General heart construction for twin torsion pairs on triangulated categories
aa r X i v : . [ m a t h . C T ] N ov GENERAL HEART CONSTRUCTION FOR TWIN TORSIONPAIRS ON TRIANGULATED CATEGORIES
HIROYUKI NAKAOKA
Abstract.
In our previous article, we constructed an abelian category fromany torsion pair on a triangulated category. This generalizes the heart of a t -structure and the ideal quotient by a cluster tilting subcategory. Recently,generalizing the quotient by a cluster tilting subcategory, Buan and Marshshowed that an integral preabelian category can be constructed as a quotient,from a rigid object in a triangulated category with some conditions. In thisarticle, by considering a pair of torsion pairs, we make a simultaneous genral-ization of these two constructions. Introduction and preliminaries
For any category K , we write abbreviately K ∈ K , to indicate that K is an objectof K . For any K, L ∈ K , let K ( K, L ) denote the set of morphisms from K to L .If M , N are full subcategories of K , then K ( M , N ) = 0 means that K ( M, N ) = 0for any M ∈ M and N ∈ N . For each K ∈ Ob( K ), similarly K ( M , K ) = 0 meansthat K ( M, K ) = 0 for any M ∈ M . We denote the full subcategory of K consistingof those K ∈ K satisfying K ( M , K ) = 0 by M ⊥ . Dually, K ( K, N ) = 0 means K ( K, N ) = 0 for any N ∈ N , and these form a full subcategory ⊥ N ⊆ K . If K isadditive and N ⊆ K is a full additive subcategory, then K / N is defined to be theideal quotient of K by N . Namely, K / N is an additive category defined by- Ob( K / N ) = Ob( K ),- For any X, Y ∈ K ,( K / N )( X, Y ) = K ( X, Y ) / { f ∈ K ( X, Y ) | f factors through some N ∈ N } . Throughout this article, we fix a triangulated category C . Any subcategory of C is a full, additive subcategory closed under isomorphisms and direct summands.For any object T ∈ C , add( T ) denotes the full subcategory of C consisting ofdirect summands of finite direct sums of T . When M , N are subcategories of C and C ∈ C , then the abbreviations Ext ( M , N ) = 0 and Ext ( M , C ) = 0and Ext ( C, N ) = 0 are defined similarly as above. For any pair of subcategories M , N ⊆ C , we define M ∗ N ⊆ C as the full subcategory consisting of those C ∈ C admitting some distinguished triangle M → C → N → M [1]with M ∈ M and N ∈ N .By definition [IY], a torsion pair ( X , Y ) on C is a pair of (full additive) subcat-egories X , Y ⊆ C satisfying(i) C ( X , Y ) = 0,(ii) C = X ∗ Y . Supported by JSPS Grant-in-Aid for Young Scientists (B) 22740005.
Previously in [N], we showed that if we are given a torsion pair ( X , Y ) on C , then(1.1) ( ( X ∗ W ) ∩ ( W ∗ Y [1]) ) / W becomes an abelian category, where W = ( X [1] ∩ Y ). This generalizes the followingtwo constructions.(1) The heart of a t -structure . A t -structure is nothing other than a torsionpair ( X , Y ) satisfying X [1] ⊆ X . In this case, (1 .
1) agrees with the heart[BBD].(2)
Ideal quotient by a cluster tilting subcategory . A full additive sub-category
T ⊆ C is a cluster tilting subcategory if and only if ( T [ − , T ) isa torsion pair on C . In this case, (1 .
1) agrees with the ideal quotient of C by T , which was shown to become abelian in [KZ] (originally in [BMR], or[KR] in 2-CY case).Recently, generalizing the second case of a cluster tilting subcategory, Buan andMarsh showed that if T is a rigid (i.e. Ext ( T, T ) = 0) object in a Hom-finiteKrull-Schmidt triangulated category C (over a field k ) with a Serre functor, then(1.2) C / X T (where X T = (add( T )) ⊥ )becomes an integral preabelian category. (In the notation of [BM], X T is writtenas X T = (add( T )) ⊥ [1]. This is only due to the difference in the definition of M ⊥ .In [BM], for any M ⊆ C , M ⊥ is defined to be the full subcategory of C consistingof those C ∈ C satisfying Ext ( M , C )=0. )As in [R] (and as quoted in [BM]), an additive category A is preabelian if anymorphism in A has a kernel and a cokernel. A preabelian category is left semi-abelian if and only if for any pullback diagram(1.3) A BC D (cid:3) α / / β (cid:15) (cid:15) γ (cid:15) (cid:15) δ / / in A , “ δ is a cokernel morphism” implies “ α is epimorphic” [R]. A right semi-abelian category is characterized dually, using pushout diagrams. A semi-abelian category is defined to be a preabelian category which is both left semi-abelian andright semi-abelian.A preabelian category A is left integral if for any pullback diagram (1 . δ isepimorphic” implies “ α is epimorphic”. A right integral category is defined dually,using pushout diagrams. An integral category is defined to to be a preabelian cate-gory which is both left integral and right integral. Thus a preabelian category A issemi-abelian whenever it is integral. Moreover if A is integral, then the localizationof A by regular (= epimorphic and monomorphic) morphisms are shown to becomeabelian. In [BM], using this fact, Buan and Marsh made an abelian category outof their integral preabelian category C / X T .In this article, we generalize simultaneously Buan and Marsh’s C / X T and our H , using a pair of torsion pairs. Starting from torsion pairs, we need no assumptionon C , except it is triangulated. ENERAL HEART CONSTRUCTION FOR TWIN TORSION PAIRS 3 Definition and basic properties
As before, C is a fixed triangulated category. Any subcategory of C is assumedto be full, additive, closed under isomorphisms and direct summands. Definition 2.1.
