Proc. Indian Acad. Sci. (Math. Sci.) Vol. 127, No. 4, September 2017, pp. 625–656. DOI 10.1007/s12044-017-0345-4

Homological in n-abelian categories

DEREN LUO

College of , Hunan Institute of Science and Technology, Yueyang 414006, Hunan, People’s Republic of China E-mail: [email protected]

MS received 15 March 2015; revised 19 July 2015; published online 16 August 2017

Abstract. In this paper, we study the homological theory in n-abelian categories. First, we prove some useful properties of n-abelian categories, such as (n+2)×(n+2)-lemma, 5-lemma and n-Horseshoes lemma. Secondly, we introduce the notions of right(left) n- derived of left(right) n-exact functors, n-(co)resolutions, and n-homological dimensions of n-abelian categories. For an n-exact , we show that the long n- theorem holds as a generalization of the classical long exact sequence ∗ ∗ theorem. As a generalization of Ext (−, −), we study the n-derived nExt (−, −) of hom-functor Hom(−, −). We give an between the abelian of m m equivalent classes of m-fold n-extensions nE (A, B) of A, B and nExtA(A, B) using n-Baer sum for m, n ≥ 1.

∗ Keywords. n-Abelian ; n-; nExt -correspondence; n-Baer sum; n-cluster tilting.

2010 Mathematics Subject Classification. 18G50; 18G15; 18E25.

1. Introduction Recently, a new of categories called 2-cluster tilting that appeared in were introduced by Buan et al.[4], and the class of n-cluster tilting subcategories was developed by Iyama and Yoshino [12]. And then, from the viewpoint of higher Auslander-Reiten theory, Iyama [8–10] investigated and introduced the notion of n-almost-split which are n-exact sequences in the sense of Jasso [13]. He developed the classical and theory to higher-dimensional n-abelian category and n-exact [13]. He also proved that n-cluster tilting subcategories are n-abelian categories. These new discoveries have broken new ground in category theory. , as a connected system of notions and results, was first developed for categories of modules by Cartan and Eilenberg [6] and was immediately generalized by Buchsbaum [5], Mac Lane [14] and Heller [7] to exact categories and abelian categories. Homological algebra can also construct on various nonabelian categories, such as pre- abelian category [20], all of their derived functors are defined on right(left) exact functors of certain short exact sequences via (co) of (co)resolutions under the right(left) exact functors.

© Indian Academy of Sciences 625 626 Deren Luo

In this paper, we study the homological theory of n-abelian categories as a generalization of homological theory of abelian categories compared to higher homological theory of abelian categories via higher (co)homology of n-(co)resolutions under right(left) n-exact functors for short n-exact sequences. This paper is organized as follows. In § 2, we recall some notions and notations of n-abelian categories, and study some properties. In section 3, we study the relationship between n-exact sequences and exact sequences, and introduce the (n + 2) × (n + 2)- lemma and 5-lemma of n-abelian categories as generalizations of classical 3 × 3-lemma and 5-lemma of abelian categories. In § 4, we introduce the notions of right(left) n- derived functors of left(right) n-exact functors, n-(co), and n-homological ∗ dimensions, specially, we introduce the functor nExt (−, −) as a generalization of ∗ Ext (−, −). We study some basic properties of n-derived functors, and prove the long n-exact sequence theorem and n-Horseshoes lemma. In § 5, we study the n-extension and m-fold n-extension groups, and we prove that there is an isomorphism nEm(A, B)  m m nExtA(A, B) of the group of equivalence classes of m-fold n-extension group nE (A, B) m (this is an under n-Baer sum) and nExtA(A, B) in n-abelian cate- m gories. This proves that we can define nExtA(A, B) without mentioning projectives or injectives.

2. n-Abelian categories Let n be a positive integer and C an . We denote the category of cochain complexes of C by Ch(C) and the category of C by H(C). Also, we denote by Chn(C) the full of Ch(C) given by all complexes

d0 d1 dn−1 dn X 0 → X 1 →··· → X n → X n+1 which are concentrated in degrees 0, 1,...,n + 1. We write C(X, Y ) for the in C from X to Y ,ifX, Y ∈ obC.

2.1 n-Kernels, n- and n-exact sequences Let C be an additive category and d0 : X 0 → X 1 a in C.Ann- of d0 is a sequence of morphisms

d1 d2 dn (d1,...,dn) : X 1 → X 2 → X 3 →···→ X n+1 such that for all Y ∈ C the induced sequence of abelian groups + 0 → C(X n 1, Y ) → C(X n, Y ) →···→C(X 1, Y ) → C(X 0, Y ) is exact. Equivalently, the sequence (d1,...,dn) is an n-cokernel of d0 if for all 1 ≤ k ≤ n − 1 the morphism dk is a weak cokernel of dk−1, and dn is moreover a cokernel of dn−1. The concept of n- of a morphism is defined dually. If n ≥ 2, the n-cokernels and n-kernels are not unique in general, but they are unique up to isomorphism in H(C) [13]. d0 d1 An n-exact sequence in C is an n-kernel-n-cokernel pair, i.e., a complex X 0 → X 1 → dn−1 dn ··· → X n → X n+1 in Chn(C) such that (d0,...,dn−1) is an n-kernel of dn, and (d1,...,dn) is an n-cokernel of d0. Homological algebra in n-abelian categories 627

We recall the Comparison lemma, together with its dual, plays a central role in the sequel.

≥ Lemma 2.1 [13, Comparison lemma 2.1]. Let C be an additive category and X ∈ Ch 0(C) ≥ k+1 k : → a complex such that for all k 0 the morphism dX is a weak cokernel of dX .Iff X Y ≥ and g : X → Y are morphisms in Ch 0(C) such that f 0 = g0, then there exists a homotopy h : f → g such that h1 = 0.

Lemma 2.2. In additive category, any n-exact sequence, whose number of nonzero terms less than n + 2, are contractible.

Proof. For any n-exact sequence X, the number of nonzero terms is less than n + 2. We can split X into of three classes of n-exact sequences as follows: 0 1 i (a) Y → Y →···→Y → 0 →···→0forsomei ≤ n, Ys = 0 for any s. (b) 0 → ··· → 0 → Y i → ··· → Y j → 0 → ··· → 0forsome1≤ i < j ≤ n, Ys = 0 for any s. i n+1 (c) 0 →···→0 → Y →···→Y for some i ≥ 1, Ys = 0 for any s. Since M → 0 and 0 → M are split and split respectively, by [13, Proposition 2.6], (a), (b) and (c) are contractible n-exact sequences so is X. 

2.2 n-Pushout, n-pullback and n-bicartesian diagrams − Let f : X → Y be a morphism of complexes in Chn 1(C)

XX0 X 1 ··· X n−1 X n − . f f 0 f 1 f n 1 f n YY0 Y 1 ··· Y n−1 Y n

The mapping cone C = C( f ) ∈ Chn(C) is

− − − d 1 d0 dn 2 dn 1 X 0 →C X 1 ⊕ Y 0 →···C →C X n ⊕ Y n−1 →C Y n, where   − k+1 k := dX 0 : k+1 ⊕ k → k+2 ⊕ k+1 dC k+1 k X Y X Y f dY   − −d0 − − for each k ∈{−1, 0,...,n − 1}. In particular, d 1 = X and dn 1 = ( f n dn 1). C f 0 C Y (1) The diagram f : X → Y is called an n-pullback diagram of Y along f n if the ( −1,..., n−2) n−1 sequence dC dC is an n-kernel of dC ; (2) The diagram f : X → Y is called an n-pushout diagram of X along f 0 if the ( 0 ,..., n−1) −1 sequence dC dC is an n-cokernel of dC ; (3) The diagram f : X → Y is called an n-bicartesian (or, n-exact diagram)ifthe ( ) = ( −1, 0 ,..., n−1) sequence C f dC dC dC is an n-exact sequence. 628 Deren Luo

Lemma 2.3 [13, Proposition 2.13]. Let C be an additive category, g : X → Z a morphism − of complexes in Chn 1(C) and suppose there exists an n-pushout diagram of X along g0,

XX0 X 1 ··· X n−1 X n . f g0 YY0 = Z 0 Y 1 ··· Y n−1 Y n : → 0 = Then, there exists a morphism of complexes p Y Z such that p 1Z 0 and a homotopy h : fp→ g with h1 = 0. Moreover, these properties determine p uniquely up to homotopy.

Lemma 2.3 shows that n-pushout is unique up to homotopy equivalence. If h = 0in Lemma 2.3, we say that the morphism f : X → Y is a good n-pushout diagram of X along f 0. Dually, we can define the good n-pullback diagram. Definition-Proposition 2.14 of [13] (resp., its dual) has proved that if there exists n-pushout (resp., n-pullback), then there exists a good n-pushout (resp., good n-pullback). We consider a in an abelian category A B C

(I) (II). X Y Z There are two well-known pullback lemmas [14]: (a) If squares (I) and (II) are pullbacks, then (I + II) is a pullback (i.e., diagrams compose). (b) if (I + II) and (II) are pullbacks, then (I) is a pullback. The third possibility:‘if (I + II) and (I) are pullbacks, then (II) is a pullback’. Generally, this does not hold (see [17]). We now show that (a) and (b) hold for any positive integer n.

PROPOSITION 2.4

g f − Let C be an additive category, Z → Y and Y → X are morphisms in Chn 1(C). Then we have the following statements: f g (i) If Y → X is an n-pullback diagram of X along f n and Z → Y is an n-pullback fg diagram of Y along gn, then Z −→ X is an n-pullback diagram of X along f n gn. f fg (ii) If Y → X is an n-pullback diagram of X along f n and Z−→ X is an n-pullback g diagram of X along f n gn, then Z → Y is an n-pullback diagram of Y along gn.

