Localization theory for triangulated categories

LOCALIZA TION THEOR Y F OR TRIAN GULA TED CA TEGORIES HENNING KRAUSE Contents 1. In tro duction 1 2. Categories of fractions and lo ca lization functors 3 3. Calculus of fractions 9 4. Lo ca lization for triangulate d cate gories 14 5. Lo ca lization via Bro wn represent abilit y 24 6. W ell generated triangulated categorie s 31 7. Lo ca lization for w ell generated categories 39 8. Epilogue: Bey ond well generatedness 47 App endix A. The abelianization of a tr iangulated category 48 App endix B. Locally pr e sentable abelian categories 50 References 55 1. Introduction These notes pr o vide an in tro duction to the theory of lo ca lization for triangulated catego ries. Lo calizat ion is a ma chinery to formally in v ert morphisms in a ca tegory . W e explain this formalism in some detail and w e s h o w ho w it is applied to triangulated catego ries. There are basically t wo wa ys to approac h the lo ca lization theory for triangulated catego ries and b oth are closely related to eac h other. T o explain this, let u s ﬁx a triangulated category T . The ﬁrst appr o ac h is V er dier lo c alization . F or this one c ho oses a full triangulated sub cat egory S of T and constructs a unive rs al exact fu ncto r T → T / S whic h annih i lates the ob j e cts b elonging to S . In fact, the quotien t category T / S is obtained by formally in ve rting all morphisms σ in T suc h that the cone of σ b elongs to S . On the other h a n d , there is Bousﬁeld lo c alization . In this case one considers an exact functor L : T → T toget h er w it h a n a tur a l morphism η X : X → LX for all X in T suc h th at L ( ηX ) = η ( LX ) is inv ertible. There are tw o fu ll triangulated sub cate gories arising from su c h a lo cal ization functor L . W e hav e the sub category Ker L formed by all L -acyclic ob jects, and we ha v e th e essen tial image Im L w hic h coincides with the sub catego ry form ed by all L -lo cal ob jects. Note that L , Ker L , and Im L determine eac h other. Moreo v er, L induces an equiv alence T / Ker L ∼ − → Im L . Thus a Bousﬁeld lo c alization functor T → T is nothing but the comp osite of a V erdier quotien t functor T → T / S with a fully faithful right adjoint T / S → T . Ha ving in tro duced these b asi c ob jects, there are a num b er of immediate questions. F or example, giv en a triangulated su bcategory S of T , can w e ﬁnd a lo cali zation functor 1 2 HENNING KRAUSE L : T → T satisfying Ker L = S or Im L = S ? On the other hand, if we start with L , which prop erties of Ker L and Im L are inherited from T ? It turn s out that well generated triangulated categ ories in the sense of Neeman [33] p ro vide an excelle nt setting for studying these qu est ions. Let us discuss brieﬂy the relev ance of we ll generated categ ories. Th e concept gener- alizes that of a co mp ac tly generated triangulated category . F or example, the derived catego ry of un b ounded c h ain complexes of mo dules o v er some ﬁxed ring is compactly generated. Also, the stable h omo topy category of CW-spectra is compactly generated. Giv en an y lo caliz ation functor L on a co mp ac tly generated triangulated categ ory , it is rare that Ker L or Im L are compactly generat ed. Ho wev er, in all kno wn examples Ker L and Im L are we ll generated. Th e follo wing theorem provides a conceptual explanation; it com bines several results from Section 7. Theorem. L et T b e a wel l gener ate d triangulate d c ate gory and S a f ul l triangulate d sub c ate gory which is close d under smal l c opr o ducts. Then the fol lowing ar e e quivalent. (1) The triangulate d c ate gory S is wel l gener ate d. (2) The triangulate d c ate gory T / S is wel l gener ate d. (3) Ther e exists a c oho molo gic al f uncto r H : T → A into a lo c al ly pr esentable ab elian c ate gory such that H pr eserves smal l c opr o ducts and S = Ker H . (4) Ther e exists a smal l set S 0 of obje cts in S such that S admits no pr op er fu l l triangulate d su b c ate g o ry clos e d under smal l c opr o ducts an d c ontaining S 0 . Mor e over, in this c ase ther e exists a lo c aliza tion functor L : T → T such that K e r L = S . Note that every ab eli an Grothendiec k categ ory is lo ca lly presen table; in particular ev ery mo dule catego ry is lo ca lly presen table. Our app r o ac h for studyin g lo caliza tion fun ct ors on w ell g enerated triangulat ed cat- egories is based on th e in terplay b et w een triangulated and ab el ian structure. A w ell kno wn construction d ue to F reyd pro vides for an y triangulate d categ ory T an ab elian catego ry A ( T ) together with a univ ersal cohomological functor T → A ( T ). Ho wev er, the category A ( T ) is u sually far to o big and therefore not m anageable. If T is w ell generated, then we h a v e a canonical ﬁltration A ( T ) = [ α A α ( T ) indexed by all r egular cardinals, such that f o r eac h α th e catego ry A α ( T ) is ab elia n and lo c ally α -p r e sentable in the sense of Gabriel and Ulmer [17]. Moreo ver, eac h inclusion A α ( T ) → A ( T ) admits an exa ct righ t adjoint and the comp osite H α : T − → A ( T ) − → A α ( T ) is the u niv ersal cohomological functor in to a lo ca lly α -present able abelian catego ry . Thus w e may think of the f uncto r s T → A α ( T ) as successiv e appro ximations of T by lo cal ly presen table ab elian cate gories. F or instance, there exists for eac h ob ject X in T some cardinal α ( X ) such that the induced map T ( X, Y ) → A β ( T )( H β X, H β Y ) is bijectiv e for all Y in T and all β ≥ α ( X ). These notes are organized as follo ws. W e start oﬀ w it h an int ro duction to catego r ies of fractions and lo caliza tion fun c tors for arbitrary categ ories. Then w e apply this to triangulated categories. First we treat arbitrary triangulated catego r ies and explain the lo caliza tion in the sense of V erdier and Bousﬁeld. Then w e p ass to compactly LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 3 and w ell generated triangulated categories where Brown represen tabilit y p ro vides an indisp ensable to ol for constructing localization fun c tors. Mo dule categories and their deriv ed categ ories are used to illustrate most of the concepts; see [12] f or complement ary material from top ology . The results on well generated categorie s are based on facts from the theory of locally presen table catego ries; w e ha v e collected these in a separate app endix. Ac kno wledgemen t. The plan to w rite an introd ucti on to the theory of triangulated lo c alization to ok sh a p e du ring the “W orkshop on T riangulated Categories” in Leeds 2006. I wish to thank the organizers Thorsten Holm, Pete r Jørgensen, and Rapha ¨ el Rouquier for their skill and diligence in organizing this meeting. Most of these notes were then written d uring a three m o nths sta y in 2007 at the Cen tre de Recerca Mate m` ati ca in Barcelona as a participan t of the sp ecial program “Homotop y Theory and Higher Categories” . I am grateful to the organizers Carles Casacub erta, Joac him Ko c k, and Amnon Neeman for creating a stim ulating atmosphere and f o r sev eral helpful discussions. Finally , I w ould like to thank Xiao-W u Chen, Daniel Murfet, and Jan ˇ S ˇ to v ´ ı ˇ cek f o r their helpful comments on a preliminary v ersion of these n ot es. 2. Ca tegories of fractions and lo caliza tion functors 2.1. Cat ego ries. T hroughout w e ﬁx a unive rse of sets in the sense of Grothendiec k [19]. The mem b ers of this unive r s e will b e called smal l sets . Let C b e a cate gory . W e denote by Ob C the set of ob jects and b y Mor C the s et of morphisms in C . Giv en ob jects X , Y in C , the set of morphisms X → Y will b e d e noted b y C ( X , Y ). The identi ty morph ism of an ob ject X is denoted by id C X or just id X . If not stated otherwise, w e alwa ys assume that the morphisms b et w een tw o ﬁxed ob jects of a ca tegory form a sm all s et. A catego ry C is called smal l if the isomorphism classes of ob jects in C form a small set. In that case we d e ﬁ ne the c ar dina lity of C as card C = P X,Y ∈C 0 card C ( X, Y ) w h ere C 0 denotes a r e p r e sentat ive set of ob ject s of C , meeting eac h isomorphism class exactly once. Let F : I → C b e a fun ct or from a small (indexing) category I to a category C . Then w e write colim − − − → i ∈I F i f o r the colimit of F , pro vided it exists. Give n a cardinal α , the colimit of F is called α -c olimit if card I < α . An example of a colimit is the copro duct ` i ∈ I X i of a family ( X i ) i ∈ I of ob jects in C where the indexing set I is alw a ys assumed to b e small. W e sa y that a category C admits smal l c opr o ducts if for eve ry f amily ( X i ) i ∈ I of ob ject s in C wh ic h is indexed by a small set I the copro duct ` i ∈ I X i exists in C . Analogous terminology is used for limits and p rodu c ts. 2.2. Cat ego ries of fra c tions. Let F : C → D b e a functor. W e sa y that F mak es a morphism σ of C in v ertible if F σ is in v ertible. Th e set of all those morphisms which F in ve rts is denoted by Σ( F ). Giv en a category C and an y set Σ of morp hisms of C , w e consider the c ate gory of fr actio ns C [Σ − 1 ] together with a canonical quotient functor Q Σ : C − → C [Σ − 1 ] ha ving the follo wing prop erties. (Q1) Q Σ mak es the m o rp hisms in Σ inv ertible. 4 HENNING KRAUSE (Q2) If a fu n ct or F : C → D m a kes the m o rp hisms in Σ inv ertible, th en there is a unique functor ¯ F : C [Σ − 1 ] → D suc h that F = ¯ F ◦ Q Σ . Note that C [Σ − 1 ] and Q Σ are essen tially u n ique if they exists. Now let us sketc h the construction of C [Σ − 1 ] and Q Σ . At this stage, we ig n o re set-theoretic issues, that is, the morphisms b et w een tw o ob jects of C [Σ − 1 ] need not to form a small set. W e put Ob C [Σ − 1 ] = Ob C . T o d e ﬁ n e the morphisms of C [Σ − 1 ], consid er the quiver (i.e . oriente d graph) with set of v ertices O b C and with set of arro ws the disjoin t un io n (Mor C ) ∐ Σ − 1 , where Σ − 1 = { σ − 1 : Y → X | Σ ∋ σ : X → Y } . Let P b e the set of p a ths in this quiv er (i.e. ﬁnite sequences of c omp osable arro ws), to gether with the ob vious comp osition whic h is the concatenatio n op erati on and denoted by ◦ P . W e deﬁne Mor C [Σ − 1 ] a s the quotien t of P m o dulo the follo wing r e lations: (1) β ◦ P α = β ◦ α for all comp osable m orp hisms α, β ∈ Mor C . (2) id P X = id C X for all X ∈ Ob C . (3) σ − 1 ◦ P σ = id P X and σ ◦ P σ − 1 = id P Y for all σ : X → Y in Σ . The comp osit ion in P in duces the comp osition of morphisms in C [Σ − 1 ]. T he fu ncto r Q Σ is the iden tit y on ob jects and on Mor C the comp osite Mor C inc − − → (Mor C ) ∐ Σ − 1 inc − − → P can − − → Mor C [Σ − 1 ] . Ha ving completed the construction of the categ ory of f rac tions C [Σ − 1 ], let us men tion that it is also called quotient c ate gory or lo c alizat ion of C with resp ect to Σ. 2.3. Adjoin t f unctors. Let F : C → D and G : D → C b e a p ai r of functors and assu me that F is left adjoint to G . W e denote by θ : F ◦ G → Id D and η : Id C → G ◦ F the corresp onding adjunction morphisms. Let Σ = Σ( F ) d en ote the set of morphisms σ of C suc h that F σ is in v ertible. Recall that a morphism µ : F → F ′ b et w een tw o functors is in v ertible if for eac h ob ject X the morphism µX : F X → F ′ X is inv ertible. Prop osition 2.3.1. The fol lowing statements ar e e quivalent. (1) The functor G is ful ly faithful. (2) The morph ism θ : F ◦ G → Id D is invertible. (3) The functor ¯ F : C [Σ − 1 ] → D satisfying F = ¯ F ◦ Q Σ is an e quivalenc e. Pr o of. See [18, I.1.3].  2.4. Lo calization functors. A fun c tor L : C → C is called a lo c alization functor if there exists a morp hism η : Id C → L su c h that Lη : L → L 2 is inv ertible and Lη = η L . Note that we only require the existence of η ; the actual morphism is n o t p a r t of the deﬁn it ion of L . Ho w ev er, w e will see that η is determined by L , up to a u nique isomorphism L → L . Prop osition 2.4.1. L et L : C → C b e a functor and η : Id C → L b e a morphism. Then the fol lowing ar e e quivalent. (1) Lη : L → L 2 is invertible and Lη = η L . (2) Ther e exists a functor F : C → D and a ful ly faithful right adjoint G : D → C such that L = G ◦ F and η : Id C → G ◦ F is the adjunction morphism. LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 5 Pr o of. (1) ⇒ (2): Let D denote the fu ll sub categ ory of C formed by all ob jects X su c h that η X is in v ertible. F or eac h X ∈ D , let θ X : LX → X b e the in v erse of η X . Deﬁne F : C → D b y F X = LX and let G : D → C b e the in clusion. W e claim that F and G form an adjoin t p a ir. In f act, it is straigh tforward to c hec k that the m a ps D ( F X, Y ) − → C ( X, GY ) , α 7→ Gα ◦ η X, and C ( X , GY ) − → D ( F X, Y ) , β 7→ θ Y ◦ F β , are mutually inv erse bijections. (2) ⇒ (1): Let θ : F G → Id D denote the second adjunction morphism. T hen t h e comp osit es F F η − − → F GF θF − − → F and G ηG − − → GF G Gθ − − → G are iden tit y morphisms; see [27, IV.1]. W e kno w from Prop osition 2.3.1 that θ is in- v ertible b ecause G is fully faithful. Therefore Lη = GF η is inv ertible. Moreo v er, we ha ve Lη = GF η = ( Gθ F ) − 1 = η GF = ηL.  Corollary 2.4.2. A functor L : C → C is a lo c alizat ion functor if and only if ther e exists a fu ncto r F : C → D and a ful ly faith ful right ad joint G : D → C such that L = G ◦ F . In that c ase ther e exist a unique e quivalenc e C [Σ − 1 ] → D making the fol lowing diagr am c omm utative C [Σ − 1 ] ∼   ¯ L + + V V V V V V V V V V V V C Q Σ 3 3 h h h h h h h h h h h h F + + W W W W W W W W W W W W W W C D G 3 3 g g g g g g g g g g g g g g wher e Σ denotes the set of morph isms σ in C such that Lσ is invertible. Pr o of. Th e c haracterization of a lo caliza tion functor follo ws f ro m Prop osition 2.4.1. No w observ e that Σ equals the set of morphisms σ in C suc h that F σ is inv ertible since G is fully faithful. Th u s we can apply Prop osition 2.3 .1 to obtai n the equiv alence C [Σ − 1 ] → D making the diagram commuta tiv e.  2.5. Lo cal o b jects. Giv en a lo caliza tion functor L : C → C , we wish to describ e those ob jects X in C su c h that X ∼ − → LX . T o this end, it is con v enient to mak e the follo wing deﬁnition. An ob ject X in a ca tegory C is called lo c al with r e sp ect to a set Σ o f morphisms if for eve r y m o rp hism W → W ′ in Σ the induced map C ( W ′ , X ) → C ( W , X ) is bijectiv e. Now let F : C → D b e a functor and let Σ( F ) d e note the set of morphism s σ of C suc h that F σ is inv ertible. An ob ject X in C is called F -lo c al if it is local with resp ect to Σ( F ). Lemma 2.5.1. L et F : C → D b e a functor and X an obje ct of C . Supp ose ther e ar e two morphis ms η 1 : X → Y 1 and η 2 : X → Y 2 such that F η i is invertible and Y i is F -lo c al for i = 1 , 2 . Then ther e exists a unique isomo rphism φ : Y 1 → Y 2 such that η 2 = φ ◦ η 1 . Pr o of. Th e morphism η 1 induces a bijection C ( Y 1 , Y 2 ) → C ( X , Y 2 ) and w e tak e for φ the unique m orp hism whic h is sent to η 2 . Exc hanging th e roles of η 1 and η 2 , we obtain the in ve rs e for φ .  6 HENNING KRAUSE Prop osition 2.5.2. L et L : C → C b e a lo c alization functor and η : Id C → L a morphism such that Lη is i nve r tible. Then the f ol low ing ar e e quivalent for an obje ct X in C . (1) The obje ct X is L -lo c al. (2) The map C ( LW , X ) → C ( W , X ) induc e d by η W is b ije ctive for al l W in C . (3) The morph ism ηX : X → LX is i nv e r tible. (4) The map C ( W , X ) → C ( LW, L X ) induc e d by L is bij e ctive for al l W in C . (5) The obje ct X is isomorphic to LX ′ for som e obje ct X ′ in C . Pr o of. (1) ⇒ (2): The morp h ism η W b elongs to Σ( L ) and therefore C ( η W , X ) is bijectiv e if X is L -lo cal. (2) ⇒ (3): Put W = X . W e obtain a morphism φ : LX → X which is an in ve rs e for η X . More precisely , we hav e φ ◦ η X = id X . On the other hand , η X ◦ φ = Lφ ◦ η LX = Lφ ◦ Lη X = L ( φ ◦ η X ) = id LX. Th u s η X is inv ertible. (3) ⇔ (4): W e use the facto rization C F − → D G − → C of L from Pr op osition 2.4.1. Then w e obtain for ea ch W in C a factorizatio n C ( W , X ) − → C ( W, LX ) ∼ − → C ( F W , F X ) ∼ − → C ( LW, LX ) of the map f W : C ( W , X ) → C ( LW , LX ) indu c ed b y L . Here, the ﬁrst map is i n duced b y η X , the second follo ws from the a d junctio n , and the third is in du ce d b y G . Th us f W is b ije ctiv e for all W iﬀ the ﬁ rst map is bijectiv e for all W iﬀ η X is inv ertible. (3) ⇒ (5): T ak e X ′ = X . (5) ⇒ (1): W e use again the factorizatio n C F − → D G − → C of L from Prop osit ion 2.4.1. Fix σ in Σ ( L ) and observ e that F σ is inv ertible. Then we ha v e C ( σ , X ) ∼ = C ( σ, G ( F X ′ )) ∼ = D ( F σ, F X ′ ) and this implies that C ( σ, X ) is b ije ctiv e since F σ is in v ertible.  Giv en a fun ct or F : C → D , w e d e n o te by Im F the essential image of F , that is, the full sub catego ry of D which is formed by all ob jects isomorphic to F X for some ob ject X in C . Corollary 2.5.3. L et L : C → C b e a lo c al ization f u ncto r. Th en L induc es an e qu i v a - lenc e C [Σ( L ) − 1 ] ∼ − → I m L and Im L is the ful l sub c ate gory of C c onsisting of al l L -lo c al sub obje cts. Pr o of. W rite L as comp osite C F − → Im L G − → C of t w o functors, where F X = LX for all X in C and G is the inclusion fun c tor. Then it follo ws from Corollary 2.4.2 that F induces an equiv alence C [Σ( L ) − 1 ] ∼ − → Im L . The second assertion is an immediate consequence of Pr o p osition 2.5.2.  Giv en a lo cali zation functor L : C → C and an ob ject X in C , the morph ism X → LX is initial among all morp hisms to an ob ject in I m D and terminal among all morp hisms in Σ( L ). The follo wing statemen t mak es this p r e cise. Corollary 2. 5.4. L et L : C → C b e a lo c alization functor and η : Id C → L a morphism such that Lη is i nve r tible. Then for e ach morphism η X : X → LX the fol lowing ho lds. (1) The obje ct LX b elongs to I m L and every morphism X → Y with Y in Im L factors uniquely thr ough η X . (2) The morphism η X b elongs to Σ( L ) and factors uniquely thr ough every morphism X → Y in Σ( L ) . LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 7 Pr o of. App l y Prop osition 2.5.2.  Remark 2.5.5. (1) Let L : C → C b e a lo caliza tion functor and supp ose there are t w o morphisms η i : Id C → L suc h that Lη i is in verti b l e for i = 1 , 2. Th e n there exists a unique isomorph ism φ : L ∼ − → L su c h that η 2 = φ ◦ η 1 . This follo ws from Lemma 2.5.1. (2) Giv en an y f uncto r F : C → D , the full sub category of F -lo cal ob j ects is closed under taking all limits whic h exist in C . 2.6. Existence of lo calization functors. W e pro vide a criterion for the existence of a lo ca lization functor L ; it explains ho w L is determined by the category of L -lo cal ob jects. Prop osition 2.6.1. L et C b e a c ate gory and D a ful l sub c ate gory. Supp ose tha t every obje ct in C isomorphic to one in D b elongs to D . Then the fol lowing ar e e quivalent. (1) Ther e exists a lo c alization functor L : C → C with Im L = D . (2) F or every obje ct X in C ther e exists a morphism η X : X → X ′ with X ′ in D such that ev e r y morphism X → Y with Y in D factors uniquely thr ough η X . (3) The inclusion functor D → C admits a left adjoint. Pr o of. (1) ⇒ (2): Sup pose there exists a lo caliza tion functor L : C → C with Im L = D and let η : Id C → L b e a morphism su c h that Lη is inv ertible. Th en Pr op osition 2.5.2 sho ws that C ( ηX , Y ) is bijectiv e for all Y in D . (2) ⇒ (3): T h e morph i sms ηX provide a functor F : C → D by send i n g eac h X in C to X ′ . It is straigh tforw ard to chec k that F is a left adjoint for the inclusion D → C . (3) ⇒ (1) : Let G : D → C denote th e inclusion and F its righ t adjoin t. Then L = G ◦ F is a localizatio n fu ncto r with Im L = D b y Prop osition 2.4.1.  