Let U and V be full additive subcategories of C . We call ( U , V ) a cotorsion pair if it satisfies(i) Ext ( U , V ) = 0,(ii) C = U ∗ V [1].
Remark . ( U , V ) is a cotorsion pair if and only if ( U [ − , V ) is a torsion pair in[IY]. (Unlike [BR], it does not require the closedness under shifts.) In this sense, acotorsion pair is essentially the same as a torsion pair. However we prefer cotorsionpairs, for the sake of duality in shifts. Remark . For any cotorsion pair ( U , V ) on C , we have U = ⊥ ( V [1]) and V =( U [ − ⊥ .Cotorsion pairs generalize t -structures and cluster tilting subcategories, as fol-lows. Example 2.4. (cf. Definition 2.6 in [ZZ], Proposition 2.6 in [N])(1) A t - structure is a pair of subcategories ( X , Y ) where ( U , V ) = ( X [1] , Y ) isa cotorsion pair satisfying U [1] ⊆ U . This is also equivalent to V [ − ⊆ V .(2) A co - t - structure is a pair of subcategories ( X , Y ) where ( U , V ) = ( X [1] , Y )is a cotorsion pair satisfying U [ − ⊆ U . This is also equivalent to V [1] ⊆ V .(3) A cotorsion pair ( U , V ) is called rigid if Ext ( U , U ) = 0. This is also equiv-alent to U ⊆ V .(4) A subcategory
T ⊆ C is a cluster tilting subcategory if and only if ( T , T )is a cotorsion pair. Remark . Using a result in [AN], we can characterize a co- t -structure by thevanishing of an abelian category H defined as below. In fact, a cotorsion pair( U , V ) becomes a co- t -structure if and only if it satisfies H = 0.In [N], we showed the following. Theorem 2.6. ( Theorem 6.4 in [N])
Let ( U , V ) be a cotorsion pair on C . If wedefine full subcategories of C by W = U ∩ V , C − = U [ − ∗ W , C + = W ∗ V [1] , H = C + ∩ C − , then the ideal quotient H / W becomes abelian. In this article, we generalize this to the case of pairs of cotorsion pairs. We workon a pair of cotorsion pairs ( S , T ) , ( U , V ) on C satisfying Ext ( S , V ) = 0. Since a“pair of pairs” is a bit confusing, we use the following terminology. Definition 2.7.
A pair of cotorsion pairs ( S , T ) , ( U , V ) on C is called a twincotorsion pair if it satisfies(2.1) Ext ( S , V ) = 0 . Remark that this condition is equal to
S ⊆ U , and also to
T ⊇ V . HIROYUKI NAKAOKA
Definition 2.8.
