{E}_{2}Q′_{2}))
≅ ∂^{2} Mor(\overline{F} (J_{1}E_{1} × J_{2}E_{2}), G(\overline{E}_{1} × \overline{E}_{2}))
≅ ∂^{2} Mor(\overline{F} (J_{1} × J_{2}), G)
= ∂^{2} Mor(R^{∗_{1},∗_{2}}F, G)

It remains to show that

\begin{matrix}∂^{2} Mor(Q_{3}F, G(Q_{1} × Q_{2})) & →^{∼} ∂^{2} Mor(Q_{3}F (E_{1} × E_{2}), G(Q_{1}E_{1} × Q_{2}E_{2})), \\ (ϕ ↦ ϕ(E_{1} × E_{2})). \end{matrix}

Let ϕ ∈ ∂^{2} Mor(Q_{3}F, G(Q_{1} × Q_{2})) with ϕ(E_{1} × E_{2}) = 0. For any X_{i}^{▪} ∈ K^{∗_{i}}(A_{i}) there exists I_{i}^{▪} ∈ L_{i} which has a quasi-isomorphism s_{i} : X_{i}^{▪} → I_{i}^{▪} (i = 1, 2). Then

\begin{aligned}ϕ_{(X_{1}^{▪},X_{2}^{▪})} & = G(Q_{1}s_{1}, Q_{2}s_{2})^{−1}ϕ_{(I_{1}^{▪},I_{2}^{▪})}Q_{3}F (s_{1}, s_{2}) \\ & = 0, \end{aligned}

and hence ϕ = 0. Given ψ ∈ ∂^{2} Mor(Q_{3}F (E_{1} × E_{2}), G(Q_{1}E_{1} × Q_{2}E_{2})), for any X_{i}^{▪} ∈ K^{∗_{i}}(A_{i}), let

ϕ_{(X_{1},X_{2})} = (G(Q_{1}s_{1}, Q_{2}s_{2}))^{−1}ψ_{(I_{1}^{▪},I_{2}^{▪})}Q_{3}F (s_{1}, s_{2})

for some quasi-isomorphism s_{i} : X_{i}^{▪} → I_{i}^{▪}, with I_{i}^{▪} ∈ L_{i} (i = 1, 2). For another quasi-isomorphism s′_{i} : X_{i}^{▪} → I′_{i}^{▪}, by the assumptions 1, we have commutative diagrams

X_{i}^{▪} ⟶^{s_{i}} I_{i}^{▪}
s′_{i}↓  ↓^{t′}_{i}
I′_{i}^{▪} ⟶^{t_{i}} I′′_{i}^{▪}

where all morphisms are quasi-isomorphisms and I′′_{i}^{▪} ∈ L_{i} (i = 1, 2). Then we have

(G(Q_{1}s_{1}, Q_{2}s_{2}))^{−1}ψ_{(I_{1}^{▪},I_{2}^{▪})}Q_{3}F (s_{1}, s_{2})
= (G(Q_{1}t′_{1}s_{1}, Q_{2}t′_{2}s_{2}))^{−1}ψ_{(I′′_{1}^{▪},I′′_{2}^{▪})}Q_{3}F (t′_{1}s_{1}, t′_{2}s_{2})
= (G(Q_{1}t_{1}s′_{1}, Q_{2}t_{2}s′_{2}))^{−1}ψ_{(I′′_{1}^{▪},I′′_{2}^{▪})}Q_{3}F (t_{1}s′_{1}, t_{2}s′_{2})
= (G(Q_{1}s′_{1}, Q_{2}s′_{2}))^{−1}ψ_{(I′_{1}^{▪},I′_{2}^{▪})}Q_{3}F (s′_{1}, s′_{2})

It is not hard to see that ϕ ∈ ∂^{2} Mor(Q_{3}F, G(Q_{1} × Q_{2})). The last assertion is easy to check.

Proposition 14.6. Let A_{i} be abelian categories (i = 1, 2, 3), and let K^{∗_{i}}(A_{i}) be a quotientizing subcategory of K(A_{i}) and F : K^{∗_{1}}(A_{1}) × K^{∗_{2}}(A_{2}) → K(A_{3}) a bi-∂- functor. Assume there exists a triangulated full subcategories L_{i} of K^{∗_{i}}(A_{i}) (i = 1, 2) such that

(a) for any X_{i}^{▪} ∈ K^{∗_{i}}(A_{i}) there is a quasi-isomorphism X_{i}^{▪} → I_{i}^{▪} with I_{i}^{▪} ∈ L_{i},

(b) Q_{A_{3}}F (L_{1}^{ϕ}, K^{∗_{2}}(A_{2})) = {O},

(c) Q_{A_{3}}F (L_{1}, L_{2}^{ϕ}) = {O},

where L_{i}^{ϕ} = K^{ϕ}(A_{i}) ∩ L_{i} (i = 1, 2). Then we have 1. There is a bi-∂-functor R_{I}^{∗_{1},∗_{2}}F : D^{∗_{1}}(A_{1}) × K^{∗_{2}}(A_{2}) → D(A_{3}) such that

∂^{2} Mor(Q_{A_{3}}F, −(Q_{A_{1}}^{∗_{1}} × 1_{K^{∗_{2}} (A_{2})})) ≅ ∂^{2} Mor(R_{I}^{∗_{1},∗_{2}}F, −),

and R_{I}^{∗_{1},∗_{2}}F (−, X_{2}) is the right derived functor of F (−, X_{2}) for any X_{2} ∈ K^{∗_{2}}(A_{2}).

2. There is a bi-∂-functor R_{II}^{∗_{1},∗_{2}}R_{I}^{∗_{1},∗_{2}}F : D^{∗_{1}}(A_{1}) × D^{∗_{2}}(A_{2}) → D(A_{3}) such that

∂^{2} Mor(R_{I}^{∗_{1},∗_{2}}F, −(1_{D^{∗_{1}} (A_{1})} × Q_{A_{2}}^{∗_{2}})) ≅ ∂^{2} Mor(R_{II}^{∗_{1},∗_{2}}R_{I}^{∗_{1},∗_{2}}F, −),

and

∂ Mor(R_{I}^{∗_{1},∗_{2}}F (X_{1},?), −Q_{A_{2}}^{∗_{2}}) ≅ ∂ Mor(R_{II}^{∗_{1},∗_{2}}R_{I}^{∗_{1},∗_{2}}F (X_{1},?), −)

for any X_{1} ∈ K^{∗_{1}}(A_{1}).

3. We have an isomorphism

∂^{2} Mor(Q_{A_{3}}F, −(Q_{A_{1}}^{∗_{1}} × Q_{A_{2}}^{∗_{2}})) ≅ ∂^{2} Mor(R_{II}^{∗_{1},∗_{2}}R_{I}^{∗_{1},∗_{2}}F, −).

In particular, R_{II}^{∗_{1},∗_{2}}R_{I}^{∗_{1},∗_{2}}F ≅ R^{∗_{1},∗_{2}}F.

Proof. According to the construction of the right derived functor of a bi-∂-functor in the proof of Theorems 14.5, 12.5, it is easy (left to the reader).

Corollary 14.7. Let A_{i} be abelian categories (i = 1, 2, 3), and let K^{∗_{i}}(A_{i}) be a quotientizing subcategory of K(A_{i}) and F : K^{∗_{1}}(A_{1}) × K^{∗_{2}}(A_{2}) → K(A_{3}) a bi-∂- functor. Assume there exists a triangulated full subcategories L_{i} of K^{∗_{i}}(A_{i}) (i = 1, 2) such that

(a) for any X_{i}^{▪} ∈ K^{∗_{i}}(A_{i}) there is a quasi-isomorphism X_{i}^{▪} → I_{i}^{▪} with I_{i}^{▪} ∈ L_{i}, (b) Q_{A_{3}}F (L_{1}^{ϕ}, K^{∗_{2}}(A_{2})) = {O},

(c) Q_{A_{3}}F (K^{∗_{1}}(A_{1}), L_{2}^{ϕ}) = {O}, where L_{i}^{ϕ} = K^{ϕ}(A_{i}) ∩ L_{i} (i = 1, 2). Then we have

1. There is a bi-∂-functor R_{I}^{∗_{1},∗_{2}}F : D^{∗_{1}}(A_{1}) × K^{∗_{2}}(A_{2}) → D(A_{3}) such that

∂^{2} Mor(Q_{A_{3}}F, −(Q_{A_{1}}^{∗_{1}} × 1_{K^{∗_{2}} (A_{2})})) ≅ ∂^{2} Mor(R_{I}^{∗_{1},∗_{2}}F, −),

and R_{I}^{∗_{1},∗_{2}}F (−, X_{2}) is the right derived functor of F (−, X_{2}) for any X_{2} ∈ K^{∗_{2}}(A_{2}).

2. There is a bi-∂-functor R_{II}^{∗_{1},∗_{2}}F : K^{∗_{1}}(A_{1}) × D^{∗_{2}}(A_{2}) → D(A_{3}) such that

∂^{2} Mor(Q_{A_{3}}F, −(1_{K^{∗_{1}} (A_{1})} × Q_{A_{2}}^{∗_{2}})) ≅ ∂^{2} Mor(R_{II}^{∗_{1},∗_{2}}F, −),

and R_{II}^{∗_{1},∗_{2}}F (X_{1}, −) is the right derived functor of F (X_{1}, −) for any X_{1} ∈ K^{∗_{1}}(A_{1}).

3. We have an isomorphism

R^{∗_{1},∗_{2}}F ≅ R_{II}^{∗_{1},∗_{2}}R_{I}^{∗_{1},∗_{2}}F ≅ R_{I}^{∗_{1},∗_{2}}R_{II}^{∗_{1},∗_{2}}F.

Definition 14.8 (Hom_{A}^{▪}). For a complexes X^{▪}, Y^{▪} ∈ C(A), we define the double complex Hom_{A}^{▪▪}(X^{▪}, Y^{▪}) by

Hom_{A}^{p,q}(X^{▪}, Y^{▪}) = Hom_{A}(X^{−p}, Y^{q})
d_{I Hom_{A}^{▪▪}(X^{▪},Y^{▪})}^{p,q} = Hom_{A}^{p,q}(d_{X^{▪}}^{−p−1}, Y^{q})
d_{II Hom_{A}^{▪▪}(X^{▪},Y^{▪})}^{p,q} = (−1)^{p+q+1} Hom_{A}^{p,q}(X^{−p}, d_{Y^{▪}}^{q}),

and define the complex Hom_{A}^{▪}(X^{▪}, Y^{▪}) by

Tot^{∧} Hom_{A}^{▪▪}(X^{▪}, Y^{▪}).

Then it is easy to see that

Hom_{A}^{▪} : C(A)^{op} × C(A) → C(𝔄𝔟)

is a bifunctor.

Lemma 14.9. For complexes X^{▪}, Y^{▪} ∈ C(A), we have an isomorphism

H^{n} Hom_{A}^{▪}(X^{▪}, Y^{▪}) ≅ Hom_{K(A)}(X^{▪}, T^{n}Y^{▪}).

Proof. By the definition, for (u^{p,q})_{p+q=r} ∈ Hom_{A}^{r}(X^{▪}, Y^{▪}) we have

\begin{matrix}d_{Hom_{A}^{▪}(X,Y)}^{r}((u^{p,q})_{p+q=r}) \\ = (u^{p−1,q}d_{X}^{−p} + (−1)^{p+q+1}d_{Y}^{q} u^{p,q−1})_{p+q=r} ∈ Hom_{A}^{r+1}(X^{▪}, Y^{▪}). \end{matrix}

Put u = f, i = −p, r = n, then f^{−i,i+n} : X^{i} → Y^{i+n} for all i and we have

\begin{matrix}d_{Hom_{A}^{▪}(X,Y)}^{n}((f^{−i,i+n})_{i∈ℤ}) \\ = (f^{−i−1,i+1+n}d_{X}^{i} − (−1)^{n}d_{Y}^{i+n}f^{−i,i+n})_{i∈ℤ}. \end{matrix}

Then it is easy to see that Ker d_{Hom_{A}^{▪}(X,Y)}^{n} = Hom_{C(A)}(X^{▪}, T^{n}Y^{▪}). Put u = h, i = −p, r = n − 1, then h^{−i,i+n−1} : X^{i} → Y^{i+n−1} for all i and we have

\begin{matrix}d_{Hom_{A}^{▪}(X,Y)}^{n−1}((h^{−i,i+n−1})_{i∈ℤ}) \\ = (h^{−i−1,i+n}d_{X}^{i} + (−1)^{n}d_{Y}^{i+n}h^{−i,i+n−1})_{i∈ℤ}. \end{matrix}

Then this means Im d_{Hom_{A}^{▪}(X,Y)}^{n−1} = Htp_{C(A)}(X^{▪}, T^{n}Y^{▪}).

Lemma 14.10. For a complexes X^{▪}, Y^{▪} ∈ C(A), the following hold.

1. d_{Hom_{A}^{▪}(X^{▪},T Y^{▪})} = −T d_{Hom_{A}^{▪}(X^{▪},Y^{▪})}

2. d_{Hom_{A}^{▪}(T^{−1}X^{▪},Y^{▪})} = T d_{Hom_{A}^{▪}(X^{▪},Y^{▪})}

3. Define θ_{1}^{p,q} : Hom_{A}^{p,q}(T^{−1}X^{▪}, Y^{▪}) → Hom_{A}^{p−1,q}(X^{▪}, Y^{▪}) by the identities, then we have an isomorphism

θ_{1} : Hom_{A}^{▪} ∘(T^{−1} × 1_{C(A)}) →^{∼} T ∘ Hom_{A}^{▪} .

4. Define θ_{2}^{p,q} : Hom_{A}^{p,q}(X^{▪}, T Y^{▪}) → Hom_{A}^{p,q+1}(X^{▪}, Y^{▪}) by (−1)^{p+q}, then we have an isomorphism

θ_{2} : Hom_{A}^{▪} ∘(1_{C(A)} × T) →^{∼} T ∘ Hom_{A}^{▪} .

Lemma 14.11. For a morphism u : X^{▪} → Y^{▪} in C(A) and N^{▪} ∈ C(A), the following hold.

