Recollements of extriangulated categories
aa r X i v : . [ m a t h . R T ] D ec RECOLLEMENTS OF EXTRIANGULATED CATEGORIES
LI WANG, JIAQUN WEI, HAICHENG ZHANG
Abstract.
We give a simultaneous generalization of recollements of abelian categoriesand triangulated categories, which we call recollements of extriangulated categories. Fora recollement ( A , B , C ) of extriangulated categories, we show that cotorsion pairs in A and C induce cotorsion pairs in B under certain conditions. As an application, our mainresult recovers a result given by Chen for recollements of triangulated categories, and italso shows a new phenomena when it is applied to abelian categories. Introduction
Abelian categories and triangulated categories are two fundamental structures in alge-bra and geometry. Recollements of triangulated categories were introduced by Be˘ılinson,Bernstein and Deligne [3] in connection with derived categories of sheaves on topologicalspaces with the idea that one triangulated category may be “glued together” from twoothers. The recollements of abelian categories first appeared in the construction of thecategory of perverse sheaves on a singular space in [3]. Recollements of abelian categoriesand triangulated categories play an important role in algebraic geometry and representa-tion theory, see for instance [10], [1], [9]. In recollements of abelian categories, the relationswith tilting modules and torsion pairs have been studied in [6] and [7], respectively. Chen[5] studied the relationship of cotorsion pairs among three triangulated categories in arecollement of triangulated categories.Recently, Nakaoka and Palu [8] introduced an extriangulated category which is extract-ing properties on triangulated categories and exact categories. Recollements of abeliancategories and triangulated categories are closely related, and they possess similar proper-ties in many aspects. This inspires us to give a simultaneous generalization of recollementsof abelian categories and triangulated categories, which we call recollements of extrian-gulated categories. Then we study the relationship of cotorsion pairs in a recollementof extriangulated categories. In order to achieve this goal, we need to consider the WICCondition (cf. [8, Condition 5.8]), introduce compatible morphisms, and then define leftexact sequences, right exact sequences, left exact functors, and right exact functors inextriangulated categories.
Mathematics Subject Classification.
Key words and phrases.
Extriangulated category; Recollement, Cotorsion pair.
The paper is organized as follows: we summarize some basic definitions and propertiesof extriangulated categories and exact functors in Section 2. In Section 3, we introducethe recollement of extriangulated categories and give some basic properties. Section 4 isdevoted to giving conditions such that the glued pair with respect to cotorsion pairs in A and C is a cotorsion pair in B for a recollement ( A , B , C ) of extriangulated categories.Moreover, we show that the conserve also holds for some special cotorsion pairs in B . Asan application, we recover the corresponding results in the triangulated category case.1.1. Conventions and notation.
For an additive category C , its subcategories areassumed to be full and closed under isomorphisms. A subcategory D of C is said to be contravariantly finite in C if for each object M ∈ C , there exists a morphism f : X → M with X ∈ D such that C ( D , f ) is an epimorphism. Dually, one defines covariantly finite subcategories in C . Given an object M ∈ C , we denote by add M the additive closure of M , that is, the full subcategory of C whose objects are the direct sums of direct summandsof M . Let Q be a finite acyclic quiver, we denote by S i the one-dimensional simple (left) kQ -module associated to the vertex i of Q , and denote by P i and I i the projective coverand injective envelop of S i , respectively.2. Preliminaries
Extriangulated categories.
Let us recall some notions concerning extriangulatedcategories from [8].Let C be an additive category and let E : C op × C → Ab be a biadditive functor. Forany pair of objects A , C ∈ C , an element δ ∈ E ( C, A ) is called an E -extension . Thezero element 0 ∈ E ( C, A ) is called the split E -extension . For any morphism a ∈ C ( A, A ′ )and c ∈ C ( C ′ , C ), we have E ( C, a )( δ ) ∈ E ( C, A ′ ) and E ( c, A )( δ ) ∈ E ( C ′ , A ) . We simplydenote them by a ∗ δ and c ∗ δ , respectively. A morphism ( a, c ): δ → δ ′ of E -extensions is apair of morphisms a ∈ C ( A, A ′ ) and c ∈ C ( C, C ′ ) satisfying the equality a ∗ δ = c ∗ δ ′ .By Yoneda’s lemma, any E -extension δ ∈ E ( C, A ) induces natural transformations δ ♯ : C ( − , C ) → E ( − , A ) and δ ♯ : C ( A, − ) → E ( C, − ) . For any X ∈ C , these ( δ ♯ ) X and ( δ ♯ ) X are defined by ( δ ♯ ) X : C ( X, C ) → E ( X, A ) , f f ∗ δ and ( δ ♯ ) X : C ( A, X ) → E ( C, X ) , g g ∗ δ .Two sequences of morphisms A x −→ B y −→ C and A x ′ −→ B ′ y ′ −→ C in C are said to be equivalent if there exists an isomorphism b ∈ C ( B, B ′ ) such that the following diagram A x / / B b ≃ (cid:15) (cid:15) y / / CA x ′ / / B ′ y ′ / / C ECOLLEMENTS OF EXTRIANGULATED CATEGORIES 3 is commutative. We denote the equivalence class of A x −→ B y −→ C by [ A x −→ B y −→ C ].In addition, for any A, C ∈ C , we denote as0 = [ A ( ) −→ A ⊕ C (0 1) −→ C ] . For any two equivalence classes [ A x −→ B y −→ C ] and [ A ′ x ′ −→ B ′ y ′ −→ C ′ ], we denote as[ A x −→ B y −→ C ] ⊕ [ A ′ x ′ −→ B ′ y ′ −→ C ′ ] = [ A ⊕ A ′ x ⊕ x ′ −→ B ⊕ B ′ y ⊕ y ′ −→ C ⊕ C ′ ] . Definition 2.1.