Let ( S , T ) , ( U , V ) be a twin cotorsion pair on C . We define sub-categories of C by W = T ∩ U , C − = S [ − ∗ W , C + = W ∗ V [1] , H = C + ∩ C − . Each of C , C + , C − , H contains W as a subcategory. We denote their ideal quotientsby W by C , C + , C − , H . Thus we obtain a sequence of full additive subcategories H ⊆ C + ⊆ C , H ⊆ C − ⊆ C . For any morphism f ∈ C ( A, B ), we denote its image in C ( A, B ) by f . Remark . Since W is closed under direct summands, for any C ∈ C we have C = 0 in C ⇐⇒ C ∈ W . Example 2.10. (1) A single cotorsion pair can be regarded as a degenerated case of a twincotorsion pair. A twin cotorsion pair ( S , T ) , ( U , V ) is a single cotorsionpair (namely, ( S , T ) = ( U , V )) if and only if S = U if and only if T = V .In this case, (since W , C + , C − and H agrees with those in Theorem 2.6,) H becomes abelian as in Theorem 2.6.(2) Another extremal case is when T = U . Remark that in this case, S ⊆ T and
U ⊇ V holds. In particular, ( S , T ) is rigid.As shown in [BM], if T is a rigid object in a Hom-finite Krull-Schmidttriangulated category C (over a field k ) with a Serre functor, then ( S , T ) =(add( T )[1] , X T ) and ( U , V ) = ( X T , ( X T ) ⊥ [ − T is rigid, this pair satisfies add( T )[1] ⊆ X T . ThusExt (add( T )[1] , ( X T ) ⊥ [ − T )[1] , X T ) , ( X T , ( X T ) ⊥ [ − H = C / X T , and it was shown in [BM] that thiscategory becomes integral preabelian. (Remark that when T = U , generallywe have W = T = U and C + = C − = C .) Remark . A similar situation to (2) in Example 2.10 appears in [BR] as a TTF-triple. A
TTF - triple on C is a triplet ( X , Y , Z ) of subcategories of C , in whichboth ( X , Y ) and ( Y , Z ) are t -structures. Lemma . (1) If U [ − → A f −→ B → U is a distinguished triangle in C satisfying U ∈ U , then A ∈ C − implies B ∈ C − . (2) If S [ − → A f −→ B → S is a distinguished triangle in C satisfying S ∈ S ,then B ∈ C − implies A ∈ C − .Proof. (1) Take distinguished triangles S A [ − s A −→ A w A −→ W A → S A ( S A ∈ S , W A ∈ W ) ,S B [ − s B −→ B t B −→ T B → S B ( S B ∈ S , T B ∈ T ) . Since Ext ( S A , T B ) = 0, f induces a morphism of triangles S A [ − A W A S A S B [ − B T B S B . s A / / w A / / / / (cid:15) (cid:15) f (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) s B / / t B / / / / (cid:8) (cid:8) (cid:8) ENERAL HEART CONSTRUCTION FOR TWIN TORSION PAIRS 5
It suffices to show T B ∈ U .Take any V † ∈ V , and any v ∈ C ( T B , V † [1]). Since Ext ( W A , V † ) = 0, we have v ◦ t B ◦ f = 0. Applying this to the given distinguished triangle, we see that v ◦ t B factors through U , U [ − A B UT B V † [1] / / f / / / / t B (cid:15) (cid:15) (cid:7) (cid:7) (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) v (cid:15) (cid:15) (cid:8) and v ◦ t B = 0 follows from Ext ( U, V † ) = 0. Thus v factors through S B , S B [ − B T B S B V † [1] / / t B / / / / v (cid:15) (cid:15) (cid:0) (cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:8) which means v = 0, since Ext ( S B , V † ) = 0.(2) Take distinguished triangles S B [ − s B −→ B w B −→ W B → S B ( S B ∈ S , W B ∈ W ) ,X → A w B ◦ f −→ W B → X [1] . Then by the octahedral axiom, S [ − → X → S B [ − → S is also a distinguished triangle. This implies X ∈ S [ − A ∈ C − . S [ − X S B [ − A BW B (cid:8)(cid:8) (cid:8) . . ]]]]]]]] . . ]]]]] $ $ JJJJJJJJ (cid:11) (cid:11) (cid:24)(cid:24)(cid:24)(cid:24) f & & MMM (cid:12) (cid:12) (cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24) (cid:3) (cid:3) (cid:7)(cid:7)(cid:7)(cid:7)(cid:7)(cid:7)(cid:7) w B (cid:1) (cid:1) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) (cid:3) Dually, the following holds.
Lemma . (1) If T → A f −→ B → T [1] is a distinguished triangle in C satisfying T ∈ T ,then B ∈ C + implies A ∈ C + . (2) If V → A f −→ B → V [1] is a distinguished triangle in C satisfying V ∈ V ,then A ∈ C + implies B ∈ C + . The following Lemma is trivial.
HIROYUKI NAKAOKA
Lemma . Let S X [ − s X −→ X t X −→ T X → S X be a distinguished triangle, with S X ∈ S and T X ∈ T . If a morphism x ∈ C ( X, Y ) factors through some T ∈ T ,then x factors through T X .Similar statement also holds for a distinguished triangle U X [ − u X −→ X v X −→ U X → V X ( U X ∈ U , V X ∈ V ) and a morphism x ∈ C ( X, Y ) factoring some V ∈ V .Proof. If x factors through T ∈ T , then x ◦ s X = 0 follows from Ext ( S X , T ) = 0.Thus it factors through t X . Similarly for the latter part. (cid:3) Remark . The dual of Lemma 2.14 also holds.3.
Adjoints
In the following, we fix a twin cotorsion pair ( S , T ) , ( U , V ) on C . Since thisassumption is self-dual, we often omit proofs of dual statements. Definition 3.1.
