General heart construction on a triangulated category (I): unifying t-structures and cluster tilting subcategories

GENERAL HEAR T CONSTR UCTION ON A TRIANGULA TED CA TEGOR Y (I): UNIFYING t -STRU CTURES AND CLUSTER TIL TING SUBCA TEGORIES HIRO YUKI NAKAOKA Abstract. In the paper of Keller and R ei ten, it was shown that the quotien t of a triangulated category (with some conditions) b y a cluster tilting sub category becomes an abelian category . After that, Ko enig and Zh u show ed in detail, ho w the abelian structure is gi v en on this quotien t category , in a more abstract setting. On the other hand, as is we ll known since 1980s, the heart of any t - structure is abeli an. W e unify these tw o construct ion by using th e notion of a cotorsion pair. T o any cotorsion pair in a triangulated category , w e can naturally associate an abeli an category , w hi c h giv es bac k eac h of the ab o ve t wo abeli an catego ri es, when the cotorsion pair comes f r om a cluster tilting subcategory , or i s a t -structure, resp ectiv ely . 1. Introduction Throughout this paper , we fix a triangulated category C . F or any categor y K , w e write abbreviately K ∈ K , to indicate that K is an ob ject of K . F or an y K, L ∈ K , let K ( K , L ) deno te the set of mo rphisms fro m K to L . If M , N are full sub categories of K , then K ( M , N ) = 0 means that K ( M , N ) = 0 for an y M ∈ M and N ∈ N . Similarly , K ( K, N ) = 0 means K ( K, N ) = 0 for a ny N ∈ N . When K admits Ex t ℓ , similar ly for Ext ℓ ( M , N ) = 0 and Ext ℓ ( K, N ) = 0 for any int eger ℓ . As is well known, if ( T ≤ 0 , T ≥ 0 ) is a t -structure on C , then its hea rt H = T ≤ 0 ∩ T ≥ 0 bec omes an ab elian catego ry [BB D]. Put T ≤ n := T ≤ 0 [ − n ] , T ≥ n := T ≥ 0 [ − n ] for an integer n . By definition, a t -structure is a pair of full additive thick s ubcat- egories ( T ≤ 0 , T ≥ 0 ) o f C satisfying ( t -1) C ( T ≤ 0 , T ≥ 1 ) = 0, ( t -2) T ≤ 0 ⊆ T ≤ 1 and T ≥ 1 ⊆ T ≥ 0 , ( t -3) F or an y C ∈ C , there exists a distinguished triangle X → C → Y → X [1] ( X ∈ T ≤ 0 , Y ∈ T ≥ 1 ) . On the o ther hand, in [KZ], Koenig and Zh u show ed that for an y cluster tilting sub c ategory T of C , the quotien t category C / T carries a naturally induced abelia n The author wishes to thank Pr ofessor T oshiyuki Katsura f or his encouragemen t. The autho r wishes to than k his colleague Professor Noriyuki Abe. This w ork was neve r p ossible without his advices. The author wishes to thank Professor Bernhard K eller and Pr ofessor Osamu Iyama for their useful commen ts and advices, especiall y on the terminology . 1 2 HIR OYUKI NAKAOKA structure. Originally , an equiv alence b etw een the quotient of a triangula ted cate- gory a nd a certain mo dule ca tegory was shown in [BMR] for the quo tien t b y a full additive thick sub category ass ociated to a tilting o b ject in C , a nd then in [K R], for the quotien ts b y a cluster tilting s ubcatego ry . By definitio n ([KR], [KZ]), a full additive thic k sub category T o f C is a cluster tilting subca tegory if it satisfies ( s -1) T is functorially finite (cf. [BR]), ( s -2) C ∈ T if and only if Ext 1 ( C, T ) = 0, ( s -3) C ∈ T if and only if Ext 1 ( T , C ) = 0. By ( s - 1), it can be easily shown that a cluster tilting sub category also satisfies the following for any C ∈ C : ( s -1) ′ F or an y C ∈ C , there ex ists a distinguished triang le T → C → T ′ [1] → T [1 ] ( T , T ′ ∈ T ) . T o unify these tw o, w e in tro duce the notion of a c otorsion p air (Definition 2.1 ). T o any cotorsion pair ( U , V ) in C , we can naturally asso ciate a subfactor categ ory H := ( C + ∩ C − ) / ( U ∩ V ). As a main theorem, we show H carries an induced ab elian structure (Theorem 6 .4). In fa ct, this construction g eneralizes the ab ov e ab elian categorie s, in the following sense (Prop osition 2.6 a nd Example 3.8): - If ( U , V ) satisfies V ⊆ V [1 ], then ( U [ − 1] , V [1]) be c omes a t -structure , and H agrees with the heart of ( U [ − 1] , V ). - If ( U , V ) satisfies U = V , then T := U = V becomes a cluster tilting sub category of C , and H agrees with C / T . 