Derived Categories Yong HU [email protected] May 6, 2012

Contents 1 Trianglulated Categories 1.1 Definition and first properties . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The homotopy category . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Cohomological functors . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 8 13

2 Derived Categories 14 2.1 Localization of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Derived categories of abelian categories . . . . . . . . . . . . . . . . . . . 17 3 Derived Functors 3.1 Existence of derived functors 3.2 Examples . . . . . . . . . . 3.2.1 R Hom and Ext . . . 3.2.2 ⊗L and T or . . . .

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20 20 25 25 28 29

Trianglulated Categories Definition and first properties

Definition 1.1. Let C be a an additive category. A structure of triangluated category on C is defined by specifying data (a) and (b) which satisfy axioms (TR1)–(TR4) below. Data: (a) An additive automorphism T : C → C, called the translation functor . For n ∈ Z, we will write X[n] := T n (X) and f [n] = T n (f ) for any object X ∈ C and any morphism f in C. We call a triangular a sextuple (X, Y, Z, u, v, w) consisting of three objects X, Y, Z and three morphisms u : X → Y , v : Y → Z and w : Z → X[1]. Some other notations for triangles: u

v

w

X −→ Y −→ Z −→ X[1] 1

or B: X w

u

Zo Y A morphism of triangles from one triangle (X, Y, Z, u, v, w) to another triangle (X 0 , Y 0 , Z 0 , u0 , v 0 , w0 ) is a triple (f, g, h) of morphisms f : X → X 0 , g : Y → Y 0 and h : Z → Z 0 such that the following diagram commutes v

u

v

X −−−→   fy

w

Y −−−→  g y

u0

Z −−−→ X[1]    f [1] yh y

v0

w0

X 0 −−−→ Y 0 −−−→ Z 0 −−−→ X 0 [1] (b) The class of distinguished triangles among all triangles. Axioms: (TR1) For each object X, the triangle Id

X X −→ X −→ 0 −→ X[1]

is distinguished; any triangle isomorphic to a distinguished one is itself distinguished; any morphism u : X → Y in C can be extended to a distinguished triangle u

X −→ Y −→ Z −→ X[1] . u

v

w

v

(TR2) A triangle X −→ Y −→ Z −→ X[1] is distinguished if and only if Y −→ −T (u)

w

Z −→ X[1] −→ Y [1] is distinguished: T (X)

B: X w

Zo

u v

⇐⇒

is distinguished

<

w

Zo

Y

−T (u) v

&

is distinguished

Y

(TR3) Given two distinguished triangles u0

u

X 0 −→ Y 0 −→ Z 0 −→ X 0 [1] ,

X −→ Y −→ Z −→ X[1] and any commutative diagram u

X −−−→   fy

Y  g y

u0

X 0 −−−→ Y 0 can be extended (not uniquely in general) to a morphism of triangles: X 

/

u

g

f

X0

/Z

Y

u0

/



/

Y0 2



h

/

Z0

/

X[1] 

f [1]

X 0 [1]

(TR4) Given a diagram +3

Z0` ?

j

Y \d

X ks

u

~

>Y v◦u

y

?

/

v ?

i

X0

0

Z

~

where X u

~

Y

v◦u

y

/

v

Z

is a commutative diagram and the three triangles marked ? +3

Z0` j

?

Y

~

X ks

X

u

>Y

0

and

?

v◦u

/

v

Y \d

?

i

X0

Z

Z

~

are distinguished triangles, there exists morphisms f : Z 0 → Y 0 and g : Y 0 → X 0 such that f /Y0 Z 0 \d ? T (j)◦i

X

0

g

}

is a distinguished triangle and that (IdX , v , f ) and (u , IdZ , g) are morphisms of triangles: f

ZO 0 $ j

Y

~

? u

/

y

X

IdX y v

/

X

YO 0

g

Y 0`

/

0 X >

y

z ?

? v◦u

/



Z

X

>Z v◦u

IdZ y u

/

Z`

? v

i

 /Y

Axiom (TR4) may be also be formulated by using a rather big “octahedron diagram”:

3

Any diagram B: X _g u

Y bj

v◦u

y

j

v

4 Z0

?

?

Y0 5

v

+

?

i

X0



Z

~

where X u

v◦u

~

v /Z Y is commutative and the three triangles marked ? are distinguished triangles, can be completed to an octahedral diagram

:B X _g u

Y bj

f

v◦u

y

j

v

4 ZKS 0

?

5

? v

T (j)◦i

+

?

i

X0

+

Y0



Z

g

~x

such that (1) the face B: Z

f

g

X0 o

Y0

is a fourth distinguished triangle; (2) the remaining four faces of the octahedron X0

X u

Y

~

v◦u v

/

Z

Z0 T (j)◦i

i

}

Y [1]

! / Z[1]

T (j)

4

f

Y0

~

! / X[1]

and Z Y0

~

g

/

X0

are commutative; (3) the two compositions j

f

v

Y −→ Z 0 −→ Y 0 ,

Y −→ Z −→ Y 0

coincide; (4) the two compositions g

u[1]

i

Y 0 −→ X 0 −→ Y [1] ,

Y 0 −→ X[1] −→ Y [1]

coincide. Proposition 1.2. Let C be a triangluated category with translation functor T . (i) If (X, Y, Z , u, v, w) is a distinguished triangle, then the sequence u

v

w

X −→ Y −→ Z −→ T (X) is a complex in the category C. (ii) For any distinguished triangle u

v

w

X −→ Y −→ Z −→ X[1] and any object W ∈ C, the induced sequences · · · −→ Hom(W, X[n]) −→ Hom(W, Y [n]) −→ Hom(W, Z[n]) −→ Hom(W, X[n + 1]) −→ · · · and · · · ←− Hom(X[n], W ) ←− Hom(Y [n] , W ) ←− Hom(Z[n], W ) −→ Hom(X[n + 1] , W ) ←− · · · are exact. (iii) For a morphism of distinguished triangles u

X −−−→   fy u0

v

Y −−−→  g y v0

w

Z −−−→ X[1]    f [1] yh y w0

X 0 −−−→ Y 0 −−−→ Z 0 −−−→ X 0 [1] if f and g are isomorphisms, then so is h.

5

Proof. (i) The commutative diagram Id

X X −−− →   IdX y

X  u y

X −−−→ Y induces a morphism of distinguished triangles by (TR1) and (TR3): Id

X X −−− →   IdX y

X −−−→  u y

u

0 −−−→ X[1]     y yId

v

w

X −−−→ Y −−−→ Z −−−→ X[1] v

−T (u)

w

whence v ◦ u = 0. Since Y −→ W −→ X[1] −→ Y [1] is a distinguished triangle by (TR2), the same argument applied to this triangle yields w ◦ v = 0. (ii) It suffices to prove that the sequences Hom(W, X) −→ Hom(W , Y ) −→ Hom(W , Z) and Hom(Z, W ) −→ Hom(Y, W ) −→ Hom(X, W ) are exact and then apply (TR2) repeatedly. By (i), the two sequences are complexes. Let f ∈ Hom(W, Y ) be such that v ◦ f = 0. By (TR1)–(TR3), we have a morphism of distinguished triangles: W −−−→   fy

− Id

0 −−−→ W [1] −−−→ W [1]      f [1] 0 hy y y

v

−u[1]

w

Y −−−→ Z −−−→ X[1] −−−→ Y [1] Writing h0 = h[1] for some h ∈ Hom(W, X), the right commutative diagram yields f = u ◦ h. This proves the exactness of the sequence Hom(W, X) −→ Hom(W, Y ) −→ Hom(W, Z) . To show the exactness of the other sequence, suppose g ∈ Hom(Y, W ) is a morphism such that g ◦ u = 0. Applying (TR2) repeatedly, we get distinguished triangles −u[1]

−v[1]

