Idempotent completion of extriangulated categories
aa r X i v : . [ m a t h . C T ] J u l IDEMPOTENT COMPLETION OF EXTRIANGULATED CATEGORIES
LI WANG, JIAQUN WEI, HAICHENG ZHANG AND TIWEI ZHAO
Abstract.
Extriangulated categories were introduced by Nakaoka and Palu as a simul-taneous generalization of exact categories and triangulated categories. In this paper, weshow that the idempotent completion of an extriangulated category admits a naturalextriangulated structure. As applications, we prove that cotorsion pairs in an extri-angulated category induce cotorsion pairs in its idempotent completion under certaincondition, and the idempotent completion of a recollement of extriangulated categoriesis still a recollement. Introduction
The concept of a triangulated category was given by Verdier and Grothendieck in [16]and has been investigated in many papers such as [3, 7, 10, 15]. Exact category andtriangulated category are two fundamental structures in algebra and geometry. Recently,Nakaoka and Palu [14] introduced an extriangulated category which is extracting proper-ties on triangulated categories and exact categories. The class of extriangulated categoriescontains triangulated categories and exact categories as examples, it is also closed undertaking some ideal quotients. There have been many further researches on extriangulatedcategories, see for example, [13, 18, 9, 17]. Balmer and Schlichting [1] showed that theidempotent completion of a triangulated category admits a triangulated structure. B¨uhler[4] showed that the idempotent completion of an exact category is still an exact category.So one may ask whether the idempotent completion of an extriangulated category carriesan extriangulated structure. In this paper, we intend to give a positive answer, and thus itprovides a unified framework for dealing with the idempotent completions of triangulatedcategories and exact categories.Compared with exact categories or triangulated categories, the proof for an extriangu-lated category seems to be more complicated, because we have to construct a biadditivefunctor and its additive realization for the idempotent completion of an extriangulatedcategory.The notion of cotorsion pairs has been generalized to extriangulated categories in[14]. Recollements of triangulated categories were introduced by Beilinson, Bernstein and
Mathematics Subject Classification.
Key words and phrases.
Extriangulated category; Idempotent completion; Cotorsion pair;Recollement.
Deligne [2] in connection with derived categories of sheaves on topological spaces with theidea that one triangulated category may be “glued together” from two others. In [5], Chenand Tang showed that the idempotent completion of a right (resp. left) recollement oftriangulated categories is still a right (resp. left) recollement. As applications, we first in-troduce the right (left) recollements of extriangulated categories, and study the cotorsionpairs and recollements in extriangulated categories under the idempotent completion.The paper is organized as follows: We summarize some basic definitions and proposi-tions about idempotent completions and extriangulated categories in Section 2. In Section3, let ( C , E , s ) be a pre-extriangulated category, in the sense that it satisfies all axioms ofan extriangulated category except (ET4) and (ET4) op . We show that the idempotent com-pletion of ( C , E , s ) has a natural structure of a pre-extriangulated category. Furthermore,we extend this result to an extriangulated category. Section 4 is devoted to the cotor-sion pairs and right (left) recollements in the idempotent completion of an extriangulatedcategory. Explicitly, we prove that cotorsion pairs in an extriangulated category inducecotorsion pairs in its idempotent completion under certain condition, and the idempotentcompletion of a right (resp. left) recollement of extriangulated categories is still a right(resp. left) recollement.2. Preliminaries
Throughout this paper, we assume, unless otherwise stated, that all considered cate-gories are additive, and subcategories are full.2.1.
Idempotent completions.
An additive category C is said to be idempotent com-plete if any idempotent morphism e : A → A splits. That is to say, there are twomorphisms p : A → B and q : B → A such that e = qp and pq = id B . Remark that C is idempotent complete if and only if every idempotent morphism has a kernel, or everyidempotent morphism has a cokernel. Definition 2.1.
Let C be an additive category. The idempotent completion of C is thecategory e C defined as follows: Objects of e C are pairs e A = ( A, e a ), where A is an object of C and e a : A → A is an idempotent morphism. A morphism in e C from ( A, e a ) to ( B, e b )is a morphism α : A → B in C such that αe a = e b α = α .The assignment A ( A, id A ) defines a functor ι : C → e C . Obviously, the functor ι isfully faithful. Namely, we can view the category C as a full subcategory of e C . Remark 2.2. (a) Note that triangulated categories with bounded t-structures are idem-potent complete (cf. [11]).(b) The bounded derived category of an idempotent complete exact category is idem-potent complete (cf. [1]).
DEMPOTENT COMPLETION OF EXTRIANGULATED CATEGORIES 3 (c) Let C be the category of finite generated free modules over a ring R , its idempotentcompletion e C is equivalent to the category of finite generated projective modules over R (cf. [12]).The following result is well-known (see, for example, [1, Proposition 1.3]). Lemma 2.3.
The category e C is an additive category, the functor ι : C → e C is additiveand e C is idempotent complete. Moreover, the functor ι induces an equivalence Hom add ( e C , L ) ∼ / / Hom add ( C , L ) for each idempotent complete additive category L , where Hom add denotes the category ofadditive functors.
Extriangulated categories.
Let us recall some definitions and notations concern-ing extriangulated categories. For more definitions and details, we refer the reader to[14].Let C be an additive category and let E : C op × C → Ab be a biadditive functor, where Ab denotes the category of abelian groups. For any pair of objects A , C ∈ C , an element δ ∈ E ( C, A ) is called an E -extension . Thus formally, an E -extension is a triplet ( A, δ, C ).The zero element 0 ∈ E ( C, A ) is called the split E -extension .Let δ ∈ E ( C, A ) be any E -extension. By the functoriality of E , for any 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 simply denote them by a ∗ δ and c ∗ δ , respectively. In this terminology, we have E ( c, a )( δ ) = c ∗ a ∗ δ = a ∗ c ∗ δ in E ( C ′ , A ′ ). A morphism ( a, c ): ( A, δ, C ) → ( A ′ , δ ′ , C ′ ) of E -extensions is a pair of morphisms a ∈ C ( A, A ′ ) and c ∈ C ( C, C ′ ) satisfying the equality a ∗ δ = c ∗ δ ′ , which is simply denoted by ( a, c ): δ → δ ′ . We obtain the category E -Ext( C ) of E -extensions, with composition and identities naturally induced from those in C .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 L. WANG, J. WEI, H. ZHANG AND T. ZHAO 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 ( ) −→ C ] . For any two 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 ′ ] . Let δ ∈ E ( C, A ), δ ′ ∈ E ( C ′ , A ′ ) be any pair of E -extensions. Let C l C −→ C ⊕ C ′ l C ′ ←− C ′ and A p A ←− A ⊕ A ′ p A ′ −→ A ′ be the coproduct and product in C , respectively. Then thebiadditivity of E implies E ( C ⊕ C ′ , A ⊕ A ′ ) ∼ = E ( C, A ) ⊕ E ( C, A ′ ) ⊕ E ( C ′ , A ) ⊕ E ( C ′ , A ′ ) . (2.1)Thus we set δ ⊕ δ ′ = ( δ, , , δ ′ ) ∈ E ( C ⊕ C ′ , A ⊕ A ′ ) through this isomorphism. Inparticular, if C = C ′ and A = A ′ , then the sum δ + δ ′ = ∆ ∗ ∇ ∗ ( δ ⊕ δ ′ ) ∈ E ( C, A ) (2.2)where ∆ = (cid:18) (cid:19) : C → C ⊕ C , and ∇ = (1 ,
1) : A ⊕ A → A .In what follows, we write an element in (2.1) in the form of matrices. For example, wewill write δ = (cid:18) δ δ δ δ (cid:19) for δ = ( δ , δ , δ , δ ) ∈ E ( C ⊕ C ′ , A ⊕ A ′ ). Definition 2.4.