2. Preliminaries Definition 2.1 . Let U and V b e full additiv e thick sub categorie s of C . W e call ( U , V ) a c otorsion p air if it satisfies the following. (1) Ext 1 ( U , V ) = 0, (2) F or any C ∈ C , ther e exis ts a (not necessar ily unique) distinguished triang le U → C → V [1] → U [1 ] satisfying U ∈ U and V ∈ V . F or an y cotorsion pair ( U , V ), put W := U ∩ V . R emark 2.2 . A pa ir ( U , V ) of full additive thick sub categories of C is a cotorsio n pair if and only if ( U [ − 1] , V ) is a torsion the ory (or a t orsio n p air ) in [IY]. (Un like [BR], it does not requir e the shift-clo s edness.) In this sense, a cotorsion pair is nothing other tha n a torsion t heor y with U shifted by − 1, and th us not a new notion. How ever we prefer the ab ov e definition, just for the sake of the duality in the index of shifts. R emark 2.3 . A pa ir ( U , V ) o f full additive subcateg o ries of C is a cotor sion pa ir if and only if it satisfies the follo wing conditions for a n y C ∈ C . (1) C belongs to U if and only if Ex t 1 ( C, V ) = 0, (2) C belongs to V if and only if Ext 1 ( U , C ) = 0, (3) F or any C ∈ C , ther e exis ts a (not necessar ily unique) distinguished triang le U → C → V [1] → U [1 ] GENERAL HEAR T CONSTR UCTION ON A TRIANGULA TED CA TE GOR Y (I) 3 satisfying U ∈ U and V ∈ V . R emark 2.4 . Let ( U , V ) b e a co torsion pair in C . (1) F or an y C ∈ C a nd a n y n ∈ Z , C admits a distinguished triangle U [ n ] → C → V [ n + 1] → U [ n + 1 ] with U ∈ U , V ∈ V . (2) F or an y n ∈ Z , ea ch U [ n ] and V [ n ] is closed under extensions and direct summands. (3) W b ecomes an a dditiv e full sub category of C , clo s ed under direct sums and direct summands, satisfying Ext 1 ( W , W ) = 0 . Example 2.5. (1) If ( T ≤ 0 , T ≥ 0 ) is a t -s tructure on C , then ( U , V ) := ( T ≤− 1 , T ≥ 1 ) bec omes a co to rsion pair . In this cas e , W = 0. (2) If T is a cluster tilting s ubcatego ry of C , then ( U , V ) := ( T , T ) bec omes a co to rsion pair . In this cas e , W = T . Remark that Ext 1 ( U , V ) = 0 implies U ∩ V [1] = 0. The following pro p osition shows that t -structures and cluster tilting subcateg ories are characterized as tw o extremal examples o f co torsion pair s. Prop osition 2.6. L et ( U , V ) b e a c otorsion p air. (1) V ⊆ V [1] if and only if ( U , V ) = ( T ≤− 1 , T ≥ 1 ) for a t -structur e ( T ≤ 0 , T ≥ 0 ) . (2) V = U if and only if ( U , V ) = ( T , T ) for a cluster t ilting sub c ate gory T . Pr o of. (1 ) This immediately follows from the definition of a t -structure. (2) The ‘o nly if ’ part is trivial. T o sho w the conv erse , put T := U = V . As in [K Z] (Definition 3.1 and Lemma 3.2), it suffices to sho w T is c on trav a rian tly finite. But this immediately follows from the fact that any ob ject C ∈ C admits a distinguished triangle T → C → T ′ [1] → T [1 ] for some T ∈ U = T and T ′ ∈ V = T .  Definition 2. 7. (cf. §§ 2.1 in [K Z]) L e t ( U , V ) be a cotorsio n pair in C , and W := U ∩ V . W e denote the quotient of C by W a s C := C / W . Namely , Ob( C ) = Ob( C ), and for any A, B ∈ C , C ( A, B ) := C ( A, B ) / { f ∈ C ( A, B ) | f factors through s o me W ∈ W } . F or any morphis m f ∈ C ( A, B ), w e denote its image in C ( A, B ) by f . This defines an additiv e functor ( ) : C → C . Since U ⊇ W and V ⊇ W , w e also hav e additive full sub categories of C U := U / W a nd V := V / W . 4 HIR OYUKI NAKAOKA R emark 2.8 . Since W is close d under direct summands, for any C ∈ C we hav e C ∼ = 0 in C ⇐ ⇒ C ∈ W . Since Ext 1 ( W , W ) = 0 , Theorem 2.3 in [KZ] can b e applied. Compare with Prop osition 6.1. F act 2.9. (Theo rem 2.3 in [KZ]) Let C , ( U , V ) and W b e as ab ov e. F or an y distin- guished triangle X f − → Y g − → Z h − → X [1] , the following hold. (1) g is epimorphic in C if a nd o nly if h = 0, (2) g is monomorphic in C if and only if f = 0 . Let ( U , V ) be a cotorsion pair in the following. Prop osition 2.10. (1) F or any U ∈ U , C ∈ C and any f ∈ C ( U [ − 1] , C ) , if f = 0 , then f = 0 . Namely, we have C ( U [ − 1] , C ) = C ( U [ − 1] , C ) = Ext 1 ( U, C ) . In p articular, C ∈ V if and only if C ( U [ − 1] , C ) = 0 for any U ∈ U . (2) Dual ly, for any V ∈ V and any C ∈ C , we have C ( C, V [1]) = C ( C, V [1]) = Ext 1 ( C, V ) . In p articular, C ∈ U if and only if C ( C, V [1]) = 0 for any V ∈ V . Pr o of. (1 ) B y definition, f = 0 if and o nly if f fac to rs throug h some W ∈ W . Since C ( U [ − 1] , W ) ⊆ Ext 1 ( U , V ) = 0, this implies f = 0. W U [ − 1] C 0 ? ?         ? ? ? ? ? ? ? f / /  (2) is s hown dually .  Lemma 2.11. F or any c otorsion p air ( U , V ) , we have C ( U , V ) = 0 . Pr o of. Le t f ∈ C ( U, V ) b e a morphism, where U ∈ U and V ∈ V . By condition (2) in Definition 2.1, w e can fo rm a distinguished triangle V ′ → U ′ u − → V v − → V ′ [1] where U ′ ∈ U a nd V ′ ∈ V . Since V is extensio n-closed (Remar k 2.4), U ′ satisfies U ′ ∈ U ∩ V = W . Since v ◦ f = 0 by Ex t 1 ( U, V ′ ) = 0, f fa c tors thr o ugh U ′ . V ′ U ′ V V ′ [1] U / / / / f   v / / } } { { { { {  Since U ′ ∈ W , this means f = 0.  GENERAL HEAR T CONSTR UCTION ON A TRIANGULA TED CA TE GOR Y (I) 5 3. Definition of C + and C − Lemma 3.1. L et f : A → B b e any morphism in C . (1) L et U A u A − → A → V A [1] → U A [1] U B u B − → B → V B [1] → U B [1] b e any distinguishe d t riangles satisfying U A , U B ∈ U and V A , V B ∈ V . T hen ther e exists a morphism f U ∈ C ( U A , U B ) such that f ◦ u A = u B ◦ f U . U A U B A B f U   u B / / u A / / f    Mor e over, f U with this pr op erty is unique in C ( U A , U B ) . (2) Dual ly, for any distinguishe d t riangles U ′ A [ − 1] → A → V ′ A → U ′ A U ′ B [ − 1] → B → V ′ B → U ′ B with U ′ A , U ′ B ∈ U and V ′ A , V ′ B ∈ V , ther e exists a morphism f ′ V ∈ C ( V ′ A , V ′ B ) c omp atible with f , uniquely u p to W . Pr o of. W e o nly s ho w (1). Existence immediately follows from C ( U A , V B [1]) = 0. Moreov er if f 1 U and f 2 U in C ( U A , U B ) satisfies f 1 U ◦ u A = u B ◦ f U = f 2 U ◦ u A , then b y ( f 1 U − f 2 U ) ◦ u A = 0, there exists w ∈ C ( U A , V B ) such that f 1 U − f 2 U factors w . V B U B B V B [1] U A A / / / / / / f 1 U − f 2 U   f   w        u A / /  By Lemma 2.1 1 we have w = 0, and thus f 1 U = f 2 U .  Prop osition 3.2. L et C b e any obje ct in C . (1) F or any distinguishe d t riangles U u − → C → V [1] → U [1] U ′ u ′ − → C → V ′ [1] → U ′ [1] satisfying U, U ′ ∈ U and V , V ′ ∈ V , ther e exists a m orphism s ∈ C ( U, U ′ ) c omp atible with u and u ′ , such that s is an isomorphism. U U ′ ∼ = s   U U ′ C s   u " " E E E E u ′ < < y y y y  6 HIR OYUKI NAKAOKA (2) Dual ly, t hose V app e aring in distinguishe d triangles U [ − 1] → C → V → U ( U ∈ U , V ∈ V ) ar e isomorphic in C . Pr o of. This immediately follows fro m Lemma 3.1.  Corollary 3.3. F or any C ∈ C , the fol lowing ar e e quivalent. (1) Ther e exists a distinguishe d triangle W 0 → C → V 0 [1] → W 0 [1] such that W 0 ∈ W , V 0 ∈ V . (2) A ny distinguishe d triangle U → C → V [1] → U [1 ] ( U ∈ U , V ∈ V ) satisfies U ∈ W . Pr o of. Supp ose (1) ho lds. By Prop osition 3.2, we hav e U ∼ = W 0 in C . By Remark 2.8, this means U ∈ W . The co nverse is trivial.  Dually , we hav e the following: Corollary 3.4. F or any C ∈ C , the fol lowing ar e e quivalent. (1) Ther e exists a distinguishe d triangle U 0 [ − 1] → C → W 0 → U 0 such that U 0 ∈ U , W 0 ∈ W . (2) A ny distinguishe d triangle U [ − 1] → C → V → U ( U ∈ U , V ∈ V ) satisfies V ∈ W . Definition 3.5. (1) C + is defined to b e the full sub category o f C , co nsisting of ob jects sa tisfying equiv alen t conditions of Co rollary 3.3. (2) C − is defined to b e the full sub category o f C , co nsisting of ob jects sa tisfying equiv alen t conditions of Co rollary 3.4. R emark 3.6 . The fo llowing are sa tisfied. (1) Each of C + and C − is an additiv e full subcateg ory of C containing W . (2) C + ⊇ V [1]. (3) C − ⊇ U [ − 1]. Definition 3.7. F o r any co torsion pair ( U , V ), put H := C + ∩ C − . Since H ⊇ W , we hav e an additive full sub category H := H / W ⊆ C , which we call the he art of ( U , V ). Example 3.8. GENERAL HEAR T CONSTR UCTION ON A TRIANGULA TED CA TE GOR Y (I) 7 (1) If ( U , V ) = ( T ≤− 1 , T ≥ 1 ), where ( T ≤ 0 , T ≥ 0 ) is a t -structure, then w e hav e C − = T ≤− 1 [ − 1] = T ≤ 0 , C + = T ≥ 1 [1] = T ≥ 0 , H = H = T ≤ 0 ∩ T ≥ 0 . Thu s the definition o f the heart agrees with that of a t -structure. Thus H is ab e lian, and admits a cohomological functor H 0 : C → H (cf. [BBD]). (2) If U = V = T is a cluster tilting s ub categor y of C , then w e hav e C + = C − = H = C , H = C / T . By [K Z], H bec omes an ab elian categ o ry , and the quotient functor ( ) : C → C / T = H is cohomological. 