1. Hom_{A}^{▪}(N^{▪}, M^{▪}(u)) ≅ M^{▪}(Hom_{A}^{▪}(N^{▪}, u)).

2. Hom_{A}^{▪}(M^{▪}(u), N^{▪}) ≅ T^{−1} M^{▪}(Hom_{A}^{▪}(u, N^{▪})).

Proof. 1. The double complex Hom_{A}^{▪▪}(N^{▪}, M^{▪}(u)) has the following form

Hom_{A}^{p,q}(N^{▪}, M^{▪}(u)) = Hom_{A}(M^{−p}, X^{q+1} ⊕ Y^{q})
d_{I Hom_{A}^{▪▪}(N^{▪},M^{▪}(u))}^{p,q} = [ \begin{matrix}Hom(d_{M}^{−(p+1)},X) & 0 \\ 0 & Hom(d_{M}^{−(p+1)},Y) \end{matrix} ]
d_{II Hom_{A}^{▪▪}(N^{▪},M^{▪}(u))}^{p,q} = [ \begin{matrix}(−1)^{p+q+2} Hom(M,d_{X}^{q+1}) & 0 \\ (−1)^{p+q+1} Hom(M,u^{q}) & (−1)^{p+q+1} Hom(M,d_{Y}^{q} ) \end{matrix} ]

On the other hand, the double complex M_{II}^{▪▪}(Hom_{A}^{▪▪}(N^{▪}, u)) has the following form

M_{II}^{p,q}(Hom_{A}^{▪▪}(N^{▪}, u)) = Hom_{A}(M^{−p}, X^{q+1} ⊕ Y^{q})
d_{I M_{II}^{▪▪}(Hom_{A}^{▪▪}(N^{▪},u))}^{p,q} = [ \begin{matrix}− Hom(d_{M}^{−(p+1)},X) & 0 \\ 0 & Hom(d_{M}^{−(p+1)},Y) \end{matrix} ]
d_{II M_{II}^{▪▪}(Hom_{A}^{▪▪}(N^{▪},u))}^{p,q} = [ \begin{matrix}(−1)^{p+q+1} Hom(M,d_{X}^{q+1}) & 0 \\ Hom(M,u^{q}) & (−1)^{p+q+1} Hom(M,d_{Y}^{q} ) \end{matrix} ]

Then it is easy to see that morphisms [ \begin{matrix}(−1)^{p+q} & 0 \\ 0 & 1 \end{matrix} ] : M_{II}^{p,q}(Hom_{A}^{▪▪}(N^{▪}, u)) →

Hom_{A}^{p,q}(N^{▪}, M^{▪}(u)) induce an isomorphism between them in C^{2}(A). By Proposition 13.14, we get the statement.

2. The double complex Hom_{A}^{▪▪}(M^{▪}(u), N^{▪}) has the following form

Hom_{A}^{p,q}(M^{▪}(u), N^{▪}) = Hom_{A}(X^{−p+1} ⊕ Y^{−p}, M^{q})
d_{I Hom_{A}^{▪▪}(M^{▪}(u),N^{▪})}^{p,q} = [ \begin{matrix}− Hom(d_{X}^{−p},M) & 0 \\ Hom(u^{p},M) & Hom(d_{Y}^{−p−1},M) \end{matrix} ]
d_{II Hom_{A}^{▪▪}(M^{▪}(u),N^{▪})}^{p,q} = [ \begin{matrix}(−1)^{p+q+1} Hom(X,d_{M}^{q}) & 0 \\ 0 & (−1)^{p+q+1} Hom(X,d_{M}^{q}) \end{matrix} ]

On the other hand, the double complex T_{I}^{−1} M_{I}^{▪▪}(Hom_{A}^{▪▪}(u, N^{▪})) has the following form

(T_{I}^{−1} M_{I}^{▪▪}(Hom_{A}^{▪▪}(u, N^{▪})))^{p,q} = Hom_{A}(X^{−p+1} ⊕ Y^{−p}, M^{q})
d_{I T_{I}^{−1} M_{I}^{▪▪}(Hom_{A}^{▪▪}(u,N^{▪}))}^{p,q} = [ \begin{matrix}Hom(d_{X}^{−p},M) & 0 \\ − Hom(u^{p},M) & − Hom(d_{Y}^{−p−1},M) \end{matrix} ]
d_{II T_{I}^{−1} M_{I}^{▪▪}(Hom_{A}^{▪▪}(u,N^{▪}))}^{p,q} = [ \begin{matrix}(−1)^{p+q+1} Hom(X,d_{M}^{q}) & 0 \\ 0 & (−1)^{p+q+1} Hom(X,d_{M}^{q}) \end{matrix} ]

Then it is easy to see that morphisms [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ] : (T_{I}^{−1} M_{I}^{▪▪}(Hom_{A}^{▪▪}(u, N^{▪})))^{p,q} → Hom_{A}^{p,q}(M^{▪}(u), N^{▪}) induce an isomorphism between them in C^{2}(A). By Lemma 13.6 and Proposition 13.14, we get the statement.

Proposition 14.12. The bi-functor Hom_{A}^{▪} : C(A)^{op} × C(A) → C(𝔄𝔟) induces the bi-∂-functor

Hom_{A}^{▪} : K(A)^{op} × K(A) → K(𝔄𝔟).

Proof. By Lemmas 14.10, 14.11 and Corollary 6.16, it is easy.

Proposition 14.13. Let F : A → A′, G : A′ → A be additive functors between abelian categories such that G ⊣ F. Then we have a functorial isomorphism

Hom_{A}^{▪}(GX^{▪}, Y^{▪}) ≅ Hom_{A′}^{▪}(X^{▪}, F Y^{▪}).

Theorem 14.14. The bi-∂-functor Hom_{A}^{▪} : K^{∗_{1}}(A)^{op} × K^{∗_{2}}(A) → K(𝔄𝔟) has the right derived functor R^{∗_{1},∗_{2}} Hom_{A}^{▪} : Hom_{A}^{▪} : D^{∗_{1}}(A)^{op} × D^{∗_{2}}(A) → D(𝔄𝔟) if it satisfies the following.

1. If A has enough projectives, then R^{−,∞} Hom_{A}^{▪} exits and R^{−,∞} Hom_{A}^{▪} ≅ R_{II}^{−,∞}R_{I}^{−,∞} Hom_{A}^{▪}

2. If A has enough injectives, then R^{∞,+} Hom_{A}^{▪} exits and R^{∞,+} Hom_{A}^{▪} ≅ R_{I}^{∞,+}R_{II}^{∞,+} Hom_{A}^{▪}

3. If A satisfies the condition Ab4 with enough projectives, then R Hom_{A}^{▪} exits and R Hom_{A}^{▪} ≅ R_{II}R_{I} Hom_{A}^{▪}

4. If A satisfies the condition Ab4* with enough injectives, then R Hom_{A}^{▪} exits and R Hom_{A}^{▪} ≅ R_{I}R_{II} Hom_{A}^{▪}

5. If A satisfies the conditions Ab4 and Ab4* with enough projectives and with enough injectives, then

R Hom_{A}^{▪} ≅ R_{II}R_{I} Hom_{A}^{▪} ≅ R_{I}R_{II} Hom_{A}^{▪}

Here ∞ means “nothing”.

Proof. By Lemma 14.9, Corollary 10.9, Propositions 14.6, 14.7 can be applied.

Remark 14.15. In the above 5, for complexes X^{▪}, Y^{▪} ∈ K(A), we take P^{▪} ∈ K^{s}(Proj A) I^{▪} ∈ K^{s}(Inj A) which have quasi-isomorphisms P^{▪} → X^{▪}, Y^{▪} → I^{▪}. Then we have an isomophisms

R Hom_{A}^{▪}(X^{▪}, Y^{▪}) ≅ Hom_{A}^{▪}(P^{▪}, Y^{▪})≅ Hom_{A}^{▪}(X^{▪}, I^{▪}) ≅ Hom_{A}^{▪}(P^{▪}, I^{▪}).

Corollary 14.16. Assume that A satisfies one of the conditions of Theorem 14.14. For X^{▪} ∈ D^{∗_{1}}(A), Y^{▪} ∈ D^{∗_{2}}(A) and n ∈ ℤ, we have an isomorphism

H^{n} R^{∗_{1},∗_{2}} Hom_{A}^{▪}(X^{▪}, Y^{▪}) ≅ Hom_{D(A)}(X^{▪}, Y^{▪}[n]).

Proof. By Proposition 11.10 and Lemma 10.10, for X^{▪} ∈ D^{∗_{1}}(A), Y^{▪} ∈ D^{∗_{2}}(A), we have either a quasi-isomorphism P^{▪} → X^{▪} or a quasi-isomorphism Y^{▪} → I^{▪} with P^{▪} ∈ K^{∗_{1}}(Proj A), I^{▪} ∈ K^{∗_{2}}(Proj A). According to Corollary 10.9 and Proposition 11.12, we have one of isomorphisms

H^{n} R^{∗_{1},∗_{2}} Hom_{A}^{▪}(X^{▪}, Y^{▪}) ≅ H^{n} Hom_{A}^{▪}(P^{▪}, Y^{▪})
≅ Hom_{K(A)}(P^{▪}, Y^{▪}[n])
≅ Hom_{D(A)}(X^{▪}, Y^{▪}[n]),
H^{n} R^{∗_{1},∗_{2}} Hom_{A}^{▪}(X^{▪}, Y^{▪}) ≅ H^{n} Hom_{A}^{▪}(X^{▪}, Y^{▪})
≅ Hom_{K(A)}(X^{▪}, I^{▪}[n])
≅ Hom_{D(A)}(X^{▪}, Y^{▪}[n]).

Definition 14.17. For a complex X^{▪} of right A-modules and a complex Y^{▪} of left

A-modules we define the double complex X^{▪} ⊗^{▪▪}_{A} Y^{▪} by

\begin{aligned}X^{▪} ⊗^{p,q}_{A} Y^{▪} & = X^{p}⊗_{A}Y^{q} \\ d_{I X^{▪}⊗^{▪▪}_{A}Y^{▪}}^{p,q} & = d_{X}^{p}⊗_{A}Y^{q} \\ d_{II X^{▪}⊗^{▪▪}_{A}Y^{▪}}^{p,q} & = (−1)^{p+q}X^{p}⊗_{A}d_{Y}^{q} \end{aligned}

and define the complex X^{▪} ⊗^{▪}_{A} Y^{▪} by

TotX^{▪} ⊗^{▪▪}_{A} Y^{▪}.

Then it is easy to see that

⊗^{▪}_{A} : C(Mod A) × C(Mod A^{op}) → C(𝔄𝔟)

is a bifunctor.

Lemma 14.18. For a complexes X^{▪} ∈ C(Mod A), Y^{▪} ∈ C(Mod A^{op}), the following hold.

1. d_{(T X^{▪})⊗^{▪}_{A}Y^{▪}} = −T d_{X^{▪}⊗^{▪}_{A}Y^{▪}}

2. d_{X^{▪}⊗^{▪}_{A}T Y^{▪}} = T d_{X^{▪}⊗^{▪}_{A}Y^{▪}}

3. Define θ_{1}^{p,q} : (T X^{▪}) ⊗^{p,q}_{A} Y^{▪} → X^{▪} ⊗^{p+1,q}_{A} Y^{▪} by the identities, then we have an isomorphism

θ_{1} : ⊗^{▪}_{A} ∘ (T × 1_{C(Mod A)}) →^{∼} T ∘ ⊗^{▪}_{A}.

4. Define θ_{2}^{p,q} : X^{▪} ⊗^{p,q}_{A} T Y^{▪} → X^{▪} ⊗^{p,q+1}_{A} Y^{▪} by the (−1)^{p+q}, then we have an

isomorphism

θ_{2} : ⊗^{▪}_{A} ∘ (1_{C(Mod A^{op})} × T) →^{∼} T ∘ ⊗^{▪}_{A}.

Lemma 14.19. The following hold.

1. For a morphism u : X^{▪} → Y^{▪} in C(Mod A^{op}) and N^{▪} ∈ C(Mod A) we have

N^{▪} ⊗^{▪}_{A} M^{▪}(u) ≅ M^{▪}(N^{▪} ⊗^{▪}_{A} u).

2. For a morphism u : X^{▪} → Y^{▪} in C(Mod A) and N^{▪} ∈ C(Mod A^{op}) we have

M^{▪}(u) ⊗^{▪}_{A} N^{▪} ≅ M^{▪}(u ⊗^{▪}_{A} N^{▪}).

Proposition 14.20. The bi-functor ⊗^{▪}_{A} : C(Mod A) × C(Mod A^{op}) → C(𝔄𝔟) induces the bi-∂-functor

⊗^{▪}_{A} : K(Mod A) × K(Mod A^{op}) → K(𝔄𝔟).

Lemma 14.21. Let D_{ℤ} = Hom_{ℤ}(−, ℚ/ℤ) : Mod A → Mod A^{op} (resp., D_{ℤ} = Hom_{ℤ}(−, ℚ/ℤ) : Mod A^{op} → Mod A). Then the following hold.

1. For an sequence X → Y → Z of A-modules, X → Y → Z is exact if and only if D_{ℤ}Z → D_{ℤ}Y → D_{ℤ}X is exact.

2. For an A-module M, M is a flat A-module if and only if D_{ℤ}M is an injective A-module.

Theorem 14.22. The bi-∂-functor ⊗^{▪}_{A} : K(Mod A) × K(Mod A^{op}) → K(𝔄𝔟) has the left derived functor

⊗^{▪}_{A}^{L} : D(Mod A) × D(Mod A^{op}) → D(𝔄𝔟).

Proof. For X^{▪} ∈ K(Mod A), Y^{▪} ∈ K(Mod A^{op}), we have isomorphisms (see Proposition 15.4)

D_{ℤ}(X^{▪} ⊗^{▪}_{A} Y^{▪}) = D_{ℤ}(Tot X^{▪} ⊗^{▪▪}_{A} Y^{▪})

≅ Tot^{∧}D_{ℤ}(X^{▪} ⊗^{▪▪}_{A} Y^{▪})

≅ Tot^{∧} Hom_{A^{op}}^{▪▪}(X^{▪}, D_{ℤ}Y^{▪})

≅ Hom_{A^{op}}^{▪}(X^{▪}, D_{ℤ}Y^{▪}).