Proof. We have a commutative diagram −1 0 1 n−1 dC(g) dC(g) dC(g) dC(g) C(g) Z0 Z1 ⊕ Y 0 Z2 ⊕ Y 1 ··· Zn ⊕ Y n−1 Y n ϕ ϕ−1=1 ϕ0 ϕ1 ϕn−1 ϕn= n − Z0 − f d 1 d0 d1 dn 1 C( fg) C( fg) C( fg) − C( fg) , C( fg) Z0 Z1 ⊕ X0 Z2 ⊕ X1 ··· Zn ⊕ Xn 1 Xn ψ−1= 0 0 1 n−1 n ψ g ψ ψ ψ ψ =1Xn −1 0 1 n−1 dC( f ) dC( f ) dC( f ) dC( f ) C( f ) Y 0 Y 1 ⊕ X0 Y 2 ⊕ X1 ··· Y n ⊕ Xn−1 Xn (1) Homological algebra in n-abelian categories 629 where     k+1 ϕk = 1Z k+1 0 ,ψk = g 0 k 0 f 01Xk for k ∈{1, 2,...,n − 1}. f g (i) Y → X is an n-pullback diagram of X along f n and Z → Y is an n-pullback n ( −1 ,..., n−2 ) ( −1 ,..., n−2 ) diagram of Y along g , then dC( f ) dC( f ) and dC(g) dC(g) are n-kernels of n−1 n−1 −1 −1 −1 dC( f ) and dC(g) respectively, dC( f ) and dC(g) are . Hence dC( fg) is a : → 0 −1 = monomorphism. Indeed, let u M Z be a morphism such that dC( fg)u 0. Then 0 = = ψ0 −1 = −1 0 0 = −1 dZ u 0. But 0 dC( fg)u dC( f )g u, this implies g u 0 since dC( f ) is a −1 = −1 −1 monomorphism. Thus dC(g)u 0soisu since dC(g) is a monomorphism. So, dC( fg) is a i−1 i = ,..., − monomorphism. Thus dC( fg) is a weak kernel of dC( fg) for i  0  n 1 (we consider i+1 −1, −1, −1, n+1, n+1, n+1 u : → i+1 ⊕ i X Y Y X Y Y as 0 objects). Indeed, let vi M Z X     i+1 i+1 i u = i ψi u = be a morphism such that dC( fg) vi 0, hence we have dC( f ) vi 0. Then   wi i+1 i+1 = , i+1 i+1 i+1 + i vi = : → dZ u 0 f g u dX 0, and there exists a morphism i−1 M     t i i+ − w u 1 − Y i ⊕ Xi−1 such that di 1 = ψ since di 1 is a weak kernel of di . C( f ) ti−1 i vi C( f ) C( f ) Then

i+1 i+1 + i wi = , i wi + i−1 i−1 = vi . g u dY 0 f dX t

Then, we have        i+1 − i+1 i+1 − i+1 i+1 i u = dZ 0 u = dZ u = . dC(g) wi i+1 i wi i+1 i+1 + i wi 0 g dY g u dY   i i−1 i s : → Therefore, since dC(g) is a weak kernel of dC(g), there exists a morhpism i−1 M     h i i+ − s u 1 Zi ⊕ Y i−1 such that di 1 = . We have that C(g) hi−1 wi

i+1 + i i = , i i + i−1 i−1 = wi . u dZ s 0 g s dY h   si Set : M → Zi ⊕ Xi−1,wehave f i−1hi−1 + ti−1      i − i i i−1 s dZ 0 s d = − C( fg) f i−1hi−1 + ti−1 f i gi di 1 f i−1hi−1 + ti−1   X i+1 = u . vi 630 Deren Luo

i−1 i = ,..., − →fg This proves that dC( fg) is a weak kernel of dC( fg) for i 0 n 1, thus Z X is an n-pullback diagram of Y along f n gn. f fg (ii) Because Y → X is an n-pullback diagram of X along f n and Z → X is an n-pullback n n ( −1 ,..., n−2 ) ( −1 ,..., n−2 ) diagram of Z along f g , dC( f ) dC( f ) and dC( fg) dC( fg) are n-kernels of n−1 n−1 −1 −1 −1 dC( f ) and dC( fg) respectively, dC( f ) and dC( fg) are monomorphisms,  so is dC(g). Hence i+1 i−1 i = ,..., − u : → i+1 ⊕ i dC(g) is a weak kernel of dC(g) for i 0 n 1. Indeed, let vi M Z Y   i+1 i u = be a morphism such that dC(g) vi 0, we have   i+1 i+1 i+1 = , i+1 i+1 + i vi = , i ϕi u = . dZ u 0 g u dY 0 dC( fg) vi 0   i i−1 i s : Therefore, since dC( fg) is a weak kernel of dC( fg), there exists a morhpism i−1     h i i+ − s u 1 M → Zi ⊕ Xi−1 such that di 1 = ϕi . Thus we have C( fg) hi−1 vi

i i + i+1 = , i i i + i−1 i−1 = i vi dZ s u 0 f g s dX h f and     vi − i i − i vi + i i i i−1 g s = dY dY g s = . dC( f ) − i−1 i vi − i i i − i−1 i−1 0 h f f g s dX h   wi−1 i−2 i−1 : → Therefore, since dC( f ) is a weak kernel of dC( f ), there exists a morhpism i−2 M     t wi−1 vi − i i i−1 ⊕ i−2 i−2 = g s − i−1wi−1 = Y X such that dC( f ) i−2 − i−1 . Thus, we have dY  t  h    i − i i i+1 vi − i i . i−1 s = dZ s = u g s Then, dC(g) −wi−1 i i − i−1wi−1 vi . This proves that g s dY i−1 i = , ··· , − →g dC(g) is a weak kernel of dC(g) for i 0 n 1, thus Z Y is an n-pullback diagram of Z along gn. 

2.3 n-Abelian categories As a generalization of the notion of classical abelian categories, Jasso introduced the n-abelian categories in [13] as follows.

DEFINITION 2.5 (n-abelian category, [13, Definition 3.1])

An n-abelian category is an additive category A which satisfies the following : (A0) The category A is idempotent complete. (A1) Every morphism in A has an n-kernel and an n-cokernel. Homological algebra in n-abelian categories 631

(A2) For every monomorphism f 0 : X 0 → X 1 in A there exists an n-exact sequence:

f 0 f 1 f n−1 f n X 0 → X 1 →··· → X n → X n+1. (A2op) For every epimorphism f n : X n → X n+1 in A there exists an n-exact sequence:

f 0 f 1 f n−1 f n X 0 → X 1 →··· → X n → X n+1.

Note that 1-abelian categories are precisely abelian categories in the usual sense. For m = n, any category are both m-abelian category and n-abelian category if and only if it is a semisimple category [13, Corollary 3.10].

Lemma 2.6 [13, Proposition 3.7]. Let A be an additive category which satisfies axioms − (A1) and (A2), and let X a complex in Chn 1(A).Ifforall0 ≤ k ≤ n − 1 the morphism dk is a weak cokernel of dk−1, then dn−1 admits a cokernel in A.

Consider a pushout diagram (in abelian category)

f AB g g . f A B It is well known that coker f  coker f , and pushout of monomorphism yields monomor- phism. We can generalize this property to the n case.

PROPOSITION 2.7

Let C be an additive category which satisfies axioms (A0)and (A1), let X be a complex in n−1(C) k k−1 ≤ ≤ − Ch such that the morphism dX is a weak cokernel of dX for 1 k n 1. Let 0 : 0 → 0 n : n → n+1 n−1 f X Y be a morphism. Then there exists a cokernel dX X X of dX ( 1 ,..., n ) 0 such that dX dX is an n-cokernel of dX , and for any n-pushout diagram

− d0 d1 dn 1 dn X 0 X X 1 X ··· X n−1 X X n X X n+1 f 0 f 1 f n−1 f n − d0 d1 dn 1 dn Y 0 Y Y 1 Y ··· Y n−1 Y Y n Y X n+1

( 0 ,..., n−1) 0 n : n → n+1 n−1 of dX dX along f , there exists a cokernel dY Y X of dY such that ( 1 ,..., n ) 0 n = n n 0 dY dY is an n-cokernel of dY and dX dY f . Moreover, if dX is a monomorphism, both rows are n-exact sequences.

n : n → n+1 n−1 Proof. The existence of dX X X of dX is immediately by Lemma 2.6. Since n−1 n−2 n : n → n+1 dC is a cokernel of dC , there exists an unique morphism dY Y X such that n = n n n n−1 = n n dX dY f and dY dY 0. Since dX is an epimorphism so is dY . It remains to show n n−1 : n → n−1 = that dY is a cokernel of dY .Letu Y M be a morphism such that udY 0. Then

( n) n−1 = ( n−1 n−1) = . uf dX u dY f 0 632 Deren Luo

n n−1 v : n+1 → Since dX is a cokernel of dX , there exists a morphism X M such that n = v n uf dX . It follows that n = v n = (v n ) n n−1 = = (v n ) n−1. uf dX dY f and udY 0 dY dY n−1 n−2 = v n n Since dC is a cokernel of dC , u dY . This shows that the epimorphism dY is a n−1 cokernel of dY . k+1 k ≤ ≤ We show that the morphism dY is a weak cokernel of dY for 2 k n, this shows ( 1 , ··· , n ) 0 : k+1 → that dY dY is an n-cokernel of dY . Indeed, let u Y M be a morphism such k = k+1 k = k k = vk+2 : k+2 → that udY 0. Then uf dX udY f 0, there exists X M such that k+1 = vk+2 k+1 k+1 k uf dX since dX is a weak cokernel of dX . Then there exists morphisms vk+3 : k+2 → k+2 : k+2 → n+1 = k+2 k+1 = X M and u Y M(we set X 0) such that u dY u k+1 k k+1 k since dC is a weak cokernel of dC . This shows that dY is a weak cokernel of dY . 0 Moreover, if dX is a monomorphism, by [13, Theorem 3.8(ii)], n-pushout preserve 0 monic, then dY is a monomorphism. By (A2), both the two rows are n-exact sequences. 

Remark 2.8. In Proposition 2.7, we call Y 0 → Y 1 → ··· → Y n → X n+1 an n-exact sequence induced by n-pushout of X 0 → ··· → X n+1 along f 0, ( f 0,..., f n, 1) is a morphism induced by n-pushout of X 0 → ··· → X n+1 along f 0. Then Y 0 → Y 1 → ··· → Y n → X n+1 is uniquely determined up to equivalence of n-exact sequences by Proposition 5.3.

Proposition 2.7 has also been generalized [13, Proposition 2.12] to be a necessary and sufficient condition. The following proposition is a characterization of composition of n-exact diagrams as an application of Proposition 2.4.