4. Existence of (co)reflections Since C + ∩ C − = H ⊇ W , w e hav e additive full sub categor ies o f C C + := C + / W and C − := C − / W . H C + C − C *  7 7 o o o o o o o  t ' ' O O O O O O O  t ' ' O O O O O O O *  7 7 o o o o o o o  Definition 4. 1. (Definition 3 .1 .1 in [B]) Let A and B b e ca tegories, and F : A → B be a functor . F or any B ∈ B , a reflectio n of B along F is a pair ( R B , η B ) of R B ∈ A and η B ∈ B ( B , F ( R B )), satisfying the following universality: ( ∗ ) F or a n y A ∈ A and an y b ∈ B ( B , F ( A )), there exists a unique morphism a ∈ A ( R B , A ) such that F ( a ) ◦ η B = b . F ( A ) B F ( R B ) b   6 6 6 6 6 6 6 η B / / F ( a )          A c or efle ction is de fined dually . Construction 4.2. F or any C ∈ C , consider a diagram U ′ C [ − 1] U C V ′ C C Z C V C [1]    u ′ C   ! ! ! ! ! ! ! v ′ C   " " " " " " " w C   : : : : : : : u C 3 3 h h h h z C   ; ; v C 4 4 i i i i i i i i x C ; ; x x x x x x x y C ? ? ~ ~ ~ ~ ~ ~ ~ 8 HIR OYUKI NAKAOKA where U C u C − → C v C − → V C [1] → U C [1] U ′ C [ − 1] u ′ C − → U C v ′ C − → V ′ C → U ′ C U ′ C [ − 1] w C − → C z C − → Z C → U ′ C are distinguished triangles, sa tisfying U C , U ′ C ∈ U and V C , V ′ C ∈ V . S ince U is extension-clos ed, we ha ve V ′ C ∈ W . By the o ctahedro n axio m, V ′ C x C − → Z C y C − → V C [1] → V ′ C also becomes a distinguished triangle. Th us Z C belo ngs to C + . Prop osition 4.3. In the notation of Construction 4.2, for any C ∈ C , z C : C → Z C gives a r efle ction of C a long C + ֒ → C . Pr o of. Le t Y b e any ob ject in C + , and let y ∈ C ( C, Y ) be any morphism. It suffices to sho w that there exists a unique morphism q ∈ C ( Z C , Y ) such that q ◦ z C = y . C Y Z C y   ) ) ) ) ) ) z C + + X X X X X X q          In fa ct, q ca n b e c hosen to satisfy q ◦ z C = y . First, we show the existence. Since U ′ C [ − 1] w C − → C z C − → Z C → U ′ C is a distinguished tria ng le, it suffices to s ho w y ◦ w C = 0 . Let U Y u Y − → Y v Y − → V Y [1] → U Y [1] be a dis tinguished triangle suc h that U Y ∈ W , V Y ∈ V . By C ( U C , V Y [1]) = 0, there exists a mor phism y C ∈ C ( U C , U Y ) suc h that y ◦ u C = u Y ◦ y U . U ′ C [ − 1] U C U Y C Y u ′ C       y U   u C / / w C   ? ? ? ? y   u Y / /   Since U Y ∈ W , we have y ◦ w C = u Y ◦ y U ◦ u ′ C = 0 . By Prop osition 2.10 , we obtain y ◦ w C = 0. GENERAL HEAR T CONSTR UCTION ON A TRIANGULA TED CA TE GOR Y (I) 9 T o sho w the uniqueness, suppos e q , q ′ ∈ C ( Z C , Y ) satisfies q ◦ z C = y = q ′ ◦ z C . C Z C U Y Y V Y [1] z C - - [ [ [ [ [ [ [ y   0 0 0 0 0 0 u Y / / v Y / / q,q ′        Since v Y ◦ ( q − q ′ ) ◦ z C = 0 in C ( C, V Y [1]), it follows v Y ◦ ( q − q ′ ) ◦ z C = 0 by Prop osition 2.10. Thu s we have a morphism of triangles C Z C U ′ C U Y Y V Y [1] . z C / / / /     q − q ′   u Y / / v Y / /   Since C ( U ′ C , V Y [1]) = 0, this implies v Y ◦ ( q − q ′ ) = 0. Thus q − q ′ factors through U Y ∈ W , which means q = q ′ .  Corollary 4.4. ( Pr op osition 3.1.2 and Pr op osition 3.1.3 in [B]) (1) Sinc e ( Z C , z C ) is a r efle ction of C along C + ֒ → C , it is determine d up to a c anonic al isomorphism in C + . (2) As in [B] , if we al low the axiom of choic e, we obtain a left adjoint σ : C → C + of the inclusion C + i + ֒ → C . If we denote the adjunction by η : Id C = ⇒ i + ◦ σ , t hen t her e exists a c anonic al isomorph ism Z C ∼ = σ ( C ) in C , c omp atible with z C and η C . C Z C σ ( C ) z C / / η C   3 3 3 3 3 3 ∼ =          Dually , we hav e the following: R emark 4.5 . F o r any C ∈ C , if w e tak e a diagram U ′′ C [ − 1] K C U ′′′ C C V ′′ C V ′′′ C [1]    . . ] ] ] ] ] ] . . ] ] ] ] ] ] $ $ J J J J J J J J k C       & & M M                            ( U ′′ C , U ′′′ C ∈ U , V ′′ C , V ′′′ C ∈ V ) where all 180 ◦ comp osition of arr o ws a re distinguished triangle s , then k C : K C → C gives a coreflection of C along C − ֒ → C . Thus K C is uniquely determined up to a canonical iso morphism in C − . Moreov er if we allow the axiom o f choice, we obtain a righ t adjoint of C − ֒ → C . 