Derived Categories with Applications to Representations of Algebras

DERIVED CATEGORIES WITH APPLICATIONS TO REPRESENTATIONSOFALGEBRAS

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. Epaisse´ 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 andFunctors Deﬁnition 1.1 (Category). We deﬁne a category C by the following data: 1. A class ObC of elements called objects of C. 2. For a ordered pair (X, Y )ofobjectsasetHomC (X, Y )ofmorphisms is given such that HomC(X,Y )∩HomC (X ,Y )=φ for (X, Y ) =( X ,Y )(anelement f of HomC(X,Y )iscalledamorphism, and denote by f : X → Y ).

Date:June2000. This is a seminar note of whichIgavealecture at Chiba University in June 2000. 1 2JUN-ICHI MIYACHI

3. For each triple (X, Y, Z)ofobjectsofC a map

θ(X,Y, Z):HomC (X, Y ) × HomC(Y,Z) → HomC(X,Z) (θ is called the composition map) is given. 4. The composition map θ is associative. 5. For each object X of C,thereisamorphism 1X : X → X such that for any g : Y → X, h : X → Z we have 1Xg = g,h1X = h. X ∈ Ob C (often X ∈C)meansthatX is an obje ct of C. Example 1.2. The following often appear in this note. 1. Set is the category consi sti ng 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 consisti ng of ri ght A-modules as obje cts and A-homomorphisms as morphisms. Definition 1.3 (Opposite Category). For a category C,theopposite category Cop of C is defined by 1. Ob Cop =ObC. (for X ∈C,wedenote by Xop ∈Cop the sameobject) 2. For Xop,Yop ∈ Ob Cop, op op HomCop (X ,Y )=HomC (Y, X). op op op (for f ∈ HomC(Y,X), we denote by f ∈ HomCop (X ,Y )) 3. The composition map θop is defined by θop(f op,gop )=θ(g, f)op . Definition 1.4. Let f : X → Y be a morphism inacategory 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 =1X. 4. f is called a split epimorphism if there is g : Y → X such that fg =1Y . 5. f is called an isomorphism if there is g : Y → X such that gf =1X and fg =1Y . We ofte nwrite for an epimorphism, and for a monomorphism. Definition 1.5 (Functor). For cate gories C and C,acovariantfunctor(resp., contravariant functor) F : C→C consists of the following data: 1. A map F :ObC→Ob C. 2. For X, Y ∈ Ob C,amap

FX,Y :HomC (X, Y ) → HomC (FX,FY) (resp., FX,Y :HomC(X,Y ) → HomC (FY,FX))

such that F(gf)=F(g)F (f)(resp.,F(gf)=F(f)F (g)), F(1X )=1F (X). Here we write simply F(f)instead of FX,Y (f). Example 1.6. In a category C,forX ∈C,wedeﬁne the covariant (resp., contravariant) functor hX : C→Set (resp., hX : C→Set) X by h (Y )=HomC(X, Y ) (resp., hX (Y )=HomC(Y,X)). DERIVED CATEGORIES AND REPRESENTATIONS OF ALGEBRAS 3

Deﬁnition 1.7 (Functorial Morphism). Forcovariant (resp.,contravariant) functors F,G : C→C,afunctorial morphism α : F → G consists of thefollowing data: 1. For each X ∈C, αX : FX → GX in C is given. 2. For any morphism f : X → Y (resp., f : Y → X)inC,wehavethefollowing commutative diagram in C FX −−−−αX→ GX     F (f) G(f )

FY −−−−αY → GY. In the case that a functorial morphism α is called a functorial isomorphism if αX are isomorphisms for all X ∈C. We denote by Mor(F, G)thecollection of all functorial morphisms from F to G. Lemma 1.8 (Representable Functor). For a covariant (resp., contravariant) functor F : C→Set,thefollowing are equivalent for C ∈C. C 1. F is isomorphic to h (resp., hC ). 2. There exists c ∈ F(C) satisfying that for any X ∈Cand x ∈ F(X),thereisauniquef ∈ HomC(C,X) (resp., f ∈ HomC (X, C)) such that x = F(f)(c). Acovariant(resp., contravariant) functor F : C→Set is called representable if there exists C ∈Csuch that F is isomorphic to hC .Similarly, a contravariant functor F : C→Set is called representable if there exists C ∈Csuch that F is isomorphic to hC . Lemma 1.9 (Yoneda’s Lemma). For X ∈Cand a covariant (resp., contravariant) functor F : C→Set,wehave the bijection X θ− : FX → Mor(h ,F) (resp., θ− : FX → Mor(hX ,F)), X where θ− is deﬁned by (θx)Y (f)=F(f)(x) for x ∈ FX, Y ∈C, f ∈ h (Y ) (resp., f ∈ hX (Y )). Corollary1.10. For X, Y ∈C,wehave the bijection − X Y h :HomC (Y, X) → Mor(h ,h ) (resp., h− :HomC(X,Y ) → Mor(hX ,hY )). Deﬁnition 1.11. Let F : C→C be a functor.

1. F is called fu l l if FX,Y :HomC (X, Y ) → HomC (FX,FY)aresurjecti ve for all X,Y ∈C. 2. F is called faithful if FX,Y :HomC (X, Y ) → HomC (FX,FY)areinjectivefor all X, Y ∈C. 3. F is called dense if for any Y ∈C,thereisX ∈Csuch that Y is isomorphic to FX. Deﬁnition 1.12 (Limit, Colimit). Le t I, C be categori es and X ∈C.Wedenote by XI : I→Cthe constant functor such that XI(i)=X for all i ∈Iand XI(f)=1X for all f ∈ HomI (i, j). For a functor F : I→C,anobjectX of C is called the colimit colim F (resp., the limit lim F)ofF provided that for all Y ∈Cwe have ∼ Mor(F, YI ) = HomC(X,Y ) ∼ (resp., Mor(YI ,F) = HomC(Y,X)). 4JUN-ICHI MIYACHI

Definition 1.13 (Filtered Col imit). Asmallcategory I is called afilteredcategory provided that 1. For any i, j ∈I,thereexistsk ∈Iand morphisms i → k, j → k in I. 2. For two morphisms f, g : i → j,thereisamorphism h : j → k such that hf = hg. Foracovari ant (resp., contravariant) functor F : I→Cfrom a filtered category I to a category C,thefiltered colimit− lim→ F (resp., the filtered limit← lim− F )ofF is the colimit colim F (resp., the limit lim F). { } Definition 1.14 (Product, Coproduct). For a collection Xi i∈I of objects indexed by a set I, X is called a coproduct i∈I Xi (resp., a product i∈I Xi)of {Xi}i∈I provided that

1. There are a collection of morphisms {qi : Xi → X}i∈I (resp., {pi : X → Xi}i∈I ). 2. For any Y ∈Cand {fi : Xi → Y }i∈I (resp., {pi : Y → Xi}i∈I ), thereexists a unique morphism f : X → Y (resp., f : Y → X)withfi = fqi (resp., f = p f)foral l i. i i { } If a coproduct i∈I Xi is also a product, then it is called a biproduct of Xi i∈I and denoted by i∈I Xi.

Deﬁnition 1.15 (Bifunctor). Let C1, C2 and D be categori es. The product category C1 ×C2 is the category consisting of pairs (X1,X2)ofobjectsX1 ∈ Ob C1 and X2 ∈ ObC2 as objects, and pairs (f1,f2)ofmorphisms f1 in C1 and f2 in C2 as morphisms. A bifunctor is the functor F : C1 ×C2 →D. For bifunctors F,G : C1 ×C2 →D,abifunctorial morphism α : F → G is a functorial morphismoffunctorsC1 ×C2 →D. Then a bifunctor F : C1 ×C2 →Dconsists of the fol low ing data:

1. For X1 ∈C1, F (X1, −):C2 →Dis a functor. 2. For X2 ∈C2, F (−,X2):C1 →Dis a functor. 3. For a morphism f1 : X1 → Y1 in C1, F (f1, −):F(X1, −) → F(Y1, −)isa functorial morphism. (or equivalently, for a morphism f2 : X2 → Y2 in C1, F(−,f2):F (−,X2) → F (−,X2)isafunctorial morphism.) And a bifunctori al morphism α : F → G consists of the following data: ∈C ×C → D 1. For each (X1,X2) 1 2, α(X1,X2) : F (X1,X2) G(X1,X2)in is given. ∈C − → − 2. For X1 1, α(X1,−) : F(X1, ) G(X1, )isafunctorial mor phism . ∈C − → − 3. For X2 2, α(−,X2) : F( ,X2) G( ,X2)isafunctorial mor phism . In the case that a bifunctorial morphism α is called a bifunctorial isomorphism if ∈C ×C α(X1,X2) are isomorphisms for all (X1,X2) 1 2. Deﬁnition 1.16 (Adjoint). Le t F : C→C, G : C →Cbe covariantfunctors. 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(−, ?) : HomC (F −, ?) → HomC(−,G?). −1 In this case, let σX = t(X, F X)(1FX)andτY = t(GY, Y ) (1GY )forX ∈C, Y ∈C.(σ : 1C → GF and τ : FG→ 1C are called the adjunction arrows.) Forcontravariant functors F : C→C, G : C →C,apair(F ,G)iscalleda right adjoint pair if there is a bifunctorial isomorphism t (−, ?) : HomC (−,F?) → HomC(?,G−). There areadjuncti on arrows σ : 1C → G F and τ : 1C → F G . DERIVED CATEGORIES AND REPRESENTATIONS OF ALGEBRAS 5

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 =1G and τF◦Fσ =1F . Theorem1.18. For a covariant functor F : C→C,thefollowing hold.

1. F has a right adjoint if and only if hY ◦ F : C→Set is representable for any Y ∈C. 2. F has a left adjoint if and only if hX ◦ F : C→Set is representable for any X ∈C. Theorem1.19. Let F : C→C, G : C →Cbe covariant functors such that F G. Then the following are equivalent. 1. G is fully faithful. 2. The adjunction arrow τ : FG→ 1C is a functorial isomorphism. Sketch.

HomC(X,Y ) RRR RRR Hom(RRRτX ,Y ) GX,Y RRR RR) ∼ / HomC (GX, GY ) HomC (FGX,Y)

Theorem 1.20 (Equivalence). For a functor F : C→C,thefollowing are equivalent. 1. F is fully faithful and dense. ∼ ∼ 2. There is a functor G : C →C such that GF = 1C and FG= 1C . In this case, F is called an equivalence and we say that C and C are equivalent. Theorem1.21. Let F : C→C be a covariant functor and let G be a right adjoint of F .Thenthefollowing hold. 1. F preserves the colimit in C of any functor. 2. G preserves the limit in C of any functor.

2. AdditiveCategoriesand AbelianCategories

In a category C,anobjectU is called an initial object if for any X ∈CHomC(U, X) has only one element, V is called a terminal object if for any X ∈CHomC(X,V ) has only one element, and O is called a null object if O is initial and terminal. Deﬁnition 2.1 (Preadditive Category). Acategory C with a null object is called a preadditive category provided that HomC (X, Y )isanabeliangroupforanyX,Y ∈ C,andthatthe compositi on map θ is bilinear. Deﬁnition 2.2 (Additi ve functor). Let C, C be preadditive categories. A covariant(resp., contravariant) functor F : C→C between preadditive cate go ri es is called an additive functor provided that for X, Y ∈C,

FX,Y :HomC (X, Y ) → HomC (FX,FY) (resp., FX,Y :HomC(X,Y ) → HomC (FY,FX)) is a group morphism. 6JUN-ICHI MIYACHI

Proposition 2.3. Let {Xi}1≤i≤n be afinite collection of objects of a preadditive category C.Thenthe following are equivalent. n { } C 1. Acoproduct i=1Xi of Xi 1≤i≤n exists in . n { } C 2. Aproduct i=1Xi of Xi 1≤i≤n exists in . 3. There exist an object X ∈Cand morphisms ui : Xi → X, pi : X → Xi (1 ≤ i ≤ n)suchthat n (a)Σi=1uipi =1X . 0 if i = j (b) piuj = 1Xi if i = j. Moreover, the above coproduct is naturally isomorphic to the above product. Proposition 2.4. Let F : C→C be an additive functor between preadditive cate- { } C n gories, Xi 1≤i≤n a finite collection of objects in .Ifthecoproduct i=1Xi exists C n C in ,thenthe coproduct i=1F(Xi) exists in and is canonically isomorphic to n F ( i=1Xi). Definition 2.5 (Additi ve Category). Apreadditive 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,WedefineAdd M (resp., add M)thefullsubcategory of C consisti ng of objects which are directsummands 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 areequivalent. → 1. For any morphism f : C i∈I Xi,thereexistsafactorization

f µF C −→ Xj −−→ Xi j∈F i∈I

where F is a finite subset ofI and µF is the canonical inclusion. → 2. For any morphism f : C i∈I Xi,thereexistsafinitesubset F of I such that f = j∈F ujpj f where uj are the structural morphisms and pj are the canonical projections. 3. The functor hC : C→Ab preserves coproducts. An object C ∈Cis called a compact object (often called a small object) of C if C satisfies the above conditions. Exercise 2.8. Show thatifaright A-module C is finitely generated, then C is a compact object in Mod A. Corollary2.9.Let C be a compact object of an additive category C,andB = EndC(C).Thefollowing hold. 1. For any object X ∈C,we have isomorphisms (I) ∼ (I) HomC(C ,X) → HomB (HomC (C, C ), HomC(C,X)) ∼ (I) → HomB (HomC (C, C) , HomC(C,X)) if a coproduct C(I) exists for a set I. DERIVED CATEGORIES AND REPRESENTATIONS OF ALGEBRAS 7

2. we have isomorphisms (I) (J) ∼ (I) (J) HomC (C ,C ) → HomB (HomC (C, C ), HomC(C,C )) ∼ (I) (J) → HomB (HomC (C, C) , HomC(C,C) ) if coproducts C(I),C(J) exist for sets I,J. Proposition 2.10. Let C be an additive category, C apreadditive category and F : C→C afunctor.IfF 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 ,wedefinethefollowing. f 1. (Cok f, cok f)=colim (X −→ Y,X −→0 Y ). f 2. (Ker f,ker f)=lim (X −→ Y,X −→0 Y ). 3. (Im f, im f)=Ker(Y −→0 Cok f,cok f). 4. (Coim f,im f)=Cok(Kerf −→0 X, ker f). f g 5. PushOut(f,g)=colim (X −→ Y, X −→ Z). f 6. PullBack(f,h)=lim (X −→ 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.Thenthere exists a unique morphism f :Coimf → Im f such that we have the following commutative diagram f X −−−−→ Y    coimf im f

f Coim f −−−−→ Im f Deﬁnition 2.13 (Abelian Category). An additive category C is called an abelian category provided that 1. For any morphism f,thereexist Ker f and Cok f in C. 2. For any morphism f,theabovemorphism f is an isomorphism. Deﬁnition 2.14. In an abelian category C,weconsiderthefollowing sequence − f i 1 f i ...→ Xi−1 −−−→ Xi −→ Xi+1 → ... . We say that the above sequence is exact at Xi if Ker f i =Imf i−1.Iftheabove sequence is exact at each Xi,thenwesaythatthe 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 f g X −−−−→ Y −−−−→ Z −−−−→ O       x y z

f g O −−−−→ X −−−−→ Y −−−−→ Z 8JUN-ICHI MIYACHI where all rows areexact.Thenwehave the following induced exact sequence Ker x → Ker y → Ker z → Cok x → Cok y → Cok z. Moreover, f (resp., g)ismonic(resp.,epic)ifandonlyifsoisKer x → Ker y (resp., Cok y → Cok z). Proposition 2.16 (Five Lemma). Suppose that the following diagram is commutative X −−−−→ X −−−−→ X −−−−→ X −−−−→ X 1 2 3 4 5      f1 f2 f3 f4 f5

X1 −−−−→ X2 −−−−→ X3 −−−−→ X4 −−−−→ X5 where all rows areexact.Thenthefollowing hold.

1. If f1 is epic, and f2, f4 are monic, then f3 is monic. 2. If f5 is monic, and f2, f4 are epic, then f3 is epic. 3. If f1 is epic, f5 is monic, and f2, f4 are isomorphisms, then f3 is an isomorphism. Proposition 2.17 (Pull Back 1). Suppose that the following diagram is commutative f / X Y

x (A) y f / X Y Then the following hold.

f [ ] [ −yf ] 1. The square (A) is pull back if and only if O → X −−x→ Y ⊕X −−−−−→ Y is exact. f [ ] [ −yf ] 2. The square (A) is push out ifandonlyifX −−x→ Y ⊕X −−−−−→ Y → O is exact. Proposition 2.18 (Pull Back 2). Suppose that the following diagram is commutative / / X Y Z

(A) (B) / / X Y 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) f / X Y

x y f / X Y Then the following hold. DERIVED CATEGORIES AND REPRESENTATIONS OF ALGEBRAS 9

1. If f (resp., f)isepic(resp.,monic), then the above diagram is also push out (resp., pullback), and f (resp., f )isalsoepic(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 / X Y

EX / X 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

f g O −−−−→ X −−−−→ Y −−−−→ Z −−−−→ O     x y

f g O −−−−→ X −−−−→ Y −−−−→ Z −−−−→ O where all rows areexact.Thenwehave the following commutative diagram

β O −−−−→ X −−−−α → Y ⊕X −−−−→ Y −−−−→ O     γ g

f −g O −−−−→ X −−−−→ Y −−−−→ Z −−−−→ O

f − where α =[x ], β =[ yf ], γ =[10],whereallrowsareexact. Proposition 2.22 (Exact Sequence 3). Suppose that the following diagram is commutative

X / /Y //Z 1 1 1 / / // X2 Y2 Z2 X / /Y //Z 3 3 3 / / // X4 Y4 Z4 10 JUN-ICHI MIYACHI

where all rows areshortexactsequences. If two of squares X −−−−→ X Y −−−−→ Y Z −−−−→ Z 1 2 1 2 1 2      

X3 −−−−→ X4 Y3 −−−−→ Y4 Z3 −−−−→ Z4 are exact, thentherestisalso exact. Hint. Consider the following commutative diagram O −−−−→ X −−−−→ Y −−−−→ Z −−−−→ O 1 1 1   

O −−−−→ X ⊕X −−−−→ Y ⊕Y −−−−→ Z ⊕Z −−−−→ O 1  2 3  2 2  1   

O −−−−→ X2 −−−−→ Y4 −−−−→ Z2 −−−−→ O.

Exercise 2.23. The following hold. f g 1. For a sequence X −→ Y −→ Z,ifforanyM ∈A,

HomA(g,M) Ho mA(f,M) 0 → HomA(Z, M) −−−−−−−−→ HomA(Y, M) −−−−−−−−→ HomA(X,M) → 0 is exact, then f g O → X −→ Y −→ Z → O is split exact. f g 2. For a sequence X −→ Y −→ Z,ifforanyM ∈A,

HomA(M,f) Ho mA(M,g) 0 → HomA(M,X) −−−−−−−−→ HomA(M,Y ) −−−−−−−−→ HomA(M,Z) → 0 is exact, then f g O → X −→ Y −→ 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 sati sfies the conditi on Ab3 (resp., Ab3*) if C has coproducts (resp., products) of objects indexed by arbitrary sets. (Ab4) We say that an abelian category C sati sfies the conditi on Ab4 (resp., Ab4*) provided that C sati sfies the condition Ab3 (resp., Ab3*), and thattheco- product (resp., product) of monics (resp., epics) is monic (resp., epic). (Ab5) We say that an abelian category C sati sfies the conditi on Ab5 (resp., Ab5*) provided that C sati sfies thecondition 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. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 11

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

(Xi ∩ X )=( Xi) ∩ X i i for any subobject X of X. Example 2.26. For a ring A, Mod A sati sﬁes theconditions Ab4*, Ab5.

3. Krull-Schmidt Categories Let R be a ringwithunity and let J(R)betheJacobson radical of R.We call R asemiperfect ring if (i) R/ J(R)isasemi-simpleArtinian ring, and (ii) any idempotent of R/ J(R)canbeliftedtoan idempotent of R. Lemma 3.1 (Semiperfect Rings 1). The following hold. 1. AringR is semiperfect if and only if R has a complete set of orthogonal primitive idempotents ei (1 ≤ i ≤ n)suchthat each eiRei is a local ring. 2. AringR 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 ei (1 ≤ i ≤ n)beacompletesetoforthogonal primitive idempotents. 1. If f (1 ≤ i ≤ m)isanothercompletesetoforthogonal primitive idempotents, i ∼ then m = n and there is a permutation π such that Rfi = Reπ(i) for all i. 2. If f is an idempotent of R,thent here are a permutationπ and an integer t ≤ ≤ ∼ t − ∼ n (1 t n)suchthat Rf = i=1Reπ(i) and R(1 f) = i=t+1Reπ(i). 3. If I is a two-sided ideal of R,thenR/I is also semiperfect.

Proposition 3.3. Let C be an additive category, and let X ∈C, B =EndC(X).If X is a direct summand of a ﬁnite coproduct of copies of X,wehave ∼ HomC (X ,Y) → HomB (HomC (X, X ), HomC (X, Y )) (f → HomC(X,f)) for all Y ∈C. → → ≤ ≤ n Proof. There are qi : X X and pi : X X (1 i n)suchthat i=1piqi = 1X .Letφ ∈ HomB(H om C(X, X ), HomC(X,Y )), for any g ∈ HomC (X, X ), we have

n φ(g)=φ piqig i=1 n = φ(pi)qig i=1 n =HomC X, φ(pi)qi (g). i=1 Then HomC(X,−)issurjective. Let f ∈HomC (X ,Y)suchthatHomC (X, f)=0. n n Then fpi =0foralli,andhence f = f i=1piqi = i=1fpiqi =0. Deﬁnition 3.4. Let C be an additive category. An object X of C is called inde- ∼ composable if X = X1 ⊕ X2 implies X1 = O or X2 = O. Deﬁnition 3.5 (Pre-Krull-Schmidt Category). An additive category C is called a pre-Krull-Schmidt category provided that EndC (X)isasemiperfectringforeach X ∈C. 12 JUN-ICHI MIYACHI

Proposition 3.6. Let C be a pre-Krull-Schmidt category. For any X ∈C,there are indecomposable objects X (1 ≤ i ≤ n)suchthat i ∼ n X = Xi. i=1 Proof. Gi ven X ∈C,sinceEndC(X)isasemiperfectring,thereisanaturalnumber nX such that EndC (X)has a complete set of orthogonal primitive idempotents ei (1 ≤ i ≤ nX). If X is not indecomposable, then we have a decomposition X = X1⊕X2 with Xi = O (i =1, 2). By Lemma 3.2,2, Proposition 3.3, we have nXi subcategory of C.ForX, Y ∈C,letI(X,Y )bethesubgroup of HomC(X, Y ) generated by morphisms which factor through some object of I.Wedeﬁne the category CI as follows. 1. Ob CI =ObC. I 2. HomCI (X, Y )=HomC(X,Y )/ (X, Y ). This category is called the stable category of C by I. Remark 3.12. For X ∈C,IfX is a direct summand of some object of I,then ∼ X = O in CI. Theorem3.13. Let C be an additive category, I an additive full subcategory of C. If C is a Krull-Schmidt category, then so is CI . DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 13

Proof. By Proposition 2.3, CI is clearly an additive category. For an indecom- ∈C n C posable object X I ,wehaveaK-S-decomposition X = i=1Xi in .Let −p→i −q→i C ≤ ≤ ei : X Xi X be the canonical morphism in (1 i n), and ei be the ∈C image of ei in I.Ifthenumberofei such that ei =0isgreaterthan 1, then by Lemma 3.2 this contradicts indecomposability of X.Thuswemayassume that e =0and e =0fori ≥ 2, and then q p factors through V ∈Ifor i ≥ 2. Then X 1 i i i i ∼ i is a direct summand of Vi. Therefore, by Lemma 3.2,3, EndCI (X) = EndCI (X1)is alocalring.Wecomplete the proof by Proposition 3.7. Example 3.14. Let R be a commutative complete local ring, A a ﬁnite R-algebra, and pr oj A the full subcategory of mod A consisting of ﬁnitely generated projective right A-modules. Then the stable category modA of mod A by pr oj A is a Krull- Schmidt category. 14 JUN-ICHI MIYACHI

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,calledthetranslation,and(2)acollection T of sextuples (X, Y, Z, u, v, w), called triangle (distinguished triangle).These data are subject to the following fouraxioms: (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, 1X, 0, 0) is a tri angl e for all X ∈C. (TR2) A tri angl e (X,Y, Z,u, v, w)isatriangle if and only if (Y,Z,TX,v,w,−Tu)is a tri angl e. (TR3) For any triangles (X, Y, Z,u, v, w), (X,Y,Z,u,v,w)andmorphisms f : X → X, g : Y → Y with gu = uf,thereexists h : Z → Z such that (f,g,h)isahomomorphism of triangles. (TR4) (Octahedral axiom) For any two consecutive morphisms u : X → Y and v : Y → Z,ifweembedu, 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 → TX     v f

X −−−−vu→ Z −−−−k → Y −−−−k → TX       j g Tu

j X X −−−−→ TY     j (Ti)j

TY −−−−Ti→ TZ and the third column is a tri angl e. Sometimes, we write X[i]forT 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 : FTC → TC F such that (FX,FY,FZ,F(u),F(v),αX F(w)) is a tri angle in C whenever (X, Y, Z,u, v, w)i s a tri angle in C.Moreove r, if a ∂- functor F is an equivalence, then we say that C is triangle equivalent to C,and ∼t denote by C = C For (F, α), (G, β):C→C ∂-functors, a functorial morphism φ : F → G is called a ∂-functorialmorphismif (TC φ)α = βφTC. We denote by ∂(C, C)thecollection of all ∂-functors from C to C,anddenote by ∂ Mor(F, G)thecollection of ∂-functorial morphisms from F to G.

