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Proc. Indian Acad. Sci. (Math. Sci.) Vol. 127, No. 4, September 2017, pp. 625–656. DOI 10.1007/s12044-017-0345-4 Homological algebra in n-abelian categories DEREN LUO College of Mathematics, 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 functors of left(right) n-exact functors, n-(co)resolutions, and n-homological dimensions of n-abelian categories. For an n-exact sequence, we show that the long n-exact sequence theorem holds as a generalization of the classical long exact sequence ∗ ∗ theorem. As a generalization of Ext (−, −), we study the n-derived functor nExt (−, −) of hom-functor Hom(−, −). We give an isomorphism between the abelian group 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 category; n-derived functor; nExt -correspondence; n-Baer sum; n-cluster tilting. 2010 Mathematics Subject Classification. 18G50; 18G15; 18E25. 1. Introduction Recently, a new class of categories called 2-cluster tilting subcategories that appeared in representation theory 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 sequences which are n-exact sequences in the sense of Jasso [13]. He developed the classical abelian category and exact category theory to higher-dimensional n-abelian category and n-exact category theory [13]. He also proved that n-cluster tilting subcategories are n-abelian categories. These new discoveries have broken new ground in category theory. Homological algebra, 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)homology 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)resolution, 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 abelian group 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 additive category. We denote the category of cochain complexes of C by Ch(C) and the homotopy category of C by H(C). Also, we denote by Chn(C) the full subcategory 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 morphisms in C from X to Y ,ifX, Y ∈ obC. 2.1 n-Kernels, n-cokernels and n-exact sequences Let C be an additive category and d0 : X 0 → X 1 a morphism in C.Ann-cokernel 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-kernel 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 direct sum 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 epimorphism and split monomorphism 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).
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