By Lemma 14.9, we can apply Proposition 11.12 to the above isomorphism. Then

by Lemma 14.21, for P^{▪} ∈ K^{s}(Proj A) and Y^{▪} ∈ K(Mod A^{op}), P^{▪} ⊗^{▪}_{A} Y^{▪} are acyclic either if P^{▪} is acyclic or if Y^{▪} is acyclic. According to the left derived version of Theorem 12.5, we complete the proof.

Remark 14.23. In the above, for complexes X^{▪} ∈ K(Mod A), Y^{▪} ∈ K(Mod A^{op}), we take P^{▪} ∈ K^{s}(Proj A) Q^{▪} ∈ K^{s}(Proj A^{op}) which have quasi-isomorphisms P^{▪} → X^{▪}, Q^{▪} → Y^{▪}. Then we have an isomophisms

X^{▪}⊗^{▪}_{A}^{L}Y^{▪} ≅ P^{▪}⊗^{▪}_{A}Y^{▪}≅ X^{▪}⊗^{▪}_{A}Q^{▪} ≅ P^{▪}⊗^{▪}_{A}Q^{▪}.

For X ∈ Mod A and Y ∈ Mod A^{op}, we denote Tor_{A}^{i}(X, Y) = H^{i}(X⊗^{▪}_{A}^{L}Y).

15. Bimodule Complexes

Throughout this section, k is a commutative ring, A, B, C are k-algebras, _{A}U_{B} an A-B-bimodule, _{B}V_{C} an B-C-bimodule, _{A}W_{C} an A-C-bimodule and _{C}S_{A} a C-A-bimodule.

Proposition 15.1. The following hold.

1. Hom_{A^{op}⊗_{k}B}(_{A}U_{B}, Hom_{C}(_{B}V_{C},_{A}W_{C})) ≅ Hom_{A^{op}⊗_{k}C}(_{A}U⊗_{B}V_{C},_{A}W_{C}).

2. Hom_{A^{op}⊗_{k}B}(_{A}U_{B}, Hom_{C}(_{B}V_{C},_{A}W_{C})) ≅ Hom_{B⊗_{k}C^{op}}(_{B}V_{C}, Hom_{A}(_{A}U_{B},_{A}W_{C})).

3. (_{A}U⊗_{B}V_{C})⊗_{A^{op}⊗_{k}C}(_{C}S_{A}) ≅ (_{A}U_{B})⊗_{A^{op}⊗_{k}B}(_{B}V ⊗_{C}S_{A}).

4. If _{A}U_{B} is A^{op}⊗_{k}B-projective, V_{C} is C-projective, then _{A}U⊗_{B}V_{C} is A^{op}⊗_{k}Cprojective.

5. If _{B}V is B-flat, _{A}W_{C} is A^{op}⊗_{k}C-injective, then Hom_{C}(_{B}V_{C},_{A}W_{C}) is A^{op}⊗_{k}B-injective.

6. If _{B}V_{C} is B^{op}⊗_{k}C-projective, _{A}W is A-injective, then Hom_{C}(_{B}V_{C},_{A}W_{C}) is A^{op}⊗_{k}B-injective.

7. If _{A}U_{B} is A^{op}⊗_{k}B-flat, V_{C} is C-flat , then _{A}U⊗_{B}V_{C} is A^{op}⊗_{k}C-flat.

Corollary 15.2. The following hold.

1. _{A}U is A-projective, V_{C} is C-projective, then _{A}U⊗_{k}V_{C} is A^{op}⊗_{k}C-projective.

2. _{B}V is B-flat, _{A}W is A-injective, then Hom_{k}(_{B}V_{k},_{A}W_{k}) is A^{op}⊗_{k}B-injective.

3. _{A}U is A-flat, V_{C} is C-flat, then _{A}U⊗_{k}V_{C} is A^{op}⊗_{k}C-flat.

Proposition 15.3. Let _{A}M be an A-module, _{B}N a B-module, then the following hold.

1. Hom_{A^{op}⊗_{k}B}(_{A}U_{B}, Hom_{k}(_{B}B_{k},_{A}V_{k})) ≅ Hom_{A}(_{A}U,_{A}V).

2. Hom_{A}(_{A}M,_{A}W) ≅ Hom_{A^{op}⊗_{k}C}(_{A}M⊗_{k}C_{C},_{A}W_{C}).

3. U⊗_{B}N ≅ (_{A}U_{B})⊗_{A^{op}⊗_{k}B}(_{B}N⊗_{k}A_{A}).

4. If B is k-projective, _{A}U_{B} is A^{op}⊗_{k}B-projective, then _{A}U is A-projective.

5. C is k-flat, _{A}W_{C} is A^{op}⊗_{k}C-injective, then _{A}W is A-injective.

6. B is k-flat, _{A}U_{B} is A^{op}⊗_{k}B-flat, then _{A}U is A-flat.

Proposition 15.4. For U^{▪} ∈ K(Mod A^{op}⊗_{k}B), V^{▪} ∈ K(Mod B⊗_{k}C^{op}), W^{▪} ∈ K(Mod A^{op}⊗_{k}C), S^{▪} ∈ K(Mod C⊗_{k}A^{op}), the following hold.

1. We have an isomorphism

Hom_{A^{op}⊗_{k}B}^{▪}(_{A}U_{B}^{▪}, Hom_{C}^{▪}(_{B}V_{C}^{▪},_{A}W_{C}^{▪})) ≅ Hom_{A^{op}⊗_{k}C}^{▪}(_{A}U^{▪} ⊗^{▪}_{B} V_{C}^{▪},_{A}W_{C}^{▪}).

2. We have an isomorphism

Hom_{A^{op}⊗_{k}B}^{▪}(_{A}U_{B}^{▪}, Hom_{C}^{▪}(_{B}V_{C}^{▪},_{A}W_{C}^{▪})) ≅ Hom_{B⊗_{k}C^{op}}^{▪}(_{B}V_{C}^{▪}, Hom_{A}^{▪}(_{A}U_{B}^{▪},_{A}W_{C}^{▪})).

3. We have an isomorphism

(_{A}U^{▪} ⊗^{▪}_{B} V_{C}^{▪}) ⊗^{▪}_{A^{op}⊗_{k}C} (_{C}S_{A}^{▪}) ≅ (_{A}U_{B}^{▪}) ⊗^{▪}_{A^{op}⊗_{k}B} (_{B}V^{▪} ⊗^{▪}_{C} S_{A}^{▪}).

Proof. 1. Let α be the trifunctorial isomorphism in 1 of Proposition 15.1. For every (p, q, r) ∈ ℤ^{3}, define

ϕ_{p,q,r} = (−1)^{\frac{r(2q+r+1)}{2}} α : Hom_{A^{op}⊗_{k}B}(_{A}U_{B}^{−p}, Hom_{C}(_{B}V_{C}^{−q},_{A}W_{C}^{r})) →
Hom_{A^{op}⊗_{k}C}(_{A}U^{−p}⊗_{B}V_{C}^{−q},_{A}W_{C}^{r}),

then (ϕ_{p,q,r}) induces the isomorphism between triple complexes. By taking Tot^{∧}, we have the assertion.

2. Let β be the trifunctorial isomorphism in 2 of Proposition 15.1. For every (p, q, r) ∈ ℤ^{3}, define

ϕ′_{p,q,r} = (−1)^{\frac{(p+q)(p+q+2r+1)}{2}} β : Hom_{A^{op}⊗_{k}B}(_{A}U_{B}^{−p}, Hom_{C}(_{B}V_{C}^{−q},_{A}W_{C}^{r})) →
Hom_{B^{op}⊗_{k}C}(_{B}V_{C}^{−q}, Hom_{A}(_{A}U_{B}^{−p},_{A}W_{C}^{r})),

then (ϕ_{p,q,r}) induces the isomorphism between triple complexes. By taking Tot^{∧}, we have the assertion.

3. Let γ be the trifunctorial isomorphism in 3 of Proposition 15.1. For every (p, q, r) ∈ ℤ^{3}, define

\begin{matrix}ψ_{p,q,r} = (−1)^{\frac{r(2q+r−1)}{2}} γ : (_{A}U^{p}⊗_{B}V_{C}^{q})⊗_{A^{op}⊗_{k}C}(_{C}S_{A}^{r}) → \\ (_{A}U_{B}^{p})⊗_{A^{op}⊗_{k}B}(_{B}V^{q}⊗_{C}S_{A}^{r}), \end{matrix}

then (ϕ_{p,q,r}) induces the isomorphism between triple complexes. By taking Tot, we have the assertion.

Proposition 15.5. The following hold.

1. For a functor − ⊗^{▪}_{A} U_{B}^{▪} : K(Mod A) → K(Mod B) and its right adjoint Hom_{B}^{▪}(_{A}U_{B}^{▪}, −) : K(Mod B) → K(Mod A), there exist the left derived functor

− ⊗^{▪} _{A}^{L}U_{B}^{▪} : D(Mod A) → D(Mod B) and the right derived functor

R Hom_{B}^{▪}(_{A}U_{B}^{▪}, −) : D(Mod B) → D(Mod A) such that

R Hom_{B}^{▪}(− ⊗^{▪} _{A}^{L}U_{B}^{▪},?) ≅ R Hom_{A}^{▪}(−, R Hom_{B}^{▪}(_{A}U_{B}^{▪},?)).

In particular, we have − ⊗^{▪} _{A}^{L}U_{B}^{▪} ⊣ R Hom_{B}^{▪}(_{A}U_{B}^{▪},?).

2. For a right adjoint pair of functors Hom_{A}^{▪}(−,_{A}U_{B}^{▪}) : K(Mod A^{op}) → K(Mod B), Hom_{B}^{▪}(−,_{A}U_{B}^{▪}) : K(Mod B) → K(Mod A^{op}), there exist the right derived functors R Hom_{A}^{▪}(−,_{A}U_{B}^{▪}) : D(Mod A^{op}) → D(Mod B), R Hom_{B}^{▪}(−,_{A}U_{B}^{▪}) : D(Mod B) → D(Mod A^{op}) such that

R Hom_{B}^{▪}(−, R Hom_{A}^{▪}(?,_{A}U_{B}^{▪})), ≅ R Hom_{A}^{▪}(?, R Hom_{B}^{▪}(−,_{A}U_{B}^{▪})).

In particular, (R Hom_{A}^{▪}(?,_{A}U_{B}^{▪}), R Hom_{B}^{▪}(−,_{A}U_{B}^{▪})) is a right adjoint pair.

Proof. 1. Let X^{▪} ∈ K(Mod A^{op}), Y^{▪} ∈ K(Mod B). According to Theorems 14.14, 14.22 and Remark 14.15, we may assume X^{▪} ∈ K^{s}(Proj A), Y^{▪} ∈ K^{s}(Inj B). It is clear for the existence of the derived functor. By Proposition 15.4 we have isomorphisms

R Hom_{B}^{▪}(X^{▪} ⊗^{▪} _{A}^{L}U_{B}^{▪}, Y^{▪}) ≅ Hom_{B}^{▪}(X^{▪} ⊗^{▪}_{A} U_{B}^{▪}, Y^{▪})
≅ Hom_{A}^{▪}(X^{▪}, Hom_{B}^{▪}(_{A}U_{B}^{▪}, Y^{▪}))
≅ R Hom_{A}^{▪}(X^{▪}, R Hom_{B}^{▪}(_{A}U_{B}^{▪}, Y^{▪})).

We get the last assertion by taking cohomologies of the above isomorphisms.

2. Let X^{▪} ∈ K(Mod A^{op}), Y^{▪} ∈ K(Mod B). According to Theorem 14.14 and Remark 14.15, we may assume X^{▪} ∈ K^{s}(Proj A^{op}), Y^{▪} ∈ K^{s}(Proj B). It is clear for the existence of the derived functor. By Proposition 15.4 we have isomorphisms

R Hom_{B}^{▪}(Y^{▪}, R Hom_{A}^{▪}(X^{▪},_{A}U_{B}^{▪})) ≅ Hom_{B}^{▪}(Y^{▪}, Hom_{A}^{▪}(X^{▪},_{A}U_{B}^{▪}))
≅ Hom_{A}^{▪}(X^{▪}, R Hom_{B}^{▪}(Y^{▪},_{A}U_{B}^{▪}))
≅ R Hom_{A}^{▪}(X^{▪}, R Hom_{B}^{▪}(Y^{▪},_{A}U_{B}^{▪})).

We get the last assertion by taking cohomologies of the above isomorphisms.

Remark 15.6. The above derived functors −⊗^{▪} _{A}^{L}U_{B}^{▪} : D(Mod A) → D(Mod B) and R Hom_{B}^{▪}(_{A}U_{B}^{▪}, −) : D(Mod B) → D(Mod A) are the derived functors of ∂-functors

− ⊗^{▪}_{A} U_{B}^{▪} and Hom_{B}^{▪}(_{A}U_{B}^{▪}, −), respectively. But they are not the derived functors of bi-∂-functors in general!

We denote by Res_{A} : Mod A^{op}⊗_{k}B → Mod A the forgetful functor, and use the same symbol Res_{A} : K(Mod A^{op}⊗_{k}B) → K(Mod A) for the induced ∂-functor.

Proposition 15.7. If B is k-projective or C is k-flat, then

R Hom_{A}^{▪} : D(Mod B^{op}⊗_{k}A)^{op} × D(Mod C^{op} ⊗_{k} A) → D(Mod C^{op} ⊗_{k} B)

exists, and we have a commutative diagram

D(Mod B^{op}⊗_{k}A)^{op} × D(Mod C^{op} ⊗_{k} A) ⟶^{Res_{A}^{op}×Res_{A}} D(Mod A)^{op} × D(Mod A)
R^{Hom}_{A}^{▪}↓  ↓^{R}^{Hom}_{A}^{▪}
D(Mod C^{op} ⊗_{k} B)  ⟶^{Res_{k}}  D(Mod k)

Proof. Assume B is k-projective. According to Proposition 15.4, for _{B}X_{A}^{▪} ∈ K(Mod B ⊗_{k} A), Y_{A}^{▪} ∈ K(Mod A), we have

Hom_{B^{op}⊗_{k}A}^{▪}(_{B}X_{A}^{▪}, Hom_{k}^{▪}(_{k}B_{B},_{k}Y_{A}^{▪})) ≅ Hom_{A}^{▪}(X_{A}^{▪}, Y_{A}^{▪}).

