EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 2, 2020, 323-345 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On the Category of Weakly U-Complexes Gustina Elfiyanti1,2,∗, Intan Muchtadi-Alamsyah1, Fajar Yuliawan1, Dellavitha Nasution1 1 Algebra Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, Indonesia 2 Mathematics Department, Faculty of Sciences and Technology, UIN Jakarta, Jakarta, Indonesia Abstract. Motivated by a study of Davvaz and Shabbani which introduced the concept of U- complexes and proposed a generalization on some results in homological algebra, we study the category of U -complexes and the homotopy category of U -complexes. In [8] we said that the category of U-complexes is an abelian category. Here, we show that the object that we claimed to be the kernel of a morphism of U -complexes does not satisfy the universal property of the kernel, hence we can not conclude that the category of U -complexes is an abelian category. The homotopy category of U-complexes is an additive category. In this paper, we propose a weakly chain U-complex by changing the second condition of the chain U-complex. We prove that the homotopy category of weakly U-complexes is a triangulated category. 2020 Mathematics Subject Classifications: 18E05,18G35, 18G80 Key Words and Phrases: U-complexes, weakly U-complexes, homotopy category of weakly U-complexes, triangulated category. 1. Introduction The notion of U-complexes was introduced by Davvaz and Shabani-Solt in [6] as a generalization of chain complexes of R-modules. They established some results in homological algebra such as Lambek Lemma, Snake Lemma and Connecting homomorphism and Exact Triangle. Their study was motivated by results from Freni and Sureau in [12] and Davvaz and Parnian-Garamaleky in [5]. Freni and Sureau introduced a notion of exact sequences of hypergroups by defining the kernel of a hypergroup homomorphism as the inverse image of U where U is the intersection of all ultra-closed subhypergroups of its codomain (note that a hypergroup does not always has zero element). Inspired by this, Davvaz and Parnian-Garamaleky proposed a generalization of exact sequences ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i2.3673 Email addresses: gustina.elfiyanti@uinjkt.ac.id (G. Elfiyanti), {ntan, fajar.yuliawan, dellavitha}@math.itb.ac.id (I. Muchtadi-Alamsyah, F. Yuliawan, D. Nasution) http://www.ejpam.com 323 c© 2020 EJPAM All rights reserved. G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 324 of R-modules, called U-exact sequences, by replacing the kernel of any differential with the preimage of a submodule U of its codomain. Then, Anvariyeh and Davvaz studied application of U -exactness and U -split exact sequences [1]. Further results on U -exactness given by Anvariyeh and Davvaz in [2] and Madanshekaf in [16]. Recently some authors continued working on U-exactness. Mahatma and Muchtadi- Alamsyah defined U -projective resolutions and U -extension modules [17]. Baur et al. then computed the U -projective resolution of modules over kQ where Q is quiver of type An and Ãn [3]. Fitriani, Surojo and Wijayanti introduced X-sub-exact sequence as a generalization of U-exact sequence [9]. By using the concept of X-sub-exact sequence, they studied X-sub-linearly independent [10]. Furthermore, the authors generalized the U-generator and M -subgenerator related to category σ [M ] [11]. In [7] and [8], we study the category of U-complexes and its homotopy category of U-complexes. We proved that the category of U-complexes and its homotopy category are additive categories. In this article we provide a corrigendum to the result in [8] which stated that the category of U-complexes is an abelian category. Then, we introduce a generalization of chain U -complexes, called weakly chain U -complexes, by changing the second condition of in the definition of the chain U -complexes. We show that the homotopy category of weakly U-complexes is a triangulated category. The paper is organized as follows. In section 2, we give the definition of additive category, triangulated category and we review the category of complexes. In