Deﬁnition 4.3. Gi ven a tri angulatedcategory C with a translation TC,wedeﬁne the opposite tri angulatedcategory Cop the following op −1 1. TCop (X )=TC (X). DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 15

op op op op −1 2. X → Y → Z → TCop X is a distinguished triangle if TC X → Z → Y → X is a distinguished triangle in C. Deﬁnition 4.4. Acovariant additi ve functor H : C→C from a tri angulated category to an abelian category is called a covariant cohomological functor,ifwhenever (X , Y,Z,u,v,w)isatriangleinC,thelongse quence

H(T i(u)) H(T i(v)) H(T i (w)) ...→ H(T i(X)) −−−−−−→ H(T i(Y )) −−−−−−→ H(T i(Z)) −−−−−−→ H(T i+1(X)) → ... is exact. If H is a cohomological functor, then we often write Hi(X)forH(T i(X)), i ∈ Z.Onedeﬁnes a contravariantcohomological 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 =0and T (u)w =0. 2. For any W ∈C, HomC (W, −):C→Ab (resp., HomC (−,W):C→Ab)isa 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),iftwooff, 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 −−−−1X→ X −−−−→ O −−−−→ TX     u

X −−−−u → Y −−−−v → Z −−−−w → TX. 2. Let (X, Y, Z,u, v, w)beatriangle. Then, since by 1, vu =0,wehave HomC(W, v) ◦ HomC (W, u)=0.Converse ly, let g ∈ HomC (W, Y )suchthat HomC(W, v)(g)=vg =0. Then by (TR3) there exists f ∈ HomC (W, Y ) which makes the following diagram commutes W −−−−1W→ W −−−−→ O −−−−→ TW         f g Tf

X −−−−u → Y −−−−v → Z −−−−w → TX. Thus g =HomC(W, u)(f)andthesequence HomC (W, X) → HomC(W, Y ) → HomC(W, Z)isexact.Itfollows by (TR2) that HomC (W, −)isacohomological 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 HomC (TY , −) → HomC (TX , −) →HomC(Z , −) → HomC (Y , −) →HomC(X , −)

↓ Ho mC (Tg,−) ↓ HomC(Tf,−) ↓ HomC(h,−) ↓ HomC(g,−) ↓ HomC(f,−) HomC(TY,−) → HomC (TX,−) → HomC (Z, −) → HomC (Y,−) → HomC (X, −).

Thus, since by 5 lemma HomC(h, −)isanisomorphism,itfollowsbyYonedalemma 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 ,thenG is also a ∂-functor. 16 JUN-ICHI MIYACHI

Proof. For X ∈C,wehaveafunctorialisomorphisms ∼ HomC(−,GTX) = HomC (F −,TX) ∼ −1 = HomC (T F−,X) ∼ −1 = HomC (FT −,X) ∼ −1 = HomC (T −,GX) ∼ = HomC (−,TGX). ∼ Then we have a functorial isomorphism β : GTC → TC G.Foratri angle (X,Y,Z,u, v,w)ofC,let(GX, GY, Z,Gu,v,w)beatri angl e of C.Sincethere is a morphism of triangles

/ / / αG / FGX FGY FZ FTGX TFGXr rrr rrr yrrr / / / X Y Z TX. Then we have a commutative diagram

/ / / GX GY Z TGXII II −1 II β II I$ / / / β / GX GY GZ GTX TGX. ∼ Since HomC (M,G−) = HomC (FM,−), (GX, GY, GZ, Gu, Gv, Gw)induces a long exact sequence. We apply HomC(M,−)totheabovediagram, then by 5 lemma, we have an isom orphism from (GX, GY, Z,Gu,v,w)to(GX, GY, GZ, Gu, Gv, βX Gw). Proposition 4.7. The following hold. C { } 1. If i∈I Xi (resp., i∈I Xi)exists in for Xi i∈I ,thenthere is an isomorphism ∼ α : TXi → T Xi i∈I i∈I ∼ (resp., β : TX → T X ). ∈ i ∈ i i I i I ∈ 2. For a col lectionof triangles (Xi,Yi,Zi,ui,vi,wi) (i I), if i∈I Xi, i∈I Yi, C i∈I Yi (resp., i∈I Xi, i∈I Yi, i∈I Yi)existin ,then ( Xi, Yi, Zi, ui, vi,α wi) i∈I i∈I i∈I i∈I i∈I i∈I (resp., ( Xi, Yi, Zi, ui, vi,β wi)) i∈I i∈I i∈I i∈I i∈I i∈I is a triangle. Proof. 1. We have isomorphisms ∼ −1 HomC(T Xi, −) = HomC ( Xi,T −) i∈I i∈I ∼ −1 = HomC(Xi,T −) i∈I ∼ = HomC(TXi, −) i∈I ∼ = HomC ( TXi, −). i∈I DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 17 2. Let ( i∈I Xi, i∈I Yi,Z , i∈I ui,v,w)beatri angl e. Then we have a commutative diagram

u v α v X −−−−−i∈I →i Y −−−−→i∈I i Z −−−−−→i∈I i T X i∈ I i i∈ I i i∈I i i ∈I i 

u −−−−−i∈I →i −−−−v → −−−−w → i∈I Xi i∈I Yi Z T i∈I Xi.

Applying HomC(M,−)totheabove,by5lemma,wecompletetheproof.

Proposition 4.8. The following hold. ⊕ 0 1. Atriangle(X,Y, Z,u, v, 0) is isomorphic to (X, Z X, Z, [ 1 ] , [ 10] , 0). 2. For a morphism of triangles

X −−−−u → Y −−−−v → Z −−−−w → TX     f g

X −−−−u → Y −−−−v → Z −−−−w → TX there exists g : Z → Z such that

v [ f ] [ −g v ] (Tu)w Y −−→ Z⊕Y −−−−−→ Z −−−−→ TY

is a triangle.

0 Proof. 1. Since HomC (Z, Z) −→ HomC (Z, TX), by Proposition 4.5, there is s : Z → Y such that vs =1Z . Then we have a commutati ve diagram

µ X −−−−→ Z⊕X −−−−π → Z −−−−0 → TX   α

X −−−−u → Y −−−−v → Z −−−−0 → TX

0 su where µ =[1 ], π =[10], α =[ ]. − −1 2. Since T −1Z −−−−−T w→ X −u→ Y −v→ Z is tri angle, we have a commutative diagram

− −1 T −1Z −−−−−T w→ X −−−−u → Y −−−−v → Z     u x

− −1 β T −1Z −−−−−−uT w→ Y −−−−α → M −−−−→ Z       v y −w

Z Z −−−−w → TX     w 0

TX −−−−Tu→ TY 18 JUN-ICHI MIYACHI

y µ By 1, Y −→x M −→ Z −→0 TY is isomorphic to Y −→ Z⊕Y −→π Z −→0 TY. Then we have a commutative diagram

− −1 T −1Z −−−−−T w→ X −−−−u → Y −−−−v → Z     u µ

− −1 β T −1Z −−−−−−uT w→ Y −−−−α → Z⊕Y −−−−→ Z       v π −w

Z Z −−−−w → TX     w 0

TX −−−−Tu→ TY v 0 − where µ =[1 ], π =[10], α = f , β =[ g v ], and v f = g v, f u = u , w = wg.Since(f − f)u =0,thereish : Z → Y such that f = f + hv.Hence we have a commutative diagram

β (Tu)w Y −−−−α → Z⊕Y −−−−→ Z −−−−→ TY   φ

β (Tu)w Y −−−−α → Z⊕Y −−−−→ Z −−−−→ TY

v − 10 where α =[f ], β =[ g v ], φ =[h 1 ].

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

X −−−−u → Y     x y X −−−−u → Y can be embeddedinadiagram

X u /Y v /Z w /TX .

x y z Tx X u /Y v /Z w /TX

x y z − −Tx −Tw TX Tu /TY Tv /TZ /T 2X which is commutative without the right and bottom corner, - anti-commutative, where all rows and columns are triangles. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 19

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

X u /Y v /Z w /TX

y γ y u β X /Y α /A /TX

y δ Tu y Y Y /TY

y (Tv )y TY Tv /TZ

X x /X x /X x /TX

u ε β X ux /Y α /A /TX

v η Tx Z Z w /TX

w (Tx)w TX Tx /TX

η (Tx)w X ε /A /Z /TX/

δ z X u /Y v /Z w /TX

(Tv )y z Tε −Tγ TZ TZ /TA

−Tγ −Tz Tη TA /TZ In parti cular, we have u = δε, v = ηα, y = δα, z = ηγ, zη = vδ and (Tx)w = (Tβ)(Tε)w = −(Tβ)(Tγ)z = −(Tw)z. Then itiseasytogetthe 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,andsuppose that C is closed under extensions in A.LetS be a collection of exact sequences in A O → X −→u Y −→v Z → O. u is called an admissible monomorphism,andv is called an admissible epimorphism. Apair(C, S)iscalledanexact 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). 20 JUN-ICHI MIYACHI

(EX3) Given the fol low ing com mutative diagram in A O −−−−→ X −−−−→ Y −−−−→ Z −−−−→ O    

O −−−−→ X −−−−→ Y −−−−→ Z −−−−→ O where all rows are exact, if thetoprow is in S and X ∈C,thenthe bottom row is in S. (EX4) Given the fol low ing com mutative diagram in A O −−−−→ X −−−−→ Y −−−−→ Z −−−−→ O    

O −−−−→ X −−−−→ Y −−−−→ Z −−−−→ O where al l rowsare exact, if the bottom row is in S and Z ∈C,thenthe top row is in S. An object X in C is called S-projective (resp., S-injective)ifforany admissi ble epimorphisms (resp., monomorphisms) v : Y → Z,HomC (X, v) (resp., HomC (v, X)) is surjective. Deﬁnition 5.2 (Frobenius Category). An exact category (C, S)iscalled a Frobe- nius category if (C, S)hasenough S-projectives and enough S-injectives and if S-projectives coincide with S-injectives. Let Q be the fullsubcategoryofC consisting of S-projective objects. A stable category C is the category CQ. Proposition 5.3. In a Frobenius category (C, S),weconsiderthefollowing commutative diagram O −−−−→ X −−−−→ I −−−−→ X −−−−→ O   1    f f −−−−→ −−−−→ −−−−→ −−−−→ O X I X1 O where I, I are S-injective, with all rows in S.Thentheimage f is uniquely determined by f in C. µ Remark 5.4. For al l X ∈Cwe choose theelementsO → X −−→X I(X) −π−→X TX → O in S,withI(X)beingS-injective. According to Proposition 5.3, an object TX is uniquely determined up to isomorphism in C independently of choice of the above sequence, but f is depend on their choice. Then we can understand the induced µ π functor T : C →Conly if we know O → X −−→X I(X) −−X→ TX → O in S for all X ∈C. Proposition 5.5. T is an auto-equivalence of C. Deﬁnition 5.6 (Tri angle). In a Frobenius category (C, S), let u : X → Y be an morphism in C.BytakingM(u)=PushOut(u, µX), we have thefollowing commutative diagram in C µ O −−−−→ X −−−−X→ I(X) −−−−πX→ TX −−−−→ O     u x

O −−−−→ Y −−−−v → M(u) −−−−w → TX −−−−→ O DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 21 with all rows in S. Then in C the sequence u v w X −→ Y −→ M(u) −→ TX is called a standard triangle.LetT be a collection of sextuples which are isomorphic to standard triangles in C. Lemma 5.7. In a Frobenius category (C, S),letw :M(u) → TX be the morphism in Deﬁnition 5.6. Consider the following commutative diagram in C − O −−−−→ X −−−−u→ Y −−−−v → M(u) −−−−→ O     x w µ O −−−−→ X −−−−→ I(X) −−−−π → TX −−−−→ O u v w with all rows in S,thenthesextupleX −→ Y −→ M(u) −→ TX is isomorphic to u v w the above triangle X −→ Y −→ M(u) −→ TX in C. Proof. By Proposition 2.21, we have the following commutative diagram β γ O −−−−→ X −−−−→ I(X) ⊕ Y −−−−→ M(u) −−−−→ O     δ w

µ − O −−−−→ X −−−−→ I(X) −−−−π→ TX −−−−→ O µ − S where β =[u ], γ =[ xv], δ =[10], with allrowsin .Sincetheright 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 C u v w X −−−−→ Y −−−−→ M(u) −−−−→ TX   η

β γ w X −−−−→ I(X) ⊕ Y −−−−→ M(u) −−−−→ TX   ε

u v w X −−−−→ Y −−−−→ M(u) −−−−→ TX where ε =[01], all verti cal arrows are isomorphisms in C. Proposition 5.8. In a Frobenius category (C, S),theimage of any element O → u v w X −→u Y −→v Z → O of S can be embedded in a triangle X −→ Y −→ Z −→ TX in C. − Proof. Since I(X)isS-injective and O → X −−→u Y −→v Z → O ∈S,wehavea commutative diagram − O −−−−→ X −−−−u→ Y −−−−v → Z −−−−→ O     w

µ O −−−−→ X −−−−→ I(X) −−−−π → TX −−−−→ O. By Lemma 5.7, we get the statement. Theorem 5.9. Let (C, S) be a Frobenius category. Then (C, T ) is a triangulated category. 22 JUN-ICHI MIYACHI

Proof. we show that (C, S)satisﬁestheaxioms of a triangulated category. (TR1) Itistrivial. (TR2) Let (X,Y,Z,u,v,w)beastandard triangle. Then we have a commutative diagram

µ O −−−−→ X −−−−X→ I(X) −−−−πX→ TX −−−−→ O     u x

O −−−−→ Y −−−−v → Z −−−−w → TX −−−−→ O     µY α

β γ O −−−−→ I(Y ) −−−−→ I(Y )⊕TX −−−−→ TX −−−−→ O     πY δ

TY TY

s 1 where α =[w ], β =[0 ], γ =[01], δ =[πY u ], πY s +u w =0,andwith all rows in S.SincesxµX =[10] αxµX =[10] βµY u = µY u,wehaveacommutative diagram

µ O −−−−→ X −−−−X→ I(X) −−−−πX→ TX −−−−→ O       u sx −u µ O −−−−→ Y −−−−Y → I(Y ) −−−−πY → TY −−−−→ O. Then we have u = −Tu,andwehaveacommutative diagram in C

Y −−−−v → Z −−−−w → TX −−−−u → TY  

Y −−−−v → Z −−−−α → I(Y )⊕TX −−−−δ → TY

0 C − where ε =[1 ]isanisomorphism in .Hence a sextuple (Y, Z,TX,v,w, Tu)isa triangle . The reverse implication is similar by Lemma 5.7. (TR3) Let (Xi,Yi,Zi,ui,vi,wi), be standard triangles (i =1, 2), and µ π O −−−−→ X −−−−Xi→ I(X ) −−−−Xi→ TX −−−−→ O i  i i   ui xi

vi wi O −−−−→ Yi −−−−→ Zi −−−−→ TXi −−−−→ O commutative diagrams with all rows in S.Letf : X1 → X2, g : Y1 → Y2 be C S morphisms satisfying u2f = gu1 in .SinceI(X1)is -injective, there exists → − S t : I(X1) Y2 such that gu1 u2f = tµX1 .SinceI(X2)is -injective,wehavea commutative diagram

µ X −−−−X1→ I(X ) 1  1   f r

µX2 X2 −−−−→ I(X2). DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 23

Then we have equations

x2rµX1 = x2µX2 f = v2u2f

v2gu1 = v2u2f + v2tµX1

=(x2r + v2t)µX1 Since µ X −−−−X1→ I(X ) 1  1   u1 x1

v1 Y1 −−−−→ Z1 is push out, there exists h :M(u1) → M(u2)suchthatv2g = hv1 and x2r + v2t = hx1. Then there is f : TX1 → TX2 such that f w1 = w2h.Since f πX1 = f w1x1 = w2hx1 = w2(x2r + v2t)=w2x2r = πX2 r we have f = Tf,andhence a morphism (f,g,h)from(X1,Y1,Z1,u1,v1,w1)to (X2,Y2,Z2,u2,v2,w2). (TR4) Let (X,Y,Z,u,i,i), (X,Z,Y ,vu,k,k)and(Y,Z,X,v,j,j)betrian- gles in C.Wehaveacommutative diagram in C

/ µX / πX / / O X I(X) TX O

u EX x / i / i / / O Y Z TX O

v EX f / k / k / / O Z Y TX O S where all rows are in .Sincei is an admissi ble monomorphism, µ Y = µZ i is also an admissible monomorphism and there is an admissible epimorphism π Y such that πZ =(T i)π Y . Then we have a commutative diagram in C

µX / πX / X I(X) TXFF F FFT u u EX x FF FF " i / µZ / π Y / T i / Y Z I(Z ) T Y TZ

v EX f EX z j k / g1 / 1 / T i / Z Y X T Y TZ Therefore, we have a triangle in C f g1 (T i)j Z −→ Y −→ X −−−−→1 TZ. C Since j1g1fx =(T u)πX =(T u)k fx and j1g1k =0=(T u)k k in ,and X −−−−u → Y −−−−v → Z     µX k

f I(X) −−−−x → Z −−−−→ Y 24 JUN-ICHI MIYACHI

S is push out, we have j1g1 =(T u)k .Since I(Y )andI(Z )are -injective, there exist α : TY → T Y and β : X → X such that

/ / // Y I(Y ) TY / / // Z X TY / / // Y I(Z ) T Y / / // Z X T Y is commutative in C, where α is an isomorphism in C. Then by Proposition 2.22, β is −1 −1 −1 an isomorphism in C,andTi =(T i)α, Tu = α T u.Letg = β g1, j = β g1k, −1 j = α j1β,thenwehavethe octahedral diagram of Deﬁnition 4.1.

Example 5.10. Let A be a self-injective algebra over a ﬁeld k,andO → Ω → µ A⊗kA −→ A → O an exact sequence, where µ is the multiplication map. Then mod A is a Frobenius category, its stable category modA is a tri angulatedcategory with a translation functor HomA(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 n n n → n+1 complex)isacollection X =(X ,dX : X XX )n∈Z of objects and morphisms B n+1 n n n n → n+1 of such that dX dX =0.Acomplex X =(X ,dX : X XX )n∈Z is called bounded below (resp., bounded above, bounded)ifXn = O for suﬃciently small (resp., large, large andsmall)n. n n Acomplex X =(X ,dX )iscalledastalk complex if there exists an integer n0 i such that X = O if i = n0.WeidentifyobjectsofB with a stalk complexes of degree 0. A morphism f of complexes X to Y is a collection of morphisms f n : Xn → Y n which commute with the maps of complexes

n+1 n n n f dX = dY f . − We denote by C(B)(resp.,C+(B), C (B), Cb(B)) the category of complexes (resp., bounded below complexes, bounded above complexes, bounded complexes) of B.Anauto-equivalence T : C(B) → C(B)iscalledtranslation if (TX)n = Xn+1 n − n+1 n n and (TdX ) = dX for any complex X =(X ,dX ). 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). DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 25

Deﬁnition 6.3. For u ∈ HomC(B)(X ,Y ), the mapping cone of u is a complex M (u)with n n+1⊕ n M (u)=X Y , −dn+1 0 n X n+1⊕ n → n+2⊕ n+1 dM(u) = n+1 n : X Y X Y . u dX Moreover,for1X ∈ HomC(B)(X ,X ), let I (X )=M(1X).

f g Deﬁnition 6.4. Let SC∗(B) be the collection of exact sequences O → X −→ Y −→ ∗ Z → O of complexes of C (B)suchthat

f n gn O → Xn −→ Y n −→ Zn → O are split exact for all n ∈ Z, where ∗ =nothing,+, −,b.Inthiscase,wecallf (resp., g)aterm-split monomorphism (resp., a term-split epimorphism). Proposition 6.5. For u ∈ HomC(B)(X ,Y ),wehave an exact sequence in C(A) µ O → Y −→u M(u) −→πu TX → O 0 S where µu =[1 ], πu =[10].Moreover, the above sequence belongs to C(B). Lemma 6.6. For X ∈ C(B),wehave I(X) ∼ (Xn+1⊕Xn, 00: Xn+1⊕Xn → Xn+2⊕Xn+1). = 1Xn+1 0 Proof.

n d Xn+1⊕Xn −−−−−M (1X→) Xn+2⊕Xn+1     αn αn+1

n Xn+1⊕Xn −−−−δ → Xn+2⊕Xn+1 n n 1Xn+1 dX n 00 where α = and δ = 1 0 . 01Xn Xn+1

Lemma 6.7. The category (C(B), SC(B)) is an exact category. ∗ Proposition 6.8. The category (C (B), SC∗(B)) is a Frobenius category, where ∗ = nothing, +, −,b. Proof. Let X ∈ C(B), then by Lemma 6.6 we have ∼ n I (X ) = I (X )[−n]. n∈Z where Xn is a stalk complex of degree 0 (Note that the above biproduct exists by Ex- n ercise 6.18). It is easy to see that I (X )[−n]isSC(B)-projective and SC(B)-injective, and then I (X )isSC(B)-projective and SC(B)-injective. For any SC(B)-injective complex Y ,byPropositi on 6.5, Y is a direct summand of I(Y ), and hence Y is SC(B)-projective. Similarly, any SC(B)-projectivecomplex is SC(B)-injective. Ac- cording to Proposition 6.5, it is easy to see that C(B)hasenough SC(B)-injectives and enough SC(B)-projectives. ∗ Deﬁnition 6.9 (Homotopy Category). A homotopy category K (B)ofB is the sta- ∗ ble category of (C (B), SC∗(B))bythefullsubcategory IC∗(B) of SC∗(B)-injective objects, where ∗ =nothing,+, −,b. 26 JUN-ICHI MIYACHI

Remark 6.10. For u ∈ HomC(B)(X ,Y ), we have a commutative diagram µ O −−−−→ X −−−−X→ I(X) −−−−πX→ TX −−−−→ O     u x

µ O −−−−→ Y −−−−u → M(u) −−−−πu → TX −−−−→ O 10 S where x =[0 u ], with allrowsin C(B).BytheproofofProposition 6.8, the definition of M(u)coincideswith the one of Definition 5.6. ∗ Proposition 6.11. Acategory 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) u v w belongs to SC∗(B),thenitcan be embedded in a triangle X −→ Y −→ Z −→ TX in ∗ K (B),where∗ = nothing, +, −,b. Proof. By Proposition 5.8. Definition 6.13 (Homotopy Relation). Two morphisms f,g ∈ HomC(B)(X ,Y ) is said to be homotopic (denotebyf g)ifthereisacollection of morphisms h h =(hn), hn : Xn → Y n−1 such that n − n n−1 n n+1 n f g = dY h + h dX ∈ Z ∈ ∗ B for all n .ForX ,Y C ( ), HtpC(B)(X ,Y )isthesubgroup of HomC(B)(X , Y )consisti ng of morphisms which are homotopic to 0. Proposition 6.14. For a morphism f ∈ HomC(B)(X ,Y ) (∗ = nothing, +, −,b), the following areequivalent. ∈ 1. f HtpC(B)(X ,Y ). µ 2. f factors through X −−→X I(X). π − 3. f factors through I(T −1Y ) −−−−T 1→Y Y . 4. f ∈IC(B)(X ,Y ). I In particular, HtpC(B)(X ,Y )= C(B)(X ,Y ). Proof. 1 ⇔ 2. Let h =(hn)beahomotopy morphism f 0, and let φ =(φn): h n X → Y , φ =[hn+1 f n ]. Then for all n ∈ Z we have −dn 0 n n n+1 n X φ dM(1 ) =[h f ] n−1 X 1Xn dX − n+1 n n n n−1 =[ h dX + f f dX ] n n n−1 n−1 =[dY h dY f ] n−1 n−1 = dY φ , n n 0 hn+1 f n φ µX =[ ] 1Xn = f n. n Conversely, let φ =(φ ):X → Y be a morphism such that f = φµX .Bythe same calculati on in the above,thereisahomotopy morphism h =(hn), hn : Xn → Y n+1 such that n n−1 n n+1 n f = dY h + h dX ∈ Z ∈ for all n .Thusf HtpC(B)(X ,Y ). DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 27

2 ⇔ 4. Since I (X )isIC(B)-injective, 2 ⇒ 4istri vial. Conversely, since µX : X → I(X)isanadmissiblemonomorphism, it is also trivial. 3 ⇔ 4. The same as 2 ⇔ 4.

Corollary6.15. 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) HomC(B)(I (X ),Y ) → HomC(B)(X ,Y ) → HomK(B)(X ,Y ) → 0, −1 HomC(B)(X ,I (T Y )) → HomC(B)(X ,Y ) → HomK(B)(X ,Y ) → 0. Then itiseasy.

Corollary6.16. Let B be another additive category, and F : C(B) → C(B) an additive functor. If F satisﬁes the conditions

(a) there exists anfunctorialisomorphism α : FTC(B) → TC(B)F , (b) for any morphism u : X → Y in C(B),wehave a commutative diagram

Fu Fµu αXFπu FX −−−−→ FY −−−−→ F M (u) −−−−−→ T B FX  C( )  s

Fu µFu πFu FX −−−−→ FY −−−−→ M (Fu) −−−−→ TC(B)FX , then F induces a ∂-functor F : K(B) → K(B ). Remark 6.17. By the proof of Proposition 6.14, X belongs to IC(B) if and only if X is a direct summand of I (X ). Hence it is easy to see that any object of IC(B) is isomorphic to I(Z)forsomeZ ∈ C(B). ≥ → n−1 → n Exercise 6.18 (Biproduct). Let n 0, and let Xi : ... Xi Xi be com- − plexes of C (B)indexedbyi ∈ N. Prove the following. i ∼ i B i B 1. i∈NT Xi = i∈NT Xi in C( ). Thus i∈NT Xi exists in C( ). i ∼ i B i B 2. i∈NT Xi = i∈NT Xi in K( ). Thus i∈NT Xi exists in K( ). ≥ 0 → 1 → → n b B Let n 0, and let Yi : Yi Yi ... Yi be complexes of C ( )indexed by i ∈ Z.Provethefollowing. i ∼ i B i B 1. i∈ZT Yi = i∈ZT Yi in C( ). Thus i∈ZT Yi exists in C( ). i ∼ i B i B 2. i∈ZT Yi = i∈I T Yi in K( ). Thus i∈ZT Yi exists in K( ). Proposition 6.19. Let R be a commutative complete local ring, A a ﬁnite R- algebra, and B is a Krull-Schmidt subcategory of mod A.ThenKb(B) is also a Krull-Schmidtcategory.

b 0 1 n Proof. Let X ∈ C (B). Since we may assume that X = X → X → ... → X , n i EndCb(B)(X )isasubringof i=0 EndA(X ), and hence EndCb(B)(X )issemiper- b B fect. It is clear that any idempotent of EndCb(B)(X )splits. Therefore C ( )isa Krull-Schmidt category. According to Theorem 3.13, we complete the proof. 28 JUN-ICHI MIYACHI

A n n Definition 6.20. Let be an abelian category. For a complex X =(X ,dX : n → n+1 A A ∈ Z X XX )n∈Z of ,wedefine an objects of for all n n n Z (X )=Ker dX n n−1 B (X )=ImdX n n−1 C (X )=CokdX Hn(X)=Zn(X)/ Bn(X) this is called nth cohomology, and define complexes n Z (X )=(Z (X ), 0)n∈Z n B (X )=(B (X ), 0)n∈Z n C (X )=(C (X ), 0)n∈Z n H (X )=(H (X ), 0)n∈Z. n n n ∈ Z AcomplexX =(X ,dX )iscalledanacyclic complex if H (X )=0forall n . Remark 6.21. Since we have a commutative diagram O −−−−→ Bn(X) −−−−→ Xn −−−−→ Cn(X) −−−−→ O    

O −−−−→ Zn(X) −−−−→ Xn −−−−→ Bn+1(X) −−−−→ O, where all rows are exac t, by snake lemma, we have a short exact sequence O → Hn(X) → Cn(X) → Bn+1(X) → O. Exercise 6.22. Let A be an abelian category, and P aprojectiveobjectofA.For X ∈ C(A), show that ∼ i HomK(A)(P, X [i]) = HomA(P, H (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).Thenwehave the induced long exact sequence ...→ Hn(X) → Hn(Y ) → Hn(Z) → Hn+1(X) → ... Proof. According to snake lemma, we have a commutati ve diagram Cn(X) −−−−→ Cn(Y ) −−−−→ Cn(Z) −−−−→ O      

O −−−−→ Zn+1(X) −−−−→ Zn+1(Y ) −−−−→ Zn+1(Z) where all rows are exact. Then we get the exact sequence Hn(X) → Hn(Y ) → Hn(Z) → Hn+1(X) → Hn+1(Y ) → Hn+1(Z), by snake lemma.