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 C be an additive category and s be an additive realization of E . • We call a sequence A x −→ B y −→ C a conflation if its equivalence class is equal to s ( δ ) for some δ ∈ E ( C, A ) and we say that this conflation realizes δ . In this case, we callthe pair ( A x −→ B y −→ C, δ ) an E - triangle , and write it in the following way. A x −→ B y −→ C δ DEMPOTENT COMPLETION OF EXTRIANGULATED CATEGORIES 5 • Let
D ⊆ C be a full additive subcategory, which is closed under isomorphisms. Thesubcategory D is said to be extension-closed , if for any conflation A x −→ B y −→ C whichsatisfies A, C ∈ D , then B ∈ D . • For any morphism ( a, c ) : δ → δ ′ with δ ∈ E ( C, A ) and δ ′ ∈ E ( C ′ , A ′ ), there is acommutative diagram with rows being E -triangles: A a (cid:15) (cid:15) x / / B y / / b (cid:15) (cid:15) C c (cid:15) (cid:15) δ / / ❴❴❴ A ′ x ′ / / B ′ y ′ / / C ′ δ ′ / / ❴❴❴ We call ( a, b, c ) a morphism of E -triangles , or a realization of ( a, c ) : δ → δ ′ . Definition 2.5. ([14, Definition 2.12]) We call the triplet ( C , E , s ) an extriangulatedcategory 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 square in 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 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 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).
L. WANG, J. WEI, H. ZHANG AND T. ZHAO
Example 2.6. (a) Exact categories, triangulated categories and extension-closed subcat-egories of an extriangulated category are extriangulated categories (cf. [14]).(b) Let C be an extriangulated category. An object P in C is called projective if forany E -triangle A x −→ B y −→ C δ and any morphism c in C ( P, C ), there exists b in C ( P, B ) such that yb = c . We denote the full subcategory of projective objects in C by P . Dually, the injective objects are defined, and the full subcategory of injective objectsin C is denoted by I . Then C / ( P ∩ I ) is an extriangulated category which is neitherexact nor triangulated in general (cf. [14, Proposition 3.30]).
Definition 2.7.
We call the triplet ( C , E , s ) a pre-extriangulated category if it satisfiesall except (ET4) and (ET4) op .There is a natural way to produce pre-extriangulated categories. In detail, let ( C , E , s )be an extriangulated category and let b E be an additive subfunctor of E . Then for any a ∈ Hom C ( A, A ′ ), c ∈ Hom C ( C ′ , C ), and δ ∈ b E ( C, A ), we have a ∗ δ ∈ b E ( C, A ′ ) and c ∗ δ ∈ b E ( C ′ , A ). Define b s to be the restriction of s to b E , that is, it is defined by b s ( δ ) = s ( δ )for any b E -extension δ . By [8, Claim 3.8], the triplet ( C , b E , b s ) satisfies (ET1), (ET2),(ET3) and (ET3) op , that is, ( C , b E , b s ) is a pre-extriangulated category.In fact, the following example shows that the category of finitely generated modules overan Artin algebra always admits a pre-extriangulated structure which is not extriangulated. Example 2.8.
Let Λ be an Artin algebra, and C = modΛ the category of finitely gen-erated Λ-modules. Let E ( C, A ) = Ext ( C, A ) for any
A, C ∈ modΛ, and define b E ( C, A )to be the subgroup of Ext ( C, A ), which is generated by the corresponding almost splitsequences in modΛ. Then ( C , b E , b s ) is a pre-extriangulated category but not an extrian-gulated category, see [6] for more details. Proposition 2.9. [14, Proposition 3.3]
Let ( C , E , s ) be a pre-extriangulated category. Forany E -triangle A x −→ B y −→ C δ , the following sequences of natural transformationsare exact. C ( C, − ) → C ( B, − ) → C ( A, − ) δ ♯ → E ( C, − ) → E ( B, − ) , C ( − , A ) → C ( − , B ) → C ( − , C ) δ ♯ → E ( − , A ) → E ( − , B ) . Let us give two preliminary lemmas.
Lemma 2.10.
Let ( C , E , s ) be a pre-extriangulated category. Consider the following com-mutative diagram in C A p (cid:15) (cid:15) u / / B v / / b (cid:15) (cid:15) C q (cid:15) (cid:15) δ / / ❴❴❴ A u / / B v / / C δ / / ❴❴❴ DEMPOTENT COMPLETION OF EXTRIANGULATED CATEGORIES 7 in which the rows are E -triangles. Suppose that any two of the triplet ( p, b, q ) are idempo-tents, then the third morphism can be replaced by an idempotent morphism such that thediagram above is still commutative.Proof. First of all, suppose that the first and third morphisms are idempotents. Then b u = bup = up = up and similarly vb = qv . Set h = b − b , then there exists a morphism s : C → B such that sv = h since hu = 0. It follows that h = svh = sv ( b − b ) = 0.Now, let r = b + h − bh and note that bh = hb , then r = b + 2 bh − b h = b + h +2 bh − b + h ) h = b + h − bh = r . However, ru = ( b + h − bh ) u = bu = up and vr = v ( b + h − bh ) = qv + vh − qvh = qv . Hence, we obtain an idempotent morphism r such that the above diagram is commutative.Now, suppose the first two morphisms are idempotents. It should be noted that ( q ) ∗ δ = q ∗ p ∗ δ = p ∗ q ∗ δ = ( p ) ∗ δ = p ∗ δ . Similar to the previous proof, set η = q − q and σ = q + η − ηh , we can obtain an idempotent morphism σ such that ( p, b, σ ) realizes ( p, σ ) : δ → δ .Similarly, we can prove the case that the second and third morphisms are idempotents. (cid:3) Lemma 2.11.