section 3, we recall some results in [6], [7] and [8] that will be needed in the next section. Section 4 is the central section of our paper. In this section we introduce weakly U-complexes and show that the homotopy category of weakly U-complexes is a triangulated category. Convention: Throughout this paper, unless otherwise specified, we use the following notations: R denotes a ring with identity. Chain complexes and its generalizations are over R-Mod, the category of R modules. C (R) , U-C (R) , CU (R) denote the category of complexes, U-complexes and weakly U-complexes respectively. We denote 0 and 1 for the zero and identity morphisms respectively . 2. Preliminaries In this section we recall some basic concepts that will be needed in the following sections. For more detail we refer to [4], [13], [14], [15], [18] and [19]. 2.1. Additive and Triangulated Categories In this section we review the definition of additive category and triangulated category. Definition 1 ([14]). A category A is called an additive category if the following conditions hold: A1 For every pair of objects X,Y the set of morphisms HomA (X,Y ) is an abelian group and the composition of following morphisms is bilinear over the integers. HomA (Y,Z)×HomA (X,Y )→ HomA (X,Z) (1) G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 325 A2 A contains a zero object 0 (i.e for every objects X in A each morphism set HomA (X, 0) and HomA (0, X) has precisely one element). A3 For every pair of objects X,Y in A there exists a coproduct X ⊕ Y . A category satisfying (A1) and (A2) is called a preadditive category. If A is a pre- additive category, then by using the following proposition we can replace the condition A3 above with the existence of a biproduct in A. Proposition 1 ([4]). Given two objects A,B of a preadditive category C, the following conditions are equivalent: (i) the product (P, pA, pB) of A,B exists; (ii) the coproduct (P, sA, sB) of A,B exists; (iii) the biproduct (P, pA, pB, sA, sB) of A,B exists, i.e. there exists an object P and morphisms pA : P −→ A, pB : P −→ B, sA : A −→ P, sB : B −→ P (2) with the properties pAsA = 1, pBsB = 1, pAsB = 0, pBsA = 0 (3) sApA + sApB = 1 (4) Moreover, under these conditions sA = ker pB, sB = ker pA, pA = co ker sB, pB = co ker sA. Let T be an additive category and Σ : T −→ T be an additive automorphism. A triangle in T is a sequence of objects and morphism in T of the form X Y Z ΣXu v w (5) A morphism of triangles is a triple (f, g, h) of morphisms in T such that the following diagram is commutative in T . X Y Z ΣX X ′ Y ′ Z ′ ΣX ′ u f v g w h Σf u′ v′ w′ (6) The triple (f, g, h) is called an isomorphism of triangles if the morphisms f, g and h are isomorphisms in T . Definition 2 ([14]). A triangulated category is an additive category T together with an additive automorphism Σ, the translation or shift functor, and a colllection of distinguished triangles satisfying the following axioms: G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 326 TR0 Any triangle isomorphic to a distinguised triangle is again a distinguised triangle. TR1 For every object X in T , the triangle X X 0 ΣX1 (7) is a distinguised triangle. TR2 For every morphism f : X −→ Y in T there is a distinguised triangle of the form X Y Z ΣX f (8) TR3 If X Y M(α(f)) ΣX f α(f) β(f) (9) is a distinguised triangle then the following rotated triangle is also a distinguised triangle. Y M(α(f)) ΣX ΣY α(f) β(f) −Σf (10) TR4 Given distinguished triangles X Y Z ΣXu v w and X ′ Y ′ Z ′ ΣX ′u′ v′ w′ then each commutative diagram X Y Z ΣX X ′ Y ′ Z ′ ΣX ′ u f v g w Σf u′ v′ w′ (11) can be completed to a morphism of triangles (but not necessarily uniquely). TR5 (Octahedral axiom) Given the following distinguised triangles X Y Z ′ ΣX Y Z X ′ ΣY X Z Y ′ ΣX u v vu (12) then there exists a distinguished triangle Z ′ Y ′ X ′ ΣZ ′ making the following diagram commutative G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 327 X Y Z ′ ΣX X Z Y ′ ΣX Y Z X ′ ΣY Z ′ Y ′ X ′ ΣZ ′ u 1 v 1 vu u 1 Σu v 1 (13) 2.2. The Category of Complexes 2.2.1. Chain Complexes A complexes (over R-Mod) is a family X = ( Xn, d X n ) n∈Z · · · Xn+1 Xn Xn−1 · · · dXn+1 dXn (14) where Xn are R-modules, and dXn : Xn −→ Xn−1 are R-modules homomorphisms such that dXn+1d X n = 0 for all n ∈ Z. The morphism dXn is called the differential of X on degree n. A morphism f : X → Y of complexes is a family f = (fn)n∈Z of morphisms hn : Xn −→ Yn such that fnd X n+1 = dYn+1fn+1 for all n ∈ Z. · · · Xn+1 Xn Xn−1 · · · · · · Yn+1 Yn Xn−1 · · · dXn+1 fn+1 dXn fn fn−1 dYn+1 dYn (15) The chain complexes together with morphism of complexes form a category C(R), the category of complexes. This category is an abelian category. 