2.7. Lo calization functors preserving copro duc ts. W e c haracterize the fact that a lo caliza tion fun ct or pr eserves small copro ducts. Prop osition 2.7.1. L et L : C → C b e a lo c alization functor and supp ose the c ate gory C admits sma l l c opr o ducts. Then the fol lo wing ar e e quivalent. (1) The functor L pr eserves smal l c opr o ducts. (2) The L -lo c al obje cts ar e close d under taking smal l c opr o ducts in C . (3) The right adjoint of the quotient functor C → C [Σ( L ) − 1 ] pr eserves smal l c opr o d- ucts. Pr o of. (1) ⇒ (2): Let ( X i ) i ∈ I b e a family of L -local ob jects. Th us the natural morphisms X i → LX i are in vertible by Prop osition 2.5.2 and they ind uce an isomorphism a i X i ∼ − → a i LX i ∼ − → L ( a i X i ) . It follo ws that ` i X i is L -lo cal. (2) ⇔ (3): W e can iden tify C [Σ( L ) − 1 ] = I m L b y Corollary 2.5.3 and then the righ t adjoin t of the quotien t fu ncto r iden tiﬁes with the inclusio n Im L → C . T h us the righ t adjoin t pr ese rves small copro ducts if and only if the inclusion Im L → C preserv es small copro ducts. (3) ⇒ (1): W rite L as comp osite C − → C [Σ( L ) − 1 ] − → C o f the quotien t functor Q with its righ t adjoin t ¯ L . Then Q preserves sm all copr o ducts sin ce it is a left adj oint. It follo ws that L preserves small copro ducts if ¯ L preserv es small copro ducts.  8 HENNING KRAUSE 2.8. Colo calization functors. A functor Γ : C → C is called c olo c alization functor if its opp osite fu nct or Γ op : C op → C op is a lo caliz ation fun ctor. W e call an ob ject X in C Γ -c olo c al if it is Γ op -local wh en view ed as an ob ject of C op . Note that a colocalizatio n functor Γ : C → C induces an equ iv alence C [Σ( Γ ) − 1 ] ∼ − → Im Γ and the essenti al image Im Γ equ als the full sub catego ry of C consisting of all Γ -colo ca l ob jects. Remark 2.8.1. W e think of Γ as L tu rned upside do wn; this explains our notation. Another r e ason for the use of Γ is th e int erpr e tation of lo ca l cohomolog y as colocaliza- tion. 2.9. Example: Lo calization of mo dules. Let A b e an asso ciati ve ring and denote b y Mo d A the category of (righ t) A -mo dules. S upp ose that A is comm utativ e and let S ⊆ A b e a multiplic ativ ely closed sub se t, that is, 1 ∈ S and st ∈ S for all s, t ∈ S . W e denote b y S − 1 A = { x/s | x ∈ A and s ∈ S } the r in g of fractions. F or eac h A -mo dule M , let S − 1 M = { x/s | x ∈ M and s ∈ S } b e the lo calized mo dule. An S − 1 A -mo dule N b ecomes an A -mo d ule via restrictio n of scalars along the canonical ring homomorpism A → S − 1 A . W e obtain a pair of fu ncto rs F : Mod A − → Mod S − 1 A, M 7→ S − 1 M ∼ = M ⊗ A S − 1 A, G : Mod S − 1 A − → Mod A, N 7→ N ∼ = Hom S − 1 A ( S − 1 A, N ) . Moreo v er, for eac h pair of mo dules M o v er A a nd N ov er S − 1 A , w e h a v e n a tur a l morphisms η M : M − → ( G ◦ F ) M = S − 1 M , x 7→ x/ 1 , θ N : S − 1 N = ( F ◦ G ) N − → N , x/s 7→ xs − 1 . These natural m o rp hisms induce m utu ally inv erse bijections as follo ws: Hom A ( M , GN ) ∼ − → Hom S − 1 A ( F M , N ) , α 7→ θ N ◦ F α, Hom S − 1 A ( F M , N ) ∼ − → Hom A ( M , GN ) , β 7→ Gβ ◦ η M . It is clear that the functors F and G form an adjoin t p air, that is, F is a left adjoin t of G and G is a righ t adjoin t of F . Mo reov er, the adju ncti on morphism θ : F ◦ G → I d is inv ertible. Therefore the comp osite L = G ◦ F is a lo caliz ation functor. Let us form ulate this slight ly m ore generally . Fix a ring homomorphism f : A → B . Then it is well kno wn that the r estriction fu n ct or Mo d B → Mo d A is fully faithful if and on ly if f is an epimorph ism; see [45, Prop osition XI.1.2]. Thus the f uncto r Mo d A → Mod A taking a mo dule M to M ⊗ A B is a lo c alization fun c tor pro vided that f is an epimorph i sm . LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 9 2.10. Example: Lo calization of sp ectra. A sp e ctrum E is a sequence of based top o- logica l spaces E n and based homeomo r p hisms E n → Ω E n +1 . A morphism of sp ectra E → F is a sequence of based con tinuous maps E n → F n strictly compatible with the giv en structural h o meomorph i sms . Th e h omo topy groups of a sp ectrum E are the groups π n E = π n + i ( E i ) for i ≥ 0 and n + i ≥ 0. A morp hism b et we en sp ectra is a we ak e quivalenc e if it indu ces an isomorphism on h o motopy group s . The stable homotopy c ate gory Ho S is obtained from the category S of sp ectra b y formally in ve rting the weak equiv ale nces. Thus Ho S = S [Σ − 1 ] where Σ denotes the set of we ak equiv alences. W e refer to [2, 39] for details. 2.11. Notes. Th e catego r y of fractions is introdu ce d by Gabriel and Zisman in [18], but the idea of formally in v erting elemen ts can b e traced bac k m uch further; see for instance [36]. T h e appropriate context for lo calization functors is the theory of monads; see [27]. 3. Calculu s of fr a ctions 3.1. Calculus of fractions. Let C b e a category and Σ a set of morph isms in C . Th e catego ry of fractio n s C [Σ − 1 ] admits an elemen tary description if some extra assumptions on Σ are satisﬁed. W e sa y that Σ adm its a c alculus of left fr actions if the follo wing holds. (LF1) If σ , τ are comp o sable morphisms in Σ, then τ ◦ σ is in Σ. The iden tit y morphism id X is in Σ for all X in C . (LF2) Eac h pair of morp hisms X ′ σ ← − X α − → Y with σ in Σ can b e completed to a comm utativ e square X α / / σ   Y σ ′   X ′ α ′ / / Y ′ suc h that σ ′ is in Σ. (LF3) Let α, β : X → Y b e morp hisms in C . If there is a morphism σ : X ′ → X in Σ with α ◦ σ = β ◦ σ , then there exists a morphism τ : Y → Y ′ in Σ with τ ◦ α = τ ◦ β . No w assume that Σ admits a calculus of left fractions. Then one obtains a new catego ry Σ − 1 C as follo ws. The ob j e cts are those of C . Giv en ob jec ts X an d Y , we call a pair ( α, σ ) of morphisms X α / / Y ′ Y σ o o in C with σ in Σ a left fr action . The morp hisms X → Y in Σ − 1 C are equiv alence classes [ α, σ ] of suc h le ft fractions, where t w o diagrams ( α 1 , σ 1 ) and ( α 2 , σ 2 ) are equiv alen t if there exists a comm utativ e d ia gram Y 1   X α 3 / / α 1 > > ~ ~ ~ ~ ~ ~ ~ ~ α 2 @ @ @ @ @ @ @ @ Y 3 Y σ 1 _ _ @ @ @ @ @ @ @ @ σ 2   ~ ~ ~ ~ ~ ~ ~ ~ σ 3 o o Y 2 O O 10 HENNING KRAUSE with σ 3 in Σ. The comp ositio n of t w o equ iv alence classes [ α, σ ] and [ β , τ ] is by deﬁnition the equiv alene class [ β ′ ◦ α, σ ′ ◦ τ ] wh ere σ ′ and β ′ are obtained from condition (LF2) as in the f ollo wing comm utativ e diagram. Z ′′ Y ′ β ′ = = | | | | | | | | Z ′ σ ′ ` ` B B B B B B B B X α > > } } } } } } } } Y σ a a B B B B B B B B β = = | | | | | | | | Z τ ` ` @ @ @ @ @ @ @ W e obtain a canonical fu n ct or P Σ : C − → Σ − 1 C b y taking t h e iden tit y map on ob jects and by sending a morp hism α : X → Y to the equiv ale nce class [ α, id Y ]. Let us compare P Σ with th e qu otient functor Q Σ : C → C [Σ − 1 ]. Prop osition 3.1.1. The functor F : Σ − 1 C → C [Σ − 1 ] which is the id entity map on obje cts and which takes a morphism [ α, σ ] to ( Q Σ σ ) − 1 ◦ Q Σ α is an isomorphism. Pr o of. Th e functor P Σ in ve rts all morp hisms in Σ and factors therefore through Q Σ via a functor G : C [Σ − 1 ] → Σ − 1 C . It is straigh tforward to c heck that F ◦ G = Id and G ◦ F = Id.  F rom no w on, w e will identi fy Σ − 1 C with C [Σ − 1 ] whenev er Σ admits a calculus of left fractions. A set of morphisms Σ in C ad mits a c alculus of right fr actions if the dual conditions of (LF1) – (LF3) are satisﬁed. Moreo ve r, Σ is cal led a multiplic ative system if it admits b oth, a calculus of left fractions and a calculus of righ t fractions. Note that all results ab out sets of morphisms admitting a calculus of left fractions ha ve a dual v ersion for sets of m orp hisms admitting a calculus of righ t fr a ctions. 3.2. Calculus of fractions and a djo int functors. Giv en a cat egory C and a set of morphisms Σ , it is an interest ing question to ask wh e n the quotien t functor C → C [Σ − 1 ] admits a righ t adjoin t. It turn s out that this problem is closely related to the prop ert y of Σ to admit a calculus of left fr a ctions. Lemma 3.2.1. L et F : C → D and G : D → C b e a p air of adjoint functors. Assume that the right adjoint G is ful ly fa ithful and let Σ b e the set of morphisms σ in C such that F σ is invertible. Then Σ adm its a c alculus of left fr actions. Pr o of. W e need to chec k the conditions (LF1) – (LF3). Observe ﬁrst that L = G ◦ F is a lo caliza tion fun ct or so that we can app ly Pr o p osition 2.5.2. (LF1): This condition is clear b ecause F is a fu ncto r. (LF2): Let X ′ σ ← − X α − → Y b e a pair of morphisms with σ in Σ. This can be completed to a comm utativ e square X α / / σ   Y σ ′   X ′ α ′ / / Y ′ LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 11 if w e tak e for σ ′ the morph ism η Y : Y → LY in Σ, b ecause the map C ( σ , L Y ) is s u rject ive b y Prop osition 2.5.2. (LF3): Let α, β : X → Y b e morphisms in C and supp ose there is a morphism σ : X ′ → X in Σ w it h α ◦ σ = β ◦ σ . Then we take τ = η Y in Σ and h a v e τ ◦ α = τ ◦ β , b ecause the map C ( σ , LY ) is in j e ctiv e b y Pr o p osition 2.5.2.  Lemma 3.2.2. L et C b e a c ate gory and Σ a set of morphisms admitt ing a c alculus of left fr actions. Then the fol lowing ar e e quivalent for an obje ct X in C . (1) X is lo c al with r esp e ct to Σ . (2) The quotient functor induc es a b i j e ction C ( W , X ) → C [Σ − 1 ]( W , X ) for al l W . Pr o of. (1) ⇒ ( 2): T o sho w that f W : C ( W , X ) → C [Σ − 1 ]( W , X ) is surjectiv e, c ho ose a left fraction W α − → X ′ σ ← − X with σ in Σ. Then there exists τ : X ′ → X with τ ◦ σ = id X since X is local. Thus f W ( τ ◦ α ) = [ α, σ ]. T o sho w that f W is injectiv e, supp ose that f W ( α ) = f W ( β ). Th e n w e ha ve σ ◦ α = σ ◦ β for some σ : X → X ′ in Σ. Th e morphism σ is a section b ec ause X is lo cal, and therefore α = β . (2) ⇒ ( 1): Let σ : W → W ′ b e a morphism in Σ. Then w e h av e C ( σ , X ) ∼ = C [Σ − 1 ]([ σ , id W ′ ] , X ). Thus C ( σ , X ) is bijectiv e since [ σ, id W ′ ] is in v ertible.  Prop osition 3.2.3. L et C b e a c ate gory, Σ a set of morphisms admitting a c alculus of left fr actio ns, and Q : C → C [Σ − 1 ] the quotient functor . Then th e fol lowing ar e e quivalent. (1) The functor Q has a right adjoint (which is then ful ly faithful). (2) F or e ach obje ct X i n C , ther e exist a morph ism η X : X → X ′ such that X ′ is lo c al with r e sp e ct to Σ and Q ( η X ) is invertible. Pr o of. (1) ⇒ (2): Denote b y Q ρ the righ t adjoint of Q and by η : Id C → Q ρ Q the adjunction morphism. W e tak e for eac h ob ject X in C the morphism η X : X → Q ρ QX . Note that Q ρ QX is lo c al by Prop osition 2.5.2. (2) ⇒ (1): W e ﬁx ob j e cts X and Y . Then we hav e tw o natural bijections C [Σ − 1 ]( X, Y ) ∼ − → C [Σ − 1 ]( X, Y ′ ) ∼ ← − C ( X, Y ′ ) . The ﬁ rst is indu c ed by η Y : Y → Y ′ and is bijectiv e since Q ( η Y ) is inv ertible. The second map is bijectiv e b y Lemma 3.2.2, since Y ′ is lo ca l with resp ect to Σ. T h us w e obtain a r ig ht adjoint for Q by send in g eac h ob ject Y of C [Σ − 1 ] to Y ′ .  3.3. A criterion for the fra c tions to form a small set. Let C b e a catego r y and Σ a set of morphisms in C . Supp ose that Σ admits a calculus of left fractions. F r o m the construction of C [Σ − 1 ] we cannot exp ect that for an y g iven pair of ob jects X and Y the equiv alence classes of fractions in C [Σ − 1 ]( X, Y ) form a small set. The situation is diﬀeren t if the category C is small. Then it is clear that C [Σ − 1 ]( X, Y ) is a small set for all ob jects X , Y . The follo wing criterion ge n e ralizes this simple observ ation. Lemma 3.3.1. L et C b e a c ate gory and Σ a set of morphisms in C which admits a c alculus of left fr actions . L et Y b e an obje ct in C and supp ose that ther e exists a smal l set S = S ( Y , Σ) of obje cts in C such that for every morphism σ : Y → Y ′ in Σ ther e is a morphis m τ : Y ′ → Y ′′ with τ ◦ σ in Σ and Y ′′ in S . Then C [Σ − 1 ]( X, Y ) is a smal l set for every obje ct X in C . 12 HENNING KRAUSE Pr o of. Th e condition on Y imp li es that every fraction X α → Y ′ σ ← Y is equ iv alen t to one of the form X α ′ → Y ′′ σ ′ ← Y with Y ′′ in S . C l early , the fractions of the form ( α ′ , σ ′ ) with σ ′ ∈ C ( Y , Y ′′ ) and Y ′′ ∈ S form a small set.  3.4. Calculus of fractions for sub categories. W e p ro vide a criterion su c h that the calculus of fractions for a set of morphisms in a category C is co mp a tible with the passage to a sub category of C . Lemma 3.4.1. L et C b e a c ate gory and Σ a set of morphisms admitting a c alculus of left fr actio ns. Supp ose D is a ful l sub c ate gory of C such that for every morphism σ : Y → Y ′ in Σ with Y in D ther e is a morp hism τ : Y ′ → Y ′′ with τ ◦ σ in Σ ∩ D . Then Σ ∩ D admits a c alculus of left fr actions and the induc e d functor D [(Σ ∩ D ) − 1 ] → C [Σ − 1 ] is ful ly faithful. Pr o of. It is s tr aightfo rward to c heck (LF1) – (LF3) for Σ ∩ D . Now let X , Y b e ob jects in D . Then we need to s h o w that the indu c ed map f : D [(Σ ∩ D ) − 1 ]( X, Y ) − → C [Σ − 1 ]( X, Y ) is bijectiv e. Th e map sends the equiv alence class of a fraction to the equiv alence class of the same fr a ction. If [ α, σ ] b elongs to C [Σ − 1 ]( X, Y ) an d τ is a morphism with τ ◦ σ in Σ ∩ D , then [ τ ◦ α, τ ◦ σ ] b elongs to D [(Σ ∩ D ) − 1 ]( X, Y ) and f sends it to [ α, σ ]. T h us f is surjectiv e. A similar argument sho ws that f is inj e ctiv e.  Example 3.4.2. Let A b e a comm utativ e no etherian ring and S ⊆ A a m ultiplicativ ely closed subset. Denote b y Σ the set of morphisms σ in Mod A suc h that S − 1 σ is in v ertible. Then Σ is a multiplic ativ e system and one can sh o w directly that for the sub category mo d A of ﬁnitely generated A -mo dules and T = Σ ∩ mo d A the du a l of th e condition in Lemma 3.4.1 h o lds. Thus the in d uced functor (mo d A )[ T − 1 ] − → (Mod A )[Σ − 1 ] is f u lly faithful. 3.5. Calculus of fractions and coproducts. W e p ro vide a criterion for the quotient functor C → C [Σ − 1 ] to p reserv e sm a ll copro ducts. Prop osition 3.5.1. L et C b e a c ate gory which admits smal l c opr o ducts. Supp ose that Σ is a set of morphisms in C which adm its a c alculus of left fr actions. If ` i σ i b elongs to Σ for every family ( σ i ) i ∈ I in Σ , then th e c ate gory C [Σ − 1 ] a dmits smal l c opr o ducts and the quotient functor C → C [Σ − 1 ] pr eserves smal l c opr o ducts. Pr o of. Let ( X i ) i ∈ I b e a family of ob ject s in C [Σ − 1 ] whic h is indexed by a small set I . W e claim that the copro duct ` i X i in C is also a copro duct in C [Σ − 1 ]. T h us we need to sho w that for eve ry ob ject Y , the canonic al map (3.5.1 ) C [Σ − 1 ]( a i X i , Y ) − → Y i C [Σ − 1 ]( X i , Y ) is b ije ctiv e. LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 13 T o c hec k surjectivit y of (3.5.1), let ( X i α i → Z i σ i ← Y ) i ∈ I b e a family o f le ft fractions. Using (LF2), w e obtain a comm u tativ e diagram ` i X i ‘ i α i / / ` i Z i   ` i Y π Y   ‘ i σ i o o Z Y σ o o where π Y : ` i Y → Y is the su mmati on morp hism and σ ∈ Σ. It is easily chec k ed that ( X i → Z σ ← Y ) ∼ ( X i α i → Z i σ i ← Y ) for all i ∈ I , and t h e refore (3.5.1) send s ` i X i → Z σ ← Y to the family ( X i α i → Z i σ i ← Y i ) i ∈ I . T o c heck injectivit y of (3.5.1), let ` i X i α ′ → Z ′ σ ′ ← Y and ` i X i α ′′ → Z ′′ σ ′′ ← Y b e left fraction suc h that ( X i α ′ i → Z ′ σ ′ ← Y ) ∼ ( X i α ′′ i → Z ′′ σ ′′ ← Y ) for all i . W e m a y assume that Z ′ = Z = Z ′′ and σ ′ = σ = σ ′′ since we can c ho ose morphisms τ ′ : Z ′ → Z a n d τ ′′ : Z ′′ → Z with τ ′ ◦ σ ′ = τ ′′ ◦ σ ′′ ∈ Σ . Thus there are morphisms β i : Z → Z i with β i ◦ α ′ i = β i ◦ α ′′ i and β i ◦ σ ∈ Σ f or all i . Eac h β i b elongs to the satur atio n ¯ Σ of Σ wh ich is the set of all morp h isms in C w h ic h b ecome in ve r tible in C [Σ − 1 ]. Note that a morph i sm φ in C b el ongs to ¯ Σ if and only if there are morphisms φ ′ and φ ′′ suc h that φ ◦ φ ′ and φ ′′ ◦ φ b elong to Σ. Therefore ¯ Σ is also closed u nder taking copro ducts. Mo reov er, ¯ Σ admits a calculus of left fractions, and w e obtain therefore a comm utativ e diagram ` i X i / / ` i Z ‘ i β i   π Z / / Z τ   ` i Z i / / Z ∗ with τ ∈ ¯ Σ. Th u s τ ◦ σ ∈ ¯ Σ, and w e hav e ( a i X i α ′ → Z σ ← Y ) ∼ ( a i X i α ′′ → Z σ ← Y ) since π Z ◦ ` i α ′ i = α ′ and π Z ◦ ` i α ′′ i = α ′ . Th e r efore the map (3.5.1) is also injectiv e, and this complet es the p r oof.  Example 3.5.2. Let C b e a catego ry which admits small copro ducts a n d L : C → C b e a lo calization functor. Then a morphism σ in C b elongs to Σ = Σ( L ) if and only if the induced map C ( σ , L X ) is inv ertible for eve r y ob ject X in C . Thus Σ is closed u nder taking s m a ll copro ducts and therefore the qu o tient fu ncto r C → C [Σ − 1 ] preserve s sm all copro ducts. 3.6. Notes. The calculus of fractions for categories has b een dev elop ed b y Gabriel and Zisman in [18 ] as a to ol for h o motopy theory . 14 HENNING KRAUSE 4. Localiza tion f or triangula ted ca tegories 4.1. T riangulated categories. L e t T b e an additiv e catego ry with an equiv alence S : T → T . A triangle in T is a sequence ( α, β , γ ) of morphism s X α − → Y β − → Z γ − → S X , and a morphism b et w een t w o triangles ( α, β , γ ) and ( α ′ , β ′ , γ ′ ) is a triple ( φ 1 , φ 2 , φ 3 ) of morphisms in T making the follo wing diagram commuta tiv e. X α / / φ 1   Y β / / φ 2   Z γ / / φ 3   S X S φ 1   X ′ α ′ / / Y ′ β ′ / / Z ′ γ ′ / / S X ′ The category T is called triangulat e d if it is equip ped with a set of distinguished triangles (calle d exact triangles ) satisfying the f o llo wing conditions. (TR1) A triangle isomorphic to an exact triangle is exact. F or eac h ob ject X , the triangle 0 → X id − → X → 0 is exact. Each morphism α ﬁts into an exact triangle ( α, β , γ ). (TR2) A triangle ( α, β , γ ) is exact if and only if ( β , γ , − S α ) is exact. (TR3) Giv en t wo exact triangles ( α, β , γ ) and ( α ′ , β ′ , γ ′ ), eac h pair of morphisms φ 1 and φ 2 satisfying φ 2 ◦ α = α ′ ◦ φ 1 can b e completed to a morph ism X α / / φ 1   Y β / / φ 2   Z γ / / φ 3   S X S φ 1   X ′ α ′ / / Y ′ β ′ / / Z ′ γ ′ / / S X ′ of triangles. (TR4) Giv en exact triangles ( α 1 , α 2 , α 3 ), ( β 1 , β 2 , β 3 ), and ( γ 1 , γ 2 , γ 3 ) with γ 1 = β 1 ◦ α 1 , there exists an exact triangle ( δ 1 , δ 2 , δ 3 ) making the follo wing diagram comm u- tativ e. X α 1 / / Y α 2 / / β 1   U α 3 / / δ 1   S X X γ 1 / / Z γ 2 / / β 2   V γ 3 / / δ 2   S X S α 1   W β 3   W δ 3   β 3 / / S Y S Y S α 2 / / S U Recall that an idemp oten t en d omo rp hism φ = φ 2 of an ob ject X in an add i tive catego ry splits if there exists a factorization X π − → Y ι − → X of φ with π ◦ ι = id Y . Remark 4.1.1. S u pp o se a triangulated category T admits co untable coprodu ct s. T hen ev ery idemp oten t end o morp hism splits. More precisely , let φ : X → X b e an idemp o tent LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 15 morphism in T , and denote by Y a homotop y co limit of th e sequence X φ − → X φ − → X φ − → · · · . The m orp hism φ factors through the canonica l morph ism π : X → Y vi a a morp hism ι : Y → X , and w e ha v e π ◦ ι = id Y . Thus φ splits; see [33, Pr op osition 1. 