10 HIR OYUKI NAKAOKA Lemma 4 .6. In the notation of Construction 4. 2, if C b elongs to C − , then Z C b elongs to H . Dual ly, C ∈ C + implies K C ∈ H ( in the notation of R emark 4.5 ) . Pr o of. W e only show the former half. The latter is shown dually . Let U [ − 1] u − → C v − → V → U U Z [ − 1] u Z − → Z C v Z − → V Z → U Z be distinguished triang les satisfying U, U Z ∈ U and V , V Z ∈ V . By assumption, V ∈ W . It suffices to show V Z ∈ W . In the no tation o f Construction 4.2, co mplete z C ◦ u : U [ − 1] → Z C int o a distin- guished triangle U [ − 1] z C ◦ u − → Z C → Q → U. By the o ctahedro n a xiom, we a lso have a disting uished triang le V → Q → U ′ C → V [1] , and th us Q ∈ U . U [ − 1] C V Z C Q U ′ C    u . . ] ] ] ] ] ] ] ] . . ] ] ] ] ] ] ] z C ◦ u $ $ J J J J J J J J z C       & & M M                             Since C ( U [ − 1 ] , V Z ) = 0, w e obtain a morphism o f tria ngles U [ − 1] Z C Q U U Z [ − 1] Z C V Z U Z .       / / / / / / v Z / / / / / /    Since C ( Q, V Z ) = 0 b y Lemma 2.11, it fo llo ws v Z = 0. F or a n y V † ∈ V a nd a n y v † ∈ C ( V Z , V † [1]) we have v † ◦ v Z = v † ◦ 0 = 0 in C ( Z C , V † [1]). B y Propos ition 2.10, we obtain v † ◦ v Z = 0. T hus v † factors through U Z , which means v † = 0 since C ( U Z , V † [1]) = 0. Z C V Z U Z V † [1] v Z / / / / v †   0 } } { { { { { { { {  Thu s we hav e C ( V Z , V [1]) = 0 , namely V Z ∈ W .  GENERAL HEAR T CONSTR UCTION ON A TRIANGULA TED CA TE GOR Y (I) 11 5. Existence of (co-)kernels Lemma 5.1. L et A f − → B b e any morphism in C . T ake a diagr am U A [ − 1] A V A B M f u A   v A   # # G G G G G G G f / / m f # # G G G G G G G G  wher e U A [ − 1] u A − → A v A − → V A → U A U A [ − 1] → B m f − → M f → U A ar e distinguishe d triangles, satisfying U A ∈ U A , V A ∈ V . Then we have the fol low- ing. (1) A ∈ C − = ⇒ m f ◦ f = 0 , (2) B ∈ C − = ⇒ M f ∈ C − . Pr o of. (1 ) This immediately follows from V A ∈ W . (2) T ake distinguished triangles U B [ − 1] → B → V B → U B U M [ − 1] → M f → V M → U M ( U B , U M ∈ U , V B , V M ∈ V ) . By assumption, V B ∈ W . It suffices to show V M ∈ W . Since C ( U B [ − 1] , V M ) = 0 , there exists a morphism of triangles U B [ − 1] B V B U B U M [ − 1] M f V M U M m U   m f   m V     u B / / u M / / v B / / v M / / / / / /    Let V † be any o b ject in V , a nd v † ∈ C ( V M , V † [1]) b e an y morphism. It suffices to show v † = 0. Since v † ◦ m V ∈ C ( V B , V † [1]) = 0, we hav e v † ◦ v M ◦ m f = 0. B M f U A V B ∈ W V M V † [1] U M v B / / v M / / 0 / / _ _ _ m f     m V   v †   / / 0 } } { { { {    Thu s v † ◦ v M factors thro ugh U A , which means v † ◦ v M = 0, since C ( U A , V † [1]) = 0. Thu s v † factors through U M , and v † = 0 follows from C ( U M , V † [1]) = 0.  Dually , we hav e the following: 12 HIR OYUKI NAKAOKA R emark 5.2 . F o r any morphism A f − → B in C , consider a dia gram V B [1] B U B A L f v B   u B   v B ◦ f # # G G G G G G G G f / / ℓ f # # G G G G G G G G  where U B u B − → B v B − → V B [1] → U B [1] V B → L f ℓ f − → A → V B [1] are distinguished triang les satisfying U B ∈ U , V B ∈ V . Then, A ∈ C + implies L f ∈ C + , and B ∈ C + implies f ◦ ℓ f =0. Prop osition 5.3. L et A f − → B b e any morphism i n C . The n m f : B → M f in L emma 5.1 satisfies the fol lowing pr op erty : ( ∗ ) F or any C ∈ C and any morphism g ∈ C ( B , C ) satisfyi ng g ◦ f = 0 , t her e exists a morphism c ∈ C ( M f , C ) such that c ◦ m f = g . A B C M f f / / g / / m f   : : : : : : c B B        Mor e over if C ∈ C + , then c ∈ C ( M f , C ) satisfying c ◦ m f = g is unique in C ( M f , C ) . The dual statement also holds for L f in R emark 5.2. Pr o of. Fir st w e sho w the existence. By Prop