2.2.2. The Homotopy Category of Complexes Let X and Y be two objects in C(R). A morphsim f ∈ HomC(R) (X,Y ) is called homotopic to zero (or null homotopic) if there exists a family h = (hn)n∈Z of morphisms hn : Xn → Yn+1 · · · Xn+1 Xn Xn−1 · · · · · · Yn+1 Yn Xn−1 · · · dXn+1 fn+1 dXn fn hn hn−1 fn−1 dYn+1 dYn (16) satisfying fn = dYn+1hn + hn−1d X n for all n ∈ Z (17) G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 328 The morphism h is called a (chain) homotopy map. Two morphisms f, g ∈HomC(R) (X,Y ) are called homotopy equivalent, denoted by f ∼ g, if and only if f − g is homotopic to zero. A complex X is called homotopic to zero if the identity morphism on X is homotopic to zero. The homotopy relation is an equivalence relation on the class of morphism in C (R). Moreover if Ht (X,Y ) is the set of morphisms from X to Y which are homotopic to zero, then the collection of all Ht (X,Y ) form and ideal in C (R). This implies the composition of two equivalence classes (modulo homotopy) can be defined as the equivalence classes of composition of two representative morphisms from each equivalence class. The quotient category of C (R) modulo this ideal is called homotopy category. Definition 3 ([15]). The homotopy category of complexes, denote by K (R), has the same object as the category C (R). The morphisms in K (R) are the equivalence classes of morphism in C (R) modulo homotopy, i.e. HomK(R) (X,Y ) = HomC(R) (X,Y ) /Ht (X,Y ) (18) and the composition of two equivalence classes (modulo homotopy) is defined as the equiva- lence classes of composition of two representative morphisms from each equivalence class, i.e. ḡ ◦ f̄ = g ◦ f for all f̄ ∈ HomC(R) (X,Y ) /Ht (X,Y ) and ḡ ∈ HomC(R) (Y, Z) /Ht (X,Y ). Proposition 2 ([14]). The homotopy category K (R) is an additive category. 2.2.3. Triangulated Structure of the Homotopy Category of Complexes In this section we recall a method to get a triangulated structure on K (R). At first we need an additive automorphism on K (R), then we find a suitable set distinguished triangles in K (R). The additive automorphism can be defined on the level of the cateogry C (R) as follow. Definition 4 ([14]). A translation functor or (left) shift Σ in C (R) is defined by shifting any complex one degree to the left. More precisely, for an object X = ( Xn, d X n ) n∈Z in C (R), define ΣX = ( (ΣX)n , d ΣX n ) n∈Z with (ΣX)n = Xn−1 and dΣX n = −dXn−1. For a morphism f : X −→ Y in C (R), set Σf = ((Σf)n)n∈Z where (Σf)n = fn−1. This functor is an addi- tive functor, i.e. for every pair objects X,Y in C (R) the map Hom(X,Y ) −→ Hom(ΣX,ΣY ) is a morphism of abelian groups. Moreover it is an automorphism of the category C (R), where the inverse is given by (right) shift. To find the set of distinguished triangles in K (R) we need the following construction of the mapping cone. Definition 5 ([14]). Let f : X −→ Y be a morphism in C (R). The mapping cone of f is the object M (f) in C (R) defined by M (f)n = Xn−1 ⊕ Yn and dM(f) n = ( −dXn−1 0 fn−1 dYn ) (19) G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 329 The following morphisms are canonical morphisms in C (R) α (f) : Y −→M (f) , where α (f) = ( 0 1 ) (20) β (f) : M (f) −→ ΣX, where β (f) = ( 1 0 ) (21) The morphisms above are also well-defined in K (R). Hence a distinguished triangle in K (R) can be defined as follow. Definition 6 ([14]). A standard triangle in K (R) is a sequence X Y M(f) ΣX f α(f) β(f) (22) A distinguished triangle in K (R) is a triangle which is isomorphic (in K (R)) to a standard triangle. With this class of distinghuished triangles we can prove the following proposition. Proposition 3. [14]The homotopy category K (R) of complexes is a triangulated category. 