Remark 6.24. Let u be a morphism of HomC(A)(X ,Y ). According to Proposi- ti on 6.5, we haveanexactsequence in C(A) µ O → Y −→u M(u) −→πu TX → O. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 29

Then we have the induced long exact sequence

n n+1 ...→ Hn(X) −→δ Hn(Y ) → Hn(M(u)) → Hn+1(X) −δ−−→ ... . Moreover, it is not hard to see that δn =Hn(u)foralln ∈ Z (c f. Proposition 10.4). Lemma 6.25. For n ∈ Z,ThefunctorHn : C(A) →Afactors through K(A).

Proof. According toRemark6.17, all objects of IC(B) are acyclic. Then by Propo- sition 6.14, it is trivial. u v w Proposition 6.26. If X −→ Y −→ Z −→ TX is a triangle in K(A),thenwehave the induced long exact sequence Hn (u) Hn(v) Hn(w) ... → Hn(X) −−−−→ Hn(Y ) −−−−→ Hn(Z) −−−−→ Hn+1(X) → ... . Proof. According to Remark 6.10, for a representative u we have a commutative diagram O −−−−→ X −−−−→ I(X) −−−−→ TX −−−−→ O     u

O −−−−→ Y −−−−v → M(u) −−−−w → TX −−−−→ O

where all rowsbelongtoSC(A),withv = µu, w = πu.ByProposition 6.23, we have the induced long exact sequence ...→ Hn(X) → Hn(Y ) → Hn(M (u)) → Hn+1(X) → ... . By Remark 6.24, Proposition 6.25, we get the statement.

7. Quotient Categories Deﬁnition 7.1 (Multi plicativeSystem). A multiplicative system in a category C is a collection S of morphisms in C which satisﬁes the following axioms:

(FR1) (1) 1X ∈ S for every X ∈C. (2) For s, t ∈ S,ifst is deﬁned, then st ∈ S.

(FR2) (1) Everydiagram in C X −−−−s → Y   f

X with s ∈ S,canbecompleted 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 30 JUN-ICHI MIYACHI

with t ∈ S,canbecompleted to a commutative square X −−−−s → Y     f g

X −−−−t → Y with s, t ∈ S. (FR3) For f, g ∈ HomC (X, Y )thefollowi ng are equival ent. (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 systemofacategory C. Definition 7.2 (Saturated Multiplicative System). Amultiplicative system S in a category C is called saturated if it satisfies the following axiom: (FR0) For a morphism s in C,ifthereexistf,g such that sf, gs ∈ S,thens ∈ S. Definition 7.3. For a morphism f : X → Y ,wesetsource(f)=X and sink(f)= Y . For a multi plicative system S,eachX ∈C, SX is a category defined by 1. Ob(SX )={s ∈ S | source(s)=X}, { ∈ | } ∈ X 2. HomSX (s, s )= f HomC (si nk(s), si nk(s )) s = fs for s, s Ob(S ), and SX is a category defined by

1. Ob(SX )={t ∈ S | si nk(t)=X}, { ∈ | } ∈ 2. HomSX (t, t )= f HomC (source(t), source(t )) t = t f for t, t Ob(SX ). Lemma 7.4. For any X ∈C, SX satisfies the following axioms: ∈ ∈ ∈ X (L1) For any f1 HomSX (s, s1 ), f2 HomSX (s, s2 ),thereexists S and ∈ ∈ g1 HomSX (s1 ,s ), g2 HomSX (s2 ,s ) such that g1f1 = g2f2. ∈ ∈ X ∈ (L2) For any f1,f2 HomSX (s, s ),thereexists S and g HomSX (s ,s ) such that gf1 = gf2. (L3) SX is connected. Definition 7.5. For X, Y ∈C,wedefine a covariant functor hX ◦ si nk : SY → Set X Y where h ◦ si nk(s)=HomC (X,si nk(s)) for s ∈ S ,and acontravariant functor

hY ◦ source : SX → Set

where hY ◦ source(t)=HomC(source(t),Y)fort ∈ SX . Lemma 7.6. Let X,Y ∈C.Deﬁne a relation , on the collection Y {(f,s)|s ∈ S ,f ∈ HomC(X,sink(s))} ∼ ∈ ∈ as follows: (f1,s1) (f2,s2) if and only if there exist h1 HomSY (s1,s),h2 ∼ HomSY (s2,s) such that (h1f1,s)=(h2f2,s).Then is an equivalence relation and we have X Y colim h ◦ sink = {(f,s)|s ∈ S ,f ∈ HomC(X, sink(s))}/ ∼ . SY Y (Write the equivalence class [(f,s)] for (f, s),wheref ∈ HomC(X, si nk(s)), s ∈ S .) DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 31

Remark 7.7 (Set-Theoretic Remark). In the above, wedealt with SY as a small category (i.e. Ob(SY )isaset). 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 SY satisfying the following (this category is called a coﬁnal subcategory):

For any Y ∈C, SY satisﬁes the following axiom: (Co) For any s ∈ SY ,thereexists a morphism f : s → s with s ∈ S’Y .

Then colim hX ◦ si nk exists, and we have S’Y colim hX ◦ si nk = colim hX ◦ si nk . SY S’Y Lemma 7.8. For any X, Y, Z ∈C we have a well-deﬁned mapping colim hX ◦ si nk ×colim hY ◦ si nk → colim hX ◦ si nk SY SZ SZ which is deﬁned as follows: with eachpair([(f,s)], [(g,t)]),sinceby(FR2)there exist s ∈ S with source(s )=sink(t), g ∈ HomC(sink(s), si nk(s )) such that g s = sg,weassociate the equivalence class [(gf,st)]. Sketch. X A Y C Z AA CC AAf CCg AA s CC t A C! Y BB Z BBg B BB s B! Z

Deﬁnition 7.9 (Quotient Category). We deﬁne a category S−1 C,calledthequo- tient category of C,asfollows: − 1. Ob(S 1 C)=Ob(C). 2. For X, Y ∈ Ob(C), the morphism set is given by X − ◦ HomS 1 C (X, Y )=colim h sink . SY 3. For X, Y, Z ∈ Ob(S−1 C), the law of composition is given by

× − → − HomS−1 C (X, Y ) HomS 1 C(Y,Z) HomS 1 C (X, Z), (([(f, s)], [(g, t)]) → [(gf, st)]), ∈ − where [(g ,s)] HomS 1 C(sink(s), si nk(t)) with g s = s g. −1 4. The identity of X ∈ Ob(S C)isgivenbytheequival ence class [(1X , 1X )]. Deﬁnition 7.10 (Quoti ent Functor). We have a functor Q : C→S−1 C,calledthe quotient functor,suchthat

Q(X)=X for X ∈C,Q(f)=[(f, 1Y )] for f ∈ HomC (X, Y ). Proposition 7.11 (Basic Properti es). The following hold. 1. Q : C→S−1 C sends null objects to null objects. 32 JUN-ICHI MIYACHI

2. For f, g ∈ HomC (X, Y ) the following are equivalent. (1) Q(f) = Q(g). (2) There exists s ∈ SY such that sf = sg. (3) There exists t ∈ SX 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 −1 Y − { | ∈ ∈ } HomS 1 C (X, Y )= Q(s) Q(f) s S ,f HomC (X, si nk(s)) −1 = {Q(g)Q(t) |t ∈ SX ,g ∈ HomC (source(t),Y)}.

4. For f ∈ HomC(X,Y ) the following are equivalent. (1) Q(f) is an isomorphism. (2) There existmorphisms g, h in C with gf, fh ∈ S. 5. Assume S is saturated. Then for any f ∈ HomC (X, Y ) the following hold. (1) Q(f) is an isomorphism if and only if f ∈ S. (2) If there exists s ∈ SY with sf ∈ S,thenf ∈ S. (3) If there exists t ∈ SX with ft ∈ S,thenf ∈ S. 6. For any X,Y ∈C we have a bijection X ζ = ζX,Y : colim h ◦ sink → colim hY ◦ source SY SX which associates with each [(f,s)] the equivalence class of (t, g) with ft = sg, and its inverse X η = ηX,Y : colim hY ◦ source → colim h ◦ si nk SX SY 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 afunctorsuch that F(s) is an isomorphism for all s ∈ S.Thenthereexists auniquefunctor F : S−1 C→C such that F = FQ. C DD DD DDF Q DD D! − _ _ _/ S 1 C C F Sketch. Since every object of S−1 C is of theformQX for X ∈C,wecandeﬁne F : S−1 C→C as follows. Let F (QX )=F (X)forQX ∈ S−1 C and F([(f, s)]) = − 1 ∈ − (Fs) Ff for [(f,s)] HomS 1 C (QX, QY ). Then we have F = FQ 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(FQ,GQ), (ζ → ζQ). Proof. By the proof of the above lemma, it is easy. − Corollary7.14. Let S˜ = {f|Q(f) is an isomorphism in S 1 C}.ThenS˜ is a sat- −1 urated multiplicative system, and S˜ C is equivalent to S−1 C. − Remark 7.15. Considering S 1 C,byCorollary 7.14, we may assume S is a saturated multi plicati ve system. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 33

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: Y (a) For any s ∈ S with Y ∈D,thereexistsf ∈ HomC (si nk(s),Y ) with Y ∈D such that fs ∈ S. (b) For any t ∈ SX with X ∈D,thereexists g ∈ HomC(X , source(t)) with X ∈Dsuch that tg ∈ S. − Then the canonical functor (S ∩D)−1D→S 1 C is fully faithfu l, so that (S ∩D)−1D can be considered as a full subcategory of S−1 C. Proof. By the samereasonofRemark 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.

−1 Proof.It is easy to see that S C sati sfies the condition of Definiti on 2.1. For n n −1 C X = i=1Xi,byProposition 2.3 we have FX = i=1FXi in S . Therefore, − S 1 C is an additive category and Q is an additive category by Definition2.5, Proposition 2.10.

8. Quotient Categories of Triangulated Categories Deﬁnition 8.1 (Compatible with Tri angl e). Amultiplicative system S in a triangulated category C is said to be compatible with thetriangulation if it satisﬁes the following axioms: (FR4) For a morphism u in C, u ∈ S if and only if Tu ∈ S. (FR5) For triangles (X, Y, Z,u, v, w), (X,Y,Z,u,v,w)andmorphisms f : X → X, g : Y → Y in S with gu = uf,thereexistsh : Z → Z in S such that (f,g,h)isahomomorphism of triangles. Throughout this section, C is a triangulated category, we assume that S is a saturated multi plicati vesystemofC which is compatible with the triangulation, and Q : C → S−1 C is a quotient functor.

−1 −1 − C→ C Lemma 8.2. There exists a unique auto-functor TS 1 C : S S such that QTC = TS−1 C Q.

We si mply write T for TS−1 C. Lemma 8.3. Let (X, Y, Z,u, v, w), (X,Y,Z,u,v,w) be triangles in C,andlet ∈ − ∈ − α HomS 1 C (QX, QX ), β HomS 1 C(QY, QY ) such that (Qu )α = βQ(u). ∈ − Then there exists γ HomS 1 C (QZ, QZ ) such that

Qu Qv Qw QX −−−−→ QY −−−−→ QZ −−−−→ TQX         α β γ Tα

Qu Qv Qw QX −−−−→ QY −−−−→ QZ −−−−→ TQX is commutative. → → → → Proof. There are f : X X1, s : X X1, g : Y Y , t : Y Y1 such −1 −1 ∈ → that α =(Qs) Qf, β =(Qt) Qg and s, t S. Then there are u1 : X1 Y2 , 34 JUN-ICHI MIYACHI

→ ∈ s1 : Y Y2 such that u1s = s1u , s1 S.Since(Qu )α = βQ(u), there are → → ∈ s2 : Y2 Y3 , t1 : Y1 Y3 such that s2u1f = t1gu, s2s1 = t1t and s2s1 S.

u / X A X A X Y @ Y AA AA @@ AAf AAu @@g AA s AA @@ t A A @ X1 @ Y Y1 @@ @u1 @@ s1 t1 @ Y2 Y3

s2 Y3 → → Therefore there are h : Z Z1, s3 : Z Z1 such that we have acommutative diagram X −−−−u → Y −−−−v → Z −−−−w → TX         f t1g h Tf

s u X −−−−2 1→ Y −−−−v → Z −−−−w → TX 1 2 1 1     s s2s1 s3 Ts

X −−−−u → Y −−−−v → Z −−−−w → TX. −1 −1 −1 with s3 ∈ S.Since (Qs2s1) (Qt1)Qg =(Qt1t) (Qt1)Qg =(Qt) Qg = β,let −1 γ =(Qs3) h,thenγ sati sﬁes the statement.

− Deﬁnition 8.4 (Tri angulation). Asextuple(QX ,QY,QZ,λ,µ,ν)inS 1 C is called a triangle if there exists a triangle (X, Y, Z, u, v, w)inC such that (QX ,QY,QZ,λ,µ,ν)isisomorphic 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.

−1 Proof. Since for any morphism α : QX → QY in S C there are f : X → Y1, s : Y → Y1, t : X1 → Y , g : X1 → Y such that we have commutative diagram

Qt X X ←−−−− X   1    Qf α Qg

Qs Y1 ←−−−− Y Y with s, t ∈ S,itiseasy.

Proposition 8.6. Let A be an abelian category and H : C→Aacohomological functor such that H(s) are isomorphisms for all s ∈ S.Thenthere exists a unique − cohomological functor H : S 1 C→Asuch that H = HQ. Proposition 8.7. Let D be another triangulated category and F =(F, θ):C→D a ∂-functor such that Fsare isomorphisms for all s ∈ S.Thenthereexistsaunique DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 35

∂-functor F =(F,θ):S−1 C→Dsuch that F = FQ and θ = θQ. C DD DD DDF Q DD D! − _ _ _/ S 1 C C F 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(FQ,GQ), (ζ → ζQ). Proof. By Proposition 7.13, it remains to check the following commutativity. For X ∈C, ψ ∈ ∂ Mor(FQ,GQ), we have θQ FT −1 QX FQTC X −−−−→ TDFQX S C     ψTC TD ψ −−−−→ GTS−1 CQX GQTC X TDGQX. ηQ

9. Ep´ aisse Subcategories Definition 9.1 (Epaiss´ e Subcategory). An épaisse subcategory U of a triangulated category C is a tri angulated fullsubcategory of C such that if u ∈ HomC(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,thenX, Y ∈U. Proposition 9.2. For a triangulated fu l l subcategory U of C,thefollowing are equivalent. 1. U is an épaisse subcategory of C. 2. U is closed under direct summands. Proof. 1 ⇒ 2. Let X, Y ∈Csuch that X⊕Y ∈U.ByProposition 4.8, we have atriangle (T −1Y,X,X⊕Y,0,µ,π). Since 0 : T −1Y → X factors through O ∈U, T −1Y, X ∈U,andhence Y, X ∈U. 2 ⇒ 1. Let (X, Y, Z,u, v, w)beatriangle such that Z ∈Uand u factors through Y ∈U,thenwehaveamorphism of tri angl es X −−−−u → Y −−−−→ Z −−−−→ TX     u

X −−−−u → Y −−−−→ Z −−−−→ TX. By Proposition4.8,wehaveatri angle (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)thecollection of morphisms u in U.suchthatM(u) ∈U. Lemma 9.4. Let U be an épaisse subcategory of a triangulated category C.Fora morphism f in C,thefollowing are equivalent. 1. f factors through some object of U. 36 JUN-ICHI MIYACHI

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,thenwehaveacommutativediagram X X     u f

U −−−−x → Y −−−−s → Z −−−−z → TU. where the bottom row is atriangle, and with s ∈ Φ(U). We have sf =0.Con- versely, if sf =0withs ∈ 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 ´epaisse subcategory of a triangulated category C. Then Φ(U) is a saturated multiplicative system which is compatible with the triangulation. Proof. We use the followingdiagramtocheck the axioms of a multiplicative system.

X −−−−u → Y −−−−i → Z −−−−i → TX     v z

X −−−−w → Z −−−−k → Y −−−−k → TX       j y Tu

j X X −−−−→ TY     j x

TY −−−−Ti→ TZ 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

j j Y −−−−u → Z −−−−→ X −−−−→ TY     r l p q Y −−−−ru→ U −−−−→ V −−−−→ TY with V ∈U.Since x =(Ti)j =(Ti)ql, x factors through V ∈U.Sinceuv ∈ Φ(U), we have Diagram A with Y ∈U. Therefore, X,Z ∈U,andhence u ∈ Φ(U). (FR1) (1) Since O ∈U,itistrivial.(2)Ifu, v ∈ Φ(U), then we have Diagram A with Z,X ∈U. Then Y ∈U,andhence vu ∈ Φ(U). (FR2) (1) Given v ∈∈ Φ(U), i,wehaveDiagramAwithX ∈U. Then z ∈ Φ(U). (2) Given z ∈ Φ(U), k,wehaveDiagramAwithX ∈U. Then v ∈ Φ(U). (FR3) By Lemma 9.4. (FR4) Itistrivial. (FR5) By Proposition 4.9, it is easy. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 37

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,whereQ : 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)betheimage of a triangle (X,Y,Z,u, v, w) of C.IftwoobjectsofQX, QY, QZ are O,thentherestone is clearly O. Then it is easy to see that Ψ(S)isatri angulated full subcategory of C.IfZ ∈ 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)−1C.Inthiscase,we say that 0 →U→ C→C/U→0isanexactsequence of tri angulatedcategories. 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,thefollowingareequivalent. 1. For every f : X → Y ∈ Φ(U), HomC(f,M):HomC (Y, M) → HomC(X,M) is bijective. 2. HomC(U,M)=0. 3. For every X ∈C, QX,M :HomC (X, M) → HomC/U (QX, QM ) is bijective.

Proof. 1 ⇒ 2. For every object U ∈U, O → U −→1 U → O is a tri angle. Then ∼ 0=HomC(O, M) = HomC(U, M). 2 ⇒ 3. Every morphism of HomC/U (QX, QM)isrepresented by a diagram X B BB BBf s BB B! XM where s is contained inatri angl e U → X −→s X → TU with U ∈U. Then there exists f : X → M in C such that f = f s,because HomC(U, M)=0.Hence QX,M is surjective. Let U → X → X → TU be a triangle with U ∈U.Ifamorphism g : X → M sati sﬁes gs =0,thenthereexistu : TU → M such that g = ut. Therefore g =0,becauseu ∈ HomC(U, M)=0.HenceQX,M is injective. 3 ⇒ 1. Let f : X → Y be a morphism in Φ(U). Then we have the following commutative diagram Hom(f,M) HomC (Y, M) −−−−−−−→ HomC (X, M)     QY,M QX,M

Hom(Qf,QM) HomC/U (QY, QM) −−−−−−−−−→ HomC/U (QX, QM )

According to 3, QX,M and QY,M are bijective. Since QU =0,Hom(Qf, QM)is bijective. Hence Hom(f,M)isbijective. 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 Q 9.8 . Let 0 →U→C−→C/U→0beanexactsequence of triangulated categories. 38 JUN-ICHI MIYACHI

The right (resp., left)adjoint of Q is called a section functor.Ifthereexists a section functor S,then{C/U; Q, S} is called a localization (resp., colocalization)of Q C,and0→U→C−→C/U→0iscalled 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 commutati ve diagram Hom(f,SV ) HomC(Y,SV ) −−−−−−−→ HomC(X,SV )    

Hom(Qf,V ) HomC/U (QY, V ) −−−−−−−→ HomC/U (QX, V ) Therefore Hom(f,SV )isanisomorphism.ByProposition9.8,SV is U-local.

Proposition 9.11. Let {C/U; Q, S} be a localization of C,andτ : QS → 1C/U and σ : 1C → SQ adjunction arrows. Then the following hold. 1. τ is an isomorphism (i.e. S is fully faithful ). 2. For every object X ∈C,thetriangleU → X −σ−X→ SQX → TU satisﬁes that U is in U. Proof. 1. For every X ∈Cand Y ∈C/U,wehaveacommutativediagram

HomC (X, SY ) HomC (X, SY )     QX,SY

Hom(QX,τY ) HomC/U (QX, QSY ) −−−−−−−−→ HomC/U (QX, Y ). By Proposition 9.8 and Lemma 9.10, Q is an isomorphism. Then Hom(QX, X,SY ∼ τY )isanisomorphism.ForanyZ ∈C/U,thereexists X ∈Csuch that Z = QX . Hence τ is an isomorphism. 2. It suﬃces to show that for any X ∈C, QσX is an isomorphism. By the QσX τQX property of adjunction arrows, we have QX −−−→ QSQX −−−→ QX = 1QX ,and hence QσX is an isomorphism. ∼ Corollary9.12. Let M ∈C.ThenM 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.ThenF induces an equivalence between C/ Ker F and C. Proof. By the universal property of Q : C→C/ Ker F ,wehave the following commutative diagram C H HH HH HFH Q HH H# C / / Ker F C F If f : X → Y is a morphism in C,thenFf is an isomorphism if and only if Qf is an isomorphism. For every object M ∈C, FM → FSFM is an isomorphism, and then QM → QSF M is an isomorphism. Therefore Q → QSF is an isomorphism. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 39

∼ By the universal property of Q and QSF = QSF Q,wehave1C/ Ker F = QSF . ∼ Since, F QS = FS = 1C , 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. HomC(U, V)=0. 3. For every X ∈C,thereexists a triangle U → X → V → TU with U ∈U and V ∈V. Proposition 9.15. Let C be a triangulated category, (U, V) astablet-structure in C.Thenthefollowing hold. 1. For X ∈C, HomC(X,V)=0if and only if X is isomorphic to an object of U. 2. For Y ∈C, HomC(U,Y)=0if and only if Y is isomorphic to an object of V. 3. Let U˜ be the full subcategory of C consisting objects which are isomorphic to objects of U.ThenU˜ is an épaisse subcategory of C. 4. Let V˜ be the full subcategory of C consisting objects which are isomorphic to objects of V.ThenV˜ is an épaisse subcategory of C. Proof. 1. For X ∈C,wehaveatri angle

τX σX UX −−→ X −−→ VX → TUX. ∼ −1 −1 If HomC (X, V)=0,thenσX =0andUX = X⊕T VX . Therefore, T VX = O ∼ −1 and X = UX ,becauseof HomC(UX,T VX)=0. 2. Similarly. 3, 4. By 1, 2, it is trivial.

Proposition 9.16. Let C be a triangulatedcategory. If {V ; Q, S} is a localization of C,thenS is fully faithful, (U,SV) is a stable t-structure, where U =KerQ. Conversely, if (U, V) is a stable t-structure in C,thenthe canonical inclusion S : V→Chas a left adjoint Q such that {V; Q, S} is a localization. Proof. Let {V; Q, S} be a localization of C. Then, by ∼ HomC(U,SV) = HomC(QU, V) =0 and Proposition 9.13, it is clear that S is fully faithful and (U,SV)isastable t-structure. Conversely, let (U, V)beastable t-structure in C.ForX ∈C,let UX → X → VX → TUX be tri angl e such that UX ∈Uand VX ∈V.Since HomC(U, V)=0andU and V arestable under translations, for any V ∈V,we have an isomorphism ∼ HomC(VX ,V) = HomC (X, V ). AccordingtoTheorem 1.18, S : V→Chas a left adjoint Q such that {V; Q, S} is alocalization.

Remark 9.17. Similarly, if (U, V)isastable t-structure in C,thenthere is a functor Q : C→Usuch 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 (SU, Ker Q)isastable t-structure in C. 40 JUN-ICHI MIYACHI

10. Derived Categories Throughout this section, A is an abelian category. ∗ Definition 10.1. For X ,Y ∈ K (A), a morphism u ∈ HomC(A)(X ,Y )iscalled a quasi-isomorphism if Hn(u)areisomorphisms for all n ∈ Z, where ∗ =nothing, +, −,b. ∗ ∗ K ,φ(A)isafullsubcategory of K (A)consisti ng of complexes of which allho- mologies are O, where ∗ =nothing,+, −,b. Proof. It is easy to see that K∗,φ(A)isanépaisse subcategory of K∗(A). By Propo- sition 6.26, it is easy. ∗ Definition 10.2 (Derived Category). The derived category D (A)ofanabelian category A is K∗(A)/ K∗,φ(A), where ∗ =nothing,+, −,b. Remark 10.3. For two mor phism s f, g : X → Y in C(A), f = g ⇒ f g ⇒ h ∼ n ∼ n f = g in D(A) ⇒ H (f) = H (g)forall 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) Qu Qv Qw QX −−→ Y −−→ QZ −−→ TQX. Proof. According to Remark 6.10, we have a commutative diagram in C(A) OO    

O −−−−→ X −−−−→ I(X) −−−−→ TX −−−−→ O     u x

O −−−−→ Y −−−−v → M(u) −−−−w → TX −−−−→ O     v s

Z Z    

OO where all rows and columns are exact. Then X −→u Y −v→ Z −→w TX is a triangle in K(A). Since I(X) ∈ Kφ(A), by Proposition 6.23, s is a quasi-morphism, and hence we have a commutative diagram in D(A)

Qu Qv Qw QX −−−−→ QY −−−−→ Q M(u) −−−−→ TQX   Qs

− Qu Qv Qw(Qs) 1 QX −−−−→ QY −−−−→ QZ −−−−−−−→ TQX.