Let s be a correspondence which associates an equivalence class s ( δ ) =[ A x −→ B y −→ C ] to any E -extension δ ∈ E ( C, A ) . This s is called a realization of E if forany morphism ( a, c ) : δ → δ ′ with s ( δ ) = [∆ ] and s ( δ ′ ) = [∆ ], there is a commutativediagram as follows: ∆ (cid:15) (cid:15) A a (cid:15) (cid:15) x / / B y / / b (cid:15) (cid:15) C c (cid:15) (cid:15) ∆ A x ′ / / B y ′ / / C. A realization s of E is said to be additive if it satisfies the following conditions:(a) For any A, C ∈ C , the split E -extension 0 ∈ E ( C, A ) satisfies s (0) = 0.(b) s ( δ ⊕ δ ′ ) = s ( δ ) ⊕ s ( δ ′ ) for any pair of E -extensions δ and δ ′ .Let s be an additive realization of E . If s ( δ ) = [ A x −→ B y −→ C ], then the sequence A x −→ B y −→ C is called a conflation , x is called an inflation and y is called a deflation .In this case, we say A x −→ B y −→ C δ is an E -triangle. We will write A = cocone( y )and C = cone( x ) if necessary. We say an E -triangle is splitting if it realizes 0. Definition 2.2. ([8, Definition 2.12]) We call the triplet ( C , E , s ) an extriangulated cat-egory if it satisfies the following conditions:(ET1) E : C op × C → Ab is a biadditive functor.(ET2) s is an additive realization of E .(ET3) Let δ ∈ E ( C, A ) and δ ′ ∈ E ( C ′ , A ′ ) be any pair of E -extensions, realized as s ( δ ) = [ A x −→ B y −→ C ], s ( δ ′ ) = [ A ′ x ′ −→ B ′ y ′ −→ C ′ ]. For any commutative squarein C A a (cid:15) (cid:15) x / / B b (cid:15) (cid:15) y / / CA ′ x ′ / / B ′ y ′ / / C ′ there exists a morphism ( a, c ): δ → δ ′ which is realized by ( a, b, c ).(ET3) op Dual of (ET3).(ET4) Let δ ∈ E ( D, A ) and δ ′ ∈ E ( F, B ) be E -extensions realized by A f −→ B f ′ −→ D and L. WANG, J. WEI, H. ZHANG B g −→ C g ′ −→ F , respectively. Then there exist an object E ∈ C , a commutative diagram A f / / B g (cid:15) (cid:15) f ′ / / D d (cid:15) (cid:15) A h / / C g ′ (cid:15) (cid:15) h ′ / / E e (cid:15) (cid:15) F F (2.1)in C , and an E -extension δ ′′ ∈ E ( E, A ) realized by A h −→ C h ′ −→ E , which satisfy thefollowing compatibilities:(i) D d −→ E e −→ F realizes E ( F, f ′ )( δ ′ ),(ii) E ( d, A )( δ ′′ ) = δ ,(iii) E ( E, f )( δ ′′ ) = E ( e, B )( δ ′ ).(ET4) op Dual of (ET4).Let C be an extriangulated category, and D , D ′ ⊆ C . We write D ∗ D ′ for the fullsubcategory of objects X admitting an E -triangle D −→ X −→ D ′ with D ∈ D and D ′ ∈ D ′ . A subcategory D of C is extension-closed , if D ∗ D = D . An object P in C is called projective if for any conflation A x −→ B y −→ C and any morphism c in C ( P, C ), there exists b in C ( P, B ) such that yb = c . We denote the full subcategory ofprojective objects in C by P ( C ). Dually, the injective objects are defined, and the fullsubcategory of injective objects in C is denoted by I ( C ). We say that C has enoughprojectives if for any object M ∈ C , there exists an E -triangle A −→ P −→ M satisfying P ∈ P ( C ). Dually, we define that C has enough injectives . In particular, if C is a triangulated category, then C has enough projectives and injectives with P ( C ) and I ( C ) consisting of zero objects. Example 2.3. (a) Exact categories, triangulated categories and extension-closed subcat-egories of triangulated categories are extriangulated categories. (cf. [8])(b) Let C be an extriangulated category. Then C / ( P ( C ) ∩ I ( C )) is an extriangulatedcategory which is neither exact nor triangulated in general (cf. [8, Proposition 3.30]). Proposition 2.4. [8, Proposition 3.3]
Let C be an extriangulated category. For any E -triangle A −→ B −→ C δ , the following sequences of natural transformations areexact. C ( C, − ) → C ( B, − ) → C ( A, − ) δ ♯ → E ( C, − ) → E ( B, − ) , C ( − , A ) → C ( − , B ) → C ( − , C ) δ ♯ → E ( − , A ) → E ( − , B ) . Let C be an extriangulated category. A finite sequence X n d n −→ X n − d n − −→ · · · d −→ X d −→ X ECOLLEMENTS OF EXTRIANGULATED CATEGORIES 5 in C is said to be an E -triangle sequence , if there exist E -triangles X n d n −→ X n − f n − −→ K n − , K i +1 g i +1 −→ X i f i −→ K i , < i < n −
1, and K g −→ X d −→ X such that d i = g i f i for any 1 < i < n .Two morphism sequences η : A f −→ B g −→ C and η : A ′ f ′ −→ B ′ g ′ −→ C ′ in C are saidto be isomorphic , denoted by η ≃ η , if there are isomorphisms x : A → A ′ , y : B → B ′ and z : C → C ′ in C such that yf = f ′ x and zg = g ′ y . Lemma 2.5. (1)
Let η and η be morphism sequences in C such that η ≃ η . Then η is a conflation if and only if η is a conflation. (2) Let A f −→ B g −→ C be an E -triangle in C . Then f is an isomorphism if andonly if C ∼ = 0 . Similarly, g is an isomorphism if and only if A ∼ = 0 .Proof. It is easily proved by [8, Corollary 3.6] and [8, Proposition 3.7]. (cid:3)
Exact functors.
In what follows, we will always assume that C is an extriangulatedcategory. In addition, we assume the following conditions for the rest of this paper (see[8, Condition 5.8]). Condition 2.6. (WIC) (1) Let f : X → Y and g : Y → Z be any composable pair ofmorphisms in C . If gf is an inflation, then f is an inflation.(2) Let f : X → Y and g : Y → Z be any composable pair of morphisms in C . If gf is a deflation, then g is a deflation. Remark 2.7. If C is a triangulated category or weakly idempotent complete exact cat-egory ([4, Proposition 7.6]), then Condition 2.6 is satisfied. Definition 2.8.
A morphism f in C is called compatible , if “ f is both an inflation anda deflation” implies f is an isomorphism. That is, the class of compatible morphisms isthe following class consisting of the morphisms in C { f | f is not an inflation , or f is not a deflation , or f is an isomorphism } . It is clear that all morphisms are compatible in an exact category. While, the compatiblemorphisms in a triangulated category C are just the isomorphisms in C . Definition 2.9.
A sequence A f −→ B g −→ C in C is said to be right exact if there existsan E -triangle K h −→ B g −→ C and a deflation h : A → K which is compatible, suchthat f = h h . Dually one can also define the left exact sequences.A 4-term E -triangle sequence A f −→ B g −→ C h −→ D is called right exact (resp. leftexact ) if there exist E -triangles A f −→ B g −→ K and K g −→ C h −→ D such that g = g g and g (resp. g ) is compatible.For the convenience of statements, given a morphism f : A → B in C , we denote byΦ f the set consisting of all pairs ( h , h ) such that h : A → K is a deflation, h : K → B is an inflation and f = h h . L. WANG, J. WEI, H. ZHANG
Lemma 2.10.