osition 2.10, g ◦ f ◦ u A = 0 means g ◦ f ◦ u A = 0. Th us there exists c ∈ C ( M f , C ) such that c ◦ m f = g . U A [ − 1] A B C M f u A   f / / f ◦ u A ! ! C C C C C C C C g 5 5 k k k k k k k m f B B B B B B c K K       T o sho w the uniqueness, let U C u C − → C v C − → V C [1] → U C [1] be a distinguished triangle with U C ∈ W , V C ∈ V . Suppose c, c ′ ∈ C ( M f , C ) satisfy c ◦ m f = g = c ′ ◦ m f . GENERAL HEAR T CONSTR UCTION ON A TRIANGULA TED CA TE GOR Y (I) 13 By Prop osition 2.10, v C ◦ ( c − c ′ ) ◦ m f = 0 means v C ◦ ( c − c ′ ) ◦ m f = 0. Thus v C ◦ ( c − c ′ ) factors through U A , and thus v C ◦ ( c − c ′ ) = 0 follows fro m C ( U A , V C [1]) = 0. Th us c − c ′ factors through U C ∈ W , which means c = c ′ . B M f U A U C C V C [1] m f / / / /   0   c − c ′   u C / / v C / /    Corollary 5.4. In H , any morphism has a c okernel and a kernel. Pr o of. W e only show the construction of the cokernel. F or any A, B ∈ H and any f ∈ C ( A, B ), define m f : B → M f as in Lemma 5.1. Since A, B ∈ C − , it follo ws m f ◦ f = 0 , M f ∈ C − by Lemma 5.1. By Propos ition 4.3, there exists z M : M f → Z M which gives a reflection z M : M f → Z M of M f along C + ֒ → C . By Lemma 4.6, Z M satisfies Z M ∈ H . W e claim that z M ◦ m f : B → Z M is the cokernel of f . Let S b e any ob ject in H , and let s ∈ C ( B , S ) b e an y morphism sa tisfying s ◦ f = 0. A B S M f Z M # # F F F f / / m f # # F F F z M # # F F s : : v v v v v v v  0 . . It suffices to show that there uniquely exists t ∈ H ( Z M , S ) such that t ◦ z M ◦ m f = s. This follo ws immediately from P rop o sition 4.3 and Prop osition 5.3.  6. Abelianess of the hear t In this section, as the main theo rem, we show that the heart H bec omes an ab elian catego r y , for any cotor sion pa ir ( U , V ). Although prop ositions and lemmas in this s ection co uld b e applied fo r ob jects in C + or C − (with cer tain mo difications of the sta tement), w e mainly consider ob jects in H . Prop osition 6.1. L et B , C ∈ H , and let A f − → B g − → C h − → A [1] b e a distinguishe d triangle in C . L et m g : C → M g b e as in Le mma 5.1 . Then the fol lowing ar e e quivalent. (1) g is epimorphic in H . (2) g is epimorphic in C + . (3) M g satisfies C ( M g , C + ) = 0 . (4) M g satisfies C ( M g , V [1]) = 0 . 14 HIR OYUKI NAKAOKA The dual statement also holds for monomorphisms. Pr o of. Fir st, we show the equiv alence o f (1) and (2). Ob viously , (2 ) implies (1). T o show the conv erse, let S be a ny o b ject in C + , and s ∈ C ( C , S ) b e any mor phism. Let k S : K S → S be the morphism defined in Remar k 4.5, which gives a co reflection of S along C − ֒ → C . By Lemma 4.6, K S ∈ H . By Remark 4.5, there exists j ∈ C − ( C, K S ) suc h that k S ◦ j = s , B C S K S g / / s / / j B B       k S   : : : : : :  and w e ha ve j = 0 ⇐ ⇒ k S ◦ j = 0 j ◦ g = 0 ⇐ ⇒ k S ◦ ( j ◦ g ) = 0 . By (1), we ha ve j = 0 ⇐ ⇒ j ◦ g = 0 . Thu s s = 0 if and only if s ◦ g = 0, i.e., g is epimo r phic in C + . Second, we s ho w tha t (2) implies (3). By Prop osition 5.3 , for an y S ∈ C + we hav e an isomorphism − ◦ m g : C ( M g , S ) ∼ = − → { s ∈ C ( C , S ) | s ◦ g = 0 } . Third, w e show that (3) implies (4). This immediately follows fro m Prop osition 2.10, since V [1] ⊆ C + . Finally , we show that (4) implies (3). Suppose C ( M g , V [1]) = 0 . F or any S ∈ C + , let U S u S − → S v S − → V S [1] → U S [1] be a distinguished tr iangle satisfying U S ∈ W , V S ∈ V . By C ( M g , V [1]) = 0, a n y morphism s ∈ C ( M g , S ) factors through U S ∈ W , which means s = 0 .  