Deﬁnition 10.5. Afullsubcategory A of A is called a thick abelian full subcategory if A is an abelianexact full subcategory which is closed under extensions. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 41

∗ ∗ For ∗ =nothing, +, −,b,wedenote by KA (A)thefullsubcategory of K (A) consisting of complexes X ∈ K(A)withHn(X) ∈A for all n ∈ Z. ∗ ∗ ∗,φ Moreover, we set DA (A)=KA (A)/ K (A), where ∗ =nothing, +, −,b. Deﬁnition 10.6 (Truncati ons). For a complex X =(Xi,di), we deﬁne the following truncations: n n+1 n+2 σ>nX : ...→ 0 → Im d → X → X → ... , n−2 n−1 n σ≤nX : ...→ X → X → Ker d → 0 → ... , n−1 n+1 n+2 σ ≥nX : ...→ 0 → Cok d → X → X → ... , n−2 n−1 n−1 σ n(X ) → O O → σ nX )= i H (X )ifi>n

i O if in i i H (X )ifin. − Then the morphism Y → σ≤n(Y )isaquasi-isomorphism, and σ≤n(Y ) ∈ K (A). For the other cases, similarly. Let Inj A (resp., Proj A)bethefullsubcategory 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 HomK(A)(X ,I )=0. (resp., HomK(A)(P ,X )=0) Corollary 10.9. The following hold. 42 JUN-ICHI MIYACHI

1. If A has enough injectives, then we have an isomorphism ∼ HomK(A)(X ,I ) = HomD(A)(X ,I ) for X ∈ K(A), I ∈ K+(Inj A). 2. If A has enough projectives, then we have an isomorphism ∼ HomK(A)(P ,Y ) = HomD(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 collectionofobjectsofA such that every object X ∈Ais a image of an epimorphism from some object of L.Thenfor 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.Thenfor − − any X ∈ KA (A),thereexistsP ∈ 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 Xi = O for all i>0. − By thebackward induction on n,weconstruct a complex P ∈ K (Proj A). as follows. Let Zn and Zn be Zn(P )andZn(X), respectively. Assumewehave a commutative diagram Zn / /P n

Zn / /Xn We take a pull back M n of Xn−1 → Zn ← Zn,andtakeanepimorphism from − n ∼ n P n 1 → M n.ThenbyProposition 2.19, H (P ) = H (X )and the induced morphism Zn(P ) → Zn(X)isepic. n−1 / / − // − /n / / Z P n 1 M n 1 Z P n

PB Zn−1 / /Xn−1 Xn−1 /Zn / /Xn

− i 2. Given an object X ∈ KA (A), we may assume that X = O for all i>0. By − thebackward induction on n,weconstruct a complex P ∈ K ((Proj A) ∩A). as follows. Let B n and Bn (resp., Cn and Cn)beZn(P )andZn(X)(resp.,Cn(P ) and Cn(X), respectively. Assume we have a commutati ve diagram P n //Cn

Xn //Cn − where P n,Cn ∈A. Then B n ∈A.WetakeapullbackCn 1 of Cn−1 → Bn ← n n−1 ∼ n−1 B . Then by Propositi on 2.19, H (P ) = H (X ). Since A is closed under − extensions, Cn 1 ∈A. Therefore we can take an epimorphism from P n−1 → DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 43

− Cn 2,withP n−1 ∈ (Proj A) ∩A.SinceP n−1 is projective, we have a morphism P n−1 → Xn−1,andwehaveacommutative diagram − //n−1 // n−1 / / //n P n 1 C B P n C

PB Xn−1 //Cn−1 //Bn−1 / /Xn //Cn

Proposition 10.11. The following hold. 1. If A has enough projectives, then − ∼t − K (Proj A) = D (A). 2. If A has enough injectives, then t + ∼ + K (Inj A) = D (A). 3. Let A be a thick abelian full subcategories of A such that A has enough A-projectives in A.Thenwehave

∗ ∼t ∗ D (A ) = DA (A) where ∗ = −,b. 4. Let A be a thick abelian full subcategories of A such that A has enough A-injectives in A.Thenwehave

∗ ∼t ∗ D (A ) = DA (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 Propositi on 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 aquasi-isomorphisminK (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 complexesofwhich homologies are bounded. Corollary 10.13. The following hold. 1. If A has enough projectives, then t −,b ∼ b K (Proj A) = D (A). 2. If A has enough injectives, then t +,b ∼ b K (Inj A) = D (A). 44 JUN-ICHI MIYACHI

Example 10.14. For a coherent ring A,letmod A be the fullsubcategory of Mod A consisting of right coherent A-modules. Then mod A is an thick abelian full subcategory of Mod A. Therefore, we have ∗ ∼t ∗ D (mod A) = Dmod A(Mod A) ∗ − ∗ ∗ where = ,b.WeoftenwriteDc (Mod A)forDmod A(Mod A). n Deﬁnition 10.15 (Yoneda Ext). For X, Y ∈Aand n ∈ N,letExactA(X, Y )be the set of exactsequences in A of the form

Σ:O → Y → Xn−1 → ...→ X0 → X → O. n For Σ1, Σ2 ∈ ExactA(X, Y ), we write Σ1 → Σ2 if we have a commutative diagram Σ :O −−−−→ Y −−−−→ X − −−−−→ ... −−−−→ X −−−−→ X −−−−→ O 1 n 1 0    ...

Σ2 :O −−−−→ Y −−−−→ X n−1 −−−−→ ... −−−−→ X 0 −−−−→ X −−−−→ O.

And, we deﬁne Σ1 ∼ Σm if there are Σi (2 ≤ i ≤ m − 1) such that Σi Σi+1 (1 ≤ i ≤ m − 1), where means → or ←. Then ∼ is a equivalentrelationon n n n ExactA(X, Y ). We denote ExactA(X, Y )/ ∼ by ExtA(X, Y ). Proposition 10.16. For X, Y ∈Aand n ∈ N,Wehave a bifunctorial isomorphism n ∼ n ExtA(X, Y ) → HomD(A)(X, T Y ). n Proof. Let Σ ∈ ExactA(X, Y )hastheform

Σ:O → Y → Xn−1 → ...→ X0 → X → O. Then we have a commutative diagram Y [n − 1] :O −−−−−→ Y −−−−−→ O

M : O −−−−−→ Xn−1 −−−−−→ Xn−2 −−−−−→ ... −−−−−→ X1 −−−−−→ X0 −−−−−→ O

X : O −−−−−→ X −−−−−→ O φ(Σ) Therefore, we have a triangle Y [n − 1] → M → X −−−→ Y [n]. It is easy to see n that φX,Y :ExtA(X,Y ) → HomD(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 → I0 → I1 → ... be an injective resolution, that is, Y → I is a quasi-isomorphism. Then by Corol- lary 10.9, it is easy to see that n ∼ n ExtA(X,Y ) = HomD(A)(X, T Y ) ∼ n = HomD(A)(X, T I ) ∼ n = HomK(A)(X, T I ) ∼ n = H (HomA(X,I )). n The last term is ExtA(X, Y )inthesenseofstandard homological algebra. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 45

Definition 10.18. Let A be an abelian category with enough injectives. A com- ∗ plex X ∈ K (A)issaidtohavefinite injective dimension if there is n ∈ Z such that HomD∗(A)(M, X [i]) = 0 for all M ∈Aand i>n, where ∗ =nothing, +, −,b. ∗ ∗ ∗ We denote by K (A)fid the fullsubcategory of K (A)consisting of X ∈ K (A) which have finite injective dimension. ∗ Moreover, for a thick abelian full subcategory A of A,wedenote by KA (A)fid ∗ ∗ the fullsubcategoryofKA (A)consisti ng of X ∈ KA (A) which have finite injective dimension. Proposition 10.19. Let A be an abelian category with enough injectives. Then ∗ ∗ ∗ K (A)fid and KA (A)fid are quotientizing subcategories of K (A).

∗ u Proof. For u : X → Y in K (A)fid,letX −→ Y → M (u) → X [1] bea triangle. For M ∈A,byapplying HomK∗(A)(M,−)tothetri angle, we have ∗ ∗ ∗ M (u) ∈ K (A)fid. Therefore , K (A)fid is a triangulated full subcategory of K (A). ∗,φ φ ∗ By Proposition 9.2, K (A)fid = K (A) ∩ K (A)fid is an épaisse subcategory of ∗ ∗ K (A). According to Proposition 7.16, K (A)fid is a quotienti zing subcategory of ∗ ∗ K (A). In the case of KA (A)fid,similarly.

∗ ∗ ∗,φ Definition 10.20. For ∗ =nothing,+, −,b, D (A)fid = K (A)fid/ K (A)fid and ∗ ∗ ∗,φ DA (A)fid = KA (A)fid/ K (A)fid. Proposition 10.21. Let A be an abelian category with enough injectives. Then the following areequivalent for X ∈ K+(A). 1. For any integer n1 ∈ Z there is n2 ∈ Z such that HomD(A)(Y ,X [i]) = 0 for + j all i>n2 and all complexes Y ∈ K (A) with H (Y )=O for jn.We take I ∈ K+(Inj A) which has a quasi-isomorphism X → I in K+(A). For i>n, we have isomorphism s i ∼ i HomD(A)(Z (I )[−i],X ) = HomK(A)(Z (I ),I [i]) =0.

This means that the canonical morphisms Ii−1 → Zi(I)issplit epic, and Bi(I )= i b Z (I ). Then σ≤nI → I is an isomorphism in K(A)andσ≤nI ∈ K (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 triangleequivalent

t b ∼ + K (Inj A) = D (A)ﬁd. Proposition 10.23. Let B be an additive full subcategory of an abelian category A b which is closed under direct summands, and K˜ (B) the triangulated full subcategory − − of K (B) consisting of objects which are isomorphic to an object of Kb(B) in K (B). b − Then K˜ (B) is an ´epaisse subcategory of K (B). 46 JUN-ICHI MIYACHI

Proof. Let X ∈ Kb (B), and Y is direct summand of X in K−(B). Then Y is a direct summand of X⊕I (Y )inC−(B). Then we have a split exact se quence in C−(B) O → Y → X⊕I(Y ) → Z → O. − Since X ∈ Kb(B), there is n ∈ Z such that we have a split exact sequence in C (B) O → τ≤nY → τ≤nI (Y ) → τ≤nZ → O. i i i Since H (τ≤nI (Y )) = O for i = n and B (τ≤nI (Y )) ∈B,H(τ≤nY )=O for i = n i ∼ − b and B (τ≤nY ) ∈B.HenceY = σ>n−2Y in K (B)withσ>n−2Y ∈ K (B). 11. Homotopy Limits Throughout this section C is a triangulated category with arbitrary coproducts. Deﬁnition 11.1. Atriangulated full subcategory L of C is called localizing if (L1) Every direct summand of anobjectinL 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 {XI }i∈I be a collection of objects of C.Itsuﬃcestoshow

fi 1. Any collection of morphisms Xi −→ Y in C/L can be lifted to a morphism f X −→ Y in C/L. i i −→f C L −q→i −→f C L 2. a morphism iXi Y in / such that all Xi iXi Y =0in / , then f =0.

fi 1. A morphism Xi −→ Y in C/L is a diagram in C

Xi ?? ? f ?? i si ?? ? Xi Y → → → ∈L where Xi Xi Xi TXi is a tri angle, with X i .Thuswegetadiagram iXi CC C f CC i si CC C! iXi Y. → → → Since iXi iXi iXi T iXi is a triangle, we have a morphism f X −→ Y in C/L. i i −→f C L 2. Given a morphism iXi Y in / ,itcorresponds to a diagram iXiD Y DD DDf DD t D" Y DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 47 ∈ L → → C L with t Φ( ). If the composite Xi iXi Y =0in / ,thenwehavea diagraminC qi / Xi iXiE Y EE EEf EE t E" Y which corresponds to 0 in C/L. Then by Proposition 7.11 and Lemma 9.4, every −q→i −f→ ∈L Xi iXi Y factors through Zi .Thusf factorizes as / iXi iZiE Y EE EE EE t E" Y . L ∈L Since is localizing, iZi and f =0. Corollary 11.3. Let A be an abeliancategorysatisfying the condition Ab4. Then D(A) has arbitrary coproducts. Proof. According to Corollary 6.15, K(A)hasarbitrary coproducts. For a collection −f→i ∈ n ∼ n of quasi-isomorphisms Xi Yi (i I), H ( ifi) = i H (fi)isisomorphic for all n ∈ Z.ThusKφ(A)islocalizing, and Lemma 11.2 can be applied.

Deﬁnition 11.4. For a se quence {Xi → Xi+1}i∈N (resp., {Xi+1 → Xi}i∈N)of morphisms in C,thehomotopycolimit(resp., limit) of the sequence is the third (resp., second) term of the tri angle 1− shift Xi −−−−−→ Xi → hlim Xi → T Xi i i −→ i −1 1− shift (resp., T Xi → hlim Xi → Xi −−−−−→ Xi) i ←− i i

fi where the above shift morphism is the coproduct (resp., product) of Xi −→ Xi+1 fi (resp., Xi+1 −→ Xi)(i ∈ N). Exercise 11.5. In the category Ab,provethefol lowing.

fi 1. For a sequence of morphisms {X −→ X } ∈N,ifthereisn ∈ N such that f i+1 i i i ≥ −1−−−−−shift→ are epimorphisms for all i n,then iXi iXi is epic. − { −f→i } −1−−−−shift→ 2. For a sequence of morphisms Xi Xi+1 i∈N, iXi iXi is monic. Lemma 11.6. The following hold.

fi 1. Assume A satisﬁes the condition Ab3. For a sequenceofmorphisms {Xi −→ Xi+1}i∈N,ifthereisn ∈ N such that fi are split monomorphisms for all i ≥ n,thenwehaveasplitexactsequence 1− shift O → Xi −−−−−→ Xi → lim Xi → O. i i −→ 48 JUN-ICHI MIYACHI

2. Assume A satisfies the condition Ab3*.Forasequenceofmorphisms{Xi+1 fi −→ Xi}i∈N,ifthereisn ∈ N such that fi are split epimorphisms for all i ≥ n, then we have a split exact sequence 1− shift O → lim Xi → Xi −−−−−→ Xi → O. ←− i i Proof. 1, For any M ∈A,wehaveacommutative diagram 1− shift HomA( X ,M) −−−−−→ HomA( X ,M) i i i i   −1−−−−− shift→ i HomA(Xi,M) i HomA(Xi,M). By Exercise 11.5, the bottom horizontal morphism is epic. 2. Similarly. Proposition 11.7. The following hold. A → 1. Assume satisfies the condition Ab3. and Xi Xi+1 asequenceofcom- plexes in C(A) satisfying that for each j ∈ Z there is n ∈ N such that j → j ≥ Xi Xi+1 are split monomorphisms for all i n.Thenwehave an exact sequence in C(A) → −1−−−−− shift→ → → O Xi Xi lim Xi O i i −→ ∼ which belongs to S A .Inparticular,lim X hlim X in K(A). C( ) −→ i = −→ i A → 2. Assume satisfies the condition Ab3*. and Xi+1 Xi asequenceofcom- plexes in C(A) satisfying that for each j ∈ Z there is n ∈ N such that j → j ≥ Xi Xi+1 are split epimorphisms for all i n.Thenwehaveanexact sequence in C(A) → → −1−−−−− shift→ → O lim Xi Xi Xi O ←− i i ∼ which belongs to S A .Inparticular,lim X hlim X in K(A). C( ) ←− i = ←− i Proof. 1. By Lemma 11.6, for any j we have a split exactsequence − O → Xj −1−−−−shift→ Xj → lim Xj → O. i i i i −→ i Then we have an exact sequence in C(A) → −1−−−−− shift→ → → O Xi Xi lim Xi O i i −→ which belongs to SC(A). The last assertion follows by Proposition 6.12. 2. Similarly. Remark 11.8. The above− lim→ Xi and← lim− Xi are the filtered colimit and the filtered limitinC(A), but are not the filtered colimit and the filtered limit in K(A)(see Lemma 16.17). A { → Remark 11.9. 1. If sati sfies the condition Ab5, then for a sequence Xi X }i∈N of morphisms in D(A), we have exact sequences i+1 O → Hn(X) → Hn(X) → Hn(hlim X) → O i i i i −→ i for all n ∈ N. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 49

A { → } 2. If satisfies the condition Ab5 and Xi Xi+1 i∈N asequence of morphisms in C(A), then by Proposition 2.25, 3 we have an exact sequence in C(A) O → X → X → lim X → O i i i i −→ i and we have a quasi-isomorphism hlim X → lim X. −→ i −→ i A { → } 3. Assume sati sfies the condition Ab4*, and let Xi+1 Xi i∈N be a sequence of morphisms in D(A)satisfyingthat for any n ∈ Z there is k ∈ N such that n ∼ n H (X ) = H (X )forall i>k. Then we have exact sequences i+1 i O → Hn(hlim X) → Hn(X) → Hn(X) → O ←− i i i i i for all n ∈ N. Proposition 11.10. For an abelian category A,thefollowing 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 HomK(A)(P , Kφ(A)) = 0. 2. If A satisfies the condition Ab4* with enough injectives, then every object of φ K(A) is quasi-isomorphic to a complex P of injectives with HomK(A)(K (A), I)=0. Proof. 1. For a complex X ∈ K(A), we have morphisms of complexes σ≤iX → → ∈ − A σ≤i+1X X .Accordingto Lemma 10.10, there is Pi K (Proj )whichhasa → A quasi-isomorphism Pi σ≤iX ,andwehave a commutative diagram in K( ) P −−−−→ P i i+1  

σ≤iX −−−−→ σ≤i+1X By Remark 11.9, we have a quasi-isomorphism hlim P → hlim σ≤ X . −→ i −→ i ∼ By Proposition 11.7, hlim σ≤ X lim σ≤ X = X in K(A). By the construction, −→ i = −→ i hlim P is a complex of projectives. Since we have an exac t sequence −→ i φ A → φ A → φ A HomK(A)(TPi , K ( )) HomK(A)(P , K ( )) HomK(A)(TPi , K ( )), i i φ we have Hom K(A)(P , K (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 Ks(Proj A)ofK(A)con- φ sisting of complexes P of projectivessuchthat HomK(A)(P , K (A)) = 0. Then (Ks(Proj A), Kφ(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 Ks(Inj A)ofK(A)consisti ng of complexes φ φ s I of injectives such that HomK(A)(K (A),I )=0. Then (K (A), K (Inj A)) is a stable t-structure in K(A). They are often called K-projective complexes (resp., K-injective complexes). 50 JUN-ICHI MIYACHI

Proposition 11.12. The following hold. 1. If A satisfies the condition Ab4 with enough projectives, then we have an isomorphism ∼ HomK(A)(P ,Y ) = HomD(A)(P ,Y ) for P ∈ Ks(Proj A), Y ∈ K(A). 2. If A satisfies the condition Ab4* with enough injectives, then we have an isomorphism ∼ HomK(A)(X ,I ) = HomD(A)(X ,I ) for X ∈ K(A), I ∈ Ks(Inj A). Theorem 11.13. The following hold. 1. If A satisfies the condition Ab4 with enough projectives, then we have a triangle equivalence t s ∼ K (Proj A) = D(A). 2. If A satisfies the condition Ab4* with enough injectives, then we have a triangle equivalence t s ∼ K (Inj A) = D(A). Proof. 1. By Proposition 11.10, (Ks(Proj A), Kφ(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, inthecasethatA sati sfies − − the condition Ab4 with enough projectives, we can define D (A)=K (Proj A), and D(A)=Ks(Proj A). Then we can bypass Remark 7.7. Similarly, in t he case that A sati sfies the condition Ab4* with enough injectives, we can define D+(A)=K+(Inj A), and D(A)=Ks(Inj A). Proposition 11.15. The following hold.

fi 1. Let C be a triangulatedcategorywithcoproducts. For a sequence {Xi −→ } ∈ N Xi+1 i∈N,ifthereisn such that fi are split monomorphisms for all i ≥ n,thenthe structural morphism X → hlim X is a split epimorphism. i i −→ i fi 2. Let C be a triangulated category with products. For a sequence {Xi+1 −→ } ∈ N ≥ Xi i∈N,ifthereisn such that fi are split epimorphisms for all i n, then the structural morphism hlim X → X is a split monomorphism. ←− i i i Proof. 1. For any M ∈Cand n ∈ N,wehaveacommutative diagram n 1− shift n HomC ( T X ,M) −−−−−→ HomC( T X ,M) i i i i  

− n −1−−−−shift→ n i HomC(T Xi,M) i HomC (T Xi,M). n −1−−−−− shift→ n Then by Exercise 11.5.2, HomC ( iT Xi,M) HomC( iT Xi,M)isepic, − and then T n X −1−−−−shift→ T n X is split monic. Therefore, hlim X → T X = i i i i −→ i i i 0, and hence X → hlim X is splitepic. i i −→ i DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 51

2. Similarly. Lemma 11.16. Let C be a triangulatedcategory with coproducts. For a sequence fi {Xi −→ Xi+1}i∈N where Xi = X and fi =1X for all i,wehave an isomorphism in C hlim X ∼ X. −→ i = p q Proof. Let X −→ X −→ hlim X −→r TX be a triangle, and α = (1 ) : i i i i −→ i i i i X i → iXi X.Byeasycalculation, the following hold. (a) αp =0. → → (b) If a morphism φ : iXi Y sati sﬁes φp =0,thenthere is a unique f : X Y such that φ = fα. By the property of hlim and the above, there exist h : X → hlim X and k : −→ −→ i hlim X → X such that α = kq and q = hα.Sinceq = hkq and α = khα,wehave −→ i hk =1andkh =1by(b)andProposition 11.15. Proposition 11.17. Let C be a triangulated category with coproducts. Let e : X → X be a morphism in C such that e2 = e.Thene splits in C. Proof. We consider three sequences (A) X −→e X −→e .... − − (B) X −−1 →e X −1−→e .... [ 00] [ 00] ⊕ −−−01→ ⊕ −−−01→ (C) X X X X .... e 1−e ⊕ → Then we have an isomorphism α = 1−ee :(A) (B) (C)ofsequences. Thus hlim (C) ∼ Y ⊕ Z in C, where Y =hlim(A), Z =hlim(B). For a sequence (C), −→ = −→ −→ we have a commutative diagram 1−shift (X ⊕ X)i −−−−→ (X ⊕ X)i −−−−→ hlim (C) i  i  −→  

⊕ − ⊕ −1−−−−−−→(1 shift) ⊕ ( iXi) ( iXi) ( iXi) ( iXi). By Lemma 11.16, we have hlim (C) ∼ X in C.Ontheotherhand, we have a −→ = commutative diagram (1−shift)⊕(1−shift) β ( X ) ⊕ ( X ) −−−−−−−−−−−−→ ( X ) ⊕ ( X ) −−−−→ Y ⊕ Z i i  i i i i  i i     α α γ

− (0 1) ⊕ −1−−−→shift ⊕ −−−−−−i →i i(X X)i i(X X)i X a 0 εη where β =[0 b ]andγ =[ ]areisomorphisms.AccordingtoProposition 11.15, a 0 − − (0 1) and β =[0 b ]areepic,thena and b are epic. Since (1 e)εb =(1 1−e − e)(0 1) [ e ]=0,wehave(1 e)ε =0.Similarly, we have eη =0.Hence we have amorphism δ : X → Y such that εδ = e and δε =1Y . Corollary 11.18. Let L be a triangulated full subcategory L of C.IfL satisfies the condition (L2) every coproduct of objects in L is in L, 52 JUN-ICHI MIYACHI then L is a localizing subcategory of C. Corollary 11.19. Let R be a commutative complete local ring, A afiniteR- algebra. Then Db(mod A) is a Krull-Schmidt category. Proof. For a complex Y ∈ Db(mod A), we may assume that Hm(Y ) =0,H n(Y ) = 0, and Hi(Y )=0forin−1(Y ), then we have a triangle in D (mod A) → → → Y1 Y Y2 TY1 . Then we have an exact sequence → → HomDb(mod A)(X ,Y1 ) HomDb(mod A)(X ,Y ) HomDb(mod A)(X ,Y2 ). By the assumption, HomDb(mod A)(X ,Y1 )andHomDb(mod A)(X ,Y2 )arefinitely generated R-modules. Then HomDb(mod A)(X ,Y )isafinitely generated R-module. Similarly, for a triangle σ≤n−1(X ) → X → σ>n−1(X ) → Tσ≤n−1(X ), we have the same result. In particular, EndDb(mod A)(X )isasemiperfectring. For an ∈ idempotent e EndDb(mod A)(X ), by Example 10.14, we may consider an idem- b ⊂ b potent in Dc (Mod A) D (Mod A). By Proposition 11.17, there exist a complex Y ∈ Db(Mod A)andmorphismsp : X → Y , q : Y → X such that qp = e and i i b ∈ pq =1Y .Sinceevery H (Y )isadirectsummand of H (X ), Y Dc (Mod A). According to Proposi tion 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 tri angulated full subcategory K∗(A)ofiscalledaquotientizing subcategory (often called localizing subcategory)ifthecanonical functor ∗ ∗ K (A)/ K ,φ(A) → D(A) ∗ ∗ ∗ is fully faithful, where K ,φ(A)=Kφ(A) ∩ K (A). If K (A)isaquotientizing subcategory of K(A), we denote by D∗(A)thequotient category K∗(A)/ K∗,φ(A) ∗ ∗ ∗ and by QA : K (A) → D (A)thecanonical quotientfunctor. ∗ Deﬁnition 12.2 (Right Deri ved Functor). Let K (A)beaquotientizing subcate- ∗ gory of K(A)andF : K (A) → K(B)a∂-functor. The right derived functor of F is a ∂-functor R∗F : D∗(A) → D(B) together with afunctorial morphism of ∂-functors ∗ ∗ ξ ∈ ∂ Mor(QB F,R FQA) with the following property: ∗ ∗ For G ∈ ∂(D (A), D(B)) and ζ ∈ ∂ Mor(QB F,GQA), there exists a unique mor- ∗ phism η ∈ ∂ Mor(R F, G)suchthat ∗ ζ =(ηQA)ξ. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 53