6.8] for details. 4.2. Exact functors. An exact functor T → U b et w een triangulated catego r ies is a pair ( F , µ ) consisting of a functor F : T → U and an isomorphism µ : F ◦ S T → S U ◦ F suc h that for every exact triangle X α → Y β → Z γ → S T X in T the triangle F X F α − → F Y F β − → F Z µX ◦ F γ − − − − − → S U ( F X ) is exact in U . W e ha v e the follo w i n g u se fu l lemma. Lemma 4.2.1. L et F : T → U and G : U → T b e an a djoint p air of functors b etwe en triangulate d c ate g o ries. If one of b oth functors is exact, then also the other is exact. Pr o of. See [33, Lemma 5.3.6].  4.3. Multiplicativ e systems. L e t T b e a triangulated c ategory a n d Σ a set of mor- phisms whic h is a multiplica tiv e system. Recall this means that Σ admits a calculus of left and righ t fracti ons. Then w e say that Σ is c omp atible with the triangulation if (1) giv en σ in Σ, the morphism S n σ b elongs to Σ for all n ∈ Z , and (2) giv en a morphism ( φ 1 , φ 2 , φ 3 ) b et w een exact triangles with φ 1 and φ 2 in Σ, there is also a morph i sm ( φ 1 , φ 2 , φ ′ 3 ) with φ ′ 3 in Σ. Lemma 4.3.1. L et T b e a triangulate d c ate gory and Σ a multiplic ative system of mor- phisms which is c omp atible with the triangulation. Th en the quotient c ate gory T [Σ − 1 ] c arr ies a unique triangulate d structur e such that the quotient functor T → T [Σ − 1 ] is exact. Pr o of. Th e equiv alence S : T → T induces a unique equiv ale nce T [Σ − 1 ] → T [Σ − 1 ] whic h comm utes w ith the quotien t f uncto r Q : T → T [Σ − 1 ]. This fol lo ws from the fact that S Σ = Σ. No w tak e as exact triangles in T [Σ − 1 ] all those isomorphic to imag es of exact triangles in T . I t is straigh tforward to ve rify the axioms (TR1) – (TR4); see [48, I I.2 .2.6]. The f u ncto r Q is exact by construction. In particular, w e ha v e Q ◦ S T = S T [Σ − 1 ] ◦ Q .  4.4. Cohomological func tors. A f uncto r H : T → A from a triangulated category T to an ab elian ca tegory A is c oho molo gic al if H sends ev ery exact triangle in T to an exact sequence in A . Example 4.4.1. F or eac h ob ject X in T , the represen table functors T ( X, − ) : T → Ab and T ( − , X ) : T op → Ab into the category Ab of ab elian groups are cohomolog ical functors. Lemma 4.4.2 . L et H : T → A b e a c ohom olo gic al functor. Then the set Σ of morph isms σ in T such that H ( S n σ ) is invertible for al l n ∈ Z forms a multip lic ative system which is c omp atible with the triangulation of T . 16 HENNING KRAUSE Pr o of. W e need to verify that Σ admits a ca lculus of left and right fractio ns . In fact, it is suﬃcien t to c heck conditions (LF1) – (LF3), b ecause th en the dual conditions are satisﬁed as w ell since the deﬁnition of Σ is self-dual. (LF1): This condition is clear b ecause H is a fu ncto r. (LF2): Let α : X → Y and σ : X → X ′ b e m orp hisms with σ in Σ. W e complet e α to an exact triangle and apply (TR3) to obtain the follo wing morphism b et w een exact triangles. W / / X α / / σ   Y / / σ ′   S W W / / X ′ α ′ / / Y ′ / / S W Then th e 5-le mma sho ws that σ ′ b elongs to Σ. (LF3): Let α, β : X → Y b e morph isms in T and σ : X ′ → X in Σ suc h that α ◦ σ = β ◦ σ . Comp lete σ to an exact triangle X ′ σ → X φ → X ′′ → S X ′ . Then α − β factors through φ via s ome morph ism ψ : X ′′ → Y . No w complete ψ to an exact triangle X ′′ ψ → Y τ → Y ′ → S X ′′ . Then τ b elongs to Σ and τ ◦ α = τ ◦ β . It remains to chec k that Σ is compatible with the triangulat ion. C o n d it ion (1) is cle ar from the deﬁnition of Σ. F or condition (2 ), observe that giv en an y morphism ( φ 1 , φ 2 , φ 3 ) b et w een exact triangle s with φ 1 and φ 2 in Σ, we ha v e that φ 3 b elongs to Σ. This is an immediate consequence of the 5-lemma.  4.5. T riangulated and t hic k sub categories. Let T b e a triangulat ed category . A non-empt y full sub category S is a triangulate d sub c ate g ory if the follo wing conditions hold. (TS1) S n X ∈ S for all X ∈ S and n ∈ Z . (TS2) Let X → Y → Z → S X b e an exact triangl e in T . If t w o ob jects from { X, Y , Z } b elong to S , then also the third. A triangulated sub cate gory S is thick if in addition the follo wing condition holds. (TS3) Let X π − → Y ι − → X b e morp hisms in T such that id Y = π ◦ ι . If X b elongs to S , then also Y . Note that a triangulated sub categ ory S of T inherits a canonical triangulated structure from T . Next observe that a triangulated sub catego ry S of T is thick p ro vided that S a dm it s coun table copro ducts. This follo ws from the fact that in a triangulated category with coun table copro ducts all idemp oten t end o morp hisms split. Let T b e a triangulated categ ory and let F : T → U b e an additiv e fu ncto r. The kernel Ker F of F is by deﬁnition the full sub catego ry of T whic h is formed by all ob jects X su c h that F X = 0. If F is an exact functor int o a triangulated catego ry , then Ker F is a thick sub catego ry of T . Also, if F is a cohomologi cal functor into an ab elian catego ry , then T n ∈ Z S n (Ker F ) is a thic k sub cat egory of T . 4.6. V erdier lo c alization. Let T b e a triangulated categ ory . Giv en a triangulated sub catego ry S , we den ote by Σ( S ) the set of morp hisms X → Y in T w h ic h ﬁt in to an exact triangle X → Y → Z → S X with Z in S . Lemma 4.6.1. L et T b e a triangulate d c ate gory and S a triangulate d sub c ate gory. Then Σ( S ) is a multipl ic ative system which is c omp atible w ith the triangulation of T . LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 17 Pr o of. Th e pr o of is similar to that of Lemma 4.4.2; see [48, I I.2.1. 8] for details.  The lo c alization of T with resp ect to a triangula ted su bcategory S is by d eﬁnitio n the qu o tient category T / S := T [Σ( S ) − 1 ] together with the quotien t f unctor T → T / S . Prop osition 4.6.2. L et T b e a triangulate d c ate gory and S a ful l triangulate d sub c ate- gory. Then the c ate gory T / S and the quotient functor Q : T → T / S have the fol lowing pr op erties. (1) The c ate gory T / S c arries a unique triangulate d structur e such that Q is exact. (2) A morphism in T is annihilate d by Q if and only if it factors thr ough a n obje ct in S . (3) The kernel Ker Q is the smal lest thick sub c ate gory c onta ining S . (4) Every exact fu ncto r T → U annihila ting S factors uniquely thr ough Q via an exact functor T / S → U . (5) Every c ohomolo gic al functor T → A a nnihilating S factors uniquely thr ough Q via a c ohomolo gic al functor T / S → A . Pr o of. (1) follo ws from Lemma 4.3.1. (2) Let φ b e a morph ism in T . W e ha v e Qφ = 0 iﬀ σ ◦ φ = 0 f or some σ ∈ Σ( S ) iﬀ φ factors through some ob ject in S . (3) Let X b e an ob ject in T . Then QX = 0 if a nd only if Q (id X ) = 0. Thus part (2) imp li es that the k ern el of Q conists of all direct factors of ob jects in S . (4) An e xact fun c tor F : T → U ann ihila ting S inv erts ev ery morphism in Σ ( S ). Thus there exists a unique fun c tor ¯ F : T / S → U suc h that F = ¯ F ◦ Q . The fu n ct or ¯ F is exact b ecause an exact triangle ∆ in T / S is up to isomorphism of the form Q ∆ ′ for some exact triangle ∆ ′ in T . Thus ¯ F ∆ ∼ = F ∆ ′ is exact. (5) Analogous to (4).  4.7. Lo calization of sub categories. Let T b e a triangula ted cat egory wit h tw o full triangulated sub cate gories T ′ and S . Then we put S ′ = S ∩ T ′ and ha v e Σ T ′ ( S ′ ) = Σ T ( S ) ∩ T ′ . Thus w e can form the follo wing comm utativ e diagram of exact fun c tors S ′ inc   inc / / T ′ inc   can / / T ′ / S ′ J   S inc / / T can / / T / S and ask when the fun ct or J is fully faithful. W e h a v e the follo wing crite r ion. Lemma 4.7.1. L et T , T ′ , S , S ′ b e as ab ove. Supp ose that e ith er (1) every morphism fr om an obje ct in S to an obje ct in T ′ factors thr ough some obje ct in S ′ , or (2) every morphism fr om an obje ct in T ′ to an obje ct in S factors thr ough some obje ct in S ′ . Then the induc e d functor J : T ′ / S ′ → T / S is ful ly faithful. 18 HENNING KRAUSE Pr o of. Su pp o se that cond ition (1) holds. W e apply the criterion fr om Lemma 3.4.1. Th u s we tak e a m o rp hism σ : Y → Y ′ from Σ ( S ) with Y in T ′ and need to ﬁnd τ : Y ′ → Y ′′ suc h th at τ ◦ σ b elongs to Σ( S ) ∩ T ′ . T o t h is end complete σ to an exact triangle X φ − → Y σ − → Y ′ → S X . Then X b elongs to S and by our assump ti on we hav e a factoriza tion X φ ′ − → Z φ ′′ − → Y of φ w it h Z in S ′ . C o mp le te φ ′′ to an exact triangle Z φ ′′ − → Y ψ − → Y ′′ → S Z . Then (TR3) yie lds a morp hism τ : Y ′ → Y ′′ satisfying ψ = τ ◦ σ . In particular, τ ◦ σ lies in Σ( S ) ∩ T ′ since Z b elongs to S ′ . The pr o of using condition (2) is dual.  4.8. Orthogonal sub categories. Let T b e a triangulate d ca tegory and S a triangu- lated s ubcategory . Then we deﬁne t wo f ull sub catego ries S ⊥ = { Y ∈ T | T ( X , Y ) = 0 for all X ∈ S } ⊥ S = { X ∈ T | T ( X , Y ) = 0 for all Y ∈ S } and call them ortho gonal sub c ate gories with resp ect to S . Note that S ⊥ and ⊥ S are thic k sub categories of T . Lemma 4.8.1. L et T b e a triangulate d c ate gory and S a triangulate d sub c ate gory. Then the fol lowing ar e e quivalent for an obje ct Y in T . (1) Y b elongs to S ⊥ . (2) Y is Σ( S ) -lo c al, that is, T ( σ , Y ) is bije ctive for al l σ in Σ( S ) . (3) The quotient functor induc es a bije ction T ( X, Y ) → T / S ( X , Y ) for al l X in T . Pr o of. (1) ⇒ (2): S upp ose T ( X , Y ) = 0 for all X in S . Then ev ery σ in Σ( S ) induces a bijection T ( σ , Y ) b ecause T ( − , Y ) is cohomologica l. T h us Y is Σ ( S )-lo c al. (2) ⇒ (1): S upp ose t h at Y is Σ( S )-local. If X b elongs to S , then the morphism σ : X → 0 b elongs to Σ ( S ) and ind uces therefore a b ije ction C ( σ , Y ). Thus Y b elongs to S ⊥ . (2) ⇔ (3): Apply Lemma 3.2.2.  4.9. Bousﬁeld lo calization. Let T b e a triangulated catego ry . W e wish to stud y exact lo c alization fu ncto rs L : T → T . T o b e more precise, w e assu me that L is an exact functor and that L is a localization f uncto r in the sense that there exists a morphism η : Id C → L with Lη : L → L 2 b eing in v ertible and Lη = η L . Note th at there is an isomorphism µ : L ◦ S ∼ − → S ◦ L since L is exa ct, and th ere exists a u nique c hoice su c h that µX ◦ η S X = S η X for all X in T . Th is follo ws from Lemma 2.5 .1 . W e observ e that the k ernel of an exact localization functor is a thic k sub category of T . The follo wing fundamen tal result c haracte rizes the thick sub catego ries of T whic h are of this form. Prop osition 4.9.1. L et T b e a triangulate d c ate gory and S a thick sub c ate gory. Then the fol lowing ar e e quivalent. (1) Ther e exists an exact lo c alization functor L : T → T with Ker L = S . (2) The inclusion functor S → T admits a right adjoint. (3) F or e ach X in T ther e exists an exact triangle X ′ → X → X ′′ → S X ′′ with X ′ in S and X ′′ in S ⊥ . (4) The quotient functor T → T / S admits a right adjoint. LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 19 (5) The c omp osite S ⊥ inc − − → T can − − → T / S is an e quivalenc e. (6) The inclusion functor S ⊥ → T admits a left adjoint and ⊥ ( S ⊥ ) = S . Pr o of. Let I : S → T and J : S ⊥ → T denote the inclusions and Q : T → T / S the quotien t fun ctor. (1) ⇒ (2): Supp ose that L : T → T is an exact lo c alization functor with Ker L = S and let η : Id T → L b e a morp hism such that Lη is in verti b l e. W e obtain a right adjoin t I ρ : T → S for the in c lusion I b y completing f or eac h X in T the morphism η X to an exact triangle I ρ X θX − − → X ηX − − → LX → S ( I ρ X ). Note that I ρ X b elongs to S sin c e Lη X is inv ertible. Moreo ver, T ( W , θ X ) is bijectiv e for all W in S since T ( W , LX ) = 0 b y Lemma 4.8.1. Here w e use that LX is Σ ( L ) -lo cal by Prop ositio n 2.5.2 and that Σ( L ) = Σ ( S ). Th u s I ρ pro vides a righ t adjoin t for I since T ( W , I ρ X ) ∼ = T ( I W , X ) for all W in S and X in T . In particular, we see that the exact triangle d e ﬁ ning I ρ X is, u p to a unique isomorphism, uniquely determined by X . Therefore I ρ is well deﬁn ed. (2) ⇒ (3): Supp ose that I ρ : T → S is a righ t adjoint of the inclusion I . W e ﬁ x an ob ject X in T and complet e the adju nct ion morph ism θ X : I ρ X → X to an exact triangle I ρ X θX − − → X → X ′′ → S ( I ρ X ). Clearly , I ρ X b elongs to S . W e ha v e T ( W, X ′′ ) = 0 for all W in S since T ( W, θ X ) is b i jectiv e. Thus X ′′ b elongs to S ⊥ . (3) ⇒ (4) : W e app ly Prop ositio n 3.2.3 to obtain a righ t adjoin t for the quotien t functor Q . T o this end ﬁ x an ob ject X in T and an exact triangle X ′ → X η − → X ′′ → S X ′′ with X ′ in S and X ′′ in S ⊥ . Th e morph i sm η belongs to Σ ( S ) by d e ﬁ niti on, and the ob ject X ′′ is Σ( S )-local by Lemma 4.8.1. No w it follo ws from Prop ositi on 3.2.3 that Q adm its a right adjoint. (4) ⇒ (1) : Let Q ρ : T / S → T denote a r i ght adjoint of Q . This functor is fu ll y faithful b y Prop osition 2.3.1 and exact b y Lemma 4.2.1. Thus L = Q ρ ◦ Q is an exact fu ncto r with Ker L = Ker Q = S . Moreo ve r, L is a lo caliz ation fun c tor by Corollary 2.4.2. (4) ⇒ (5) : Let Q ρ : T / S → T denote a righ t adjoin t of Q . The comp osite Q ◦ J : S ⊥ → T / S is fully faithful b y Lemma 4.8.1. Giv en an ob ject X in T / S , w e h a v e Q ( Q ρ X ) ∼ = X b y Prop o sition 2.3.1, and Q ρ X b elongs to S ⊥ , since T ( W , Q ρ X ) ∼ = T / S ( QW , X ) = 0 for all W in S . Th us Q ◦ J is d en s e and th erefore an equiv alence. (5) ⇒ (6): Sup p ose Q ◦ J : S ⊥ → T / S is an equiv alence and let F : T / S → S ⊥ b e a quasi-in v erse. W e ha ve f or all X in T and Y in S ⊥ T ( X , J Y ) ∼ − → T / S ( QX, Q J Y ) ∼ − → S ⊥ ( F QX , F QJ Y ) ∼ − → S ⊥ ( F QX , Y ) , where the ﬁrst bijection follo ws fr o m Lemma 4.8.1 and the others are clear fr o m the c hoice of F . Thus F ◦ Q is a left adj oint for the inclusion J . It remains to show that ⊥ ( S ⊥ ) = S . Th e inclusion ⊥ ( S ⊥ ) ⊇ S is clear. No w let X b e an ob ject of ⊥ ( S ⊥ ). Then w e h a v e T / S ( QX, QX ) ∼ = S ⊥ ( F QX , F QX ) ∼ = T ( X , J ( F QX )) = 0 . Th u s QX = 0 and therefore X b elongs to S . (6) ⇒ (3): Supp ose that J λ : T → S ⊥ is a left adjoin t of the inclusion J . W e ﬁx an ob ject X in T and complete the adjunction morp hism µX : X → J λ X to an exact triangle X ′ → X µX − − → J λ X → S X ′ . Clearly , J λ X b elongs to S ⊥ . W e ha ve T ( X ′ , Y ) = 0 for all Y in S ⊥ since T ( µX, Y ) is bijectiv e. Thus X ′ b elongs to ⊥ ( S ⊥ ) = S .  20 HENNING KRAUSE The follo win g diagram d ispla ys the functors whic h arise from a localizatio n functor L : T → T . W e use the co nv en tion that F ρ denotes a right adjoin t of a fun ctor F . S I =inc / / T Q =can / / I ρ o o T / S Q ρ o o ( L = Q ρ ◦ Q and Γ = I ◦ I ρ ) 4.10. Acyclic a nd lo cal ob jects. Let T b e a triangulated ca tegory and L : T → T an exact lo caliz ation functor. An ob ject X in T is by deﬁnition L -acyclic if L X = 0. Recall that an ob j ect in T is L -lo cal if and only if it b elongs to the essen tial image Im L of L ; see P roposition 2.5 .2 . Th e exactness of L implies that S := Ker L is a thic k sub catego ry and that Σ( L ) = Σ( S ). Therefore L -lo ca l and Σ( S )-local ob jects coincide. The follo wing r e su lt sa ys that acyclic and lo ca l ob jects form an orthogonal pair. Prop osition 4.10.1. L et L : T → T b e an exact lo c aliza tion functor . Then we have Ker L = ⊥ (Im L ) and (Ker L ) ⊥ = Im L. Mor e explictly, the fol lowing hol ds. (1) X ∈ T is L -acyclic if and only if T ( X, Y ) = 0 for every L -lo c al obje ct Y . (2) Y ∈ T is L -lo c al if and only if T ( X , Y ) = 0 f or every L -acyclic obje ct X . Pr o of. (1) W e write L = G ◦ F wh ere F is a fu ncto r and G a fully faithful righ t adjoint; see Corolla ry 2 .4.2. Sup pose ﬁ rst w e hav e giv en ob jects X, Y su c h that X is L -acyclic and Y is L -local. Observe that F X = 0 since G is faithful. Thus T ( X , Y ) ∼ = T ( X , GF Y ) ∼ = T ( F X, F Y ) = 0 . No w su p pose that X is an ob j e ct w ith T ( X, Y ) = 0 for all L - lo cal Y . Then T ( F X , F X ) ∼ = T ( X , GF X ) = 0 and therefore F X = 0. Thus X is L -acyclic. (2) This is a reformulatio n of Lemma 4.8.1 .  4.11. A functorial triangle. Let T b e a triangulated ca tegory and L : T → T an exact lo c alization fun c tor. W e den ote by η : Id T → L a morphism such that Lη is in vertible. It follo ws f r o m Proposition 4.9.1 and its pro of th at we obtain an exact fun ctor Γ : T → T b y completing for eac h X in T the morph ism η X to an exact triangle (4.11. 1) Γ X θX − → X ηX − → LX − → S ( Γ X ) . The exactness of Γ follo ws from Lemma 4.2.1. Observ e that Γ X is L -acyclic and th a t LX is L -local. In f a ct, the exact triangle (4.11. 1 ) is essent ially determined by these prop erties. This is a consequence of the follo w ing basic pr o p erties of L and Γ . Prop osition 4.11.1. The functors L, Γ : T → T have the fol lowing pr op erties. (1) L induc es an e quivalenc e T / Ker L ∼ − → Im L . (2) L induc es a left adjoint for the inclusion Im L → T . (3) Γ induc es a right adjoint for the inclusion Ker L → T . Pr o of. (1) is a reform ulation of Corollary 2.5.3 , and (2) follo ws from Corollary 2.4.2. (3) is an immed iate consequence of the construction of Γ via Prop osition 4.9.1.  LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 21 Prop osition 4.11.2. L et L : T → T b e an exact lo c alization fu nc tor and X an obje c t in C . Given any exact triangle X ′ → X → X ′′ → S X ′ with X ′ L -acyclic and X ′′ L -lo c al, ther e ar e unique isomorphisms α and β m aking the fol lowing diagr am c ommutative. (4.11. 2) X ′ / / α   X / / X ′′ / / β   S X ′ S α   Γ X θX / / X ηX / / LX / / S ( Γ X ) Pr o of. Th e morphism θ X in duces a bijection T ( X ′ , θ X ) since X ′ is acyclic. Th u s X ′ → X factors un i qu e ly through θ X via a morphism α : X ′ → Γ X . An application of (TR3) giv es a morphism β : X ′′ → LX making the d ia gram (4. 11.2) comm utativ e. Now apply L to this diagram. Th en Lβ is an isomorp h ism since LX ′ = 0 = LΓ X , and Lβ is isomorphic to β since X ′′ and L X are L -lo cal. Th us β is an isomorphism, and therefore α is an isomorphism.  4.12. Lo calization versus colo calization. F or exact functors on triangulated cate- gories, w e ha v e the follo wing symmetry p rinciple relating localizatio n and colocalization. Prop osition 4.12.1. L et T b e a triangulate d c ate gory. (1) Supp ose L : T → T is an exact lo c aliza tion functor and Γ : T → T the functor which is deﬁne d in terms of the exact triangle (4.11.1) . Then Γ is an exact c olo c alizatio n functor with Ker Γ = Im L and Im Γ = Ker L . (2) Supp ose Γ : T → T is an exact c olo c alization fu ncto r and L : T → T the functor which is deﬁne d in terms of the e x act triangle (4.11.1 ) . Then L is an exact lo c alization functor with Ker L = Im Γ and Im L = Ker Γ . Pr o of. It suﬃces to pr ov e (1) b ecause (2) is the dual statemen t. So let L : T → T b e an exact lo cal ization functor. It follo ws from the constru c tion o f Γ t h at it is of th e f o rm Γ = I ◦ I ρ where I ρ denotes a right adjoint of the fully faithful inclusion I : Ker L → T . Th u s Γ is a colo c alization fu ncto r b y Corolla ry 2.4.2. The exactness of Γ f o llo ws from Lemma 4.2.1, and th e iden tities Ker Γ = Im L and Im Γ = Ker L are easily deriv ed fr o m the exact tr iangle (4.1 1.1 ).  