Lemma 6.2. L et B , C ∈ H , and let A f − → B g − → C h − → A [1] b e a distinguishe d triangle in C . T ake a diagr am U A [ − 1] A V A U B [ − 1] B V B C f U / / f / / f V / / u A   v A   u B   v B   g / /   wher e U A [ − 1] u A − → A v A − → V A → U A U B [ − 1] u B − → B v B − → V B → U B GENERAL HEAR T CONSTR UCTION ON A TRIANGULA TED CA TE GOR Y (I) 15 ar e distinguishe d triangles satisfying U A , U B ∈ U , V A ∈ V and V B ∈ W . If g is an epimorphism in H , then t her e exists a morphism s ∈ C ( V B , V A ) such that s ◦ v B ◦ f = v A . Pr o of. Le t C m g − → M g be as in Lemma 5.1. Remark h factors m g . C A [1] M g h / / m g   : : : : : : ∃ B B       By Prop osition 6.1, C ( M g [ − 1] , V A ) ∼ = C ( M g , V A [1]) = 0, and th us v A ◦ h [ − 1] = 0. C [ − 1] M g [ − 1] A V A h [ − 1] / / m g [ − 1]   v A   0 / / = = { { { { { { { { { {   Thu s there exists q ∈ C ( B , V A ) suc h that q ◦ f = v A . C [ − 1] A V A U B [ − 1] B V B h [ − 1] / / f / / 0 $ $ I I I I I I I I I v A   f V / / q { { x x x x x x x x x u B   v B     Since C ( U B [ − 1] , V A ) = 0 , we hav e q ◦ u B = 0, and th us there exis ts s ∈ C ( V B , V A ) such that s ◦ v B = q . Thus we o bta in s ◦ v B ◦ f = q ◦ f = v A .  Lemma 6.3. In t he notation of L emma 6.2, if g is epimorph ic in H , then A b elongs to C − . Pr o of. W e use the notation o f L e mma 6.2. By the sa me lemma, there exists s ∈ C ( V B , V A ) suc h that s ◦ v B ◦ f = v A . It suffices to show V A ∈ W . By assumption V B ∈ W and thus fo r any V † ∈ V and any v † ∈ C ( V A , V † [1]), we have v † ◦ s = 0. Thus v † ◦ v A = v † ◦ s ◦ v B ◦ f = 0, and v † factors through U A , w hich implies v † = 0 . A B V A U A V B V † [1] v A   f / / v B     s o o v † E E E " " E E 0        0 3 3 f f f     16 HIR OYUKI NAKAOKA Theorem 6.4. F or any c otorsion p air ( U , V ) in C , its he art H = ( C + ∩ C − ) / ( U ∩ V ) is an ab elia n c ate gory. Pr o of. Since H is an a dditiv e catego r y with (co- )k ernels (Cor ollary 5.4 ), it r emains to sho w the follo wing: (1) If g is epimorphic in H , then g is a cok ernel of so me morphism in H . (2) If g is monomorphic in H , then g is a k ernel of so me morphis m in H . Since (2) ca n b e sho wn dually , w e only sho w (1). Complete g into a dis ting uished triang le A f − → B g − → C h − → A [1] in C . By Lemma 6.3, we ha ve A ∈ C − . By Prop o sition 4.3, there exists z A : A → Z A which gives a reflectio n z A : A → Z A of A along C + ֒ → C . Moreover by Lemma 4.6, Z A belo ngs to H . Since B b elongs to C + , there exis ts b ∈ C ( Z A , B ) such that b ◦ z A = f . A B Z A f / / z A   : : : : : : b B B        W e claim tha t g = cok( b ). Let S be an y ob ject in H , and s ∈ C ( B , S ) b e any mor phism. By Pr opos ition 4.3, s ◦ f = 0 if and only if s ◦ b = 0. In par ticula r, g ◦ b = 0. So it suffices to show for any s s atisfying s ◦ f = 0, there uniquely ex ists c ∈ C ( C, S ) such that c ◦ g = s . Uniqueness immedia tely follows from the fact that g is e pimo rphic. So it rema ins to sho w the existence of c . Since A ∈ C − , ther e exists a distinguished triangle U A [ − 1] u A − → A v A − → V A → U A with U A ∈ U , V A ∈ W . By assumption we have s ◦ f ◦ u A = 0, a nd this mea ns s ◦ f ◦ u A = 0 by Prop osition 2 .10. So s ◦ f factors thro ugh V A , and we obtain a morphism of tria ngles A B C A [1] V A S T V A [1] v A   s   t   − v A [1]   f / / i / / g / / j / / h / / / /    where V A i − → S j − → T → V A [1] is a distinguished triangle. B y the same ar gumen t as in the pr oof of Lemma 6.2 (since C ( M g , V A [1] = 0)), we have v A [1] ◦ h = 0. Th us t factors through S , namely there exists c ∈ C ( C, S ) s uc h t hat j ◦ c = t . Since GENERAL HEAR T CONSTR UCTION ON A TRIANGULA TED CA TE GOR Y (I) 17 j ◦ ( s − c ◦ g ) = j ◦ s − t ◦ g = 0, there exists s ′ ∈ C ( B , V A ) suc h that i ◦ s ′ = s − c ◦ g . V A B S C T s ′ { { w w w w w w w w g / / i / / s   j / / c { { w w w w w w w w w t    Since V A ∈ W , this means s = c ◦ g .  7. Existence of enough projectives/injective s Lemma 7.1. F or any c otorsion p air ( U , V ) , the fol lowing ar e e quivalent. (1) U ⊆ V . (2) W = U . (3) C + = C . Pr o of. Le ft to the reader.  Corollary 7. 2. If a c otorsion p air ( U , V ) satisfies U ⊆ V , then we have U [ − 1] ⊆ H . Pr o of. This immediately fo llows from U [ − 1] ⊆ C − (Remark 3.6) and C = C + (Lemma 7.1).  