−w[1]

X[1] −→ Y [1] −→ Z[1] −→ X[2] and

− Id

0 −→ W [1] −→ W [1] −→ 0 . By (TR3), we have a morphism of distinguished triangles −u[1]

−v[1]

−w[1]

X[1] −−−→ Y [1] −−−→ Z[1] −−−→ X[2]         g[1]y y yh[1] y 0

− Id

−−−→ W [1] −−−→ W [1] −−−→ 6

0

which yields g = h ◦ v for some h ∈ Hom(Z, W ). (iii) Let kZ 0 denote the functor Hom(Z 0 , −). Consider the commutative diagram kZ 0 (X) −−−→ kZ 0 (Y ) −−−→ kZ 0 (Z) −−−→ kZ 0 (X[1]) −−−→ kZ 0 (Y [1])           g∗ y g[1] f [1] f∗ y h∗ y ∗y ∗y kZ 0 (X 0 ) −−−→ kZ 0 (Y 0 ) −−−→ kZ 0 (Z 0 ) −−−→ kZ 0 (X 0 [1]) −−−→ kZ 0 (Y 0 [1]) whose rows are exact by (ii). If f and g are isomorphisms, then the usual 5-lemma in the category of abelian groups shows that h∗ is an isomorphism. In particular, h : Z 0 → Z admits a right inverse. Similarly, one can prove that h has a left inverse. Hence, h is an isomorphism as well. Remark 1.3. (1) By Prop. 1.2 (iii), for any morphism u : X → Y in C, the distinguished triangles extending u are all isomorphic. Since the morphism h in Axiom (TR3) is not necessarily unique, the isomorphisms are not uniquely determined by u : X → Y in general. This fact is the source of many difficulties (e.g., gluing problems in sheaf theory). (2) In Axiom (TR3), if HomC (Y, X 0 ) = 0 and HomC (T (X), Y 0 ) = 0, then the morphism h is uniquely determined (cf. [KS06, p.246, Prop. 10.1.17]). Proposition 1.4 ([KS06, p.247, Prop. 10.1.19]). Let C be a triangulated category which admits direct sums indexed by a set I. Then direct sums indexed by I commute with the translation functor, and a direct sum of distinguished triangles indexed by I is a distinguished triangle. Corollary 1.5. Let C be a triangulated category. u v w u0 v0 w0 (i) If X −→ Y −→ Z −→ X[1] and X 0 −→ Y 0 −→ Z 0 −→ X 0 [1] are distinguished triangles, then u⊕u0

v⊕v 0

w⊕w0

X ⊕ X 0 −→ Y ⊕ Y 0 −→ Z ⊕ Z 0 −→ X[1] ⊕ X 0 [1] is a distinguished triangle. (ii) For any objects X, Y ∈ C, 0

X −→ X ⊕ Y −→ Y −→ X[1] is a distinguished triangle. Proof. Immediate from Prop. 1.4. Definition 1.6. Let C , C0 be triangular categories. A triangular functor or a morphism of triangular categories from C to C0 is an additive functor F : C → C0 which commutes with the translation functors and sends distinguished triangles to distinguished triangles. A triangulated subcategory B of a triangulated category C is a full subcategory B of C which is itself a triangulated category such that the inclusion B → C is a morphism of triangulated categories. 7

Remark 1.7. What we call triangulated functor is called exact functor in [Del77, C.D.]. If we adopt the generalized definition of exactness for functors between arbitrary categories given in [KS06, p.81, Definition 3.3.1], then triangulated functors between triangulated categories are all exact functors by [KS06, p.247, Prop. 10.1.18]. Proposition 1.8. Let C be a triangulated category and B ⊆ C a triangulated subcategory. u v w (i) A triangle X −→ Y −→ Z −→ X[1] in B is distinguished in C if and only if it is distinguished in B. u v w (ii) Let X −→ Y −→ Z −→ X[1] be a distinguished triangle in C. If two of the three objects X, Y, Z are objects of B, then the third is isomorphic to an object of B. Proof. (i) The “if” part is clear from the definition of triangulated subcategory. Assume the triangle is distinguished in C. There is distinguished triangle v0

u

w0

X −→ Y −→ Z 0 −→ X[1] in B by (TR1). This triangle is isomorphic to the original triangle by (TR3) and Prop. 1.2 (iii). (ii) By (TR2), we may assume X and Y are objects in B, the result follows from our proof of the “only if” part of (i).

1.2

The homotopy category

(1.9) Let A be an additive category. Recall that a (cochain) complex in A is a sequence of objects and morphisms in A: X • = (X n )n∈Z :

dn−1

dn

dn+1

· · · −→ X n −→ X n+1 −→ · · ·

such that dn ◦ dn−1 = 0 for all n ∈ Z. A morphism of complexes f = (f n ) : X • = (X n ) → Y • = (Y n ) consists of a sequence of morphisms f n : X n → Y n such that for each n ∈ Z, the diagram dn X n −−−→ X n+1     n+1 f ny yf dn

Y n −−−→ Y n+1 is commutative. The category of complexes in A will be denoted C(A). This is again an additive category. A morphism of complexes f : X • −→ Y • is said to be homotopic to 0, denoted f ∼ 0, if there is a sequence k = (k n ) of morphisms k n : X n → Y n−1 , n ∈ Z, sometimes referred to as a contraction of f , such that n : X n −→ Y n . f n = k n+1 ◦ dnX • + dYn−1 • ◦ k

If f : X • → Y • is homotopic to 0, then f ◦ h ∼ 0 for all morphisms h : W • → X • and h0 ◦ f ∼ 0 for all morphisms h0 : Y • → Z • . If f1 , f2 : X • → Y • are morphisms homotopic to 0, then f1 + f2 ∼ 0. 8

Two morphisms of complexes f, g : X • → Y • are called homotopic, denoted f ∼ g, if f − g is homotopic to 0. Two complexes X • , Y • ∈ C(A) are called homotopic or homotopically equivalent to 0 if there are morphisms f : X • → Y • and g : Y • → X • such that f ◦ g ∼ IdY • and g ◦ f ∼ IdX • . For any complexes X • , Y • ∈ C(A), set Htp(X • , Y • ) := { f ∈ HomC(A) (X • , Y • ) | f ∼ 0 } . The homotopy category K(A) of A is the category defined by Ob(K(A)) := Ob(C(A)) , HomK(A) (X • , Y • ) := HomC(A) (X • , Y • )/Htp(X • , Y • ) . The category K(A) is also an additive category. A complex X • ∈ C(A) is called bounded below if X n = 0 for all n  0. We define in the similar way the notion of complexes bounded above. A complex is bounded if it is bounded below and bounded above. We denote by C+ (A), C− (A) and Cb (A) respectively the full subcategory of C(A) consisting of complexes bounded below, complexes bounded above and bounded complexes. Similarly, denote by K+ (A), K− (A) and Kb (A) respectively the full subcategory of K(A) whose objects are complexes bounded below, complexes bounded above and bounded complexes. (1.10) With notation as in (1.9), we have an additive automorphism T : C(A) −→ C(A) ;

X • 7−→ X • [1] ,

defined as follows: (T (X • ))n := X n+1 (dnT (X • ) :

T (X • )n −→ T (X • )n+1 ) := −dn+1 X• :

X n+1 −→ X n+2




and (T (f ) :

T (X • ) −→ T (Y • )) := (f n+1 : X n+1 = T (X • )n −→ T (Y • )n = Y n+1 )n∈Z

for any f ∈ HomC(A) (X • , Y • ). For each n ∈ Z, T n will be referred to as the translation by n functor. Clearly, for each n ∈ Z, T n induces automorphisms of the subcategories C+ (A), C− (A) and Cb (A). Similarly, the translation functor X • 7→ X • [1] induces an automorphism of the category K(A), which preserves the subcategories K+ (A), K− (A) and Kb (A). (1.11) Let A be an additive category and let f : X • → Y • be a morphism in C(A). The mapping cone of f , denoted Con(f ), is the defined as the following complex: For each n ∈ Z, (Con(f ))n := X n+1 ⊕ Y n and the differential map dn = dnCon(f ) :