Let ( C , E , s ) be a pre-extriangulated category. A morphism sequence △ : A ⊕ A ′ x x ′ ! −→ B ⊕ B ′ y y ′ ! −→ C ⊕ C ′ δ δ ! is an E -triangle for δ ⊕ δ , where δ ∈ E ( C, A ) and δ ∈ E ( C ′ , A ′ ) , if and only if △ : A x −→ B y −→ C δ and △ : A ′ x ′ −→ B ′ y ′ −→ C ′ δ are E -triangles for δ and δ , respectively.Proof. Suppose △ and △ are E -triangles. Then △ is an E -triangle since s ( δ ) ⊕ s ( δ ) = s ( δ ⊕ δ ). Conversely, suppose that the direct sum of △ and △ is an E -triangle for δ ⊕ δ ∈ E ( C ⊕ C ′ , A ⊕ A ′ ). Then we have respectively two E -triangles for δ and δ : A u −→ D v −→ C δ (2.3)and A ′ u ′ −→ E v ′ −→ C ′ δ . (2.4)It follows that we have a commutative diagram with rows being E -triangles: A ⊕ A ′ ( id A (cid:15) (cid:15) x x ′ ! / / B ⊕ B ′ ( b b ′ ) (cid:15) (cid:15) y y ′ ! / / C ⊕ C ′ ( id C (cid:15) (cid:15) δ δ ! = δ / / ❴❴❴ A u / / D v / / C δ / / ❴❴❴❴ where the middle morphism exists since (id A , ∗ δ = (id C , ∗ δ . Then we have a morphism b : B → D such that b x = u and y = vb . Similarly, we also have a morphism L. WANG, J. WEI, H. ZHANG AND T. ZHAO b : B ′ → E such that b x ′ = u ′ and y ′ = v ′ b . Hence, we have the following commutativediagram of E -triangles in C A ⊕ A ′ x x ′ ! / / B ⊕ B ′ b b ! (cid:15) (cid:15) y y ′ ! / / C ⊕ C ′ δ / / ❴❴❴ A ⊕ A ′ u u ′ ! / / B ⊕ B ′ v v ′ ! / / C ⊕ C ′ δ / / ❴❴❴ . This implies that b and b are isomorphisms by [14, Corollary 3.6]. Then, △ and △ areisomorphic to the E -triangles (2 .
3) and (2 . (cid:3) Main resultsTheorem 3.1.
The idempotent completion of an extriangulated category has a naturalstructure of an extriangulated category.
Before proving Theorem 3.1, we need some preparations.(
Step 1 ) First of all, we are going to construct a biadditive functor F : e C op × e C → Ab.
For any objects e C = ( C, e c ) and e A = ( A, e a ) in e C we define F ( e C, e A ) = { e ω = ( ω, e c , e a ) | ω ∈ E ( C, A ) satisfies e ∗ c ω = ω = ( e a ) ∗ ω } . For two elements f ω , f ω ∈ F ( e C, e A ), define f ω + f ω = ^ ω + ω = ( ω + ω , e c , e a ) . (3.1)In fact, e ∗ c ( ω + ω ) = e ∗ c ω + e ∗ c ω = ω + ω and ( e a ) ∗ ( ω + ω ) = ω + ω since E is abiadditive functor. So the addition (3 .
1) is well-defined. Thus, we have an abelian group F ( e C, e A ).For e A ′ = ( A ′ , e a ′ ) we construct a group homomorphism between F ( e C, e A ) and F ( e C, e A ′ ).For any α ∈ e C ( e A, e A ′ ) and e ω ∈ F ( e C, e A ), we define a homomorphism α • by F ( e C, e A ) α • (cid:15) (cid:15) e ω = ( ω, e c , e a ) ❴ (cid:15) (cid:15) F ( e C, e A ′ ) α • e ω = ( α ∗ ω, e c , e a ′ ) . Indeed, since α ∗ ω = α ∗ e c ∗ ω = e c ∗ α ∗ ω and α ∗ ω = α ∗ ( e a ) ∗ ω = ( αe a ) ∗ ω = ( e ′ a α ) ∗ ω =( e ′ a ) ∗ α ∗ ω , we obtain that α • e ω really belongs to F ( e C, e A ′ ). Similarly, we can define β • suchthat β • e ω ∈ F ( f C ′ , e A ) for β ∈ e C ( f C ′ , e C ) . DEMPOTENT COMPLETION OF EXTRIANGULATED CATEGORIES 9
On the one hand, if there is another homomorphism α ′ ∈ e C ( e A, e A ′ ). It is easy to see( α + α ′ ) • ( e ω ) = (( α + α ′ ) ∗ ω, e c , e ′ a ) = ( α ∗ ω + α ′∗ ω, e c , e ′ a ) = α • e ω + α ′• e ω (3.2)and so is ( β + β ′ ) • ( e ω ) = β • e ω + β ′• e ω for β ′ ∈ e C ( f C ′ , e C ) . On the other hand, for any morphism m : e A → e A ′ and n : f C ′ → e C . We have thefollowing commutative diagram which follows from the fact that E is biadditive. F ( e C, e A ) n • (cid:15) (cid:15) m • / / F ( e C, e A ′ ) n • (cid:15) (cid:15) F ( f C ′ , e A ) m • / / F ( f C ′ , e A ′ )From what has been discussed above, we have the following Lemma 3.2.
Suppose C is equipped with a biadditive functor E : C op × C → Ab . Thenwe have a biadditive functor F : e C op × e C → Ab.