Let η : A f −→ B g −→ C be a right exact sequence in C . (1) If f is an inflation, then η is a conflation. (2) If A = 0 , then g is an isomorphism.Proof. (1) Since η is right exact, there is an E -triangle K h −→ B g −→ C and acompatible morphism h such that ( h , h ) ∈ Φ f . Since f is an inflation, by Condition2.6(1), we obtain that h is an inflation. Since h is also a deflation, we get that h is anisomorphism. Then by Lemma 2.5(1), we have that η is a conflation.(2) Noting that 0 → B is an inflation, we obtain that 0 → B g → C is an E -triangle.Then by Lemma 2.5(2), we get that g is an isomorphism. (cid:3) We omit the dual statement of Lemma 2.10.
Remark 2.11. (1) A sequence η : A f −→ B g −→ C is both left exact and right exact ifand only if η is a conflation.(2) If C is an abelian category, then A f −→ B g −→ C is right exact if and only if A f −→ B g −→ C −→ A f −→ B g −→ C is left exact if and only if0 −→ A f −→ B g −→ C is exact. If C is a triangulated category with the suspensionfunctor [1]. Then η : A f −→ B g −→ C is right exact if and only if A f −→ B g −→ C −→ A [1]is a triangle if and only if η is left exact.Let us give the notion of right (left) exact functors in extriangulated categories. Definition 2.12.
Let ( A , E A , s A ) and ( B , E B , s B ) be extriangulated categories. An addi-tive covariant functor F : A → B is called a right exact functor if it satisfies the followingconditions(1) If f is a compatible morphism in A , then F f is compatible in B .(2) If A a −→ B b −→ C is right exact in A , then F A
F a −→ F B
F b −→ F C is right exactin B . (Then for any E A -triangle A f −→ B g −→ C δ , there exists an E B -triangle A ′ x −→ F B
F g −→ F C such that
F f = xy and y : F A → A ′ is a deflation andcompatible. Moreover, A ′ is uniquely determined up to isomorphism.)(3) There exists a natural transformation η = { η ( C,A ) : E A ( C, A ) −→ E B ( F op C, A ′ ) } ( C,A ) ∈A op ×A such that s B ( η ( C,A ) ( δ )) = [ A ′ x −→ F B
F g −→ F C ].Dually, we define the left exact functor between two extriangulated categories.The extriangulated functor between two extriangulated categories has been defined in[2]. For our requirements, we modify the definition as the following
Definition 2.13.
Let ( A , E A , s A ) and ( B , E B , s B ) be extriangulated categories. We sayan additive covariant functor F : A → B is an exact functor if the following conditionshold.
ECOLLEMENTS OF EXTRIANGULATED CATEGORIES 7 (1) If f is a compatible morphism in A , then F f is compatible in B .(2) There exists a natural transformation η = { η ( C,A ) } ( C,A ) ∈A op ×A : E A ( − , − ) ⇒ E B ( F op − , F − ) . (3) If s A ( δ ) = [ A x −→ B y −→ C ], then s B ( η ( C,A ) ( δ )) = [ F ( A ) F ( x ) −→ F ( B ) F ( y ) −→ F ( C )]. Proposition 2.14.
Let ( A , E A , s A ) and ( B , E B , s B ) be extriangulated categories. An ad-ditive covariant functor F : A → B is exact if and only if F is both left exact and rightexact.Proof. We only need to prove the sufficiency. Let A f −→ B g −→ C δ be an E A -triangle.Since F is both left exact and right exact, we obtain that F A
F f −→ F B
F g −→ F C is bothleft exact and right exact. By Remark 2.11(1),
F A
F f −→ F B
F g −→ F C is a conflation. (cid:3)
Remark 2.15.
If the categories A and B are abelian, Definition 2.12 coincides with theusual right exact functor in abelian categories, and Definition 2.13 coincides with theusual exact functor. If the categories A and B are triangulated, by Remark 2.11 andProposition 2.14, we know that F is a left exact functor if and only if F is a trianglefunctor if and only if F is a right exact functor. Lemma 2.16.
Let ( A , E A , s A ) and ( B , E B , s B ) be extriangulated categories and F : A → B be a functor which admits a right adjoint functor G . (1) If A has enough projectives and F is an exact functor which preserves projectives,then E B ( F X, Y ) ∼ = E A ( X, GY ) for any X ∈ A and Y ∈ B . (2) If B has enough injectives and G is an exact functor which preserves injectives, then E B ( F X, Y ) ∼ = E A ( X, GY ) for any X ∈ A and Y ∈ B .Proof. (1) For any X ∈ A , there exists an E A -triangle M / / P / / X / / ❴❴❴ (2.2)with P ∈ P ( A ). Since F is an exact functor, and F preserves projectives, we obtain thefollowing E B -triangle F M / / F P / / F X / / ❴❴❴ (2.3)with F P ∈ P ( B ). Applying the functors Hom A ( − , GY ) and Hom B ( − , Y ) to (2.2) and(2.3), respectively, we get the following commutative diagramHom A ( P, GY ) (cid:15) (cid:15) / / Hom A ( M, GY ) (cid:15) (cid:15) / / E A ( X, GY ) (cid:15) (cid:15) / / B ( F P, Y ) / / Hom B ( F M, Y ) / / E B ( F X, Y ) / / . By Five-Lemma, we obtain that E B ( F X, Y ) ∼ = E A ( X, GY ).(2) It is similar to (1). (cid:3)
L. WANG, J. WEI, H. ZHANG
The following lemma is well-known. For the convenience of the reader we give a shortproof.
Lemma 2.17.
Let A and B be two categories and F : A → B be a functor which admitsa right adjoint functor G . Let η : Id A ⇒ GF be the unit and ǫ : F G ⇒ Id B be the counit. (1) Id F X = ǫ F X F ( η X ) for any X ∈ A . (2) Id GY = G ( ǫ Y ) η GY for any Y ∈ B .Proof. (1) Let η : Hom B ( F − , − ) ⇒ Hom A ( − , G − ) be the adjoint isomorphism. Forany X ∈ A , consider the following commutative diagramHom B ( F GF X, F X ) ( F η X ) ∗ (cid:15) (cid:15) η GF X,F X / / Hom A ( GF X, GF X ) η ∗ X (cid:15) (cid:15) Hom B ( F X, F X ) η X,F X / / Hom A ( X, GF X ) . It follows that η X,F X ( ǫ F X F ( η X )) = ( η GF X,F X ( ǫ F X )) η X = η X = η X,F X (Id
F X ). Hence, wehave that Id
F X = ǫ F X F ( η X ). The proof of (2) is similar. (cid:3) Recollements
Let us introduce the concepts of recollements of extriangulated categories.
Definition 3.1.