4.13. Recollemen ts. A r e c ol lement is by deﬁnition a d ia gram of exact fu ncto rs (4.13. 1) T ′ I / / T Q / / I λ o o I ρ o o T ′′ Q λ o o Q ρ o o satisfying the f ollo wing conditions. (1) I λ is a left adjoint and I ρ a right adjoint of I . (2) Q λ is a left adjoint and Q ρ a right adjoint of Q . (3) I λ I ∼ = Id T ′ ∼ = I ρ I and QQ ρ ∼ = Id T ′′ ∼ = QQ λ . (4) Im I = Ker Q . Note that the isomorphisms in (3) are sup posed to b e the adjunction morphisms resulting from (1) and (2). A recollemen t giv es rise to v arious lo caliz ation and colo cal ization functors for T . First observe that the fun c tors I , Q λ , a n d Q ρ are fully fait h f ul; see Proposition 2 .3.1 . Therefore Q ρ Q and I I λ are lo cali zation functors and Q λ Q and I I ρ are colocalization 22 HENNING KRAUSE functors. This follo ws from Corollary 2.4.2. Note that the lo c alization functor L = Q ρ Q has the additional p roperty that the inclusion Ker L → T admits a left adjoin t. Moreo v er, L determines the reco llement u p to an equiv alence. Prop osition 4.13.1. L et L : T → T b e an exact lo c alizatio n functor and supp ose tha t the inc lusion Ker L → T admits a left adjoint. Then L induc es a r e c ol lement of the fol lowing form . Ker L inc / / T / / o o o o Im L o o o o Mor e over, any r e c ol lement f or T is, up to e quivalenc es, of this f orm for some exact lo c alization functor L : T → T . Pr o of. W e apply Prop osition 4.9.1 and its dual assertion. Observe ﬁrst that any lo ca l- izatio n functor L : T → T induces the follo wing diagram. Ker L I =inc / / T Q = L / / I ρ = Γ o o Im L Q ρ =inc o o The fun ctor I admits a left adjoin t if and only if Q admits a left adjoin t. Th us the diagram can b e completed t o a r ecollemen t if and only if the inclusion I admits a left adjoin t. Supp ose no w t h e re is giv en a r e collemen t of the form (4.1 3.1 ). Th e n L = Q ρ Q is a lo c alization fu ncto r and the inclusion Ker L → T admits a left adjoint . T he fun ct or I induces an equiv alence T ′ ∼ − → K e r L and Q ρ induces an equiv alence T ′′ ∼ − → Im L . It is straigh tforw ard to form ulate and chec k the v arious compatibilit ies of these equiv alence s.  As a ﬁnal remark, let u s men tion that for a ny recollemen t of the form (4.13.1), the functors Q λ and Q ρ pro vide t wo (in general diﬀeren t) em b eddings of T ′′ in to T . If w e iden tify T ′ = Im I , then Q ρ iden tiﬁes T ′′ with ( T ′ ) ⊥ and Q λ iden tiﬁes T ′′ with ⊥ ( T ′ ); see Prop osition 4.10.1. 4.14. Example: T he derived category of a mo dule category . Let A b e an associa- tiv e ring. W e denote b y K (Mo d A ) the category of c hain complexes of A -mo dules w hose morphisms are the homotop y classes of c hain maps . T h e functor H n : K (Mo d A ) → Mo d A taking the cohomology of a complex in degree n is cohomologic al. A morph ism φ is ca lled quasi-isomorphism if H n φ is an isomorphism for all n ∈ Z , and we denote the set of all quasi-isomorphisms by qis. T hen D ( A ) := D (Mod A ) := K (Mo d A )[qis − 1 ] is by d eﬁnitio n the derive d c ate gory of Mo d A . Th e k ernel o f the quotien t functor Q : K (Mo d A ) → D (Mod) is the full sub ca tegory K ac (Mod A ) whic h is formed b y all acyclic complexes. Note that Q admits a left adjoin t Q λ taking eac h complex to its K-pro jectiv e r e solution and a righ t adjoint Q ρ taking eac h complex to its K-injectiv e resolution. Th u s we obtain the follo wing reco llement . (4.14. 1) K ac (Mod A ) inc / / K (Mod A ) Q / / o o o o D (Mod A ) Q λ o o Q ρ o o LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 23 It follo ws that for eac h pair of c hain complexes X , Y the set of morphisms D (Mod A )( X, Y ) is small, since Q λ induces a bijection with K (Mod A )( Q λ X, Q λ Y ). The a d jo ints of Q are discus s e d in more detail in Section 5.8. 4.15. Example: A derived category w ithout small morphism sets. F or an y ab elia n ca tegory A , the derive d c ate gory D ( A ) is b y deﬁnition K ( A )[qis − 1 ]. Here, K ( A ) denotes the categ ory of chain complexes in A whose morphisms are the homotop y classes of c hain maps, and qis denotes the set of quasi-isomorphisms. Let us iden tify ob jects in A with chain complexes concen trated in degree zero. W e giv e an example of an ab elian category A and an o b ject X in A such that the set Ext 1 A ( X, X ) ∼ = D ( A )( X, S X ) is not small . This exa mp l e is tak en from F reyd [14, pp. 131] and has b een p oin ted out to me by Neeman. Let U denote the set of all cardinals o f small s e ts. This s et is not small. Consid e r the free asso ciativ e Z - algebra Z h U i w hic h is generated by the elemen ts of U . No w let A = Mod A denote the catego ry of A -mo dules, where it is assumed that the u nderlying set of eac h mo dule is small. Let Z denote the trivial A -mo dule, that is, z u = 0 for all z ∈ Z an d u ∈ U . W e claim that the set Ext 1 A ( Z , Z ) is not small. T o see this, d e ﬁ ne for eac h u ∈ U an A -mo dule E u = Z ⊕ Z b y ( z 1 , z 2 ) x = ( ( z 2 , 0) if x = u , (0 , 0) if x 6 = u , where ( z 1 , z 2 ) ∈ E u and x ∈ U . Then we h a v e short exact sequences 0 → Z [ 1 0 ] − − → E u [ 0 1 ] − − − → Z → 0 which yield pairwise diﬀeren t element s of Ext 1 A ( Z , Z ) as u ru ns though the element s in U . 4.16. Example: The recollemen t induced by an idempotent. Recollemen ts can b e d e ﬁ n ed for ab elian categorie s in the same wa y as for triangulated categories. A t yp ical example arises for an y mo dule category from an idemp oten t elemen t of the un derlying ring. Let A b e an asso ciativ e ring and e 2 = e ∈ A an id e mp oten t. T h en the fun c tor F : Mod A → Mod eAe taking a mo dule M to M e and restriction along p : A → A/ AeA induce the f o llo win g recollemen t. Mo d A/ AeA p ∗ / / Mo d A F / / o o o o Mo d eAe −⊗ eAe eA o o Hom eAe ( Ae, − ) o o Note that we can describ e adjoin ts of F since F = Hom A ( eA, − ) = − ⊗ A Ae. The recollemen t for Mo d A induces the f ollo wing recollemen t of triangulate d cate gories for D ( A ). Ker D ( F ) inc / / D ( A ) D ( F ) / / o o o o D ( eAe ) −⊗ L eAe eA o o R Hom eAe ( Ae, − ) o o 24 HENNING KRAUSE The functor F is exact and D ( F ) tak es b y d eﬁ n it ion a complex X to F X . The func- tor D ( p ∗ ) : D ( A/ AeA ) → D ( A ) id e ntiﬁes D ( A/ AeA ) with Ker D ( F ) if and only if T or A i ( A/ AeA, A/ AeA ) = 0 for all i > 0. 4.17. Notes. T riangulated catego ries were introdu c ed indep enden tly in algebraic geom- etry by V erd ie r in his th` ese [48], and in algebraic top ol ogy b y Pupp e [38]. Grothendiec k and h is sc ho ol used the formalism of triangulated and derive d catego ries for studying homologica l prop erti es of ab elia n categories. Early examples are Grothendiec k d uali ty and lo cal cohomology for categ ories of shea v es. The basic example of a triangulated catego ry from top olo gy is the stable homotop y cat egory . Lo ca lizations of triangulate d ca tegories are discussed in V erdier’s th ` ese [48]. In par- ticular, he int ro duced the lo caliza tion (or V er dier quotient ) of a triangulated category with resp ect to a triangulated sub ca tegory . In t h e context of stable homotop y theory , it is more common to think of lo cal ization fun ctors as endofun ct ors; see for instance the wo rk of Bousﬁ eld [8], which explains the term B ousﬁeld lo c alizat ion . The standard reference for recollemen ts is [6]. Resolutions of unb ounded complexes were ﬁrst studied b y Sp a ltenstein in [44]; see also [5]. 5. Localiza tion via Br own represent ability 5.1. Bro wn represen tatbility. Let T b e a triangulated catego ry and sup pose that T has small copro ducts. A lo c alizing sub c ate gory of T is by d e ﬁ n it ion a thic k sub category whic h is closed und e r taking small copro ducts. A lo ca lizing su b category of T is gener ate d b y a ﬁxed set of ob jects if it is the smallest lo calizing su bcategory of T which conta ins this s e t. W e sa y that T is p erfe ctly gener ate d b y some small set S of ob ject s of T p ro vided the follo wing h o lds. (PG1) There is n o prop er lo ca lizing sub cat egory of T which con tains S . (PG2) Giv en a family ( X i → Y i ) i ∈ I of morp hisms in T such that the induced map T ( C , X i ) → T ( C, Y i ) is surjectiv e for all C ∈ S and i ∈ I , the in duced map T ( C , a i X i ) − → T ( C, a i Y i ) is surjectiv e. W e sa y that a triangulated catego ry T with small pro ducts is p erfe c tly c o gener ate d if T op is p erfectly generated. Theorem 5.1.1. L et T b e a triangulate d c ate gory with smal l c opr o ducts and supp ose T is p erfe ctly gener ate d. (1) A functor F : T op → Ab is c ohomo lo gic al and sends smal l c opr o ducts in T to pr o ducts if and only i f F ∼ = T ( − , X ) for some obje ct X in T . (2) An exact functor T → U b etwe en triangulate d c ate g ories pr eserves smal l c opr o d- ucts if and only if it has a right adjoint. Pr o of. F or a pro of of (1) see [24, Theorem A]. T o p ro v e (2), su pp o se that F p r e serves small copro ducts. Then one deﬁnes the righ t adjoint G : U → T b y send ing an ob ject X in U to the ob ject in T represen ting U ( F − , X ). Thus U ( F − , X ) ∼ = T ( − , GX ). Con ve r s ely , giv en a righ t adjoint of F , it is automatic that F preserves small copro ducts.  LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 25 Remark 5.1.2. (1) In the presence of (PG2), condition (PG1) is equiv alen t to the follo wing: F or an o b ject X in T , w e ha v e X = 0 if T ( S n C, X ) = 0 for all C ∈ S and n ∈ Z . (2) A p erfectly generated triangulated category T h as small pro ducts. In fact, Bro wn represen tabilit y implies that for any family of ob jects X i in T th e functor Q i T ( − , X i ) is r e p r ese nted by an ob ject in T . 5.2. Lo calization functors via Bro w n represen tability . The existence of lo ca l- izatio n f u ncto rs is b asically equiv alen t to the existence of certain right adjoints; see Prop ositio n 4.9.1. W e combine this observ atio n with Bro wn’s represen tabilit y theorem and obtain the follo wing. Prop osition 5.2.1. L et T b e a triangulate d c ate gory which admits smal l c opr o ducts and ﬁx a lo c alizing sub c ate gory S . (1) Supp ose S is p erfe c tly gener ate d. Then ther e exists an exact lo c alization functor L : T → T with Ker L = S . (2) Supp ose T is p erfe ctly gener ate d. Then ther e exists an exact lo c alization functor L : T → T with Ker L = S if and onl y if the morphisms b etwe en any two obje cts in T / S form a smal l set. Pr o of. Th e e xistence of a lo cali zation functor L with Ker L = S is equiv alen t to the existence of a right adjoin t for the in c lusion S → T , and it is equiv ale nt to th e existence of a right adjoin t for the quotien t f unctor T → T / S . Both fun ct ors pr e serve sm a ll copro ducts since S is closed u nder taking small copro ducts; s e e Prop osition 3.5.1. No w apply Theorem 5.1.1 for the existence of right adjoint s.  5.3. Compactly generated triangulated categories. Let T b e a triangulated cat - egory with small coprod u ct s. An ob ject X in T is calle d c omp act (or smal l ) if every morphism X → ` i ∈ I Y i in T factors through ` i ∈ J Y i for some ﬁnite subset J ⊆ I . Note that X is compact if and only if the represen table functor T ( X, − ) : T → Ab p reserv es small copro ducts. The compact ob jects in T form a thic k sub category whic h we denote b y T c . The triangulated catego ry T is called c omp actl y g ener ate d if it is p erfectly generated b y a small set of compact ob jects. Observe that cond i tion (PG2) is automatically satisﬁed if ev ery ob ject in S is compact. A compactly generated triangulated category T is p erfectly cogenerate d . T o see this, let S b e a set of compact generato rs. T hen the ob jects r e p r e senting Hom Z ( T ( C , − ) , Q / Z ), where C runs through all ob jects in S , form a set of p erfect cogenerato rs for T . The follo wing prop osition is a reformulat ion of Bro wn r e pr e senta bility f o r compactly generated triangulated categories. Prop osition 5.3.1. L et F : T → U b e an exact fu nc tor b etwe en triangulate d c ate gories. Supp ose that T has smal l c op r o ducts and tha t T is c omp actly gener ate d. (1) The functor F admits a right adjoint if and only if F pr eserves smal l c op r o ducts. (2) The functor F admits a left adjoint if and only if F p r eserves smal l pr o ducts. 5.4. Righ t adjoin t functors preserving copro duc ts. The foll owing lemma pro vides a c haracteriza tion of the fact that a right adjoin t functor preserv es small copro ducts. This w ill b e us e fu l in the co ntext of compactly generated categorie s. 26 HENNING KRAUSE Lemma 5.4.1. L et F : T → U b e an exact fu nctor b etwe en triangulate d c ate gories which has a right adjoint G . (1) If G pr eserves smal l c opr o ducts, then F pr eserves c omp actness. (2) If F pr eserves c omp actness and T is g ener ate d by c omp act obje cts, then G pr e - serves smal l c opr o ducts. Pr o of. Let X b e an ob ject in T and ( Y i ) i ∈ I a family of ob jects in U . (1) W e ha v e (5.4.1 ) U ( F X, a i Y i ) ∼ = T ( X , G ( a i Y i )) ∼ = T ( X , a i GY i ) . If X is compact, th en the isomorphism shows that a morphism F X → ` i Y i factors through a ﬁ nite copro duct. It follo ws that F X is compact. (2) Let X b e compact. Th e n the canonical morph ism φ : ` i GY i → G ( ` i Y i ) induces an isomorp h ism T ( X , a i GY i ) ∼ = a i T ( X , GY i ) ∼ = a i U ( F X, Y i ) ∼ = U ( F X, a i Y i ) ∼ = T ( X , G ( a i Y i )) , where the last isomorph ism uses that F X is compact. It is easily chec k ed that the ob jects X ′ in T s uc h that T ( X ′ , φ ) is an isomorphism form a lo calizing sub category of T . Thus φ is an isomorph ism b ecause the compact ob ject s generate T .  5.5. Lo calization functors preserving copro ducts. The follo wing result pro vides a c haracteriza tion of the fact that an exact localization fun ct or L preserv es small co- pro ducts; in that case one calls L smashing . The example giv en b elo w explains this terminology . Prop osition 5.5.1. L et T b e a c ate gory with smal l c opr o ducts and L : T → T an exact lo c alization functor. Then the f ol low ing ar e e quivalent. (1) The functor L : T → T pr eserves smal l c opr o ducts. (2) The c olo c alization fu ncto r Γ : T → T with Ker Γ = Im L pr eserves smal l c opr o d- ucts. (3) The right adjoint of the inclusion functor Ker L → T pr e s erves smal l c opr o ducts. (4) The right adjoint of the quotient functor T → T / Ker L pr eserves sm al l c opr o d- ucts. (5) The sub c ate gory Im L of al l L -lo c al obje cts is close d under taking smal l c op r o d- ucts. If T is p erfe ctly gener ate d, in addition the f ol low ing is e quivalent. (6) Ther e exists a r e c ol lement of the fol lowing form. (5.5.1 ) Im L inc / / T / / o o o o Ker L o o o o Pr o of. (1) ⇔ (4) ⇔ (5) follo ws from Prop osition 2.7.1. (1) ⇔ (2) ⇔ (3) is easily dedu ced from the fun ctorial triangle (4.11. 1 ) r e lating L and Γ . (5) ⇔ (6): Assume that T is p erfectly generated. Then we can apply Bro wn’s repre- sen tabilit y theorem and consider the sequence Im L I − → T Q − → Ker L LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 27 where I denotes the in clusion and Q a righ t adjoint of the inclusion Ker L → T . Note that Q induces an equiv alence T / Im L ∼ − → Ker L ; see Prop ositions 4.11.1 and 4.12.1. The f uncto r s I and Q ha ve left adjoin ts. Thus the pair ( I , Q ) giv es r ise to a recollemen t if and only if I and Q b oth admit r i ght adjoints. It follo ws from Prop osition 4.9.1 that this happ ens if and only if Q admits a righ t adj oint. No w Bro wn ’s r ep r e sentabilit y theorem imp li es that this is equiv alen t to th e fact that Q pr ese rves small coprodu ct s. And Proposition 3.5.1 sho ws that Q preserv es small c opro ducts if and only if Im L is closed under taking small copro ducts. Th is ﬁnishes the p roof.  Remark 5.5.2. (1) Th e implication (6) ⇒ (5) holds without any extra assumption on T . (2) S u pp o se an exact lo ca lization fun ct or L : T → T preserves small copro ducts. If T is compactly generated, then Im L is compactly ge nerated. This follo ws from Lemma 5.4.1, b ecause the left adjoint of the inclusion Im L → T send s the compact generators of T to compact generators for Im L . A similar argumen t sh o ws that Im L is p erfectly generated provided that T is p erfectly generated. Example 5.5.3. L et S b e the stable homotop y catego r y of sp ectra and ∧ its smash pro duct. Then a n exact localization functor L : S → S pr eserves small copro ducts if and only if L is of th e form L = − ∧ E for some sp ectrum E . W e sket ch the argumen t. Let S denote th e sphere sp ectrum. T here exist s a natural morphism η X : X ∧ LS → LX for eac h X in S . S upp ose that L preserv es small copro ducts. Then the sub cat egory of ob jects X in S suc h that η X is inv ertible con tains S and is closed und er f o rm in g small copro duts and exact triangles. Thus L = − ∧ E for E = LS . Let L : T → T b e a n exact lo ca lization fun c tor whic h induces a r ecollemen t of the form (5.5.1). T h en the sequence Ker L → T → Im L of left adjoint f unctors ind u ce s a sequence (Ker L ) c − → T c − → (Im L ) c of exact fun ctors, by Lemma 5.4.1. This sequen ce is of some interest. The fu nct or (Ker L ) c → T c is fully f a ithful a n d iden tiﬁes ( Ker L ) c with a thic k sub catego r y of T c , whereas the functor T c → (Im L ) c shares some formal prop erties with a quotien t fu nct or. A t ypical example arises from ﬁ nite lo ca lization; see Th e orem 5.6.1. Ho w eve r , there are examples w h ere T is co mp actly generated bu t 0 = (Ker L ) c ⊆ Ker L 6 = 0; see [25] for details. 5.6. Finite lo calization. A common type of lo caliza tion for triangulated categories is ﬁnite localization. Here, we explain the basic result and refer to our discussion of w ell generated categories for a more general app roa ch and fur t h er details. Let T b e a compactly generated triangulated category and supp ose w e ha ve giv en a sub catego ry S ′ ⊆ T c . Let S denote the lo caliz ing sub catego ry generated by S ′ . Then S is compactly generated and therefore the inclusion functor S → T admits a righ t adjoin t by Bro wn’s rep r e sentabili ty theorem. In particular, we hav e a lo cali zation functor L : T → T with Ker L = S and the morp hisms b et w een an y pair of ob jects in T / S form a small set; see Prop osition 5.2.1. W e observ e that the compact ob jects of S ident ify with the sm a llest thic k su bcategory of T c con taining S ′ . Th is fol lows from 28 HENNING KRAUSE Corollary 7.2.2. Th u s w e obtain the f o llo wing comm utativ e diagram of exact fu nct ors. S c inc   inc / / T c inc   can / / T c / S c J   S I =inc / / T Q =can / / T / S Theorem 5.6.1. L et T and S b e as ab ove. Then the quotient c ate gory T / S is c omp actly gener ate d. The induc e d exact f u ncto r J : T c / S c → T / S is ful ly f aithful and the c ate gory ( T / S ) c of c omp act obje cts e quals the f u l l sub c ate gory c onsisting of al l dir e ct facto rs of obje c t s in the image of J . Mor e over, the inclusion S ⊥ → T induc e s the fol lowing r e c ol lement. S ⊥ inc / / T / / o o o o S o o o o Pr o of. Th e inclusion I preserv es compactness and therefore the right adj oint I ρ pre- serv es small copro ducts by Lemma 5.4.1. Thus Q ρ preserv e small copro ducts by Prop o- sition 5.5.1, and th erefore Q preserv es compactness, aga in b y Lemma 5.4.1. It follo ws that J induces a functor T c / S c → ( T / S ) c . In particular, Q sends a set of compact generators of T to a set of compact generators for T / S . Next we apply Lemma 4.7.1 to sho w that J is fully faithful. F or this, one needs to c hec k that ev ery morphism from a compact ob ject in T to an ob ject in S fact ors through some ob ject in S c . This follo ws f rom Theorem 7.2.1. T he image of J is a full triangulated sub catego ry of T c whic h generates T / S . Another app lic ation of Corolla r y 7. 