If _{B}X_{A}^{▪} ∈ K^{s}(Proj B^{op} ⊗_{k} A), then by the above isomorphism, we have Res_{A}X^{▪} ∈ K^{s}(Proj A). Therefore we get the assertion by Theorem 14.5.

Assume C is k-flat. For _{C}Y_{A}^{▪} ∈ K(Mod C^{op} ⊗_{k} A), X_{A}^{▪} ∈ K(Mod A), we have

Hom_{C^{op⊗_{k}A}}^{▪}(_{C}C ⊗_{k} X_{A}^{▪},_{C}Y_{A}^{▪}) ≅ Hom_{A}^{▪}(X_{A}^{▪}, Y_{A}^{▪}).

If _{C}Y_{A}^{▪} ∈ K^{s}(Inj C^{op} ⊗_{k} A), then by the above isomorphism, we have Res_{A}Y^{▪} ∈ K^{s}(Inj A). By the same reason as the above.

Proposition 15.8. If either A or C is k-flat, then

⊗^{▪}_{A}^{L} : D(Mod A^{op}⊗_{k}B) × D(Mod B^{op} ⊗_{k} C) → D(Mod A^{op} ⊗_{k} C)

exists, and we have a commutative diagram

D(Mod A^{op}⊗_{k}B) × D(Mod B^{op} ⊗_{k} C) ⟶^{Res_{B}×Res_{B^{op}}} D(Mod B) × D(Mod B^{op})
⊗^{▪}_{B}^{L} ↓  ↓⊗^{▪}_{B}^{L}
D(Mod A^{op} ⊗_{k} C)  ⟶^{Res_{k}}  D(Mod k)

Proof. Assume A is k-flat. For X^{▪} ∈ K(Mod A^{op}⊗_{k}B), Y^{▪} ∈ K(Mod B^{op}), by Proposition 15.4, we have an isomorphism

(_{A}X_{B}^{▪}) ⊗^{▪}_{A^{op}⊗B} (_{B}Y^{▪}⊗_{k}A_{A}) ≅ X^{▪} ⊗^{▪}_{B} Y^{▪}.

If _{A}X_{B}^{▪} ∈ K^{s}(Proj A^{op} ⊗_{k} B), then by the proof of Theorem 14.22, X^{▪} ⊗^{▪}_{B} Y^{▪} is acyclic if either X^{▪} or Y^{▪} is acyclic. Therefore we get the assertion by Theorem 14.5. In case of C being k-flat, similarly.

Example 15.9. Let F be a field, k = A = F [[x]], B = C = F [[x]]/(x^{2}). Let AM_{B} = F [[x]]/(x^{2}), _{A}N_{C} = F, R′ Hom_{A}^{▪} the right derived functor of

Hom_{A}^{▪} : K^{−}(Mod B^{op}⊗_{k}A)^{op} × K^{+}(Mod C^{op}⊗_{k}A) → K(Mod C^{op}⊗_{k}B).

Then we have

R′ Hom_{A}^{▪}(M, N) ≅ Hom_{A}(F [[x]]/(x^{2}), F).

Let R Hom_{A}^{▪} be the right derived functor of Hom_{A}^{▪} : K^{−}(Mod A)^{op} × K^{+}(Mod A) → K(Mod k). Then we have

R Hom_{A}^{▪}(M, N) ≅ X^{0} → X^{1}

where X^{0} → X^{1} = Hom_{A}(F [[x]], F) ⟶^{Hom_{A}(x^{2},F)} Hom_{A}(F [[x]], F). Then

R′ Hom_{A}^{▪}(M, N) ≇ R Hom_{A}^{▪}(M, N)

in D(Mod k).

16. Tilting Complexes

Throughout this section, A, B, C are rings. We recall that mod A is the category of finitely presented right A-modules, and that proj A is the full subcategory of mod A consisting of finitely generated projective right A-modules.

Definition 16.1 (Perfect Complex). A complex X^{▪} ∈ D(Mod A) is called a perfect complex if X^{▪} is isomorphic to a complex of K^{b}(proj A) in D(Mod A). We denote by D(Mod A)_{perf} the triangulated full subcategory of D(Mod A) consisting of perfect complexes.

Lemma 16.2. For X^{▪} ∈ K^{b}(Proj A), the following are equivalent.

1. X^{▪} is a compact object in K^{b}(Proj A).

2. X^{▪} is isomorphic to an object of K^{b}(proj A).

Proof. 2 ⇒ 1. By Lemma 16.3.

1 ⇒ 2. Let X^{▪} = X^{0} ⟶^{d^{0}} X^{1} → … → X^{n}, with X^{i} ∈ Proj A. By adding P ⟶^{1} P to X^{▪}, we may assume that X^{0} is a free A-module A^{(I)}. If I is a finite set, then by 2 ⇒ 1 X^{0} is also compact, and hence τ_{≥1}X^{▪} is compact. by induction on n, we get the assertion. Otherwise, Since we have Hom_{K(Mod A)}(X^{▪}, A^{(I)}) ≅ Hom_{K(Mod A)}(X^{▪}, A)^{(I)}, the canonical morphism X^{▪} → A^{(I)} factors through a direct summand μ : A^{m} ↪ A^{(I)} for some m ∈ ℕ. Then there is a homotopy morphism h : X^{1} → A^{(I)} such that 1_{A^{(I)}} − μg = hd^{0} with some g : A^{(I)} → A^{m}. Let A^{(I)} =

A^{m}⊕A^{(J)} be the canonical decomposition, then A^{(J)} ⟶^{d^{0}|_{A^{(J)}}} X^{1} ⟶^{ph} A^{(J)} = 1_{A^{(J)}}, where p : A^{(I)} → A^{(J)} is the canonical projection. Therefore X^{▪} ≅ M^{▪}(1_{A^{(J)}})[−1] ⊕ X′^{▪}, where X′^{▪} : A^{m} → X′^{1} → … → X^{n} with X′^{1} being a direct summand of X^{1}. Then we reduce the case of X^{0} being a finitely generated free A-module.

Lemma 16.3. For a complex X^{▪} ∈ K(Proj A), the following hold.

1. X^{▪} is a compact object in K(Mod A).

2. There is a complex P^{▪} ∈ K^{b}(proj A) such that X^{▪} ≅ P^{▪} in K(Mod A).

Proof. 2 ⇒ 1. We may assume X^{▪} ∈ K^{b}(proj A). Let {Y_{i}^{▪}}_{i∈I} be a collection of complexes of K(Mod A). Since a finitely generated A-module is a compact object in Mod A (see Exercise 2.8), we have isomorphisms

Hom_{A}^{▪}(X^{▪}, ∐_{i∈I}Y_{i}^{▪}) ≅ Tot Hom_{A}^{▪▪}(X^{▪}, ∐_{i∈I}Y_{i}^{▪})
≅ ∐_{i∈I} Tot Hom_{A}^{▪▪}(X^{▪}, Y_{i}^{▪})
≅ ∐_{i∈I} Hom_{A}^{▪}(X^{▪}, Y_{i}^{▪}).

By taking cohomology, we have an isomorphism

Hom_{K(Mod A)}(X^{▪}, ∐_{i∈I}Y_{i}^{▪}) ≅ ∐_{i∈I} Hom_{K(Mod A)}(X^{▪}, Y_{i}^{▪}).

1 ⇒ 2. Since C^{▪}(X^{▪}) = ⨁_{i∈ℤ} C^{i}(X^{▪})[−i], we have isomorphisms in 𝔄𝔟

∐_{i∈ℤ} Hom_{K(Proj A)}(X^{▪}, C^{i}(X^{▪})[−i]) ≅ Hom_{K(Proj A)}(X^{▪}, ⨁_{i∈ℤ} C^{i}(X^{▪})[−i])
≅ ∏_{i∈ℤ} Hom_{K(Proj A)}(X^{▪}, C^{i}(X^{▪})[−i]).

Then it is easy to see Hom_{K(Proj A)}(X^{▪}, C^{i}(X^{▪})[−i]) = 0 for all but finitely many i ∈ ℤ. Therefore for all but finitely many i ∈ ℤ, we have exact sequences

Hom_{A}(X^{i+1}, C^{i}(X^{▪})) → Hom_{A}(C^{i}(X^{▪}), C^{i}(X^{▪})) → O.

This means that the canonical morphisms C^{i}(X^{▪}) → X^{i+1} are split monomorphisms. Then there are m ≤ n such that X^{▪} ≅ σ′_{≥m}σ_{≤n}X^{▪}. σ′_{≥m}σ_{≤n}X^{▪} ∈ K(Proj A). Then we may assume X^{▪} : X^{0} → X^{1} → … → X^{n} with H^{0}(X^{▪}) ≠ 0, H^{n}(X^{▪}) ≠ 0. By the proof of Lemma 16.2, we get the statement.

Proposition 16.4. Let X^{▪} be a complex of D^{∗}(Mod A), where ∗ = nothing, −. Then the following are equivalent.

1. X^{▪} is a perfect complex.

2. X^{▪} is a compact object in D^{∗}(Mod A).

Proof. Since K^{s}(Proj A) ≅^{t} D(Mod A), it is trivial by Lemma 16.3.

Lemma 16.5. Let T^{▪} ∈ K^{b}(proj A) with Hom_{K(Mod A)}(T^{▪}, T^{▪}[i]) = 0 for i ≠ 0, and B = End_{K(Mod A)}(T). Then there exists a fully faithful ∂-functor F : K^{−}(Proj B) → K^{−}(Proj A) such that

1. F B ≅ T^{▪}.

2. F preserves coproducts.

3. F has a right adjoint G : K^{−}(Proj A) → K^{−}(Proj B).

Skip. This lemma is important. But the proof is out of the methods of derived categories.

Lemma 16.6. If T^{▪} satisfies the condition (G), then F : K^{−}(Proj B) → K^{−}(Proj A) is an equivalence. (G) For X^{▪} ∈ K^{−}(Proj A), X^{▪} = O whenever Hom_{K^{−}(Proj A)}(T^{▪}, X^{▪}[i]) = 0 for all

i.

Proof. Let X^{▪} ∈ K^{−}(Proj A) such that GX^{▪} = O. Then Hom_{K^{−}(Proj A)}(T^{▪}, X^{▪}[i]) ≅ Hom_{K^{−}(Proj B)}(B, GX^{▪}[i]) = 0 for all i. Therefore Ker G = {O}. By the left version of Proposition 9.13, G and F are equivalences.

Definition 16.7. Let C be a triangulated category. A subcategory B of C is said to generates C as a triangulated category if C is the smallest triangulated full subcategory which is closed under isomorphisms and contains B.

Remark 16.8. Let C be a triangulated category. For a subcategory B of C, we can construct the smallest triangulated full subcategory EB which is closed under isomorphisms and contains B as follows.

Let E^{0}B = B. For n > 0, let E^{n}B be the full subcategory of C consisting of objects X there exist U, V ∈ E^{n−1}B satisfying that either of (X, U, V, ∗, ∗, ∗) or (U, V, X, ∗, ∗, ∗) is a triangle in C. Then it is easy to see that EB = ⋃_{n≥0}E^{n}B is the smallest triangulated full subcategory which is closed under isomorphisms and contains B

Theorem 16.9. Let T^{▪} be a complex of K^{b}(proj A) such that

(a) Hom_{K(Mod A)}(T^{▪}, T^{▪}[i]) = 0 for i ≠ 0,

(b) add T_{A}^{▪} generates K^{b}(proj A). Then F : K^{−}(Proj B) → K^{−}(Proj A) is an equivalence.

Proof. It suffices to show that T^{▪} satisfies the condition of Lemma 16.6. Since add T_{A}^{▪} generates K^{b}(proj A), if Hom_{K^{−}(Proj A)}(T^{▪}, X^{▪}[i]) = 0 for all i, then Hom_{K^{−}(Proj A)}(A, X^{▪}[i]) = 0 for all i. Thus X^{▪} = O.

Lemma 16.10. For X^{▪} ∈ D^{−}(Mod A), the following are equivalent.

1. X^{▪} ∈ D^{b}(Mod A).

2. For any Y^{▪} ∈ D^{−}(Mod A), there is n such that Hom_{D(Mod A}(Y^{▪}, X^{▪}[i]) = 0 for all i < n.

Proof. 1 ⇒ 2. We may assume X^{▪} ∈ C^{b}(Mod A), Y^{▪} ∈ K^{−}(Proj A). Then Hom_{D(Mod A)}(Y^{▪}, X^{▪}[i]) ≅ Hom_{K(Mod A)}(Y^{▪}, X^{▪}[i]).

2 ⇒ 1. Since Hom_{D(Mod A^{b})}(A, X^{▪}[i]) ≅ H^{i}(X^{▪}), it is easy.

For an additive category B and m ≤ n, we write K^{[m,n]}(B) for the full subcategory of K(B) consisting of complexes X^{▪} with X^{i} = O for i < m, n < i.

Lemma 16.11. For X^{▪} ∈ D^{b}(Mod A), the following are equivalent.

1. X^{▪} is isomorphic to an object of K^{b}(Proj A).

2. For any Y^{▪} ∈ D^{b}(Mod A), there is n such that Hom_{D(Mod A}(X^{▪}, Y^{▪}[i]) = 0 for all i > n.

Proof. 1 ⇒ 2. It is trivial.

2 ⇒ 1. We may assume X^{▪} ∈ K^{−}(Proj A). Let M = ∏_{i∈ℤ} C^{i}(X^{▪}). By the same reason as the proof of Lemma 16.3, Hom_{K^{−}(Mod A)}(X^{▪}, M[i]) = 0 for all i > n if and only if X^{▪} is isomorphic to an object in K^{[−n,∞)}(Proj A).

Theorem 16.12. Let A, B be rings. The following are equivalent.

1. D^{−}(Mod A) ≅^{t} D^{−}(Mod B).

2. D^{b}(Mod A) ≅^{t} D^{b}(Mod B).

3. K^{b}(Proj A) ≅^{t} K^{b}(Proj B).

4. K^{b}(proj A) ≅^{t} K^{b}(proj B).

5. There exists T^{▪} ∈ K^{b}(proj A) with B ≅ Hom_{K^{b}(proj A)}(T^{▪}) such that (a) Hom_{K(Mod A)}(T^{▪}, T^{▪}[i]) = 0 for i ≠ 0,

(b) add T_{A}^{▪} generates K^{b}(proj A).