In other words, we can simply write the above using functor categories. For triangulated categories C, C,the∂-functor category ∂(C, C)isthecategory (?) consisti ng of ∂-functors from C to C as objects and ∂-functorial morphisms as morphisms. Then we have ∗ ∼ ∗ ∂ Mor(QBF, −QA) = ∂ Mor(R F,−) ∗ as functors from ∂(D (A), D(B)) to Set (See Lemma1.8). Deﬁnition 12.3. Let K∗(A)beaquotientizing subcategory of K∗(A)andF : ∗ ∗ K (A) → K(B)a∂-functor. When F has a right derived functor R∗F : D (A) → ∗ D(B), we deﬁne Ri F =Hi 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 rightderivedfunctor R∗F : D (A) → D(B).Thenfor any exact sequence O → X → Y → Z → O in C(A) we have a long exact sequence ...→ Ri F (X) → Ri F(Y ) → Ri F (Z) → Ri+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, φ (b) QB F(L )={O}, where Lφ = Kφ(A) ∩L.Thenthere exists the right derived functor (R∗F,ξ) such ∗ that ξI : QBFI → R FI is a quasi-isomorphism for I ∈L. Proof. Let E : L→K∗(A)betheembedding functor, then by the assumption (a) ∗ and Proposition 7.16 the canonical functor E : L/Lφ → D (A)isanequival ence. ∗ Let J : D (A) →L/Lφ be a quasi-inverse of E.Bytheassumption (b) and φ Proposition 8.7 there is a ∂-functor F : L/L → D(B)suchthatQBFE = FQL, φ ∗ ∗ where QL : L→L/L is the canonical quotient. Put R F = FJ.SinceQAE = EQL,wehave ∗ ∼ ∂ Mor(QBFE,GQAE) = ∂ Mor(FQL,GEQL) ∼ = ∂ Mor(FJE,GE) ∼ = ∂ Mor(FJ,G). It remains to show that ∗ ∼ ∗ ∂ Mor(QB F,GQA) = ∂ Mor(QB FE,GQAE)(φ → φE ). ∗ ∗ Let φ ∈ ∂ Mor(QBF,GQA)withφE =0.Forany X ∈ K (A)thereexistsI ∈L ∗ −1 which has a quasi-isomorphism s : X → I . Then φX =(GQAs) φI QB Fs =0, ∗ ∗ and hence φ =0.Givenψ ∈ ∂ Mor(QB FE,GQAE), for any X ∈ K (A), let ∗ −1 φX =(GQAs) ψI QBFs for some quasi-isomorphism s : X → I ,withI ∈L. For another quasi-isomorphism s : X → I ,bytheassumption (a), we have a commutative diagram X −−−−s → I     s t I −−−−t → I 54 JUN-ICHI MIYACHI

where all morphisms are quasi-isomorphisms and I ∈L. Then we have ∗ −1 ∗ −1 (GQAs) ψI QB Fs =(GQAt s) ψI QB Ft s ∗ −1 =(GQAts ) ψI QB Fts ∗ −1 =(GQAs ) ψI QB Fs . ∗ It is not hardtoseethatφ ∈ ∂ Mor(QB F,GQA). 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 ∈Ahas 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 → FX → FY → FZ → 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 sati sﬁes 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),thereexistsaquasi-isomorphism X → I with I ∈L, ∗∗ (b) for any X ∈ K (A),thereexists a quasi-isomorphism X → I with I ∈ ∗∗ L∩K (A),and φ (c) QB F(L )={O}, φ φ where L = K (A) ∩L.ThenbothF and F|K∗∗ (A) have the right derived func- ∗ ∗∗ tors (R F, ξ) and (R (F |K∗∗(A)),ζ),respectively, and thecanonical ∂-functorial morphism ∗∗ ∗ φ : R (F |K∗∗ (A)) → R F|K∗∗ (A) is an isomorphism.

Proof. By Theorem 12.5bothF 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) =(φQA )ζ.Forany I ∈L∩K (A), by Theorem 12.5 both ξI and ζI are isomorphisms, so that φQI is an isomorphism. Thus, since by assumption (b), the canonical functor Q : L∩K∗∗(A) → D∗∗(A)isdense,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 theassumptions1,2of Theorem 12.5 are satisﬁed, † (b) K (B) has a triangulated fu l l 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. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 55

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),andthecanonical 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), respective ly. Let X ∈Lbe acyclic. Then, since Q(F (X)) = 0, F (X) is acyclic and Q(G(F (X))) = 0. Thus, againbyTheorem 12.5 GF has a right ∗ derived functor (R∗(GF ),η). Also, for any X ∈ D (A), since we have a quasi - ∗ ∼ ∗ ∼ † isomorphism 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.LetI ∈L. Then ξI ,ζFI and ηI are isomorphisms, ∗ so that φQI 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 athick ∗ abelian full subcategory of A.LetK (A)beaquotientizing subcategory of K(A). ∗ A ∂-functor F : DA (A) → D(B)iscalled way-outright(resp., way-out left)provided ∗ that for any n1 ∈ Z there exists n2 ∈ Z such that if X ∈ D (A)isacomplex with i i H (X )=O for all in2), then H (FX )=O for all in1). Moreover, F is called way-out inbothdirections if F is way-out right and way-out left. Lemma 12.10. For a complex X ∈ C(A),wehavetriangles in D(A) 1. n τ≥n−1X → τ≥nX → X [−n] → τ≥n−1X [1]. 2. n n H (X )[−n] → σ>n−1X → σ>nX → H (X )[1 − n]. Proof. By 10.6, we have an exact sequence O → Y → σ>n−1X → σ>nX → O, n−1 → n n−1 → n ∼ where Y : Y Y =ImdX Ker dX . Then itiseasytoseeY = Hn(X)[−n].

Proposition 12.11. Let A, B be abelian categories, A athick abelian full subcat- ∗ ∗ ∗ ∗ ∗ ∗ egory of A.LetF ,G : DA (A) → D(B) be ∂-functors, and η ∈ ∂ Mor(F ,G ), where ∗ = nothing, +, −,b.Thenthefollowing hold. 1. If ηb(X) are isomorphisms for all X ∈A,thenηb is an isomorphism. 2. Assume that F and G are way-out right. If η+(X) are isomorphisms for all X ∈A,thenη+ is an isomorphism. 3. Assume that F and G are way-out in both directions. If η(X) are isomorphisms for all X ∈A,thenη is an isomorphism. 56 JUN-ICHI MIYACHI

4. Let I be a collectionofobjectsofA 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,thenη∗(X) are isomorphisms for all X ∈A.

b Proof. 1. Let X ∈ D (A). For n 0, σ>nX = O,andthenη(σ>nX )isan b isomorphism. By Lemma 12.10, η(σ>n−1X )isanisomorphism.SinceX ∈ D (A), we get the statement. 2. Let X ∈ D+(A). For any n ∈ Z,weshowthatHn(η(X)) is an isomorphism. ∗ Put n1 = n +1. Then there exists n2 ∈ Z such that if Y ∈ D (A)isacomplex i i i with H (Y )=O for all in2 X )=O for in2 X )=H (Fσ>n2 X )=O, n n−1 H (Gσ>n2 X )=H (Gσ>n2 X )=O. → → → Consideringatriangle σ≤n2 X X σ>n2 X σ≤n2 X [1], we have acommutative diagram

n ∼ n H (Fσ≤ X ) −−−−→ H (FX )  n2    n −−−−∼ → n H (Gσ≤n2 X ) H (GX ). where allhorizontalarrows 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>0X )isanisomorphism. Consi dering σ≤0X → X → σ>0X → σ≤0X [1], η(X )isanisomorphism. 4. For X ∈A,bythedualofLemma 10.10, there is a resolution I with each Ii ∈I.Byreplacingσ by τ in 1 and 2, we have the statement.

Proposition 12.12. Let A, B be abelian categories, A, B thick abelian full sub- ∗ ∗ categories of A, B,respectively. Let F : DA (A) → D(B) be a ∂-functor,where∗ = nothing, +, −,b.Thenthefollowing hold. b b b 1. If F (X) ∈ DB (B) for all X ∈A,thenF (X) ∈ DB (B) for all X ∈ DA (A). b 2. Assume that F and G are way-out right. If F (X) ∈ DB (B) for all X ∈A, b + then F (X) ∈ DB (B) for all X ∈ DA (A). b 3. Assume that F and G are way-out in both directions. If F (X) ∈ DB (B) for b all X ∈A,thenF (X) ∈ DB (B) for all X ∈ DA (A). 4. Let I be a collectionofobjectsofA 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) ∈ DB (B) for all I ∈I,thenF(X) ∈ DB (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 bigradedobject p,q A p,q p,q → p+1,q p,q p,q → p,q+1 (C )p,q∈Z of together with dI : C C and dII : C C such DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 57 that q p,q p,q p,q → p+1,q C =(C ,dI : C C ) p p,q p,q p,q → p,q+1 C =(C ,dII : C C ) p,q+1 p,q p+1,q p,q are complexes satisfying dI dII + dII dI =0. A morphism f of double complexes X to Y is a collection of morphisms fp,q : Xp,q → Y p,q such that f q : Xq → Y q and f p : Xp → Y p are morphisms of complexes for all p, q ∈ Z. We denote by C2(A)thecategories of doublecomplexes of A.Auto-equival ences 2 2 p,q p+1,q TI,TII : C (A) → C (A)arecalled the translations if (TIX ) = X and p,q (TId# X ) − p+1,q p,q p,q+1 p,q − p,q+1 = d# X and (TIIX ) = X and (TIId# X ) = d# X for any complex p,q p,q p,q X =(X ,dI X ,dII X ), where # = I,II. r p Moreover, an r-tuple complex C is an r-tuple gradedobject(C )p∈Zr of A p p → p+ei ≤ ≤ together with di : C C (1 i r)suchthat 2 ≤ ≤ di =0(1 i r), didj + djdi =0foralli, j, where ei =(0,... ,0, 1, 0,... ,0). ith component

2 Proposition 13.2. Let CI(A) (resp., CII(A))bethefullsubcategory of C (A) consisting of complexes X such that Xp,q = O for all q =0 (resp., p =0 ). Then we ∼ ∼ 2 ∼ ∼ have C(A) = CI(A) = CII(A) and C (A) = C(CI(A)) = C(CII(A)).Inparticular, C2(A) is an abeliancategory.

2 Proof. We deﬁne a functor FI : C (A) → C(CI(A)) as follows. For anydouble p,q p,q p,q p,q p,q p,q ,q ,q ,q complex X =(X ,dI X ,dII X ), FI(X ,dI X ,dII X )=(X ,dII X ) where X = p,q − q p,q → p,q p,q (X , ( 1) dI X ). For a morphism f : X Y , FI(f) = f . Then itiseasy to see that FI is an equivalence.

2 By the above, wecandeal wit h C (A)asC(CI(A)) or C(CII(A)).

p,q p,q p,q Deﬁnition 13.3 (Truncati ons). For a double complex X =(X ,dI ,dII ), we deﬁne the following truncations:   O if pn Im dI if p = n ≤n Ker dI if p = n  p,q  X if p>n O if p>n O if qn Im dII if q = n ≤n Ker dII if q = n  p,q  X if q>n O if q>n Xp,q if p ≤ n O if pn X if p n p,q ≤ II p,q X if q n II p,q O if qn n Xp,q if q ≥ n

p,q Lemma 13.4. For a double complex X =(X ,dI,dII),thefollowinghold. 58 JUN-ICHI MIYACHI

1. We have exact sequences in C2(A) → # → → # → O σ≤nX X σ>nX O for all n ∈ Z with #=I,II. 2. We have exact sequences in C2(A) → # → → # → O τ≥nX X τ≤n−1X O for all n ∈ Z with #=I,II. p,q p,q p,q Definition 13.5 (Total Complexes). For a double complex X =(X ,dI ,dII ), we define the total complexes Tot C =(Xn,dn), where Xn = Cp,q,dn = dp,q + dp,q p+q=n p+q=n I II ∧ TotC =(Y n,dn), where Y n = Cp,q,dn = dp,q + dp,q. p+q=n p+q=n I II r p p ≤ ≤ Moreover, for an r-tuple complex X =(X ,di )(1 i r), we define the total complexes r r Tot C =(Xn,dn), where Xn = Cp,dn = dp |p|=n |p|=n i i=1 ∧ r r TotC =(Y n,dn), where Y n = Cp,dn = dp, |p|=n |p|=n i i=1 where |p| = |(p1,... ,pr)| = p1 + ...+ pr. Lemma 13.6. The following hold. 1. If A satisfies the condition Ab4, then the functor Tot : C2(A) → C(A) preserves translations, exact sequences and coproducts. ∧ 2. If A satisfies the condition Ab4*, then the functor Tot : C2(A) → C(A) preserves translations, exact sequences and products. p,q Lemma 13.7. For a double complex X =(X ,dI,dII),thefollowinghold. 1. If Xp,q = O for qnand Xq are acyclic complexes in C(A) for all q,thenTot X is acyclic in C(A) . 4. Assume A satisfies the condition Ab4*. If Xp,q = O for qnand Xp are acyclic ∧ complexes in C(A) for all p,thenTotX is acyclic in C(A) . 2 Proof. 1. Let nX = n − m.ByLemma13.4,wehave an exact sequence in C (A) → II → → II → O τ≥n−1X X τ≤nX O. Then by Lemma 13.6, we have the exact sequence in C(A) → II → → n − → O Tot τ≥n−1X Tot X X [ n] O. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 59

II By theassumpti on of induction on nX ,Totτ≥n−1X is acyclic. Then Tot X is acyclic because Xn[−n]isacyclic. 2 2. Let nX = n − m.ByLemma13.4, we have an exact sequence in C (A) → II → → II → O σ≤n−1X X σ>n−1X O. Then by Lemma 13.6, we have the exact sequence in C2(A) → II → → II n → O Tot σ≤n−1X Tot X Tot σ>n−1X O. II By theassumption of induction on nX ,Totσ≤n−1X is acyclic. It is easy to see II n ∼ − that Totσ>n−1X = M (1X n )[ n]isacyclic. Then Tot X is acyclic. 3. By Lemma 13.4, we have the canonical morphisms in C2(A) II −f→r II τ≥−rX τ≥−(r+1)X , II → which are term-split monomorphisms for all p, q.SinceTotfr :Totτ≥−rX II A Tot τ≥−(r+1)X is term-split monomorphisms in ,byProposition 11.7, we have II ∼ II ∼ hlim Tot τ≥− X lim Tot τ≥− X Tot X . −→ r = −→ r = By 1, Tot τ II X is acyclicforall r. Then hlim Tot τ II X is acyclic, and hence ≥−r −→ ≥−r so is Tot X. 4. Dual of 3. 5. By Lemma 13.4, we have the canonical morphisms in C2(A) II −g→r II σ≤rX σ≤r+1X . By Exercise 11.5, we haveanexactsequence in C(A) II II II O → Tot σ≤ X → Tot σ≤ X → lim Tot σ≤ X → O. r r r r −→ r Then we have II ∼ II ∼ hlimTotσ≤ X lim Totσ≤ X TotX . −→ r = −→ r = By 2, Tot σII X is acyclic for all r. Then hlim Tot σII X is acyclic, and hence ≤r −→ ≤r so is TotX. 6. Dual of 5. Deﬁnition 13.8. We deﬁne an embedding functor emI : C(A) → C2(A)asfollows. For a complex X ∈ C(A), Xp if q =0 emI(X)p,q = O othewise.

f g Deﬁnition 13.9 (Proper Exact). An exact sequence O → X −→ Y −→ Z → O in C(A)iscalled proper exact if theinduced sequence O → Z (X) → Z (Y ) → Z(Z) → O is also exact. In this case, f (resp., g)iscalledaproper monomorphism (resp., a proper epimorphism). AcomplexX ∈ C(A)iscalledaproper projective complex (resp., a proper injective complex)if ∼ n X = P [−n]⊕ M (1Qn )[−n − 1] n∈Z n∈Z where P n,Qn are projective (resp., injective) objects of A. 60 JUN-ICHI MIYACHI

Lemma 13.10. For a complex Z ∈ C(A),thefollowing hold. 1. M is proper projective if and only if for every proper epimorphism g : X → Y in C(A), HomC(A)(M ,g) is surjective. 2. M is proper injective if and only if for every proper monomorphism f : X → Y in C(A), HomC(A)(f, M ) is surjective. Proof. 1. If M is proper projective, then it suﬃces to check the case that M = P [−n]orM(1P )[−n], where P is a projective object of A, n ∈ Z.Foranyproper epimorphism g : X → Y ,wehavecommutative diagrams − HomC(A)(P [ n],g) Hom A (P [−n],X ) −−−−−−−−−−−−→ Hom A (P [−n],Y ) C( )  C( )   

n n Ho mA(P,Z (g)) n HomA(P, Z (X )) −−−−−−−−−−→ HomA(P, Z (Y )),

− HomC(A)(M (1P )[ n],g) Hom A (M (1 )[−n],X ) −−−−−−−−−−−−−−−→ Hom A (M (1 )[−n],Y ) C( )  P C( )  P  

n−1 n−1 HomA(P,g ) n−1 HomA(P, X ) −−−−−−−−−−→ HomA(P,Y ). Since P is projective, thebottomarrows of the above diagrams are surjective. Then the top arrows are also surjective . Conve rse ly, let M be a complex sati sfying the surjective condition. For any epimorphism g : X → Y in A,wehaveacommutative diagram − Ho mC(A)(M ,g[ n]) Hom A (M ,X[−n]) −−−−−−−−−−−−−→ Hom A (M ,Y[−n]) C( )  C( )   

n n Ho mA(C (M ),g) n HomA(C (M ),X) −−−−−−−−−−−→ HomA(C (M ),Y). Since g[−n]:X[−n] → Y [−n]isproper epic, the top arrow is surjective. Then Cn(M )areprojective o b jects of A.Itiseasytoseethat ∼ n M = M (p )[−n − 1], n∈Z where pn :Cn(M) → Bn(M )arethecanonical epimorphisms. We have a commutative diagram

n Hom A (M (p ),M (g)) n C( ) n Hom A (M (p ), M (1 )) −−−−−−−−−−−−−−−→ Hom A (M (p ), M (1 )) C( )  X C( )  X  

n n Ho mA(B (M ),g) n HomA(B (M ),X) −−−−−−−−−−−→ HomA(B (M ),Y). Since M(pn)isproperprojective,thetoparrowissurjective,andthenthebottom arrow is surjective. Therefore Bn(M )isaprojectiveobjectofA.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),thereare proper projective complexes DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 61

(resp., proper injective complexes) Xn (n ≥ 0)whichhasaproperprojective resolution (resp., a proper injective resolution) in C(A) ...→ X−1 → X0 → M → O (resp., O → M → X0 → X1 → ...). Proof. For a complex M ∈ C(A), we have exactsequences in C(A) O → Z(M ) → M → B(M )[1] → O O → B(M ) → Z(M) → H(M ) → O. n n n For each B (M ), there is a projective objectQ which has a epimorphism Q → n − → B (M ). Then we have an epimorphism n∈Z M (1Qn )[ n] B (M )[1] and − → n its lift n∈Z M (1Qn )[ n] M .Foreach H (M ), there is a projective ob- n ject P n which has a epimorphism Pn → H (M ). Then we have a morphism n − → n − → → n∈ZP [ n] B (M )anditslift n∈ZP [ n] Z (M ) M . Then itis n − ⊕ − → easy to see that n∈ZP [ n] n∈Z M (1Qn )[ n] M is proper epimorphism. Then by induction we complete the proof. The above proper resolution ... → X−1 → X0 (resp., X0 → X1 → ...)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),letπ : P → M be a proper projective resolution. Then Tot π :TotP → M is a quasi-isomorphism in K(A),andTot P ∈ Ks(Proj A). 2. Assume that Asatisﬁes the condition Ab4* withenoughinjectives.Givena complex M ∈ C(A),letµ : M → I be a proper injective resolution. Then ∧ ∧ Totµ : M → TotI ∧ is a quasi-isomorphism in K(A),andTotI ∈ Ks(Inj A). Proof. 1. We can consider π : P → M as π : P → emIM in C2(A). Then we have a commutative diagram

I σ≤ π σI P −−−−n→ σI emIM ≤n ≤n    αn βn

σI π I −−−−≤n→ I I σ≤n+1P σ≤n+1em M where αn, βn are term-split monomorphisms. Therefore we have a commutative diagram

σI π I ≤n Tot σ P −−−−→ σ≤ M ≤n n   Tot αn βn

σI π I −−−−≤n→ Tot σ≤n+1P σ≤n+1M 62 JUN-ICHI MIYACHI where Tot αn, βn are term-split monomorphisms, and all horizontal morphisms are σI π I −−−→≤n I I quasi-isomorphisms, because Tot(σ≤nP σ≤nem M )isacyclic by Lemma 13.7. According to Proposition 11.7, we have isomorphisms in D(A) I Tot P =lim−→ Tot σ≤nP ∼ I hlim Tot σ≤ P = −→ n ∼ hlim σ≤ M = −→ n ∼ = −lim→ σ≤nM = M . On the other hand, since Tot σI P ∈ K−(Proj A), Tot P ∼ hlim Tot σI P ∈ ≤n = −→ ≤n Ks(Proj A). 2. Similarly

Deﬁnition 13.13. For a morphism u : X → Y of double complexes, we deﬁne the complexes MI (u), MII(u)asfollows. p,q p+1,q ⊕ p,q MI (u) = X Y − p+1,q p,q dI X 0 dIM (u) = p+1,q p,q I u dI Y − p+1,q p,q dII X 0 dII M (u) = p,q , I 0 dII Y

p,q p,q+1 ⊕ p,q MII(u) = X Y − p,q+1 p,q dI X 0 d = p,q IM (u) 0 d II I Y − p,q+1 p,q dII X 0 dII M (u) = p+1,q p,q . II u dII Y Proposition 13.14. For a morphism u : X → Y of double complexes, the following hold. 1. Tot MI (u)=M(Tot u). 2. Tot MII(u)=M(Tot u).

14. Derived Functors of Bi-∂-functors

Deﬁnition 14.1 (Bi-∂-functor). For tri angulated categories C1, C2 and D, abi- ∂-functor (F,θ1,θ2):C1 ×C2 →Dis a bifunctor F : C1 ×C2 →Dtogether − →∼ − − →∼ with bifunctorial isomorphisms θ1 : F (TC1 , ?) TDF ( , ?), θ2 : F( ,TC2 ?) TDF (−, ?) such that ∈C − − C →D (a) For each object X1 1, F(X1, )=(F (X1, ),θ2(X1,−)): 2 is a ∂-functor. ∈C − − C →D (b) For each object X2 2, F( ,X2)=(F ( ,X2),θ1(−,X2)): 1 is a ∂-functor.

For bi-∂-functors (F, θ1,θ2), (G, η1,η2):C1 ×C2 →D,abifunctorial morphism → × φ : F G is called a bi-∂-functorialmorphismif (TDφ)θ1 = η1φ(TC1 1C2 ), × (TD φ)θ2 = η2φ(1C1 TC2 ). 2 We denote by ∂ (C1 ×C2, D)thecollection of all bi-∂-functors from C1 ×C2 to D,anddenote by ∂2 Mor(F, G)thecollection of all bi-∂-functorial morphisms from F to G. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 63

Proposition 14.2. Let C1, C2 and C3 be triangulated categories, Ui ´epaisse subcategories of Ci,andQi : Ui →Ci/Ui the canonical quotients (i =1, 2). Assume a bi-∂-functor F =(F,θ1,θ2):C1 ×C2 →C3 satisﬁes that

(a) F (U1, C2)={O}. (b) F (C1, U2)={O}.

Then there exists a unique bi-∂-functor F =(F,θ1, θ2):C1/U1 ×C2/U2 →C3 such that F = F(Q1 × Q2) and θi = θi(Q1 × Q2) (i =1, 2). C ×C 1 2L LL LL × LF Q1 Q2 LL LLL _ _ _/& C1/U1 ×C2/U2 C3 F

Sketch. We deﬁne the functor F : C1/U1 ×C2/U2 →C3 as follows. For Xi ∈Ci,let F (Q1X1,Q2X2)=F(X1,X2). For [(fi,si)] : Xi → Yi in Ci/Ui,letF([(f1,s1)], −1 −1 Q2X2)=F(s1,X2) F (f1,X2), F (Q1X1, [(f2,s2)]) = F(X1,s2) F (X1,f2). Let Ti, T i are translations of Ci, Ci/Ui,respectively (i =1, 2, 3). Then wedeﬁne

F(T 1Q1X1,Q2SX2) F(Q1T1X1,Q2X2) F (T1X1,X2) SSS SSS SSS θ (X ,X ) SSS 1 1 2 θ1(Q1X1,Q2X2) S) T3F(Q1X1,Q2X2) T3F (X1,X2)

F(Q1X1, T 2Q2SX2) F(Q1X1,Q2T2X2) F (X1,T2X2) SSS SSS SSS θ (X ,X ) SSS 2 1 2 θ2(Q1X1,Q2X2) S) T3F(Q1X1,Q2X2) T3F (X1,X2) Then it is not hard to see that F sati sﬁes the assertions(lefttothereader).

Proposition 14.3. Let C1, C2 and C3 be triangulated categories, Ui ´epaisse subcategories of Ci,andQi : Ui →Ci/Ui the canonical quotients (i =1, 2). For bi-∂-functors F =(F,θ1,θ2),G =(G, η1,η2):C1/U1 ×C2/U2 →C3,wehave a bijective correspondence 2 ∼ 2 ∂ Mor(F,G) → ∂ Mor(F (Q1 × Q2),G(Q1 × Q2)), (ζ → ζ(Q1 × Q2)).