3. A Generalization of the Category of Complexes In this section we review some results in [6], [7] and [8]. In the first subsection we also provide a corrigendum to [8]. 3.1. The Category of U-Complexes A chain U-complex (over R-Mod) is a family X = ( Xn, U X n , d X n ) n∈Z · · · Xn+1 Xn Xn−1 Xn−2 · · · dXn+1 dXn dXn−1 (23) where Xn and UXn are R-modules, UXn is a submodule of Xn, and dXn : Xn −→ Xn−1 are R-modules homomorphisms such that for all n ∈ Z: (i) dXn d X n+1 (Xn+1) ⊆ UXn−1, and (ii) Im ( dXn ) ⊇ UXn−1 A morphism of U-complexes f : X → Y is a family f = (fn : Xn −→ Yn)n∈Z of R- modules homomorphisms such that every rectangle commutes and fn ( UXn ) ⊆ UYn for all n ∈ Z. The morphism f is called an isomorphism of U -complexes if each fn is an R-module isomorphism and the sequence of R-module morphisms f−1 = ( f−1 n : Yn −→ Xn ) n∈Z is also a morphism of U-complexes. The following are examples of chain U-complexes and morphisms of U-complexes. G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 330 Example 1. (i) Every chain complex is a chain U-complex with Un = 0 for all n ∈ Z. (ii) Suppose we have the following sequence of R-modules and R-modules homomorphsim · · · Xn+1 Xn Xn−1 Xn−2 · · ·dn+1 dn dn−1 (24) Then the families X = (Xn, dn+1dn+2 (Xn+2) , dn)n∈Z and Y = (Xn, dn (Xn) , dn)n∈Z are chain U-complexes. A morphsim f : X −→ Y defined by fn = 1 is a morphism of U-complexes, but generally it is not an isomorphism of U-complexes. Suppose f = (fn : Xn −→ Yn)n∈Z and g = (gn : Yn −→ Zn)n∈Z are morphisms of U-complexes, then it is clear that gf = (gnfn : Xn −→ Zn)n∈Z is also a morphism of U-complexes. We define the category of U-complexes, denote by U-C (R), as a category whose objects are chain U-complexes and the morphisms are morphism of U-complexes. This category is an additive category. In [8], we also stated that U-C (R) is an abelian category by claiming the kernel of a morphism U-complexes f : X −→ Y is K = ( Kn, U K n , d K n ) n∈Z with Kn = ker fn = {x ∈ Xn | fn (x) = 0} , UKn = ( dKn+1d K n+2 ) (Kn+2) (25) and dKn is the resitriction of dXn on Kn. But in the following example we can see that generally it does not satisfy the universal property of kernel. Hence we can not conclude that U-C (R) is an abelian category by defining the kernel as in (25). Example 2. Suppose X be the chain U-complex defined by X0 = X−1 = Z and zero otherwise, dX0 = 1 and zero otherwise, UX−1 = Z and zero otherwise. Let Y be the chain U-complex defined by shifting X one degree to the left. If f : X −→ Y is defined by f0 = 1 and zero otherwise, then we have K = X as follow: K : 0 0 0 0 ⊆ Z 0 X : 0 0 0 ⊂ Z Z ⊆ Z 0 Y : 0 0 ⊂ Z Z ⊆ Z 0 0 k 1 f 1 1 1 (26) Let L = X, then l : L −→ X defined by l−1 = 1 and zero otherwise is a morphism of G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 331 U-complexes, moreover fl = 0. K : 0 0 0 0 ⊂ Z 0 L : 0 0 0 ⊂ Z Z ⊆ Z 0 X : 0 0 0 ⊂ Z Z ⊆ Z 0 Y : 0 0 ⊂ Z Z ⊆ Z 0 0 k g l 1 1 1 f 1 1 1 (27) The morphism g : L −→ X defined by g−1 = 1 and zero otherwise is the only morphism such that kg = l, but g is not a morphism of U-complexes since g ( UL−1 ) = Z 6⊆ UK−1 = 0. Hence K is not the kernel of f . 3.2. The Homotopy Category of U-Complexes A morphism f : X −→ Y in U -C (R) is called homotopic to zero (or null homotopic) if there exists a chain homotopy map h = (hn : Xn −→ Yn+1)n∈Z such that fn = dYn+1hn + hn−1d X n and hn ( UXn ) ⊆ UYn+1 (28) We call two morphisms f, g : X −→ Y in U -C (R) homotopic (or homotopy equivalent), if f − g is null homotopic. We write f ∼ g if they are homotopy equivalent. The homotopy relation ∼ is also an equivalence relation on the class of morphisms in U -C (R). Furthemore the collections of homotopy equivalence classes of morphisms of U -complexes form an ideal in U-C (R). Lemma 1. Suppose X and Y are any objects in U-C (R). Then the collections of all Ht (X,Y ) = { f ∈ HomCU (R) (X,Y ) | f ∼ 0 } (29) forms an ideal in U-C (R). Proof. Let f, g ∈ Ht (X,Y ), α ∈ HomU−C(R) (Y,Z) and β ∈ HomU−C(R) (W,X). Suppose r = (rn : Xn → Yn+1)n∈Z and s = (sn : Xn → Yn+1)n∈Z be homotopy maps such that fn = dYn+1rn + rn−1d X n and gn = dYn+1sn + sn−1d X n . Then βn (fn − gn)αn = βn ( dYn+1rn + rn−1d X n − dYn+1sn − sn−1d X n ) αn = βnd Y n+1 (rn − sn)αn + βn (rn−1 − sn−1) dXn αn = dZn+1βn+1 (rn − sn)αn + βn (rn−1 − sn−1)αn−1d W n−1 Set t = (tn = βn+1 (rn − sn)αn : Wn → Zn+1)n∈Z, then tn is a homotopy map. Hence βn (fn − gn)αn ∼ 0. Therefore we can define the homotopy category of chain U-complexes as the quotient of U-C (R) modulo this ideal. G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 332 Definition 7. The homotopy category of U-complexes, denote by U-K (R), has the same object as the category U-C (R). The morphisms in U-K (R) are the equivalence classes of morphism in U-C (R) modulo homotopy, i.e. HomU−K(R) (X,Y ) = HomU−C(R) (X,Y ) /Ht (X,Y ) (30) The homotopy category of U-complexes is also an additive category [7]. To check whether the homotopy category of U-complexes U-K (R) carries a triangulated structure, we need to construct a mapping cone in U-C (R). Let f : X −→ Y be a morphism in U-C (R). Suppose M (f)n = Xn−1 ⊕ Yn, UM(f) n = UXn−1 ⊕ UYn and dM(f) n = ( −dXn−1 0 fn−1 dYn ) (31) For any (x, y) ∈ Xn−1 ⊕ Yn, observe that dM(f) n d M(f) n+1 (x, y) = ( dXn d X n−1 0 dYn fn − dXn fn−1 dYn d Y n+1 )( x y ) ∈ ( UXn−2 UYn−1 ) = U M(f) n−1 (32) and dM(f) n (x, y) = ( −dXn−1(x) fn−1(x) + dYn (y) ) (33) In the following example we note that in general Im ( d M(f) n ) does not contain U M(f) n−1 . Hence we can not define the mapping cone in U-C (R) as in (31). Example 3. Suppose we have the following morphism of chain U-complexes X : 0 0 R R 0 Y : 0 R R⊕R 0 0 f 1 f0 dY1 dY0 (34) where dX0 = 1, f0 = ( 0 1 ) , dY1 = ( 1 0 ) , dY0 = ( 0 0 ) , UX−1 = R, UY0 = R⊕0. Then M(f) is 0 R⊕R R⊕(R⊕R) 0 0∂ (35) For any (x, y) ∈ R⊕R, observe that ∂ ( x y ) = −1 0 0 1 1 0 (x y ) = −xy z  (36) Since Im(∂) = −R⊕ (R⊕R) 6⊇ R⊕(R⊕0) = UX0 ⊕UY1 we conclude that M(f) is not an object in U-C (R). G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 333 Observe that for M (f) in the construction (31) we have d M(f) n ( U M(f) n ) ⊆ U M(f) n−1 . Furthermore for any chain U -complex X, it also satisfies dXn ( UXn ) ⊆ UXn−1. This motivate us to define a weakly chain U-complex. 4. A Generalization of the Category of U-Complexes In this section we propose a generalization of chain U-complex, called weakly chain U-complex. Then, we prove that the homotopy category of weakly U-complexes carries triangulated structure. Let X = ( Xn, U X n , d X n ) n∈Z be a family of R-modules and R-modules homomorphisms where UXn is a submodule of Xn. We define a weakly chain U-complex (over R-Mod) by replacing the second condition of chain U -complex i.e dn (Xn) ⊇ Un−1 with dn(Un) ⊆ Un−1. It is easy to check that every chain complex and chain U-complex are weakly chain U-complexes. We define a morphism of weakly chain U -complexes analog to the definition of morphism of U-complexes, i.e. f : X −→ Y is a morphism of weakly chain U-complexes if f = (fn : Xn −→ Yn)n∈Z is a family of R-modules homomorphisms such that every rectangle commutes and fn ( UXn ) ⊆ UYn for all n ∈ Z. We denote the category of weakly chain U-complexes as CU (R). Proposition 4. The category CU (R) of weakly chain U-complexes is an additive category. Proof. The structure of an abelian group of HomCU (R) (X,Y ) and the billinearity of composition of morphisms are inherited from HomC(R) (X,Y ). The zero object in C (R) is also a zero object in CU (R). A biproduct of two objects X and Y is quintuple (X ⊕ Y, pX , pY , sX , sY ) where X ⊕ Y, pX , pY , sX and sY are defined as follow: X ⊕ Y = ( X ⊕ Y,UX⊕Y , dX⊕Y ) = ( Xn ⊕ Yn, UX⊕Yn , dX⊕Yn ) n∈Z (37) where UX⊕Yn = ( UXn UYn ) and dX⊕Yn = ( dXn 0 0 dYn ) (38) (sX)n = ( 1 0 ) , (sY )n = ( 0 1 ) , (39) (pX)n = ( 1 0 ) , (pY )n = ( 0 1 ) (40) Analog to the definition of homotopy equivalent in the category of U-complexes, we call two morphisms f, g ∈ HomCU (R) (X,Y ) are homotopy equivalent if f − g is homotopic to zero (or null homotopic), i.e., there exists a chain map s = (sn : Xn −→ Yn+1)n∈Z such that fn − gn = dYn+1sn + sn−1d X n and sn ( UXn ) ⊆ UYn+1 (41) G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 334 We called a weakly chain U-complex X is homotopic to zero (or null homotopic) if the identity morphism on X is homotopic to zero. It is clear that the homotopy relation is an equivalence relation on the class of morphisms in CU (R) and the collection of homotopy equivalence classes of morphisms in CU (R) form an ideal in CU (R). We define the homotopy category KU (R) of weakly chain U -complexes as the quotient of CU (R) modulo this ideal. Since composition and addition are well defined on the homotopy classes, it follows that KU (R) inherits the bilinear composition from CU (R). Therefore we have the following result. Proposition 5. The homotopy category KU (R) of weakly chain U-complexes is an additive category. Next, we will show that KU (R) is a triangulated category. We construct a translation functor Σ on CU (R) analog to translator functor on K (R). Definition 8. The translation functor shift Σ of X is an object ΣX = ( ΣXn, U ΣX n , dΣX n ) n∈Z defined by ΣXn = Xn−1, U ΣX n = UXn−1, and dΣX n = −dXn−1 (42) and for a morphism f = (fn)n∈Z in KU (R) we set Σf = (Σfn)n∈Z where Σfn = fn−1. (43) The functor Σ above is an additive automorphism in CU (R). Moreover it is compatible with homotopies, hence we have a well-defined induced functor Σ on KU (R). A triangle and morphism of triangles in CU (R) is defined analog to the definition of triangle and morphism of triangles in homotopy category C (R) of complexes. Lemma 2. Let f : X −→ Y be a morphism in CU (R) then M (f) = ( M (f)n , U M(f) n , dM(f) n ) n∈Z (44) where M (f)n = Xn−1 ⊕ Yn, UM(f) n = UXn−1 ⊕ UYn , and dM(f) n = ( −dXn−1 0 fn−1 dYn ) (45) is an object in KU (R) Proof. From (31) we know that d M(f) n d M(f) n+1 ( M (f)n+1 ) ⊆ U M(f) n−1 . Now let (a, b) ∈ UXn−1 ⊕ UYn , note that dM(f) n (a, b) = ( −dXn−1 0 fn−1 dYn )( a b ) = ( −dXn−1 (a) fn−1 (a) + dYn (b) ) ∈ UM(f) n−1 (46) The object M (f) above is called the mapping cone of f . G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 335 Lemma 3. The mapping cone M (1) of identity morphims on X is homotopic to zero. Proof. The mapping cone of identity morphism on X is M (1) = ( Xn−1 ⊕Xn, U X n−1 ⊕ UXn , dM(1) n ) n∈Z (47) where dM(1) n = ( −dXn−1 0 1 dXn ) : Xn−1 ⊕Xn −→ Xn−2 ⊕Xn−1 (48) Look at the following diagram. · · · Xn ⊕Xn+1 Xn−1 ⊕Xn Xn−2 ⊕Xn−1 · · · · · · Xn ⊕Xn+1 Xn−1 ⊕Xn Xn−2 ⊕Xn−1 · · · d M(1) n+1 d M(1) n 1 sn sn−1 d M(1) n+1 d M(1) n (49) Suppose sn : Xn−1 ⊕Xn → Xn ⊕Xn+1 is defined by sn = ( 0 1 0 0 ) . It is clear that sn ( U M(1) n ) ⊆ UM(1) n+1 and d M(1) n ( U M(1) n ) ⊆ UM(1) n−1 . Observe that d M(1) n+1 sn + sn−1d M(1) n = ( −dXn 0 1 dXn+1 )( 0 1 0 0 ) + ( 0 1 0 0 )( −dXn−1 0 1 dXn ) = ( 1 0 0 1 ) (50) Hence, in the homotopy category KU (R) of weakly chain U-complexes, the identity morphism on M (1) is equal to the zero map. As a censequence, in the KU (R), the mapping cone M (1) is isomorphic to zero complex. Lemma 4. If f : X −→ Y is a morphism in CU (R), then the following canonical morphisms are also morphisms in CU (R): α (f) : Y −→M (f) where α (f) = ( 0 1 ) (51) and β (f) : M (f) −→ ΣX, where β (f) = ( 1 0 ) (52) Furthermore, X Y M(f) ΣX f α(f) β(f) (53) is a short exact sequence of chain complexes. G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 336 Proof. We only need to show that α (f) dan β (f) satisfy the second condition of morphsim of U-complexes. Suppose x ∈ UYn and (v, w) ∈ UM(f) n , then α (f)n (x) = ( 0 x ) ∈ ( UXn−1 UYn ) = UM(f) n (54) and β (f)n ( v w ) = v ∈ UΣX n (55) The morphisms α (f) and β (f) above are also well-defined in KU (R). This bring us to the following definition. Definition 9. A distinguished triangle in KU (R) is a triangle which is isomorphic (in KU (R) to the following standard triangle X Y M(f) ΣX f α(f) β(f) (56) We use this class of distinguished triangles to prove that the homotopy category of weakly U-complexes has a triangulated structure. Theorem 1. The homotopy category KU (R) of weakly U-complexes is a triangulated category. Proof. By Definition 9 and