6. There exists T^{▪} ∈ K^{b}(proj A) with B ≅ Hom_{K^{b}(proj A)}(T^{▪}) such that (a) Hom_{K(Mod A)}(T^{▪}, T^{▪}[i]) = 0 for i ≠ 0,

(b) For X^{▪} ∈ K^{−}(Proj A), X^{▪} = O whenever Hom_{K^{−}(Proj A)}(T^{▪}, X^{▪}[i]) = 0 for all i.

Proof. By Theorem 16.9, Lemmas 16.6, 16.10, 16.11 and 16.2.

Remark 16.13. Since the functors Hom_{A}(−, A) : K^{b}(proj A) → K^{b}(proj A^{op}) and Hom_{A}(−, A) : K^{b}(proj A^{op}) → K^{b}(proj A) induce a duality between them, the condition 5 of Theorem 16.12 induces the left version of the condition 5. Therefore,

D^{−}(Mod A) ≅^{t} D^{−}(Mod B) if and only if D^{−}(Mod A^{op}) ≅^{t} D^{−}(Mod B^{op}).

Definition 16.14. A complex T_{A}^{▪} ∈ K^{b}(proj A) is called a tilting complex for A provided that

1. Hom_{K(Mod A)}(T^{▪}, T^{▪}[i]) = 0 for i ≠ 0.

2. add T_{A}^{▪} generates K^{b}(proj A).

We say that B is derived equivalent to A if there is a tilting complex T_{A}^{▪} such that B ≅ End_{K(Mod A)}(T^{▪}).

Lemma 16.15. Let U be a collection of objects K^{[r,s]}(proj A) for some r ≤ s. Consider a sequence of triangles

U_{1}^{▪} → U_{0}^{▪} → X_{1}^{▪} →
U_{2}^{▪}[1] → X_{1}^{▪} → X_{2}^{▪} →
…
U_{n}^{▪}[n − 1] → X_{n−1}^{▪} → X_{n}^{▪} →
…,

with all U_{i}^{▪} ∈ U. Then hlim_{⟶} X_{n}^{▪} ∈ K̃^{−}(proj A).

Proposition 16.16. For rings A, B, the following are equivalent.

1. B is derived equivalent to A.

2. K^{−}(proj A) ≅^{t} K^{−}(proj B).

In this case, we have K^{−,b}(proj A) ≅^{t} K^{−,b}(proj B).

Proof. 1 ⇒ 2. According to Theorem 16.12, we have K^{b}(proj A) ≅^{t} K^{b}(proj B). By

Lemma 16.15, it is easy to see that K^{−}(proj A) ≅^{t} K^{−}(proj B)

2 ⇒ 1. Let X^{▪} ∈ K^{−}(proj A). Since ⨁_{n∈ℕ}τ_{≤−n}X^{▪} exists in K^{−}(proj A), if X^{▪} is a compact object in K^{−}(proj A), then

Hom_{K(Mod A)}(X^{▪}, τ_{≤−n}X^{▪}) = 0

for all but finitely many n. If Hom_{K(Mod A)}(X^{▪}, τ_{≤−n}X^{▪}) = 0, then by Proposition 4.8, X^{▪} ≅ τ_{≥−n+1}X^{▪}⊕τ_{≤−n}X^{▪}[−1]. According to Proposition 10.23 X^{▪} ∈ K̃^{b}(proj A). As a consequence, X^{▪} is a compact object in K^{−}(proj A) if and only if X^{▪} is isomorphic to an object in K^{b}(proj A). Since compactness is a categor-

ical property, we have K^{b}(proj A) ≅^{t} K^{b}(proj B). By Theorem 16.12, we get the statement.

The last assertion is trivial by Lemma 16.10.

Lemma 16.17. For P^{▪} ∈ C^{−,b}(Proj A), we have isomorphisms in K^{−,b}(Proj A)

hlim_{⟶} τ_{≥−i}P^{▪} ≅ \underrightarrow{lim}_{C^{−,b}(Proj A)}τ_{≥−i}P^{▪}
≅ \underrightarrow{lim}_{K^{−,b}(Proj A)}τ_{≥−i}P^{▪}.

Proof. According to Proposition 11.7, we have the first isomorphism. For Y^{▪} ∈ C^{−,b}(Proj A), there is n ∈ ℤ such that H^{i}(Y^{▪}) = 0 for all i ≤ n. Applying Hom_{K(Mod A)}(−, Y^{▪}[j]) to a triangle τ_{≥−i+1}P^{▪} → τ_{≥−i}P^{▪} → P^{−i}[i] → τ_{≥−i+1}P^{▪}[1], we have an exact sequence

Hom_{K(Mod A)}(P^{−i}[i], Y^{▪}[j]) → Hom_{K(Mod A)}(τ_{≥−i}P^{▪}, Y^{▪}[j]) ⟶^{μ_{i}}

Hom_{K(Mod A)}(τ_{≥−i+1}P^{▪}, Y^{▪}[j]) → Hom_{K(Mod A)}(P^{−i}[i], Y^{▪}[j + 1]).

By Exercise 6.22, we have

Hom_{K(Mod A)}(P^{−i}, Y^{▪}[j − i + 1]) ≅ Hom_{A}(P^{−i}, H^{j−i+1}(Y^{▪})) = 0

for i ≥ j − n + 1, and then μ_{i} are epic for i ≥ j − n + 1. By Exercise 11.5, we have an exact sequence

0 → Hom_{K(Mod A)}(\underrightarrow{lim}_{C^{−,b}(Proj A)}τ_{≥−i}P^{▪}, Y^{▪}) → ∏_{i} Hom_{K(Mod A)}(τ_{≥−i}P^{▪}, Y^{▪}) →
∏_{i} Hom_{K(Mod A)}(τ_{≥−i}P^{▪}, Y^{▪}) → 0.

Hence

\underrightarrow{lim}_{C^{−,b}(Proj A)}τ_{≥−i}P^{▪} ≅ \underrightarrow{lim}_{K^{−,b}(Proj A)}τ_{≥−i}P^{▪}.

Proposition 16.18. Let A, B be coherent rings. The following are equivalent. 1. B is derived equivalent to A.

2. D^{−}(mod A) ≅^{t} D^{−}(mod B).

3. D^{b}(mod A) ≅^{t} D^{b}(mod B).

Proof. 1 ⇒ 2, 3. By Proposition 16.16.

3 ⇒ 1. By Corollary 10.13, D^{b}(mod A) and D^{b}(mod B) are triangle equivalent to K^{−,b}(proj A) and K^{−,b}(proj B), respectively. Then we have an equivalence F : K^{−,b}(proj A) → K^{−,b}(proj B) and its quasi-inverse G : K^{−,b}(proj B) → K^{−,b}(proj A). We may assume that G(B) = Q^{▪} : … → Q^{−1} → Q^{0} and F (A) = P^{▪} : … → P^{n−1} → P^{n}. By Proposition 11.7, τ_{≤−1}P^{▪} ≅ \underrightarrow{lim}τ_{≥−i}τ_{≤−1}P^{▪}. Since G is an equivalence, by Lemma 16.17 Gτ_{≤−1}P^{▪} ≅ \underrightarrow{lim}_{K^{−,b}(Proj A)}Gτ_{≥−i}τ_{≤−1}P^{▪} in K^{−,b}(proj B). Since Gτ_{≤−1}P^{▪} ∈ K^{−,b}(proj B), there exists k ∈ ℤ such that H^{k}(Gτ_{≤−1}P^{▪}) ≠ 0 and H^{j}(Gτ_{≤−1}P^{▪}) = 0 for all j > k. Let C_{k}^{▪} = C^{k}(Gτ_{≤−1}P^{▪}), then C_{k}^{▪} = C^{k}(Gτ_{≤−1}P^{▪}) ∈ K^{−,b}(proj A) and Hom_{K(proj A)}(Gτ_{≤−1}P^{▪}, C_{k}^{▪}[−k]) ≠ 0, because A is coherent. Therefore, there is m ≥ 1 such that Hom_{K(proj A)}(Gτ_{≥−m}τ_{≤−1}P^{▪}, C_{k}^{▪}[−k]) ≠ 0. Then we have m < 0, because Q^{▪} ∈ K^{[−∞,0]}(proj A). Since

Hom_{K(proj A)}(P^{▪}, τ_{≤−1}P^{▪}) ≅ Hom_{K(proj A)}(A, Gτ_{≤−1}P^{▪}) ≅ H^{0}(Gτ_{≤−1}P^{▪}) = 0,

by Proposition 4.8 τ_{≥0}P^{▪} ≅ P^{▪}⊕τ_{≤−1}P^{▪}[−1]. According to Proposition 10.23, P^{▪} ∈ K̃^{b}(proj A), and hence P_{A}^{▪} is a tilting complex.

17. Two-sided Tilting Complexes

17.1. The Case of Flat k-algebras. Throughout this subsection, k is a commutative ring, A, B, C are k-algebras which are k-flat modules. See Propositions 15.7, 15.8.

Theorem 17.1. Let A_{i} be an k-algebra with a tilting complex T_{i}^{▪} whose endomor-

phism is isomorphic to B_{i} (i = 1, 2). Then T_{1}^{▪} ⊗^{▪}_{k} T_{2}^{▪} is a tilting complex for A_{1}⊗_{k}A_{2} whose endomorphism is isomorphic to B_{1}⊗_{k}B_{2}.

Proof. Since T_{i}^{j} is A_{i}-projective, by Proposition 15.1 we have isomorphisms for all i, j, k, l

Hom_{A_{1}⊗_{k}A_{2}}(T_{1}^{i}⊗_{k}T_{2}^{j}, T_{1}^{k}⊗_{k}T_{2}^{l}) ≅ Hom_{A_{1}}(T_{1}^{i}, Hom_{A_{1}}(T_{2}^{j}, T_{1}^{k}⊗_{k}T_{2}^{l}))
≅ Hom_{A_{1}}(T_{1}^{i}, A_{1})⊗_{A_{1}^{op}}T_{1}^{k}⊗_{k}T_{2}^{l}⊗_{A_{2}} Hom_{A_{2}}(T_{2}^{j}, A_{2})
≅ Hom_{A_{1}}(T_{1}^{i}, T_{1}^{k})⊗_{k} Hom_{A_{2}}(T_{2}^{j}, T_{2}^{l}).

This induces an isomorphism between quadruple complexes. Since T_{i}^{▪} are bounded complexes, we have an isomorphism between complexes

Hom_{A_{1}⊗_{k}A_{2}}^{▪}(T_{1}^{▪} ⊗^{▪}_{k} T_{2}^{▪}, T_{1}^{▪} ⊗^{▪}_{k} T_{2}^{▪}) →^{∼} Hom_{A_{1}}^{▪}(T_{1}^{▪}, T_{1}^{▪}) ⊗^{▪}_{k} Hom_{A_{2}}^{▪}(T_{2}^{▪}, T_{2}^{▪}).

Since A_{i}, B_{i} are k-flat, we have isomorphisms in D(Mod k)

Hom_{A_{1}}^{▪}(T_{1}^{▪}, T_{1}^{▪}) ⊗^{▪}_{k} Hom_{A_{2}}^{▪}(T_{2}^{▪}, T_{2}^{▪}) ≅ Hom_{A_{1}}^{▪}(T_{1}^{▪}, T_{1}^{▪}) ⊗^{▪} _{k}^{L} Hom_{A_{2}}^{▪}(T_{2}^{▪}, T_{2}^{▪})

≅ End_{K(Mod A_{1})}(T_{1}^{▪}) ⊗^{▪} _{k}^{L} End_{K(Mod A_{2})}(T_{2}^{▪})

≅ B_{1} ⊗^{▪} _{k}^{L}B_{2}

≅ B_{1}⊗_{k}B_{2}.

Thus Hom_{K(Mod A_{1}⊗_{k}A_{2})}(T_{1}^{▪} ⊗^{▪}_{k} T_{2}^{▪}, T_{1}^{▪} ⊗^{▪}_{k} T_{2}^{▪}[i]) = 0 for i ≠ 0. It is easy to see that

End_{K(Mod A_{1}⊗_{k}A_{2})}(T_{1}^{▪} ⊗^{▪}_{k} T_{2}^{▪}) ≅ B_{1}⊗_{k}B_{2} as k-algebras.

Let F = A_{1}⊗_{k}− : K^{b}(proj A_{2}) → K^{b}(proj A_{1}⊗_{k}A_{2}). Since add T_{2}^{▪} generates

K^{b}(proj A_{2}), add A_{1}⊗_{k}T_{2}^{▪} generates K^{b}(proj A_{1}⊗_{k}A_{2}). Let G = −⊗^{▪}_{k}T_{2}^{▪} : K^{b}(proj A_{1}) → K^{b}(proj A_{1}⊗_{k}A_{2}). Since add T_{1}^{▪} generates K^{b}(proj A_{1}), the triangulated full sub category generated by add T_{1}⊗_{k}T_{2}^{▪}, contains add A_{1}⊗_{k}T_{2}^{▪}. Therefore add T_{1}⊗_{k}T_{2}^{▪} generates K^{b}(proj A_{1}⊗_{k}A_{2}).

Proposition 17.2. Let A be a k-algebra which is derived equivalent to a k-algebra B. Then we have isomorphisms

HH_{k}^{▪}(A, A) = Ext_{A^{op}⊗_{k}A}^{▪}(A, A)
≅ Ext_{B^{op}⊗_{k}B}^{▪}(B, B)
= HH_{k}^{▪}(B, B).

Here HH_{k}^{▪} means Hochschild homology.