PROPOSITION 2.9

g f − Let A be an n-abelian category, Z → Y and Y → X are morphisms in Chn 1(A).If f g Y → X is an n-pullback diagram of X along f n and Z → Y is an n-pullback diagram of g f fg Y along gn, then Z → Y and Y → X are n-exact diagrams if and only if Z−→ Xisan n-exact diagram.

Proof.

(⇒) This is obvious by the dual of Proposition 2.4. ⇐ →f n ( −1 ,..., n−2 ) ( ) Because Y X is a n-pullback diagram of X along f , dC( f ) dC( f ) is a n−1 n-kernel of dC( f ) in the bottom row of diagram (1). The middle row of diagram (1)is n−1 an n-exact sequence by the definition of n-exact diagram and dC( f ) is an epimorphism n−1 op since dC( fg) is an epimorphism. By axiom (A2 )ofn-abelian category, the bottom row g of diagram (1)isann-exact sequence, thus Y → X is an n-exact diagram. →g n−1 We show that Z Y is an n-exact diagram. It is enough to show that dC(g) is an op ( −1 ,..., n−2 ) epimorphism by axiom (A2 )ofn-abelian category since dC(g) dC(g) is an n-kernel Homological algebra in n-abelian categories 633

n−1 : n → n−1 = ( n n−1) = of dC(g). Indeed, let u y M be a morphism such that udC(g) ug udY 0. ( ) : n ⊕ n−1 → ( ) n−2 = . Then, u 0 Y X M is a morphism such that u 0 dC( f ) 0 Therefore, n−1 n−2 : n → since dC( f ) is the cokernel of dC( f ), there exists a morphism h X M such that ( n n−1) = ( ). n = , n−1 = = n−1 = h f dX u 0 Thus hf u hdX 0. Hence h 0 since hdC( fg) ( n−1 n n) = ( n) = n−1 = hdX hf g 0 ug 0 and dC( fg) is an epimorphism. Thus, u 0. This proves our assertion. 

2.4 n-Cluster tilting subcategories Recall that a full subcategory D of an abelian category A is cogenerating if for every object X ∈ A there exists an object Y ∈ D and a monomorphism X → Y . The concept of generating subcategory is defined dually. Let A be an abelian category and D a generating–cogenerating full subcategory of A. D is called an n-cluster-tilting subcategory of A if D is functorially finite in A and

i D ={X ∈ A|∀ i ∈{1,...,n − 1} ExtA(X, D) = 0} i ={X ∈ A|∀ i ∈{1,...,n − 1} ExtA(D, X) = 0}.

Note that A itself is the unique 1-cluster-tilting subcategory of A.

Lemma 2.10 (Theorem 3.16 of [13]). Let A be an abelian category and D an n-cluster tilting subcategory of A. Then, D is an n-abelian category.

Let C be a small additive category. A C- is a contravariant functor G : C → Mod Z. The category Mod C of C-modules is an abelian category. Morphisms in Mod C are natural transformations of contravariant functors. As a consequence of Yoneda’s lemma, functors C(−, N) are projective objects in Mod C.Thecategory of coherent C- modules, denoted by modC, is the full subcategory of Mod C whose objects are the C- modules G such that there exists a morphism f : X → Y in C and an exact sequence of functors f ∗ C(−, X) −→ C(−, Y ) → G → 0. Note that modC is closed under cokernels and extensions in Mod C. Moreover, modC is closed under kernels in Mod C if and only if C has weak kernels, in which case mod C is an abelian category [1]. Let C be a small projectively generated n-abelian category. Let P be the category of projective objects in C. P has weak kernels since C is a projectively generated n-abelian category, thus modP is an abelian category. Lemma 2.10 says that any n-cluster tilting subcategory of abelian category is an n-abelian category. The following lemma says that certain n-abelian categories can be realized as n-cluster tilting subcategories of abelian categories.

Lemma 2.11 (Lemma 3.22 of [13] and Theorem 1.3 of [18]). Let C be a small projectively generated n-abelian category. Let P be the category of projective objects in C and F : C → mod P the functor defined by F X := C(−, X)|P .Also, let FC := {M ∈ mod P|∃ X ∈ 634 Deren Luo

C such that M  FX} be the essential of F. Then, the following statements hold:

(i) F is a fully faithful functor. (ii) FC is an n-cluster tilting subcategory of mod P.

In the rest of this paper, all functors ‘F’ are the functor defined in Lemma 2.11 unless otherwise specified.

3. (n + 2) × (n + 2)-Lemma and 5-lemma In this section, we study the relationship between exact sequences and n-exact sequences in n-cluster tilting subcategories of abelian categories and prove the (n +2)×(n +2)-lemma and 5-lemma in n-abelian categories as generalizations of the classical 3 × 3-lemma and 5-lemma in abelian categories.

PROPOSITION 3.1

d0 d1 Let A be an abelian category, D a full additive subcategory of A. Let X : X 0 → X 1 → dn−1 dn ··· → X n → X n+1 be a complex in Chn(D). Then we have

(i) If D is a contravariantly finite generating subcategory of A and (d0,..., d0 d1 dn−1 dn dn−1) is an n-kernel of dn, then 0 → X 0 −→ X 1 −→··· −→ X n −→ X n+1 is an exact sequence of A; (ii) If D is a covariantly finite cogenerating subcategory of A and (d1,...,dn) is an d0 d1 dn−1 dn n-cokernel of d0, then X 0 −→ X 1 −→··· −→ X n −→ X n+1 → 0 is an exact sequence of A; (iii) If D is a functorially finite generating–cogenerating subcategory of A and X is an d0 d1 dn−1 dn n-exact sequence of D, then 0 → X 0 → X 1 → ··· → X n → X n+1 → 0 is an exact sequence of A.

Proof. We only prove (i); (ii) follows from the dual of (i) and (iii) follows from (i), (ii). We need to show that d0 is a monomorphism and Ker dk = Im dk−1 for k = 1, 2,...,n in A. First, d0 is a monomorphism in A. Indeed, let u : M → X 0 be a morphism in A such that d0u = 0. Because D is a contravariantly finite generating subcategory of A, there exists a right D-approximation π : D → M in A (note that, π is an epimorphism in A since D is a generating subcategory of A). Then, d0(uπ) = 0. We have that uπ = 0 since uπ is a morphism in D and d0 is a monomorphism in D.Also,u = 0 since π is an epimorphism in A.Sod0 is a monomorphism in A. Second, Ker dk = Im dk−1 for k = 1, 2,...,n in A, this is an adaptation of the proof of [19, 1.6.11]. It is trivial that dkdk−1 = 0. We have an inclusion ι : Ker dk → X k and a right D-approximation π : E → Ker dk in A. Therefore, since dkιπ = 0 and E ∈ D and dk−1 is a weak kernel of dk, there exists a morphism σ : E → X k−1 such that ιπ = dk−1σ.ItfollowsthatKer dk = Im ιπ ⊂ Im dk−1.  Homological algebra in n-abelian categories 635

PROPOSITION 3.2

d0 d1 Let D be an n-cluster tilting subcategory of an abelian category A and X : X 0 → X 1 → dn−1 dn ··· → X n → X n+1 a complex in Chn(D). Then we have d0 d1 dn−1 dn (i)0→ X 0 −→ X 1 −→···−→ X n −→ X n+1 is an exact sequence of A if and only if (d0,...,dn−1) is an n-kernel of dn in D. d0 d1 dn−1 dn (ii) X 0 −→ X 1 −→···−→ X n −→ X n+1 → 0 is an exact sequence of A if and only if (d1,...,dn) is an n-cokernel of d0 in D. d0 d1 dn−1 dn (iii)0→ X 0 −→ X 1 −→···−→ X n −→ X n+1 → 0 is an exact sequence of A if and only if X is an n-exact sequence of D.

Proof. We only prove (i); (ii) follows from the dual of (i) and (iii) follows from (i), (ii). We only need to prove the ‘only if’ part, the ‘if’ part is a corollary of Proposition 3.1. Put Li := Im di−1 in A . Then we have exact sequences

f i gi + f n gn + 0 → Li −→ Xi −→ Li 1 → 0 (1 ≤ i ≤ n − 1) and 0 → Ln −→ X n −→ Ln 1, where L1 = X 0, Ln+1 = X n+1.LetM ∈ D. Applying A(M, −),wehave

j 1 j 2 ExtA(M, L ) = 0 (1 ≤ j ≤ n − 1) and ExtA(M, L ) = 0 (1 ≤ j ≤ n − 2) and

1 i−1 2 i−2 ExtA(M, L )  ExtA(M, L ) i−2 2  ···ExtA (M, L ) = 0 (3 ≤ i ≤ n).

Now, we show that di−1 is a weak kernel of di for i = 1,...,n. Indeed, let u : M → Xi in D be any morphism such that di u = 0, then gi u = 0 since f i+1 is a n+1 = v : → i monomorphism( f 1Xn+1 ). Then, there exists a morphism M L such that f i v = u.Ifi = 1, this proves d0 is a kernel of d1 since d0 is a monomorphism. For i > 1, we have an exact sequence

− − 0 → A(M, Li 1) → A(M, Xi 1) → A(M, Li ) → 0

1 i−1 since ExtA(M, L ) = 0fori = 2 and

1 i−1 2 i−2 ExtA(M, L )  ExtA(M, L ) i−2 2  ···ExtA (M, L ) = 0 (3 ≤ i ≤ n).

There exists a morphism t : M → Xi−1 such that gi−1t = v. Then di−1t = f i gi−1t = f i v = u. This proves that di−1 is a weak kernel of di for i = 1, ··· , n and d0 is a kernel of d1, so that (d0,...,dn−1) is an n-kernel of dn. 

As an application, since every small projectively generated n-abelian category can be embedded in an abelian category by Lemma 2.11, we have the following Corollary. 636 Deren Luo

COROLLARY 3.3

Let A be a small projectively generated n-abelian category. Let P be the category of projective objects in A and X a complex in Chn(A). Then we have

Fd0 Fd1 Fdn−1 Fdn (i)0→ FX0 −→ FX1 −→··· −→ FXn −→ FXn+1 is an exact sequence of mod P if and only if (d0,...,dn−1) is an n-kernel of dn in A. Fd0 Fd1 Fdn−1 Fdn (ii) FX0 −→ FX1 −→··· −→ FXn −→ FXn+1 → 0 is an exact sequence of mod P if and only if (d1,...,dn) is an n-cokernel of d0 in A. Fd0 Fd1 Fdn−1 Fdn (iii)0→ FX0 −→ FX1 −→··· −→ FXn −→ FXn+1 → 0 is an exact sequence of mod P if and only if X is an n-exact sequence of A.