For any C ∈ C , define K C ∈ C and k C ∈ C ( K C , C ) as follows:1. Take a distinguished triangle S [ − → C a −→ T → S ( S ∈ S , T ∈ T )2. then, take a distinguished triangle U → T b −→ V [1] → U [1] ( U ∈ U , V ∈ V )3. and then, take a distinguished triangle V → K C k C −→ C b ◦ a −→ V [1] . By the octahedral axiom, S [ − → K C k C −→ C → S is also a distinguished triangle. S [ − K C UC TV [1] (cid:8)(cid:8) (cid:8) . . ]]]]]]] . . ]]]]]] $ $ JJJJJJJJ k C (cid:11) (cid:11) (cid:24)(cid:24)(cid:24)(cid:24) a & & MMM (cid:12) (cid:12) (cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24) (cid:3) (cid:3) (cid:7)(cid:7)(cid:7)(cid:7)(cid:7)(cid:7)(cid:7)(cid:7) b (cid:1) (cid:1) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) Claim 3.2. (1)
For any C ∈ C , we have K C ∈ C − . (2) If C ∈ C + , then K C ∈ H .Proof. We use the notation in Definition 3.1.(1) Remark that we have a distinguished triangle V → U → T b −→ V [1] . Since
T ⊇ V ∋ V and T is closed under extensions, it follows U ∈ U ∩ T = W .Since S [ − → K C → U → S is a distinguished triangle, we obtain K C ∈ C − .(2) Since V → K C → C → V [1] is a distinguished triangle, this immediatelyfollows from Lemma 2.13. (cid:3) Dually, we define as follows.
ENERAL HEART CONSTRUCTION FOR TWIN TORSION PAIRS 7
Definition 3.3.
For any C ∈ C , define Z C ∈ C and z C ∈ C ( C, Z C ) as follows:1. Take a distinguished triangle V → U a −→ C → V [1] ( U ∈ U , V ∈ V )2. then, take a distinguished triangle T [ − → S [ − b −→ U → T ( S ∈ S , T ∈ T )3. and then, take a distinguished triangle S [ − a ◦ b −→ C z C −→ Z C → S. By the octahedral axiom, V → T → Z C → V [1] is also a distinguished triangle. S [ − UT C Z C V [1] (cid:8)(cid:8) (cid:8) b (cid:16) (cid:16) !!!!!!! (cid:16) (cid:16) """"""" (cid:29) (cid:29) ::::::: a hhhh z C (cid:29) (cid:29) ;; iiiiiiiii ; ; xxxxxxxx ? ? ~~~~~~~ Claim 3.4. (1)
For any C ∈ C , we have Z C ∈ C + . (2) If C ∈ C − , then Z C ∈ H .Proof. This is the dual of Claim 3.4. (cid:3)
Proposition 3.5.
For any C ∈ C , let K C k C −→ C be as in Definition 3.1. Then forany X ∈ C − , k C ◦ − : C − ( X, K C ) → C ( X, C ) is bijective.Proof. Take a distinguished triangle S X [ − s X −→ X w X −→ W X → S X ( S X ∈ S , W X ∈ W ) . First we show the injectivity. Suppose x ∈ C ( X, K C ) satisfies k C ◦ x = 0. Bydefinition, this means that k C ◦ x factors through some object in W . Thus byLemma 2.14, k C ◦ x factors through w X . This gives a morphism of triangles S X [ − X W X S X V K C C V [1] . s X / / w X / / / / (cid:15) (cid:15) x (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / k C / / / / (cid:8) (cid:8) (cid:8) Since Ext ( S X , V ) = 0, it follows x ◦ S X = 0, and thus x factors through W X , S X [ − X W X K Cs X / / w X / / + + x (cid:20) (cid:20) ))))) (cid:0) (cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:8) (cid:8) HIROYUKI NAKAOKA which means x = 0.Second, we show the surjectivity. Take any y ∈ C ( X, C ). In the notation ofDefinition 3.1, S [ − K C UC TV [1] (cid:8)(cid:8) (cid:8) . . ]]]]]]] . . ]]]]]] $ $ JJJJJJJJ k C (cid:11) (cid:11) (cid:24)(cid:24)(cid:24)(cid:24) a & & LLL b ◦ a (cid:12) (cid:12) (cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24) (cid:3) (cid:3) (cid:7)(cid:7)(cid:7)(cid:7)(cid:7)(cid:7)(cid:7)(cid:7) b (cid:1) (cid:1) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) since Ext ( S X , T ) = 0, y induces a morphism of triangles S X [ − X W X S X S [ − C T S. s X / / w X / / / / (cid:15) (cid:15) y (cid:15) (cid:15) ∃ t (cid:15) (cid:15) (cid:15) (cid:15) / / a / / / / (cid:8) (cid:8) (cid:8) Since Ext ( W X , V ) = 0, we obtain b ◦ a ◦ y = b ◦ t ◦ w X = 0 ◦ w X = 0 . Thus y factors through k C . V K C C V [1] X / / k C / / b ◦ a / / (cid:6) (cid:6) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) (cid:28) y (cid:28) (cid:28) :::::: (cid:8) (cid:8) (cid:3) Dually, we have the following.
Proposition 3.6.