Proof. Let T^{▪} be a tilting complex for A whose endomorphism ring is isomorphic to B. By Theorem 17.1, T^{▪}⊗_{k}T^{▪∗} is a tilting complex for A^{op}⊗_{k}A whose endomorphism ring is isomorphic to B^{op}⊗_{k}B, where T^{▪∗} = Hom_{A}(T^{▪}, A) According to Lemma 16.5, there is a equivalence F : D^{b}(Mod B^{op}⊗_{k}B) → D^{b}(Mod A^{op}⊗_{k}A) which sends B^{op}⊗_{k}B to T^{▪}⊗_{k}T^{▪∗}. Let X^{▪} ∈ D^{b}(Mod B^{op}⊗_{k}B) such that F X^{▪} ≅ A in D^{b}(Mod A^{op}⊗_{k}A). Then we have

H^{n}(X^{▪}) ≅ Hom_{D^{b}(Mod B^{op}⊗_{k}B)}(B^{op}⊗_{k}B, X^{▪}[n])
≅ Hom_{D^{b}(Mod A^{op}⊗_{k}A)}(T^{▪}⊗_{k}T^{▪∗}, A[n])
≅ Hom_{D^{b}(Mod A)}(T^{▪}, T^{▪∗∗}[n])
≅ Hom_{D^{b}(Mod A)}(T^{▪}, T^{▪}[n]).

Hence X^{▪} ≅ B in D^{b}(Mod B^{op}⊗_{k}B).

Definition 17.3. Let A be a k-algebra which is derived equivalent to a k-algebra B, and P^{▪} a tilting complex for A whose endomorphism ring is isomorphic to B. Then we have a triangle equivalence F : D^{b}(Mod B) → D^{b}(Mod A) which sends B to P^{▪}. We take a complex Q^{▪} of K^{b}(proj B) which is isomorphic to F^{−1}A in K^{b}(Mod B), and P^{▪∗} = Hom_{A}(P^{▪}, A), Q^{▪ ⋆} = Hom_{B}(Q^{▪}, B). Then a tilting complex B⊗_{k}P^{▪} induces the triangle equivalence

D^{b}(Mod B^{op}⊗_{k}B) → D^{b}(Mod B^{op}⊗_{k}A).

D^{b}(Mod A^{op}⊗_{k}A) D^{b}(Mod B^{op}⊗_{k}A) D^{b}(Mod A^{op}⊗_{k}B) D^{b}(Mod B^{op}⊗_{k}B)
A  T^{▪}  T^{∨▪}  B

Let T^{▪} be the image of B of D^{b}(Mod B^{op}⊗_{k}B). Also, a tilting complex A⊗_{k}Q^{▪⋆} induces the triangle equivalence

D^{b}(Mod A^{op}⊗_{k}A) → D^{b}(Mod A^{op}⊗_{k}B).

Let T^{∨▪} be the image of A of D^{b}(Mod A^{op}⊗_{k}A).

Proposition 17.4. The following hold.

1. Res_{A}T^{▪} ≅ P^{▪} in D^{b}(Mod A).

2. Res_{B^{op}}T^{▪} ≅ Q^{▪⋆} in D^{b}(Mod B^{op}).

3. Res_{B}T^{∨▪} ≅ Q^{▪} in D^{b}(Mod B).

4. Res_{A^{op}}T^{∨▪} ≅ P^{▪∗} in D^{b}(Mod A^{op}).

Proof. 1. We have isomorphisms of functors K^{−}(proj A) → Mod k

Hom_{D(Mod A)}(−, Res_{A}T^{▪}) ≅ Hom_{D(Mod B^{op}⊗_{k}A)}(B^{op}⊗_{k}−, T^{▪})
≅ Hom_{D(Mod A^{op}⊗_{k}A)}(P^{▪∗}⊗_{k}−, A)
≅ Hom_{D(Mod A)}(−, P^{▪∗∗})
≅ Hom_{D(Mod A)}(−, P^{▪}).

2. We have isomorphisms of functors K^{−}(proj A) → Mod k

Hom_{D(Mod B^{op})}(−, Res_{B^{op}}T^{▪}) ≅ Hom_{D(Mod B^{op}⊗_{k}A)}(−⊗_{k}A, T^{▪})
≅ Hom_{D(Mod B^{op}⊗_{k}B)}(−⊗_{k}Q^{▪}, B)
≅ Hom_{D(Mod B^{op})}(−, Q^{▪⋆}).

3, 4. Similarly.

Lemma 17.5. There is an isomorphism ϕ : P^{▪} → Res_{A}T^{▪} in D(Mod A) such that ϕf = λ(f)ϕ for all f ∈ End_{D(Mod A)}(P^{▪}), where λ : B → End_{D(Mod A)}(Res_{A}T^{▪}) is the left multiplication morphism.

Proof. For f ∈ End_{D(Mod A)}(P^{▪}), by the above isomorphisms, we have a commutative diagram

Hom_{D(Mod A)}(−, Res_{A}T^{▪}) ⟶^{∼} Hom_{D(Mod Bop⊗kA)}(B^{op}⊗_{k}−, T^{▪}) ⟶^{∼} Hom_{D(Mod A)}(−, P^{▪})
Hom(−,λ(f))↓  ↓^{Hom(fop⊗−,A)}  ↓^{Hom(−,f)}
Hom_{D(Mod A)}(−, Res_{A}T^{▪}) ⟶^{∼} Hom_{D(Mod Bop⊗kA)}(B^{op}⊗_{k}−, T^{▪}) ⟶^{∼} Hom_{D(Mod A)}(−, P^{▪}).

Theorem 17.6. For ∗ = nothing, +, −, b, the ∂-functor

− ⊗^{▪} _{B}^{L∗}T^{▪} : D^{∗}(Mod B) → D^{∗}(Mod A)

is an triangle equivalence, and its quasi-inverse is

R^{∗} Hom_{A}(T^{▪}, −) ≅ − ⊗^{▪} _{A}^{L∗}T^{∨▪} : D^{∗}(Mod A) → D^{∗}(Mod B).

Proof. For a complex X^{▪} ∈ D^{∗}(Mod B), X^{▪} ⊗^{▪} _{B}^{L∗}T^{▪} ≅ X^{▪} ⊗^{▪}_{B} Q^{▪⋆} in D(Mod k). Then

− ⊗^{▪} _{B}^{L∗}T^{▪} is a way-out in both directions. Similarly, R^{∗} Hom_{A}(T^{▪}, −), − ⊗^{▪} _{A}^{L∗}T^{∨▪} are way-out in both directions. Therefore we have the above functors between the above derived categories. By Proposition 17.4, We have

Res_{A}T^{▪} ⊗^{▪} _{A}^{L∗}Res_{A^{op}}T^{∨▪} ≅ P^{▪} ⊗^{▪} _{A}^{L∗}P^{▪∗}
≅ End_{D^{∗}(Mod A)}(P^{▪})
= B

By Lemma 17.5, we have T^{▪} ⊗^{▪} _{A}^{L∗}T^{∨▪} ≅ B in D(Mod B^{op}⊗_{k}B). Similarly we have

T^{∨▪} ⊗^{▪} _{B}^{L∗}T^{▪} ≅ A in D(Mod A^{op}⊗_{k}A). Then − ⊗^{▪} _{B}^{L∗}T^{▪} is an equivalence. By adjoint-

ness, we have R^{∗} Hom_{A}(T^{▪}, −) ≅ − ⊗^{▪} _{A}^{L∗}T^{∨▪}.

Definition 17.7 (Biperfect Complex). A complex X^{▪} ∈ D(Mod A^{op}⊗_{k}B) is called a biperfect complex if Res_{A}X^{▪} ∈ D(Mod A)_{perf} and Res_{B^{op}}X^{▪} ∈ D(Mod B^{op})_{perf}. We denote by D(Mod A^{op}⊗_{k}B)_{biperf} the triangulated full subcategory of D(Mod A^{op}⊗_{k}B) consisting of biperfect complexes.

Definition 17.8. A bimodule complex _{B}T_{A}^{▪} ∈ K(Mod B^{op}⊗_{k}A) is called a twosided tilting complex provided that

1. _{B}T_{A}^{▪} is a biperfect complex.

2. There exists a biperfect complex _{A}T_{B}^{∨▪} such that

(a) _{B}T^{▪} ⊗^{▪} _{A}^{L}T_{B}^{∨▪} ≅ B in D^{b}(Mod B^{op}⊗_{k}B),

(b) _{A}T^{∨▪} ⊗^{▪} _{B}^{L}T_{A}^{▪} ≅ A in D^{b}(Mod A^{op}⊗_{k}A).

We call _{A}T_{B}^{∨▪} the inverse of _{B}T_{A}^{▪}.

R^{∗} Hom_{A}(T^{▪}, −) : D^{∗}(Mod A) → D^{∗}(Mod B) is called a standard equivalence, where ∗ = nothing, +, −, b.

Theorem 17.9. The following are equivalent.

1. D(Mod A) ≅^{t} D(Mod B).

2. D^{+}(Mod A) ≅^{t} D^{+}(Mod B).

3. A is derived equivalent to B.

4. There exists a two-sided tilting complex _{B}T_{A}^{▪}.

Proof. By Theorem 17.6 and the dual of Lemma 16.10.

Corollary 17.10. Let _{B}T_{A}^{▪} and _{C}S_{B}^{▪} be two-sided tilting complexes. Then _{C}S_{B}^{▪} ⊗^{▪} _{B}^{L}T_{A}^{▪} is a two-sided tilting complex.

17.2. The Case of Projective k-algebras. Throughout this subsection, k is a commutative ring, A, B, C are k-algebras which are k-projective modules. See Propositions 15.7, 15.8.

Lemma 17.11. Let X, P be B-A-bimodules such that Res_{A}P is a finitely generated projective A-module. Then the following hold.

1. For a right A-module M and a left A-module N, we have B-module morphisms

M⊗_{A} Hom_{A}(X, A) → Hom_{A}(X, M) (m⊗f ↦ (x ↦ mf(x))),

X⊗_{A}N → Hom_{A}(Hom_{A}(X, A), N) (x⊗n ↦ (f ↦ f(x)n)).

2. We have a functorial isomorphism of functors Mod A → Mod B

−⊗_{A} Hom_{A}(P, A) ≅ Hom_{A}(P, −).

3. We have a functorial isomorphism of functors Mod A^{op} → Mod B^{op}

P ⊗_{A}− ≅ Hom_{A}(Hom_{A}(P, A), −).

Lemma 17.12. Let X^{▪} ∈ D^{∗}(Mod B^{op}⊗_{k}A) with Res_{A}X^{▪} ∈ D^{∗}(Mod A)_{perf}, where ∗ = nothing, +, −, b. Then the following hold.

1. We have a ∂-functorial isomorphism of ∂-functors D^{∗}(Mod A^{op}⊗_{k}A) → D^{∗}(Mod A^{op}⊗_{k}B)

− ⊗^{▪} _{A}^{L∗}R Hom_{A}^{▪}(X^{▪}, A) ≅ R^{∗} Hom_{A}^{▪}(X^{▪}, −).

2. We have a ∂-functorial isomorphism of ∂-functors D^{∗}(Mod A^{op}⊗_{k}A) → D^{∗}(Mod B^{op}⊗_{k}A)

X^{▪} ⊗^{▪} _{A}^{L∗}− ≅ R^{∗} Hom_{A}^{▪}(R Hom_{A}^{▪}(X^{▪}, A), −).

Proof. 1. By Lemma 17.11, we have a ∂-functorial morphism of ∂-functors D^{∗}(Mod A^{op}⊗_{k}A) → D^{∗}(Mod A^{op}⊗_{k}B)

ϕ : − ⊗^{▪} _{A}^{L}R Hom_{A}^{▪}(X^{▪}, A) → R Hom_{A}^{▪}(X^{▪}, −).

Let Q^{▪} ∈ K^{b}(proj A) which has a quasi-isomorphism Q^{▪} → Res_{A}X^{▪}. By Lemma 17.11, we have a ∂-functorial isomorphism of ∂-functors D^{∗}(Mod A) → D^{∗}(Mod k)

ψ : − ⊗^{▪}_{A} Hom_{A}^{▪}(Q^{▪}, A) →^{∼} Hom_{A}^{▪}(Q^{▪}, −).

Since

Res_{k}ϕ ≅ ψ,

and H^{▪}(ψ) is an isomorphism, ϕ is a functorial isomorphism.

2. Similarly.

Corollary 17.13. Let T^{▪} and T^{∨▪} be a two-sided tilting complex and its inverse. Then we have isomorphisms in D^{b}(Mod B^{op}⊗_{k}A)

\begin{aligned}T^{∨▪} & ≅ R Hom_{A}^{▪}(T^{▪}, A) \\ & ≅ R Hom_{B}^{▪}(T^{▪}, B). \end{aligned}

Theorem 17.14. For a bimodule complex _{B}T_{A}^{▪}, the following are equivalent. 1. _{B}T_{A}^{▪} is a two-sided tilting complex.

2. _{B}T_{A}^{▪} satisfies that (a) _{B}T_{A}^{▪} is a biperfect complex,

(b) The right multiplication morphism ρ_{A} : A → R Hom_{B}^{▪}(T^{▪}, T^{▪}) is an isomorphism in D(Mod A^{op}⊗_{k}A),

(c) The left multiplication morphism λ_{B} : B → R Hom_{A}^{▪}(T^{▪}, T^{▪}) is an isomorphism in D(Mod B^{op}⊗_{k}B).

3. _{B}T_{A}^{▪} satisfies that (a) _{B}T_{A}^{▪} is a biperfect complex,

(b) Hom_{D^{b}(Mod B^{op})}(T^{▪}, T^{▪}[i]) = 0 for i ≠ 0,

(c) Hom_{D^{b}(Mod A)}(T^{▪}, T^{▪}[i]) = 0 for i ≠ 0,

(d) The right multiplication morphism ρ_{A} induces a ring isomorphism A → End_{D^{b}(Mod B^{op})}(T^{▪})^{op}, (e) The left multiplication morphism λ_{B} induces a ring isomorphism B → End_{D^{b}(Mod A)}(T^{▪}).

Proof. 1 ⇒ 3. By Corollary 17.13, Lemma 17.12, we have isomorphisms in D(Mod B^{op}⊗_{k}B)

R Hom_{A}^{▪}(T^{▪}, T^{▪}) ≅ _{B}T^{▪} ⊗^{▪} _{A}^{L}R Hom_{A}^{▪}(T^{▪}, A)

≅ _{B}T^{▪} ⊗^{▪} _{A}^{L}T_{B}^{∨▪}

≅ B.

And we have an isomorphism in D(Mod A^{op}⊗_{k}A)

R Hom_{B}^{▪}(T^{▪}, T^{▪}) ≅ R Hom_{B}^{▪}(T^{▪}, B) ⊗^{▪} _{B}^{L}T_{A}^{▪}

≅ _{A}T^{∨▪} ⊗^{▪} _{B}^{L}T_{A}^{▪}

≅ A.