2.2 shows that ev ery compact ob jec t of T / S is a d irec t factor of some ob ject in the image of J . Let L : T → T d e n ote the lo caliza tion functor with Ker L = S . Then S ⊥ equals the full sub categ ory of L -lo cal ob jects. This sub category is closed un der small copro ducts since S is generated by compact ob jects. T h us the existence of the recolle ment follo ws from Prop osition 5.5.1.  5.7. Cohomological lo calization via lo calization of graded mo dules. Let T b e a triangulated category whic h admits sm a ll copro ducts. Supp ose that T is ge n erated b y a sm a ll set of compact ob jects. W e ﬁ x a graded 1 ring Λ and a graded cohomological functor H ∗ : T − → A in to the category A of graded Λ-mo dules. Thus H ∗ is a functor which s en d s eac h exact triangle in T to an exact sequen ce in A , and we hav e an isomorph ism H ∗ ◦ S ∼ = T ◦ H ∗ where T denotes the shift functor for A . In addition, w e assume that H ∗ preserv es small pro ducts and copro ducts. Theorem 5.7.1. L et L : A → A b e an exact lo c aliza tion functor for the c ate gory A of gr ad e d Λ -mo dules. Then ther e exists an exact lo c alization functor ˜ L : T → T such that the fol lowing squar e c ommutes up to a natur al isomorp hism. T ˜ L / / H ∗   T H ∗   A L / / A 1 All graded rings and mo dules are graded ove r Z . Morphisms b et ween graded mo dules are degree zero maps. LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 29 Mor e pr e ci sely, the adjunction morph isms Id A → L and I d T → ˜ L induc e for e ach X in T the fol lowing isomo rphisms. H ∗ ˜ LX ∼ − → L ( H ∗ ˜ LX ) = LH ∗ ( ˜ LX ) ∼ ← − LH ∗ ( X ) An obje ct X in T is ˜ L -acyclic if and only if H ∗ X is L -acyclic. If an obje ct X in T is ˜ L - lo c al, then H ∗ X is L -lo c al. The c onverse holds, pr ovide d tha t H ∗ r e ﬂ e cts isomorphisms. Pr o of. W e recall that T is p erfect ly cog enerated b eca use it is compactl y generated. Thus Bro wn’s representabili ty theorem provi des a co mp a ct ob ject C in T suc h that H ∗ X ∼ = T ( C , X ) ∗ := a i ∈ Z T ( C , S i Y ) for all X ∈ T . No w consider the essen tial image Im L of L which equals the fu l l sub catego r y formed b y all L -local ob jects in A . Because L is exact, this sub category is c oher ent , that is, for an y exact sequence X 1 → X 2 → X 3 → X 4 → X 5 with X 1 , X 2 , X 4 , X 5 ∈ A , we ha ve X 3 ∈ A . This is an immediate consequence of the 5-le mm a . In addition, Im L is closed und er taking small pro ducts. The L -lo ca l ob jects form an ab elian Grothendiec k catego ry and therefore Im L admits an injectiv e cogenerator, say I ; see [16]. Using again Bro wn’s representabili ty theorem, there exists ˜ I in T suc h th at (5.7.1 ) A ( H ∗ − , I ) ∼ = T ( − , ˜ I ) and therefore A ( H ∗ − , I ) ∗ ∼ = T ( − , ˜ I ) ∗ . No w consider the sub cat egory V of T which is formed by all ob jects X in T su c h that H ∗ X is L -lo cal. This is a triangulat ed sub category w h ic h is closed under taking small pro ducts. Observe that ˜ I b elongs to V . T o pr o v e this, take a f ree presenta tion F 1 − → F 0 − → H ∗ C − → 0 o v er Λ and apply A ( − , I ) ∗ to it. Using the isomorphism (5.7.1), we see that H ∗ ˜ I b elongs to Im L b ecause Im L is coherent and closed under taking small pro ducts. No w let U denote the sm a llest triangulated sub category of T con taining ˜ I and closed under taking small p roducts. Observe that U ⊆ V . W e claim th at U is p erfectly cogen- erated by ˜ I . Thus, give n a family of morph i sm s φ i : X i → Y i in U suc h that T ( Y i , ˜ I ) → T ( X i , ˜ I ) is surjectiv e for all i , w e need to sh ow that T ( Q i Y i , ˜ I ) → T ( Q i X i , ˜ I ) is sur- jectiv e. W e argue as follo ws. If T ( Y i , ˜ I ) → T ( X i , ˜ I ) is surjectiv e, then the isomorphism (5.7.1) imp li es that H ∗ φ i is a monomorphism s i n c e I is an inj e ctiv e cogenerator for Im L . Th u s the pro duct Q i φ i : Q i X i → Q i Y i induces a monomorp hism H ∗ Q i φ i = Q i H ∗ φ i and therefore T ( Q i φ i , ˜ I ) is surjectiv e. W e conclude fr o m Bro wn’s represen tabilit y th e o- rem that the inclusion fu nct or G : U → T has a left ad j o int F : T → U . Thus ˜ L = G ◦ F is a localizatio n fu ncto r by Corollary 2.4.2. Next we sho w that an ob ject X ∈ T is ˜ L -acyclic if and only if H ∗ X is L -acyclic. This follo ws from Prop osition 4.10.1 and the isomorphism (5.7.1), b ecause we h av e ˜ LX = 0 ⇐ ⇒ T ( X , ˜ I ) = 0 ⇐ ⇒ A ( H ∗ X, I ) = 0 ⇐ ⇒ LH ∗ X = 0 . 30 HENNING KRAUSE No w d enot e b y η : Id A → L and ˜ η : Id T → ˜ L the adju ncti on morphisms and consider the follo wing comm utativ e square. (5.7.2 ) H ∗ X ηH ∗ X / / H ∗ ˜ η X   LH ∗ X LH ∗ ˜ ηX   H ∗ ˜ LX ηH ∗ ˜ LX / / LH ∗ ˜ LX W e claim that LH ∗ ˜ η X a nd ηH ∗ ˜ LX are inv ertible for eac h X in T . T he morphism ˜ η X induces an exac t triangle X ′ → X ˜ η X − − → ˜ LX → S X ′ with ˜ LX ′ = 0 = ˜ LS X ′ . Applying the cohomolog ical fu ncto r LH ∗ , we see that LH ∗ ˜ η X is an isomorphism , since LH ∗ X ′ = 0 = LH ∗ S X ′ . Thus LH ∗ ˜ η is inv ertible. The morphism η H ∗ ˜ LX is inv ertible b eca u s e H ∗ ˜ LX is L -lo cal . T his f o llo ws from the fact that ˜ LX b elo ngs to U . The commuta tive square (5.7.2) implies that H ∗ ˜ η X is inv ertible if and only if η H ∗ X is in v ertible. T h us if X is ˜ L -local, then H ∗ X is L -local. The con v erse holds if H ∗ reﬂects isomorphisms.  Remark 5.7.2. (1) The localizatio n f u ncto r ˜ L is essen tially uniquely determined b y H ∗ and L , b e cause Ker ˜ L = Ker LH ∗ . (2) Sup pose that C is a generator of T . If L p reserv es small copro ducts, then it follo ws that ˜ L preserve s small copro ducts. In fact, the assump ti on implies that H ∗ ˜ L preserv es small copro ducts, since LH ∗ ∼ = H ∗ ˜ L . But H ∗ reﬂects isomorphisms b ecause C is a generator of T . Th us ˜ L preserv es small copro ducts. 5.8. Example: Resolutions of c hain complex es. Let A b e an asso ciati ve ring. Then the deriv ed cat egory D ( A ) of un b ounded chain complexes of mo dules o ver A is compactly generated. A compact generator is the ring A , view ed as a complex concent rated in degree zero. Let us b e more p reci se, b ecause w e wan t to giv e an explicit construction of D ( A ) whic h implies that the morph isms b et w een any tw o ob jects in D ( A ) f orm a small set. Moreo v er, we com b i n e Bro wn repr e sentabili ty with Pr o p osition 4.9.1 to pro vide descriptions of the adjoints Q λ and Q ρ of the quotien t functor Q : K (Mod A ) → D ( A ) whic h app ear in the recollemen t (4.14.1). Denote by Lo c A the lo calizing sub ca tegory of K (Mo d A ) whic h is generated b y A . Then Lo c A is a compactly generated triangulated categ ory and (Lo c A ) ⊥ = K ac (Mod A ) since K (Mod A )( A, S n X ) ∼ = H n X. Bro wn rep r ese ntabilit y p ro vides a right adjoin t f or the inclusion L o c A → K (Mo d A ) and therefore the comp osit e F : Lo c A inc − − → K (Mod A ) can − − → D ( A ) is a n equiv alence b y Prop ositio n 4.9.1. The right adjoin t of the inclusion Lo c A → K (Mod A ) annihilates the acyclic complexes and in duces therefore a fu ncto r D ( A ) → Loc A (whic h is a quasi- in ve rs e for F ). The comp osite with the inclusion Lo c A → K (Mo d A ) is the left adjoint Q λ of Q and tak es a co mp le x to its K- p r oje ctive r esolution . No w ﬁx an injectiv e cogenerator I for the c ategory of A -mod ules, for instance I = Hom Z ( A, Q / Z ). W e d e note by Colo c I th e smallest thic k sub catego ry of K (Mo d A ) LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 31 closed u nder small pro ducts and con taining I . Then I is a p erfect cogenerator for Colo c I and ⊥ (Coloc I ) = K ac (Mod A ) since K (Mod A )( S n X, I ) ∼ = Hom A ( H n X, I ) . Bro wn rep r ese ntabilit y provides a left adjoint for the inclusion Colo c I → K (Mo d A ) and therefore the composite G : Coloc I inc − − → K (Mo d A ) can − − → D ( A ) is an equiv alence by Prop ositio n 4.9.1. Th e le ft adjoin t of the inclusion Colo c I → K (Mod A ) ann ihila tes the acyclic c omplexes and induces therefore a functor D ( A ) → Colo c I (whic h is a quasi- in ve rs e for G ). The comp o sition with the inclusion Coloc I → K (Mo d A ) is the right adjoin t Q ρ of Q and tak es a complex to its K-i nje ctive r esolution . 5.9. Example: Homological epimorphisms. Let f : A → B b e a r in g homomor- phism and f ∗ : D ( B ) → D ( A ) the fu ncto r giv en b y restriction of scalars. Clearly , f ∗ preserv es small pro ducts and copro ducts. Th us Bro wn represen tabilit y implies the exis- tence of left and righ t adjoin ts for f ∗ since D ( A ) is co mp a ctly generate d. F or instance, the left adjoin t is the derived tensor functor − ⊗ L A B : D ( A ) → D ( B ) w hic h preserves compactness. The fun ct or f ∗ is fully faithful if and only if ( f ∗ − ) ⊗ L A B ∼ = Id D ( B ) iﬀ B ⊗ A B ∼ = B and T or A i ( B , B ) = 0 for all i > 0. In that case f is called homolo gic al epimorphism and the exact fun c tor L : D ( A ) → D ( A ) sending X to f ∗ ( X ⊗ L A B ) is a lo cal ization functor. T ake for instance a commuta tiv e ring A and let f : A → S − 1 A = B b e the localiza- tion with resp ect to a m ultiplicativ ely closed su b set S ⊆ A . Th en the induced exac t lo c alization functor L : D ( A ) → D ( A ) tak es a c hain complex X to S − 1 X . Note that L preserv es small copro ducts. In particular, L giv es rise to the follo wing recolleme nt. D ( B ) f ∗ / / D ( A ) / / −⊗ L A B o o R Hom A ( B , − ) o o U o o o o The triangulated category U is equiv alen t to t h e k ernel of − ⊗ L A B , an d one can sho w that Ker( − ⊗ L A B ) is the lo calizing sub catego ry of D ( A ) generated by the complexes of the form · · · → 0 → A x − → A → 0 → · · · ( x ∈ S ) . 5.10. Notes. Th e Bro wn represen tabilit y theorem in homotop y theory is due to Bro wn [9]. Generalizatio ns of the Brown represen tabilit y theore m for triangulated c ategories can b e found in work of F ranke [13], Keller [21], and Neeman [31, 33]. T h e v ersion used here is tak en from [24]. T h e ﬁnite localizatio n theorem for compactly generated triangulated categ ories is due to Neeman [30]; it is based on previous w ork of Bousﬁeld, Ra v enel, Thomason-T robaugh, Y ao, and others. T he cohomologica l lo caliza tion functors comm uting with lo caliza tion functors of graded mo dules ha v e b een u s ed to set up lo cal cohomolog y functors in [7]. 6. Wel l genera ted triangula ted ca tegories 6.1. Regular cardinals. A cardinal α is called r e gular if α is n o t the su m of fewer than α cardinals, all smaller than α . F or example, ℵ 0 is regular b ecause the sum of ﬁnitely many ﬁn i te cardinals is ﬁn ite. Also, the s ucc essor κ + of ev ery inﬁnite cardinal κ 32 HENNING KRAUSE is regular. In particular, there are arb it rarily large regular cardinals. F or more details on r eg ular cardinals, see for ins tance [26]. 6.2. Lo calizing sub categories. Let T b e a triangulated category and α a regular cardinal. A coprod uct in T is calle d α -c opr o duct if it has less than α factors. A full sub catego ry of T is called α -lo c alizing if it is a thic k su bcategory and closed u nder taking α - copro ducts. Giv en a sub cate gory S ⊆ T , w e denote b y Lo c α S the smallest α -localizing su bcategory of T whic h con tains S . Note that Lo c α S is small pro vided that S is small. A full sub category of T is c alled lo c alizing if it is a thic k sub cat egory and cl osed und e r taking small copro ducts. The smallest localizing sub cat egory cont aining a sub category S ⊆ T is Lo c S = S α Lo c α S where α ru ns through all regular cardinals. W e call Lo c S the lo calizing sub category generated by S . 6.3. W ell generated triangulated categories. Let T b e a triangulated category whic h admits small copro ducts and ﬁx a regular cardinal α . An ob ject X in T is called α -smal l if ev ery morphism X → ` i ∈ I Y i in T factors through ` i ∈ J Y i for s o me subset J ⊆ I with card J < α . The triangulat ed c ategory T is calle d α -wel l gener ate d if it is p erfectly generated by a small set of α -small ob jects, and T is ca lled wel l gener ate d if it is β -w ell generat ed for some r egular ca r d inal β . Supp ose T is α -w ell generated by a small set S of α -small ob jects. Giv en any regular cardinal β ≥ α , w e denote by T β the β -localizing sub cate gory Lo c β S generated b y S and call the ob jects of T β β -c omp act . Cho osing a representa tiv e for eac h isomorp h ism class, one can show that the β -compact ob jects form a small set of β -small p erfect generators for T . Moreo v er, T β do es not dep end on the c hoice of S . F or a pro of we refer to [23, Lemma 5]; see al so Prop osition 6.8.1 and Remark 6 .10.2. Note that T = S β T β , where β runs th r o u g h all regular cardinals greater or equal than α , b ecause S β T β is a triangulated su bcategory con taining S and closed u nder small copro ducts. Remark 6.3.1. T he α -small ob ject s of T form an α -lo ca lizing sub categ ory . Example 6.3.2. A triangulated category T is ℵ 0 -w ell generated if and only if T is compactly generated. In that case T ℵ 0 = T c . Example 6.3.3. Let A b e the catego ry of shea ve s of ab elian groups on a n o n - compact, connected manifold of dimension at least 1. T hen the deriv ed catego ry D ( A ) of un- b ounded c hain complexes is we ll generated, b ut the only compact ob ject in D ( A ) is the zero ob ject; see [34]. F or more examples of well generated but not compactly generated triangulated categories, see [32]. 6.4. Filtered categories. Let α b e a regular cardinal. A catego ry C is called α -ﬁlter e d if th e follo wing h o lds. (FIL1) There exists an ob ject in C . (FIL2) F or ev ery family ( X i ) i ∈ I of few er than α ob jects there exists an ob ject X with morphisms X i → X for all i . (FIL3) F or eve r y family ( φ i : X → Y ) i ∈ I of few er than α morphism s th ere exists a morphism ψ : Y → Z with ψ φ i = ψ φ j for all i and j . One drops the cardin a l α and calls C ﬁlter e d in case it is ℵ 0 -ﬁltered. LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 33 Giv en a fun ctor F : C → D , we use th e term α -ﬁlter e d c olimit f o r the colimit co lim − − − → X ∈C F X pro vided that C is a small α -ﬁltered ca tegory . Lemma 6.4.1 . L et i : C ′ → C b e a ful ly faithful functor with C a smal l α -ﬁlter e d c ate gory. Supp ose that i is c oﬁnal in the sense tha t for any X ∈ C ther e is an o bje ct Y ∈ C ′ and a morphism X → iY . Then C ′ is a smal l α -ﬁlter e d c ate gory, and fo r any functor F : C → D into a c ate gory which admits α -ﬁlter e d c olim its, the natur al morph ism colim − − − → Y ∈C ′ F ( iY ) − → colim − − − → X ∈C F X is an isomorphism . Pr o of. See [19, Prop osition 8.1.3].  A fu ll sub category C ′ of a small α -ﬁltered catego r y C is called c oﬁna l if for any X ∈ C there is an ob j e ct Y ∈ C ′ and a morphism X → Y . 6.5. Comma categories. Let T b e a triangulated category wh ic h admits small co- pro ducts a nd ﬁ x a full sub catego r y S . Giv en an ob ject X in T , let S /X denote the catego ry whose ob jects are pairs ( C, µ ) with C ∈ S and µ ∈ T ( C, X ). The morphism s ( C, µ ) → ( C ′ , µ ′ ) are the morph isms γ : C → C ′ in T making the follo wing diagram comm utativ e. C γ / / µ   6 6 6 6 6 6 6 C ′ µ ′          X Analogously , one deﬁn e s for a morphism φ : X → X ′ in T the category S /φ w hose ob jects are commuting squares in T of th e form C γ / / µ   C ′ µ ′   X φ / / X ′ with C, C ′ ∈ S . Lemma 6.5.1. L et α b e a r e gular c ar dinal and S an α -lo c alizing sub c ate gory of T . Then the c ate gories S /X and S /φ ar e α -ﬁlter e d for e ach obje ct X and e ach morphism φ in T . Pr o of. Straight forward.  6.6. The comma category of a n e xact triangle. Let T b e a triangulated category . W e consider the category of pairs ( φ 1 , φ 2 ) of co mp osable morph isms X 1 φ 1 − → X 2 φ 2 − → X 3 in T . A morphism µ : ( φ 1 , φ 2 ) → ( φ ′ 1 , φ ′ 2 ) is a triple µ = ( µ 1 , µ 2 , µ 3 ) of morph isms in T making the follo w ing d ia gram co mmutativ e. X 1 φ 1 / / µ 1   X 2 φ 2 / / µ 2   X 3 µ 3   X ′ 1 φ ′ 1 / / X ′ 2 φ ′ 2 / / X ′ 3 34 HENNING KRAUSE A p a ir ( φ 1 , φ 2 ) of comp osa ble morphisms is called exact if it ﬁts in to an exact triangle X 1 φ 1 − → X 2 φ 2 − → X 3 φ 3 − → S X 1 . Lemma 6.6.1. L et µ : ( γ 1 , γ 2 ) → ( φ 1 , φ 2 ) b e a morp hism b etwe en p airs of c omp osa ble morphis ms and supp ose that ( φ 1 , φ 2 ) is exact. Then µ factors thr ough an e xa ct p air of c omp osable morphism s which b elong to the smal lest ful l triangulate d sub c ate gory c on- taining γ 1 and γ 2 . Pr o of. W e pro ceed in t wo steps. Th e ﬁ rst step pro vides a facto rization of µ through a pair ( γ ′ 1 , γ ′ 2 ) of comp osable morp hisms suc h that γ ′ 2 γ ′ 1 = 0. T o ac hiev e this, complete γ 1 to an exac t triangle C 1 γ 1 − → C 2 ¯ γ 2 − → ¯ C 3 → S C 1 . Note that φ 2 µ 2 factors throu gh ¯ γ 2 . No w complete  γ 2 ¯ γ 2  to an exact triangle C 2 h γ 2 ¯ γ 2 i − − − → C 3 ∐ ¯ C 3 [ δ ¯ δ ] − − − → C ′ 3 → S C 2 and observe that µ 3 factors th r o u g h δ via a morp h ism µ ′ 3 : C ′ 3 → X 3 . Thus we obtain the follo w ing factoriza tion of µ with ( δ γ 2 ) γ 1 = − ¯ δ ¯ γ 2 γ 1 = 0. C 1 γ 1 / / id   C 2 γ 2 / / id   C 3 δ   C 1 γ 1 / / µ 1   C 2 δγ 2 / / µ 2   C ′ 3 µ ′ 3   X 1 φ 1 / / X 2 φ 2 / / X 3 F or the second step we ma y assu m e that γ 2 γ 1 = 0. W e complete γ 2 to an exact triangle ¯ C 1 ¯ γ 1 − → C 2 γ 2 − → C 3 → S ¯ C 1 . Clearly , γ 1 factors through ¯ γ 1 via a morphism ρ : C 1 → ¯ C 1 and µ 2 ¯ γ 1 factors through φ 1 via a morphism σ : ¯ C 1 → X 1 . Thus w e obtain the follo wing factoriza tion of µ C 1 γ 1 / / [ ρ id ]   C 2 γ 2 / / id   C 3 h id 0 i   ¯ C 1 ∐ C 1 [ ¯ γ 1 0 ] / / [ σ µ 1 − σρ ]   C 2 [ γ 2 0 ] / / µ 2   C 3 ∐ S C 1 [ µ 3 0 ]   X 1 φ 1 / / X 2 φ 2 / / X 3 where the m iddle ro w ﬁts into an exact triangle.  The follo wing statemen t is a reformula tion of the p revio us one in terms of coﬁnal sub catego ries. Prop osition 6.6.2. L et T b e a triangulate d c ate gory and S a ful l triangulate d sub c ate- gory. Supp ose tha t X 1 φ 1 − → X 2 φ 2 − → X 3 φ 3 − → S X 1 is an exact triangle in T and d enote by S / ( φ 1 , φ 2 ) the c ate gory whose obje cts ar e the c om mutative diagr ams in T of the fol lowing LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 35 form. C 1 γ 1 / /   C 2 γ 2 / /   C 3   X 1 φ 1 / / X 2 φ 2 / / X 3 such e ach C i b elongs to S . Then the ful l sub c ate gory forme d by the diagr ams such tha t ther e exists an e xa ct triangle C 1 γ 1 − → C 2 γ 2 − → C 3 γ 3 − → S C 1 is a c oﬁnal sub c ate gory of S / ( φ 1 , φ 2 ) . 