For any C ∈ C , let C z C −→ Z C be as in Definition 3.3. Then,for any Y ∈ C + , − ◦ z C : C + ( Z C , Y ) → C ( C, Y ) is bijective. In the terminology of [B], Proposition 3.6 means that for any C ∈ C , Z C z C −→ C gives a reflection of C along the inclusion functor C + ֒ → C . As a corollary, weobtain the following. Corollary 3.7. ( Proposition 3.1.2 and Proposition 3.1.3 in [B])
The inclusionfunctor i + : C + ֒ → C has a left adjoint τ + : C → C + . If we denote the adjunctionby η : Id C = ⇒ i + ◦ τ + , then there exists a natural isomorphism Z C ∼ = τ + ( C ) in C ,compatible with z C and η C . C Z C τ + ( C ) z C / / η C (cid:25) (cid:25) ∼ = (cid:5) (cid:5) (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11) (cid:8) In particular, Z C is determined up to an isomorphism in C . ENERAL HEART CONSTRUCTION FOR TWIN TORSION PAIRS 9
Dually the following holds.
Corollary 3.8.
The inclusion functor i − : C − ֒ → C has a right adjoint τ − : C → C − . For any C ∈ C , there is a natural isomorphism K C ∼ = τ − ( C ) in C − . Lemma 3.9.
For any C ∈ C , the following are equivalent. (1) τ + ( C ) = 0 . (2) Z C ∈ W (3) C ∈ U .Proof. The equivalence between (1) and (2) follows from Remark 2.9 and Corollary3.7.Suppose (2) holds. Then, in the notation of the following diagram,(3.1) S [ − UT C Z C V [1] (cid:8)(cid:8) (cid:8) b (cid:16) (cid:16) !!!!!!! (cid:16) (cid:16) """"""" (cid:29) (cid:29) ::::::: a hhhh z C (cid:29) (cid:29) ;; c iiiiiiiii e ; ; xxxxxxxx d ? ? ~~~~~~~ we have d = 0 since Ext ( Z C , V ) = 0. Thus it follows c = 0 and C becomes a directsummand of U , which means C ∈ U .Conversely, suppose (3) holds. Then in the notation in (3 . a and e are isomorphisms. Since T ∈ W , we obtain Z C ∈ W . (cid:3) Dually we have the following.
Lemma 3.10.
For any C ∈ C , the following are equivalent. (1) τ − ( C ) = 0 . (2) K C ∈ W (3) C ∈ T . By the same argument as in [AN], we can also show the following. Since we donot use it in this article, we only introduce the results.
Proposition 3.11.
The inclusion functor i U : U = U / W ֒ → C has a right adjoint σ U : C → U . For any C ∈ C , σ U ( C ) is naturally isomorphic to U C ∈ U appearingin a distinguished triangle V C → U C u C −→ C v C −→ V C [1] ( U C ∈ U , V C ∈ V ) . Moreover for any C ∈ C , the following are equivalent. (1) σ U ( C ) = 0 . (2) U C ∈ W . (3) C ∈ C + . Dually, the inclusion functor i T : T = T / W ֒ → C admits a left adjoint σ T : C →T . We have a natural isomorphism σ T ( C ) ∼ = T C in C for any C ∈ C , and σ T ( C ) = 0 ⇐⇒ T C ∈ W ⇐⇒ C ∈ C − , where S C [ − → C → T C → S C is a distinguished triangle with S C ∈ S , T C ∈ T . H is preabelian Now we construct the (co-)kernel of a morphism in H . Definition 4.1.
For any A ∈ C − , B ∈ C and f ∈ C ( A, B ), define M f ∈ C and m f ∈ C ( B, M f ) as follows.1. Take a distinguished triangle S A [ − s A −→ A w A −→ W A → S A ,
2. then, take a distinguished triangle S A [ − f ◦ s A −→ B m f −→ M f → S A .S A [ − AW A B M fs A (cid:15) (cid:15) w A (cid:15) (cid:15) $ $ HHHHHH f / / m f % % KKKKKKK (cid:8)
Proposition 4.2.