Since −⊗_{B}^{L}T_{A}^{▪} is an equivalence, we have a commutative diagram

Hom_{D(Mod B)}(B_{B}, B_{B}) ⟶^{∼} Hom_{D(Mod A)}(B⊗_{B}^{L}T_{A}^{▪}, B⊗_{B}^{L}T_{A}^{▪})
↑  ↓
B  ⟶^{λ_{B}}  Hom_{D(Mod A)}(T_{A}^{▪}, T_{A}^{▪})

where all vertical arrows are isomorphisms. Then λ_{B} is an isomorphism. Similarly, ρ_{A} is an isomorphism.

3 ⇒ 2. It is easy to see that we have a morphism λ_{A} : A → R Hom_{B}^{▪}(T^{▪}, T^{▪}) in D(Mod A^{op}⊗_{k}A). By taking cohomologies, we get the condition (b) of 2. Similarly, we get the condition (c) of 2.

2 ⇒ 1. Let T^{∨▪} = R Hom_{A}^{▪}(T^{▪}, A), then we have isomorphisms

BT^{▪} ⊗^{▪} _{A}^{L}T_{B}^{∨▪} = _{B}T^{▪} ⊗^{▪} _{A}^{L}R Hom_{A}^{▪}(T^{▪}, A)
≅ R Hom_{A}^{▪}(T^{▪}, T^{▪})
≅ B.

By Proposition 15.5 2, we have an isomorphism

R Hom_{A}^{▪}(T^{▪}, A) ≅ R Hom_{A}^{▪}(T^{▪}, R Hom_{B}^{▪}(T^{▪}, T^{▪}))
≅ R Hom_{B}^{▪}(T^{▪}, R Hom_{A}^{▪}(T^{▪}, T^{▪}))
≅ R Hom_{B}^{▪}(T^{▪}, B).

Then we have

AT^{∨▪} ⊗^{▪} _{B}^{L}T_{A}^{▪} ≅ R Hom_{B}^{▪}(T^{▪}, B) ⊗^{▪} _{B}^{L}T_{A}^{▪}
≅ R Hom_{B}^{▪}(T^{▪}, T^{▪})
≅ A.

Theorem 17.15. Let (A_{i}, B_{i}) be derived equivalent k-algebras, T_{i}^{▪} two-sided tilting complexes in D^{b}(Mod B_{i}^{op}⊗_{k}A_{i}) and their inverses T_{i}^{∨▪} (i = 0, 1, 2). Then we have the following commutative diagrams.

1.

D^{b}(Mod A_{0}^{op}⊗_{k}A_{1}) × D^{b}(Mod A_{1}^{op}⊗_{k}A_{2}) ⟶^{−⊗^{▪}_{A_{1}}^{L} −} D^{b}(Mod A_{0}^{op}⊗_{k}A_{2})
F_{0}×F_{1}↓  ↓^{F_{2}}
D^{b}(Mod B_{0}^{op}⊗_{k}B_{1}) × D^{b}(Mod B_{1}^{op}⊗_{k}B_{2}) ⟶^{−⊗^{▪}_{B_{1}}^{L} −} D^{b}(Mod B_{0}^{op}⊗_{k}B_{2}).

2.

D^{b}(Mod A_{1}^{op}⊗_{k}A_{2}) × D^{b}(Mod A_{0}^{op}⊗_{k}A_{2}) ⟶^{R Hom_{A_{2}}^{▪} (−,−)} D^{b}(Mod A_{0}^{op}⊗_{k}A_{1})
F_{1}×F_{2}↓  ↓^{F_{0}}
D^{b}(Mod B_{1}^{op}⊗_{k}B_{2}) × D^{b}(Mod B_{0}^{op}⊗_{k}B_{2}) ⟶^{R Hom_{B_{2}}^{▪} (−,−)} D^{b}(Mod B_{0}^{op}⊗_{k}B_{1}).

Here F_{0} = T_{0}^{▪} ⊗^{▪} _{A_{0}}^{L} −⊗^{▪}_{A_{1}}^{L}T_{1}^{∨▪}, F_{1} = T_{1}^{▪} ⊗^{▪} _{A_{1}}^{L} −⊗^{▪}_{A_{2}}^{L}T_{2}^{∨▪}, F_{2} = T_{0}^{▪} ⊗^{▪} _{A_{0}}^{L} −⊗^{▪}_{A_{2}}^{L}T_{2}^{∨▪}.

Proof. 1. We have isomorphisms

(T_{0}^{▪} ⊗^{▪} _{A_{0}}^{L} − ⊗^{▪}_{A_{1}}^{L}T_{1}^{∨▪}) ⊗^{▪} _{B_{1}}^{L}(T_{1}^{▪} ⊗^{▪} _{A_{1}}^{L} − ⊗^{▪}_{A_{2}}^{L}T_{2}^{∨▪})

≅ (T_{0}^{▪} ⊗^{▪} _{A_{0}}^{L}−) ⊗^{▪} _{A_{1}}^{L}(T_{1}^{∨▪} ⊗^{▪} _{B_{1}}^{L}T_{1}^{▪}) ⊗^{▪} _{A_{1}}^{L}(− ⊗^{▪} _{A_{2}}^{L}T_{2}^{∨▪})

≅ (T_{0}^{▪} ⊗^{▪} _{A_{0}}^{L}−) ⊗^{▪} _{A_{1}}^{L}(A_{1}) ⊗^{▪} _{A_{1}}^{L}(− ⊗^{▪} _{A_{2}}^{L}T_{2}^{∨▪})

≅ (T_{0}^{▪} ⊗^{▪} _{A_{0}}^{L}−) ⊗^{▪} _{A_{1}}^{L}(− ⊗^{▪} _{A_{2}}^{L}T_{2}^{∨▪}).

2. Let X^{▪} ∈ Mod A_{0}^{op}⊗_{k}A_{2}, Y^{▪} ∈ Mod A_{1}^{op}⊗_{k}A_{2}, Z^{▪} ∈ Mod A_{0}^{op}⊗_{k}A_{2}. Since we have an adjoint isomorphism

Hom_{D(Mod A_{0}^{op}⊗_{k}A_{2})}(X^{▪} ⊗^{▪} _{A_{1}}^{L}Y^{▪}, Z^{▪}) ≅ Hom_{D(Mod A_{0}^{op}⊗_{k}A_{1})}(X^{▪}, R Hom_{A_{2}}^{▪}(Y^{▪}, Z^{▪})),

we get the assertion by 1.

17.3. The Case of Finite Dimensional k-algebras. Throughout this subsection, we assume k is a field, and all algebra are finite dimensional k-algebras. We denote D_{k} = Hom_{k}(−, k).

Definition 17.16 (Nakayama Functor). A triangle auto-equivalence ν_{A} = − ⊗^{▪} _{A}^{L}D_{k}A : D^{b}(mod A) → D^{b}(mod A) is called a Nakayama functor.

Proposition 17.17. Let (A, B) be derived equivalent k-algebras, T^{▪} a two-sided tilting complex in D^{b}(Mod B^{op}⊗_{k}A) and its inverse T^{∨▪}. Then we have a commutative diagram

D^{−}(Mod A) ⟶^{ν_{A}} D^{−}(Mod A)
F ↓  ↓^{F}
D^{−}(Mod B) ⟶^{ν_{B}} D^{−}(Mod B),

where F is a standard equivalence.

Proof. By Proposition 17.2, the standard equivalence G : D^{b}(Mod A^{op}⊗_{k}A) → D^{b}(Mod B^{op}⊗_{k}B) sends A to B.

By the case of A_{1} = A^{op}⊗_{k}A, B_{1} = B^{op}⊗_{k}B and A_{0} = A_{2} = B_{0} = B_{2} = k in Theorem 17.15 2, we have GD_{k}A ≅ D_{k}B.

By the case of A_{1} = A_{2} = A, B_{1} = B_{2} = B and A_{0} = B_{0} = k in Theorem 17.15 1, we have F ν_{A} = ν_{B}F.

Corollary 17.18. Let A be a finite dimensional k-algebra which is derived equivalent to B. If A is a symmetric algebra, then B is a symmetric algebra.

Lemma 17.19. Let A, B be finite dimensional self-injective k-algebras, _{A}P_{B} is A^{op}⊗_{k}B-projective, _{B}V_{A} is A-projective and B-projective. The following hold.

1. B^{op}⊗_{k}A is a self-injective algebra.

2. Hom_{A}(P, A) is B^{op}⊗_{k}A-projective.

3. Hom_{A}(V, A) is A-projective and B-projective.

4. _{A}P ⊗_{B}V_{A} is A^{op}⊗_{k}A-projective.

5. X⊗_{A}P_{B} is B-projective for any X ∈ Mod A.

6. Y ⊗_{B}V_{A} is A-projective for any Y ∈ Proj B.

Proof. By Propositions 15.1, 15.3.

Proposition 17.20. Let A and B be derived equivalent self-injective k-algebras.

Then K(mod A) ≅^{t} K(mod B), and there are bimodules _{A}M_{B} and _{B}N_{A} such that

−⊗_{A}M : mod A → mod B and − ⊗_{B}N : mod B → mod A

induce an equivalence \underline{mod}A ≅ \underline{mod}B.

Proof. Let T^{▪} be a two-sided tilting complex in D^{b}(Mod B^{op}⊗_{k}A) and T^{∨▪} its inverse. By taking a B^{op}⊗_{k}A-projective resolution of T^{▪}, and its shifting and truncation, we may assume T^{▪} is isomorphic to

S^{▪} : S^{−n} → S^{−n+1} → … → S^{0}

where S^{i} are B^{op}⊗_{k}A-projective (−n < i ≤ 0), and S^{−n} is A-projective and Bprojective. Then T^{∨▪} ≅ Hom_{A}(S^{▪}, A) in D^{b}(Mod A^{op}⊗_{k}B), and we have

S^{▪} ⊗^{▪}_{A} Hom_{A}(S^{▪}, A) ≅ B in K^{b}(Mod B^{op}⊗_{k}B),
Hom_{A}(S^{▪}, A) ⊗^{▪}_{B} S^{▪} ≅ A in K^{b}(Mod A^{op}⊗_{k}A).

These imply that K(mod A) ≅^{t} K(mod B). Let M = Ω^{n}(Hom_{A}(S^{−n}, A)), the nth syzygy as as a B^{op}⊗_{k}A-module and N = Ω^{−n}(S^{−n}), the −nth syzygy as as a A^{op}⊗_{k}B-module. Since Hom_{A}(S^{−n}, A) is A-projective and B-projective, M is Aprojective and B-projective. Similarly, N is A-projective and B-projective. Then

−⊗_{A}M : mod A → mod B and − ⊗_{B}N : mod B → mod A

induce triangle functors between \underline{mod}A and \underline{mod}B. By Lemma 17.19, all terms

but the term Hom_{A}(S^{−n}, A)⊗_{A}S^{−n} of a double complex Hom_{A}(S^{▪}, A) ⊗^{▪▪}_{B} S^{▪} are A^{op}⊗_{k}A-projective. Therefore A is a direct summand of ⨁_{p=q} Hom_{A}(S^{p}, A)⊗_{B}S^{q} as a A^{op}⊗_{k}A-module. For each X ∈ mod A, we have isomorphisms in \underline{mod}A

X ≅ X⊗_{A}A
≅ X⊗_{A} Hom_{A}(S^{−n}, A)⊗_{B}S^{−n}
≅ ω^{−n}(X⊗_{A}M⊗_{B}S^{−n})
≅ ω^{n}ω^{−n}(X⊗_{A}M⊗_{B}N)
≅ X⊗_{A}M⊗_{B}N,

86 JUN-ICHI MIYACHI

where ω is the loop space functor on \underline{mod}A. Similarly, for each Y ∈ mod B, we have an isomorphism in \underline{mod}B

Y ≅ Y ⊗_{B}N⊗_{A}M.

Proposition 17.21. Let A and B be indecomposable symmetric k-algebras, X^{▪} a biperfect complex in D^{b}(Mod B^{op}⊗_{k}A), and X^{∨▪} = Hom_{k}^{▪}(X^{▪}, k). If X^{▪} ⊗^{▪}_{A}^{L}X^{∨▪} ≅ B in D^{b}(Mod B^{op}⊗_{k}B), then X^{▪} is a two-sided tilting complex. Proof. Since we have isomorphisms

\begin{aligned}X^{∨▪} & ≅ R Hom_{A}^{▪}(X^{▪}, A) \\ & ≅ R Hom_{B}^{▪}(X^{▪}, B), \end{aligned}

by Proposition 15.5, lemma 17.12, F = − ⊗^{▪} _{B}^{L}X^{▪} : D^{b}(mod B) → D^{b}(mod A) and F^{∨} = − ⊗^{▪} _{A}^{L}X^{∨▪} : D^{b}(mod A) → D^{b}(mod B) are both left and right adjoint to one another. By adjunction arrows of adjoint pairs (F, F^{∨}), (F^{∨}, F), we have morphisms α : A → X^{∨▪} ⊗^{▪} _{B}^{L}X^{▪}, β : X^{∨▪} ⊗^{▪} _{B}^{L}X^{▪} → A. By Proposition 1.17, we have a split monomorphism αF : X^{▪} → X^{▪} ⊗^{▪} _{A}^{L}X^{∨▪} ⊗^{▪} _{B}^{L}X^{▪}, a split epimorphism βF : X^{▪} ⊗^{▪} _{A}^{L}X^{∨▪} ⊗^{▪} _{B}^{L}X^{▪} → X^{▪}. Since D^{b}(mod B^{op}⊗_{k}A) is a Krull-Schmidt category by Corollary 11.19, X^{▪} ⊗^{▪} _{A}^{L}X^{∨▪} ≅ B implies that αF, βF are isomorphisms in D^{b}(mod B^{op}⊗_{k}A). If βα is not an isomorphism, then this contradicts (βα)F is an isomorphism. Therefore A is a direct summand of X^{∨▪} ⊗^{▪} _{B}^{L}X^{▪} in D^{b}(mod A^{op}⊗_{k}A).