Proof. We only prove (i); (ii) follows from the dual of (i) and (iii) follows from (i), (ii). ⇐: The exact sequence at FXi for any k ∈{1, 2,...,n}. Indeed, it is trivial that Fdk Fdk−1 = 0. We have an inclusion ι : Ker Fdk → FXk and an epic right FA- approximation π : FE → Ker Fdk. Then there exists a morphism δ : E → X k such that Fδ = ιπ since F is fully faithful. Therefore, since F(dkδ) = 0 and dk−1 is a weak kernel of dk, there exists a morphism σ : E → X k−1 such that δ = dk−1σ . It follows that

Ker Fdk = Im ιπ = Im Fδ = Im F(dk−1σ) ⊂ Im Fdk−1.

Fd0 is a monomorphism. Indeed, let u : G → FX0 be a morphism in mod P such that (Fd0)u = 0. There exists an epic right FA-approximation v : FM → G. Therefore, there exists a morphism s : M → X 0 such that Fs = uv. We have that Fd0 Fs = 0, so sd0 = 0. s = 0 since d0 is a monomorphism, so is Fs. This implies u = 0 since v is an epimorphism. It follows that Fd0 is a monomorphism. ⇒:Fork ∈{1, 2,...,n},letu : M → X k be a morphism such that dku = 0. It follows that Fdk Fu = 0, therefore, since F is fully faithful and Fdk−1 is a weak kernel of Fdk (Proposition 3.2(i)), there exists a morphism v : M → X k−1 such that Fdk−1 Fv = Fu, so dk−1v = u. This proves that dk−1 is a weak kernel of dk. Hence d0 is a monomorphism and follows from Fd0 which is a monomorphism and F is fully faithful. 

Lemma 3.4. Let A be an abelian category. Then (i) Ch(A) is an abelian category; (ii) If 0 → A• → B• → C• → 0 is a short exact sequence of cochain complexes, then, whenever two of the three complexes A•, B•, C• are exact complexes, so is the third; → • → • →···→ • → (iii) If 0 A1 A2 Ar 0 is an exact sequence of cochain complexes for ≥ − •, •,..., • r 3, then, whenever r 1 of the r complexes A1 A2 Ar are exact complexes, so is the remainder.

Proof. The proof follows immediately from [19, Theorem 1.2.3, Exercise 1.3.1]. 

The following theorem is a generalization of 3 × 3-lemma of the abelian category. Homological algebra in n-abelian categories 637

Theorem 3.5 ((n + 2) × (n + 2)-lemma). Let A be a small projectively generated n- abelian category, P the category of projective objects in A. Let

A1,1 A1,2 ··· A1,n+2

A2,1 A2,2 ··· A2,n+2 ......

An+2,1 An+2,2 ··· An+2,n+2 be a commuting diagram in A such that all columns are n-exact sequences. Then n + 1 of the n + 2 rows are n-exact sequences and implies the remainder.

Proof. By Lemma 2.11, Corollary 3.3 and Lemma 3.4, the remainder row is an exact sequence of mod P under the functor F. Then the remainder row in A is an n-exact sequence by Corollary 3.3 again. 

Theorem 3.6 (5-lemma). Let A be a small projectively generated n-abelian category, P the category of projective objects in A. Let

d0 d1 d2 d3 X 0 X X 1 X X 2 X X 3 X X 4 f 0 f 1 f 2 f 3 f 4 d0 d1 d2 d3 Y 0 Y Y 1 Y Y 2 Y Y 3 Y Y 4 be a commuting diagram. If f 1 and f 3 are , f 0 is an epimorphism, f 4 is a monomorphism and one of the following conditions holds, then f 2 is also an isomorphism. i i i−1 i−1 = , , , (i) dX and dY are weak cokernels of dX and dY respectively for i 1 2 3 4. i i i+1 i+1 = , , , (ii) dX and dY are weak kernels of dX and dY respectively for i 0 1 2 3.

4 : 4 → 5 Proof. We only prove (i). By axiom (A1), there exists a weak cokernel dX X X 3 i : i → i+1 = ,..., of dX , inductively, there exist morphisms dX X X for i 5 n such that ( 1 , 2 ,..., n ) 0 dX dX dX is an n-cokernel of dX by Lemma 2.6. Similarly, there exist morphisms i : i → i+1 = ,..., ( 1 , 2 ,..., n ) 0 dY Y Y for i 4 n such that dY dY dY is an n-cokernel of dY .By the property of weak cokernels we obtain a commutative diagram

− d0 d1 dn 1 dn X 0 X X 1 X ··· X n−1 X X n X X n+1 − + , f 0 f 1 f n 1 f n f n 1 − d0 d1 dn 1 dn Y 0 Y Y 1 Y ··· Y n−1 Y Y n Y Y n+1

Applying F, by Corollary 3.3, we obtain a commutative diagram with exact rows

− Fd0 Fd1 Fdn 1 Fdn FX0 X FX1 X ··· FXn−1 X FXn X FXn+1 0 − + , Ff0 Ff1 Ffn 1 Ffn Ffn 1 − Fd0 Fd1 Fdn 1 Fdn FY0 Y FY1 Y ··· FYn−1 Y FYn Y FYn+1 0 638 Deren Luo

We need to show that Ff0 is an epimorphism in modP, Ff4 is a monomorphism in modP and Ff1, Ff3 are isomorphisms in modP. Indeed, let u : FY0 → G be a morphism in modP such that uFf0 = 0. There exists an object M ∈ A and a monomorphism left FM-approximation v : G → FM since FA is a n-cluster tilting subcategory of mod P. Therefore, since F is a fully faithful functor, there exists a morphism s : Y 0 → M such that Fs = vu. We have that FsFf 0 = 0, thus sf0 = 0. s = 0 since f 0 is a epimorphism, so is Fs. This implies u = 0 since v is a monomorphism. It follows that Ff0 is an epimorphism. Similarly, if Ff4 is a monomorphism, Ff1, Ff3 are isomorphisms. By 5-lemma of abelian categories, Ff3 is an isomorphism, so is f 3 since F is fully faithful. 

4. (Co)homology of n-abelian categories In this section, we introduce the right(resp., left) derived functors of covariant or con- travariant left(resp., right) n-exact functors and study their basic properties.

4.1 n-Exact functors and n-resolutions Let A be an n-abelian category and B an abelian category, and let G : A → B be a d0 d1 dn−1 dn covariant additive functor. Let X : X 0 → X 1 →··· → X n → X n+1 in Chn(A) be an n-exact sequence. We say that G is (i) Left n-exact if 0 → GX0 → GX1 →···→ GXn → GXn+1 is an exact sequence of B. (ii) Right n-exact if GX0 → GX1 →···→ GXn → GXn+1 → 0 is an exact sequence of B. (iii) n-Exact if 0 → GX0 → GX1 →···→ GXn → GXn+1 → 0 is an exact sequence of B. The notions of contravariant additive left n-exact functors, right n-exact functors and n-exact functors are defined dually. For example, the hom-functors A(M, −) and A(−, M) are covariant left n- and contravariant left n-exact functor respectively by the definitions of n-kernel and n- cokernel. We say that an n-abelian category A has enough projectives if for every object M ∈ A, there exists projective objects P1, P2,...,Pn ∈ A and an n-exact sequence

N → Pn →···→ P1 → M The notion of having enough injectives is defined dually. Let A be an n-abelian category which has enough projectives, M ∈ A.Wehaven-exact sequences

j1 dn n M → Pn →···→ P1 → M π 2 →j2 →···→d2n →1  n M P2n Pn+1 n M π 3 →j3 →···→d3n →2 2 n M P3n P2n+1 n M . . Homological algebra in n-abelian categories 639

Connecting them, let din+1 = ji πi . We call the sequence

d3n d2n+1 d2n dn+1 dn d1 ···→ P3n →···→ P2n+1 → P2n →···→ Pn+1 → Pn →···→ P1 → M (2)

→d1 k a projective n-resolution of M, also denoted simply as P• M. We call n M the k-th n- syzygy of M for k ≥ 0. It is easily seen that n defines a functor between stable categories n : A → A, and moreover, if A is Frobenious, n : A → A is self-equivalence [13, −k Proposition 5.8]. The notions of injective n-coresolution and k-th n-cosyzygy n M of M are defined dually. We cannot see that din+1 is a weak kernel (resp. weak cokernel) of din (resp. din+2)for any i. But, it is easy to see that din is a weak cokernel of din+1, din+2 is a weak kernel of din+1.

4.2 Right(left) n-derived functors Let A be an n-abelian category and B an abelian category, and let G : A → B be a contravariant left n-exact functor. We can construct the right n-derived functors nRi G for i ≥ 0 as follows. Let M ∈ A. Choose a projective n-resolution P• → M as (2) and define i nR G(M) := Hin+1(GP•) := Ker Gdin+2/Im Gdin+1 for i = 0, 1,.... 0 (Observe that GP• is exact at Pj for any j = in + 1.) Note that nR G(M)  GM.

Lemma 4.1. Let A be an n-abelian category and B an abelian category, and let G : A → B be a contravariant left n-exact functor. Then (i) The objects nR i G(M) of B are well defined up to natural isomorphism. That is, if Q• → M is a second projective n-resolution, then there is a canonical isomorphism: i in+1 in+1 nR G(M) := H (GP•)  H (GQ•). (ii) If f : M → N is any in A, there is a natural map nR i G( f ) : nR i G(N) → nR i G(M) for each i ≥ 0. (iii) nR i G(P) = 0 for all P and i > 0. (iv) Each nR i G is an additive functor from A to B.

Proof. We can prove (i), (ii), and (iv) similar to that of [19, proof of Lemma 2.4.1, Lemma 2.4.4, Theorem 2.4.5] by using Comparison Lemma 2.1. For (iii), the assertion follows by choosing the projective n-resolution of P to be ···→ 1 0 → P → P. 