For any pair of objects e A , e B ∈ e C , an element e ω ∈ F ( e C, e A ) is called an F - extension . Amorphism ( α, β ): e ω → e ω ′ is a pair of morphisms α ∈ e C ( e A, e A ′ ) and β ∈ e C ( e C, f C ′ ) suchthat α • e ω = β • e ω ′ . We obtain the category F -Ext( e C ) of F -extensions. For any E -extension δ ∈ E ( C, A ), the assignment δ e δ = ( δ, id C , id A )defines a fully faithful functor ˜ i from E -Ext( C ) to F -Ext( e C ). In addition, for any γ ∈ C ( A, A ′ ), γ • e δ = ( γ ∗ δ, id C , id A ) = γ ∗ δ . Similarly, for any θ ∈ C ( C ′ , C ), we have θ • e δ = θ ∗ δ . Namely, F | C op × C = E .( Step 2 ) Secondly, we need to construct an additive realization c for F . For any E -extension δ ∈ E ( C, A ), there is a correspondence which associates an equivalence class s ( δ ) = [ A x −→ B y −→ C ]. If e A = ( A, e a ) ∈ e C , take e A ′ = ( A, id A − e a ). It is easy tocheck that e A ⊕ e A ′ ∼ = ( A, id A ). Similarly, for e C = ( C, e c ), take f C ′ = ( C, id C − e c ), we have e C ⊕ f C ′ ∼ = ( C, id C ). For any e ω ∈ F ( e C, e A ), since (cid:18) e ω
00 0 (cid:19) belongs to F ( e C ⊕ f C ′ , e A ⊕ e A ′ ) ∼ = E ( C, A ), there exists an E -triangle in C ∆ : e A ⊕ e A ′ α −→ B β −→ e C ⊕ f C ′ e ω
00 0 ! . By Lemma 2.10, there exists an idempotent morphism b : B → B such that the followingis the morphism of E -triangles e A ⊕ e A ′ e a
00 0 ! (cid:15) (cid:15) α / / B b (cid:15) (cid:15) β / / e C ⊕ f C ′ e ω
00 0 ! / / ❴❴❴ e c
00 0 ! (cid:15) (cid:15) e A ⊕ e A ′ α / / B β / / e C ⊕ f C ′ e ω
00 0 ! / / ❴❴❴ . Consider the sequence ∆ : e A bα −→ ( B, b ) βb −→ e C in e C , we have the canonical inclusions e A (cid:127) _ (cid:15) (cid:15) bα / / ( B, b ) (cid:127) _ (cid:15) (cid:15) βb / / e C (cid:127) _ (cid:15) (cid:15) e A ⊕ e A ′ α / / ( B, id B ) β / / e C ⊕ f C ′ (3.3)and (id e A , • e ω = (id e C , • (cid:18) e ω
00 0 (cid:19) . We set c ( e
0) = e c ( e ω ) = [∆ ]. Then c ( e ω ⊕ e ω ′ ) = c ( e ω ) ⊕ c ( e ω ′ ). Thus c is a correspondence which associates to e ω ∈ F ( e C, e A ) an equivalenceclass [ e A x −→ e B y −→ e C ] in e C . In this case, we call the sequence an F -triangle and write itin the following way ∆ : e A x −→ e B y −→ e C e ω . Let ∆ , ∆ be as above, then we have a sequence of maps ∆ i −→ ∆ π −→ ∆ such that πi = id ∆(2) and π is also a morphism of F -triangles from (cid:18) e ω
00 0 (cid:19) to e ω .Let e ω ∈ F ( e C, e A ) and e ω ′ ∈ F ( f C ′ , e A ′ ) be any pair of F -extensions, with∆(1) : e A x −→ e B y −→ e C e ω and ∆(2) : e A ′ x ′ −→ f B ′ y ′ −→ f C ′ e ω . For any morphism ( a, c ) : e ω → e ω ′ in F -Ext( e C ), we have a sequence of maps ∆(1) i −→ ∆ π −→ ∆(1) and ∆(2) i ′ −→ ∇ π ′ −→ ∆(2) such that πi = id ∆(1) and π ′ i ′ = id ∆(2) , where ∆and ∇ are E -triangles for δ and δ ′ , respectively. The morphism ( a, c ) induces a morphism( i a ′ aπ a , i c ′ cπ c ) : δ → δ ′ . We can apply (ET3) to extend the map ( i a ′ aπ a , i c ′ cπ c ) to amorphism of E -triangles α : ∆ → ∇ . Then π ′ αi is a morphism from ∆(1) to ∆(2). Thus c is an additive realization of F . In conclusion, we have the following Lemma 3.3.
Let e C , F , c be as above. Then c is an additive realization for F . We show that the triplet ( e C , F , c ) is also compatible to (ET3). DEMPOTENT COMPLETION OF EXTRIANGULATED CATEGORIES 11
Proposition 3.4.
Let e ω ∈ F ( e C, e A ) and e ω ′ ∈ F ( f C ′ , e A ′ ) be any pair of F -extensions, realizedas e A x −→ e B y −→ e C and e A ′ x ′ −→ f B ′ y ′ −→ f C ′ , respectively. For any commutative square in e C ∆ a,b ) (cid:15) (cid:15) e A a (cid:15) (cid:15) x / / e B b (cid:15) (cid:15) y / / e C e ω / / ❴❴❴ ∆ e A ′ x ′ / / f B ′ y ′ / / f C ′ e ω ′ / / ❴❴❴ there exists a morphism ( a, c ) : e ω → e ω ′ which is realized by ( a, b, c ) .Proof. By the construction of F -triangles, there exists a sequence of maps ∆ i −→ ∆ π −→ ∆ and ∆ i ′ −→ ∇ π ′ −→ ∆ such that πi = id ∆ and π ′ i ′ = id ∆ , where ∆ and ∇ are E -triangles in C . The map ( a, b ) induces a map ( i ′ a aπ a , i ′ b bπ b ) from ∆ to ∇ . Using (ET3),we complete the map to a morphism of E -triangles ϕ : ∆ → ∇ . Then π ′ ϕi is a morphismof F -triangles from ∆ to ∆ . This finishes the proof. (cid:3) We can check that ( e C , F , c ) is compatible to (ET3) op in the same way. Proposition 3.5.
The idempotent completion of a pre-extriangulated category has a nat-ural structure of a pre-extriangulated category.Proof.
It follows from Lemmas 3.2, 3.3 and Proposition 3.4. (cid:3)
Let ( C , E , s ) be a pre-extriangulated category. We say an E -triangle is splitting if itrealizes a split E -extension. The E -triangle A x −→ B y −→ C δ sometimes is denoted bythe triplet ( x, y, δ ). Lemma 3.6.
Consider the following commutative diagram A x (cid:15) (cid:15) u / / B y (cid:15) (cid:15) v / / C z (cid:15) (cid:15) δ / / ❴❴❴ A ′ a / / B ′ b / / C ′ where the vertical morphisms are isomorphism. Then ( a, b, x ∗ ( z − ) ∗ δ ) is an E -triangle.Proof. First of all, by [14, Proposition 3.7], we know that A ′ ux − −→ B zv −→ C ′ x ∗ ( z − ) ∗ δ isan E -triangle in C . Note that a = yux − and b = zvy − . It follows that A ′ a −→ B ′ b −→ C ′ x ∗ ( z − ) ∗ δ is an E -triangle in C . (cid:3) Lemma 3.7.
Consider the E -triangle ∆ : A ⊕ X a u x ! −→ B ⊕ Y b v y ! −→ C ⊕ Z c w z ! in a pre-extriangulated category. If ( a, b, c ) is a direct sum of two split E -triangles (id A , , and (0 , id C , . Then ( x, y, z ) is an E -triangle. Dually, If ( x, y, z ) is a direct sum of twosplit E -triangles (id X , , and (0 , id Z , . Then ( a, b, c ) is an E -triangle.Proof. Suppose that ( a, b, c ) is a direct sum of two split E -triangles (id A , ,
0) and(0 , id C , A ⊕ X u u x −→ A ⊕ C ⊕ Y v y ! −→ C ⊕ Z w z ! . Consider the following commutative diagram∆ (cid:15) (cid:15) A ⊕ X m (cid:15) (cid:15) u u x / / A ⊕ C ⊕ Y n (cid:15) (cid:15) v y ! / / C ⊕ Z l (cid:15) (cid:15) ∆ A ⊕ X x / / A ⊕ C ⊕ Y y ! / / C ⊕ Z where m = (cid:18) u (cid:19) , n = v ! , l = (cid:18) (cid:19) . It is easy to check that the threemorphisms yield an isomorphism of ∆ and ∆ . It is a straightforward verification that m ∗ (cid:18) w z (cid:19) = l ∗ (cid:18) z (cid:19) . By Lemma 3.6, we know that ∆ is a conflation and realizes (cid:18) z (cid:19) . By Lemma 2.11, we obtain that ( x, y, z ) is an E -triangle. For the dual case, it isproved similarly. (cid:3) Now we are ready to prove Theorem 3.1.