Deﬁnition 14.4 (Right Deri ved Functor). Let Ai be abelian categories, and let ∗i ∗1 ∗2 K (Ai)beaquotientizing subcategory of K(Ai)andF : K (A1) × K (A2) → K(A3)abi-∂-functor (i =1, 2, 3). The right derived functor of F is a bi-∂-functor

∗1,∗2 ∗1 ∗2 R F : D (A1) × D (A2) → D(A3) together with afunctorial morphism of bi-∂-functors 2 ∗ ,∗ ∗ ∗ ξ ∈ ∂ Mor(QA F, R 1 2 F (Q 1 × Q 2 )) 3 A1 A2 with the following property: ∗ ∗ ∗ ∗ ∈ 2 1 A × 2 A A ∈ 2 1 × 2 For G ∂ (D ( 1) D ( 2), D( 3)) and ζ ∂ Mor(QA3 F, G(QA QA )), ∗ ∗ 1 2 there existsaunique morphism η ∈ ∂2 Mor(R 1, 2 F, G)suchthat ∗ ∗ ζ =(η(Q 1 × Q 2 ))ξ. A1 A2 64 JUN-ICHI MIYACHI

In other words,

2 ∗ ∗ ∼ 2 ∗ ,∗ ∂ Mor(QA F, −(Q 1 × Q 2 )) ∂ Mor(R 1 2 F,−) 3 A1 A2 =

2 ∗1 ∗2 as functors from ∂ (D (A1) × D (A2), D(A3)) to Set (See Lemma 1.8).

Theorem 14.5 (Existence Theorem). Let Ai be abelian categories (i =1, 2, 3), ∗i ∗1 ∗2 and let K (Ai) be a quotientizing subcategory of K(Ai) and F : K (A1) × K (A2) → K(A3) abi-∂-functor. Assume there exist triangulated full subcategories Li of ∗i K (Ai) (i =1, 2)suchthat ∗ ∈ i A → ∈L (a) for any Xi K ( i) there is a quasi-isomorphism Xi Ii with Ii i, Lφ L { } (b) QA3 F( 1 , 2)= O , L Lφ { } (c) QA3 F( 1, 2 )= O , Lφ φ A ∩L where i = K ( i) i (i =1, 2). Then there exists the right derived functor ∗ ,∗ ∗ 1 2 → (R F,ξ) such that ξ(I ,I ) : QA3 F (I1,I2) R F (I1,I2) is a quasi-isomorphism ∈L ×L 1 2 for (I1,I2) 1 2.

∗i ∗i Proof. Let Qi : K (Ai) → D (Ai)bethecanonical quotients, and let Ei : Li → ∗i K (Ai)betheembedding functors, then by the assumption 1 and Proposition ∗ L Lφ → i A 7.16 the canonical functor Ei : i/ i D ( i)areequivalences (i =1, 2). Let ∗ i A →L Lφ Ji : D ( i) i/ i be quasi-inverses of Ei (i =1, 2). By the assumption 2, 3 and Proposition 14.2 there is a bi-∂-functor L Lφ ×L Lφ → A F : 1/ 1 2/ 2 D( 3) × × L →L Lφ such that Q3F(E1 E2)=F(Q 1 Q 2), where Q i : i i/ i are the canonical ∗ ∗ 1, 2 × × × quotients. Put R F = F(J1 J2). Since (Q1E1 Q2E2)=(E1Q 1 E2Q 2), we have 2 ∂ Mor(Q3F(E1 × E2),G(Q1E1 × Q2E2)) ∼ 2 × × = ∂ Mor(F(Q 1 Q 2),G(E1Q 1 E2Q 2)) ∼ 2 = ∂ Mor(F(J1E1 × J2E2),G(E1 × E2)) ∼ 2 = ∂ Mor(F(J1 × J2),G) ∗ ∗ = ∂2 Mor(R 1, 2 F, G) It remains to show that 2 ∼ 2 ∂ Mor(Q3F,G(Q1 × Q2)) → ∂ Mor(Q3F (E1 × E2),G(Q1E1 × Q2E2)), (φ → φ(E1 × E2)). ∗ ∈ 2 × × ∈ i A Let φ ∂ Mor(Q3F, G(Q1 Q2)) with φ(E1 E2)=0.ForanyXi K ( i) ∈L → there exists Ii i which has a quasi-isomorphism si : Xi Ii (i =1, 2). Then −1 φ = G(Q s ,Q s ) φ Q F (s ,s ) (X1,X2) 1 1 2 2 (I1,I2) 3 1 2 =0,

2 and hence φ =0.Givenψ ∈ ∂ Mor(Q3F(E1 × E2),G(Q1E1 × Q2E2)), for any ∗ ∈ i A Xi K ( i), let −1 φ =(G(Q s ,Q s )) ψ Q F(s ,s ) (X1,X2) 1 1 2 2 (I1,I2) 3 1 2 → ∈L for some quasi-isomorphism si : Xi Ii,withIi i (i =1, 2). For another → quasi-isomorphism s i : Xi I i,bytheassumptions 1, we have commutative DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 65 diagrams X −−−−si → I i i   s i t i

−−−−ti → I i I i ∈L where all morphisms are quasi-isomorphisms and I i i (i =1, 2). Then we have −1 (G(Q s ,Q s )) ψ Q F (s ,s ) 1 1 2 2 (I1,I2) 3 1 2 −1 =(G(Q t s ,Q t s )) ψ Q F (t s ,t s ) 1 1 1 2 2 2 (I 1,I 2) 3 1 1 2 2 −1 =(G(Q t s ,Q t s )) ψ Q F (t s ,t s ) 1 1 1 2 2 2 (I 1,I 2) 3 1 1 2 2 −1 =(G(Q s ,Q s )) ψ Q F(s ,s ) 1 1 2 2 (I 1,I 2) 3 1 2 2 It is not hardtoseethatφ ∈ ∂ Mor(Q3F, G(Q1 × Q2)). The last asserti on is easy to check.

∗i Proposition 14.6. Let Ai be abelian categories (i =1, 2, 3), and let K (Ai) be ∗1 ∗2 aquotientizing subcategory of K(Ai) and F : K (A1) × K (A2) → K(A3) abi-∂- ∗i functor. Assume there exists a triangulated full subcategories Li of K (Ai) (i = 1, 2)suchthat ∗ ∈ i A → ∈L (a) for any Xi K ( i) there is a quasi-isomorphism Xi Ii with Ii i, ∗ Lφ 2 A { } (b) QA3 F( 1 , K ( 2)) = O , L Lφ { } (c) QA3 F( 1, 2 )= O , Lφ φ A ∩L where i = K ( i) i (i =1, 2). Then we have ∗ ∗ ∗ ∗ 1, 2 1 A × 2 A → A 1. There isabi-∂-functor RI F : D ( 1) K ( 2) D( 3) such that

2 ∗1 ∼ 2 ∗1,∗2 ∂ Mor(QA F, −(Q × 1 ∗2 A )) ∂ Mor(R F,−), 3 A1 K ( 2) = I ∗ ∗ 1, 2 − − ∈ and RI F( ,X2) is the right derived functor of F ( ,X2) for any X2 ∗2 K (A2). ∗ ∗ ∗ ∗ ∗ ∗ 1, 2 1, 2 1 A × 2 A → A 2. There isabi-∂-functor RII RI F : D ( 1) D ( 2) D( 3) such that

2 ∗1,∗2 ∗2 ∼ 2 ∗1,∗2 ∗1,∗2 ∂ Mor(R F, −(1 ∗1 A × Q )) ∂ Mor(R R F, −), I D ( 1) A2 = II I and ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∂ Mor(R 1, 2 F(X , ?), −Q 2 ) ∼ ∂ Mor(R 1, 2 R 1, 2 F(X , ?), −) I 1 A2 = II I 1

∗1 for any X1 ∈ K (A1). 3. We have an isomorphism

2 ∗ ∗ ∼ 2 ∗1,∗2 ∗1,∗2 ∂ Mor(QA F, −(Q 1 × Q 2 )) = ∂ Mor(R R F, −). 3 A1 A2 II I

∗1,∗2 ∗1,∗2 ∼ ∗1,∗2 In particular, RII RI F = R 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).

∗i Corollary 14.7. Let Ai be abelian categories (i =1, 2, 3), and let K (Ai) be a ∗1 ∗2 quotientizing subcategory of K(Ai) and F : K (A1) × K (A2) → K(A3) abi-∂- ∗i functor. Assume there exists a triangulated full subcategories Li of K (Ai) (i = 1, 2)suchthat ∗ ∈ i A → ∈L (a) for any Xi K ( i) there is a quasi-isomorphism Xi Ii with Ii i, 66 JUN-ICHI MIYACHI

∗ Lφ 2 A { } (b) QA3 F( 1 , K ( 2)) = O , ∗ 1 A Lφ { } (c) QA3 F(K ( 1), 2 )= O , Lφ φ A ∩L where i = K ( i) i (i =1, 2). Then we have ∗ ∗ ∗ ∗ 1, 2 1 A × 2 A → A 1. There isabi-∂-functor RI F : D ( 1) K ( 2) D( 3) such that ∗ ∗ ,∗ 2 1 ∗ ∼ 2 1 2 ∂ Mor(QA F, −(Q × 1 2 A )) ∂ Mor(R F,−), 3 A1 K ( 2) = I ∗ ∗ 1, 2 − − ∈ and RI F( ,X2) is the right derived functor of F ( ,X2) for any X2 ∗2 K (A2). ∗ ∗ ∗ ∗ 1, 2 1 A × 2 A → A 2. There isabi-∂-functor RII F : K ( 1) D ( 2) D( 3) such that 2 ∗2 ∼ 2 ∗1,∗2 ∂ Mor(QA F, −(1 ∗1 A × Q )) ∂ Mor(R F,−), 3 K ( 1) A2 = II ∗ ∗ 1, 2 − − ∈ and RII F(X1, ) is the right derived functor of F (X1, ) for any X1 ∗1 K (A1). 3. We have an isomorphism

∗1,∗2 ∼ ∗1,∗2 ∗1,∗2 ∼ ∗1,∗2 ∗1,∗2 R F = RII RI F = RI RII F. Definition 14.8 (HomA). For a com plexes X ,Y ∈ C(A), we define the double complex HomA(X ,Y )by p,q −p q HomA (X ,Y )=HomA(X ,Y ) p,q p,q −p−1 q d =HomA (d ,Y ) IHomA(X ,Y ) X p,q p+q+1 p,q −p q d =(−1) HomA (X ,d ), II Ho mA(X ,Y ) Y and define the complex HomA(X ,Y )by ∧ Tot Hom A(X ,Y ). Then itiseasytoseethat op HomA : C(A) × C(A) → C(Ab) is a bifunctor. Lemma 14.9. For complexes X,Y ∈ C(A),wehaveanisomorphism n ∼ n H HomA(X ,Y ) = HomK(A)(X ,T Y ). p,q r Proof. By the definition,for(u )p+q=r ∈ HomA(X ,Y )wehave r p,q d ((u ) ) Ho mA(X,Y ) p+q=r p−1,q −p − p+q+1 q p,q−1 ∈ r+1 =(u dX +( 1) dY u )p+q=r HomA (X ,Y ). Put u = f, i = −p, r = n,thenf −i,i+n : Xi → Y i+n for all i and wehave n −i,i+n d ((f ) ∈Z) Ho mA(X,Y ) i −i−1,i+1+n i − − n i+n −i,i+n =(f dX ( 1) dY f )i∈Z. n n Then itiseasytosee that Ker d =Hom A (X ,T Y ). Put u = h, i = HomA(X,Y ) C( ) −p, r = n − 1, then h−i,i+n−1 : Xi → Y i+n−1 for all i and we have n−1 −i,i+n−1 d ((h )i∈Z) HomA(X,Y ) −i−1,i+n i − n i+n −i,i+n−1 =(h dX +( 1) dY h )i∈Z. n−1 n Then this means Im d =Htp A (X ,T Y ). HomA(X,Y ) C( ) Lemma 14.10. For a complexes X,Y ∈ C(A),thefollowing hold. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 67

− 1. dHomA(X ,T Y ) = TdHomA(X ,Y ) −1 2. dHomA(T X ,Y ) = TdHo mA(X ,Y ) p,q p,q −1 → p−1,q 3. Deﬁne θ1 :HomA (T X ,Y ) HomA (X ,Y ) by the identities, then we have an isomorphism −1 ∼ θ1 :HomA ◦(T × 1C(A)) → T ◦ HomA . p,q p,q → p,q+1 − p+q 4. Deﬁne θ2 :HomA (X ,TY ) HomA (X ,Y ) by ( 1) ,thenwehave an isomorphism ∼ θ2 :HomA ◦(1C(A) × T ) → T ◦ HomA . Lemma 14.11. For a morphism u : X → Y in C(A) and N ∈ C(A),thefollowing hold. ∼ 1. HomA(N , M (u)) = M (H om A(N ,u)). ∼ −1 2. HomA(M (u),N ) = T M (HomA(u, N )). Proof. 1. The double complex HomA(N , M (u)) has the following form p,q −p q+1 ⊕ q HomA (N , M (u)) = HomA(M ,X Y ) −(p+1) p,q Hom(dM ,X)0 d = −(p+1) IHomA(N ,M (u)) 0Hom(d ,Y ) M − p+q+2 q+1 p,q ( 1) Ho m(M,dX )0 dII Ho m (N ,M (u)) = − p+q+1 q − p+q+1 q A ( 1) Hom(M,u )(1) Hom(M,dY ) On the other hand, the double complex MII(HomA(N ,u)) has the following form p,q −p q+1 ⊕ q MII (HomA(N ,u)) = HomA(M ,X Y ) − −(p+1) p,q Hom(dM ,X)0 d = −(p+1) IMII(Ho mA(N ,u)) 0Hom(d ,Y ) M − p+q+1 q+1 p,q ( 1) Hom(M,dX )0 d = q p+q+1 q II M (Ho mA(N ,u)) Hom(M,u )(−1) Ho m(M,d ) II Y (−1)p+q 0 p,q Then itiseasytoseethat morphisms :M (H om A(N ,u)) → 01 II p,q 2 HomA (N , M (u)) induce an isomorphism between them in C (A). By Proposition 13.14, we get the statement. 2. The double complex HomA(M (u),N )hasthefollowing form p,q −p+1 ⊕ −p q HomA (M (u),N )=Hom A(X Y ,M ) − −p p,q Hom(dX ,M)0 dIHom (M(u),N) = p −p−1 A Ho m(u ,M)Hom(dY ,M ) − p+q+1 q p,q ( 1) Ho m(X,dM )0 dII Hom (M(u),N) = − p+q+1 q A 0(1) Ho m(X,dM ) −1 On the other hand, the double complex TI MI (HomA(u, N )) has the following form −1 p,q −p+1 ⊕ −p q (TI MI (H om A(u, N ))) =Hom A(X Y ,M ) −p p,q Ho m(dX ,M)0 d −1 = − p − −p−1 I TI MI (HomA(u,N )) Ho m(u ,M) Ho m(dY ,M) − p+q+1 q p,q ( 1) Ho m(X,dM )0 d − = II T 1 M (Hom (u,N )) 0(−1)p+q+1 Hom(X,dq ) I I A M 10 −1 p,q → Then itiseasytoseethat morphisms 0 −1 :(TI MI (HomA(u, N ))) p,q 2 HomA (M (u),N )induce an isomorphism between them in C (A). By Lemma 13.6 and Proposition 13.14, we get the statement. 68 JUN-ICHI MIYACHI

op Proposition 14.12. The bi-functor HomA : C(A) × C(A) → C(Ab) induces the bi-∂-functor op HomA : K(A) × 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 →Abe additive functors between abelian categories such that G F .Thenwehave a functorial isomorphism ∼ HomA(GX ,Y ) = HomA (X ,FY ).

∗1 op ∗2 Theorem 14.14. The bi-∂-functor HomA : K (A) × K (A) → K(Ab) has the ∗1,∗2 ∗1 op ∗2 right derived functor R HomA :HomA : D (A) × D (A) → D(Ab) if it satisfies the following. 1. If A has enough projectives, then −,∞ −,∞ ∼ −,∞ −,∞ R HomA exits and R HomA = RII RI HomA 2. If A has enough injectives, then ∞,+ ∞,+ ∼ ∞,+ ∞,+ R HomA exits and R HomA = RI RII HomA 3. If A satisfies the condition Ab4 with enough projectives, then ∼ R HomA exits and R HomA = RIIRI HomA 4. If A satisfies the condition Ab4* with enough injectives, then ∼ R HomA exits and R HomA = RIRII HomA 5. If A satisfies the conditions Ab4 and Ab4* with enough projectives and with enough injectives, then ∼ ∼ R HomA = RIIRI HomA = RIRII HomA 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 ∈ Ks(Proj A) I ∈ Ks(Inj A) which have quasi-isomorphisms P → X, Y → I. Then we have an isomophisms ∼ ∼ ∼ R HomA(X ,Y ) = HomA(P ,Y )= HomA(X ,I ) = HomA(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,wehave an isomorphism

n ∗1,∗2 ∼ H R HomA(X ,Y ) = HomD(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

n ∗1,∗2 ∼ n H R HomA(X ,Y ) = H HomA(P ,Y ) ∼ = HomK(A)(P ,Y [n]) ∼ = HomD(A)(X ,Y [n]),

n ∗1,∗2 ∼ n H R HomA(X ,Y ) = H HomA(X ,Y ) ∼ = HomK(A)(X ,I [n]) ∼ = HomD(A)(X ,Y [n]). DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 69

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 p,q p q X ⊗ A Y = X ⊗AY p,q p q d = d ⊗AY X I X ⊗AY p,q p+q p q d =(−1) X ⊗Ad Y II X ⊗AY and define the complex X ⊗A Y by TotX ⊗A Y . Then itiseasytoseethat op ⊗A : C(Mod A) × C(Mod A ) → C(Ab) is a bifunctor. Lemma 14.18. For a complexes X ∈ C(Mod A),Y ∈ C(Mod Aop ),thefollowing hold.

− 1. d = Td (TX )⊗AY X ⊗AY 2. d = Td X ⊗ATY X ⊗AY p,q p+1,q p,q ⊗ → ⊗ 3. Deﬁne θ1 :(TX ) A Y X A Y by the identities, then we have an isomorphism

∼ θ1 : ⊗A ◦ (T × 1C(Mod A)) → T ◦ ⊗A.

p,q p,q+1 p,q ⊗ → ⊗ − p+q 4. Deﬁne θ2 : X A TY X A Y by the ( 1) ,thenwehave an isomorphism

∼ θ2 : ⊗A ◦ (1C(Mod Aop ) × T ) → T ◦ ⊗A. Lemma 14.19. The following hold. 1. For a morphism u : X → Y in C(Mod Aop ) 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 Aop ) we have ∼ M (u) ⊗A N = M (u ⊗A N ). op Proposition 14.20. The bi-functor ⊗A : C(Mod A) × C(Mod A ) → C(Ab) induces the bi-∂-functor op ⊗A : K(Mod A) × K(Mod A ) → K(Ab).

op Lemma 14.21. Let DZ =HomZ(−, Q/Z):Mod A → Mod A (resp., DZ = op HomZ(−, Q/Z):Mod A → 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 DZZ → DZY → DZX is exact. 2. For an A-module M, M is a ﬂat A-module if and only if DZM is an injective A-module. 70 JUN-ICHI MIYACHI

op Theorem 14.22. The bi-∂-functor ⊗A : K(Mod A) × K(Mod A ) → K(Ab) has the left derived functor ⊗L × op → A : D(Mod A) D(Mod A ) D(Ab). Proof. For X ∈ K(Mod A), Y ∈ K(Mod Aop), we have isomorphisms (see Proposi - ti on 15.4) DZ(X ⊗A Y )=DZ(Tot X ⊗A Y ) ∧ ∼ = TotDZ(X ⊗A Y ) ∧ ∼ = Tot HomAop (X ,DZY ) ∼ = HomAop (X ,DZY ). By Lemma 14.9, we can apply Proposition 11.12 to the above isomorphism. Then s op by Lemma 14.21, for P ∈ K (Proj A)andY ∈ K(Mod A ), 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,wecomplete the proof. Remark 14.23. In the above, for complexes X ∈ K(Mod A), Y ∈ K(Mod Aop), we take P ∈ Ks(Proj A) Q ∈ Ks(Proj Aop) which have quasi-isomorphisms P → X, Q → Y . Then we have an isomophisms ⊗L ∼ ⊗ ∼ ⊗ ∼ ⊗ X AY = P AY = X AQ = P AQ . ∈ ∈ op i i ⊗L For X Mod A and Y Mod A ,wedenote TorA(X, Y )=H(X AY ). 15. Bimodule Complexes

Throughout this section, k is a commutative ring, A, B, C are k-algebras, AUB an A-B-bimodule, BVC an B-C-bimodule, AWC an A-C-bimodule and C SA a C- A-bimodule. Proposition 15.1. The following hold. ∼ 1. HomAop ⊗ B (AUB, HomC (B VC , AWC)) = HomAop ⊗ C (AU⊗BVC , AWC ). k ∼ k 2. HomAop ⊗ B(AUB, HomC (BVC , AWC)) = HomB⊗ Cop (B VC , HomA(AUB , AWC )). k ∼ k ⊗ ⊗ op ⊗ op ⊗ 3. (AU BVC ) A ⊗kC (CSA) = (AUB ) A ⊗kB(BV C SA). op op 4. If AUB is A ⊗kB-projective, VC is C-projective, then AU⊗B VC is A ⊗kC- projective. op 5. If BV is B-flat, AWC is A ⊗kC-injective, then HomC (B VC , AWC ) is op A ⊗kB-injective. op 6. If BVC is B ⊗kC-projective, AW is A-injective, then HomC (B VC , AWC ) is op A ⊗kB-injective. op op 7. If AUB is A ⊗kB-flat, VC is C-flat , then AU⊗BVC is A ⊗kC-flat. Corollary 15.2. The following hold. op 1. AU is A-projective, VC is C-projective, then AU⊗kVC is A ⊗kC-projective. op 2. B V is B-flat, AW is A-injective, then Homk(B Vk, AWk) is A ⊗kB-injective. op 3. AU is A-flat, VC is C-flat, then AU⊗kVC is A ⊗kC-flat.

Proposition 15.3. Let AM be an A-module, B N a B-module, then the following hold. ∼ op 1. HomA ⊗kB (AUB, Homk(BBk, AVk)) = HomA(AU, AV ). DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 71 ∼ 2. Hom ( M, W) = Hom op ⊗ ( M⊗ C , W ). A A∼ A A k C A k C A C ⊗ ⊗ op ⊗ 3. U B N = (AUB) A ⊗kB (B N kAA). op 4. If B is k-projective, AUB is A ⊗kB-projective, then AU is A-projective. op 5. C is k-flat, AWC is A ⊗kC-injective, then AW is A-injective. op 6. B is k-flat, AUB is A ⊗kB-flat, then AU is A-flat. op op Proposition 15.4. For U ∈ K(Mod A ⊗kB), V ∈ K(Mod B⊗k C ), W ∈ op op K(Mod A ⊗kC), S ∈ K(Mod C⊗kA ),thefollowinghold. 1. We have an isomorphism ∼ Hom op ( U , Hom ( V , W )) Hom op ( U ⊗ V , W ). A ⊗kB A B C B C A C = A ⊗kC A B C A C 2. We have an isomorphism ∼ Hom op ( U , Hom ( V , W )) Hom op ( V , Hom ( U , W )). A ⊗kB A B C B C A C = B⊗k C B C A A B A C 3. We have an isomorphism ∼ ⊗ ⊗ op ⊗ op ⊗ (AU B VC ) A ⊗kC (CSA) = (AUB ) A ⊗kB (B V C SA). Proof. 1. Let α be the trifunctorial isomorphism in 1 of Proposition 15.1. For every (p, q, r) ∈ Z3,define

r(2q+r+1) −p −q r − 2 op ⊗ → φp,q,r =( 1) α :HomA kB (AUB , HomC (BVC , AWC )) −p −q r op ⊗ ⊗ HomA k C (AU B VC , AWC ), ∧ then (φp,q,r)induces the isomorphism between triple complexes. By taking Tot, we have the assertion. 2. Let β be the tri functori al isomorphism in2ofProposition 15.1. For every (p, q, r) ∈ Z3,deﬁne

(p+q)(p+q+2r+1) −p −q r − 2 op ⊗ → φ p,q,r =( 1) β :HomA kB (AUB , HomC (B VC , AWC )) −q −p r op ⊗ HomB kC (BVC , HomA(AUB , AWC )), ∧ then (φp,q,r)induces the isomorphism between triple complexes. By taking Tot, we have the assertion. 3. Let γ be the trifunctorial isomorphism in3ofProposition15.1.Forevery (p, q, r) ∈ Z3,deﬁne

− r(2q+r 1) p q r − 2 ⊗ ⊗ op ⊗ → ψp,q,r =( 1) γ :(AU B VC ) A kC (C SA) p q r ⊗ op ⊗ ⊗ (AUB ) A kB(BV C SA), then (φp,q,r)induces the isomorphism between triple complexes. By taking Tot, we have the assertion.