The notions of right n-derived functors nRi for covariant left n-exact functors by using injective n-resolutions, left n-derived functors nLi for covariant right n-exact functors by using projective n-coresolutions, and left n-derived functors nLi for contravariant right n-exact functors by using injective n-coresolutions are defined dually. Specially, for contravariant (resp., covariant) left n-exact functor A(−, B) (resp., A(A, −)), we have 640 Deren Luo

DEFINITION 4.2 (nExt functors)

Let A be an n-abelian category, the hom-functor A(−, B)(resp., A(A, −)) is a contravari- ant (resp., covariant) left n-exact additive functor. We define the right n-derived functors

i i i i nExtA(−, B) = nR A(−, B) resp., nExtA(A, −) = nR A(A, −)

0 0 In particular, nExtA(−, B) = A(−, B), nExtA(A, −) = A(A, −).

PROPOSITION 4.3

Let A be an n-abelian category which has enough projectives and enough injectives, A, B ∈ A. We have i i i (i) nExtA(A, −)(B)  nExtA(−, B)(A) = nExtA(A, B). (ii) If A is an n-cluster tilting subcategory of a projectively generated injectivity cogen- erated abelian category D and B is an abelian category, then for any left n-exact functor G : D → B and any right n-exact functor S : D → B, we have m mn nR G(A)  R G(A), nLm S(A)  LmnS(A) ∀A ∈ A, m ≥ 0, mn+i R G(A)=0, Lmn+i S(A) = 0 ∀A ∈ A, m ≥ 0, 1≤i ≤ n−1, i where R G and Li S are the classical right and left derived functors respectively. In particular, m mn mn+i nExtA(A, B)  ExtD (A, B), Ext (A, B) = 0 ∀A, B ∈ A, m ≥ 0, 1 ≤ i ≤ n − 1.   D   m( , )  m( , ) m( , )  (iii) nExtA A i∈I Bi i∈I nExtA A Bi and nExtA i∈I Ai B i∈I m nExtA(Ai , B) for any m ≥ 0.

Proof. (i) follows from [19, Theorem 2.7.6]. (ii) follows immediately from the definition of right(resp. left) n-derived functors and Proposition 3.2, and (iii) follows from [19, proof of Corollary 2.6.11] and its dual. 

The following proposition is a generalization of the classical ‘Horseshoes lemma’ of projective generated abelian categories.

PROPOSITION 4.4 (n-Horseshoes lemma)

Let A be a small n-abelian category which has enough projectives, P the category of α0 α1 αn projective objects in A. The X : X 0 → X 1 → ··· → X n+1 is an n-exact sequence of i i A. Then, there exist projective n-resolutions P• → X for i ={0, 1,...,n + 1} such that the following diagram commute:

0 1 n n+1 0 P• P• ··· P• P• 0 , (3)

X 0 X 1 ··· X n X n+1 where the upper rows is a of complexes. Homological algebra in n-abelian categories 641

Proof. Applying F to X, by Corollary 3.3,0→ FX0 → FX1 →···→ FXn+1 → 0is an exact sequence of mod P, we split it to exact sequences of mod P,

+ 0 → Li → FXi → Li 1 for i ∈{0, 1,...,n}, where L0 = FX0, Ln+1 = FXn+1. Step 1. FA is closed under n-th syzygy, dually, closed under n-th cosyzygy. Indeed, for any Y ∈ FA, there is an object X ∈ A such that FX  Y . Since A is a small n-abelian category which has enough projectives, there is an n-exact sequence n X → Pn−1 → ···→ P0 → X such that 0 → Fn X → FPn−1 →···→ FP0 → FX  Y → 0isan exact sequence in mod P (see Corollary 3.3) where Pi are projective objects. This proves that FA closed under n-th syzygy. 0 0 n+1 n+1 Step 2. Giving two projective n-resolutions P• → X and P• → X , by Corollary 0 0 n+1 n+1 3.3, FP• → FX and FP• → FX are projective resolutions in mod P. For any Li , i ∈{1, 2,...,n}, take a projective resolution

Fdi Fdi Fdi i → i : ···→ i →2 i →1 i →0 i → , FP• L FP2 FP1 FP0 L 0

i ∈ P ∈{, ,...}. P where Pj j 0 1 By Horseshoes lemma of abelian category mod , there exists a projective resolution

Ffi f i Ffi i →0 i : ···→3 ( i ⊕ i+1) →2 FQ• FX F P2 P2 Ffi Ffi ( i ⊕ i+1) →1 ( i ⊕ i+1) →0 i → F P1 P1 F P0 P0 FX 0

i i i+1 such that 0 → FP• → FQ• → FP• → 0 is a split short exact sequence of projective i i i+1 0 complexes, so is 0 → P• → Q• → P• → 0. Connecting them, we have 0 → P• → 1 2 n n+1 Q• → Q• →···→ Q• → P• → 0 is a split exact sequence of projective complexes. i i Step 3. Q• → X is a projective n-resolution for any i ∈{1, 2,...,n}. Indeed, we can i i split FQ• → FX to

ki j : → i →j i → i S 0 Tj FQjn−1 FQjn−2 πi → ···→ i →j−1 i → , FQ( j−1)n Tj−1 0

i = i ∈{, ,...} = where T0 FX for j 1 2 .For j 1, since F is fully faithful, there is f i i →0 i A i = πi k ( i , A)  an epimorphism Q0 X in such that Ff0 0.WehaveExtmod P T1 F n+k ( i , A) = = , ,..., − Extmod P FX F 0fork 1 2 n 1 (see Proposition 4.3 (ii)). It provides i ∈ A ∈ A that T1 F (see Lemma 2.11 and Step 1). Then there exists an object G1 such that i  → i → ··· → i → i T1 FG1. Then there is an n-exact sequence G1 Qn−1 Q0 X (see 2 3 i i Corollary 3.3). We can discuss S , S ,..., inductively. This proves that Q• → X is a projective n-resolution.  642 Deren Luo

Theorem 4.5 (Long n-exact sequence theorem). Let A be a small n-abelian category, P the category of projective objects in A. B is an abelian category, G : A → B is an α0 α1 αn additive functor, and X : X 0 → X 1 →···→ X n+1 an n-exact sequence of A. Then we have (i) If A has enough injectives and G is a covariant left n-exact functor, we have an exact sequence + ∂ 0 → GX0 → GX1 →···→ GXn 1 →n nR1G(X 0) ∂1 ∂i−1 ∂i →···→nR1G(X n+1) →···n →n nR i G(X 0) →···→nR i G(X n+1) →···n . (ii) If A has enough projectives and G is a contravariant left n-exact functor, we have an exact sequence + ∂ + 0 → GXn 1 → GXn →···→ GX0 →n nR1G(X n 1) 1 i−1 i ∂ ∂ + ∂ →···→nR1G(X 0) →···n →n nR i G(X n 1) →···→nR i G(X 0) →···n . (iii) If A has enough projectives and G is a covariant right n-exact functor, we have an exact sequence ∂i ∂i−1 ∂1 n 0 n+1 n n 0 ···→ nLi G(X ) →···→nLi G(X ) → ···→ nL1G(X ) ∂ n+1 n 0 n+1 →···→nL1G(X ) → GX →···→GX →0. (iv) If A has enough injectives and G is a contravariant right n-exact functor, we have an exact sequence ∂i ∂i−1 ∂1 n n+1 0 n n n+1 ···→ nLi G(X ) →···→nLi G(X ) → ···→ nL1G(X ) ∂ 0 n n+1 0 →···→nL1G(X ) → GX →···→ GX → 0.

Proof. If n = 1, then the result is the classical long exact sequence theorem. Let n ≥ 2, we only prove (ii). Applying G to the first arrow of (3) gives a split exact sequence of complexes

n+1 n 1 0 GP• → GP• →···→ GP• → GP• (4) since the first arrow of (3) is a split exact sequence of complexes. Since Ch(B) is an abelian category, we can split (4) to short split exact sequences of complexes

i+1 i i 0 → L → GP• → L → 0 (1 ≤ i ≤ n),

n+1 n+1 1 0 j where L = GP• and L = GP• . Thus Hkn+i (GP• ) = 0fori ∈{2, 3,...,n}, k ∈ {0, 1, 2,...} and all j ∈{0, 1,...,n + 1} by the definition of left n-exact functor. n+1 n n For exact sequence 0 → L → GP• → L → 0, by the long exact sequence theorem, we have exact sequences

n+1 n n 0 → H1(L ) → H1(GP• ) → H1(L ) → 0(5) n n+1 n 0 → Hkn(L ) → Hkn+1(L ) → Hkn+1(GP• ) n → Hkn+1(L ) → 0fork = 1, 2,... (6) Homological algebra in n-abelian categories 643 and

n Hkn+i (L ) = 0fori = 2,...,n − 1. (7)

Inductively, let s ∈{2, 3,...,n − 1} and suppose that for all i ≥ s + 1 we have exact sequences

i+1 i i 0 → H1(L ) → H1(GP• ) → H1(L ) → 0, (8) i i+1 i 0 → Hkn(L ) → Hkn+1(L ) → Hkn+1(GP• ) i → Hkn+1(L ) → 0fork = 1, 2,... (9) and

i i+1 Hkn+ j (L )  Hkn+ j+1(L ) for j = 2,...,n − 1. (10)

s+1 s s For i = s, i.e., the exact sequence 0 → L → GP• → L → 0, by the long exact sequence theorem, we have exact sequences

s+1 s s s+1 0 → H1(L ) → H1(GP• ) → H1(L ) → H2(L ) → 0, s s+1 s 0 → Hkn(L ) → Hkn+1(L ) → Hkn+1(GP• ) s s+1 → Hkn+1(L ) → Hkn+2(L ) → 0fork = 1, 2,... and

s s+1 Hkn+ j (L )  Hkn+ j+1(L ) for j = 2,...,n − 1.

But, ⎧ ⎨ ( ) (10) (10) (10) ( n) =7 ( > ) ( s+1)  ( s+2)  ···  Hn−s+1 L 0 s 2 H2 L H3 L ( ) ⎩ n 7 Hn−1(L ) = 0 (s = 2), ( ) s+1 10 s+2 H + (L )  H + (L ) nk 2 nk 2 ⎧ ⎨ ( ) (10) (10) ( n) =7 ( > )  ···  Hnk+n−s+1 L 0 s 2 ( ) ⎩ n 7 Hnk+n−1(L ) = 0 (s = 2).