Con(f )n = X n+1 ⊕ Y n −→ X n+2 ⊕ Y n+1 = Con(f )n+1 9

is represented by the matrix 

0 dnX • [1] (f [1])n dnY •



 n+1  −dX • 0 = . f n+1 dnY •

The mapping cylinder of f , denoted Cyl(f ), is the complex defined as follows: For each n ∈ Z, Cyl(f )n := X n ⊕ X n+1 ⊕ Y n , and the differential map dnCyl(f ) :

X n ⊕ X n+1 ⊕ Y n −→ X n+1 ⊕ X n+2 ⊕ Y n+1

is represented by the matrix   n   n dX • − IdnX • 0 dX • − IdnX • 0  0 0  =  0 −dn+1 dnX • [1] 0  X• n+1 n n 0 f dnY • 0 (f [1]) dY • Clearly, if f is a morphism in C+ (A) (resp. C− (A), resp. Cb (A)), then both Con(f ) and Cyl(f ) lie in C+ (A) (resp. C− (A), resp. Cb (A)). Note that we have canonical morphism of complexes π : Y • −→ Con(f ) ; y 7−→ (0, y) , π : Cyl(f ) −→ Con(f ) ; (x, x0 , y) 7−→ (x0 , y) , δ : Con(f ) −→ X • [1] ; (x , y) 7−→ x , f : X • −→ Cyl(f ) ; α : Y • −→ Cyl(f ) ;

x 7−→ (x , 0, 0) , y 7−→ (0, 0, y)

and β : Cyl(f ) −→ Y • ;

(x, x0 , y) 7−→ f (x) + y .

Proposition 1.12. Let A be an additive category and let f : X • → Y • be a morphism of complexes in A. Then the following diagram of objects and morphisms in C(A) is a commutative diagram with exact rows: (1.12.1)

/

0

0

/

X•

f

X•

f

/

Y•

/

Con(f )

δ

/

X • [1]

α



Cyl(f ) /

π



π

/

Con(f )

/

0

β

Y•

where the morphisms π , δ , f , π, α and β are defined as in (1.11). Moreover, one has β ◦ α = IdY • and α ◦ β ∼ IdCyl(f ) . 10

/

0

Proof. [GM03, p.155, Lemma III.3.3] or [Wei94, §1.5]. Definition 1.13. Let A be an additive category and let T : X • → X • [1] denote the automorphism of C(A) (or of K(A)) given by translation of complexes. A triangle of the form f π δ X • −→ Y • −→ Con(f ) −→ X • [1] in C(A) (or K(A)) as defined in Prop. 1.12 is called a mapping cone triangle. A distinguished triangle in K(A) is a triangle isomorphic (in K(A)) to a mapping cone triangle. Proposition 1.14. Let A be an additive category. The homotopy category K(A) of A with the class of distinguished triangles defined in Definition 1.13 is a triangle category. Proof. [KS06, p.276, Prop. 11.2.8] or [Wei94, p.376, Prop. 10.2.4]. Remark 1.15. Let A be an additive category. (1) Using Prop. 1.12 one can prove that the homotopy category K(A) has the following universal property (cf. [Wei94, p.370, Prop. 10.1.2]): For any functor F : C(A) → D which sends homotopy equivalences of complexes to isomorphisms, there is a unique factorization through the canonical functor C(A) → K(A): /

F

C(A) can.

#

=D

∃!

K(A) (2) As a triangulated category, K(A) together with the canonical functor C(A) → K(A), is uniquely determined by C(A) and the class of mapping cone triangles. More precisely, for any functor F from C(A) to a triangulated category T which commutes∗ with the translation functors of C(A) and T, and which sends mapping cone triangles in C(A) to distinguished triangles in T, there is a unique morphism of triangulated categories (cf. Definition 1.6) F : K(A) → T such that the following diagram is commutative: F /T C(A) = can.

#

∃! F

K(A) (cf. [Del77, C.D., §2.4]) (1.16) Let A be an additive category and let C be an additive full subcategory of C(A) and let K denote the full subcategory of K(A) whose objects are complexes in C. The category K is a “quotient category” of C in the sense that HomK (X • , Y • ) = HomK(A) (X • , Y • ) = ∗

HomC(A) (X • , Y • ) HomC (X • , Y • ) = Htp(X • , Y • ) Htp(X • , Y • )

Throughout these notes, we will never distinguish isomorphisms with identities for functors between categories. So commutativity for a diagram of functors between categories means commutativity up to isomorphism of functors.

11

for all X • , Y • ∈ Ob(K). The category K is an additive category and the natural functor C → K is an additive functor. The proof of Prop. 1.14 can actually show that if C is closed under the translation functor T and the formation of mapping cones, then K is a triangular category. In particular, K+ (A), K− (A) and Kb (A) are triangulated subcategories of the homotopy category K(A). (1.17) Now let A be an abelian category. The category C(A) of complexes in A is again an abelian category ([Wei94, p.7, Thm. 1.2.3]). Let X • = (X n )n∈Z be a complex in A. The cohomology of the complex X • are defined by dn

H n (X • ) :=

Ker(X n −→ X n+1 ) dn−1

∀ n ∈ Z.

,

Im(X n−1 −→ X n )

We say X • is acyclic or exact at the term X n if H n (X • ) = 0. A complex X • is the abelian category A is exact (or acyclic, or an exact sequence) if it is acyclic at all terms. A morphism f : X • → Y • in C(A) is called a quasi-isomorphism if the induced homomorphisms H n (f ) : H n (X • ) −→ H n (Y • ) , ∀ n ∈ Z are all isomorphisms. One can prove that f is a quasi-isomorphism if and only if the mapping cone complex Con(f ) is exact ([Wei94, p.19, Coro. 1.5.4]). If g : X • → Y • is another morphism such that f ∼ g, then H n (f ) = H n (g), ∀ n ∈ Z. So the class of quasiisomorphisms is closed under homotopy equivalence of morphisms, and the cohomology H n (X • ) , n ∈ Z are uniquely determined (up to isomorphism) by the homotopy class of the complex X • . The functors H n : C(A) → A, n ∈ Z thus factor through the canonical functor C(A) → K(A). Proposition 1.18. Let A be an abelian category and let f

g

0 −→ X • −→ Y • −→ Z • −→ 0 be an exact sequence in C(A). Then there is a quasi-isomorphism γ : Con(f ) → Z • such that the diagram f

π

0 −−−→ X • −−−→ Cyl(f ) −−−→ Con(f ) −−−→ 0     β γ Idy y y f

0 −−−→ X • −−−→

Y•

g

−−−→

Z•

−−−→ 0

is commutative. Proof. [GM03, p.157, Prop. III.3.5], [Wei94, §1.5]. Remark 1.19. The morphism γ in Prop. 1.18 may not be a homotopic equivalence in general (cf. [Wei94, p.23, Exercise 1.5.5]). 12

1.3

Cohomological functors

Definition 1.20. Let C be a triangulated category with translation functor T and A an abelian category. A cohomological functor from C to A is an additive functor F : C → A such that for every distinguished triangle X → Y → Z → X[1] in C, the sequence F (X) −→ F (Y ) −→ F (Z) is exact in A. For a cohomological functor F , we denote F n := F ◦ T n , n ∈ Z. Example 1.21. By Prop. 1.2 (ii), for any object W in the triangulated category C, the two functors HomC (W , ·) : C −→ Ab and