For any A ∈ C − , B ∈ C and f ∈ C ( A, B ) , let B m f −→ M f be asin Definition 4.1. Then, we have the following. (1) m f ◦ f = 0 . (2) m f induces a bijection − ◦ m f : C ( M f , Y ) ∼ = −→ { β ∈ C ( B, Y ) | β ◦ f = 0 } for any Y ∈ C + . (3) If B ∈ C − , then M f ∈ C − .Proof. (1) is trivial, since there is a morphism of triangles S A [ − A W A S A S A [ − B M f S A . f (cid:15) (cid:15) (cid:15) (cid:15) s A / / / / w A / / m f / / / / / / (cid:8) (cid:8) (cid:8) (3) follows from Lemma 2.12.We show (2). Take a distinguished triangle V Y → W Y w Y −→ Y v Y −→ V Y [1] ( V Y ∈ V , W Y ∈ W ) . To show the injectivity, suppose x ∈ C ( M f , Y ) satisfies x ◦ m f = 0. By definition x ◦ m f factors through some object in W . By (the dual of) Lemma 2.14, x ◦ m f factors through w Y , and thus we obtain a morphism of triangles S A [ − B M f S A V Y W Y Y V Y [1] . f ◦ s A / / m f / / / / (cid:15) (cid:15) (cid:15) (cid:15) x (cid:15) (cid:15) (cid:15) (cid:15) / / w Y / / v Y / / (cid:8) (cid:8) (cid:8) Since Ext ( S A , V Y ) = 0, x factors through W Y , which means x = 0. ENERAL HEART CONSTRUCTION FOR TWIN TORSION PAIRS 11
To show the surjectivity, suppose y ∈ C ( B, Y ) satisfies y ◦ f = 0. By the sameargument as above, we see that y ◦ f factors W Y . This implies y ◦ f ◦ s A = 0, sinceExt ( S A , W Y ) = 0. Thus y factors m f . S A [ − B M f S A Y f ◦ s A / / m f / / . . / / y (cid:27) (cid:27) (cid:4) (cid:4) (cid:10)(cid:10)(cid:10)(cid:10)(cid:10) (cid:8)(cid:8) (cid:3) Dually, we have the following:
Remark . For any A ∈ C , B ∈ C + and any f ∈ C ( A, B ), take a diagram V B [1] BW B AL f v B (cid:15) (cid:15) w B (cid:15) (cid:15) v B ◦ f % % KKKKKKK f / / ℓ f % % KKKKKKKK (cid:8) where V B → W B w B −→ B v B −→ V B [1] V B → L f ℓ f −→ A v B ◦ f −→ V B [1]are distinguished triangles satisfying W B ∈ W , V B ∈ V .Then, the following holds.(1) f ◦ ℓ f = 0.(2) ℓ f induces a bijection ℓ f ◦ − : C ( X, L f ) ∼ = −→ { α ∈ C ( X, A ) | f ◦ α = 0 } for any X ∈ C − .(3) If A ∈ C + , then L f ∈ C + . Corollary 4.4.
For any twin cotorsion pair, H is preabelian.Proof. First we construct a cokernel. For any
A, B ∈ H and any f ∈ C ( A, B ), let m f : B → M f be as in Definition 4.1. Since A, B ∈ C − , it follows m f ◦ f = 0 , M f ∈ C − by Proposition 4.2. By Proposition 3.6, there exists z M f : M f → Z M f which givesa reflection z M f : M f → Z M f of M f along C + ֒ → C . By Claim 3.4, Z M f satisfies Z M f ∈ H .Then z M f ◦ m f : B → Z M f gives a cokernel of f . In fact for any H ∈ H , thereis a bijection − ◦ z M f ◦ m f : C ( Z M f , H ) ∼ = −→ C ( M f , H ) ∼ = −→ { β ∈ C ( B, H ) | β ◦ f = 0 } . A B HM f Z M f FFF f / / m f FFF z Mf FF β : : vvvvvvv (cid:8) / / A kernel of f ∈ H ( A, B ) is constructed dually. Let L ℓ f −→ A be as in Remark4.3, and let K L f k Lf −→ L f be as in Definition 3.1. Then ℓ f ◦ k L f gives a kernel of f . (cid:3) Corollary 4.5.
Let f ∈ H ( A, B ) be a morphism in H . The following are equivalent. (1) f ∈ H ( A, B ) is epimorphic. (2) Z M f ∈ W . (3) M f ∈ U .Proof. (1) is equivalent to (2), since cok f ∼ = Z M f in H . Also (2) is equivalent to(3) by Lemma 3.9. (cid:3) Corollary 4.6.
Let f ∈ H ( A, B ) be a morphism in H . If a distinguished triangle A f −→ B g −→ C → A [1] admits a factorization B CU g / / b (cid:26) (cid:26) c D D (cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10) (cid:8) for some U ∈ U , then f ∈ H ( A, B ) is epimorphic.Proof. By the definition of B m f −→ M f , there is a commutative diagram made ofdistinguished triangles as follows (Definition 4.1). S A [ − AW A B M f C (cid:8)(cid:8) (cid:8) s A (cid:16) (cid:16) !!!!!!! w A (cid:16) (cid:16) """"""" (cid:29) (cid:29) ::::::: f hhhh m f (cid:29) (cid:29) ;; g iiiiiiiiii e ; ; xxxxxxx d ? ? ~~~~~~~~ By Corollary 4.5, it suffices to show M f ∈ U . Take any V † ∈ V and v ∈ C ( M f , V † [1]). Since v ◦ e = 0 by Ext ( W A , V † ) = 0, there exists v ′ ∈ C ( C, V † [1]) ENERAL HEART CONSTRUCTION FOR TWIN TORSION PAIRS 13 satisfying v ′ ◦ d = v . W A M f CV † [1] B e / / d / / v (cid:27) (cid:27) v ′ (cid:4) (cid:4) (cid:10)(cid:10)(cid:10)(cid:10)(cid:10) m f (cid:3) (cid:3) (cid:7)(cid:7)(cid:7)(cid:7)(cid:7) g (cid:26) (cid:26) (cid:8)(cid:8) Since g factors through U ∈ U , it follows v ◦ m f = v ′ ◦ d ◦ m f = v ′ ◦ g = 0. Thus v factors through S A , S A [ − B M f S A V † [1] f ◦ s A / / m f / / - - / / v (cid:27) (cid:27) (cid:4) (cid:4) (cid:10)(cid:10)(cid:10)(cid:10)(cid:10) (cid:8)(cid:8) which means v = 0, since Ext ( S A , V † ) = 0. (cid:3) Remark . Duals of Corollary 4.5 and 4.6 also hold for monomorphisms in H .5. H is semi-abelian Lemma 5.1.