Proof of Theorem 3.1.
By Proposition 3.5, it is enough to show that e C is compatible to (ET4) and (ET4) op .Let e ω ∈ F ( e D, e A ) and e ω ′ ∈ F ( e F , e B ) be two F -extensions respectively realized by e A f −→ e B f ′ −→ e D e ω (3.4)and e B g −→ e C g ′ −→ e F e ω ′ . (3.5)There exist e X, e Y and e Z such that e A ⊕ e X , e D ⊕ e Y , e F ⊕ e Z ∈ C . Clearly, e X id e X −→ e X −→ (3.6)and 0 −→ e Y id e Y −→ e Y (3.7) DEMPOTENT COMPLETION OF EXTRIANGULATED CATEGORIES 13 are split F -triangles. Considering the direct sum of (3 . .
6) and (3 . F -triangle: e A ⊕ e X f
00 10 0 −→ e B ⊕ e X ⊕ e Y f ′ ! −→ e D ⊕ e Y f ω (3.8)where f ω = (cid:18) e ω
00 0 (cid:19) . Observe that f ω ∈ E ( e D ⊕ e Y , e A ⊕ e X ) fits into an E -triangle of C which is, via ι , an F -triangle of e C . These two triangles are isomorphic since e C is apre-extriangulated category. Then (3 .
8) is an E -triangle for f ω in C .Similarly, we consider the following F -triangle in e C by the direct sum of (3 . e Y , , , id e Z , e B ⊕ e Y g
00 10 0 −→ e C ⊕ e Y ⊕ e Z g ′ ! −→ e F ⊕ e Z e ξ (3.9)where e ξ = (cid:18) e ω ′
00 0 (cid:19) . Adding (3 .
6) to (3 . e B ⊕ e X ⊕ e Y h −→ e C ⊕ e X ⊕ e Y ⊕ e Z h ′ −→ e F ⊕ e Z f ω , (3.10)where h = g , h ′ = (cid:18) g ′ (cid:19) and f ω = e ω ′
00 00 0 ! .By (ET4), we have the following commutative diagram since (3 .
8) and (3 .
10) are E -triangles in C . e A ⊕ e X f
00 10 0 / / e B ⊕ e X ⊕ e Y h (cid:15) (cid:15) f ′ ! / / e D ⊕ e Y α =( α α ) (cid:15) (cid:15) f ω / / ❴❴❴ e A ⊕ e X gf
00 10 00 0 / / e C ⊕ e X ⊕ e Y ⊕ e Z h ′ (cid:15) (cid:15) i =( i i i i ) / / H β = β β ! (cid:15) (cid:15) f ω / / ❴❴❴❴ e F ⊕ e Z f ω (cid:15) (cid:15) ✤✤✤ e F ⊕ e Z f ω (cid:15) (cid:15) ✤✤✤ By some direct calculations, we obtain ( α f ′ , , α ) = ( i g, i , i ) β i = g ′ , β i = 1( α α ) • f ω = ( α α ) • ( f ω f ω ) T = (cid:18) α • g ω α • g ω (cid:19) = f ω f
00 10 0 • ( f ω f ω ) T = f • g ω g ω = β • f ω = β • e ω ′ (cid:18) f ′ (cid:19) • f ω = (cid:18) f ′• e ω ′
00 0 (cid:19) = f ω = (cid:18) g ω g ω g ω g ω (cid:19) . Thus f ω = (cid:18) g ω
00 0 (cid:19) , where f ω ∈ F ( e F , e D ). We have F -triangles e D p −→ e K p −→ e F g ω (3.11)and e Y ! −→ e Y ⊕ e Z ( ) −→ e Z . (3.12)Consider the direct sum of (3 .
11) and (3 . s = ( s , s , s ) T : H → e K ⊕ e Y ⊕ e Z with its inverse s ′ = ( s ′ , s ′ , s ′ ) such that sα = s α s α s α s α s α s α ! = p
00 10 0 ! βs ′ = (cid:18) β s ′ β s ′ β s ′ β s ′ β s ′ β s ′ (cid:19) = (cid:18) p (cid:19) . Let us modify the diagram above. Firstly, we consider the following commutativediagram: e A ⊕ e X / / e C ⊕ e X ⊕ e Y ⊕ e Z i / / H s (cid:15) (cid:15) f ω / / ❴❴❴❴❴ e A ⊕ e X / / e C ⊕ e X ⊕ e Y ⊕ e Z si / / e K ⊕ e Y ⊕ e Z f ω / / ❴❴❴ where f ω = s ′• f ω = (cid:18) s ′• g ω s ′• g ω s ′• g ω (cid:19) . Obviously, the second row is an E -trianglefor ω ∈ E ( e K ⊕ e Y ⊕ e Z, e A ⊕ e X ) by the isomorphism. Now we have a new commutative DEMPOTENT COMPLETION OF EXTRIANGULATED CATEGORIES 15 diagram in e C as follows: e A ⊕ e X f
00 10 0 / / e B ⊕ e X ⊕ e Y h (cid:15) (cid:15) f ′ ! / / e D ⊕ e Y sα (cid:15) (cid:15) f ω / / ❴❴❴❴ e A ⊕ e X / / e C ⊕ e X ⊕ e Y ⊕ e Z h ′ (cid:15) (cid:15) si / / e K ⊕ e Y ⊕ e Z βs ′ (cid:15) (cid:15) f ω / / ❴❴❴ e F ⊕ e Z f ω (cid:15) (cid:15) ✤✤✤ e F ⊕ e Z f ω (cid:15) (cid:15) ✤✤✤ Calculating again, we have s i g = p f ′ , s α = s α = 0 , s α = 1 p s i = g ′ , s i = 0 , s i = 1 , s i = 0( sα ) • f ω = f ω , e ω = p • s ′• f ω , s ′• f ω = 0( βs ′ ) • f ω = f
00 10 0 • f ω , p • e ω ′ = f • s ′• f ω . Obviously, we have si = s i s i s i s i ! and f ω = (cid:18) s ′• g ω s ′• g ω (cid:19) . Putting all thesetogether, we obtain the following commutative diagram in e C : e A f / / e B g (cid:15) (cid:15) f ′ / / e D p (cid:15) (cid:15) e ω / / ❴❴❴ e A gf / / e C g ′ (cid:15) (cid:15) s i / / e K p (cid:15) (cid:15) s ′ • g ω / / ❴❴❴ e F e ω ′ (cid:15) (cid:15) ✤✤✤ e F g ω (cid:15) (cid:15) ✤✤✤ which is compatible to (ET4). It should be noted that e A gf −→ e C s i −→ e K s ′ • g ω (3.13) is an F -triangle in e C . Indeed, we have the following morphisms of F -triangles △ (1) (cid:127) _ (cid:15) (cid:15) e X (cid:127) _ (cid:15) (cid:15) ( ) / / e X ⊕ e Y (cid:127) _ (cid:15) (cid:15) (0 1) / / e Y (cid:127) _ (cid:15) (cid:15) / / ❴❴❴❴❴ △ (2) (cid:127) _ (cid:15) (cid:15) e X ⊕ e A (cid:127) _ (cid:15) (cid:15) gf / / e X ⊕ e Y ⊕ e C (cid:127) _ (cid:15) (cid:15) s i s i ! / / e Y ⊕ e K (cid:127) _ (cid:15) (cid:15) s ′• g ω ! / / ❴❴❴❴ △ (3) e A ⊕ e X σ / / e C ⊕ e X ⊕ e Y ⊕ e Z µ / / e K ⊕ e Y ⊕ e Z f ω / / ❴❴❴ where σ = gf
00 10 00 0 , µ = s i s i s i s i ! , f ω = (cid:18) s ′• g ω s ′• g ω (cid:19) , and all ֒ → are thecanonical embeddings. By Lemma 3.7, we know that △ (2) is an F -triangle in e C . It followsthat (3 .