HomC (· , W ) : C −→ Ab

are cohomological functors. Theorem 1.22. Let A be an abelian category and K(A) its homotopy category. Then the functors H n : K(A) −→ A ; n ∈ Z are cohomological. Proof. One can prove the theorem using Prop. 1.18 as in [GM03, p.158, Thm. III.3.6]. See also [KS06, p.304, Coro. 12.3.4]. Corollary 1.23. Let A be an abelian category and K(A) its homotopy category. Then for any distinguished triangle u

v

w

X • −→ Y • −→ Z • −→ X • [1] in K(A), the sequence H n (u)

H n (v)

H n (w)

· · · −→ H n (X • ) −→ H n (Y • ) −→ H n (Z • ) −→ H n+1 (X • ) −→ · · · is exact. (1.24) Let A be an abelian category and let f

g

0 −→ X • −→ Y • −→ Z • −→ 0 be an exact sequence in C(A). Then the usual cohomology theory yields a long exact sequence (1.24.1)

H n (f )

H n (g)

δn

· · · −→ H n (X • ) −→ H n (Y • ) −→ H n (Z • ) −→ H n+1 (X • ) −→ · · ·

On the other hand, let γ : Con(f ) −→ Z • be the quasi-isomorphism given in Prop. 1.18. The cohomology of the exact sequence (cf. (1.12.1)) π

δ

0 −→ Y • −→ Con(f ) −→ X • [1] −→ 0 13

yields maps H n (δ) : H n (Con(f )) −→ H n (X • [1]) = H n+1 (X • ) . Let δen : H n (Z • ) → H n+1 (X • ) denote the composition map H n (γ)−1

H n (δ)

H n (Z • ) −→ H n (Con(f )) −→ H n+1 (X • ) . The next proposition ensures that the map δen is essentially the same as the map δ n arisen in (1.24.1). Proposition 1.25 ([KS06, p.305, Prop. 12.3.6]). With notation and hypotheses as in (1.24), for each n ∈ Z, the two maps δ n , δen : H n (Z • ) −→ H n+1 (X • ) , coincide up to a sign.

2 2.1

Derived Categories Localization of categories

Definition 2.1. Let C be a category and let S be a collection of morphism in C. A localization of C with respect to S in a category S −1 C together with a functor q : C → S −1 C having the following properties: (1) q(s) is an isomorphism in S −1 C for every s ∈ S. (2) Any functor F : C → D such that F (s) is an isomorphism for all s ∈ S factors in a unique way through q: F /D C < q

"

∃! −1

S C This universal property implies that the localization S −1 C is unique up to equivalence, if it exists. Example 2.2. Let A be an abelian category. (1) Let S0 be the collection of homotopy equivalences in C(A). The universal property stated in Remark 1.15 (1) shows that the canonical functor C(A) → K(A) is the localization C(A) → S0−1 C(A). (2) Let Se (resp. S) denote the collection of quasi-isomorphism in C(A) (resp. K(A)). The derived category D(A) of A will be denoted to be the localization S −1 K(A) (cf. Definition 2.11). Since Se ⊇ S0 , it follows that Se−1 C(A) = S −1 (S −1 C(A)) = S −1 K(A) . 0

Therefore we could have defined the derived category D(A) to be the localization Se−1 C(A). However, in order to prove that existence of the localization Se−1 C(A), we must first prove that S −1 K(A) exists, by giving an explicit construction (cf. [Wei94, §10.3], [Del77, C.D., Chap. 1, §2.3]). 14

Definition 2.3. Let C be a category. A multiplicative system in C is a collection S of morphisms in C satisfying the following axioms: (FR1) S is closed under composition, i.e., for any f, g ∈ S, one has g ◦ f ∈ S whenever the composition is defined, and S contains all identity morphisms. (FR2) Any diagram of the form Z   yt g

X −−−→ Y with t ∈ S can be completed to a commutative diagram of the form f

(2.3.1)

W −−−→   sy

Z   yt

g

X −−−→ Y with s ∈ S. (The slogan for this is “t−1 g = f s−1 for some f and s”.) Any diagram of the form f W −−−→ Z   sy X with s ∈ S can be completed to a commutative diagram of the form (2.3.1) with t ∈ S. (The slogan for this is “f s−1 = t−1 g for some g and t”.) (FR3) (Cancellation) If f, g : X → Y are morphisms in C with the same source and target, then the following two conditions are equivalent: (a) s ◦ f = s ◦ g for some s ∈ S with source Y ; (b) f ◦ t = g ◦ t for some t ∈ S with target X. Theorem 2.4. Let C be a category and S a multiplicative system in C. Then the localization q : C → S −1 C exists and it has the following properties: (i) For any morphism f, g : X → Y in C, one has q(f ) = q(g) in S −1 C if and only if s ◦ f = s ◦ g for some morphism s ∈ S with target X. (ii) If C has a zero object 0, then q(0) is a zero object in S −1 C and for any object X ∈ C, q(X) ∼ = 0 in S −1 C if and only if S contains the zero morphism 0 : X → X. (iii) If C is an additive category, then so is S −1 C and the canonical functor q : C → S −1 C is an additive functor. Proof. See [Wei94, §10.3], [Har66, §I.3], or [Del77, C.D., Chap. 1, §2.3]. Definition 2.5. Let C be a category and S a multiplicative system in C. A localizing subcategory of C (with respect to S) is a full subcategory D such that S ∩ D is a multiplicative system of D and the natural functor (S ∩ D)−1 D → S −1 C is fully faithful. For legibility, we shall write S −1 D instead of (S ∩ D)−1 D. 15

Lemma 2.6. Let C be a category, S a multiplicative system in C and D a full subcategory of C. Assume that S ∩ D is a multiplicative system in D and one of the following conditions is satisfied: (i) Whenever s : X 0 → X is a morphism in S with X ∈ Ob(D), there is a morphism f : X 00 → X 0 in C with X 00 ∈ Ob(D) such that s ◦ f : X 00 → X is in S. (ii) Whenever t : X → X 0 is a morphism in S with X ∈ Ob(D), there is a morphism g : X 0 → X 00 in C with X 00 ∈ Ob(D) such that g ◦ t : X → X 00 is in S. Then D is a localizing subcategory of C with respect to S. Proof. [Har66, p.32, Prop. I.3.3] or [Del77, C.D., Chap. I, §2, Thm. 4.2].† Definition 2.7. Let C be a triangulated category with translation T . A multiplicative system S in C is called compatible with the triangulation if the following two axioms are satisfied: (FR4) s ∈ S ⇐⇒ T (s) ∈ S. (FR5) Given distinguished triangles u0

u

X 0 −→ Y 0 −→ Z 0 −→ X 0 [1] ,

X −→ Y −→ Z −→ X[1] and any commutative diagram u

X −−−→   fy

Y  g y

u0

X 0 −−−→ Y 0 with f, g ∈ S can be extended (not uniquely in general) to a morphism of triangles: X 

/

u

g

f

X0

/Z

Y

u0

/



/

Y0



/

X[1]

h

/

Z0



f [1]

X 0 [1]

with h ∈ S. Proposition 2.8 ([Har66, p.32, Prop. I.3.2]). Let C be a triangulated category and S a multiplicative system in C which is compatible with the triangulation. Then the localization S −1 C has a unique structure of triangulated category such that the natural functor q : C → S −1 C is a morphism of triangulated categories. Moreover, q : C → S −1 C has the following universal property: For any morphism of triangulated categories F : C → T such that F (s) is an isomorphism for every s ∈ S, there is a unique factorization F

C q

"

<

/

T

∃! F

S −1 C †

If the reader uses the 1994 hardback edition of [Wei94], he should be warned of an error in Lemma 10.3.13 on pages 383-384 of that book. See the Errata on the webpage: http://www.math.rutgers.edu/˜weibel/Hbook-corrections.html

16

with F : S −1 C → T a morphism of triangulated categories. Proposition 2.9 ([Wei94, p.385, Prop. 10.4.1], [Har66, p.35, Prop. I.4.2]). Let C be a triangulated category, A an abelian category and H : C → A a cohomological functor. Let S be the collection of all morphisms s in C such that H n (s) are isomorphisms for all n ∈ Z. Then S is a multiplicative system in C compatible with the triangulation.