6.7. A Kan extension. Let T b e a triangulated categ ory with s ma ll copro ducts and S a s m a ll full sub category . Supp ose that the ob ject s of S are α -small and that S is closed under α -copro ducts. W e denote b y Add α ( S op , Ab) the category of α -pro d uct preserving fun ctors S op → Ab. This is a lo cal ly presentable ab elia n categ ory in the sense of [17] and w e r efe r to th e App endix B for basic facts on lo call y presen table categorie s. Dep ending on the choi ce o f S , we can th i n k of Add α ( S op , Ab) as a lo cal ly presen table appro ximation of the tr iangulated cate gory T . In order to m ake this precise, we need to in tro duce v arious functors. Let h T : T → A ( T ) denote the ab elia nization of T ; see App endix A. Sometimes w e write b T instead of A ( T ). The inclusion functor f : S → T induces a functor f ∗ : A ( T ) − → Add α ( S op , Ab) , X 7→ A ( T )(( h T ◦ f ) − , X ) , and we observe that the co mp osite T h T − → A ( T ) f ∗ − → Add α ( S op , Ab) is the restricte d Y oneda fu ncto r send ing eac h X ∈ T to T ( − , X ) | S . The next prop osition discusses a left adjoint for f ∗ . T o t h is end, w e denote for an y catego ry C b y h C the Y oneda fu nct or send ing X in C to C ( − , X ). Prop osition 6.7.1. The functor f ∗ admits a left ad joint f ∗ which makes the fol low ing diagr am c ommutative. S h S / / f =inc   Add α ( S op , Ab) f ∗   T h T / / A ( T ) Mor e over, the functor f ∗ has the fol lowing pr op erties. (1) f ∗ is f u l ly faithful and identiﬁes Add α ( S op , Ab) with the ful l sub c ate gory forme d by al l c olimits of obje cts in {T ( − , X ) | X ∈ S } . (2) f ∗ pr eserves smal l c opr o ducts if and only if (PG2) hol ds for S . (3) Supp ose that S is a triangulate d sub c ate gory of T . Then for X in T , the adjunc- tion morph ism f ∗ f ∗ ( h T X ) → h T X identiﬁes with the c anonic al morphism colim − − − → ( C,µ ) ∈S /X h T C − → h T X. Pr o of. Th e fun ct or f ∗ is constru c ted as a left Kan extension. T o explain this, it is con- v enien t to ident ify Add α ( S op , Ab) with the cat egory Lex α ( b S op , Ab) of left exact functors 36 HENNING KRAUSE b S op → Ab wh i ch preserve α -pro ducts. T o be m o re precise, the Y oneda functor h : S → b S induces an equiv alence Lex α ( b S op , Ab) ∼ − → Add α ( S op , Ab) , F 7→ F ◦ h, b ecause ev ery add it ive fun ct or S op → Ab extends u niquely to a left exa ct fu nct or b S op → Ab; s e e L e mma A.1. Using this iden tiﬁcation, the existence of a f ully faithful left adjoin t Lex α ( b S op , Ab) → A ( T ) for f ∗ and its b a sic prop erties follo w from Lemma B.6, b eca use the in clusion f : S → T in d uces a fully faithful and righ t exact fun ctor b f : b S → b T = A ( T ). This functor preserves α -copro ducts and iden tiﬁes b S with a fu ll sub ca tegory of α -presenta ble ob jects, since the ob jects f r o m S are α -small in T . (2) Let Σ = Σ( f ∗ ) denote the set of morph isms of A ( T ) which f ∗ mak es inv ertible. I t follo ws from Prop osition 2.3.1 that f ∗ induces an equiv alence A ( T )[Σ − 1 ] ∼ − → Lex α ( b S op , Ab) , and therefore f ∗ preserv es small copro ducts if and only if Σ is closed un der taking small copro ducts, b y Prop osition 3.5.1. It is n ot hard to c hec k th at f ∗ is exact, and therefore a morphism in A ( T ) b elongs to Σ if and only if its k ernel and cok ernel are ann ih i lated b y f ∗ . No w observe that an ob jec t F in A ( T ) with presentat ion T ( − , X ) → T ( − , Y ) → F → 0 is ann i h i lated by f ∗ if and only if T ( C, X ) → T ( C, Y ) is surjectiv e for all C ∈ S . It follo ws hat f ∗ preserv es small copro ducts if and only if (PG2) holds for S . (3) Let F = f ∗ ( h T X ) = T ( − , X ) | S . Then Lemma B.7 imp li es that F = colim − − − → ( C,µ ) ∈S /X h S C , since S /F = S /X . Th us f ∗ F = colim − − − → ( C,µ ) ∈S /X h T C .  Corollary 6.7.2. L et T b e a triangulate d c ate gory with smal l c opr o ducts. Supp ose T is α -wel l gener ate d and denote by T α the ful l sub c ate gory form e d by al l α -c omp act obje cts. Then the functor T → A ( T ) taking an obje ct X to colim − − − → ( C,µ ) ∈T α /X T ( − , C ) pr eserves smal l c opr o ducts. 6.8. A criterion for well generatedness. Let T b e a triangulated categ ory whic h admits small copro ducts. The follo wing result pro vides a useful criterio n for T to b e we ll generated in terms of cohomologi cal fun ct ors in to lo c ally present able ab elian categorie s. Prop osition 6.8.1. L et T b e a triangulate d c ate gory with smal l c opr o ducts and α a r e gular c ar dinal. L et S 0 b e a smal l set of o bje cts and denote by S the ful l sub c ate gory forme d by al l α -c opr o ducts of obje cts in S 0 . Then the fol lowing ar e e quivalent. (1) The obje cts of S 0 ar e α -sma l l and (PG2) holds for S 0 . (2) The obje cts of S ar e α -smal l and (PG2) holds for S . (3) The functor H : T → Add α ( S op , Ab) taking X to T ( − , X ) | S pr eserves smal l c opr o ducts. Pr o of. It is clear that (1) and (2) are equiv alen t, and it follo ws from Prop osition 6.7.1 that (2) implies (3). T o prov e that (3) implies (2), assume that H preserve s small copro ducts. Let φ : X → ` i ∈ I Y i b e a morphism in T with X ∈ S . W rite ` i ∈ I Y i = LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 37 colim − − − → J ⊆ I Y J as α -ﬁltered coli mit of copro ducts Y J = ` i ∈ J Y i with card J < α . Then w e ha ve colim − − − → J ⊆ I T ( X , Y J ) ∼ = colim − − − → J ⊆ I Hom S ( S ( − , X ) , H Y J ) ∼ = Hom S ( S ( − , X ) , colim − − − → J ⊆ I H Y J ) ∼ = Hom S ( S ( − , X ) , a i ∈ I H Y i ) ∼ = Hom S ( S ( − , X ) , H ( a i ∈ I Y i )) ∼ = T ( X , a i ∈ I Y i ) . Th u s φ f actors through some Y J , and it follo ws that X is α - small. No w Prop ositio n 6.7.1 implies that (PG2) holds for S .  6.9. Cohomological functors via ﬁltered colimits. The foll owing theorem sho ws that cohomologi cal functors on well generated triangulated categorie s can b e computed via ﬁltered colimits. This ge n e ralizes a fact whic h is w ell kno wn for compact ly generated triangulated cate gories. W e say that an ab elian cat egory h a s exact α -ﬁlter e d c olimits pro vided that every α -ﬁltered colimit of exac t sequences is exact. Theorem 6.9.1. L et T b e a triangulate d c ate gory with smal l c opr o ducts. Supp ose T is α -wel l gener ate d and denote by T α the ful l sub c ate gory form e d by al l α -c omp act obje cts. L et A b e an ab elian c ate gory which has smal l c opr o ducts and exact α -ﬁlter e d c olimits. If H : T → A i s a c ohomolo gic al functor which pr eserves sma l l c opr o ducts, then we have for X in T a natur al isomorphism (6.9.1 ) colim − − − → ( C,µ ) ∈T α /X H C ∼ − → H X. Pr o of. Th e left h a n d term of (6.9.1) d e ﬁ nes a functor ˜ H : T → A and we need to sho w that the canonical morphism ˜ H → H is inv ertible. First observe that ˜ H is cohomological . This is a consequence of Prop osition 6.6. 2 and Lemma 6.4.1, b ecause for any exact triangle X 1 → X 2 → X 3 → S X 1 in T , the sequence ˜ H X 1 → ˜ H X 2 → ˜ H X 3 can b e written as α -ﬁltered col imit of exact sequences in A . Next we claim that ˜ H preserves small copro ducts. T o this end consider the exact functor ¯ H : A ( T ) → A whic h extends H ; see L e mma A.2. Note that ¯ H p reserv e sm all copro ducts b ecause H has this prop ert y . W e ha v e f or X in T ˜ H X = colim − − − → ( C,µ ) ∈T α /X ¯ H  T ( − , C )  ∼ = ¯ H  colim − − − → ( C,µ ) ∈T α /X T ( − , C )  . No w the assertion follo ws from Corollary 6.7.2. T o complete the pro of, consider the f ull su b category T ′ consisting of those ob jects X in T suc h that the morphism ˜ H X → H X is an isomorphism. Clearly , T ′ is a triangulated sub ca tegory since b oth fu n ct ors are c ohomological, it is closed und er ta king small copro ducts since they are preserved by b oth functors, and it con tains T α . Thus T ′ = T .  38 HENNING KRAUSE Remark 6.9.2. F or an alternativ e pro of of the fact that ˜ H is cohomol ogical, one uses Lemma B.5. 6.10. A univ ersal prop ert y. Let T b e a triangulated category wh ic h admits sm all copro ducts and is α -well generated. W e denote by A α ( T ) the fu ll sub category of A ( T ) whic h is formed b y all colimits of ob jects T ( − , X ) with X in T α . Obs erve t h a t A α ( T ) is a locally present able ab elian category with exact α -ﬁltered colimits. This f ollo ws from Prop ositio n 6.7.1 a n d the discussion in App endix B, b ecause A α ( T ) can b e iden tiﬁed with a category of left exact functors. W e ha v e tw o functors H α : T − → A α ( T ) , X 7→ colim − − − → ( C,µ ) ∈T α /X T ( − , C ) , h α : T − → Add α (( T α ) op , Ab) , X 7→ T ( − , X ) | T α , whic h are r e lated by an equiv alence as follo ws. Add α (( T α ) op , Ab) f ∗ ∼   T h α 2 2 e e e e e e e e e e e e H α , , Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y A α ( T ) The functor f ∗ is induced b y the incl u sio n f : T α → T and discussed in Prop osi tion 6.7.1. In p a rticular, there it is shown that f ∗ ( h α X ) = f ∗ f ∗ ( h T X ) = H α X for all X in T . Prop osition 6.10.1. The functor H α : T → A α ( T ) has the fol lowing u niversa l pr op erty. (1) The functor H α is a c oho molo gic al functor to an ab elian c ate gory with smal l c opr o ducts and exact α -ﬁlter e d c olimits and H α pr eserves smal l c opr o ducts. (2) Given a c ohomo lo gic al functor H : T → A to an ab elian c ate gory with smal l c opr o ducts and exact α -ﬁlter e d c olimits such that H pr eserves smal l c opr o ducts, ther e exists an essential ly unique exact functor ¯ H : A α ( T ) → A which pr eserves smal l c opr o ducts and satisﬁes H = ¯ H ◦ H α . Pr o of. (1) It is clear that h α is cohomolog ical and it follo ws from Prop osition 6.7.1 that h α preserv es small copro ducts. (2) Give n H : T → A , w e denote by ˜ H : A ( T ) → A the exact functor whic h extends H , and we deﬁne ¯ H : A α ( T ) → A by sending eac h X to ˜ H X . The follo wing comm utativ e diagram illustrates this construction. T α h T α / / f =inc   A ( T α ) h A ( T α ) / / A ( f )   A α ( T ) f ∗ =inc   T h T / / A ( T ) ˜ H   A ( T ) T H / / A Let us c heck the prop erties of ¯ H . T he fu ncto r ¯ H preserv es small copro ducts since ˜ H has this prop ert y . The f u ncto r ¯ H is exact when restricted to A ( T α ). Thus it follo ws from Lemma B.5 that ¯ H is exact. Th e equalit y H = ¯ H ◦ H α is a consequence of Theorem 6.9.1 LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 39 since b oth functors coincide on T α . Supp ose now there is a s e cond functor A α ( T ) → A ha ving the pr o p erties of ¯ H . T hen b oth f u ncto rs agree on P = {T ( − , X ) | X ∈ T α } and therefore on all of A α ( T ) since ea ch o b ject in A α ( T ) is a colimit of ob jects in P and b oth functors preserve colimits.  Remark 6.10.2. The universal prop ert y can b e used to sh ow that the category T α of α -compact ob jects does not dep en d on the c hoice of a p erfectly generati n g set for T . More precisely , if T is α -w ell generated, then t wo α -lo caliz ing sub categories coincide if eac h conta ins a sm a ll set of α -small p erfect generat ors. T his f o llo ws f r o m the fact that the f u ncto r H α iden tiﬁes the α -compact ob jects with the α -pr e sentable pro j e ctiv e ob jects of A α ( T ). 6.11. Notes. W ell generate d triangulated were in tro duced and studied b y Neeman in his b ook [33] as a natural generalizat ion of compactly generated triangulated categories. F or an alternativ e approac h whic h simpliﬁes the deﬁn i tion, se e [23]. More recent ly , w ell generated categories w ith speciﬁc mo dels ha v e been s tu died; see [37, 47] for work in vo lving algebraic mo dels via diﬀerent ial graded catego ries, and [20] for top olog ical mo dels. In [43], Rosic k´ y used com binatorial mo dels and sho we d th a t there exist univer- sal cohomologic al functors in to lo cally presenta b l e categories wh i ch are full. Int eresting consequences of this fact are discussed in [35]. The description of the unive rsal cohomo- logica l functors in terms of ﬁltered colimits seems to b e new. 7. Lo caliza tion f or well genera ted ca tegories 7.1. Cohomological lo calization. Th e follo wing theorem sho ws that cohomologic al functors on well generated triangulated categories indu ce lo cal ization functors. This generalize s a fact whic h is w ell kno wn for compactly generated triangulated categories. Theorem 7.1.1. L et T b e a tria ngulate d c ate gory with smal l c opr o ducts which is wel l gener ate d. L et H : T → A b e a c ohomolo gic al functor into an ab elian c ate gory which has smal l c opr o ducts and exact α -ﬁlter e d c olimits for some r e gular c ar dinal α . Supp ose also that H pr eserves smal l c opr o ducts. Then ther e exists an exact lo c alization fu ncto r L : T → T such that for e ach obje ct X we have LX = 0 i f and only if H ( S n X ) = 0 for al l n ∈ Z . Pr o of. W e may assume t h a t T is α -well generate d. Let Σ = Σ( H ) denote the set of morphisms σ in T such H σ is inv ertible. Next w e assume that S Σ = Σ. Oth e rw ise , w e replace A by a coun table pr oduct A Z of copies of A and H b y ( H S n ) n ∈ Z . Then Σ admits a calculus of right fract ions by Lemma 4.4.2 , and w e apply the criterio n of Lemma 3 .3.1 to sho w that the morph i sm s b et w een an y t w o ob jects in T [Σ − 1 ] form a small set. Th e existence of a localization fun c tor L : T → T w it h Ker L = Ker H then follo ws from Prop osition 5.2.1. Th u s w e need to sp ec ify f o r eac h ob ject Y of T a small set of ob jects S ( Y , Σ) such that for eve ry morph ism X → Y in Σ , there exist a m o rp hism X ′ → X in Σ with X ′ in S ( Y , Σ). Su pp o se that Y b elongs to T κ . W e deﬁne by indu c tion κ − 1 = κ + α and κ n = sup { card T α /U | U ∈ T κ n − 1 } + + κ n − 1 for n ≥ 0 . Then we p ut S ( Y , Σ) = T ¯ κ with ¯ κ = ( P n ≥ 0 κ n ) + . No w ﬁx σ : X → Y in Σ. T he morphism X ′ → X in Σ with X ′ in S ( Y , Σ) is constructed as follo ws. Th e canonical morph ism π : ` ( C,µ ) ∈T α /X C → X induces an 40 HENNING KRAUSE epimorphism H π b y Theorem 6.9.1. W e can choose C ⊆ T α /X with card C ≤ card T α / Y suc h that π 0 : X 0 = ` ( C,µ ) ∈C C → X indu ce s an epimorphism H π 0 since H σ is inv ertible. More precisely , w e call t w o ob jects ( C , µ ) and ( C ′ , µ ′ ) of T α /X equiv alen t if σ µ = σ µ ′ , and we choose as ob jects of C precisely one representat ive f or eac h equiv alence class. Supp ose w e ha v e already constructed π i : X i → X with X i in T κ i for some i ≥ 0. Then we f orm the follo wing co mmutativ e diagram w it h exact ro ws. U i ι i / / σ i   X i π i / / X σ   / / S U i S σ i   V i / / X i / / Y / / S V i Note that H σ i is inv ertible. Thus we can c ho ose C i ⊆ T α /U i with card C i ≤ card T α /V i ≤ card T α /X i + ca r d T α / Y < κ i +1 suc h that ξ i : ` ( C,µ ) ∈C i C → U i induces an epimorphism H ξ i . No w complete ι i ◦ ξ i to an exact triangle and deﬁn e π i +1 : X i +1 → X b y the comm utativit y of the follo wing diagram. ` ( C,µ ) ∈C i C ι i ◦ ξ i / / X i φ i / / π i   6 6 6 6 6 6 6 X i +1 π i +1          / / S  ` ( C,µ ) ∈C i C  X Observe that X i +1 b elongs to T κ i +1 and that Ker H π i = Ker H φ i . T h e φ i induce an exact triangle (7.1.1 ) a i ∈ N X i (id − φ i ) − − − − − → a i ∈ N X i ψ − → X ′ − → S  a i ∈ N X i  suc h that X ′ b elongs to S ( Y , Σ) and the morph ism ( π i ) : ` i ∈ N X i → X factors through ψ via a morphism τ : X ′ → X . W e claim that H τ is in vertible . In fact, the lemma b elo w implies that th e π i induce the follo wing exact s equ en ce 0 − → a i ∈ N H X i (id − H φ i ) − − − − − − → a i ∈ N H X i ( H π i ) − − − → H X − → 0 . On the other hand, the exact triangle (7.1.1) induces the exact sequence H  a i ∈ N X i  H (id − φ i ) − − − − − − → H  a i ∈ N X i  H ψ − → H X ′ − → H S  a i ∈ N X i  , and a comparison sho ws that H τ is inv ertible. Here, we use again that H preserv es small copro ducts, and this completes the pr oof.  Lemma 7.1.2. L et A b e an ab elian c ate gory which admits c ountable c opr o ducts. Then a se quenc e of e p imorphisms ( π i ) i ∈ N X i φ i / / π i   7 7 7 7 7 7 7 X i +1 π i +1          Y LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 41 satisfying π i = π i +1 ◦ φ i and Ker π i = Ker φ i for al l i induc es an exact se quenc e 0 − → a i ∈ N X i (id − φ i ) − − − − − → a i ∈ N X i ( π i ) − − → Y − → 0 . Pr o of. Th e assumption U i := Ker π i = K e r φ i implies that there exists a morp hism π ′ i : Y → X i with π i π ′ i = id Y and φ i π ′ i = π ′ i +1 for all i ≥ 1. Thus we ha ve a sequence of comm uting squares U i ∐ Y h 0 0 0 id i   [ inc π ′ i ] / / X i φ i   U i +1 ∐ Y [ inc π ′ i +1 ] / / X i +1 where the horizon tal maps are isomorph isms. T aking colimits on b oth sid es, the assertion follo ws.  7.2. Lo calization with resp ect to a small set of ob jects. Let T b e a wel l generat ed triangulated category and S a lo c alizing sub categ ory whic h is generated b y a small set of ob jects. Th e follo wing result s ays that S and T / S are b o th well generated and th at the ﬁ l tration T = S α T α via α -compact ob jects indu ce s canonical ﬁ l trations S = [ α ( S ∩ T α ) and T / S = [ α T α / ( S ∩ T α ) . Theorem 7.2.1 . L et T b e a wel l gener ate d triangulate d c ate gory and S a lo c alizing sub c ate gory wh ich is gener ate d by a smal l set of obje cts. Fix a r e gular c ar dinal α such that T is α - wel l gener ate d and S is gener ate d by α -c omp act obje cts. (1) An obje ct X in T b elongs to S if and only if every morphism C → X fr om an obje ct C in T α factors thr ough some obje ct in S ∩ T α . (2) The lo c alizing sub c ate gory S and the qu otient c ate gory T / S ar e α - wel l gener ate d. (3) We have S α = S ∩ T α and a c ommutative diagr am of exact functors S α inc   inc / / T α inc   can / / T α / S α J   S inc / / T can / / T / S such that J is ful ly faithful. Mor e over, J i nd uc es a functor T α / S α → ( T / S ) α such th at every obje ct of ( T / S ) α is a dir e ct factor of an obje ct in th e image of J . This functor is an e quivalenc e if α > ℵ 0 . Pr o of. Let C = S ∩ T α . Then the in c lusion i : C → T α induces a fully f a ithful and exact functor i ∗ : Add α ( C op , Ab) → Add α (( T α ) op , Ab) whic h is left adjoint to the functor i ∗ taking F to F ◦ i ; see Lemma B.8. Note that the image Im i ∗ of i ∗ is closed und er small copro ducts. W e consider the r e stricted Y oneda fu nct or h α : T → Add α (( T α ) op , Ab ) taking X to T ( − , X ) | T α and obs erve th a t h − 1 α (Im i ∗ ) is a lo caliz ing su bcategory o f T 42 HENNING KRAUSE con taining C . Th u s we obtain a functor H making the follo win g diagram commuta tive . S inc / / H   T h α   Add α ( C op , Ab) i ∗ / / Add α (( T α ) op , Ab) Let us compare H with the restricted Y oneda fu ncto r H ′ : S − → Add α ( C op , Ab) , X 7→ S ( − , X ) | C . In fact, w e ha ve an isomorp h ism H ∼ − → i ∗ ◦ i ∗ ◦ H = i ∗ ◦ h α | S = H ′ and H ′ preserv es small copro ducts s in c e h α do es. It follo ws from Prop osition 6.8.1 that C provi d es a small set of α -small p erfect ge n erators for S . T h us S is α -we ll ge nerated and S α = Lo c α C = S ∩ T α . Next w e apply Prop ositi on 5.2.1 and obta in a lo caliz ation functor L : T → T with Ker L = S . W e use L to sho w that S = h − 1 α (Im i ∗ ). W e kno w already that S ⊆ h − 1 α (Im i ∗ ). Now let X b e an ob ject in h − 1 α (Im i ∗ ) and consider the exact t r iangle Γ X → X → LX → S ( Γ X ). Th e n T ( C, L X ) = 0 for all C ∈ C and therefore i ∗ h α LX = 0. On the other hand, h α LX = i ∗ F for some functor F and therefore 0 = i ∗ h α LX = i ∗ i ∗ F ∼ = F . Thus LX = 0 and therefore X b elongs to S . This shows S = h − 1 α (Im i ∗ ). No w w e pro v e (1) and use the description of the essent ial image o f i ∗ from Lemma B.8. W e ha v e for an o b ject X in T that X b elongs to S iﬀ h α X b elongs to Im i ∗ iﬀ ev ery morphism T α ( − , C ) → h α X with C ∈ T α factors through T α ( − , C ′ ) for some C ′ ∈ C iﬀ every morph ism C → X with C ∈ T α factors through some C ′ ∈ C . An imm ed iate consequence of (1) is the fact that J is fully f aithfu l . This follo ws f r o m Lemma 4.7.1. No w consider the quotien t functor q : T α → T α / S α . This ind u ce s an exact functor q ∗ : Add α (( T α ) op , Ab) → Add α (( T α / S α ) op , Ab) which is le ft adjoin t to the fully faithful functor q ∗ taking F to F ◦ q ; see Lemma B.8. Clearly , q ∗ ◦ h α annihilates S and induces therefore a f u ncto r K making the follo wing diagram commutat ive. T Q =can / / h α   T / S K   Add α (( T α ) op , Ab) q ∗ / / Add α (( T α / S α ) op , Ab) Note that Q admits a righ t adjoin t whic h we denote b y Q ρ . W e ident ify T α / S α via J with a fu ll triangulated sub category of T / S and consider the r e stricted Y oneda functor K ′ : T / S − → Add α (( T α / S α ) op , Ab) , X 7→ T / S ( − , X ) | T α / S α . Adjoin tness giv es the follo wing isomorphism T / S ( J q C, X ) = T / S ( QC, X ) ∼ = T ( C, Q ρ X ) for all C ∈ T α and X ∈ T / S . Th us we hav e an isomorphism K ∼ ← − K ◦ Q ◦ Q ρ = q ∗ ◦ h α ◦ Q ρ ∼ = q ∗ ◦ q ∗ ◦ K ′ ∼ − → K ′ LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 43 and K ′ preserv es small copro ducts since h α do es. It follo ws from Prop osition 6.8.1 that T α / S α pro vides a small set of α -small p erfect generators for T / S . Thus T / S is α -w ell generated and ( T / S ) α = Lo c α ( T α / S α ).  