13) is an F -triangle since △ (1) is a splitting F -triangle. (cid:3) Recall that an additive functor between two extriangulated categories is exact if thefunctor sends conflations to conflations.
Corollary 3.8.
Let C be an extriangulated category. Then its idempotent completion e C admits a smallest structure of an extriangulated category such that the canonical functor i : C → e C becomes exact. Moreover, if e C is endowed with this structure, then for eachidempotent complete extriangulated category D , the functor ι induces an equivalence Hom exact ( e C , D ) ∼ / / Hom exact ( C , D ) , where Hom exact denotes the category of exact functors between two extriangulated cate-gories.Proof.
Suppose that e C has another extriangulated structure such that i : C → e C becomes exact, then it contains all F -triangles by Lemma 2.11 and Lemma 3.3. Then therest follows from Lemma 2.3 and Lemma 2.11. (cid:3) Remark 3.9.
Recall that each exact structure or triangulated structure forms an extrian-gulated structure in the sense of [14, Example 2.13] and [14, Proposition 3.22], respectively.Let ( C , E , s ) be an extriangulated category which is neither exact nor triangulated. ByTheorem 3.1, we know that the idempotent completion ( e C , F , c ) of ( C , E , s ) also carriesan extriangulated structure. We claim that this extriangulated structure is neither exactnor triangulated.Assume that ( e C , F , c ) is an exact category. By [8, Corollary 3.18], we know that anyinflation in e C is monomorphic, and any deflation in e C is epimorphic. It follows that anyinflation in C is monomorphic, and any deflation in C is epimorphic. Thus, by again [8, DEMPOTENT COMPLETION OF EXTRIANGULATED CATEGORIES 17
Corollary 3.18], ( C , E , s ) is exact, this is a contradiction. Hence, ( e C , F , c ) is not exact.Next, we prove ( e C , F , c ) is not triangulated.First of all, we observe that each project object in C is also projective in e C . Indeed, let P be a projective object in C , i.e., E ( P, B ) = 0 for any B ∈ C . For any e B = ( B, e ) ∈ e C ,take e B ′ = ( B, id B − e ) ∈ e C , then F ( P, e B ) ⊕ F ( P, e B ′ ) ∼ = F ( P, e B ⊕ e B ′ ) ∼ = E ( P, B ) = 0, thus F ( P, e B ) = 0, that is, P is projective in e C . Similarly, we can prove each injective objectin C is also injective in e C .Suppose that ( e C , F , c ) is a triangulated category. By [14, Corollary 7.6], ( e C , F , c ) isFrobenius with P = I = 0. According to the observations above, we obtain that C hasno nonzero projective or injective objects. Now we prove C has enough projectives andinjectives. For any non-zero object C = ( C, id C ) ∈ e C , there exists an F -triangle e A = ( A, e ) −→ −→ ( C, id C ) e ω (3.14)since e C has enough projectives and P = 0 in e C . Take e A ′ = ( A, id A − e ), we consider thefollowing F -triangle obtained by the direct sum of (3.14) and the split F -triangle (id e A ′ , , A, id A ) −→ e A ′ −→ ( C, id C ) e ω
00 0 ! . (3.15)Then e A ′ ∈ C , since e ω
00 0 ! ∈ E ( C, A ). It follows that either e = 0 or e = id A . If e = 0,then e A ∼ = 0, thus by (3.14), C = 0, this contradicts the hypothesis C is nonzero. Hence, e = id A . Thus, the F -triangle (3.14) is actually an E -triangle A −→ −→ C e ω , so C has enough projectives. Similarly, it is proved that C has enough injectives. Thus, by[14, Corollary 7.6], C is triangulated, this is a contradiction. Hence, ( e C , F , c ) is not atriangulated category.4. Applications
In this section, we give two applications about the idempotent completion of extrian-gulated categories.4.1.
Cotorsion pairs.
In this subsection, we focus our attention on cotorsion pairs in theidempotent completion of an extriangulated category. Let ( C , E , s ) be an extriangualtedcategory with the idempotent completion ( e C , F , c ).Recall that the pair ( T , F ) is called a cotorsion pair in C if it satisfies the followingconditions: • E ( T , F ) = 0 . • For any C ∈ C , there exists a conflation F −→ T −→ C satisfying F ∈ F , T ∈ T . • For any C ∈ C , there exists a conflation C −→ F ′ −→ T ′ satisfying F ′ ∈ F , T ′ ∈ T . Obviously, the cotorsion pair induces a pair of subcategories ( e T , e F ) in e C , where e T = { ( T, t ) | T ∈ T , t ∈ End ( T ) is an idempotent } and e F = { ( F, f ) | F ∈ F , f ∈ End ( F ) is an idempotent } . Theorem 4.1.