Since X^{∨▪} ⊗^{▪} _{B}^{L}X^{▪} ⊗^{▪} _{A}^{L}X^{∨▪} ⊗^{▪} _{B}^{L}X^{▪} ≅ X^{∨▪} ⊗^{▪} _{B}^{L}X^{▪}, we have A ≅ X^{∨▪} ⊗^{▪} _{B}^{L}X^{▪} in

D^{b}(mod A^{op}⊗_{k}A).

Proposition 17.22. Let A be a symmetric k-algebra which has no simple projective A-module, e an idempotent of A, and P^{▪} : P^{−1} ⟶^{d^{−1}} P^{0} = Ae⊗_{k}eA ⟶^{μ} A, where μ is the multiplication morphism. Then P^{▪} is a tilting complex. Moreover, AP_{A}^{▪} is a two-sided tilting complex if and only if dim_{k} eAe = 2. Proof. Since D_{k}P^{▪} = (D_{k}A → D_{k}(Ae⊗_{k}eA)) ≅ (A → Ae⊗_{k}eA), Hom_{A}^{▪▪}(P_{A}^{▪}, P_{A}^{▪}) ≅

P^{▪} ⊗^{▪▪} _{A}D_{k}P^{▪} has the form

Ae⊗_{k}eA ⟶  A
↓ ↓
Ae⊗_{k}eAe⊗_{k}eA ⟶ Ae⊗_{k}eA,

where the left vertical arrow is monic and the bottom horizontal arrow is epic. Then H^{i}(Hom_{A}^{▪}(P_{A}^{▪}, P_{A}^{▪})) = 0 for i ≠ 0. Since

P^{▪} = (eAe⊗_{k}eA → eA)⊕((1 − e)Ae⊗_{k}eA → (1 − e)A)

and dim_{K} eAe = n ≥ 2, P^{▪} ≅ (eA^{n−1} → O)⊕((1 − e)Ae⊗_{k}eA → (1 − e)A), and then P^{▪} generates K^{b}(proj A_{A}). By the above diagram, we have an isomorphism in mod A^{op}⊗_{k}A

Ae⊗_{k}eA⊕ H^{0}(P^{▪} ⊗^{▪}_{A} D_{k}P^{▪})⊕Ae⊗_{k}eA ≅ Ae⊗_{k}eAe⊗_{k}eA⊕A.

By the Krull-Schmidt Theorem, we have

H^{0}(P^{▪} ⊗^{▪}_{A} D_{k}P^{▪}) ≅ A⊕(Ae⊗_{k}eA)^{n−2}.

18. Cotilting Bimodule Complexes

Throughout this section, unless otherwise stated, k is a commutative ring, A, B, C are k-algebras which are k-projective modules. See Propositions 15.7, 15.8.

Definition 18.1 (Cotilting Bimodule Complexes). Let A be a right coherent kalgebra and B a left coherent k-algebra. A complex _{B}U_{A}^{▪} ∈ D^{b}(Mod B^{op}⊗_{k}A) is called a cotilting B-A-bimodule complex provided that it satisfies

1. Res_{A}U^{▪} ∈ D_{c}^{b}(Mod A)_{fid} and Res_{B^{op}}U^{▪} ∈ D_{c}^{b}(Mod B^{op})_{fid}.

2. Hom_{D(Mod A)}(U^{▪}, U^{▪}[i]) = 0 for all i ≠ 0.

3. Hom_{D(Mod B^{op})}(U^{▪}, U^{▪}[i]) = 0 for all i ≠ 0.

4. the left multiplication morphism B → End_{D(Mod A)}(U^{▪}) is a ring isomorphism.

5. the right multiplication morphism A → End_{D(Mod B^{op})}(U^{▪})^{op} is a ring isomorphism.

In case of B = A, we will call a cotilting A-A-bimodule complex a dualizing A-bimodule complex.

Proposition 18.2. Let A be a right coherent k-algebra and B a left coherent kalgebra, and _{B}U_{A}^{▪} ∈ D^{b}(Mod B^{op}⊗_{k}A) a cotilting B-A-bimodule complex. Then

R^{∗} Hom_{A}^{▪}(−, U^{▪}) : D_{c}^{∗}(Mod A) → D_{c}^{†}(Mod B^{op})

and

R^{†} Hom_{B}^{▪}(−, U^{▪}) : D_{c}^{†}(Mod B^{op}) → D_{c}^{∗}(Mod A)

induce the duality, where (∗, †) = (nothing, nothing), (+, −), (−, +), (b, b).

Proof. Since Res_{A}U^{▪} ∈ D_{c}^{b}(Mod A)_{fid}, by Proposition 10.21 R^{∗} Hom_{A}^{▪}(−, U^{▪}) is way-out in both directions. Since R^{∗} Hom_{A}^{▪}(A, U^{▪}) ≅ Res_{B^{op}}U^{▪} ∈ D_{c}^{†}(Mod B^{op}),we have R^{∗} Hom_{A}^{▪}(−, U^{▪}) : D_{c}^{∗}(Mod A) → D_{c}^{†}(Mod B^{op}) by Proposition 12.12. Similarly we have R^{†} Hom_{B}^{▪}(−, U^{▪}) : D_{c}^{†}(Mod B^{op}) → D_{c}^{∗}(Mod A). Since

(R^{∗} Hom_{A}^{▪}(−, U^{▪}), R^{†} Hom_{B}^{▪}(−, U^{▪}))

is a right adjoint pair, we have adjunction arrows

η : 1_{D_{c}^{∗}(Mod A)} → R^{†} Hom_{B}^{▪}(−, U^{▪}) ∘ R^{∗} Hom_{A}^{▪}(−, U^{▪})
θ : 1_{D_{c}^{∗}(Mod B^{op})} → R^{∗} Hom_{A}^{▪}(−, U^{▪}) ∘ R^{†} Hom_{B}^{▪}(−, U^{▪}).

It is not hard that we have a commutative diagram in D_{c}^{∗}(Mod A^{op}⊗_{K}A)

A ⟶^{η(A)} R^{†} Hom_{B}^{▪}(R^{∗} Hom_{A}^{▪}(A, U^{▪}), U^{▪})
∥  ↓^{≀}
A ⟶^{ρ_{A}}  R^{†} Hom_{B}^{▪}(U^{▪}, U^{▪}),

Then η(A) is an isomorphism. By Proposition 12.11, η is an isomorphism. Similarly, θ is an isomorphism.

Lemma 18.3 (Piled Resolutions Lemma). Let A be an abelian category satisfying the condition Ab4* with enough injectives, and let C^{▪▪} be a double complex with C^{p,q} = 0 (p < 0 or q < 0), C^{▪j} → I^{▪j} quasi-isomorphisms with I^{▪j} : 0 → I^{−s,j} → I^{−s+1,j} → … ∈ K^{+}(Inj A) for all i. Then, there is a quasi-isomorphism from

Tot C^{▪▪} = Tot^{∧}C^{▪▪} to a complex J^{▪} of the following form in K^{+}(Inj A)

J^{n} = \begin{cases}O if n < −s \\ ⨁_{i+j=n}I^{i,j} if n ≥ −s \end{cases}

Proof. For a double complex C^{▪▪}, we have a sequence of morphisms {Tot τ_{≤n}^{II}C^{▪▪} → Tot τ_{≤n−1}^{II}C^{▪▪}}. By induction on n ≥ 0, we construct complexes V_{n}^{▪} and morphisms of triangles in K^{+}(A)

Tot τ_{≤n}^{II}C^{▪▪} ⟶ Tot τ_{≤n−1}^{II}C^{▪▪} ⟶ C^{▪n}[1 − n] ⟶ Tot τ_{≤n}^{II}C^{▪▪}[1]
↓  ↓  ↓  ↓
V_{n}^{▪}  ⟶  V_{n−1}^{▪}  ⟶ I^{▪j}[1 − n] ⟶  V_{n}^{▪}[1]

where all vertical arrows are quasi-isomorphisms. If n = 0, then we take V_{0}^{▪} = I^{▪0}. Let n > 0. Since V_{n−1}^{▪} → Tot τ_{≤n−1}^{II}C^{▪▪} and C^{▪n}[1 − n] → I^{▪n}[1 − n] are quasi-

isomorphisms, we choose a morphism V_{n−1}^{▪} ⟶^{g_{n−1}} I^{▪j}[1 − n] in C(A) such that

Tot τ_{≤n−1}^{II}C^{▪▪} ⟶ C^{▪n}[1 − n]
↓  ↓
V_{n−1}^{▪}  ⟶^{g_{n−1}} I^{▪j}[1 − n]

is commutative in K^{+}(A). We take V_{n}^{▪} = M^{▪}(g_{n−1})[−1] in C(A). Then V_{n}^{▪} → V_{n−1}^{▪} is a term-split epimorphism and have the above morphism of triangles. By Proposition 11.7, we have isomorphisms in D^{+}(A)

Tot C^{▪▪} = Tot^{∧}C^{▪▪}

≅ \underleftarrow{lim} Tot τ_{≤n}^{II}C^{▪▪} ≅ hlim_{⟵} Tot τ_{≤n}^{II}C^{▪▪}

≅ hlim_{⟵} V_{n}^{▪}

≅ \underleftarrow{lim} V_{n}^{▪}.

By the construction of V_{n}^{▪}, we have the required complex.

Theorem 18.4. Let A be a right noetherian k-algebra and B a left coherent kalgebra, and _{B}U_{A}^{▪} ∈ D^{b}(Mod B^{op}⊗_{k}A) a cotilting B-A-bimodule complex. If I^{▪} ∈ K^{+}(Inj A) is quasi-isomorphic to Res_{A}U^{▪} in K^{+}(Mod A), then I^{▪} contains all indecomposable injective A-modules.

Proof. According to Proposition 18.2 and Example 10.14, for any X^{▪} ∈ D_{c}^{b}(Mod A), there exists P^{▪} ∈ K^{−}(proj B^{op}) such that

\begin{aligned}Hom_{B}^{▪}(P^{▪}, U^{▪}) & ≅ R Hom_{B}^{▪}(P^{▪}, U^{▪}) \\ & ≅ X^{▪}. \end{aligned}

For any P^{i}, it is easy to see Hom_{B}(P^{i}, U^{▪}) ∈ add U_{A}^{▪}. Then Hom_{B}(P^{i}, U^{▪}) ∈ add I^{▪}. By piled resolutions lemma, we have a quasi-isomorphism X^{▪} → J^{▪} in K^{+}(Mod A) with all J^{i} ∈ Add (∐_{i∈ℤ}I^{i}). Since A is noetherian, J^{i} is A-injective. We take X^{▪} = A/𝔭 where 𝔭 is any right ideal of A. Since J^{▪} in K^{+}(Mod A), it is easy to see that we have a monomorphism A/𝔭 → J^{0}. Hence the injective envelope E(A/𝔭) is a direct summand of J^{0}.

Corollary 18.5 (Like-Corollary). Let A be a right noetherian and left coherent ring with inj dim_{A}A, inj dim A_{A} < ∞. Then any injective resolution of a right A-module A_{A} contains all indecomposable injective A-modules.

Proof. By the same technique in Proposition 18.2, R^{b} Hom_{A}(−, A) : D^{b}(mod A) → D^{b}(mod A^{op}) and R^{b} Hom_{A}(−, A) : D^{b}(mod A^{op}) → D^{b}(mod A) induce a duality. By the above proof, we get the statement.

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Index

S-injective, 20 S-projective, 20 U-colocal, 37 U-local, 37 ∂-functor, 14 ∂-functorial morphism, 14 épaisse subcategory, 35

abelian category, 7 acyclic, 28 additive functor, 5, 6 adjunction arrows, 4 admissible epimorphism, 19 admissible monomorphism, 19

bi-∂-functorial morphism, 62 bi-∂-functor, 62 bifunctor, 4 bifunctorial isomorphism, 4 bifunctorial morphism, 4 biperfect complex, 81 bounded above complex, 24 bounded below complex, 24 bounded complex, 24

category, 1 cochain complex, 24 cofinal subcategory, 31 cohomology, 28 colimit, 3 colocalization, 38 colocalization exact, 38 compatible with the triangulation, 33 complex, 24 contravariant cohomological functor, 15 contravariant functor, 2 coproduct, 4 cotilting bimodule complex, 87 covariant cohomological functor, 15 covariant functor, 2

dense, 3 derived equivalent, 77 distinguished triangle, 14 double complex, 56 dualizing bimodule complex, 87

epimorphism, 2 exact, 7 exact category, 19 exact functor, 14

faithful, 3 finite injective dimension, 45 Frobenius category, 20 full, 3 functorial isomorphism, 3 functorial morphism, 3 generates a triangulated category, 75

homotopic, 26

homotopy category, 25

indecomposable, 11 initial object, 5 inverse of a two-sided tilting complex, 81 isomorphism, 2

Krull-Schmidt category, 12

left adjoint, 4

limit, 3

localization, 38

localization exact, 38

localizing subcategory, 46, 52

mapping cone, 25 monomorphism, 2 morphism of complexes, 24 morphism of double complexes, 57 multiplicative system, 29

Nakayama functor, 84

null object, 5

opposite category, 2

perfect complex, 74

pre-Krull-Schmidt category, 11

preadditive category, 5

product, 4

product category, 4 proper epimorphism, 59 proper exact, 59 proper injective complex, 59 proper monomorphism, 59 proper projective complex, 59

quasi-isomorphism, 40 quotient category, 31 quotient functor, 31 quotientizing subcategory, 52

r-tuple complex, 57 right adjoint, 4 right adjoint pair, 4 right derived functor, 52 right derived functor of a bi-∂-functor, 63

saturated multiplicative system, 30

section functor, 38 split epimorphism, 2 split monomorphism, 2 stable t-structure, 39 stable category, 12 stalk complex, 24 standard equivalence, 81 standard triangle, 21

term-split epimorphism, 25 term-split monomorphism, 25 terminal object, 5 thick abelian full subcategory, 40 tilting complex, 76 total complexes, 58 translation, 14 triangle equivalent, 14 triangulated category, 14 two-sided tilting complex, 81

way-out in both directions, 55 way-out left, 55 way-out right, 55

J. Miyachi: Department of Mathematics, Tokyo Gakugei University, Koganei-shi, Tokyo, 184-8501, Japan

E-mail address: miyachi@u-gakugei.ac.jp