Corollary 7.2.2. L et T b e an α -wel l gener ate d triangulate d c ate gory and S a lo c alizing sub c ate gory ge ner ate d by a smal l set S 0 of α -c omp act obje cts. Then S is α -wel l g e ner ate d and S α e quals the α -lo c alizing sub c ate gory gener ate d by S 0 . Pr o of. In the preceding p roof of Theorem 7.2.1, w e ca n c ho ose for C instead of S ∩ T α the α -lo caliz ing sub categ ory of T which is generated by S 0 . Then the pr o of shows that C p ro vides a small set of α -small p erfect generators for S . Thus w e ha v e S α = C by deﬁnition.  The lo ca lization with r esp ect to a localizing su bcategory generated by a small set of ob jects can b e interpreted in v arious w ays. The follo wing remark p r o vides some indication. Remark 7.2.3. (1) Let T b e a w ell generated triangulated catego ry and φ a morphism in T . Then t h ere exists a univ ersal exact lo cali zation fu ncto r L : T → T in ve rting φ . T o see this, complete φ to an exact triangle X φ − → Y → Z → S X and let L b e the lo c alization fu ncto r such that Ker L equals the localizing s ub category generate d b y Z . Con ve r s ely , an y exact lo caliz ation functor L : T → T is the u niv ersal exact localizatio n functor inv erting some morphism φ pro vided that Ker L is generated b y a small s et S 0 of ob jects. T o see this, tak e φ : 0 → ` X ∈S 0 X . (2) Let T b e a triangulated category and L : T → T an exact localization functor such that S = Ker L is generated b y a single ob ject W . Then the ﬁrst morphism Γ X → X from the fun c torial triangle Γ X → X → LX → S ( Γ X ) is called c el lularization and the second morp hism X → LX is cal led nul liﬁc ation with resp ect to W . T he ob jects in S are built fr om W . 7.3. F unctors b et ween w ell generated categories. W e consider fu nct ors betw een w ell ge n e rated triangulated categories whic h are exact and preserv e small coprodu c ts. The follo w ing r e su lt sho ws that suc h fun ct ors are con trolle d b y their restrict ion to the sub catego ry of α -compact ob j ects for some r eg ular cardinal α . Prop osition 7.3.1. L et F : T → U b e an exact functor b etwe en α -wel l gener ate d tri- angulate d c ate gories. Supp ose that F pr eserves smal l c opr o ducts and let G b e a right adjoint. (1) Ther e exists a r e gular c ar dinal β 0 ≥ α such that F pr eserves β 0 -c omp actness. In that c ase F pr eserves β -c omp actness fo r al l r e gular β ≥ β 0 . (2) Given a r e gular c ar dinal β ≥ β 0 , the r estriction f : T β → U β of F induc es the fol lowing diagr am of functors which c ommute up to natur al isomorp hisms. T β f = F β / / inc   U β inc   T F / / h β ( T )   U h β ( U )   G / / T h β ( T )   Add β (( T β ) op , Ab) f ∗ / / Add β (( U β ) op , Ab) f ∗ / / Add β (( T β ) op , Ab) 44 HENNING KRAUSE Pr o of. (1) Cho ose β 0 ≥ α suc h th a t F ( T α ) ⊆ U β 0 . Then we get for β ≥ β 0 F ( T β ) = F (Lo c β T α ) ⊆ Lo c β F ( T α ) ⊆ Lo c β U β 0 = U β . (2) W e apply Theorem 6.9.1 to show that h β ( U ) ◦ F ∼ = f ∗ ◦ h β ( T ). In f a ct, it fol- lo ws f rom Prop o sition 6.10.1 and Lemma B.8 that b oth comp osites are cohomologica l functors, preserv e small copro ducts, and agree on T β . The isomorphism h β ( T ) ◦ G ∼ = f ∗ ◦ h β ( U ) follo ws from the adjoin tness of F and G , since T ( C, GX ) ∼ = U ( f C, X ) for ev ery C ∈ T β and X ∈ U .  7.4. The kernel of a functor b et w een well generated categories. W e show that the class of w ell generated triangulate d categories is closed under taking k ernels of exact functors wh ic h preserv e small copro ducts. Theorem 7.4.1. L et F : T → U b e an exact functor b etwe en α -wel l gener ate d trian- gulate d c ate gories and supp ose that F pr eserves smal l c opr o ducts. L et S = Ker F and cho ose a r e gu lar c ar dinal β ≥ α such that F pr eserves β -c omp actness. (1) An obje ct X in T b e longs to S if and only if every morphism C → X with C ∈ T β factors thr ough a morphism γ : C → C ′ in T β satisfying F γ = 0 . (2) Supp ose β > ℵ 0 . Then S is β -wel l gener ate d and S β = S ∩ T β . Pr o of. Let f : T β → U β b e the restriction of F a n d den ote b y I the set of morphisms in T β whic h are ann ihila ted by F . (1) Let X b e an ob ject in T . Th e n it follo ws from Prop osi tion 7. 3.1 that F X = 0 if and only f ∗ h β X = 0. No w Lemma B.8 implies that f ∗ h β X = 0 iﬀ eac h morph i sm C → X with C ∈ T β factors through some morph ism C → C ′ in I . (2) Let S ′ denote th e lo calizi n g sub category of T which is generated b y all homotop y colimits of sequences C 0 − → C 1 − → C 2 − → · · · of morphisms in I . W e claim th a t S ′ = Ker F . Clearly , w e ha v e S ′ ⊆ Ker F . No w ﬁx an ob ject X ∈ Ker F . W e hav e s e en in (1) that eac h morphism µ : C → X with C ∈ T β factors through some m o rp hism C → C ′ in I . W e obtain b y indu ct ion a sequence C = C 0 γ 0 − → C 1 γ 1 − → C 2 γ 2 − → · · · of morph isms in I suc h th a t µ factors through eac h ﬁnite comp osite γ i . . . γ 0 . Thus µ factors through the homoto py colimit of this sequence and therefore thr o u g h an ob ject of S ′ ∩ T β . Here one uses that β > ℵ 0 . W e conclude from Theorem 7.2.1 that X b elongs to S ′ . Moreo v er, w e conclude fr om this theorem th a t S ′ is β -we ll generated.  Remark 7.4.2. It is n e cessary to assume in part (2) of the preceding theorem that β > ℵ 0 . F or example, there exists a ring A with Jacobson radical r su c h that the functor F = − ⊗ L A A/ r : D ( A ) → D ( A/ r ) satisﬁes S = Ker F 6 = 0 but S ∩ D ( A ) c = 0; see [22]. Observe that Theorem 7.4.1 p ro vides a partial answe r to th e telescope conjecture for compactly generated catego r ies. This conjecture claims that the ke rn el of a lo caliza tion functor L : T → T is generated by compact ob jects pro vided that L preserves sm all co- pro ducts. P art (1) implies th a t S = Ker L is generated by morph ism s b et we en compact ob jects, and part (2) sa ys that S is generated by ℵ 1 -compact ob ject s. I am grateful to Amnon Neeman for explaining to me ho w to deduce (2) from (1). The follo wing co r ollary mak es the connection with the tel escop e conjecture more pr ecise; ju st put α = ℵ 0 . LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 45 Corollary 7.4.3. L et L : T → T b e an exact lo c alization functor which pr eserves smal l c opr o ducts. Supp ose that T is α -wel l gener ate d and let β ≥ max( α, ℵ 1 ) . Then S = Ker L is β -wel l gener ate d and S β = S ∩ T β . Pr o of. Let L : T → T b e a lo caliza tion functor wh ic h preserv es small copro ducts. W r it e L = G ◦ F as the comp o site o f a qu o tient functor F : T → U and a fully faithful right adjoin t G , where U = T / S an d S = Ker L . Then G p reserv es small copro ducts by Prop ositio n 5.5.1. The isomorphism (5.4.1) from the pro of of Lemma 5.4.1 s ho ws that F preserv es α -smallness and sends a set of p erfec t generat ors of T to a set o f p erfect generators of U . In particular, F preserv es β -co mpactness for all regular β ≥ α . No w apply Theorem 7.4.1.  7.5. The kernel of a cohomological functor on a well generated category. The follo wing result s ays that kernels of cohomological f uncto rs from w ell generated triangulated categories into lo cally presenta ble ab elian categ ories are well generated. The argument is basically the same as that for k ernels of exact fu n ct ors b et ween well generated triangulated categories. Theorem 7.5.1. L et H : T → A b e a c ohomolo gic al functor fr om a wel l g ener ate d trian- gulate d c ate gory into a lo c al ly pr esentable ab elian c ate gory and supp ose that H pr eserves smal l c opr o ducts. L et S denote the lo c alizing sub c ate gory of T c onsisting of al l obje cts X such that H ( S n X ) = 0 for al l n ∈ Z . Then S is a wel l gener ate d triangulate d c ate g ory. Pr o of. Replacing H by ( H S n ) n ∈ Z , we ma y assume that S = Ker H . C hoose a regular cardinal α suc h that T is α -w ell generated and A is lo cally α -presenta ble. Then w e ha ve H ( T α ) ⊆ A β for some regular ca r d inal β and w e assume β ≥ max( α, ℵ 1 ). The description of H in Theorem 6.9.1 sho ws that H restricts to a fun ct or h : T β → A β , and w e denote b y ¯ h : A ( T β ) → A β the ind uced exact functor. Then w e obtain the follo wing functor h ∗ : Add β (( T β ) op , Ab) ∼ − → Lex β ( A ( T β ) op , Ab) ¯ h ∗ − → Lex β (( A β ) op , Ab) ∼ − → A where the ﬁ rst equiv alence follo ws from Lemma B.1 and the second equiv alence follo ws from Lemma B.6. The fun ctor ¯ h ∗ is a left Kan extension; it tak es a ﬁltered colimit F = colim − − − → ( C,µ ) ∈ A ( T β ) /F A ( T β )( − , C ) to colim − − − → ( C,µ ) ∈ A ( T β ) /F A β ( − , ¯ hC ) . Note that h ∗ is exact and preserv es small copro ducts. T his follo ws from Lemma B.5 and the fact that ¯ h ∗ is left adjoint to the restriction fun c tor ¯ h ∗ . The comp osite h ∗ ◦ h β : T → A coincides with H on T β and th erefore h ∗ ◦ h β ∼ = H b y Theorem 6.9.1. In particular, w e ha ve for eac h X in T that H X = 0 if and only if h ∗ ( h β X ) = 0. No w w e use th e same a r gu m e nt a s in t h e pro of of Theorem 7.4.1 and sho w that Ker H is generated by all homotop y colimits of co u n table sequences of morp hisms in T β whic h are ann ihila ted by H .  7.6. Lo calization of w ell generated categories v ersus ab elian lo calization. W e demonstrate the inte rp la y b et wee n triangulated and ab elian lo ca lization. T o this end re- call from Prop osition 6.10.1 that we ha v e for eac h w ell generated category T a univ ersal cohomolog ical functor H α : T → A α ( T ) into a lo cally α -presen table ab elian ca tegory . W e sho w that eac h exac t lo caliza tion functor for T can b e extended to an exact lo cal- izatio n functor for A α ( T ) for some regular cardin al α . 46 HENNING KRAUSE Theorem 7.6.1. L et T b e a wel l gener ate d triangulate d c ate gory and L : T → T an exact lo c alization functor. Supp ose that Ker L is wel l g ener ate d. Then ther e exists a r e gular c ar dinal α and an exa ct lo c alization functor L ′ : A α ( T ) → A α ( T ) such that the fol lowing squar e c ommutes u p to a natur al isomorph ism. T L / / H α   T H α   A α ( T ) L ′ / / A α ( T ) Mor e pr e cisely, the adjunction morphisms Id T → L and Id A α ( T ) → L ′ induc e for e ach X in T the fol lowing isomo rphisms. H α LX ∼ − → L ′ ( H α LX ) = L ′ H α ( LX ) ∼ ← − L ′ H α ( X ) An obje ct X in T is L -acyclic if and onl y if H α X is L ′ -acyclic, and X is L - lo c al if and only if H α X is L ′ -lo c al. Pr o of. Ch oose a regular cardinal α > ℵ 0 suc h that T is α -well generated and S = Ker L is generated b y α -compact ob jects. Let U = T / S and write L = G ◦ F as the comp osit e of the quotien t f uncto r F : T → U with its right adjoint G : U → T . No w iden tify A α ( T ) = Add α (( T α ) op , Ab) and A α ( U ) = Add α (( U α ) op , Ab). The induced fu ncto r f : T α → U α equals, up to an equiv alence, the quotie nt fu ncto r T α → T α / S α , b y Theorem 7.2 .1 . F rom f w e obtain a pair of adjoin t fu nct ors f ∗ and f ∗ b y Lemma B.8. Both functors are exact and th e rig ht adjoin t f ∗ is fully faithful. Thus w e obtain an exact lo ca lization functor L ′ = f ∗ ◦ f ∗ for A α ( T ) b y Corollary 2.4.2. The comm utativit y H α ◦ L ∼ = L ′ ◦ H α and the assertions ab out acyclic and lo cal ob jects then follo w from Prop osition 7.3.1.  7.7. Example: The deriv ed category of an ab elian Grothendiec k c at ego ry. Let A b e an ab el ian Grothendiec k category . Then the derived categ ory D ( A ) of unb ounded c hain complexes is a well generated triangulated category . Let us sket ch an argumen t. The P op escu-Gabriel theorem says that for eac h generator G of A , the functor T = A ( G, − ) : A → Mo d A (where A = A ( G, G ) denotes the endomorph ism r i n g of G ) is fully faithfu l and admits an exact left adjoin t, say Q ; see [45, Theorem X.4.1]. Consider the cohomologic al functor H : D ( A ) → A taking a complex X to Q ( ` n ∈ Z H n X ). T h en an ap p li cation of Theorem 7.5.1 sh o ws that S = Ker H is well generated, and therefore D ( A ) / S is well generated by Th e orem 7.2.1. Next observe that K ( Q ) induces an equiv alence K (Mod A ) / (Ker K ( Q )) ∼ − → K ( A ) since K ( Q ) has K ( T ) as a f u ll y faithfu l r ight adjoint . Moreo v er, the cohomology of eac h ob ject in the kernel of K ( Q ) lies in the kernel of Q . T h us w e obtain the follo wing LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 47 comm utativ e diagram. Ker K ( Q ) inc / /   K (Mod A ) can   K ( Q ) / / K ( A ) F   S   inc / / D ( A ) H ∗   can / / D ( A ) / S ¯ H   Ker Q inc / / Mo d A Q / / A It is easily c hec ke d that the k ernel of F consists of all acyclic complexes. Thus F induces an equ iv alence D ( A ) ∼ − → D ( A ) / S . 7.8. Notes. G iven a triangulat ed c ategory T , there are tw o basic questions when one studies exact lo ca lization functors T → T . On e can ask for the existence of a lo caliza tion functor w i th some prescrib e d k ernel, and one can ask for a c lassiﬁcation, or at least some structural results, for the set of all lo c alization fun ct ors on T . W ell generated categories pro vide a su itable setting for some p a r tial answers. The fact that cohomologica l functors induce l o calizat ion fun ctors is w ell kn o wn f or compactly generated triangulated categorie s [28], but the result seems to b e new for w ell generated categories. The lo ca lization theorem whic h describ es the lo caliza tion with resp ect to a small set of ob jects is du e to Neeman [33]. T he example of the deriv ed catego ry of an ab elian Grothend ie ck category is discussed in [3, 34]. The description of the k ernel o f an exact functor b et wee n w ell generated categories seems to be new. A motiv ation f o r this is the telescop e conjecture which is due to Bousﬁeld and Ra v enel [8, 40 ] and orig inally form ulated for the stable h o motopy category of CW-sp ectra. It is in teresting to note that the e xistence of l o calizat ion functors dep ends to some exten t on axioms fr o m set theory; see for instance [11, 10]. 8. Epilogue: Be yond we ll gene ra tedness W ell generated triangulated categories we re introduced by Neeman as a class of trian- gulated categorie s which includes all compactly g enerated cate gories and b eha ve s w ell with resp ect to lo caliza tion. W e h a v e discussed in Sections 6 and 7 most of the ba- sic prop ertie s of well ge n erated categories but th e picture is still not complete b ecause some imp ortant questions remain op en. F or instance, giv en a w ell generated triangu- lated categ ory T , w e do not kno w wh en a lo calizing sub categ ory arises as the k ernel of a lo c alization functor and wh en it is generated by a small set of ob jects. Also, one might ask wh en the set of all lo cal izing su bcategories is small. Another asp ect is Bro wn repre- sen tabilit y . W e do kno w that ev ery c ohomological fun ctor T op → Ab preserving s ma ll pro ducts is represent able, but what ab out co v arian t functors T → Ab? It seems that one obtains m ore insight by studying the un iv ersal cohomologic al fu ncto rs T → A α ( T ); in particular we need t o kn ow when they are full; see [43, 35] for some recen t wo r k in this d irec tion. Instead of answering these op en qu e stions, let us b e adv enturous and mo ve a little bit b ey ond the class of we ll generated categories. In fact, there are n at ur a l examples of triangulated categorie s whic h are not we ll generated. Suc h examples arise fr o m additiv e catego ries b y taking their h o motopy catego ry of c hain complexes. More precisely , let A 48 HENNING KRAUSE b e an additiv e category and supp ose that A admits small co pr oducts. W e denote b y K ( A ) the category of c hain complexes in A wh o se morph isms are the h omotopy classes of c hain maps. T ake for instance the category A = Ab of ab elian groups. T h en one can sho w that K (Ab) is not w ell generat ed; see [33]. In fac t, more is true. The catego ry K (Ab) admits no small set of generators, that is, any lo calizing sub category generated b y a small set of ob jects is a prop e r sub categ ory . Ho we ver, it is not diﬃcult to sho w that any localizing sub category generated by a s ma ll set of ob jects is w ell generated. So w e may think of K (Ab) as lo c al ly wel l gener ate d . In fact, d isc us sio ns w it h Jan ˇ S ˇ to v ´ ı ˇ cek suggest that K ( A ) is lo cally w ell generated whenev er A is lo c ally ﬁnitely pr esented; s ee [46]. Recall that A is lo c al ly ﬁnitely pr esente d if A has ﬁltered colimits and th ere exists a small set of ﬁnitely presented ob jects A 0 suc h th at ev ery ob ject can b e written as the ﬁltered colimit of ob j ects in A 0 . On the other hand, K ( A ) is only generated by a s m a ll set o f ob jects if A = Add A 0 for some small set of ob jects A 0 . Here, Add A 0 denotes the smallest sub category of A which con tains A 0 and is closed un der t aking small copro ducts and direct summ and s. W e refer to [46] for f u rther details. Appendix A. The abel i aniza tion of a triangula te d ca t egor y Let C b e an additiv e cate gory . W e consider fun c tors F : C op → Ab into the cat egory of ab e lian groups and ca ll a sequence F ′ → F → F ′′ of functors e xa ct if the induced sequence F ′ X → F X → F ′′ X of ab elian group s is exact for all X in C . A functor F is said to b e c oher ent if there exist s an exact sequence (called pr esentation ) C ( − , X ) − → C ( − , Y ) − → F − → 0 . The morphisms b et ween tw o coheren t functors form a sm all set by Y oneda’s lemma, and the coheren t fu ncto rs C op → Ab form an additiv e ca tegory with cok ernels. W e denote this category by b C . A basic to ol is the fully faithful Y one da functor h C : C → b C which sends an ob ject X to C ( − , X ). On e m ight think of this functor as the completion of C with resp ect to the formation of ﬁnite colimit s. T o formula te some further prop erties, w e recall that a morph ism X → Y is a we ak kernel for a morphism Y → Z if the indu c ed sequence C ( − , X ) → C ( − , Y ) → C ( − , Z ) is exact. Lemma A.1. L et C b e an additive c ate gory. (1) Given an ad ditive functor H : C → A to an additive c ate gory which ad mits c okernels, ther e i s (up to a unique isomorphism ) a unique right exact functor ¯ H : b C → A such tha t H = ¯ H ◦ h C . (2) If C has we ak kernels, then b C is an ab elian c ate gory. (3) If C has smal l c opr o ducts, then b C has sm al l c opr o ducts and the Y one da functor pr eserves smal l c opr o ducts. Pr o of. (1) Extend H to ¯ H by sending F in b C with presenta tion C ( − , X ) ( − ,φ ) − → C ( − , Y ) − → F − → 0 to the coke rn e l of H φ . LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 49 (2) Th e category b C h a s cok ernels, and it is th erefore suﬃcien t to sh o w that b C h a s k ernels. T o this end ﬁx a morphism F 1 → F 2 with the f ollo wing presen tation. C ( − , X 1 ) / /   C ( − , Y 1 ) / /   F 1 / /   0 C ( − , X 2 ) / / C ( − , Y 2 ) / / F 2 / / 0 W e construct the kernel F 0 → F 1 b y sp ecifying the follo wing p resen tatio n. C ( − , X 0 ) / /   C ( − , Y 0 ) / /   F 0 / /   0 C ( − , X 1 ) / / C ( − , Y 1 ) / / F 1 / / 0 First the morphism Y 0 → Y 1 is obtained from the wea k k ernel sequence Y 0 − → X 2 ∐ Y 1 − → Y 2 . Then the morphisms X 0 → X 1 and X 0 → Y 0 are obtained f rom the we ak k ernel s e quen ce X 0 − → X 1 ∐ Y 0 − → Y 1 . (3) F or ev ery family of fun ctors F i ha ving a p resen tatio n C ( − , X i ) ( − ,φ i ) − → C ( − , Y i ) − → F i − → 0 , the copro duct F = ` i F i has a present ation C ( − , a i X i ) ( − , ∐ φ i ) − → C ( − , a i Y i ) − → F − → 0 . Th u s copro ducts in b C are n o t computed p oin t wise.  