2.2

Derived categories of abelian categories

(2.10) Let A be an abelian category and K(A) its homotopy category. By Prop. 1.14 and Thm. 1.22, K(A) is a triangulated category and the functor H = H 0 : K(A) −→ A is a cohomological functor. The collection S of morphisms s such that H n (s) are isomorphisms for all n ∈ Z is the collection of quasi-isomorphisms in K(A). So the localization D(A) := S −1 K(A) exists and the natural functor q : K(A) → D(A) is a morphism of triangulated categories. Similarly, we have triangulated categories D∗ (A) = S −1 K∗ (A) for ∗ ∈ { +, − , b }. Let us show that the subcategories K∗ (A), ∗ ∈ { +, − , b } are localizing subcategories of K(A). Hence D∗ (A), ∗ ∈ { +, − , b } are full subcategories of D(A). We shall verify conditions (ii) of Lemma 2.6. Let f : X • → Y • be a quasi-isomorphism in K(A) with X • bounded below. Let N ∈ Z be such that X m = 0, ∀ m < N . Let Z • be the following complex: −1 N Z • : · · · → 0 → 0 → Z N −1 = Im dN = Y N −→ Z N +1 = Y N +1 −→ · · · Y • −→ Z

There is a natural morphism g : Y • → Z • given by ··· ···

N −2 / / Y N −2 d

/



0

/

Y N −1 

dN −1

/

N −1 dY •

YN /

Id

Z N −1



ZN

dN

/

Y N +1

/ ···

Id

 / Z N +1

/ ···

One verifies easily that g ◦ f : X • → Z • is a quasi-isomorphism in K+ (A). We leave it to the reader to check that K− (A) and Kb (A) are also localizing subcategories of K(A). Therefore, the categories D+ (A), D− (A) and Db (A) are full subcategories of D(A), and clearly Db (A) = D+ (A) ∩ D− (A). Definition 2.11. Let A be an abelian category. The category D(A) is called the derived category of A. The objects of D(A) that are isomorphic to objects in Db (A) (resp. D+ (A), resp. D− (A)) are called bounded objects (resp. objects bounded on the left, resp. objects bounded on the right). (2.12) Let A be an abelian category and let q : C(A) → D(A) be the canonical functor. Let f, g : X • → Y • be morphisms of complexes in A. Consider the following statements: 17

(i) f = g; (ii) f ∼ g; (iii) q(f ) = q(g) in D(A); (iv) H n (f ) = H n (g) for all n ∈ Z. Then we have the implications (i) =⇒ (ii) =⇒ (iii) =⇒ (iv) . These implications are in fact all strict (cf. [Har66, p.39]). Definition 2.13. Let C be an additive category and C ∈ Ob(C). The complex concentrated in degree 0 associated to C is the complex (C n )n∈Z with C 0 = C and C n = C for all n 6= 0. By abuse of notation, this complex will also be denoted by C. Proposition 2.14. Let A be an abelian category. Let D0 (A) denote the full subcategory of D(A) whose objects are complexes X • such that H n (X • ) = 0 for all n 6= 0. Then the functor A → D(A), sending each object A of A to its associated complex concentrated in degree 0, induces an equivalence between A and D0 (A). Proof. Easy exercise left to the reader. Proposition 2.15. Let A be an abelian category and let f

g

0 −→ X • −→ Y • −→ Z • −→ 0 be a short exact sequence in C(A). Then there is a morphism w : Z • → X • [1] in the derived category D(A) such that f

g

w

X • −→ Y • −→ Z • −→ X • [1] is a distinguished triangle in D(A). Proof. Combining Props. 1.12 and 1.18, we obtain a commutative diagram with exact rows π / δ / X • [1] /0 /Y• Con(f ) 0 α

0 0

/

/

X• X•

f

/



π

Cyl(f )

f



/

/

/

Con(f )

0

γ

β

 / Z•

g

Y•

/

0

in C(A). The maps α, β are homotopic equivalences, γ is a quasi-isomorphism, and β ◦α = IdY • . Thus, in D(A), the maps α, β and γ become isomorphisms with β ◦α = Id. Defining w = δ ◦ γ −1 : Z • → X • [1] in D(A), we get a commutative diagram in D(A): X•

f

X•

f

/

Y•

π

Y•

g

/

Con(f )

/

δ

X • [1]

γ

/

 / Z•

18

w

/

X • [1]

where the vertical maps are all isomorphisms. Hence f

g

w

X • −→ Y • −→ Z • −→ X • [1] is isomorphic to a mapping cone triangle in D(A), whence a distinguished triangle. We will now give several lemmas and another description of the derived category D (A) when A has enough injectives. +

Lemma 2.16. Let f : Z • → I • be a morphism of complex in an abelian category A. Assume that Z • is acyclic, and I • is a bounded below complex whose terms I n are all injective objects in A. Then f ∼ 0. Proof. An easy exercise. Lemma 2.17 ([Har66, p.41, Lemma I.4.5], [Wei94, p.387, Lemma 10.4.6]). Let s : I • → X • be a quasi-isomorphism of complexes in an abelian category A. If I • is a bounded below complex whose terms are all injectives in A, then there is a quasi-isomorphism X • → I • such that t ◦ s ∼ IdI • . Lemma 2.18 ([Wei94, p.388, Coro. 10.4.7]). Let A be an abelian category and I • a bounded below complex whose terms are all injectives in A. Then for any X • ∈ C(A), one has a natural isomorphism HomK(A) (X • , I • ) ∼ = HomD(A) (X • , I • ) . Lemma 2.19 ([Har66, p.42, Lemma 4.6]). Let I be a collection of objects in an abelian category A such that every object A of A admits a monomorphism A → I with I ∈ I. Then every X • ∈ Ob(K(A)) such that H n (X • ) = 0 for all n  0 admits a quasiisomorphism X • → I • , where I • is a bounded below complex of objects in I. Theorem 2.20. Let A be an abelian category and let I be the full (additive) subcategory of injective objects of A. Then the natural functor K+ (I) −→ D+ (A) is fully faithful. If A has enough injectives, then the above functor is an equivalence of categories. Proof. Let I • , J • be objects in K+ (A). By lemma 2.18, we have HomK(A) (I • , J • ) ∼ = HomD(A) (I • , J • ) . So the functor K+ (I) → D+ (A) is fully faithful. If A has enough injectives, then by Lemma 2.19, every bounded below complex is quasi-isomorphic to a bounded below complex of injective objects. So the functor K+ (A) → D+ (A) is essentially surjective, whence an equivalence of categories. Remark 2.21. There are also dual statements of the last results involving injective objects. For instance, if A is an abelian category having enough projectives, then there is a natural equivalence of categories K− (P) −→ D− (A) , where P denotes the full subcategory of projective objects in A. 19

3 3.1

Derived Functors Existence of derived functors

Let F : A → B be an additive functor between abelian categories. Since F preserves homotopy equivalences, it extends to additive functors C(A) → C(B) and K(A) → K(B). Since F commutes with translation of complexes, it even preserves mapping cones and distinguished triangles. So we get a morphism of triangulated categories F : K(A) −→ K(B) . We would like to extend F further to a functor D(A) → D(B). However, this may not be possible since F will not take quasi-isomorphisms to quasi-isomorphisms. Thus we are led to ask if there is a functor from D(A) to D(B) which is at least close to F : K(A) → K(B), and this gives rise to the notion of derived functor below. Definition 3.1. Let A, B be abelian categories, K a localizing triangluated subcategory of K(A), and F : K → K(B) a triangulated functor of triangulated categories. Let D ⊆ D(A) denote the full subcategory corresponding to K. Let q (resp. qB ) denote the localization functor K → D (resp. K(B) → D(B)). (1) A right derived functor of F : K → K(B) is a triangulated functor of triangulated categories RF : D −→ D(B) together with a morphism of functors ξ : qB ◦ F −→ RF ◦ q which has the following property: If G : D → D(B) is another triangulated functor K