Lemma 4 it is clear that axioms (TR0) and (TR2) are satisfied. Suppose X,Y are objects in KU (R) . TR1 Consider the triangle X X M(1) ΣX1 β(f) (57) From Lemma 3 we know that the mapping cone M (1) is isomorphic to a zero object in KU (R). Hence, the following is a distinguished triangle. X X 0 ΣX1 (58) TR3 Suppose X Y M(f) ΣX f α(f) β(f) be a distinguised triangle. We will show that the rotated triangle Y M(f) ΣX ΣY α(f) β(f) f (59) is a distinguished triangle by proving that it is isomorphic in KU (R) to the following standard triangle Y M(f) M (α (f)) ΣY α(f) α(α(f)) β(α(f)) (60) G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 337 To construct an isomorphism between (59) and (60), we take identity map for the first, second and fourth entries. Y M(f) ΣX ΣY Y M(f) M(α(f)) ΣY α(f) β(f) −Σf φ α(f) α(α(f)) β(α(f)) ψ (61) and define φn = −fn−1 1 0  and ψn = ( 0 1 0 ) . First we will show that φ and ψ are morphisms in KU (R). Look at the following diagram Xn Xn−1 Xn−2 Yn ⊕Xn ⊕ Yn+1 Yn−1 ⊕Xn−1 ⊕ Yn Yn−2 ⊕Xn−2 ⊕ Yn−1 −dXn −dXn−1 φn d M(α(f)) n+1 d M(α(f)) n ψn (62) where dM(α(f)) n = −dYn−1 0 0 0 −dXn−1 0 1 fn dYn  (63) It is clear that φn ( UΣX n ) ⊆ UM(α(f)) n and ψn ( U M(α(f)) n ) ⊆ UΣX n . Moreover φn ( −dXn )fndXn−dXn 0  = d M(α(f)) n+1 φn+1 (64) and −dXn−1ψn = ( 0 −dXn−1 0 ) = ψn−1d M(α(f)) n (65) Now we will show that φ and ψ give a morphism of triangle in KU (R). Note that β(α(f))nφn = ( 1 0 0 )−fn−1 1 0  = −fn−1 (66) Hence β (α (f))φ = −Σf . Observe the following diagram Xn ⊕ Yn+1 Xn−1 ⊕ Yn Xn−2 ⊕ Yn−1 Yn ⊕Xn ⊕ Yn+1 Yn−1 ⊕Xn−1 ⊕ Yn Yn−2 ⊕Xn−2 ⊕ Yn−1 d M(f) n+1 d M(f) n φnβ(f)n α(α(f))n hn hn−1 d M(α(f)) n+1 d M(α(f)) n (67) G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 338 where d M(α(f)) n+1 −dYn 0 0 0 −dXn 0 1 fn dYn+1  and dM(f) n = ( −dXn−1 0 fn−1 dYn ) (68) Let hn = 0 −1 0 0 0 0  : M (f)n −→M (α (f))n+1 (69) then it is clear that hn ( U M(f) n ) ⊆ UM(α(f)) n+1 and φnβ (f)n − α (α (f))n = −fn−1 0 0 0 0 −1  = d M(α(f)) n+1 hn−1 + hnd M(f) n (70) Thus φβ (f) ∼ α (α (f)). We also have β(f) = ψα (α(f)) since β (f)n − ψnα (α (f))n = ( 1 0 ) − ( 0 1 0 )0 0 1 0 0 1  = ( 0 0 ) (71) Now we will show −Σfψ ∼ β (α (f)). Consider the folowing diagram Yn ⊕Xn ⊕ Yn+1 Yn−1 ⊕Xn−1 ⊕ Yn Yn−2 ⊕Xn−2 ⊕ Yn−1 Yn Yn−1 Yn−2 d M(α(f)) n+1 d M(f) n −Σfnψn β(α(f))n gn −dYn −dYn−1 (72) Let gn = ( 0 0 −1 ) then it is clear that gn ( U M(α(f)) n ) ⊆ UΣY n+1 and −Σfnψn − β (α (f))n = ( −1 −fn−1 0 ) = −dYn gn + gn−1d M(α(f)) n (73) We get −Σfψ ∼ β (α (f)). Next, we will show that ψφ = 1 and φψ ∼ 1. Note that ψφ = ( 0 1 0 )−Σf 1 0  = 1 (74) Let pn : M (α (f))n −→M (α (f))n+1 = 0 0 −1 0 0 0 0 0 0  (75) G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 339 Then it is clear that pn ( U M(α(f)) n ) ⊆ U M(α(f)) n+1 and φnψn − 1 = −1 −fn−1 0 0 0 0 0 0 −1  = d M(α(f)) n+1 pn−1 + pnd M(f) n (76) Thus φψ ∼ 1. So the following is a distinguished triangle. Y M(f) Σ ΣY α(f) β(f) −Σf (77) TR4 Suppose we have a diagram X Y M(u) ΣX X ′ Y ′ M(u′) ΣX ′ u f α(u) g β(u) Σf u′ α(u′) β(u′) (78) where the left square commutes in KU (R), i.e. there exist homotopy map sn : Xn −→ Y ′n+1 such that gnun − u′nfn = dY ′ n+1sn + sn−1d X n and sn ( UXn ) ⊆ UY ′n+1 for all n ∈ Z. Define h = (hn) : M (u) −→M ( u′ ) where hn = ( fn−1 0 sn−1 gn ) (79) Observe that hnα (u)n = ( 0 gn ) = α ( u′ ) gn (80) and β ( u′ ) n hn = ( fn−1 0 ) = (Σf)n β (u)n (81) and for any (a, b) ∈ UM(u) n = UXn−1 ⊕ UYn we have h ( a b ) = ( fn−1 (a) gn (b) + sn−1 (a) ) ∈ ( UX ′ n−1 UY ′ n ) = UM(u′) n (82) Hence (f, g, h) is a morphism of triangle in KU (R). TR5 Assume that we have the following diagram in KU (R). X Y M(u) ΣX X Z M(vu) ΣX Y Z M(v) ΣY M(u) M(vu) M(v) ΣM(u) u α(u) v β(u) vu u α(vu) β(vu) Σu α(u) v α(vu) α(v) β(v) Σα(u) (83) G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 340 We define the missing morphisms as follows. f : M (u) −→M (vu) where fn = ( 1 0 0 vn ) (84) g : M (vu) −→M (v) where gn = ( un−1 0 0 1 ) (85) h : M (v) −→ ΣM (u) where hn = Σα (u)β (v) = ( 0 0 1 0 ) (86) X Y M(u) ΣX X Z M(vu) ΣX Y Z M(v) ΣY M(u) M(vu) M(v) ΣM(u) u α(u) v β(u) f vu u α(vu) β(vu) g Σu α(u) v α(vu) α(v) β(v) Σα(u) f g h (87) It easy to