Let A , B and C be three extriangulated categories. A recollement of B relative to A and C , denoted by ( A , B , C ), is a diagram A i ∗ / / B i ∗ w w i ! g g j ∗ / / C j ! w w j ∗ g g (3.1)given by two exact functors i ∗ , j ∗ , two right exact functors i ∗ , j ! and two left exact functors i ! , j ∗ , which satisfies the following conditions:(R1) ( i ∗ , i ∗ , i ! ) and ( j ! , j ∗ , j ∗ ) are adjoint triples.(R2) Im i ∗ = Ker j ∗ .(R3) i ∗ , j ! and j ∗ are fully faithful.(R4) For each X ∈ B , there exists a left exact E B -triangle sequence i ∗ i ! X θ X / / X ϑ X / / j ∗ j ∗ X / / i ∗ A (3.2)with A ∈ A , where θ X and ϑ X are given by the adjunction morphisms.(R5) For each X ∈ B , there exists a right exact E B -triangle sequence i ∗ A ′ / / j ! j ∗ X υ X / / X ν X / / i ∗ i ∗ X (3.3)with A ′ ∈ A , where υ X and ν X are given by the adjunction morphisms. ECOLLEMENTS OF EXTRIANGULATED CATEGORIES 9
Remark 3.2. (1) If the categories A , B and C are abelian, then Definition 3.1 coincideswith the definition of recollement of abelian categories (cf. [10], [9], [7]).(2) If the categories A , B and C are triangulated, then Definition 3.1 coincides with thedefinition of recollement of triangulated categories (cf. [3]).Now, we collect some properties of recollement of extriangulated categories, which willbe used in the sequel. Lemma 3.3.
Let ( A , B , C ) be a recollement of extriangulated categories as (3.1). (1) All the natural transformations i ∗ i ∗ ⇒ Id A , Id A ⇒ i ! i ∗ , Id C ⇒ j ∗ j ! , j ∗ j ∗ ⇒ Id C are natural isomorphisms. (2) i ∗ j ! = 0 and i ! j ∗ = 0 . (3) i ∗ preserves projective objects and i ! preserves injective objects. (3 ′ ) j ! preserves projective objects and j ∗ preserves injective objects. (4) If i ! (resp. j ∗ ) is exact, then i ∗ (resp. j ∗ ) preserves projective objects. (4 ′ ) If i ∗ (resp. j ! ) is exact, then i ∗ (resp. j ∗ ) preserves injective objects. (5) If B has enough projectives, then A has enough projectives and P ( A ) = add ( i ∗ ( P ( B ))) ;if B has enough injectives, then A has enough injectives and I ( A ) = add ( i ! ( I ( B ))) . (6) If B has enough projectives and j ∗ is exact, then C has enough projectives and P ( C ) = add ( j ∗ ( P ( B ))) ; if B has enough injectives and j ! is exact, then C has enoughinjectives and I ( C ) = add ( j ∗ ( I ( B ))) . (7) If B has enough projectives and i ! is exact, then E B ( i ∗ X, Y ) ∼ = E A ( X, i ! Y ) for any X ∈ A and Y ∈ B . (7 ′ ) If C has enough projectives and j ! is exact, then E B ( j ! Z, Y ) ∼ = E C ( Z, j ∗ Y ) for any Y ∈ B and Z ∈ C . (8) If i ∗ is exact, then j ! is exact. (8 ′ ) If i ! is exact, then j ∗ is exact.Proof. (1) The proof follows from the fact that i ∗ , j ! and j ∗ are fully faithful.(2) For any X ∈ C , since j ∗ i ∗ = 0, we have thatHom A ( i ∗ j ! X, i ∗ j ! X ) ∼ = Hom B ( j ! X, i ∗ i ∗ j ! X ) ∼ = Hom C ( X, j ∗ i ∗ i ∗ j ! X )= 0 , which follows that i ∗ j ! = 0. Similarly, i ! j ∗ = 0.(3) Let P ∈ P ( B ), we need to show that i ∗ P ∈ P ( A ). Let X f / / Y g / / Z / / ❴❴❴ (3.4) be an arbitrary E A -triangle and h ∈ Hom A ( i ∗ P, Z ). Applying i ∗ to (3.4), we have an E B -triangle i ∗ X i ∗ f / / i ∗ Y i ∗ g / / i ∗ Z / / ❴❴❴ . (3.5)Applying the functors Hom A ( i ∗ P, − ) and Hom B ( P, − ) to (3.4) and (3.5), respectively, weobtain the following commutative diagramHom( i ∗ P, Y ) η P,Y (cid:15) (cid:15) / / Hom( i ∗ P, Z ) η P,Z (cid:15) (cid:15) / / E A ( i ∗ P, X ) l (cid:15) (cid:15) ✤✤✤ Hom(
P, i ∗ Y ) / / Hom(
P, i ∗ Z ) / / E B ( P, i ∗ X ) . Since P is projective in B and then E B ( P, i ∗ X ) = 0, there exists a morphism t : P → i ∗ Y such that i ∗ g ( t ) = η P,Z ( h ). It follows that g ( η − P,Y ( t )) = η − P,Z i ∗ g ( t ) = h. Hence, i ∗ P ∈ P ( A ). It is proved dually that i ! preserves injective objects. The proofs of(3 ′ ), (4) and (4 ′ ) are similar.(5) For any X ∈ A , there exists a deflation f : P → i ∗ X with P ∈ P ( B ). Applyingthe functor i ∗ , we obtain a deflation i ∗ P → X with i ∗ P ∈ P ( A ). Hence A has enoughprojectives. Since i ∗ preserves projectives, we have that add ( i ∗ ( P ( B ))) ⊆ P ( A ). Con-versely, for Q ∈ P ( A ), as above, there exists a deflation i ∗ P → Q with P ∈ P ( B ), whichimplies that Q is a direct summand of i ∗ P . That is, P ( A ) ⊆ add ( i ∗ ( P ( B ))). Similarly,the second statement in (5) can be proved.(6) Since j ∗ (resp. j ! ) is exact, by (4) (resp. (4 ′ )), we obtain that j ∗ preserves projectives(resp. injectives). Then, using the similar proof of (5), we can prove (6).(7) By (5) we obtain that A has enough projectives. Since i ! is exact, we get that i ∗ preserves projectives. Then according to Lemma 2.16(1), we prove (7).(7 ′ ) By using Lemma 2.16(1) immediately, we obtain the proof.(8) Let X f −→ Y g −→ Z be an E C -triangle. Since j ! is right exact, there is an E B -triangle X ′ h −→ j ! Y j ! g −→ j ! Z and a compatible morphism h such that ( h , h ) ∈ Φ j ! f .Noting that j ∗ j ! X j ∗ j ! f −→ j ∗ j ! Y j ∗ j ! g −→ j ∗ j ! Z is an E C -triangle since j ∗ j ! ∼ = Id C , and j ∗ j ! f = ( j ∗ h )( j ∗ h ), we obtain that j ∗ h is an inflation and then j ∗ h is an isomorphismsince j ∗ h is a deflation and compatible. So j ∗ X ′ ∼ = j ∗ j ! X . Set M = cocone( h ), byLemma 2.5(2), we have that j ∗ M = 0. By (R2), there is an object N ∈ A such that i ∗ N = M . Since i ∗ is exact, i ∗ X ′ i ∗ h −→ i ∗ j ! Y i ∗ j ! g −→ i ∗ j ! Z is an E A -triangle, which implies i ∗ X ′ = 0 by i ∗ j ! = 0. Similarly, since i ∗ M −→ i ∗ j ! X i ∗ h −→ i ∗ X ′ is an E A -triangle, weget that i ∗ M = 0. Then M = i ∗ N ∼ = i ∗ ( i ∗ i ∗ N ) ∼ = i ∗ i ∗ M = 0. Hence, h is an isomorphismand j ! X j ! f −→ j ! Y j ! g −→ j ! Z is an E B -triangle. For the natural transformation, it isimplied from the assumption that j ! is right exact. The proof of (8 ′ ) is similar. (cid:3) ECOLLEMENTS OF EXTRIANGULATED CATEGORIES 11
Proposition 3.4.