Let β ∈ H ( B, C ) be any morphism. If β is a cokernel morphism,namely, if there exists a morphism f ∈ H ( A, B ) such that β = cok f , then thereexist g ∈ H ( B, C ′ ) and an isomorphism η ∈ H ( C, C ′ ) such that (i) η is compatible with β and g , B CC ′ β / / g (cid:26) (cid:26) ∼ = η (cid:4) (cid:4) (cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10) (cid:8) (ii) g admits a distinguished triangle S [ − s −→ B g −→ C ′ → S with S ∈ S .Proof. Take a morphism f ∈ H ( A, B ) such that β = cok f . As shown in Corollary4.4, cok f is given by z M f ◦ m f . S A [ − AW A B M fs A (cid:15) (cid:15) v A (cid:15) (cid:15) (cid:31) (cid:31) ???????? f / / m f ! ! CCCCCCCC (cid:8) S [ − UT M f Z M f V [1] (cid:8)(cid:8) (cid:8) (cid:16) (cid:16) !!!!!!! (cid:16) (cid:16) """"""" (cid:29) (cid:29) ::::::: hhh z Mf (cid:29) (cid:29) ;; iiiiiiii < < xxxxxxxx > > ~~~~~~~ Thus there exists an isomorphism η ∈ H ( C, Z M f ) compatible with z M f ◦ m f and β . It suffices to show g = z M f ◦ m f satisfies condition (ii). If we complete g into adistinguished triangle Q [ − → B z Mf ◦ m f −→ Z M f → Q, then by the octahedral axiom, we have a distinguished triangle S A → Q → S → S A [1] , which implies Q ∈ S . BM f S A Z M f Q S (cid:8)(cid:8) (cid:8) m f (cid:16) (cid:16) """"""" (cid:16) (cid:16) !!!!!!! (cid:29) (cid:29) ::::::: z Mf hh (cid:29) (cid:29) :: iiiiiiiii < < yyyyyyyy > > }}}}}}}} (cid:3) Lemma 5.2.
Suppose X ∈ C − , B ∈ H and x ∈ C ( X, B ) admit a distinguishedtriangle X x −→ B → U → X [1] with some U ∈ U . Then, the unique morphism ζ ∈ H ( Z X , B ) satisfying ζ ◦ z X = x ( Proposition 3.6 ) becomes epimorphic.Proof. As in Proposition 3.6 (or, dual of the proof of Proposition 3.5), we see thatthere exists b ∈ C ( Z X , B ) satisfying ζ = b and b ◦ z X = x . If we complete b into adistinguished triangle C [ − → Z X b −→ B c −→ C, then c factors through U . XZ X B CU z X (cid:15) (cid:15) x $ $ JJJJJ b : : ttttt c : : ttttt $ $ JJJJJ O O (cid:8) (cid:8) Thus Lemma 5.2 follows from Corollary 4.6. (cid:3)
Lemma 5.3.
Let (5.1)
A BC D (cid:3) α / / β (cid:15) (cid:15) γ (cid:15) (cid:15) δ / / be a pullback diagram in H . If there exist X ∈ C − , x B ∈ C ( X, B ) , x C ∈ C ( X, C ) which satisfies the following conditions, then α is epimorphic. ENERAL HEART CONSTRUCTION FOR TWIN TORSION PAIRS 15 (i)
The following diagram is commutative.
X BC D x B / / x C (cid:15) (cid:15) γ (cid:15) (cid:15) δ / / (cid:8) (ii) There exists a distinguished triangle X x B −→ B → U → X [1] with U ∈ U .Proof. Take X z X −→ Z X as in Definition 3.3. By the adjointness, there exist ζ B ∈H ( Z X , B ) and ζ C ∈ H ( Z X , C ) satisfying ζ B ◦ z X = x B , ζ C ◦ z X = x C . By Lemma 5.2, ζ B is epimorphic. From γ ◦ x B = δ ◦ x C , it follows γ ◦ ζ B = δ ◦ ζ C . X BC DZ Xx B & & x C (cid:24) (cid:24) z X (cid:31) (cid:31) ????? ζ B ζ C (cid:13) (cid:13) γ (cid:15) (cid:15) δ / / (cid:8)(cid:8)(cid:8) Since (5 .