Thus we have exact sequences

s+1 s s 0 → H1(L ) → H1(GP• ) → H1(L ) → 0, s s+1 0 → Hkn(L ) → Hkn+1(L ) s s → Hkn+1(GP• ) → Hkn+1(L ) → 0fork = 1, 2,...,

This finishes the induction step. 644 Deren Luo

2 1 1 For exact sequence 0 → L → GP• → L → 0, by the long exact sequence theorem, we have exact sequences

2 1 1 2 0 → H1(L ) → H1(GP• ) → H1(L ) → H2(L ) → 0, 2 1 0 → Hkn+1(L ) → Hkn+1(GP• ) 1 2 → Hkn+1(L ) → Hkn+2(L ) → 0fork = 1, 2,... and

2 Hkn+i (L ) = 0fori = 3, 4,...,n. (11)

But

( ) ( ) ( ) 2 10 3 10 10 n Hkn+2(L )  Hkn+3(L )  ···  H(k+1)n(L ) for k = 0, 1,..., ( ) ( ) ( ) i 10 i−1 10 10 2 Hkn(L )  Hkn−1(L )  ···  Hkn−i+2(L ) (11)  0fork = 1, 2,..., i = 1, 2,...,n − 1

Thus we have exact sequences

2 1 1 n 0 → H1(L ) → H1(GP• ) → H1(L ) → Hn(L ) → 0, (12) i+1 i i 0 → Hkn+1(L ) → Hkn+1(GP• ) → Hkn+1(L ) → 0 for k = 1, 2,..., i = 1,...,n − 1, (13) 2 1 1 0 → Hkn+1(L ) → Hkn+1(GP• ) → Hkn+1(L ) n → H(k+1)n(L ) → 0fork = 1, 2,.... (14)

Connecting them in order (5)(8)(12)(6)(13)(14)(6)(13)(14)(6)(13)(14)...,wehavethe desired exact sequence. 

j The morphisms ∂n induced in Theorem 4.5 are called the j-th n-connecting morphisms. Specially, ∂n is called the n-connecting morphism.

COROLLARY 4.6

Let A be a small n-abelian category which has enough projective objects, P the category 1 of projective objects in A.IfnExtA(A, B) = 0, then every n-exact sequence starting at A and ending with B is contractible.

0 dξ Proof. Given an n-exact sequence ξ : B → X 1 → ··· → X n → A. By Theorem 4.5, there is an exact sequence of groups

0 → A(A, B) → A(X n, B) →···→A(X 1, B) ∂ n 1 → A(B, B) −→ nExtA(A, B) = 0.

1 0 So the identity morphism 1B lifts to a morphism σ : X → B and σ dξ = 1B. This proves that ξ is a contractible n-exact sequence.  Homological algebra in n-abelian categories 645

COROLLARY 4.7

Let A be an n-abelian category which has enough projectives. The following are equivalent: (i) A is a projective object. (ii) A(A, −) is an n-exact functor. i (iii) nExtA(A, B) = 0 for all i = 0 and all B. 1 (iv) nExtA(A, B) = 0 for all B.

Proof. It is clear that (i) ⇔ (ii), (iii) ⇒ (iv). (i) ⇒ (iii), (iv) holds by Lemma 4.1(iii). If wearegivingann-resolution P• → A, applying A(−, B),wehave

0 →A(A, B) A(P1, B) d∗ d∗ n+1 n+2 . →···→ A(Pn, B) ∗ A(Pn+ , B) A(Pn+ , B) i π∗ 1 2

A(n A, B) i∗ is surjective since the upper row is an exact sequence and π ∗ is an injective. Let B = n A, we have that i : n A → Pn is a split monomorphism. It follows that A is a direct summand of P1 by [13, Proposition 2.6], so (iv) ⇒ (i). 

By the definition of n-derived functors, we have

PROPOSITION 4.8 ( shifting theorem)

Let A be an n-abelian category which has enough projectives and enough injectives. Then, for i ≥ 1, j ≥ 0, we have i − j i+ j (i) If G is a covariant left n-exact functor nR G(n A)  nR G(A). i j i+ j (ii) If G is a contravariant left n-exact functor, nR G(n A)  nR G(A). i j i+ j (iii) If G is a covariant right n-exact functor, nL G(n A)  nL G(A). i − j i+ j (iv) If G is a contravariant right n-exact functor, nL G(n A)  nL G(A). (v) Specially, i j i+ j i − j nExtA(n A, B)  nExtA (A, B)  nExtA(A,n B).

4.3 n-Homological dimension

DEFINITION 4.9

Let A be an n-abelian category which has enough projective objects, M ∈ A.Then- projective dimension npd M is the minimum integer m (if it exists) such that there is a projective n-resolution of M,

···→0 → Pmn+1 → Pmn →···→ P2 → P1 → M. (15)

The notion of n-injective dimension nid M is defined dually. 646 Deren Luo

PROPOSITION 4.10

The following are equivalent for any n-abelian category A. (i) npdA ≤ m. d (ii) nExtA(A, B) = 0 for all d > m and all B ∈ A. m+1 (iii) nExtA (A, B) = 0 for all B ∈ A.

(iv) If A → Pmn →···→ P1 → A is any m-fold n-exact sequence with Pi projective, then A is also projective.

∗ Proof. Since nExt (A, B) may be computed using a projective n-resolution of A,itis clear that (iv) ⇒ (i) ⇒ (ii) ⇒ (iii). If we give a projective n-resolution as (iv), then + nExtm 1(A, B)  nExt1(A , B) by Proposition 4.8, and A is projective if and only if nExt1(A , B) = 0 for all B ∈ A by Corollary 4.7, so (iii) ⇒ (iv). 

5. n-Extensions and m-fold n-extensions We assume that m, n be two positive integers in the rest of this section. 1 In any abelian category A, we can define ExtA(A, B) even if it has no projectives and no injectives, to be the set of equivalence classes of extensions under Baer sum (R. Baer in 1934) by using pushout and pullback [19, Definition 3.4.4]. Baer’s description of 1 ExtA(A, B) as extension E(A, B) has been generalized by Yoneda [15] to a description m m of ExtA(A, B) for all m ≥ 1. Elements of Yonedas ExtA(A, B) are a certain equivalence classes of m-fold exact sequences and add them by a generalized Baer sum ([5], [14,p. 82–p. 87], [19, Vista 3.4.6], [15,16]). In this section, let A be an n-abelian category which has enough projectives. We prove m m nE (A, B)  nExtA(A, B) under n-Baer sum where nEm(A, B) is the equivalence class of m-fold n-extensions of A by B.

5.1 n-Extension groups

DEFINITION 5.1

Let A be an n-abelian category, A, B ∈ A.Ann-extension ξ of A by B is an n-exact sequence B → X 1 → ··· → X n → A in A.Andtwon-extensions ξ,ξ of A by B are equivalent if there is a commutative diagram

0 dξ ξ : BX1 ··· X n−1 X n A f 1 f n−1 f n . (16) 0 dξ ξ : BY1 ··· Y n−1 Y n A

0 ξ is contractible if it is equivalent to an n-exact sequence ξ with dξ a split monomorphism. We simplify the 0 n-exact sequence 0 → 0 →···→0 by 0 if no confusion appears. Homological algebra in n-abelian categories 647

0 Thus dξ is a split monomorphism if and only if ξ is a contractible n-exact sequence [13, Proposition 2.6]. The following lemma shows the equivalent to be an equivalence relation.

Lemma 5.2 (Proposition 4.10 of [13]). Let A be an n-abelian category. If there exists an equivalence of n-exact sequences f : ξ → ξ , then there exists an equivalence of n-exact sequences g : ξ → ξ such that f and g are mutually inverse isomorphisms in H(A).

We denote the equivalence class of an n-exact sequence ξ by [ξ], and we define

nE(A, B) ={[ξ]|ξ is an n-extension of A by B} and ξ ≡ ξ if [ξ]=[ξ ]. Before giving the additive group structure of nE(A, B) with of n-Baer sum, we give some properties of the equivalence classes of n-exact sequences.

PROPOSITION 5.3

0 1 n−1 n dξ dξ dξ dξ Let A be an n-abelian category, ξ : B → B1 → ··· → Bn → A an n-extension of A by B and f 0 : B → C a morphism in A. Taking an n-pushout along f 0, by Proposition 2.7, there is a morphism between n-exact sequences

0 1 n dξ dξ dξ ξ : BB1 ··· Bn A f f 0 f 1 f n . 0 1 n dξ dξ dξ po 1 po n po ξpo : CT ··· T A

0 Then, [ξpo] is unique as determined by [ξ] and f .

: ξ → ξ : ξ → ξ Proof. Let g be an equivalence and t po is a morphism induced by n-pushout of ξ along f 0 : B → C.By[13, Proposition 2.13] of n-pushout, there exists a morphism

0 1 n dξ dξ dξ ξ : po 1 po ··· n po po CT T A p p1 pn 0 1 n dξ dξ dξ po 1 po n po ξpo : CT ··· T A such that there exists homotopy h : fg → pt with h1 = 0. We have

n n n n n n n n n−1 n n n n (dξ − dξ p )t = dξ − dξ ( f g + dξ h ) = dξ − dξ g = 0, po po po po n n n n−1 (dξ − dξ p )dξ = 0, po po po

n n−1 n n n but ( t dξ ) is an epimorphism, so that dξ = dξ p . This proves [ξpo]=[ξ ].  po po po po

The following property is a generalization of [14, Lemma 1.3]. 648 Deren Luo

PROPOSITION 5.4

Let A be an n-abelian category. A morphism f : X → Y of two n-exact sequences X, Y factors over an n-exact sequence Z,

d0 d1 dn XX0 X X 1 X ··· X n X X n+1 g f 0 g1 gn d0 d1 dn ZYf 0 Z Z 1 Z ··· Z n Z X n+1 p p1 pn f n+1 d0 d1 dn YY0 Y Y 1 Y ··· Y n Y Y n+1 in such a way that the upper-left n-squares and lower right n-squares are n-exact diagrams. [Z] does not depend on the choices of n-extensions in the equivalence classes [X], [Y ].