Let ( A , B , C ) be a recollement of extriangulated categories as (3.1). (1) If i ! is exact, for each X ∈ B , there is an E B -triangle i ∗ i ! X θ X / / X ϑ X / / j ∗ j ∗ X / / ❴❴❴ where θ X and ϑ X are given by the adjunction morphisms. (2) If i ∗ is exact, for each X ∈ B , there is an E B -triangle j ! j ∗ X υ X / / X ν X / / i ∗ i ∗ X / / ❴❴❴ where υ X and ν X are given by the adjunction morphisms.Proof. We only prove (1) since the proof of (2) is similar. By (R4), for each X ∈ B ,there is a left exact E B -triangle sequence i ∗ i ! X θ X / / X ϑ X / / j ∗ j ∗ X h / / i ∗ A such that i ∗ i ! X θ X −→ X h −→ M and M h −→ j ∗ j ∗ X h −→ i ∗ A are E B -triangles, h iscompatible and ϑ X = h h . Since i ! is exact, we obtain that i ! i ∗ i ! X i ! θ X −→ i ! X i ! ϑ X −→ i ! j ∗ j ∗ X is left exact. By Lemma 3.3(2), i ! j ∗ j ∗ X = 0, so i ! h is an isomorphism. Thus, i ! M = 0. Itfollows that A ∼ = i ! i ∗ A = 0. By Lemma 2.5(2), h is an isomorphism and thus i ∗ i ! X θ X −→ X ϑ X −→ j ∗ j ∗ X is an E B -triangle. This finishes the proof. (cid:3) In what follows, let us give an example of recollement of an extriangulated categorywhich is neither abelian nor triangulated.
Example 3.5.
Let A be the path algebra of the quiver 1 α −→ mod A is as follows P ψ (cid:31) (cid:31) ❄❄❄❄❄ S ϕ ? ? ⑧⑧⑧⑧⑧ S . Then the triangular matrix algebra B = A A A ! is given by the quiver · β (cid:30) (cid:30) ❂❂❂❂❂❂❂ · α @ @ ✁✁✁✁✁✁✁ δ (cid:30) (cid:30) ❂❂❂❂❂❂❂ ·· γ @ @ ✁✁✁✁✁✁✁ with the relation βα = γδ . It is well-known that mod B can be identified with themorphism category of A -modules. That is, each B -module can be written as a triple (cid:0) XY (cid:1) f such that f : Y → X is a homomorphism of A -modules. In the following, we write (cid:0) XY (cid:1) instead of (cid:0) XY (cid:1) . The Auslander-Reiten quiver of B is given by (cid:0) P (cid:1) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄ (cid:0) S (cid:1) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄ (cid:0) S S (cid:1) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄ (cid:0) S (cid:1) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:0) P S (cid:1) ϕ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ / / (cid:0) P P (cid:1) / / (cid:0) S P (cid:1) ψ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄ (cid:0) S (cid:1)(cid:0) S S (cid:1) ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:0) S (cid:1) ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:0) P (cid:1) ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ By [9, Example 2.12], we have a recollement of abelian categories mod A i ∗ / / mod B i ∗ s s i ! j j j ∗ / / mod A j ! s s j ∗ j j (3.6)such that i ∗ ( (cid:0) XY (cid:1) f ) = Coker( f ), i ∗ ( X ) = (cid:0) X (cid:1) , i ! ( (cid:0) XY (cid:1) f ) = X , j ! ( Y ) = (cid:0) YY (cid:1) , j ∗ ( (cid:0) XY (cid:1) f ) = Y and j ∗ ( Y ) = (cid:0) Y (cid:1) .Let X = mod A and X = add ( S ⊕ P ). Observe that X is an extriangulatedcategory which is neither abelian nor triangulated. Indeed, X is an extension-closedsubcategory of mod B . On the one hand, P is a non-zero injective object. This implies X is not triangulated. On the other hand, Ker ψ does not belong to X . This implies X is not abelian. In addition, let X = add ( (cid:18) S (cid:19) ⊕ (cid:18) P (cid:19) ⊕ (cid:18) S (cid:19) ⊕ (cid:18) P P (cid:19) ⊕ (cid:18) S P (cid:19) ψ ⊕ (cid:18) S S (cid:19) ⊕ (cid:18) P (cid:19) ⊕ (cid:18) S (cid:19) ) , and it is also an extriangulated category.We claim that X i ∗ / / X i ∗ v v i ! h h j ∗ / / X j ! w w j ∗ g g (3.7)is a recollement of an extriangulated category which is neither abelian nor triangulated.In fact, one can check that i ∗ , j ∗ are exact functors, i ∗ , j ! are right exact functors and i ! , j ∗ are left exact functors.(R1) For any (cid:0) XX ′ (cid:1) f ∈ X and Y ∈ X ,Hom X ( j ! Y, (cid:18) XX ′ (cid:19) f ) ∼ = Hom mod A ( Y, j ∗ (cid:18) XX ′ (cid:19) f ) = Hom X ( Y, j ∗ (cid:18) XX ′ (cid:19) f )and Hom X ( j ∗ (cid:18) XX ′ (cid:19) f , Y ) ∼ = Hom mod B ( (cid:18) XX ′ (cid:19) f , j ∗ Y ) = Hom X ( (cid:18) XX ′ (cid:19) f , j ∗ Y ) . It follows that ( j ! , j ∗ , j ∗ ) is an adjoint triple. Similarly, so is ( i ∗ , i ∗ , i ! ). ECOLLEMENTS OF EXTRIANGULATED CATEGORIES 13 (R2) Note that Im i ∗ = Ker j ∗ = add ( (cid:0) P (cid:1) ⊕ (cid:0) S (cid:1) ⊕ (cid:0) S (cid:1) ).(R3) Since the functors i ∗ , j ! and j ∗ in (3.6) are fully faithful, it follows that i ∗ , j ! and j ∗ in (3.7) are also fully faithful.(R4) Note that i ! is exact. For any (cid:0) XY (cid:1) f ∈ X , by [9, Proposition 2.6], there exists anexact sequence 0 −→ i ∗ i ! (cid:18) XY (cid:19) f −→ (cid:18) XY (cid:19) f −→ j ∗ j ∗ (cid:18) XY (cid:19) f −→ mod B , which also provides a left exact E -triangle sequence (cid:18) X (cid:19) −→ (cid:18) XY (cid:19) f −→ (cid:18) Y (cid:19) −→ X .(R5) For (cid:0) XY (cid:1) f ∈ X , by [9, Proposition 2.6], there exists an exact sequence0 −→ (cid:18) X ′ (cid:19) f −→ j ! j ∗ (cid:18) XY (cid:19) f −→ (cid:18) XY (cid:19) f −→ i ∗ i ∗ (cid:18) XY (cid:19) f −→ mod B with X ′ ∈ mod A , which provides a right exact E -triangle sequence (cid:18) X ′ (cid:19) −→ YY ! −→ XY ! f −→ (cid:18) Coker f (cid:19) in X .4. Glued cotorsion pairs
First of all, let us recall the definition of cotorsion pairs in an extriangulated category.