check that fn ( U M(u) n ) ⊆ UM(vu) n , g ( U M(vu) n ) ⊆ UM(v) n and hn ( U M(v) n ) ⊆ U ΣM(v) n . Moreover fnα(u)n − α(vu)nvn = ( 1 0 0 vn )( 0 1 ) − ( 0 1 ) vn = ( 0 0 ) (88) β(vu)nfn − 1β(u)n = ( 1 0 )(1 0 0 vn ) − 1 ( 1 0 ) = ( 0 0 ) (89) gnα(vu)n − α(v)n = ( un−1 0 0 1 )( 0 1 ) − ( 0 1 ) = ( 0 0 ) (90) β(v)ngn − un−1β(vu)n = ( 1 0 )(un−1 0 0 1 ) − un−1 ( 1 0 ) = ( 0 0 ) (91) Hence (f, g, h) is a morphism of triangles in KU (R). Now we need to show that the bottom line M(u) M(vu) M(v) ΣM(u) f g h (92) is a distinguished triangle di KU (R). For this we construct an isomorphism to the standard triangle M(u) M(vu) M(f) ΣM(u) f α(f) β(f) (93) Since only the third entries in triangles are different, it suffices to find morphisms σ : M (v) −→ M (f) and τ : M (f) −→ M (v) such that the diagrams commute in G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 341 KU (R), i.e. β (f)σ = h, hτ = β (f) , σg = α (f) and τα (f) = g, up to homotopy. Moreover, we have to show that they are isomorphisms in KU (R). Set σn =  0 0 1 0 0 0 0 1  and τn = ( 0 1 un−1 0 0 0 0 1 ) (94) Look at the following diagram M(u) M(vu) M(u) ΣM(u) M(u) M(vu) M(f) ΣM(u) f g h σ f α(f) β(f) τ (95) By definition we can check that σn ( U M(v) n ) ⊆ U M(f) n and τn ( U M(f) n ) ⊆ U M(v) n . Moreover τnα (f)n − gn = ( 0 1 un−1 0 0 0 0 1 ) 0 0 0 0 1 0 0 1 − (un−1 0 0 1 ) = ( 0 0 0 0 ) (96) and β (f)n σn − hn = ( 1 0 0 0 0 1 0 0 ) 0 0 1 0 0 0 0 1 − (0 0 1 0 ) = ( 0 0 0 0 ) Thus τα (f) = g and β (f)σ = h. Next we will show that α (f) ∼ σg. Consider the following diagram. Xn ⊕ Zn+1 Xn−1 ⊕ Zn Xn−2 ⊕ Zn−1 Xn−1 ⊕ Yn ⊕Xn ⊕ Zn+1 Xn−2 ⊕ Yn−1 ⊕Xn−1 ⊕ Zn Xn−3 ⊕ Yn−2 ⊕Xn−2 ⊕ Zn−1 d M(vu) n+1 d M(vu) n α(f))n σngn sn sn−1 d M(f) n+1 d M(f) n (97) Note that dM(vu) n = ( −dXn−1 0 (vu)n−1 dZn ) (98) dM(f) n =  dXn−2 0 0 0 −un−2 −dYn−1 0 0 1 0 −dXn−1 0 0 vn−1 (vu)n−1 dZn  (99) G. Elfiyanti et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 323-345 342 Define rn : M (vu)n →M (f)n+1 by rn =  1 0 0 0 0 0 0 0 . Then rn ( U M(vu) n ) ⊆ UM(f) n+1 and α (f)n − σngn =  0 0 0 0 1 0 0 1 −  0 0 1 0 0 0 0 1 (un−1 0 0 1 ) =  0 0 −un−1 0 1 0 0 0  = d M(f) n+1 rn + rn−1d M(vu) n (100) Therefore we obtain α (f) ∼ σg. Now we will show that β (f) ∼ hτ . Look at the following diagram Xn−1 ⊕ Yn ⊕Xn ⊕ Zn+1 Xn−2 ⊕ Yn−1 ⊕Xn−1 ⊕ Zn Xn−3 ⊕ Yn−2 ⊕Xn−2 ⊕ Zn−1 Xn−1 ⊕ Yn Xn−2 ⊕ Yn−1 Xn−3 ⊕ Yn−2 d M(f) n+1 d M(f) n βfn hnτn sn sn−1 d ΣM(u) n+1 d ΣM(u) n (101) with dΣM(u) n = ( −dXn−2 0 un−2 dYn−1 ) (102) Let sn = ( 0 0 −1 0 0 0 0 0 ) then it is clear that sn ( U M(f) n ) ⊆ UΣM(u) n+1 and β (f)n − hnτn = ( 1 0 0 0 0 1 0 0 ) − ( 0 0 1 0 )( 0 1 un−1 0 0 0 0 1 ) = ( 1 0 0 0 0 0 −un−1 0 ) = d ΣM(u) n+1 sn + sn−1d M(f) n (103) Hence we have β (f) ∼ hτ . Last we will show that τ and σ are isomorphisms in the homotopy category KU (R). Note that τnσn − 1 = ( 0 0 0 0 ) (104) Let tn : M (f)n −→M (f)n+1 defined by tn =  0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0  (105) REFERENCES 343 then it is clear that tn ( U M(f) n ) ⊆ UM(f) n+1 and σnτn − 1n =  −1 0 0 0 0 0 un−1 0 0 0 −1 0 0 0 0 0  = d M(f) n+1 tn + tn−1d M(f) n Thus τσ = 1 and στ ∼ 1 which mean that σ and τ are isomorphism of triangle in KU (R). Hence, M(u) M(vu) M(v) ΣM(u) f g h is a distinguished triangle in KU (R) and we have proved the octahedral axiom for KU (R). 5. Conclusion Category of U -complexes is a generalization of category of complexes defined by replacing the objects with chain U-complexes and the morphisms with morphisms of U-complexes. It is an additive category. The homotopy category of U-complexes is also an additive category. Let X = ( Xn, U X n , d X n ) n∈Z be a chain U -complex, then dXn ( UXn ) ⊆ UXn−1. We introduce a weakly chain U-complex by replacing the second condition of chain U-complex with dXn ( UXn ) ⊆ UXn−1. The category of weakly U-complexes is again an additive category and its homotopy category is a triangulated category. Every chain complex is a chain U-complex with Un = 0foralln ∈ mathbbZ. From the first and the second condition of chain U -complex we know that chain U -complexes is also a weakly U-complexes. 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