/

F

D(B)

q



RF

D G

/

/



qB

D(B)

and if ζ : qB ◦ F → G ◦ q is a morphism of functors, then there is a unique morphism η : RF → G such that ζ(A) = ηq(A) ◦ ξ(A) , ∀ A ∈ K . (qB ◦ F )(A)

ξ(A)

/

(RF ) ◦ q(A) ηq(A)

(qB ◦ F )(A)

ζ(A)

/



G ◦ q(A)

(2) Similarly, a left derived functor of F : K → K(B) is a triangulated functor LF : D → D(B) together with a morphism of functors ξ : LF ◦ q −→ qB ◦ F 20

which has the following universal property: If G : D → D(B) is another triangulated functor and if ζ : G ◦ q −→ qB ◦ F is a morphism of functors, then there is a unique morphism of functors η : G → LF such that ζ(A) = ξ(A) ◦ ηq(A) , G ◦ q(A)

ζ(A)

∀ A ∈ K. /

(qB ◦ F )(A)

η



LF ◦ q(A)

ξ(A)

/

(qB ◦ F )(A)

(3.2) With notation as above, since LF is R(F op ), where F op : Kop → K(Bop ), we can translated any statement about RF into a dual statement about LF . Note that the universal property guarantees that if RF exists, then it is unique up to unique isomorphism of functors. When K is K+ (A), K− (A), Kb (A), etc, we will write R+ F , R− F , Rb F etc for the corresponding derived functors. Remark 3.3. Keep the notations as above. (1) We will write Ri F for H i ◦ RF . It will follow from later results (cf. Coro. 3.10) that if F is the functor K(A) → K(B) that comes from a left exact additive functor F : A → B, and if A has enough injectives, then Ri F are the usual right derived functors ‡ of F . (2) If ϕ : F → G is a morphism of functors from K to K(B) and if RF and RG both exist, then there is a unique morphism of functors Rϕ : RF −→ RG compatible with the ξ’s. (3) If K0 is another localizing category of K(A) with K0 ⊆ K, and if both RF and R(F |K0 ) exist, then there is a natural morphism of functors R(F |K0 ) −→ RF |D0 where D0 denotes the full subcategory of D(A) corresponding to K0 . We do not know if the above morphism is an isomorphism in general, but it will be the case in all the applications we have in mind. Example 3.4. Let F : A → B be an exact functor between abelian categories. Since a morphism of complexes f : X • → Y • is a quasi-isomorphism if and only if its mapping cone Con(f ) is an exact complex, it follows easily that the exact functor F preserves quasi-isomorphisms. Hence F extends trivially to a triangulated functor F : D(A) → D(B). Thus, F is its own left and right derived functor. Definition 3.5. Let A, B be abelian categories, K ⊆ K(A) a localizing subcategory and F : K → K(B) a triangulated functor. A complex X • ∈ K is called F-acyclic if the complex F (X • ) is acyclic, i.e., if H i (F (X • )) = 0 , ∀ i ∈ Z. ‡

Recall that the definition of the usual right derived functor does not need assumption on the left exactness. For a left exact functor F , one has R0 F ∼ = F.

21

Example 3.6. With notation as in Definition 3.5, assume that every exact complex X • ∈ K is F -acyclic (this is the case if F is exact). Then for any quasi-isomorphism s : X • → Y • , F (Con(s)) is acyclic. Since F preserves distinguished triangles, the cohomology sequence shows that H n (F (s)) : H n (F (X • )) −→ H n (F (Y • )) is an isomorphism for every n ∈ Z. Therefore, F (s) is a quasi-isomorphism. By the universal property of localization D = S −1 K, there is a unique functor q −1 F : D −→ D(B) such that qB ◦ F = (q −1 F ) ◦ q. The functor q −1 F is both the left and the right derived functor of F . Theorem 3.7 ([Wei94, p.53, Thm. I.5.1], [Wei94, p.393, Thm. 10.5.9]). Let A , B be abelian categories, K ⊆ K(A) a localizing triangulated subcategories with corresponding derived subcategory D ⊆ D(A), and F : K → K(B) a triangulated functor. Suppose that there is a triangulated subcategory L ⊆ K such that (i) Every object of K admits a quasi-isomorphism into an object of L, and (ii) Every exact complex I • ∈ L is F -acylic. Then the right derived functor (RF , ξ) of F exists and for any I • ∈ L, the map ξ(I • ) : qB ◦ F (I • ) −→ RF ◦ q(I • ) is an isomorphism in D(B). Corollary 3.8. Let A, B, K, F and so on be as in Thm. 3.7. Assume there is a triangulated subcategory L ⊆ K satisfying conditions (i) and (ii) in Thm. 3.7. Let K0 ⊆ K be another localizing triangulated subcategory K(A) such that every object of K0 admits a quasi-isomorphism into an object of L ∩ K0 . Then RF and R(F |K0 ) both exist and the natural morphism R(F |K0 ) −→ RF |D0 is an isomorphism, D0 denoting the derived category of K0 . Proof. The existence of the two derived functors follows from Thm. 3.7. To prove the isomorphism, note that every X • ∈ Ob(D0 ) is isomorphic to one coming from an object of L ∩ K0 . So we may assume X • = q(I • ) with I • ∈ Ob(L ∩ K0 ). Then the desired isomorphism follows from the second assertion of Thm. 3.7. Corollary 3.9. Let A, B be abelian categories and let F : K+ (A) → K(B) be a triangulated functor. Assume that A has enough injectives. Then R+ F exists and for any bounded below complex I • of injectives in A, qB ◦ F (I • ) ∼ = R+ F (I • ) .

22

Proof. Let L ⊆ K+ (A) be the triangulated subcategory of bounded below complexes of injectives in A. By Lemma 2.19, condition (i) of Thm. 3.7 holds. By Lemma 2.18, every quasi-isomorphism in L is an isomorphism. So if I • ∈ Ob(L) is an exact complex, then I• ∼ = 0 in L. Hence condition (ii) of Thm. 3.7 also holds. Corollary 3.10. Let F : A → B be a left exact additive functor between abelian categories. Assume A has enough injectives. Then for any X ∈ Ob(A), we have (Ri F )(X) ∼ = H i (R+ F (X)) , where Ri F denotes the usual derived right functor of F and X is identified with its associated complex concentrated in degree 0. Proof. The complex X is quasi-isomorphic to any injective resolution of the object X. The result then follows by taking cohomology of the isomorphism in Corollary 3.9. Corollary 3.11. Let F : A → B be an additive functor between abelian category. Assume that there is a collection P of objects of A having the following properties: (i) Every object of A admits an injection into an object in P. (ii) If 0 → X → Y → Z → 0 is a short exact sequence in A with X ∈ P, then Y ∈ P if and only if Z ∈ P. (iii) If 0 → X → Y → Z → 0 is a short exact sequence in A with X, Y, Z ∈ P, then 0 −→ F (X) −→ F (Y ) −→ F (Z) −→ 0 is exact in B. Then the right derived functor R+ F of the functor F : K+ (A) → K(B) exists. Proof. Let L ⊆ K+ (A) be the triangulated subcategory of complexes of objects in P. Again by Lemma 2.19, condition (i) of Thm. 3.7 is satisfied. To verify condition (ii) of the theorem, let Z • ∈ Ob(L) be an exact complex. We may use the second property of P repeatedly and the fact that Z • ∈ K+ (A) to show that Ker(dnZ • ) ∈ P for all n ∈ Z. Then by the third property of P, F (Z • ) is acyclic. Definition 3.12. Let F : A → B be an additive functor between abelian categories. Assume that the right derived functor R+ F exists (e.g. A has enough injectives) and write Ri F = H i ◦ R+ F . (1) We say F has cohomological dimension n if Ri F = 0 for all i > n but n R F 6= 0. (2) An object X ∈ A is called F -acyclic if Ri F (X) = 0 for all i > 0. Remark 3.13. Note that if F : A → B is a left exact functor between abelian categories, then the collection of F -acyclic objects satisfies conditions (ii) and (iii) of Coro. 3.11. So in this case, if A has enough F -acyclic objects, then the two conditions of Thm. 3.7 are verified and hence R+ F exists.