Definition 4.1. [8, Definition 4.1] Let C be an extriangulated category and T , F ⊆ C bea pair of subcategories of C . The pair ( T , F ) is called a cotorsion pair in C if it satisfiesthe following conditions:( a ) E ( T , F ) = 0 . ( b ) For any C ∈ C , there exists a conflation F −→ T −→ C such that F ∈ F , T ∈ T .( c ) For any C ∈ C , there exists a conflation C −→ F ′ −→ T ′ such that F ′ ∈ F , T ′ ∈ T . Remark 4.2.
Let ( U , V ) be a cotorsion pair in an extriangulated category C . Then • M ∈ U if and only if E ( M, V ) = 0; • N ∈ V if and only if E ( U , N ) = 0; • U and V are extension-closed; • U is contravariantly finite and V is covariantly finite in C ; • P ⊆ U and I ⊆ V . Definition 4.3.
Let ( A , B , C ) be a recollement of extriangulated categories as (3.1).Given cotorsion pairs ( T , F ) and ( T , F ) in A and C , respectively, set T = { B ∈ B | i ∗ B ∈ T and j ∗ B ∈ T } and F = { B ∈ B | i ! B ∈ F and j ∗ B ∈ F } . In this case, we call ( T , F ) the glued pair with respect to ( T , F ) and ( T , F ).For a subcategory X of C , we define full subcategories X ⊥ = { M ∈ C | E ( X , M ) = 0 } . Recall that an extriangulated category C is called Frobenius if C has enough projectivesand enough injectives and moreover the projectives coincide with the injectives. Notethat each triangulated category is a Frobenius extriangulated category.Now, we can give our main result of this paper as the following Theorem 4.4.
Let ( A , B , C ) be a recollement of extriangulated categories as (3.1), and ( T , F ) and ( T , F ) be two cotorsion pairs in A and C , respectively. Let ( T , F ) be theglued pair with respect to ( T , F ) and ( T , F ) . Assume that B has enough projectives and i ! , j ! are exact. (1) If one of the following conditions holds: (i) E B ( T , F ) = 0 . (ii) i ∗ is exact. (iii) For any morphism f : i ∗ A → j ! T with A ∈ A and T ∈ T , the induced map f ∗ : B ( j ! T, F ) → B ( i ∗ A, F ) is surjective for any F ∈ F . (iv) T ⊆ j ! T or i ∗ F ⊆ T ⊥ . (v) A and B are Frobenius extriangulated categories.Then ( T , F ) is a cotorsion pair in B . (2) If ( U , V ) is a cotorsion pair in B such that i ∗ i ! U ⊆ U and i ∗ i ∗ U ⊆ U , then ( i ∗ U , i ! V ) is a cotorsion pair in A . (3) If ( U , V ) is a cotorsion pair in B such that j ∗ j ∗ V ⊆ V or j ! j ∗ U ⊆ U , then ( j ∗ U , j ∗ V ) is a cotorsion pair in C . In what follows, we say that a commutative diagram is exact if every sub-diagram ofthe form X → Y → Z is a conflation. Before proving Theorem 4.4, we give the following Lemma 4.5.
Keep the notation as Definition 4.3. (1) If i ! is exact, then ( T , F ) satisfies ( b ) in Definition 4.1. (2) If i ! and j ! are exact, then ( T , F ) satisfies ( c ) in Definition 4.1. ECOLLEMENTS OF EXTRIANGULATED CATEGORIES 15
Proof. (1) For any M ∈ B , there exists an E C -triangle F −→ T −→ j ∗ M with F ∈ F and T ∈ T , since ( T , F ) is a cotorsion pair in C . Since i ! is exact, byLemma 3.3(8 ′ ), j ∗ is exact. Applying j ∗ to the above E C -triangle, we obtain an E B -triangle j ∗ F −→ j ∗ T −→ j ∗ j ∗ M . Consider the following commutative diagram j ∗ F / / H (cid:15) (cid:15) / / M η M (cid:15) (cid:15) η ∗ M δ / / ❴❴❴❴ j ∗ F / / j ∗ T / / j ∗ j ∗ M δ / / ❴❴❴ (4.1)where η M is the unit of the adjoint pair ( j ∗ , j ∗ ). Applying j ∗ to (4.1) and using Lemma2.17(1), we obtain that j ∗ H ∼ = T . We also have an E A -triangle F −→ T −→ i ∗ H with F ∈ F and T ∈ T since ( T , F ) is a cotorsion pair in A . Then i ∗ F −→ i ∗ T −→ i ∗ i ∗ H is an E B -triangle, since i ∗ is exact. For H ∈ B , by (R5), there existsa commutative diagram i ∗ A ′ / / j ! j ∗ H υ H / / h " " ❊❊❊❊❊❊❊❊ H h / / i ∗ i ∗ HK h ? ? ⑦⑦⑦⑦⑦⑦⑦⑦ in B such that i ∗ A ′ −→ j ! j ∗ H h −→ K and K h −→ H h −→ i ∗ i ∗ H are E B -trianglesand h is compatible, moreover, j ! j ∗ H v H −→ H h −→ i ∗ i ∗ H is right exact. By [8, Proposition3.15], we have the following exact commutative diagram K t (cid:15) (cid:15) K h (cid:15) (cid:15) i ∗ F / / T l (cid:15) (cid:15) / / H h (cid:15) (cid:15) i ∗ F / / i ∗ T / / i ∗ i ∗ H. (4.2)Consider the following commutative diagram j ! j ∗ H / / h " " ❊❊❊❊❊❊❊❊ T l / / i ∗ T K t ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ (4.3)where h is a deflation and compatible, and t is an inflation. Thus, the first row of (4.3)is right exact. Applying the right exact functor i ∗ to (4.3) and using Lemma 2.10(2),we obtain that i ∗ T ∼ = i ∗ i ∗ T ∼ = T . Applying j ∗ to the E B -triangle in the second row of(4.2), we have that j ∗ T ∼ = j ∗ H ∼ = T . Hence, T ∈ T . Applying (ET4) op yields an exact commutative diagram i ∗ F / / F (cid:15) (cid:15) / / j ∗ F (cid:15) (cid:15) i ∗ F / / T (cid:15) (cid:15) / / H (cid:15) (cid:15) M M. (4.4)Applying j ∗ to the E B -triangle in the first row of (4.4), we obtain that j ∗ F ∼ = j ∗ j ∗ F ∼ = F .Similarly, applying i ! to the E B -triangle in the first row of (4.4), by the dual of Lemma2.10, we have that i ! F ∼ = i ! i ∗ F ∼ = F . That is, F ∈ F . So the second column in (4.4)gives a desired E B -triangle. Therefore, ( T , F ) satisfies ( b ) in Definition 4.1.