Prop osition 7.3. L et ( U , V ) b e a c otorsion p air satisfyi ng U ⊆ V . If an obje ct P ∈ H lies in the image of U [ − 1] ( i. e. P ∈ U [ − 1 ] / W ) , then P is pr oje ctive in H . Pr o of. Le t B and C b e an y ob jects in H and let p ∈ C ( P, C ) b e a ny morphism. Let g ∈ C ( B , C ) b e any morphism whic h is epimorphic in H , a nd take a distin- guished triangle A f − → B g − → C h − → A [1] . By P r opos ition 6.1 and Le mma 7.1, g is epimorphic in C . B y F act 2.9, this is equiv alen t to h = 0. Thus h fa c tors thro ugh some W ∈ W . Since C ( P, W ) ⊆ Ext 1 ( U , V ) = 0, w e ha ve h ◦ p = 0, and p factors through B a s desir ed.  Corollary 7.4. If a c otorsion p air ( U , V ) satisfies U ⊆ V , then its he art H has enough pr oje ctives. Pr o of. By definition, for an y C ∈ H , there exists a distinguished triangle U [ − 1] u − → C v − → V → U with U ∈ U , V ∈ V . Since v = 0 by C ( C, V ) = 0 (Lemma 2.11), u is epimorphic in C , and th us in H . Th us Co r ollary 7.4 follo ws from Cor ollary 7.2 and P ropo sition 7.3.  Dually , we hav e the following. Corollary 7.5. If a c otorsion p air ( U , V ) satisfies V ⊆ U , then i ts he art H has enough inje ctives. 18 HIR OYUKI NAKAOKA 8. Example arising from an n -cluster til ting subca tegor y Definition 8.1. ( §§ 5.1 in [KR]) A full additiv e thick subcatego ry T ⊆ C is an n - cluster tilting s ub c ate gory if it satis fie s the following. (1) T is functorially finite. (2) An o b ject C ∈ C belongs to T if and only if E xt ℓ ( C, T ) = 0 (0 < ∀ ℓ < n ) (3) An o b ject C ∈ C belongs to T if and only if E xt ℓ ( T , C ) = 0 (0 < ∀ ℓ < n ) Definition 8 .2. Let T be an n + 1-cluster tilting subcategor y . F or any pair of int eger s i and j satisfying i ≤ j , put T [ i,j ] := { C ∈ C | Ext ℓ ( C, T ) = 0 ( j + 1 ≤ ∀ ℓ ≤ n ) } , T [ i,j ] := { C ∈ C | Ext ℓ ( T , C ) = 0 ( − j ≤ ∀ ℓ ≤ − i ) } . In pa rticular, if j − i ≥ n , then T [ i,j ] = T [ i,j ] = C . Example 8. 3. Let T b e an n + 1-cluster tilting subcateg o ry . F or any integer  satisfying 0 ≤  ≤ n − 1, put T  := T [0 , ] , T  := T [ , n − 1] . Then it can be shown that ( T  , T  ) is a cotorsion pa ir (Theorem 3 .1 in [IY]). W e can calculate its hea rt H  as follo ws. C + = T [ , n ] , C − = T [ − 1 , ] , W = T [  ] , H  = ( T [ , n ] ∩ T [ − 1 , ] ) / T [  ] . Thu s ( T [ , n ] ∩ T [ − 1 , ] ) / T [  ] is an ab elian categor y for each  ∈ [0 , n − 1], as shown in Corollary 6.4 in [IY]. If n = 1 and  = 0 , this is nothing other tha n the case of a cluster tilting sub category . Moreov er in Co rollary 6,4 in [IY], it was shown that these ( T [ , n ] ∩ T [ − 1 , ] ) / T [  ] are m utually equiv alent for  ∈ [0 , n − 1]. W e abbrevia te these equiv alent ab elian categorie s by H . Since ( T 0 , T 0 ) satisfies T 0 = T ⊆ T 0 , we see that H has enough pro jectives b y Corollar y 7 .4. Dually , since ( T n − 1 , T n − 1 ) satisfies T n − 1 = T [ n − 1] ⊆ T n − 1 , we see H has enough injectives b y Corollary 7.5. References [BBD] Be ˘ ılinson, A. A.; Bernstein, J.; Deligne, P .: F aisc e aux p ervers (F rench) [Perv erse sheav es] Analysis and top ology on si ngular spaces, I (Luminy , 1981), 5– 171, Ast ´ erisque, 100 , Soc. Math. F rance, Paris, 1982. [BR] Beligiannis, A.; Reiten, I.: Homolo gic al and homotopic al asp e c ts of torsion the ories (English summary), M em. Amer. Math. So c. 188 (2007), no. 883, viii+207 pp. [B] Borceux, F.: H and b o ok of c ate goric al algebra 1, Basic c ate gory the ory , Encyclop edia of Math- ematics and its Applications, 50 . Cambridge Universit y Press, Cambridge, 1994. xvi+345 pp. [BMR] Buan, A. B.; Mar sh, R. J.; Reiten, I.: Cluster-tilte d algebr as , T rans. Amer. Math. Soc. 359 (2007), no. 1, 323–332. [IY] Iya ma, O.; Y oshino, Y.: Mutation in triangulate d c ate gories and rigid Cohen-Mac aulay mo dules (English summ ary), Inv en t. Math. 1 72 (2008), no. 1, 117–168. [KR] Keller, B.; Reiten, I.: Cluster-tilted algebr as ar e Gor enste in and stably Calabi-Y au , Adv. Math. 211 (2007), no. 1, 123–151. [KZ] Ko enig, S.; Zh u, B.: F r om triangulate d ca te gories to ab elian c ate gories: cluster tilting in a gener al fr amework (English summary), Math. Z. 25 8 (2008), no. 1, 143–160. GENERAL HEAR T CONSTR UCTION ON A TRIANGULA TED CA TE GOR Y (I) 19 Gradua te School of Mat hematic al Sciences, The University of Tokyo 3-8-1 Komaba, Meguro, Tokyo, 153 -8914 Jap an E-mail addr ess : deutsch e@ms.u-tokyo. ac.jp