The assigmen t C 7→ b C is fu ncto rial in the follo wing we ak s e ns e . Giv en a functor F : C → D , there is (up to a unique iso morp hism) a u nique righ t exa ct functor b F : b C → b D extending the composite h D ◦ F : C → b D . No w let T b e a triangulat ed catego ry . Then w e write A ( T ) = b T and call this cat - egory the ab elianization of T , because th e Y oneda functor T → A ( T ) is the univ ersal cohomolog ical functor for T . Lemma A.2. L et T b e a triangulate d c ate gory. Then the c ate gory A ( T ) is ab elian and the Y one da functor h T : T → A ( T ) is c ohom olo gic al. (1) Given a c ohomol o gic al functor H : T → A to an ab elian c ate gory, ther e is (up to a unique isomorphism) a unique exact functor ¯ H : A ( T ) → A such tha t H = ¯ H ◦ h T . (2) Given an exact functor F : T → T ′ b etwe en triangulate d c ate gories, ther e is (up to a unique isomorphism) a uniqu e exact functor A ( F ) : A ( T ) → A ( T ′ ) such that h T ′ ◦ F = A ( F ) ◦ h T . Pr o of. Th e category T has wea k kernels and therefore A ( T ) is ab eli an. Note that the w eak k ernel of a m orp hism Y → Z is obtained by completing the morphism to an exact triangle X → Y → Z → S X . 50 HENNING KRAUSE (1) Let H : T → A b e a cohomological functor and let ¯ H : A ( T ) → A b e the righ t exact fun ct or extending H whic h exists b y Lemma A.1. Th e n ¯ H is exact b eca us e H is cohomolog ical. (2) Let F : T → T ′ b e exact. Then H = h T ′ ◦ F is a cohomolog ical functor and w e let A ( F ) = ¯ H b e the exact fu ncto r w hic h extends H .  The assignmen t T 7→ A ( T ) from triangulated categories to ab elian cate gories pr e - serv es v ario u s prop erties of exact functors b et we en triangulated catego r ies. Let us men - tion some of them. Lemma A.3. L et F : T → T ′ and G : T ′ → T b e exact functors b etwe en triangulate d c ate gories. (1) F i s ful ly faithful if and only if A ( F ) is fu l ly faithful. (2) If F induc es a n e quivalenc e T / Ker F ∼ − → T ′ , then A ( F ) induc es an e quivalenc e A ( T ) / (Ker A ( F )) ∼ − → A ( T ′ ) . (3) F pr eserves smal l (c o)pr o ducts if and only if A ( F ) pr eserves smal l (c o)pr o ducts. (4) F i s left adjoint to G if and only if A ( F ) is left adjoint to A ( G ) . Pr o of. Straight forward.  Notes. The ab elia n ization of a triangulated category app ears in V erdier’s th` ese [48] and in F reyd’s work on the stable homotopy category [15]. Note that their constru ction is sligh tly diﬀeren t fr o m the one giv en here, w hic h is based on coherent f uncto r s in the sense of Auslander [4]. Appendix B. Lo call y present a ble abelian ca tegories Fix a r e gular cardinal α and a small add it ive categ ory C whic h adm it s α -copro ducts. W e denote by Add( C op , Ab) the cat egory of additiv e fun c tors C op → Ab in to the category of ab elia n groups. This is an ab elian category w hic h admits small (co)produ ct s. In fact, (co)k ernels and (co)produ ct s are computed p oin twise in Ab. Giv en functors F and G in Add( C op , Ab), we write Hom C ( F , G ) for the set of morph isms F → G . The most imp ortan t ob jects in Add( C op , Ab) are the r epr esentable functors C ( − , X ) with X ∈ C . Recall that Y oneda’s lemma pr o vid e s a bijecti on Hom C ( C ( − , X ) , F ) ∼ − → F X for all F : C op → Ab and X ∈ C . W e denote b y Add α ( C op , Ab) the full sub categ ory of Add( C op , Ab) w h ic h is f o rm e d by all fu nct ors preserving α -pro ducts. This is an exact ab el ian sub catego ry , b ecause kernels and cok ernels of morph ism b et w een α -pro duct preserving functors preserve α -pro ducts. In p a rticular, Add α ( C op , Ab) is an ab elian category . No w supp ose that C admits cok ernels. Then Lex α ( C op , Ab) denotes the full su bcate- gory of Add ( C op , Ab) whic h is formed by all left exact functors preserving α -pro ducts. This catego ry is lo cal ly presen table in the sense of Gabriel and Ulmer and we refer to [17, § 5] for an extensiv e treatme nt. In this app endix we collect some basic facts. First observ e that α -ﬁltered colimits in Lex α ( C op , Ab) are computed p oin t wise. This follo ws from the f a ct that in Ab taking α -ﬁltered colimits comm utes with taking α - limits; see [17, Satz 5.12]. In particular, Lex α ( C op , Ab) h as small copro ducts because ev ery small copro duct is the α -ﬁltered colimit of its su bcopro ducts with less than α factors. LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 51 Next we sho w that one can identify Add α ( C op , Ab) w it h a category of le ft exact functors. T o this end consider the Y oneda fun c tor h C : C → b C ta king X to C ( − , X ). Lemma B.1. L et C b e a smal l additive c ate gory with α -c opr o ducts. Then the Y one da functor induc es an e q uivalenc e Lex α ( b C op , Ab) ∼ − → Add α ( C op , Ab) by taking a functor F to F ◦ h C . Pr o of. Use that ev ery a d diti ve functor C op → Ab extends u niquely to a left exact functor b C op → Ab; see L emm a A.1.  F rom now on we assume that C admits α -copro ducts and cok ernels. Give n an y add i- tiv e f u ncto r F : C op → Ab, we consider the catego r y C /F whose ob ject s are pairs ( C, µ ) consisting of an ob ject C ∈ C and an elemen t µ ∈ F C . A morp hism ( C, µ ) → ( C ′ , µ ′ ) is a morp hism φ : C → C ′ suc h that F φ ( µ ′ ) = µ . Lemma B.2. L et F : C op → Ab b e an addit ive functor. (1) The c ano nic al morphism colim − − − → ( C,µ ) ∈C /F C ( − , C ) − → F in Add( C op , Ab) is an isomo rphism. (2) The functor F b elongs to L e x α ( C op , Ab) if and only if the c ate gory C /F is α - ﬁlter e d. Pr o of. (1) is easy . F or (2), see [17, Satz 5.3].  The r e p r e sentable functors in Lex α ( C op , Ab) share the foll owing ﬁ n it eness prop ert y . Recall that an ob j ect X f r o m an additive category A with α -ﬁltered colimits is α - pr esentable if the representable functor A ( X, − ) : A → Ab preserve s α -ﬁltered co limits. Next observe that the in clusion Lex α ( C op , Ab) → Add( C op , Ab) preserves α -ﬁltered col- imits. T h is follo w s from the fact that in Ab taking α -ﬁltered colimit s comm utes with taking α -limits. This has the follo wing consequence. Lemma B.3. F or e ach X in C , the r epr esentable functor C ( − , X ) is an α -pr esentable obje ct of Lex α ( C op , Ab) . Pr o of. Combine Y on ed a’s lemma with the fact that the inclusion Lex α ( C op , Ab) → Add( C op , Ab) preserv es α -ﬁltered col imits.  There is a general resu lt for the category L e x α ( C op , Ab) wh ic h sa ys that taking α - ﬁltered colimits comm utes w it h taking α -limits; see [1 7 , Korollar 7.1 2]. Here we need the follo wing sp ecial case. Lemma B.4. Supp ose the c ate g ory L e x α ( C op , Ab) is ab elian. Then an α -ﬁlter e d c olimit of exact se quenc es is again exact. Pr o of. W e need to sho w that taking α -ﬁlte red colimits comm utes with taking k ernels and cok ernels. A cok ernel is nothing but a colimit and therefore taking colimits and cok ernels co mmute. The state ment ab out kernels follo ws from th e fact that the inclusion Lex α ( C op , Ab) → Add( C op , Ab) preserv es kernels and α -ﬁltered co limits. Thus w e can compute ke rn e ls and α -ﬁltered col imits in Add( C op , Ab ) and therefore in the category 52 HENNING KRAUSE Ab of ab elian groups. In Ab it is well kno wn that taking k ernels a nd ﬁltered colimit s comm ute.  Lemma B.5. Supp ose that C is ab elian. Then Lex α ( C op , Ab) is ab elian and the Y one da functor h C : C → Lex α ( C op , Ab) is exa ct. Given a n ab elian c ate gory A which a dmits smal l c opr o ducts and exact α -ﬁlter e d c olimits, and given a f u ncto r F : Lex α ( C op , Ab) → A pr eserving α -ﬁlter e d c olimits, we have tha t F is exact if and only if F ◦ h C is exact. Pr o of. W e use the analogue of Lemma B.2 for morphisms whic h sa ys that eac h morphism φ in Lex α ( C op , Ab) can b e written as α -ﬁltered colimit φ = colim − − − → i ∈C /φ φ i of morphisms b et w een repr esentable fun ctors. Thus one computes Cok er φ = colim − − − → i ∈C /φ Cok er φ i and Ker φ = colim − − − → i ∈C /φ Ker φ i , and w e see that Lex α ( C op , Ab ) is ab elian; see Lemma B.4. The formula for kernels and cok ernels sho ws that e ac h exact sequence can b e written as α -ﬁlt ered colimit of exact sequences in the image of the Y oneda em b edding. The criterion for the exactness of a functor Lex α ( C op , Ab ) → A is an immediate consequence.  Let A b e a co co mp lete additiv e cat egory . W e denote by A α the full sub cat egory whic h is formed b y all α -presen table ob jects. F ollo wing [17], the category A is cal led lo c al ly α -pr e sentable if A α is small and eac h ob ject is an α -ﬁltered colimit of α -presen table ob jects. W e call A lo c al ly pr esentable if it is lo cally β -presen table for some cardinal β . Note that we ha ve for eac h lo cally p resen table category A a ﬁltratio n A = S β A β where β runs through all regular cardinals. W e ha ve already seen that Lex α ( C op , Ab) is lo c ally α -presentable, and the next lemma implies that, up to an equiv alence, all lo ca lly α -presen table categories are of this form. Let f : C → A b e a fully faithful and r ig ht exact functor in to a co c omplete additiv e catego ry . Supp ose that f preserve s α -copro ducts and that eac h ob ject in the image of f is α -presen table. Then f indu ce s the fun c tor f ∗ : A − → Lex α ( C op , Ab) , X 7→ A ( f − , X ) , and the follo w i n g lemma d isc u sses its left adjoint. Lemma B.6. Ther e is a ful ly faithful functor f ∗ : Lex α ( C op , Ab) → A which sends e ach r epr esentable functor C ( − , X ) to f X and identiﬁes Lex α ( C op , Ab) with the f ul l sub c ate gory of A for me d by al l c olimits o f obje cts in th e image of f . The functor f ∗ is a left adjoint of f ∗ . Pr o of. Th e functor is the left K a n extension of f ; it tak es F = colim − − − → ( C,µ ) ∈C /F C ( − , C ) in Lex α ( C op , Ab) to co lim − − − → ( C,µ ) ∈C /F f C in A . W e refer to [17 , Satz 7.8] for details.  Supp ose no w that C is a triangulated catego r y . The follo w ing lemma c haracte rizes the cohomological functors C op → Ab. Lemma B.7. L e t C b e a smal l triangulate d c ate gory and supp ose C adm its α -c opr o ducts. F or a functor F in Add α ( C op , Ab ) the fol lowing ar e e q uivalent . (1) The c ate gory C /F is α -ﬁlter e d. LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 53 (2) F i s an α -ﬁlter e d c olimit of r epr esentable functors. (3) F i s a c ohom olo gic al functor. Pr o of. Th e implications (1) ⇒ (2) ⇒ (3) are cl ear. So we pro v e (3 ) ⇒ (1). It is con- v enien t to iden tify Add α ( C op , Ab) w it h Lex α ( b C op , Ab) and this identiﬁes F with the left exact functor ¯ F : b C → Ab which extends F . In fact, ¯ F is exact since F is coho- mologica l, b y Lemma A.2. No w w rit e ¯ F as α -ﬁltered colimit of representable fu ncto rs ¯ F = colim − − − → ( M ,ν ) ∈ b C / ¯ F b C ( − , M ); see Lemma B.2. The exactness of ¯ F implies that the r e pr e- sen table fu nct ors C ( − , C ) with C ∈ C form a full su bcategory of b C / ¯ F whic h is coﬁnal. W e ident ify this su bcategory w i th C /F and conclude from L e mma 6.4.1 that C /F is α -ﬁltered.  Next we discuss the functorialit y of the assignment C 7→ Add α ( C op , Ab). Lemma B.8. L et f : C → D b e an exact functo r b etwe en sma l l triangulate d c ate gories which admit α -c opr o ducts. Supp ose that f pr eserves α -c opr o ducts. Then the r estriction functor f ∗ : Add α ( D op , Ab) − → Add α ( C op , Ab) , F 7→ F ◦ f , has a left adjoint f ∗ which sends C ( − , X ) to D ( − , f X ) for al l X in C . Mor e over, the fol lowing ho lds. (1) The functors f ∗ and f ∗ ar e exact. (2) Supp ose f induc es an e quivalenc e C / Ker f ∼ − → D . Then f ∗ is ful ly faithful. (3) Supp ose f is ful ly faithful. Then f ∗ is ful ly faithful. Mor e over, a c ohom olo g- ic al functor F : D op → Ab is in the e ss ential image of f ∗ if and only if every morphis m D ( − , D ) → F factors thr ough D ( − , f C ) for some obje ct C in C . (4) A c ohom olo gic al functor F : C op → Ab b elongs to the kernel of f ∗ if and only if every mo rphism C ( − , C ) → F factors thr ough a morphism C ( − , γ ) : C ( − , C ) → C ( − , C ′ ) such that f γ = 0 . Pr o of. Th e l eft adjoint of f ∗ is the left Kan extension. W e can describ e it exp l icitly if w e iden tify Ad d α ( C op , Ab) with Lex α ( b C op , Ab); see Lemma B.1. Giv en a functor F in Lex α ( b C op , Ab) written as α -ﬁltered colimit F = colim − − − → ( C,µ ) ∈ b C /F b C ( − , C ) of representa ble functors, w e p ut f ∗ F = colim − − − → ( C,µ ) ∈ b C /F b D ( − , b f C ) . Th u s f ∗ mak es the f ollo win g diagram commuta tive . C f   h C / / b C b f   h b C / / Lex α ( b C op , Ab) = / / f ∗   Add α ( C op , Ab) f ∗   D h D / / b D h b D / / Lex α ( b D op , Ab) = / / Add α ( D op , Ab) 54 HENNING KRAUSE W e c h e ck that f ∗ is a left adjoin t for f ∗ . F or a representa b le fu ncto r F = b C ( − , X ) w e ha ve Hom b D ( f ∗ b C ( − , X ) , G ) = Hom b D ( b D ( − , b f X ) , G ) ∼ = G ( b f X ) = f ∗ G ( X ) ∼ = Hom b C ( b C ( − , X ) , f ∗ G ) for all G in Lex α ( b D op , Ab). C l early , this isomorphism extends to every colimit of repr e - sen table fun c tors. (1) Th e exactness of f ∗ is clea r b ecause a sequence F ′ → F → F ′′ in Add α ( C op , Ab ) is exact if and only if F ′ X → F X → F ′′ X is exact for all X in C . F or the exactness of f ∗ w e ident ify again Add( C op , Ab ) with Lex α ( b C op , Ab) and apply Lemm a B.5. Thus w e need to c hec k that the comp o sition of f ∗ with the Y oneda functor h b C is exact. But w e ha ve that f ∗ ◦ h b C = h b D ◦ b f , and n o w the exactness follo ws from that of f . Finally , we use the f act that taking α -ﬁltered colimits in Add α ( D op , Ab) is exact by Lemma B.4. (2) It is well known that for an y epimorphism f : C → D of additiv e categ ories inducing a bijection Ob C → Ob D , the restriction functor Add( D op , Ab) → Add( C op , Ab) is fully faithful; see [29, Corollary 5.2]. Giv en a triangulat ed sub category C ′ ⊆ C , the quotien t functor C → C / C ′ is an epimorphism. Thus the assertion follo ws since Add α ( C op , Ab) is a fu ll sub category of Add( C op , Ab). (3) W e k eep our iden tiﬁcation Add α ( C op , Ab) = L ex α ( b C op , Ab ) and consider the ad- junction morphism η : Id → f ∗ ◦ f ∗ . W e claim that η is an isomorphism. Because f is fully faithful, η F is an isomorphism for eac h r e p r ese ntable f uncto r F = b C ( − , X ). It follo ws that η F is an isomorphism for all F since f ∗ and f ∗ b oth preserv e α -ﬁltered colimits and eac h F can b e expr e ssed as α -ﬁltered colimit of r ep r ese ntable fu nct ors. No w Prop osition 2.3.1 imp li es that f ∗ is f u lly faithful. Let F b e a cohomological fu ncto r in Add α ( D op , Ab) and apply Lemma B.7 to write the functor as α -ﬁltered colimit F = colim − − − → ( D,µ ) ∈D /F D ( − , D ) of represen table fu ncto rs . Supp ose ﬁrst that eve ry morphism D ( − , D ) → F factors through D ( − , f C ) f o r some C ∈ C . Then Im f /F is a c oﬁn a l su bcategory of D /F and therefore F = colim − − − → ( D,µ ) ∈ Im f /F D ( − , D ) by Lemma 6.4.1. Thus F b elongs to the essen tial image of f ∗ since D ( − , f C ) = f ∗ C ( − , C ) for all C ∈ C and the essent ial image is closed under taking colimits. No w supp ose that F belongs to the essent ial image of f ∗ . Th e n F = f ∗ G ∼ = f ∗ f ∗ f ∗ G = f ∗ f ∗ F for some G . The fu ncto r f ∗ F is cohomological and therefore f ∗ F = colim − − − → ( C,µ ) ∈C /f ∗ F C ( − , C ), again b y Lemma B.7. Thus F ∼ = colim − − − → ( C,µ ) ∈C /f ∗ F D ( − , f C ) and w e use Lemma B.3 to conclude that eac h morph ism D ( − , D ) → F f a ctors through D ( − , f C ) for some ( C , µ ) ∈ C /f ∗ F . (4) Let F b e a cohomologica l fun ctor in Add α ( C op , Ab) and apply Lemma B.7 to write the functor a s α -ﬁltered co limit F = colim − − − → ( C,µ ) ∈C /F C ( − , C ) of represen table func- tors. No w f ∗ F = c olim − − − → ( C,µ ) ∈C /F D ( − , f C ) = 0 if and only if for eac h D ∈ D , we ha ve colim − − − → ( C,µ ) ∈C /F D ( D, f C ) = 0. This happ ens iﬀ for eac h ( C, µ ) ∈ C /F , w e ﬁnd a morph ism γ : C → C ′ in C /F inducing a map D ( f C, f C ) → D ( f C , f C ′ ) whic h annihilates the LOCALIZA TION TH EO R Y FOR TRIANGULA TED CA TEGORIES 55 iden tit y m o rp hism. But this means that f γ = 0 and that µ : C ( − , C ) → F f actors through C ( − , γ ).  Notes. Lo cally presentable categ ories w ere introduced and studied by Gabriel and Ul- mer in [17]; see [1] for a mo dern treatment . In [33], Neeman in itiated the u se of lo cally presen table ab elian categories for stu dying triangulated categories. Referenc es [1] J. Ad ´ amek and J. R osick´ y, L o c al ly pr esentable and ac c essible c ate gories , Cam bridge Univ. Press, Cam bridge, 1994. [2] J. F. Adams, Stable homotopy and gener al i se d homolo gy , Univ. Chicago Press, Chicago, Ill., 1974. [3] L. 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T abuada, Homotopy theory of well-generated algebraic triangulated categories, J. K-theory , in press. [48] J.-L. V erdier, Des cat´ egories d´ eri v´ ees des cat ´ egories ab´ eliennes, Ast´ erisque No. 23 9 (1996), xii+253 pp. (1997). Henning Krause, Institut f ¨ ur Ma thema ti k , Universit ¨ at P aderborn, 33095 P aderborn, Germany. E-mail addr ess : hkrause@math. upb.de

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