(2) For any M ∈ B , there exists an E C -triangle j ∗ M −→ F −→ T with F ∈ F and T ∈ T , since ( T , F ) is a cotorsion pair in C . Since j ! is exact, we obtain an E B -triangle j ! j ∗ M −→ j ! F −→ j ! T . Consider the following exact commutative diagram j ! j ∗ M ǫ M (cid:15) (cid:15) / / j ! F (cid:15) (cid:15) / / j ! T δ / / ❴❴❴ M / / H / / j ! T ǫ M ) ∗ δ / / ❴❴❴ (4.5)where ǫ M is the counit of the adjoint pair ( j ! , j ∗ ). Applying j ∗ to (4.5) and using Lemma2.17(2), we obtain that j ∗ H ∼ = F . We also have an E A -triangle i ! H −→ F −→ T with F ∈ F and T ∈ T , since ( T , F ) is a cotorsion pair in A . Then i ∗ i ! H −→ i ∗ F −→ i ∗ T is an E B -triangle since i ∗ is exact. Consider the following exact commutativediagram i ∗ i ! H ǫ H (cid:15) (cid:15) / / i ∗ F (cid:15) (cid:15) / / i ∗ T δ / / ❴❴❴ H / / F / / i ∗ T ǫ H ) ∗ δ / / ❴❴❴ (4.6)where ǫ H is the counit of the adjoint pair ( i ∗ , i ! ). Applying i ! to (4.6) and using Lemma2.17(2), we obtain that i ! F ∼ = F . Applying j ∗ to the E B -triangle in the second row of(4.6) and using j ∗ i ∗ = 0, we obtain that j ∗ F ∼ = j ∗ H ∼ = F . Thus, F ∈ F . Applying (ET4)yields an exact commutative diagram M / / H (cid:15) (cid:15) / / j ! T (cid:15) (cid:15) M / / F (cid:15) (cid:15) / / T (cid:15) (cid:15) i ∗ T i ∗ T . (4.7) ECOLLEMENTS OF EXTRIANGULATED CATEGORIES 17
Applying j ∗ to the E B -triangle in the third column of (4.7), we obtain that j ∗ T ∼ = j ∗ j ! T ∼ = T . Similarly, applying i ∗ to the E B -triangle in the third column of (4.7), by Lemma2.10(2), we have that i ∗ T ∼ = i ∗ i ∗ T ∼ = T . That is, T ∈ T . So the second row in (4.7) givesa desired E B -triangle. Therefore, ( T , F ) satisfies ( c ) in Definition 4.1. (cid:3) Now we are in the position to prove Theorem 4.4.
Proof of Theorem 4.4.
By Lemma 3.3(8 ′ ), (5) and (6), we have that j ∗ is exact and A , C has enough projectives.(1) Take any objects T ∈ T and F ∈ F . By Lemma 4.5, we only need to prove E B ( T, F ) = 0.(i) Immediately.(ii) Since i ∗ is exact, by Proposition 3.4(2), there is an E B -triangle j ! j ∗ T −→ T −→ i ∗ i ∗ T . Applying Hom B ( − , F ) to this E B -triangle, we get an exact sequence E B ( i ∗ i ∗ T, F ) −→ E B ( T, F ) −→ E B ( j ! j ∗ T, F ) . By Lemma 3.3(7), we obtain that E B ( i ∗ i ∗ T, F ) ∼ = E A ( i ∗ T, i ! F ) = 0, since i ∗ T ∈ T and i ! F ∈ F . Similarly, E B ( j ! j ∗ T, F ) ∼ = E C ( j ∗ T, j ∗ F ) = 0. It follows that E B ( T, F ) = 0.(iii) By (R5), there exists a commutative diagram i ∗ A ′ / / j ! j ∗ T / / " " ❊❊❊❊❊❊❊❊ T / / i ∗ i ∗ T / / K ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ in B such that i ∗ A ′ −→ j ! j ∗ T −→ K and K −→ T −→ i ∗ i ∗ T are E B -triangles.Applying Hom( − , F ) to the two E B -triangles above, we have two exact sequences E B ( i ∗ i ∗ T, F ) → E B ( T, F ) → E B ( K, F )and Hom B ( j ! j ∗ T, F ) f ∗ → Hom B ( i ∗ A ′ , F ) → E B ( K, F ) → E B ( j ! j ∗ T, F ) . By Lemma 3.3(7 ′ ), we obtain that E B ( j ! j ∗ T, F ) ∼ = E C ( j ∗ T, j ∗ F ) = 0. Similarly, E B ( i ∗ i ∗ T, F ) ∼ = E A ( i ∗ T, i ! F ) = 0. By hypothesis, f ∗ is surjective, so we obtain that E B ( K, F ) = 0. Itfollows that E B ( T, F ) = 0.(iv) Since i ! is exact, by Proposition 3.4(1), there is an E B -triangle i ∗ i ! F −→ F −→ j ∗ j ∗ F . Applying Hom( T, − ) to the E B -triangle above, we have an exact sequence E B ( T, i ∗ i ! F ) −→ E B ( T, F ) −→ E B ( T, j ∗ j ∗ F ) . Since i ! is exact, we obtain that j ∗ is exact and then j ∗ preserves projectives by Lemma3.3(4). Thus, by Lemma 2.16(1), we have that E B ( T, j ∗ j ∗ F ) ∼ = E C ( j ∗ T, j ∗ F ) = 0. If T ⊆ j ! T , i.e., there exists T ∈ T such that T ∼ = j ! T . Then E B ( T, i ∗ i ! F ) ∼ = E B ( j ! T , i ∗ i ! F ) ∼ = E C ( T , j ∗ i ∗ i ! F ) = 0. If i ∗ F ⊆ T ⊥ , then E B ( T, i ∗ i ! F ) = 0 since i ! F ∈ F . Hence, E B ( T, F ) = 0.(v) Since A and B are Frobenius extriangulated categories, projectives coincide withinjectives, by Lemma 3.3(4), i ∗ preserves injectives. Then by Lemma 2.16(2), we havethe isomorphism E A ( i ∗ X, Y ) ∼ = E B ( X, i ∗ Y ) for any X ∈ B and Y ∈ A . Using the similarproof of (iv), we prove (v).(2) For any X ∈ A , there exists an E B -triangle V −→ U −→ i ∗ X with V ∈ V and U ∈ U . Then we have an E A -triangle i ! V −→ i ! U −→ X , since i ! is exact. Since i ∗ i ! U ∈ i ∗ i ! U ⊆ U , we obtain that i ! U ∈ i ∗ U . That is, ( i ∗ U , i ! V ) satisfies ( b ) in Definition4.1. Dually, we can prove that ( i ∗ U , i ! V ) also satisfies ( c ) in Definition 4.1. For any U ∈ U and V ∈ V , by Lemma 3.3(7), we have that E A ( i ∗ U, i ! V ) ∼ = E B ( i ∗ i ∗ U, V ) ∼ = E B ( U ′ , V ) = 0for some U ′ ∈ U . Hence, ( i ∗ U , i ! V ) is a cotorsion pair in A .(3) The proof is analogous to (2). (cid:3) Applying the main theorem to recollements of triangulated categories, we have thefollowing
Corollary 4.6. [5, Theorem 3.1]
Let ( A , B , C ) be a recollement of triangulated categories.Let ( T , F ) be a glued pair with respect to cotorsion pairs ( T , F ) and ( T , F ) in A and C , respectively. Then ( T , F ) is a cotorsion pair in B . We finish this section with a straightforward example illustrating Theorem 4.4.