23

Proposition 3.14. Let F : A → B be an additive functor between abelian categories. Assume R+ F exists (e.g. A has enough injectives) and F has finite cohomological dimension. (i) A has enough F -acyclic objects and the collection of F -acyclic objects satisfies the two conditions of Thm. 3.7. (ii) The right derived functor RF : D(A) → D(B) of F : K(A) → K(B) exists and the restriction of RF to D+ (A) is isomorphic to R+ F : D+ (A) −→ D(B) , the right derived functor of F : K+ (A) → D(B). Proof. [Del77, C.D., Chap. 2, §2.2, Coro. 2], [Har66, p.57, Coro. I.5.3] or [Wei94, p.394, Coro. 10.5.11]. Proposition 3.15 ([Har66, p.59, Prop. I.5.4]). Let A, B, C be abelian categories and let K∗ ⊆ K(A) and K† ⊆ K(B) be localizing subcategories with corresponding derived categories D∗ ⊆ D(A) and D† ⊆ D(B). Let F : K∗ → K ∗ (B) and G : K† → K(C) be triangulated functors and assume F (K∗ ) ⊆ K† . (i) Assume that the right derived functors R∗ F , R† F and R∗ (G ◦ F ) of the functors F, G and G ◦ F : K∗ −→ K(C) all exist, and assume (R∗ F )(D∗ ) ⊆ D† . Then there is a unique morphism of functors α : R∗ (G ◦ F ) −→ (R† G) ◦ (R∗ F ) such that the diagram qC ◦ G ◦ F   ξG◦F y

ξ ◦F

G −− −→

R † G ◦ qB ◦ F   † yR G◦ξF

α◦q

A R∗ (G ◦ F ) ◦ qA −−−→ R † G ◦ R ∗ F ◦ qA

is commutative. (ii) Assume that there are triangulated subcategories L ⊆ K∗ and M ⊆ K† satisfying the two conditions of Thm. 3.7 for F and G respectively, and assume that F (L) ⊆ M. Then R∗ F, R† G and R∗ (G ◦ F ) all exist and the morphism α : R∗ (G ◦ F ) −→ (R† G) ◦ (R∗ F ) is an isomorphism. Corollary 3.16 ([Del77, C.D., Chap. 2, §2.3, Prop. 3.1]). Let F : A → B and G : B → C be left exact additive functors between abelian categories. Assume that A has enough F -acyclic objects, B has enough G-acyclic objects and F sends F -acyclic objects to G-acyclic objects. Then the derived functors R+ F, R+ G and R+ (G ◦ F ) exist and there is a natural isomorphism R+ (G ◦ F ) −→ (R+ G) ◦ (R+ F ) . 24

Proof. In view of Remark 3.13, we may apply Prop. 3.15 to the subcategory L ⊆ K+ (A) (resp. M ⊆ K+ (B)) consisting of complexes of F -acyclic (resp. G-acyclic) objects. Remark 3.17. The above corollary shows the convenience of derived functors in the context of derived categories. What used to be a spectral sequence becomes now simply a composition of functors. One can recover the usual Grothendieck spectral sequence from Coro. 3.16 by taking hypercohomology and using the spectral sequence of a double complex. Corollary 3.18 ([Del77, C.D., Chap. 2, §2.3, Prop. 3.1]). Let F : A → B and G : B → C be additive functors between abelian categories. Assume R+ F and R+ G exist and F and G has finite cohomology dimensions.§ Assume further that A has enough F -acyclic objects, B has enough G-acyclic objects and F sends F -acyclic objects to G-acyclic objects. Then the derived functors R+ F, R+ G and R+ (G ◦ F ) exist and there is a natural isomorphism R+ (G ◦ F ) −→ (R+ G) ◦ (R+ F ) . Proof. Mimic the proof of Coro. 3.16, using Prop. 3.14 as well.

3.2 3.2.1

Examples R Hom and Ext

Definition 3.19. Let A be an abelian category and X • , Y • ∈ Ob(D(A)). We define the n-th hyperext of the pair (X • , Y • ) to be Extn (X • , Y • ) := HomD(A) (X • , Y • [n]) . For objects X, Y ∈ Ob(A), we define Extn (X , Y ) to be the corresponding hyperext of their associated complexes concentrated in 0. (We will see below that if A has enough injectives, so that the usual Ext is defined, then this definition agrees with the usual definition of Ext.) Proposition 3.20. Let A be an abelian category and let f

g

0 −→ X • −→ Y • −→ Z • −→ 0 be a short exact sequence in C(A). Then for any complex V • ∈ C(A) there are long exact sequences · · · → Extn (V • , X • ) −→ Extn (V • , Y • ) −→ Extn (V • , Z • ) −→ Extn+1 (V • , X • ) → · · · and · · · → Extn (Z • , V • ) −→ Extn (Y • , V • ) −→ Extn (X • , V • ) −→ Extn+1 (Z • , V • ) → · · · §

It seems that in [Del77, C.D., Chap. 2, §2.3, Prop. 3.1] no assumption on the cohomological dimensions of F and G is required. If this is true, the proof might be more complicated.

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Proof. By Prop. 2.15, there is a morphism w : Z • → X • [1] in the derived category f

g

w

D(A) such that X • −→ Y • −→ Z • −→ X • [1] is a distinguished triangle. As D(A) is a triangulated category, the Hom functors Extn (V • , −) = HomD(A) (V • , T n (−))

and

Extn (− , V • ) = HomD(A) (− , T n (V • ))

are cohomological functors (cf. Example 1.21). So one has long exact sequences as claimed. Now we will define a functor whose cohomology gives the Ext groups. Definition 3.21. Let X • , Y • ∈ C(A) be complexes of objects of an abelian category A. We define a complex Hom• (X • , Y • ) by Y Homn (X • , Y • ) := HomA (X p , Y p+n ) , ∀ n ∈ Z p∈Z

and dn := ((dpX • )∗ + (−1)n+1 (dp+1 Y • )∗ )p∈Z ,

∀ n ∈ Z,

i.e., for f = (fp )p∈Z ∈ Homn (X • , Y • ) =

Y

HomA (X p , Y p+n ) ,

p∈Z

dn f ∈ Homn+1 (X • , Y • ) =

Y

HomA (X p , Y p+n )

p∈Z

has components (dn f )p : X p −→ Y p+n+1 given by (dn f )p = fp+1 ◦ dpX • + (−1)n+1 dYp+n • ◦ fp . (3.22) It follows easily from the definition that Ker(dnHom• (X • , Y • ) ) consists of morphisms of complexes from X • to Y • [n] and that Im(dn−1 Hom• (X • , Y • ) ) consists of those morphisms which are homotopic to 0. So the cohomology groups of the complex Hom• (X • , Y • ) are given by H n (Hom• (X • , Y • )) ∼ = HomK(A) (X • , Y • [n]) . One verifies that Hom• : K(A)opp × K(A) −→ K(Ab) defines a bi-triangulated functor (i.e., Hom• is a bifunctor and for fixed X • ∈ Ob(K(A)), Hom• (X • , −) and Hom• (− , X • ) are both triangulated functors.) (3.23) Now let A be an abelian category with enough injectives. Then for each X • ∈ Ob(K(A)) the right derived functor of Hom• (X • , −) : K+ (A) −→ K(Ab)