Proposition 15.5. The following hold. − ⊗ → 1. For a fu nctor A UB : K(Mod A) K(Mod B) and its right adjoint − → HomB(AUB, ):K(Mod B) K(Mod A),thereexistthe left derived functor − ⊗ L → AUB : D(Mod A) D(Mod B) and the right derived functor − → R HomB (AUB , ):D(Mod B) D(Mod A) such that − ⊗ L ∼ − R HomB( AUB, ?) = R HomA( , R HomB(AUB, ?)). 72 JUN-ICHI MIYACHI

− ⊗ L In particular, we have AUB R HomB (AUB , ?). op 2. For a right adjoint pair of functors HomA(−, AUB):K(Mod A ) → K(Mod B), − → op HomB( , AUB ):K(Mod B) K(Mod A ),thereexisttherightderived − op → − functors R HomA( , AUB):D(Mod A ) D(Mod B), R HomB ( , AUB ): D(Mod B) → D(Mod Aop) such that − ∼ − R HomB( , R HomA(?, AUB)), = R HomA(?, R HomB( , AUB)). − In particular, (R HomA(?, AUB ), R HomB ( , AUB ))isarightadjoint pair. Proof. 1. Let X ∈ K(Mod Aop ), Y ∈ K(Mod B). According to Theorems 14.14, 14.22 and Remark 14.15, we may assume X ∈ Ks(Proj A), Y ∈ Ks(Inj B). It is clear for the existence of the derived functor. By Proposition 15.4 we have isomorphisms ⊗ L ∼ ⊗ R HomB (X AUB,Y ) = HomB (X A UB ,Y ) ∼ = HomA(X , HomB(AUB,Y )) ∼ = R HomA(X , R HomB (AUB ,Y )). We get the last assertion by taking cohomologies of the above isomorphisms. 2. Let X ∈ K(Mod Aop), Y ∈ K(Mod B). According to Theorem 14.14 and Remark 14.15, we may assume X ∈ Ks(Proj Aop ), Y ∈ Ks(Proj B). It is clear for the existence of the derived functor. By Proposition 15.4 we have isomorphisms ∼ R HomB(Y , R HomA(X , AUB)) = HomB (Y , HomA(X , AUB)) ∼ = HomA(X , R HomB (Y , AUB )) ∼ = R HomA(X , R HomB(Y , AUB)). We get the last assertion by taking cohomologies of the above isomorphisms. −⊗L → Remark 15.6. The above derived functors AUB : D(Mod A) D(Mod B)and − → R HomB (AUB , ):D(Mod B) D(Mod A)arethederived functors of ∂-functors − ⊗ − A UB and HomB (AUB , ), respectively. But they arenotthederived functors of bi-∂-functors in general!

op We denote by ResA : Mod A ⊗kB → Mod A the forgetful functor, and use the op same symbol ResA : K(Mod A ⊗kB) → K(Mod A)for theinduced ∂-functor. Proposition 15.7. If B is k-projective or C is k-ﬂat, then op ⊗ op × op ⊗ → op ⊗ R HomA : D(Mod B kA) D(Mod C k A) D(Mod C k B) exists, and we have a commutative diagram

Resop ×Res D(Mod Bop ⊗ A)op × D(Mod Cop ⊗ A) −−−−−−−−A →A D(Mod A)op × D(Mod A) k  k    R Ho mA R HomA

op Resk D(Mod C ⊗k B) −−−−→ D(Mod k) ∈ Proof. Assume B is k-projective. According to Proposition 15.4, for B XA ⊗ ∈ K(Mod B k A), YA K(Mod A), we have ∼ Hom op ( X , Hom ( B , Y )) Hom (X ,Y ). B ⊗kA B A k k B k A = A A A ∈ s op ⊗ ∈ If B XA K (Proj B k A), then by the above isomorphism, we have ResAX Ks(Proj A). Therefore we get the assertion by Theorem 14.5. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 73

∈ op ⊗ ∈ Assume C is k-ﬂat. For C YA K(Mod C k A), XA K(Mod A), we have ∼ ⊗ ⊗ HomCop kA (CC k XA, C YA) = HomA(XA,YA). ∈ s op ⊗ ∈ If C YA K (Inj C k A), then by the above isomorphism, we have ResAY Ks(Inj A). By the same reason as the above. Proposition 15.8. If either A or C is k-ﬂat, then ⊗L op⊗ × op ⊗ → op ⊗ A : D(Mod A kB) D(Mod B k C) D(Mod A k C) exists, and we have a commutative diagram

Res ×Res op D(Mod Aop⊗ B) × D(Mod Bop ⊗ C) −−−−−−−−−→B B D(Mod B) × D(Mod Bop ) k  k    ⊗L ⊗L B B

op Resk D(Mod A ⊗k C) −−−−→ D(Mod k) op op Proof. Assume A is k-flat. For X ∈ K(Mod A ⊗kB), Y ∈ K(Mod B ), by Proposition 15.4, we have an isomorphism ⊗ ⊗ ∼ ⊗ (AXB ) Aop ⊗B (BY kAA) = X B Y . ∈ s op ⊗ ⊗ If AXB K (Proj A k B), then by the proof of Theorem 14.22, X B Y is acyclic ifeitherX 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]]/(x2). Let 2 AMB = F[[x]]/(x ), ANC = F , R HomA the right derived functor of − op ⊗ op × + op⊗ → op ⊗ HomA : K (Mod B kA) K (Mod C kA) K(Mod C kB). Then we have ∼ 2 R HomA(M, N) = HomA(F [[x]]/(x ),F). − op × + → Let R HomA be the right derive d functor of HomA : K (Mod A) K (Mod A) K(Mod k). Then we have ∼ 0 → 1 R HomA(M,N) = X X 2 0 1 HomA(x ,F ) where X → X =HomA(F [[x]],F) −−−−−−−−→ HomA(F [[x]],F). Then R HomA(M, N) R HomA(M,N) in D(Mod k). 74 JUN-ICHI MIYACHI

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 pr oj A is the fullsubcategoryof mod A consisti ng of finitely generated projective right A-modules. Definition 16.1 (Perfect Complex). Acomplex X ∈ D(Mod A)iscalled aperfect complex if X is isomorphic to a complex of Kb(pr oj A)inD(Mod A). We denote by D(Mod A)perf the triangulated full subcategory of D(Mod A)consisti ng of perfect complexes. Lemma 16.2. For X ∈ Kb (Proj A),thefollowing are equivalent. 1. X is a compact object in Kb(Proj A). 2. X is isomorphic to an object of Kb(pr oj A). Proof. 2 ⇒ 1. By Lemma 16.3. 0 1 ⇒ 2. Let X = X0 −d→ X1 → ... → Xn,withXi ∈ Proj A.Byadding P −→1 P to X,wemayassume that X0 is a free A-module A(I).IfI is a finite 0 set, then by 2 ⇒ 1 X is also compact, and hence τ≥1X is compac t. by induction (I) ∼ on n,wegettheassertion. Otherwise, Since we have HomK(Mod A)(X ,A ) = (I) (I) HomK(Mod A)(X ,A) ,thecanonical morphism X → A factors through a direct summand µ : Am → A(I) for some m ∈ N. Then there is a homotopy morphism 1 → (I) − 0 (I) → m (I) h : X A such that 1A(I) µg = hd with some g : A A .LetA = d0| m⊕ (J) (J) −−−−→A(J) 1 −→ph (J) A A be the canonical decomposition, then A X A =1A(J) , (I) → (J) ∼ − ⊕ where p : A A is the canonical projection. Therefore X = M (1A(J) )[ 1] X , where X : Am → X1 → ...→ Xn with X1 beingadirectsummand of X1. Then we reduce the case of X0 being a finitely generated free A-module.

Lemma 16.3. For a complex X ∈ K(Proj A),thefollowing hold. 1. X is a compact object in K(Mod A). b ∼ 2. There is a complex P ∈ K (proj A) such that X = P in K(Mod A). ⇒ ∈ b { } Proof. 2 1. We may assume X K (proj A). Let Yi 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 ∼ HomA(X , Yi ) = Tot Hom A(X , Yi ) i∈I i∈I ∼ = Tot Hom A(X ,Yi ) i∈I ∼ = Hom (X ,Y ). i∈I A i By taking cohomology, we haveanisomorphism ∼ HomK(Mod A)(X , Yi ) = HomK(Mod A)(X ,Yi ). i∈I i∈I ⇒ i − 1 2. Since C (X )= i∈Z C (X )[ i], we have isomorphisms in Ab i ∼ i HomK(Proj A)(X , C (X )[−i]) = HomK(Proj A)(X , C (X )[−i]) i∈Z i∈Z ∼ i = HomK(Proj A)(X , C (X )[−i]). i∈Z DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 75

i Then itiseasytoseeHomK(Proj A)(X , C (X )[−i]) = 0 for all but finitely many i ∈ Z. Therefore for all but finitely many i ∈ Z,wehaveexactsequences i+1 i i i HomA(X ,C (X )) → HomA(C (X ),C (X )) → O. This means that the canonical morphisms Ci(X) → Xi+1 are split monomor- ∼ phisms. Then there are m ≤ n such that X = σ ≥mσ≤nX . σ ≥mσ≤nX ∈ K(Proj A). Then we may assume X : X0 → X1 → ... → Xn with H0(X) =0, Hn(X) =0.Bytheproofo fLemma16.2, 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). t s ∼ Proof. Since K (Proj A) = D(Mod A), it is trivial by Lemma 16.3. b Lemma 16.5. Let T ∈ K (pr oj A) with HomK(Mod A)(T ,T [i]) = 0 for i =0 ,and − B =EndK(Mod A)(T).Thenthereexistsafullyfaithful ∂-functor F : K (Proj B) → − K (Proj A) such that ∼ 1. FB = 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 HomK (Proj A)(T ,X [i]) = 0 for all i. − ∈ − Proof. Let X K (Proj A)suchthatGX = O. Then HomK (Proj A)(T ,X [i]) ∼ { } = HomK− (Proj B)(B,GX [i]) = 0 for all i. Therefore Ker G = O .Bytheleft versi on of Proposition9.13, G and F are equival ences. 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 tri angulated category. For a subcategory B of C,we canconstruct the smallest triangulated full subcategory EB which is closed under isomorphisms and contains B as follows. Let E0B = B.Forn>0, let E nB be the full subcategory of C consisti ng of n−1 objects X there exist U, V ∈E B satisfying that either of (X,U, V, ∗, ∗, ∗)or ∗ ∗ ∗ C EB E nB (U, V, X, , , )isatri angl e in . Then itiseasytoseethat = n≥0 is thesmallest tri angulatedfull subcategory which is closed under isomorphisms and contains B Theorem 16.9. Let T be a complex of Kb (proj A) such that (a) HomK(Mod A)(T ,T [i]) = 0 for i =0 , b (b) add TA generates K (proj A). Then F : K−(Proj B) → K−(Proj A) is an equivalence. 76 JUN-ICHI MIYACHI

Proof. It suﬃces to show that T satisﬁes the condition of Lemma 16.6. Since b add TA generates K (pr oj A), if HomK− (Proj A)(T ,X [i]) = 0 for all i,then HomK− (Proj A)(A, X [i]) = 0 for all i.ThusX = O. − Lemma 16.10. For X ∈ D (Mod A),thefollowing are equivalent. 1. X ∈ Db(Mod A). − 2. For any Y ∈ D (Mod A),thereisn such that HomD(Mod A(Y ,X [i]) = 0 for all in.

Proof. 1 ⇒ 2. It is trivial. ⇒ ∈ − i 2 1. We may assume X K (Proj A). Let M = i∈Z C (X ). By the same reason as the proof of Lemma 16.3, HomK− (Mod A)(X ,M[i]) = 0 for all i>nif 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. − ∼t − 1. D (Mod A) = D (Mod B). t b ∼ b 2. D (Mod A) = D (Mod B). t b ∼ b 3. K (Proj A) = K (ProjB). t b ∼ b 4. K (pr oj A) = K (proj B). ∈ b ∼ 5. There exists T K (proj A) with B = HomKb(proj A)(T ) such that (a) HomK(Mod A)(T ,T [i]) = 0 for i =0 , b (b) add TA generates K (proj A). ∈ b ∼ 6. There exists T K (proj A) with B = HomKb(proj A)(T ) such that (a) HomK(Mod A)(T ,T [i]) = 0 for i =0 , − ∈ − (b) For X K (Proj A), X = O whenever HomK (Proj A)(T ,X [i]) = 0 for all i. Proof. By Theorem 16.9, Lemmas 16.6, 16.10, 16.11 and 16.2.

b b op Remark 16.13. Since the functors HomA(−,A):K (proj A) → K (proj A )and b op b HomA(−,A):K (pr oj A ) → K (pr oj A)induce a duality between them, the condition 5 of Theorem 16.12 induces the left version of the condition 5. Therefore, − ∼t − − ∼t − D (Mod A) = D (Mod B)ifandonly if D (Mod Aop ) = D (Mod Bop ). ∈ b Deﬁnition 16.14. AcomplexTA K (proj A)iscalledatilting complex for A provided that 1. HomK(Mod A)(T ,T [i]) = 0 for i =0. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 77

b 2. add TA generates K (proj A). We say that B is derived equivalent to A if there is a tilting complex TA such ∼ that B = EndK(Mod A)(T ). Lemma 16.15. Let U be a collectionofobjectsK[r,s](pr oj A) for some r ≤ s.Con- sider a sequence of triangles → → → U1 U0 X1 → → → U2[1] X1 X2 ... − → → → Un[n 1] Xn−1 Xn ... , − with all U ∈U.Thenhlim X ∈ K˜ (proj A). i −→ n Proposition 16.16. For rings A, B,thefollowing are equivalent. 1. B is derivedequivalent to A. − ∼t − 2. K (pr oj A) = K (proj B). t −,b ∼ −,b In this case, we have K (proj A) = K (proj B).

t b ∼ b Proof. 1 ⇒ 2. According to Theorem 16.12, we have K (proj A) = K (proj B). By − ∼t − Lemma 16.15, it is easytoseethatK (proj A) = K (pr oj B) ⇒ ∈ − − 2 1. Let X K (pr oj A). Since n∈Nτ≤−nX exists in K (pr oj A), if X is acompact object in K−(pr oj A), then HomK(Mod A)(X ,τ≤−nX )=0 for all but ﬁnitely many n.IfHomK(Mod A)(X ,τ≤−nX )=0,thenbyProposi- ∼ ti on 4.8, X = τ≥−n+1X ⊕τ≤−nX [−1]. According to Proposition 10.23 X ∈ b − K˜ (proj A). As a consequence, X is a compact object in K (proj A)ifandonly if X is isomorphic to an object in Kb(pr oj A). Since compactness is a categor- t b ∼ b ical property, we have K (proj A) = K (proj B). By Theorem 16.12, we get the statement. The last assertion is trivial byLemma16.10. Lemma 16.17. For P ∈ C−,b(Proj A),wehave isomorphisms in K−,b(Proj A) ∼ hlim τ≥− P lim −,b τ≥− P −→ i = −→ C (Proj A) i ∼ = −lim→ K−,b(Proj A)τ≥−iP . Proof. According to Proposi tion 11.7, we have the ﬁrst isomorphism. For Y ∈ − C ,b(Proj A), there is n ∈ Z such that Hi(Y )=0forall i ≤ n.Applying −i HomK(Mod A)(−,Y [j]) to a triangle τ≥−i+1P → τ≥−iP → P [i] → τ≥−i+1P [1], we have an exact sequence

−i µi HomK(Mod A)(P [i],Y [j]) → HomK(Mod A)(τ≥−iP ,Y [j]) −→ −i HomK(Mod A)(τ≥−i+1P ,Y [j]) → HomK(Mod A)(P [i],Y [j +1]). By Exercise 6.22, we have −i ∼ −i j−i+1 HomK(Mod A)(P ,Y [j − i + 1]) = HomA(P , H (Y )) = 0 78 JUN-ICHI MIYACHI for i ≥ j − n +1,andthenµi are epic for i ≥ j − n +1. ByExercise11.5, we have an exact sequence 0 → HomK(Mod A)(lim −,b τ≥−iP ,Y ) → HomK(Mod A)(τ≥−iP ,Y ) → −→ C (Proj A) i HomK(Mod A)(τ≥−iP ,Y ) → 0. i Hence ∼ −lim→ C−,b(Proj A)τ≥−iP = −lim→ K−,b(Proj A)τ≥−iP .

Proposition 16.18. Let A, B be coherent rings. The following are equivalent. 1. B is derivedequivalent to A. − ∼t − 2. D (mod A) = D (mod B). t b ∼ b 3. D (mod A) = D (mod B). Proof. 1 ⇒ 2, 3. By Proposition 16.16. 3 ⇒ 1. By Corollary 10.13, Db(mod A)andDb(mod B)aretriangle equivalent − − to K ,b(proj A)andK ,b(proj B), respectively. Then we have an equivalence F : − − − − K ,b(proj A) → K ,b(proj B)anditsquasi-inverse G : K ,b(pr oj B) → K ,b(proj A). We may assume that G(B)=Q : ... → Q−1 → Q0 and F(A)=P : ... → n−1 → n ∼ P P .ByPropositi on 11.7, τ≤−1P = −lim→τ≥−iτ≤−1P .SinceG is an equiv- ∼ −,b alence, by Lemma 16.17 Gτ≤−1P = −lim→ K−,b(Proj A)Gτ≥−iτ≤−1P in K (proj B). −,b k Since Gτ ≤−1P ∈ K (pr oj B), there exists k ∈ Z such that H (Gτ ≤−1P ) =0 j k and H (Gτ ≤−1P )=0forall j>k.LetCk =C(Gτ≤−1P ), then Ck = k ∈ −,b − C (Gτ ≤−1P ) K (pr oj A)andHomK(proj A)(Gτ≤−1P ,Ck[ k]) =0,becauseA is ≥ − coherent. Therefore , there is m 1suchthatHomK(proj A)(Gτ≥−mτ≤−1P ,Ck[ k]) =0. Then we have m<0, because Q ∈ K[−∞,0] (proj A). Since ∼ ∼ 0 HomK(proj A)(P ,τ≤−1P ) = HomK(proj A)(A, Gτ≤−1P ) = H (Gτ≤−1P )=0, ∼ by Proposition 4.8 τ≥0P = P ⊕τ≤−1P [−1]. According to Proposition 10.23, P ∈ ˜b K (proj A), and hence PA is a tilting complex.

17. Two-sided Tilting Complexes 17.1. The CaseofFlatk-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 Ai be an k-algebra with a tilting complex Ti whose endomor- ⊗ ⊗ phism is isomorphic to Bi (i =1, 2). Then T1 k T2 is a tilting complex for A1 kA2 whose endomorphism is isomorphic to B1⊗kB2. j Proof. Since Ti is Ai-projective, by Proposition 15.1 we have isomorphisms for all i, j, k, l i⊗ j k⊗ l ∼ i j k⊗ l HomA1⊗kA2 (T1 kT2 ,T1 kT2) = HomA1 (T1, HomA1 (T2 ,T1 kT2)) ∼ i k l j = Hom (T ,A )⊗ op T ⊗ T ⊗ Hom (T ,A ) A1 1 1 A1 1 k 2 A2 A2 2 2 ∼ i k ⊗ j l = HomA1 (T1,T1 ) k HomA2 (T2 ,T2). DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 79

This induces an isomorphism between quadruple complexes. Since Ti arebounded complexes, we have an isomorphism between complexes ∼ Hom (T ⊗ T ,T ⊗ T ) → Hom (T ,T) ⊗ Hom (T ,T). A1⊗k A2 1 k 2 1 k 2 A1 1 1 k A2 2 2

Since Ai, Bi are k-ﬂat, we have isomorphisms in D(Mod k) Hom (T ,T) ⊗ Hom (T ,T) ∼ Hom (T ,T) ⊗ L Hom (T ,T) A1 1 1 k A2 2 2 = A1 1 1 k A2 2 2 ∼ ⊗ L = EndK(Mod A1)(T1) k EndK(Mod A2)(T2) ∼ L = B ⊗ B ∼ 1 k 2 = B1⊗kB2. ⊗ ⊗ Thus HomK(Mod A1⊗kA2)(T1 k T2,T1 k T2[i]) = 0 for i =0.Itiseasytoseethat ⊗ ∼ ⊗ EndK(Mod A1⊗kA2)(T1 k T2) = B1 kB2 as k-algebras. ⊗ − b → b ⊗ Let F = A1 k : K (pr oj A2) K (proj A1 kA2). Since add T2 generates b ⊗ b ⊗ −⊗ b K (proj A2), add A1 kT2 generates K (pr oj A1 kA2). Let G = k T2 : K (pr oj A1) → b ⊗ b K (pr oj A1 kA2). Since add T1 generates K (pr oj A1), the triangulated full sub ⊗ ⊗ ⊗ category generated by add T1 kT2,contains add A1 kT2. Therefore add T1 kT2 b generates K (proj A1⊗kA2). Proposition 17.2. Let A be a k-algebra which is derived equivalent to a k-algebra B.Thenwehave isomorphisms HHk(A, A)=ExtAop ⊗ A(A, A) ∼ k = ExtBop ⊗ B(B,B) k =HHk(B, B). Here HHk means Hochschild homology. Proof. Let T be a tilting complex for A whose endomorphism ring is isomorphic ∗ op to B.ByTheorem 17.1, T ⊗kT is a tilting complex for A ⊗kA whose endo- op ∗ morphism ring is isomorphic to B ⊗kB, where T =HomA(T ,A)Accordingto b op b op Lemma 16.5, there isaequivalence F : D (Mod B ⊗kB) → D (Mod A ⊗kA) op ∗ b op ∼ which sends B ⊗kB to T ⊗kT .LetX ∈ D (Mod B ⊗kB)suchthatFX = A b op in D (Mod A ⊗kA). Then we have n ∼ op H (X ) Hom b op (B ⊗ B,X [n]) = D (Mod B ⊗kB) k ∼ ∗ Hom b op (T ⊗ T ,A[n]) = D (Mod A ⊗kA) k ∼ ∗∗ = HomDb(Mod A)(T ,T [n]) ∼ = HomDb(Mod A)(T ,T [n]). ∼ b op Hence X = B in D (Mod B ⊗kB). Deﬁnition 17.3. Let A be a k-algebra which is derived equivalent to a k-algebra B,andP atiltingcomplex for A whose endomorphism ring is isomorphic to B. Then we have a triangle equivalence F : Db(Mod B) → Db(Mod A) which sends B to P .Wetake a complex Q of Kb (proj B) which is isomorphic to F −1A in b ∗ K (Mod B), and P =HomA(P ,A), Q =HomB (Q ,B). Then a tilting complex B⊗kP induces the triangle equivalence b op b op D (Mod B ⊗kB) → D (Mod B ⊗kA). 80 JUN-ICHI MIYACHI

b op b op b op b op D (Mod A ⊗kA) D (Mod B ⊗kA) D (Mod A ⊗kB) D (Mod B ⊗kB) A T T ∨ B

b op Let T be the image of B of D (Mod B ⊗kB). Also, a tilting complex A⊗kQ induces the triangle equivalence

b op b op D (Mod A ⊗kA) → D (Mod A ⊗kB).

∨ b op Let T be the image of A of D (Mod A ⊗kA). Proposition 17.4. The following hold. ∼ b 1. ResAT = P in D (Mod A). ∼ b op 2. ResBop T = Q in D (Mod B ). ∨ ∼ b 3. ResBT = Q in D (Mod B). ∨ ∼ ∗ b op 4. ResAop T = P in D (Mod A ). Proof. 1. We have isomorphisms of functors K−(proj A) → Mod k ∼ op − op ⊗ − HomD(Mod A)( ,ResAT ) = HomD(Mod B ⊗kA)(B k ,T ) ∼ ∗ op ⊗ − = HomD(Mod A ⊗kA)(P k ,A) ∼ ∗∗ = HomD(Mod A)(−,P ) ∼ = HomD(Mod A)(−,P ). − 2. We have isomorphisms of functors K (pr oj A) → Mod k ∼ op − op op −⊗ HomD(Mod B )( ,ResB T ) = HomD(Mod B ⊗kA)( kA, T ) ∼ op −⊗ = HomD(Mod B ⊗kB)( kQ ,B) ∼ = HomD(Mod Bop)(−,Q ). 3, 4. Similarly.

Lemma 17.5. There is an isomorphism φ : P → ResAT in D(Mod A) such that φf = λ(f)φ for all f ∈ EndD(Mod A)(P ),whereλ : B → EndD(Mod A)(ResAT ) is the left multiplication morphism. Proof. For f ∈ EndD(Mod A)(P ), by the above isomorphisms, we have a commutati ve diagram

∼ op ∼ Ho m (−,Res T ) −−−−−→ Hom op ⊗ (B ⊗ −,T ) −−−−−→ Hom (−,P ) D(Mod A) A D(Mod B kA) k D(Mod A)

Hom(−,λ(f)) Hom(fop ⊗−,A) Ho m(−,f )

∼ op ∼ − −−−−−→ op ⊗ ⊗ − −−−−−→ − Ho mD(Mod A)( ,ResA T ) HomD(Mod B kA)(B k ,T ) HomD(Mod A)( ,P ).