Example 4.7.
Keep the notation as Example 3.5.Observe that P ( mod A ) = add ( P ⊕ S ) and I ( mod A ) = add ( P ⊕ S ). By Remark4.2, we know that mod A has only two cotorsion pairs H = ( P ( mod A ) , mod A ) and H = ( mod A, I ( mod A )) . (1) Let ( T , F ) be the glued pair with respect to H and H . Then T = add ( (cid:18) S (cid:19) ⊕ (cid:18) P (cid:19) ⊕ (cid:18) S S (cid:19) ⊕ (cid:18) S (cid:19) ⊕ (cid:18) P P (cid:19) ⊕ (cid:18) S P (cid:19) ψ ⊕ (cid:18) P (cid:19) )and F = mod B .(2) Let ( T , F ) be the glued pair with respect to H and H . Then T = mod B \ add ( (cid:18) P S (cid:19) ϕ ⊕ (cid:18) S (cid:19) )and F = mod B \ add ( (cid:18) S S (cid:19) ⊕ (cid:18) P S (cid:19) ϕ ⊕ (cid:18) S (cid:19) ) . (3) Let ( T , F ) be the glued pair with respect to H and H . Then T = mod B and F = mod B \ add ( (cid:18) S S (cid:19) ⊕ (cid:18) P S (cid:19) ϕ ⊕ (cid:18) S (cid:19) ) . ECOLLEMENTS OF EXTRIANGULATED CATEGORIES 19 (4) Let ( T , F ) be the glued pair with respect to H and H . Then T = mod B \ add ( (cid:18) S S (cid:19) ⊕ (cid:18) S (cid:19) )and F = mod B \ add ( (cid:18) S (cid:19) ⊕ (cid:18) S S (cid:19) ) . However, every glued pair above is not a cotorsion pair in mod B since Ext ( T , F ) = 0.Observe that P ( mod B ) = add ( (cid:18) S (cid:19) ⊕ (cid:18) P (cid:19) ⊕ (cid:18) S S (cid:19) ⊕ (cid:18) P P (cid:19) )and I ( mod B ) = add ( (cid:18) P P (cid:19) ⊕ (cid:18) S S (cid:19) ⊕ (cid:18) P (cid:19) ⊕ (cid:18) S (cid:19) ) . (5) Take T = P ( mod B ) and F = mod B , then ( T , F ) is a cotorsion pair in mod B .One can check that i ∗ i ∗ T = i ∗ i ! T = add ( (cid:18) S (cid:19) ⊕ (cid:18) P (cid:19) ) ⊆ T and j ∗ j ∗ F ⊆ F . Therefore, by Theorem 4.4(2) and (3), we know that ( i ∗ T , i ! F ) and( j ∗ T , j ∗ F ) are cotorsion pairs in mod A .(5 ′ ) Let T = add ( (cid:18) S (cid:19) ⊕ (cid:18) P (cid:19) ⊕ (cid:18) S S (cid:19) ⊕ (cid:18) P P (cid:19) ⊕ (cid:18) S (cid:19) )and F = add ( (cid:18) P P (cid:19) ⊕ (cid:18) S S (cid:19) ⊕ (cid:18) P (cid:19) ⊕ (cid:18) S (cid:19) ⊕ (cid:18) P (cid:19) ⊕ (cid:18) S S (cid:19) ⊕ (cid:18) S (cid:19) ) , then ( T , F ) is also a cotorsion pair in mod B . We have that i ∗ i ∗ T = i ∗ i ! T = add ( (cid:18) S (cid:19) ⊕ (cid:18) P (cid:19) ) ⊆ T . Thus, by Theorem 4.4(2), we obtain that ( i ∗ T , i ! F ) is a cotorsion pairs in mod A . But( j ∗ T , j ∗ F ) is not a cotorsion pair in mod A since Ext ( j ∗ T , j ∗ F ) = 0. References [1] L. Angeleri H¨ugel, S. Koenig, Q. Liu, On the uniqueness of stratifications of derived module cate-gories, J. Algebra (2012), 120–137.[2] R. Bennett-Tennenhaus, A. Shah, Transport of structure in higher homological algebra,arXiv:2003.02254.[3] A. A. Be˘ılinson, J. Bernstein, P. Deligne, Faisceaux pervers, in: Analysis and topology on singularspaces, I, Luminy, 1981, Ast´erisque, vol. 100, Soc. Math. France, Paris, 1982, 5–171.[4] T. B¨uhler, Exact categories, Expo. Math. (2010), 1–69. [5] J. Chen, Cotorsion pairs in a recollement of triangulated categories, Comm. Algebra (2013),2903–2915.[6] X. Ma, T. Zhao, Recollements and tilting modules, Comm. Algebra (2020), 5163–5175.[7] X. Ma, Z. Huang, Torsion pairs in recollements of abelian categories, Front. Math. China (2018),875–892.[8] H. Nakaoka, Y. Palu, Extriangulated categories, Hovey twin cotorsion pairs and model structures,Cah. Topol. G´eom. Diff´er. Cat´eg. (2) (2019), 117–193.[9] C. Psaroudakis, Homological theory of recollements of abelian categories, J. Algebra (2014),63–110.[10] V. Franjou, T. Pirashvili, Comparison of abelian categories recollements, Doc. Math. (2004), 41–56. Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal Univer-sity, Nanjing 210023, P. R. China.
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