26

exists. We denote it by RII Hom• (X • , −) : D+ (A) −→ D(Ab) . This induces a bi-triangulated functor RII Hom• (− , −) : K(A) × D+ (A) −→ D(Ab) . Let I • ∈ K+ (A) be a complex of injectives. If X • → X 0 • is a quasi-isomorphism in K(A), we have by (3.22) and Lemma 2.18 H n (Hom• (X • , I • )) ∼ = HomD(A) (X • , I • [n]) = HomK(A) (X • , I • [n]) ∼ • • ∼ = HomK(A) (X 0 , I • [n]) = HomD(A) (X 0 , I • [n]) ∼ • ∼ = H n (Hom• (X 0 , I • )) i.e., Hom• (X • , I • ) is quasi-isomorphic to Hom• (X 0 • , I • ). On the other hand, Coro. 3.9 implies that • • RII Hom• (X • , I • ) = Hom• (X • , I • ) ∼ = Hom• (X 0 , I • ) = RII Hom• (X 0 , I • )

in the derived category D(Ab). Since every object Y • ∈ D+ (A) is isomorphic to a complex I • ∈ D+ (A) whose terms are all injectives (cf. Lemma 2.19 or Thm. 2.20), we have an isomorphism RII Hom• (X • , Y • ) ∼ = RII Hom• (X 0 , Y • ) •

in D(Ab) for every Y • ∈ D+ (A) and every quasi-isomorphism X • → X 0 • in K(A). So the bi-triangulated functor RII Hom• (− , −) : K(A)opp × D+ (A) −→ D(Ab) induces a bi-triangulated functor RI RII Hom• (− , −) : D(A)opp × D+ (A) −→ D(Ab) . Definition 3.24. Let A be an abelian category with enough injectives. The bifunctor RI RII Hom• (− , −) : D(A)opp × D+ (A) −→ D(Ab) will be denoted by R Hom when no confusion can result. Remark 3.25. If A is an abelian category with enough projectives, then by the usual process of “reversing the arrows” one sees that there is a bi-triangulated functor RII RI Hom• (− , −) : D− (A)opp × D− (A) −→ D(Ab) . When Ab has enough injectives and enough projectives one can show that RI RII Hom• (− , −) ∼ = RII RI Hom• (− , −) on D− (A)opp × D+ (A). (cf. [Har66, p.66, Lemma 6.3].) 27

Theorem 3.26 (Yoneda). Let A be an abelian category with enough injectives. Then the functors Extn (− , −) : D(A)opp × D+ (A) −→ Ab and H n ◦ R Hom(− , −) : D(A)opp × D+ (A) −→ Ab are isomorphic. Proof. As in (3.23), using Thm. 2.20, we need only prove that for every X • ∈ D(A) and every I • ∈ D+ (A) whose terms are all injectives, Extn (X • , I • ) ∼ = H n (R Hom• (X • , I • )) ,

∀ n ∈ Z.

We have already seen (in (3.23)) that H n (R Hom(X • , I • )) ∼ = H n (Hom• (X • , I • )) . By (3.22),

H n (Hom• (X • , I • )) ∼ = HomK(A) (X • , I • [n])

and by Lemma 2.18, which gives HomK(A) (X • , I • [n]) ∼ = HomD(A) (X • , I • [n]) = Extn (X • , I • ) . So the theorem follows. Corollary 3.27. Let A be an abelian category with enough injectives. Then for objects X, Y ∈ Ob(A), the Ext groups Extn defined in Definition 3.19 are the usual ones. Proof. Let I • be an injective resolution of Y . Then Thm. 3.26 yields Extn (X , Y ) = H n (Hom• (X , I • )) where the right hand side is the usual definition of Ext for objects of A. 3.2.2

⊗L and T or

Definition 3.28. Let T be a site and A a sheaf of rings on T. Let AT denote the category of sheaves on A -modules on T. Then one can define a complex of sheaves H om• (F • , G • ) ∈ C(AT ) for complexes F • , G • ∈ C(AT ) in the same way as in Definition 3.21 by replacing Hom by the sheafified H om. The same argument as above yields a bi-triangulated functor RH om : D(AT )opp × D+ (AT ) −→ D(AT ) . One can also define hyperext sheaves by E xt := H n ◦ RH om . 28

We will now briefly explain the construction of the left adjoint functor of RH om. The readers are referred to [Fu11, §6.4], [AGV73, Exp. XVII] for more details. Definition 3.29. With notation as in Definition 3.28, the total tensor product F • ⊗ G • of two complexes F • , G • ∈ C(AT ) is defined as follows M (F • ⊗ G • )n := F i ⊗ G n−i i∈Z

and the differential maps dn :

M

F i ⊗ G n−i −→

i∈Z

M

F i ⊗ G n+1−i

i∈Z

are induced by dn |F i ⊗G n−i := diF ⊗ IdG • +(−1)i IdF • ⊗dn−i G• . (3.30) One can define a bi-triangulated functor − ⊗L − : D− (AT ) × D− (AT ) −→ D− (AT ) as follows: To start with, one proves that for fixed F ∈ K− (AT ) the total tensor product functor F • ⊗ − transforms quasi-isomorphisms in K− (A) to quasi-isomorphisms. On the other hand, one can show that the category AT has enough A -flat objects (i.e., every F ∈ AT is quotient of an A -flat sheaf). This allows us to define the left derived functor LII (F • ⊗ −) : D− (AT ) −→ D− (AT ) . Using the fact that every object of K− (AT ) is quasi-isomorphic to a complex whose terms are all A -flat sheaves, one proves in a dual way of (3.22) that there is a bi-triangulated functor LI LII (− ⊗ −) : D− (AT ) × D− (AT ) −→ D− (AT ) . This functor is in fact isomorphic to the functor LII LI obtained by taking derived functors in the other order. We denote − ⊗• − := LI LII (− ⊗ −) ∼ = LII LI (− ⊗ −) . We can also define hypertor sheaves by T orn (F • , G • ) := H −i (F • ⊗L G • ) and thus a functor T orn (− , −) : D− (AT ) × D− (AT ) −→ D− (AT ) . Theorem 3.31 ([Del77, C.D., Chap. 2, §3.2]). Let T be a site, A a sheaf of rings on T and AT the category of sheaves of A -modules on T. Then for all F • , G • ∈ D− (AT ) and all H ∈ D+ (AT ) there is a functorial isomorphism HomD(AT ) (F • ⊗• G • , H • ) ∼ = HomD(AT ) (F • , RH om(G • , H • )) . 29

References [AGV73] M. Artin, A. Grothendieck, and J.-L. Verdier. Th´eorie des Topos et Coho´ mologie Etale des Sch´emas (SGA 4), Tomes 1–3, volume 269, 270, 305 of Lecture Notes in Math. Springer-Verlag, 1972–1973. [Del77]

P. Deligne. Cohomologie ´etale, volume 569 of Lecture Notes in Math. Spriner, 1977.

[Fu11]

´ L. Fu. Etale Cohomology Theory. World Scientific, 2011.

[GM03]

S. I. Gelfand and Y. I. Manin. Methods of homological algebra. Springer, second edition, 2003.

[Har66]

R. Hartshorne. Residues and duality. Number 20 in Lecture Notes in Math. Springer–Verlag, 1966.

[KS06]

M. Kashiwara and P. Schapira. Categories and sheaves. Springer, 2006.

[Wei94]

C. A. Weibel. An introduction to homological algebra. Cambridge Univ. Press, 1994.

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