Let C be an extriangulated category. Then any cotorsion pair ( T , F ) in C which satisfies C ( T , F ) = 0 induces a cotorsion pair ( e T , e F ) in e C .Proof. We have to verify the three conditions in the definition of cotorsion pairs.(1) For any two objects e T = ( T, t ) ∈ e T and e F = ( F, f ) ∈ e F . Clearly, F ( e T , e F ) = 0 since E ( T, F ) = 0.(2) For any e C = ( C, e ) ∈ e C with C ∈ C . Then there exists an E -triangle in C ∆ : F x −→ T y −→ C δ satisfying F ∈ F , T ∈ T . Set C = ( C, e ), C = ( C, id C − e ), then we have ( C, id C ) ∼ = C ⊕ C . For the idempotent e ∈ C ( C, C ), since E ( T, F ) = 0, we obtain a morphism of E -triangles: F x / / f (cid:15) (cid:15) T y / / t (cid:15) (cid:15) C e (cid:15) (cid:15) δ / / ❴❴❴ F x / / T y / / C δ / / ❴❴❴ . Since yt = ey , we have yt = eyt = e y = ey . Thus there exists h : F → F such that( h, t , e ) is also a morphism of E -triangles from ∆ to ∆. It follows that h = f since h ∗ δ = e ∗ δ = f ∗ δ . On the one hand, there exists a morphism s : T → F such that xs = t − t since y ( t − t ) = 0. Hence ( t − t ) = ( t − t ) xs = ( t x − tx ) s = 0. On theother hand, we set r = t + u − tu where u = t − t . Noting that u = 0 and ut = tu , weobtain that r = t + 2 tu − t u and then replacing t by u + t , we have r = t + u − tu .It is easy to see that r makes the right hand square commutative. By Lemma 2.10, thereexists an idempotent g : F → F such that ( g, r, e ) is a morphism of E -triangles from ∆to ∆. Note that F ≃ ( F, g ) ⊕ ( F, id F − g ) =: F ⊕ F , T ≃ ( T, r ) ⊕ ( T, id T − r ) =: T ⊕ T and we have the following commutative diagram( F, id F ) x / / ∼ = (cid:15) (cid:15) ( T, id T ) y / / ∼ = (cid:15) (cid:15) ( C, id C ) ∼ = (cid:15) (cid:15) e δ / / ❴❴❴ F ⊕ F v / / T ⊕ T w / / C ⊕ C e δ ′ / / ❴❴❴ where v = (cid:18) xg x (id F − g ) (cid:19) , w = (cid:18) yr y (id T − r ) (cid:19) . In addition, by Lemma 3.6, we know that e δ ′ = (cid:18) g • e • e δ
00 (id F − g ) • (id C − e ) • e δ (cid:19) . This shows that DEMPOTENT COMPLETION OF EXTRIANGULATED CATEGORIES 19 F xf −→ T yt −→ C g • e • e δ is an F -triangle in e C . Similarly, we can prove the third condition in the definition of acotorsion pair. (cid:3) It should be noted that if C is a triangulated category, then ( T , F ) is a cotorsion pairif and only if ( T [ − , F ) is a torsion pair in [10, Definition 2.2]. In particular, we havethe following Corollary 4.2.
Let C be a triangulated category. Then any torsion pair ( T , F ) in C which satisfies F [ − ⊆ F or T [1] ⊆ T induces a torsion pair in e C . Recollements.
In this subsection, we introduce the notion of right (left) recolle-ments of extriangulated categories, and prove that the idempotent completion of a right(resp. left) recollement of extriangulated categories is still a right ( resp. left) recollement,which generalizes [5, Theorem 4].Let C and D be additive categories, and F : C → D an additive functor. Then thereis an additive functor e F : e C / / e D ( A, e ) ✤ / / α (cid:15) (cid:15) ( F A, F e ) F α (cid:15) (cid:15) ( A ′ , e ′ ) ✤ / / ( F A ′ , F e ′ ) . Combining the functor ι C : C → e C given by A ( A, id A ), one has a commutativediagram as follows: C ι C / / F (cid:15) (cid:15) e C e F (cid:15) (cid:15) D ι D / / e D . Lemma 4.3.
Let C and D be extriangulated categories, and F : C → D an exact functor.Then the induced functor e F : e C → e D is exact.Proof. Let △ be an F -triangle in e C . Then there exists an F -triangle △ ′ in e C such that △ ⊕ △ ′ is an E -triangle in C . Since F is exact, F ( △ ⊕ △ ′ ) ∼ = e F ( △ ) ⊕ e F ( △ ′ ) is an E -trianglein D . Thus e F ( △ ) is an F -triangle in e D , which shows that e F is exact. (cid:3) Now given two additive functors
F, G and a natural transformation η as η (cid:11) (cid:19) C F ' ' G D . Then there is a natural transformation e η as e η (cid:11) (cid:19) e C e F & & e G e D satisfying e η ( A,e ) = G ( e ) ◦ η A ◦ F ( e ) for any ( A, e ) ∈ e C . Lemma 4.4.
Keep the notations as above. Let C and D be extriangulated categories. If ( F, G ) is an adjoint pair of exact functors, then so is ( e F , e G ) .Proof. Assume that (
F, G ) is an adjoint pair. Then for any A ∈ C and any B ∈ D ,there is a natural isomorphism η A,B : Hom D ( F A, B ) ≃ −→ Hom C ( A, GB ) . We claim that there is a induced natural isomorphism e η A,B : Hom e D ( e F ( A, e ) , ( B, f )) −→ Hom e C (( A, e ) , e G ( B, f ))for any (
A, e ) ∈ e C and any ( B, f ) ∈ e D . Indeed, for any α ∈ Hom e D ( e F ( A, e ) , ( B, f )), wehave α ◦ F ( e ) = α = f ◦ α ⇒ η A,B ( α ) ◦ e = η A,B ( α ) = G ( f ) ◦ η A,B ( α ) ⇒ η A,B ( α ) ∈ Hom e C (( A, e ) , e G ( B, f )) . Moreover, let α ∈ Hom D ( F A, B ) and η A,B ( α ) ∈ Hom e C (( A, e ) , e G ( B, f )). Then η A,B ( α ) ◦ e = η A,B ( α ) = G ( f ) ◦ η A,B ( α ) ⇒ α ◦ F ( e ) = α = f ◦ α ⇒ α ∈ Hom e D ( e F ( A, e ) , ( B, f )) . This finishes the proof. (cid:3)
Definition 4.5.
Let C , C ′ and C ′′ be three extriangulated categories. A right recollement of C relative to C ′ and C ′′ is a diagram C ′ i ! ( ( C i ! i i j † ) ) C ′′ j † h h given by exact functors i ! , i ! , j † and j † which satisfies the following conditions:(RR1) ( i ! , i ! ) and ( j † , j † ) are adjoint pairs.(RR2) i ! j † = 0.(RR3) i ! and j † are fully faithful.(RR4) For each A ∈ C , there is an E -triangle i ! i ! A θ A / / A ϑ A / / j † j † A δ / / ❴❴❴ in C , where θ A and ϑ A are given by the adjunction morphisms. DEMPOTENT COMPLETION OF EXTRIANGULATED CATEGORIES 21
Now we show that the idempotent completion of a right recollement of extriangulatedcategories is still a right recollement.
Theorem 4.6.