Proof. Taking a good n-pushout of X along f ; d0 d1 dn XX0 X X 1 X ··· X n X X n+1 . g f 0 g1 gn d0 d1 dn ZY0 Z Z 1 Z ··· Z n Z X n+1

n : n → n+1 Then, by Proposition 2.7, there exists a cokernel dZ Z X such that Z is an 0 n-exact sequence. It is an n-exact diagram since dX is a monomorphism. By the definition of the good n-pushout, there exists a morphism p : Z → Y such that f = pg.Thelower n right n-squares is an n-exact diagram by the dual of [13, Proposition 4.8] since dY is an epimorphism. The last sentence follows from Proposition 5.3. 

Notation 5.5. Denote the n-exact sequence ξpo in Proposition 5.3 by f · ξ. By Proposition 5.3, [ f · ξ] does not depend on the choice of n-pushout of ξ along f and nor on the choice of n-extensions in the equivalence class [ξ]. Thus we can define f ·[ξ]:=[f · ξ].The notions ξ · g, [ξ]·g := [ξ]·g for g : A → A are defined dually by the dual of Proposition 5.3.

The following proposition shows that any n-pushout (resp., n-pullback) along the first (resp., last) morphism of an n-exact sequence yields a contractible n-exact sequence, which is a generalization of [14, Lemma 1.7].

d0 dn Lemma 5.6. Let C be an additive category, ξ : X 0 → X 1 → ··· → X n → X n+1 an n-exact sequence. Then d0 · ξ and ξ · dn are contractible.

Proof. The diagram 0 1 2 X 0 d X 1 d X 2 d ··· X n+1 d0 (1 d1)T (10)T (01) 2 X 1 X 1 ⊕ X 2 X 2 d ··· X n+1 is commutative. Then, the properties follow from Proposition 5.3 and its dual.  Homological algebra in n-abelian categories 649

PROPOSITION 5.7

Let A be an n-abelian category, ξ : B → X 1 →···→ X n → A an n-extension of A by f f g g B. Then, for any morphisms B → B → B , and any morphism A → A → A, we have (i) ( f f ) ·[ξ]= f ·[f · ξ]=[f · ( f · ξ)]. (ii) [ξ]·(gg ) =[ξ · g]·g =[(ξ · g) · g ]. (iii) f ·[ξ]·g := ( f ·[ξ]) · g = f · ([ξ]·g).

Proof. (i), (ii) follow immediately from Proposition 2.4,2.7,5.3 and their dual. For (iii), we only need to show ( f · ξ)· g ≡ f · (ξ · g) by Proposition 5.3 and its dual. There is a morphism ( f, f1,..., fn, g) : ξ · g → f · ξ. By Proposition 5.4, ( f, f1,..., fn, g) has factorizations

ξ · g → ( f · ξ)· g → f · ξ or ξ · g → f · (ξ · g) → f · ξ

By Proposition 5.3, ( f · ξ)· g ≡ f · (ξ · g).  ∈ A For any object M , the diagonal map is ∇M = 1M 1M : M ⊕ M → M and the sum map is

T M = 1M 1M : M → M ⊕ M. Note that the fact that the equivalent class [(∇(ξ ⊕ ξ )) ] is well defined does not depend on the choices of the representable elements of equivalent classes of [ξ] and [ξ ], and does not depend on the choices of the n-pushout and n-pullback by Proposition 5.7(iii).

Theorem 5.8. Let A be an n-abelain category, A, B two objects of A. Then nE(A, B) is an abelian group under n-Baer sum

[ξ]+[ξ ]=[(∇(ξ ⊕ ξ )) ] with zero being the class of contractible n-extensions, the inverse of any [ξ] is the n- extension (−1B) ·[ξ].

For α, αi : A → A and γ,γi : B → B for i = 1, 2, we have ([ξ]+[ξ ]) · α =[ξ]·α +[ξ ]·α, γ · ([ξ]+[ξ ]) = γ ·[ξ]+γ ·[ξ ],

[ξ]·(α1 + α2) =[ξ]·α1 +[ξ]·α2,(γ1 + γ2) ·[ξ] = γ1 ·[ξ]+γ2 ·[ξ].

Proof. The proof of this theorem is same as [14, Theorem 2.1]. 

Theorem 5.9. Let A be an n-abelian category which has enough projectives, A, B ∈ A. There is a group isomorphism of abelian groups

1  : nE(A, B) → nExtA(A, B)

1 in which the contractible n-extensions correspond to the element 0 ∈ nExtA(A, B). 650 Deren Luo

Proof. See Theorem 5.17. 

5.2 m-Fold n-extension groups m( , )  m( , ) m( , ) In the rest of this section, we show that nE A B nExtA A B , where nE A B is the equivalent classes of m-fold n-extensions of A by B. Let A be an n-abelian category, and

ξ : C → E2n →···→ En+1 → B, and

ξ : B → En →···→ E1 → A are two n-exact sequences of A. Then, splice them together as

ξ ◦ ξ : C → E2n →···→ En+1 → En →···→ E1 → A ξ ◦ ξ maybefalse,a2n-exact sequence. We call a sequence

dmn d(m−1)n+1 S : Am → Emn −→···→ E(m−1)n+1 −→

dn+1 dn d1 ···−→ En −→···→ E1 −→ A0 (17) an m-fold n-exact sequence starting at Am and ending at A0 if din+1 can be written as a composite Ein+1 → Ai → Ein for each i ∈{1, 2,...,m−1} such that ξ j : A j → E jn → ··· → E( j−1)n+1 → A j−1 are n-exact sequences for j ∈{1, 2,...,m}. Conventionally, we write S = ξm ◦ ξm−1 ◦···◦ξ1. Similarly, we write

Sd ◦ Sd−1 ◦···◦S1 for ni -fold n-exact sequences Si .

Remark 5.10. We cannot see that din+1 is a weak kernel(resp. weak cokernel) of din(resp. din+2) for any i. But, it is easily to see that, din is a weak cokernel of din+1, din+2 is a weak kernel of din+1.

DEFINITION 5.11

Let A be an n-abelian category. An m-fold n-extension of A0 by Am is an m-fold n-exact sequence (17)inA.Them-fold n-extension S of A0 by Am is similar to the m-fold n-extension S if there is a commutative diagram

S : Am Emn ··· E1 A0

fmn f1 . : ··· S Am Emn E1 A0

The m-fold n-exact sequence S is equivalent to S or S ≡m S if there exists a finite sequences of m-fold n-exact sequences S0, S1,...,Sr such that S = S0, S = Sr and Si is similar to Si+1 or Si+1 is similar to Si for i = 0, 1,...,r − 1. Specially, we write ≡ for ≡1. Homological algebra in n-abelian categories 651

It is easy to see that ‘≡m’ is an equivalence relation. We call S a contractible m-fold 1Am n-exact sequence if S is equivalent to ξm ◦ 0 ◦···◦0 ◦ ξ1 with ξm : Am → Am → 0 → 1A0 ···→0,ξ1 : 0 →···→0 → A0 → A0 for m ≥ 2. We denote the equivalence class of an m-fold n-exact sequence S by [S], and we define

nEm(A, B) ={[S]|S is an m-fold n-extension of A by B}.

There are some basic properties of m-fold n-exact sequences. We list them in the fol- lowing lemma without proof.

Lemma 5.12. Let A be an n-abelian category.

(i) For any n-extensions ξ : A → E2n → ··· → En+1 → C and ξ : D → En → ···→ E1 → B, and for any morphism σ : D → C, we have

ξσ ◦ ξ ≡2 ξ ◦ σξ .

(ii) Let S = ξm ◦ξm−1 ◦···◦ξ1 be an m-fold n-exact sequence. Assume ξi ◦···◦ξi− j ≡ j+1 μi ◦···◦μi− j for some i = 1, 2,...,m − 1, i > j ≥ 0. Then S ≡m ξm ◦···◦ξi+1 ◦ μi ◦···◦μi− j ◦ ξi− j−1 ◦···◦ξ1 (iii) Each morphism γ : S → S of m-fold n-extensions starting at α and ending with β

yields αS ≡m S β.

Similar to Notation 5.5, we define

α ·[S]:=[α · S]=[α · (ξm ◦ ξm−1 ◦···◦ξ1)] := [(α · ξm) ◦ ξm−1 ◦···◦ξ1], (18) [S]·γ := [S · γ ]=[(ξm ◦ ξm−1 ◦···◦ξ1) · γ ] := [ξm ◦···◦ξ2 ◦ (ξ1 · γ)]. (19) Similar to Proposition 5.7,wehave

PROPOSITION 5.13

Let A be an n-abelian category, S an m-fold n-extension of A by B. Then, for any mor- f f g g phisms B → B → B , and any morphism A → A → A, we have

(i) ( f f ) ·[S]= f ·[f · S]=[f · ( f · S)]. (ii) [S]·(gg ) =[S · g]·g =[(S · g) · g ]. i dmn d2 π (iii) If S : B → Emn → ···→ E1 → A, then i · S and S · π are contractible. (iv) f ·[S]·g := ( f ·[S]) · g = f · ([S]·g). (v) For [R]∈nE k(C, A), [T ]∈nE l (B, M), [T ]◦[S]=[T ◦ S],([T ]◦[S]) ◦[R]=[T ]◦([S]◦[R]).

Proof.

(i), (ii) follow immediately from Proposition 5.7. 652 Deren Luo

(iii) By Lemma 5.6, there is a morphism S → i · S and an equivalence i · S ≡m S in the following diagram:

i dmn dmn−1 π S : BEmn Emn−1 Emn−2 ··· E1 A T i (1 dmn) T (10) (01) dmn−1 π i · S : Emn Emn ⊕ Emn−1 Emn−1 Emn−2 ··· E1 A ≡ m (10) π 1 1A S : Emn Emn 00··· AA

so i · S is contractible, and S · π is contractible by dual. (iv) If m = 1, it is trivial by Proposition 5.7.Ifm ≥ 2, ( f · S) · g ≡m f · (S · g) follows from Proposition 5.12 and there is a map S · g → f · S starting at f and ending with g.If

S ≡m S ,by(18), we have S · g ≡m S · g and f · S ≡m f · S . Then, by (18),

f · (S · g) ≡m ( f · S) · g ≡m ( f · S ) · g.