1) is a pullback diagram in H , there exists ζ ∈ H ( Z X , A ) which satisfies α ◦ ζ = ζ B and β ◦ ζ = ζ C . Z X A BC D (cid:3) α / / β (cid:15) (cid:15) γ (cid:15) (cid:15) δ / / ζ (cid:31) (cid:31) ??? ζ B " " ζ C (cid:29) (cid:29) (cid:8) (cid:8) Since ζ B is epimorphic, α is also an epimorphism. (cid:3) Theorem 5.4.
For any twin cotorsion pair, H is semi-abelian.Proof. By duality, we only show H is left semi-abelian. Assume we are given apullback diagram(5.2) A BC D (cid:3) α / / β (cid:15) (cid:15) γ (cid:15) (cid:15) δ / / in H , where δ is a cokernel morphism. It suffices to show α becomes epimorphic.By Lemma 5.1, replacing D by an isomorphic one if necessary, we may assumethere exists d ∈ H ( C, D ) satisfying δ = d , which admits a distinguished triangle S [ − → C d −→ D s −→ S with S ∈ S . If we take c ∈ H ( B, D ) satisfying γ = c , and complete s ◦ c into adistinguished triangle S [ − → X x B −→ B s ◦ c −→ S, then c ◦ x B factors through d . In fact, there exists x C ∈ C ( X, C ) which gives amorphism of triangles as follows. S [ − X B SS [ − C D S / / x B / / s ◦ c / / x C (cid:15) (cid:15) c (cid:15) (cid:15) / / d / / s / / (cid:8) (cid:8) (cid:8) By Lemma 2.12, we have X ∈ C − . Thus α becomes epimorphic by Lemma 5.3. (cid:3) The case where H becomes integral In the rest, additionally we assume that ( S , T ) , ( U , V ) satisfies(6.1) U ⊆ S ∗ T or T ⊆ U ∗ V . This condition is satisfied, for example in the following cases.
Example 6.1.
A twin cotorsion pair ( S , T ) , ( U , V ) satisfies (6 .
1) in the followingcases.(1) U = S . Namely, ( S , T ) = ( U , V ) is a single cotorsion pair.(2) U = T . For example, Buan and Marsh’s triplet (add( T )[1] , X T , ( X T ) ⊥ [ − S , T ) is a co- t -structure. In this case, S ∗ T = C .(4) ( U , V ) is a co- t -structure. In this case, U ∗ V = C .Remark that the following holds. Fact 6.2. ([R]) A semi-abelian category A is left integral if and only if A is rightintegral. Theorem 6.3.
If a twin cotorsion pair ( S , T ) , ( U , V ) satisfies (6 . , then H becomesintegral.Proof. By duality, it suffices to show that
U ⊆ S ∗ T implies left integrality.Let b ∈ H ( B, D ) and c ∈ H ( C, D ) be morphisms satisfying β = b and δ = d .Since δ is epimorphic, if we take D m d −→ M d as in Definition 4.1, then M d ∈ U byCorollary 4.5. S C [ − CW C D M ds C (cid:15) (cid:15) w C (cid:15) (cid:15) $ $ HHHHHH d / / m d % % KKKKKKK (cid:8)
By assumption
U ⊆ S ∗ T , there exists a distinguished triangle S s −→ M d t −→ T → S [1]with S ∈ S , T ∈ T .If we take a distinguished triangle S B [ − s B −→ B w B −→ W B → S B ( S B ∈ S , W B ∈ W ) , ENERAL HEART CONSTRUCTION FOR TWIN TORSION PAIRS 17 then by Ext ( S B , T ) = 0, m d ◦ c ◦ s B factors through s . Namely, there exists g ∈ C ( S B [ − , S ) which makes the following diagram commutative. S B [ − S BD M dg / / s B (cid:15) (cid:15) c (cid:15) (cid:15) m d / / s (cid:15) (cid:15) (cid:8) If we complete g into a distinguished triangle S [ − → X f B −→ S B [ − g −→ S , then X ∈ S [ − ⊆ C − . Moreover there exists f C ∈ C ( X, S C [ − d ◦ s C ◦ f C = c ◦ s B ◦ f B . S [ − X S B [ − S M d [ − S C [ − D M d / / f B / / g / / (cid:15) (cid:15) f C (cid:15) (cid:15) c ◦ s B (cid:15) (cid:15) s B (cid:15) (cid:15) / / d ◦ s C / / m d / / (cid:8) (cid:8) (cid:8) Thus we have a commutative diagram
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