Let C , C ′ and C ′′ be three extriangulated categories. Assume that thereis a right recollement C ′ i ! ( ( C i ! i i j † ) ) C ′′ . j † h h Then e C admits a right recollement relative to f C ′ and f C ′′ f C ′ e i ! ( ( e C e i ! i i e j † ) ) f C ′′ . e j † h h Proof. (RR1) By Lemma 4.4, ( e i ! , e i ! ) and ( e j † , e j † ) are adjoint pairs of exact functors.(RR2) By assumption, i ! j † = 0. Thus e i ! e j † = f i ! j † = 0.(RR3) Since we can view each object of f C ′ as a direct summand of C ′ , it follows from[5, Lemma 11] that e i ! is fully faithful. Similarly, e j † is fully faithful.(RR4) Let ( A, e ) ∈ e C . Then there is a commutative diagram i ! i ! A θ A / / i ! i ! e (cid:15) (cid:15) A ϑ A / / e (cid:15) (cid:15) j † j † A δ / / ❴❴❴ j † j † e (cid:15) (cid:15) i ! i ! A θ A / / A ϑ A / / j † j † A δ / / ❴❴❴ in C . In particular, ( i ! i ! e ) • δ = ( j † j † e ) • δ . Setting A = ( A, e ) and A = ( A, id A − e ). Thenwe have the following commutative diagram i ! i ! A θ A / / i ! i ! ei ! i ! (id A − e ) ! (cid:15) (cid:15) A ϑ A / / e id A − e ! (cid:15) (cid:15) j † j † A (cid:15) (cid:15) j † j † ej † j † (id A − e ) ! (cid:15) (cid:15) e i ! e i ! A ⊕ e i ! e i ! A e θ A e θ A ! / / A ⊕ A e ϑ A e ϑ A ! / / e j † e j † A ⊕ e j † e j † A , where the vertical morphisms are isomorphisms by [5, Lemma 12]. Moreover, by Lemma3.6, e i ! e i ! A ⊕ e i ! e i ! A e θ A e θ A ! / / A ⊕ A e ϑ A e ϑ A ! / / e j † e j † A ⊕ e j † e j † A δ ′ / / ❴❴❴❴ is an E -triangle, where δ ′ = ( j † j † e, j † j † (id A − e )) • i ! i ! ei ! i ! (id A − e ) ! • δ = ( i ! i ! e ) • ( j † j † e ) • δ ( i ! i ! e ) • ( j † j † (id A − e )) • δ ( i ! i ! (id A − e )) • ( j † j † e ) • δ ( i ! i ! (id A − e )) • ( j † j † (id A − e )) • δ ! . Here, ( i ! i ! e ) • ( j † j † (id A − e )) • δ = ( j † j † (id A − e )) • ( i ! i ! e ) • δ = ( j † j † (id A − e )) • ( j † j † e ) • δ = ( j † j † ( e (id A − e ))) • δ = 0 . Similarly, ( i ! i ! (id A − e )) • ( j † j † e ) • δ = 0. Thus δ ′ = ( i ! i ! e ) • ( j † j † e ) • δ
00 ( i ! i ! (id A − e )) • ( j † j † (id A − e )) • δ ! . Set δ A = ( i ! i ! e ) • ( j † j † e ) • δ and δ A = ( i ! i ! (id A − e )) • ( j † j † (id A − e )) • δ . Then e i ! e i ! A e θ A / / A e ϑ A / / e j † e j † A δ A / / ❴❴❴ and e i ! e i ! A e θ A / / A e ϑ A / / e j † e j † A δ A / / ❴❴❴ are F -triangles in e C . Using a similar argument to that of [5, Remark 10], we know that e θ A , e θ A , e ϑ A and e ϑ A are the adjunction morphisms. This completes the proof. (cid:3) Dually, one has the notion of left recollements of extriangulated categories.
Definition 4.7. A recollement of extriangulated categories is a diagram C ′ i † = i ! / / C i † v v i ! h h j ! = j † / / C ′′ j ! w w j † g g of extriangulated categories and exact functors such that C ′ i ! / / C i ! i i j † / / C ′′ j † h h is a right recollement, and C ′ i † / / C i † u u j ! / / C ′′ j ! v v is a left recollement.Following Theorem 4.6 and its dual, we have the following Theorem 4.8.
Let C , C ′ and C ′′ be three extriangulated categories. Assume that thereis a recollement C ′ i † = i ! / / C i † v v i ! h h j ! = j † / / C ′′ . j ! v v j † h h Then e C admits a recollement relative to f C ′ and f C ′′ f C ′ e i † = e i ! / / e C e i † v v e i ! h h e j ! = e j † / / f C ′′ . e j ! v v e j † h h DEMPOTENT COMPLETION OF EXTRIANGULATED CATEGORIES 23
Acknowledgments
The third author is grateful to Panyue Zhou for his helpful suggestions and comments.This work is supported partially by the National Natural Science Foundation of China(No.s 11801273, 11901341).
References [1] P. Balmer, M. Schlichting, Idempotent completion of triangulated categories, J. Algebra (2001),819–834.[2] A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, In: Analysis and topology on singularspaces, I (Luminy, 1981), Ast´erisque , pp. 5–171. Soc. Math. France, Paris, 1982.[3] A. Beligiannis, N. Marmaridis, Left triangulated categories arising from contravariantly finite sub-categories, Comm. Algebra (1994), 5021–5036.[4] T. B¨uhler, Exact categories, Expo. Math. (2010), 1–69.[5] Q. Chen, L. Tang, Recollements, idempotent completions and t -structures of triangulated categories,J. Algebra (2008), 3053–3061.[6] P. Dr¨axler, I. Reiten, O. Smalø, Ø. Solberg, Exact categories and vector space categories, With anappendix by B. Keller, Trans. Amer. Math. Soc. 351 (1999) no. 2, 647–682.[7] D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, Lon-don Math. Soc. Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988.[8] M. Herschend, Y. Liu, H. Nakaoka, n -exangulated categories, arXiv: 1709.06689, 2017.[9] J. Hu, D. Zhang, P. Zhou, Proper classes and Gorensteinness in extriangulated categories, J. Algebra (2020), 23–60.[10] O. Iyama, Y. Yoshino, Mutations in triangulated categories and rigid Cohen-Macaulay modules,Invent. Math. (2008), 117–168.[11] J. Le, X. Chen, Karoubianness of a triangulated category, J. Algebra (2007), 452–457.[12] J. Liu, L. Sun, Idempotent completion of pretriangulated categories, Czechoslovak Math. J. (2014), 477–494.[13] Y. Liu, H. Nakaoka, Hearts of twin cotorsion pairs on extriangulated categories, J. Algebra (2019), 96–149.[14] H. Nakaoka, Y. Palu, Extriangulated categories, Hovey twin cotorsion pairs and model structures,Cah. Topol. Gom. Diffr. Catg. (2019), 117–193.[15] R. W. Thomason, The classification of triangulated categories, Compos. Math. (1997), 1–27.[16] J. L. Verdier, cat´ e gories d´ e riv´ e es, ´ e tat0, J. Algebra (1981), 262–317.[17] T. Zhao, L. Tan, Z. Huang, Almost split triangles and morphisms determined by objects in extrian-gulated categories, J. Algebra (2020), 346–378.[18] P. Zhou, B. Zhu, Triangulated quotient categories revisited, J. Algebra (2018), 196–232. Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal Univer-sity, Nanjing 210023, P. R. China.
E-mail address : [email protected] (Wang); [email protected] (Wei); [email protected](Zhang). School of Mathematical Sciences, Qufu Normal University, Qufu 273165, P. R. China
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