(v) The first formula follows from Lemma 5.12(ii). The second formula follows from the first. 

Proposition 5.13 shows that [(∇(S ⊕ S )) ] is well defined for any equivalence classes [S], [S ]∈nEm(A, B).

Theorem 5.14. Let A be an n-abelain category, A, B be two objects of A. Then nEm(A, B) is an abelian group under n-Baer sum

[S]+[S ]=[(∇(S ⊕ S )) ]

with zero being the class of contractible m-fold n-extensions, the inverse of any [S] is the n-extension (−1B ) ·[S]. For [R]∈nE k(C, A), [T ]∈nE l (B, M),

([S]+[S ]) ◦[R]=[S]◦[R]+[S ]◦[R], (20)

[T ]◦([S]+[S ]) =[T ]◦[S]+[T ]◦[S ]. (21)

Proof. The proof of this theorem is same as [14, Theorem 5.3]. 

5.3 m-Fold n-extensions and nExt m In the rest of this section, we simplify A( f, M) by f ∗ for any morphism f . Let

αmn S : B = Am → Emn −→···→ E(m−1)n+1 α (m−1)n+1 αn+1 αn α2 α1 −→ ···−→ En −→···→ E1 → A0 = A Homological algebra in n-abelian categories 653 be an m-fold n-extension of A by B with m ≥ 1, taking an n-resolution P• → A of A. Then we have a commutative diagram lifting 1A,

dmn+2 dmn+1 dmn d1 P• → A : ··· Pmn+1 Pmn ··· P1 A . f fmn+1 fmn f1 αmn+1 αmn α1 S : BEmn ··· E1 A

We have fmn+1dmn+2 = 0. Applying A(−, B) to the diagram, define  : m( , ) → m ( , ) [ ] → ∗ ( ) + ∗ . m nE A B nExtA A B by S fmn+1 1B Im dmn+1

Lemma 5.15. m is well-defined.

Proof. We show that m does not depend on the chain map f : (P• → A) → S nor on the choice of extensions in the equivalence class [S]. Since fmn+1dmn+2 = 0, fmn+1 ∈ Ker dmn+2. Replace f by any other chain map f : (P• → A) → S lifting 1A,

dn+1 dn dn−1 P• → A : ··· Pn+1 Pn Pn−1 ··· P1 A π1 i1

f, f n A hn−1 h0 . α + α + α α − n 2 n 1 hn n n 1 : ··· + − ··· S En 1 π i En En 1 E1 A 1 1

A1

, :  → There exists a morphism g1 g1 n A A1 lifting 1A such that the above diagram , ,..., , commute (solid line). Comparison lemma 2.1 says that there exist h0 h1 hn−1 hn in − = the above diagram such that g1 g1 hni1. Since Pn is projective, there is a morphism : → = π hn Pn En+1 with hn 1hn. Then, we have − = + = π + fn fn i1hn hn−1dn i1 1hn hn−1dn = αn+1hn + hn−1dn α [( − ) − ]=( − ) n+1 fn+1 fn+1 hndn+1 fn fn dn+1 − ( − − ) + = . fn fn hn−1dn dn 1 0 : → − = Then there is a morphism hn+1 Pn+1 En+2 such that fn+1 fn+1 αn+2hn+1 − hndn+1. Inductively, we can construct morphisms hi : Pi → Ei+1 for ∈{ , ,..., } = := − = α − i 0 1 hmn (h0 0 and Emn+1 B) such that f j f j j+1h j h j−1d j for j ∈{1, 2,...,mn}. Then, we have

α [( − ) − ]=( − ) mn+1 fmn+1 fmn+1 hmndmn+1 fmn fmn dmn+1 −[( − ) − ] = . fmn fmn hmn−1dmn dmn+1 0 − = α This implies fmn+1 fmn+1 hmndmn+1 since mn+1 is a monomorphism. It follows ∗ ( ) + ∗ = ∗ ( ) + ∗ that fmn+1 1B Im dmn+1 f mn+1 1B Im dmn+1. Next, replace S by any equivalent m-fold n-exact sequence S . It suffice to consider the case when there is a morphism γ : S → S and the case when there is a morphism β : S → 654 Deren Luo

S, where γ and β are morphisms starting and ending with 1. If there is a morphism γ = ∗ ∗ (1B, kmn,...,k1, 1A) : S → S ,wehave[S ] → f + (1B)+Im d + by the definition mn 1 mn 1 of m. Thus m is well defined. If there is a morphism β = (1B,βmn,...,β1, 1A) : S → S, there is a chain map φ : (P• → A) → S lifting 1A. By comparison lemma and the above paragraph, there exists a morphism hmn : Pmn → B such that φmn+1 − fmn+1 = φ∗ + ∗ = ∗ + ∗  hmndmn+1. This proves mn+1 Im dmn+1 fmn+1 Im dmn+1.

Lemma 5.16. Let A be an n-abelain category, [S], [R]∈nE m(A, B), [S ]∈nE m(C, D), h ∈ A(B, B ). Then we have

(i) m[S ⊕ S ]=m[S]⊕m[S]; (ii) m[h · S]=hm[S]; (iii) m([S ⊕ R]· A) = m([S ⊕ R]) · .

Proof.

(i) Let P• → A, P• → C be projective n-resolutions, and f : (P• → A) → S, f : (P• → C) → S are chain maps lifting 1A and 1C respectively. Then P•⊕P• → A⊕C is a projective n-resolution, f ⊕ f is a chain map lifting 1A⊕C . (ii) Let P• → A be a projective n-resolution, and f : (P• → A) → S a chain map lifting 1A. There is a chain map h := (h,...,1) : S → h · S. Then we have hf : P• → h · S. This implies m[h · S]=hm[S]. (iii) Let P• → A be a projective n-resolution, and f : (P• → A) → (S ⊕ R) · A a chain map lifting 1A, σ : (S ⊕ R) · A → (S ⊕ R) induced by the n-pullback of (S ⊕ R) along A. Thus, the left most morphism of σ is 1B⊕B. There is a chain map t : (P• ⊕ P• → A ⊕ A) → S ⊕ R lifting 1A⊕A, and there is a diagonal chain map P• : P• ⊕ P• → P•. Thus we have  ([ ⊕ ]· ) = ∗ + ∗ m S R A fmn+1 Im dmn+1 and  ([ ⊕ ]) = ( ∗ + ( ∗ ⊕ ∗ )) m S R Pmn+1 tmn+1 Im dmn+1 dmn+1 Pmn+1 = ( )∗ + ∗ . tmn+1 Pmn+1 Im dmn+1 − ∈ ∗  ([ ⊕ ]· By Comparison lemma, tmn+1 Pmn+1 fmn+1 Im dmn+1. Thus we have m S R ) =  ([ ⊕ ]) .  A m S R Pmn+1

Theorem 5.17. Let A be an n-abelain category, A, B two objects of A. Then the map m m m : nE (A, B) → nExtA(A, B) is an isomorphism of abelian groups.

Proof. The proof of this theorem is an adoption of [14, Proof of Theorem 6.4]. First, m is a group . Indeed, for any [S], [S ]∈nEm(A, B),

m([S]+[S ]) = m[(∇(S ⊕ S )) ]=∇(m[S ⊕ S ])

=∇(m[S]⊕m[S ]) = m[S]+m[S ].

Second, we construct the inverse of m. Given an n-resolution P• → A, factor dmn+1 : π m i Pmn+1 →  A → Pmn, with π the surjection and i the injection. For any fmn+1 : Homological algebra in n-abelian categories 655

Pmn+1 → B such that fmn+1dmn+2 = 0but fmn+1 = rdmn+1 ∀r ∈ A(Pmn, B). There is : m → = π π a morphism h n A B such that fmn+1 h since is a cokernel of dmn+2. Taking an n-pushout along h, we have a commutative diagram

··· Pmn+2 Pmn+1 π : m i ··· ··· . Sn n APmn P(m−1)n+1 P(m−1)n A

h fmn α S : BEmn ··· E(m−1)n+1 P(m−1)n ··· A

Define

 : m ( , ) → m( , ) ∗ + ∗ →[ ]. m nExtA A B nE A B by fmn+1 Im dmn+1 S  ∗ + ∗ = ∗ + ∗ We show that m is well defined. Indeed, let fmn+1 Im dmn+1 gmn+1 Im dmn+1. Then there is a morphism p : Pmn → B such that fmn+1 − gmn+1 = pdmn+1. There is : m → = π π a morphism k n A B such that gmn+1 k since is a cokernel of dmn+2.We have (h − k)π = fmn+1 − gmn+1 = pdmn+1 = piπ,soh − k = pi. Then, (h − k) · [Sn]=p ·[iSn] is contractible and follows from Proposition 5.13(iii). This proves that  ( ∗ + ∗ ) =  ( ∗ + ∗ ).  ( ∗ + ∗ ) m fmn+1 Im dmn+1 m gmn+1 Im dmn+1 Next, if m fmn+1 Im dmn+1 is a contractible m-fold n-exact sequence, then there is a morphism β : Emn → B such that βα = = β ∗ ∈ ∗ 1B. Then, fmn+1 fmndmn+1, it follows that fmn+1 Im dmn+1. To prove m is an isomorphism. It is enough to show that m is a homomorphism. Let  ( ∗ + ∗ ) =[ ]  ( ∗ + ∗ ) =[ ] m fmn+1 Im dmn+1 S , m gmn+1 Im dmn+1 S . Then  (( + )∗ + ∗ ) m fmn+1 gmn+1 Im dmn+1 =  [(∇( ⊕ ) )∗ + ∗ ] m fmn+1 gmn+1 Im dmn+1 ∗ ∗ = m[∇(( fmn+1 ⊕ gmn+1) + Im (dmn+1 ⊕ dmn+1) ) ] ∗ ∗ =∇m(( fmn+1 ⊕ gmn+1) + Im (dmn+1 ⊕ dmn+1) ) =∇([S]⊕[S ]) =[S]+[S ].



Acknowledgements This work is supported by Hunan Provincial Innovation Foundation for Postgraduate #CX2014B189, and partly supported by Natural Science Foundation of China #11671126.

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