Theorem 17.6. For ∗ = nothing, +, −,b,the∂-functor − ⊗ L∗ ∗ → ∗ B T : D (Mod B) D (Mod A) is an triangle equivalence, and its quasi-inverse is ∗ − ∼ − ⊗ L∗ ∨ ∗ → ∗ R HomA(T , ) = A T : D (Mod A) D (Mod B). DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 81

∈ ∗ ⊗ L∗ ∼ ⊗ Proof. For a complex X D (Mod B), X B T = X B Q in D(Mod k). Then − ⊗ L∗ ∗ − − ⊗ L∗ ∨ B T is a way-out in both directions. Similarly, R HomA(T , ), A T are way-out in both directions. Therefore we have theabove functors betwe en the above derived categories. By Proposition 17.4, We have ⊗ L∗ ∨ ∼ ⊗ L∗ ∗ ResAT A ResAop T = P A P ∼ = EndD∗(Mod A)(P ) = B ⊗ L∗ ∨ ∼ op ⊗ By Lemma17.5, we have T A T = B in D(Mod B kB). Similarl y wehave ∨ ⊗ L∗ ∼ op⊗ − ⊗ L∗ T B T = A in D(Mod A kA). Then B T is an equivalence. By adjoint- ∗ − ∼ − ⊗ L∗ ∨ ness, wehave R HomA(T , ) = A T . op Deﬁnition 17.7 (Biperfect Complex). AcomplexX ∈ D(Mod A ⊗kB)iscalled op a biperfect complex if ResAX ∈ D(Mod A)perf and ResBop X ∈ D(Mod B )perf. op We denote by D(Mod A ⊗kB)biperf the triangulated full subcategory of op D(Mod A ⊗kB)consisti ng of biperfect complexes. ∈ op ⊗ Deﬁnition 17.8. Abimodule complex B TA K(Mod B kA)iscalledatwo- sided tilting complex provided that 1. B TA is a biperfect complex. ∨ 2. There exists abiperfect complex ATB such that ⊗ L ∨ ∼ b op ⊗ (a) B T ATB = B in D (Mod B kB), ∨ ⊗ L ∼ b op⊗ (b) AT B TA = A in D (Mod A kA). ∨ We call ATB the inverse of B TA. ∗ ∗ ∗ R HomA(T , −):D (Mod A) → D (Mod B)iscalledastandard equivalence, where ∗ =nothing,+, −,b. Theorem 17.9. The following are equivalent. ∼t 1. D(Mod A) = D(Mod B). t + ∼ + 2. D (Mod A) = D (Mod B). 3. A is derived equivalent to B. 4. There exists a two-sided tilting complex BTA. Proof. By Theorem 17.6 and the dual of Lemma 16.10. ⊗ Corollary 17.10. Let BTA and C SB be two-sided tilting complexes. Then C SB L B TA is a two-sided tilting complex. 17.2. The CaseofProjectivek-algebras. Throughout this subsection, k is acommutative 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 ResAP is a ﬁnitely generated projective A-module. Then the following hold. 1. For a right A-module M and a left A-module N,wehave B-module morphisms

M⊗A HomA(X, A) → HomA(X, M)(m⊗f → (x → mf(x))),

X⊗AN → HomA(HomA(X,A),N)(x⊗n → (f → f(x)n)). 82 JUN-ICHI MIYACHI

2. We have a functorial isomorphism of functors Mod A → Mod B ∼ −⊗A HomA(P, A) = HomA(P,−). 3. We have a functorial isomorphism of functors Mod Aop → Mod Bop ∼ P ⊗A− = HomA(HomA(P,A), −). ∗ op ∗ Lemma 17.12. Let X ∈ D (Mod B ⊗kA) with ResAX ∈ D (Mod A)perf,where ∗ = nothing, +, −,b.Thenthe following hold. ∗ op 1. We have a ∂-functorial isomorphism of ∂-functors D (Mod A ⊗kA) → ∗ op D (Mod A ⊗kB) − ⊗ L∗ ∼ ∗ − A R HomA(X ,A) = R HomA(X , ). ∗ op 2. We have a ∂-functorial isomorphism of ∂-functors D (Mod A ⊗kA) → ∗ op D (Mod B ⊗kA) ⊗ L∗− ∼ ∗ − X A = R HomA(R HomA(X ,A), ). Proof. 1. By Lemma 17.11, we have a ∂-functorial morphism of ∂-functors ∗ op ∗ op D (Mod A ⊗kA) → D (Mod A ⊗kB) − ⊗ L → − φ : AR HomA(X ,A) R HomA(X , ). b Let Q ∈ K (proj A)whichhasaquasi-isomorphism Q → ResAX .ByLemma 17.11, we have a ∂-functorial isomorphism of ∂-functors D∗(Mod A) → D∗(Mod k) − ⊗ →∼ − ψ : A HomA(Q ,A) HomA(Q , ). Since ∼ Reskφ = ψ, and H( ψ)isanisomorphism,φ is a functorial isomorphism. 2. Similarly. Corollary 17.13. Let T and T ∨ be a two-sided tilting complex and its inverse. b op Then we have isomorphisms in D (Mod B ⊗kA) ∨ ∼ T = R HomA(T ,A) ∼ = R HomB(T ,B). Theorem 17.14. For a bimodule complex B TA,thefollowing are equivalent. 1. B TA is a two-sided tilting complex. 2. B TA satisﬁes that (a) B TA is a biperfect complex, → (b) The right multiplication morphism ρA : A R HomB(T ,T ) is an iso- op morphism in D(Mod A ⊗kA), → (c) The left multiplication morphism λB : B R HomA(T ,T ) is an iso- op morphism in D(Mod B ⊗kB). 3. B TA satisﬁes that (a) B TA is a biperfect complex, (b) HomDb(Mod Bop )(T ,T [i]) = 0 for i =0, (c) HomDb(Mod A)(T ,T [i]) = 0 for i =0, (d) The right multiplication morphism ρA induces a ring isomorphism A → op EndDb(Mod Bop )(T ) , DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 83

(e) The left multiplication morphism λB induces a ring isomorphism B → EndDb(Mod A)(T ). Proof. 1 ⇒ 3. By Corollary 17.13, Lemma 17.12, we have isomorphisms in op D(Mod B ⊗kB) ∼ ⊗ L R HomA(T ,T ) = B T AR HomA(T ,A) ∼ L ∨ = T ⊗ T ∼ B A B = B.

op And we have an isomorphism in D(Mod A ⊗kA) ∼ ⊗ L R HomB (T ,T ) = R HomB (T ,B) B TA ∼ ∨ L = T ⊗ T ∼ A B A = A. −⊗L Since B TA is an equivalence, we have acommutative diagram ∼ Hom (B ,B ) −−−−→ Hom (B⊗L T ,B⊗L T ) D(Mod B) B B D(Mod A)  B A B A   

−−−−λB→ B HomD(Mod A)(TA,TA) where allverti calarrows are isomorphisms. Then λB is an isomorphism. Similarly, ρA is an isomorphism. ⇒ → 3 2. It is easy to see that we have a morphism λA : A R HomB(T ,T )in op D(Mod A ⊗kA). By taking cohomologies, we get the condition (b) of 2. Similarly, we get the condition (c) of 2. ⇒ ∨ 2 1. Let T = R HomA(T ,A), then we have isomorphisms ⊗ L ∨ ⊗ L BT ATB = BT AR HomA(T ,A) ∼ = R Hom (T ,T ) ∼ A = B. By Proposition 15.5 2, we have an isomorphism ∼ R HomA(T ,A) = R HomA(T , R HomB(T ,T )) ∼ = R HomB(T , R HomA(T ,T )) ∼ = R HomB(T ,B). Then we have ∨ ⊗ L ∼ ⊗ L AT BTA = R HomB(T ,B) B TA ∼ = R Hom (T ,T ) ∼ B = A.

Theorem 17.15. Let (Ai,Bi)bederivedequivalent k-algebras, Ti two-sided tilting b op⊗ ∨ complexes in D (Mod Bi kAi) and their inverses Ti (i =0, 1, 2). Then we have the following commutative diagrams. 84 JUN-ICHI MIYACHI

1. −⊗L − Db(Mod Aop⊗ A ) × Db(Mod Aop ⊗ A ) −−−−−A1→ Db(Mod Aop⊗ A ) 0 k 1  1 k 2 0 k 2   F0×F1 F2

−⊗L − b op⊗ × b op⊗ −−−−−B1→ b op⊗ D (Mod B0 kB1) D (Mod B1 kB2) D (Mod B0 kB2). 2. R Hom (−,−) Db(Mod Aop⊗ A ) × Db(Mod Aop ⊗ A ) −−−−−−−−−−A2 → Db(Mod Aop⊗ A ) 1 k 2  0 k 2 0 k 1   F1×F2 F0

R Hom (−,−) b op ⊗ × b op⊗ −−−−−−−−−−B2 → b op⊗ D (Mod B1 kB2) D (Mod B0 kB2) D (Mod B0 kB1). Here F = T ⊗ L −⊗L T ∨, F = T ⊗ L −⊗L T ∨, F = T ⊗ L −⊗L T ∨. 0 0 A0 A1 1 1 1 A1 A2 2 2 0 A0 A2 2 Proof. 1. We have isomorphisms (T ⊗ L − ⊗L T ∨) ⊗ L (T ⊗ L − ⊗L T ∨) 0 A0 A1 1 B1 1 A1 A2 2 ∼ ⊗ L − ⊗ L ∨ ⊗ L ⊗ L − ⊗ L ∨ = (T0 A0 ) A1 (T1 B1 T1) A1 ( A2 T2 ) ∼ ⊗ L − ⊗ L ⊗ L − ⊗ L ∨ = (T0 A0 ) A1 (A1) A1 ( A2 T2 ) ∼ (T ⊗ L −) ⊗ L (− ⊗ L T ∨). = 0 A0 A1 A2 2 ∈ op ⊗ ∈ op⊗ ∈ op ⊗ 2. Let X Mod A0 kA2, Y Mod A1 kA2, Z Mod A0 kA2.Sincewe have an adjoint isomorphism L ∼ Hom op ⊗ (X ⊗ Y ,Z ) = Hom op ⊗ (X , R Hom (Y ,Z )), D(Mod A0 kA2) A1 D(Mod A0 kA1) A2 we get the assertion by 1. 17.3. The CaseofFiniteDimensionalk-algebras. Throughout this subsec- ti on, weassume k is a field, and all algebra are finite dimensional k-algebras. We denote Dk =Homk(−,k). Definition 17.16 (Nakayama Functor). Atriangle auto-equivalence νA = − ⊗ L b → b ADkA : D (mod A) D (mod A)iscalledaNakayama functor. Proposition 17.17. Let (A, B) be derived equivalent k-algebras, T atwo-sided b op ∨ tilting complex in D (Mod B ⊗kA) and its inverse T .Thenwehaveacommu- tative diagram D−(Mod A) −−−−νA → D−(Mod A)     F F

D−(Mod B) −−−−νB → D−(Mod B), where F is a standard equivalence.

b op Proof. By Proposition 17.2, the standard equivalence G : D (Mod A ⊗kA) → b op D (Mod B ⊗kB)sends A to B. By the case of A = Aop ⊗ A, B = Bop ⊗ B and A = A = B = B = k in 1 k ∼1 k 0 2 0 2 Theorem 17.15 2, we have GDkA = DkB. DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 85

By the case of A1 = A2 = A, B1 = B2 = B and A0 = B0 = k in Theorem 17.15 1, we have FνA = νB F. Corollary 17.18. Let A be a ﬁnite dimensional k-algebra which is derived equivalent to B.IfA is asymmetric algebra, then B is a symmetric algebra.

Lemma 17.19. Let A, B be ﬁnite dimensional self-injective k-algebras, APB is op A ⊗kB-projective, B VA is A-projective and B-projective. The following hold. op 1. B ⊗kA is aself-injective algebra. op 2. HomA(P,A) is B ⊗kA-projective. 3. HomA(V, A) is A-projective and B-projective. op 4. AP ⊗BVA is A ⊗kA-projective. 5. X⊗APB is B-projective for any X ∈ Mod A. 6. Y ⊗BVA 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. ∼t Then K(mod A) = K(mod B),andtherearebimodules AMB and BNA such that

−⊗AM : mod A → mod B and −⊗B N : mod B → mod A ∼ induce an equivalence modA = modB.

b op ∨ Proof. Let T be a two-sided tilting complex in D (Mod B ⊗kA)andT its in- op verse. By taking a B ⊗kA-projective resolution of T ,anditsshifting and trun- cation, we may assume T is isomorphic to S : S−n → S−n+1 → ...→ S0 i op −n where S are B ⊗kA-projective (−n

t ∼ n −n These imply that K(mod A) = K(mod B). Let M =Ω (HomA(S ,A)), the nth op −n −n syzygy as as a B ⊗kA-moduleandN =Ω (S ), the −nth syzygy as as a op −n A ⊗kB-module. Since HomA(S ,A)isA-projective and B-projective, M is A- projective and B-projective. Similarly, N is A-projective and B-projective. Then

−⊗AM : mod A → mod B and −⊗B N : mod B → mod A induce triangle functors between modA and modB.ByLemma17.19, all terms −n ⊗ −n ⊗ but the term HomA(S ,A) AS of a double complex HomA(S ,A) B S are op⊗ p ⊗ q A kA-projective. Therefore A is a direct summand of p=q HomA(S ,A) B S op as a A ⊗kA-module. For each X ∈ mod A,wehaveisomorphismsinmodA ∼ X = X⊗AA ∼ −n −n = X⊗A HomA(S ,A)⊗B S ∼ −n −n = ω (X⊗AM⊗B S ) ∼ n −n = ω ω (X⊗ M⊗ N) ∼ A B = X⊗AM⊗B N, 86 JUN-ICHI MIYACHI where ω is the loop space functor on modA.Similarly, for each Y ∈ mod B,we have an isomorphism in modB ∼ Y = Y ⊗BN⊗AM.

Proposition 17.21. Let A and B be indecomposable symmetric k-algebras, X a b op⊗ ∨ ⊗L ∨ ∼ biperfect complex in D (Mod B kA),andX =Homk(X ,k).IfX AX = B b op in D (Mod B ⊗kB),thenX is a two-sided tilting complex. Proof. Since we have isomorphisms ∨ ∼ X = R HomA(X ,A) ∼ = R HomB(X ,B), − ⊗ L b → b by Proposition 15.5, lemma 17.12, F = B X : D (mod B) D (mod A)and ∨ − ⊗ L ∨ b → b F = AX : D (mod A) D (mod B)are both left and right adjoint to one another. Byadjunctionarrows of adjoint pairs (F, F∨), (F ∨,F), we have → ∨ ⊗ L ∨ ⊗ L → morphisms α : A X B X , β : X B X A.ByProposition 1.17, we → ⊗ L ∨ ⊗ L have a split monomorphism αF : X X AX BX ,asplitepimorphism ⊗ L ∨ ⊗ L → b op⊗ βF : X AX B X X .SinceD (mod B kA)isaKrull-Schmidt cate- ⊗ L ∨ ∼ gory by Corollary 11.19, X AX = B implies that αF , βF are isomorphisms in b op D (mod B ⊗kA). If βα is not an isomorphism, then this contradicts (βα)F is an ∨ ⊗ L b op⊗ isomorphism. Therefore A is a direct summand of X B X in D (mod A kA). ∨ ⊗ L ⊗ L ∨ ⊗ L ∼ ∨ ⊗ L ∼ ∨ ⊗ L Since X B X AX BX = X B X ,wehaveA = X B X in b op D (mod A ⊗kA).

Proposition 17.22. Let A be a symmetric k-algebra which has no simple projec- −1 −1 d 0 µ tive A-module, e an idempotent of A,andP : P −−→ P = Ae⊗keA −→ A, where µ is the multiplication morphism. Then P is a tilting complex. Moreover, APA is a two-sided tilting complex if and only if dimk eAe =2. → ⊗ ∼ → ⊗ ∼ Proof. Since DkP =(DkA Dk(Ae keA)) = (A Ae keA), HomA(PA,PA) = P ⊗ ADkP has the form Ae⊗ eA −−−−→ A k   

Ae⊗keAe⊗keA −−−−→ Ae⊗keA, where theleftverticalarrow is monic and the bottom horizontal arrow is epic. i Then H (HomA(PA,PA)) = 0 for i =0.Since P =(eAe⊗keA → eA)⊕((1 − e)Ae⊗keA → (1 − e)A) and dimK eAe = n ≥ 2, ∼ n−1 P = (eA → O)⊕((1 − e)Ae⊗keA → (1 − e)A), DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 87

b and then P generates K (pr oj AA). By the above diagram, we have an isomorphism op in mod A ⊗kA 0 ∼ Ae⊗keA⊕ H (P ⊗A DkP )⊕Ae⊗keA = Ae⊗keAe⊗keA⊕A. By the Krull-Schmidt Theorem, we have 0 ∼ n−2 H (P ⊗A DkP ) = A⊕(Ae⊗keA) .

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 (Coti lti ng Bimodule Complexes). Let A be a right coherent k- ∈ b op⊗ algebra and B a left coherent k-algebra. A complex B UA D (Mod B kA)is called a cotilting B-A-bimodulecomplex provided that it satisfies ∈ b ∈ b op 1. ResAU Dc (Mod A)fid and ResBop U Dc (Mod B )fid. 2. HomD(Mod A)(U ,U [i]) = 0 for all i =0. 3. HomD(Mod Bop )(U ,U [i]) = 0 for all i =0. 4. the left multiplication morphism B → EndD(Mod A)(U )isaringisomorphism. op 5. the right multiplication morphism A → EndD(Mod Bop )(U ) is a ring isomorphism. In case of B = A,wewillcallacotiltingA-A-bimodule complex a dualizing A-bimodule complex. Proposition 18.2. Let A be a right coherent k-algebra and B aleftcoherent k- ∈ b op ⊗ algebra, and B UA D (Mod B kA) acotiltingB-A-bimodule complex. Then ∗ − ∗ → † op R HomA( ,U ):Dc (Mod A) Dc(Mod B ) and † − † op → ∗ R HomB ( ,U ):Dc(Mod B ) Dc (Mod A) induce the duality, where (∗, †)=(nothing, nothing), (+, −), (−, +), (b, b). ∈ b ∗ − Proof. Since ResAU Dc (Mod A)fid,byPropositi on 10.21 R HomA( ,U )is ∗ ∼ ∈ † op way-out in both directions. Since R HomA(A, U ) = ResBop U Dc(Mod B ),we ∗ − ∗ → † op have R HomA( ,U ):Dc (Mod A) Dc(Mod B )byProposition 12.12. Simi- † − † op → ∗ larly we have R HomB ( ,U ):Dc(Mod B ) Dc (Mod A). Since ∗ − † − (R HomA( ,U ), R HomB ( ,U )) is aright adjointpair, we have adjuncti on arrows † ∗ η : 1 ∗ → R Hom (−,U ) ◦ R Hom (−,U ) Dc (Mod A) B A ∗ † θ : 1 ∗ op → R Hom (−,U ) ◦ R Hom (−,U ). Dc (Mod B ) A B ∗ op⊗ It is not hard thatwehave a commutative diagram in Dc (Mod A K A) η(A) † ∗ A −−−−→ R Hom (R Hom (A, U ),U) B  A 

−−−−ρA→ † A R HomB (U ,U ), 88 JUN-ICHI MIYACHI

Then η(A)isanisomorphism.ByProposition 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 Cp,q =0(p<0 or q<0), Cj → Ij quasi-isomorphisms with Ij :0→ I−s,j → I−s+1,j → ... ∈ K+(Inj A) for all i.Then,thereisaquasi-isomorphism from ∧ Tot C = TotC toacomplex J of the following form in K+(Inj A) O if n<−s Jn = i,j ≥− i+j=nI if n s { II → Proof. For a double complex C ,wehaveasequence of morphisms Tot τ≤nC II } ≥ Tot τ≤n−1C .Byinduction on n 0, we construct complexes Vn and morphisms of triangles in K+(A) Tot τ II C −−−−→ Tot τ II C −−−−→ Cn[1 − n] −−−−→ Tot τ II C[1] ≤n ≤n−1  ≤n    

−−−−→ −−−−→ j − −−−−→ Vn Vn−1 I [1 n] Vn[1] 0 where allverti calarrows are quasi-isomorphisms. If n =0,thenwetakeV0 = I . → II n − → n − Let n>0. Since Vn−1 Tot τ≤n−1C and C [1 n] I [1 n]arequasi- −g−−n−→1 j − A isomorphisms, we choose a morphism Vn−1 I [1 n]inC( )suchthat Tot τII C −−−−→ Cn[1 − n] ≤n−1   

−−−−gn−1→ j − Vn−1 I [1 n] + A − A → is commutative in K ( ). We take Vn =M(gn−1)[ 1] in C( ). Then Vn Vn−1 is aterm-splitepimorphism and have the above morphism of tri angles. By Proposition 11.7, we have isomorphisms in D+(A) ∧ Tot C = TotC ∼ II = ←lim− Tot τ≤nC ∼ II hlim Tot τ≤ C = ←− n ∼ hlim V = ←− n ∼ = ←lim− Vn. By the construction of Vn,wehavetherequired complex. Theorem 18.4. Let A be a right noetherian k-algebra and B a left coherent k- ∈ b op ⊗ ∈ algebra, and B UA D (Mod B kA) acotiltingB-A-bimodule complex. If I + + K (Inj A) is quasi-isomorphic to ResAU in K (Mod A),thenI contains all indecomposable injective A-modules. ∈ b Proof. According to Propositi on 18.2 and Example 10.14, for any X Dc (Mod A), − there exists P ∈ K (proj Bop)suchthat ∼ HomB (P ,U ) = R HomB(P ,U ) ∼ = X . DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 89

i i ∈ i ∈ For any P ,itiseasytoseeHomB (P ,U ) add UA. Then HomB (P ,U ) add I . + By piled resolutions lemma, we have a quasi-isomorphism X → J in K (Mod A) i ∈ i i with all J Add i∈ZI .Since A is noetherian, J is A-injective. We take X = A/p where p is any right ideal of A.SinceJ in K+(Mod A), it is easy to see that we have a monomorphism A/p → J0.Hencetheinjective envelope E(A/p)is adirect summand of J0. Corollary 18.5 (Like-Corollary). Let A be a right noetherian and left coherent ring with inj dim AA, inj dim AA < ∞.Thenanyinjective resolution of a right A-module AA contains all indecomposable injective A-modules. b b Proof. By the sametechnique in Proposition 18.2, R HomA(−,A):D (mod A) → b op b b op b D (mod A )andR HomA(−,A):D (mod A ) → D (mod A)induce a duality. By the above proof, we ge t the statement. 90 JUN-ICHI MIYACHI

References

[BBD] A. A. Beilinson, J. Bernstein and P. Deligne, Faisceaux Pervers, Astérisque 100 (1982). [BN] M. Böckstedt and A. Neeman, Homotopy Limits in Triangulated Categories, Compositio Math. 86 (1993), 209-234. [CE] H. Cartan, S. Eilenberg, “Homological Algebra,” Princeton Univ. Press, 1956. [CW] S. Mac Lane, “Categories for the Working Mathematician,” GTM 5,Springer-Verlag, Berlin, 1972. [Ha] D. Happel, “Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras,” London Math. Soc. Lecture Notes 119,UniversityPress, Cambridge, 1987. [HS] P. J. Hilton, U. Stammbach, “A CourseinHomological Algebra,” GTM 4,Springer- Ver lag, Ber lin, 1971. [Ho] M. Hoshino, Derived Categories, Seminar Note, 1997. [HK] M. Hoshino and Y. Kato, Tilting Complex es Defined by Idempotents, preprint. [Ke1] B. Keller, Deriving DG Categories, Ann.Scient.Ec.Norm.Sup.27 (1994), 63-102. [Ke2] B. Keller, Invariance and Localization for Cyclic Homology of DG algebras, Journal of Pure and Applied Algebra, 123 (1998), 223-273. [KV] B. Keller,D.Vossieck,SousLesCatégorie Dérivées, C. R. Acad. Sci. Paris 305 (1987), 225-228. [ML] S. Mac Lane, “Homology,” Springer-Verlag, Berlin, 1963. [Mi1] J. Miyachi, Localization ofTriangulatedCategories and Derived Categories, J. Algebra 141 (1991), 463-483. [Mi2] J. Miyachi, Duality for Derived Categories and Cotilting Bimodules, J. Algebra 185 (1996), 583 - 603. [Mi3] J. Miyachi, Derived Categories and Morita Duality Theory, J. Pure and Applied Algebra 128 (1998), 153-170. [Mi4] J. Miyachi, Injective Resolutions of Noetherian Rings and Cogenerators, Proceedings of The AMS 128 (2000), no. 8, 2233-2242. [Po] N. Popescu, “Abelian Categories with Applications to Rings and Modules, ” Academic Press, London-New York, 1973. [Qu] D. Quillen, “Higher Algebraic K-theory I,” pp. 85-147, Lecture Notes in Math. 341, Springer-Verlag, Berlin, 1971. [RD] R. Hartshorne, “Residues and Duality,” Lecture Notes in Math. 20,Springer-Verlag, Berlin, 1966. [Rd1] J. Rickard, Morita Theory for Derived Categories, J. London Math. Soc. 39 (1989), 436- 456. [Rd2] J. Rickard, Derived Equivalences as Derived Functors, J. London Math. Soc. 43 (1991), 37-48. [Rd3] J. Rickard, Splendid Equivalences: Derived Categories and Permutation Modules, Proc. London Math. Soc. 72 (1996), 331-358. [Rl] C.M. Ringel, “Tame Algebras and Integral Quadratic Forms,” Lecture Notes in Math. 1099,Springer-Verlag, Berlin, 1984. [RZ] R. Rouquier and A. Zimmermann, Picard Groups for Derived Module Categories, preprint. [We] C. A. Weibel, “An Introduction to Homological Algebra,” Cambridge studies in advanced mathe matics. 38,Cambridge Univ. Press, 1995. [Sp] N. Spaltenstein, Resolutions of Unbounded Complexes, Composition Math. 65 (1988), 121-154. [Ye] A. Yekutieli, Dualizing Complexes over Noncommutative Graded Algebras, J. Algebra 153 (1992), 41-84. [Ve] J. Verdier, “Catéories Déivées, état 0”, pp. 262-311, Lecture Notes in Math. 569,Springer- Ver lag, Ber lin, 1977. Index

S-injective, 20 generates a triangulated category, 75 S-projective, 20 U-colocal, 37 homotopic, 26 U-local, 37 homotopy category, 25 ∂-functor, 14 indecomposable, 11 ∂-functorial morphism, 14 initial object, 5 épaisse subcategory, 35 inverse of a two-sided tilting complex, 81 abelian category, 7 isomorphism, 2 acyclic, 28 Krull-Schmidtcategory, 12 additive functor, 5, 6 adjunction arrows, 4 left adjoint, 4 admissible epimorphism, 19 limit, 3 admissible monomorphism, 19 localization, 38 localization exact, 38 bi-∂-functorial morphism, 62 localizingsubcategory, 46, 52 bi-∂-functor, 62 bifunctor, 4 mapping cone, 25 bifunctorial isomorphism, 4 monomorphism, 2 bifunctorial morphism, 4 morphism of complexes, 24 biperfect complex, 81 morphism of double complexes, 57 bounded above complex, 24 multiplicative system, 29 bounded below complex, 24 bounded complex, 24 Nakayama functor, 84 null object, 5 category, 1 cochain complex, 24 opposite category, 2 cofinal subcategory, 31 cohomology, 28 perfect complex, 74 colimit, 3 pre-Krull-Schmidt category, 11 colocalization, 38 preadditive category, 5 colocalization exact, 38 product, 4 compatible with the triangulation, 33 productcategory, 4 complex, 24 proper epimorphism, 59 contravariant cohomological functor, 15 proper exact, 59 contravariantfunctor, 2 proper injective complex, 59 coproduct, 4 proper monomorphism, 59 cotilting bimodule complex, 87 proper projective complex, 59 covariant cohomological functor, 15 covariant functor, 2 quasi-isomorphism, 40 quotient category, 31 dense, 3 quotient functor, 31 derived equiva l ent , 7 7 quotientizing subcategory, 52 distinguishedtriangle, 14 double complex, 56 r-tuple complex, 57 dualizing bimodule complex, 87 right adjoint, 4 right adjoint pair, 4 epimorphism, 2 right derived functor, 52 ex act, 7 right derived functor of a bi-∂-functor, 63 ex act category, 19 ex act functor, 14 saturated multiplicative system, 30 section functor, 38 faithful, 3 split epimorphism, 2 finite injective dimension, 45 split monomorphism, 2 Frobenius category, 20 stable t-structure, 39 full, 3 stable category, 12 functorial isomorphism, 3 stalk complex, 24 functorial morphism, 3 standard equivalence, 81 91 92 JUN-ICHI MIYACHI 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 DERIVED CATEGORIES AND REPRE SENTATIONS OF ALGEBRAS 93

J. Miyachi: Department of Mathematics, Tokyo Gakugei University, Koganei-shi, To kyo, 184-8501, Japan E-mail address: [email protected]