EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1913-1939 ISSN 1307-5543 – ejpam.com Published by New York Business Global Localization in the Category COMP (Gr(A−Mod)) of Complex associated to the Category Gr(A−Mod) of Graded left A−modules over a Graded Ring Ahmed Ould Chbih1,∗, Mohamed Ben Faraj Ben Maaouia2, Mamadou Sanghare3 1 Unité de recherche Géométrie, Analyse, Algèbre et Applications (G3A), Faculté des Sciences et Techniques/Universitéé de Nouakchott, Nouakchott, Mauritanie 2 Applied Mathematics, UFR-SAT/Gaston BERGER, University, Saint-Louis, Senegal 3 Université Cheikh Anta Diop, Dakar (UCAD), Sénégal Abstract. The main results of this paper are: If A = ⊕ n∈Z An is a graded duo-ring, SH is a part formed of regulars homogeneous elements of A, SH is the homogeneous multiplicatively closed subset of A generated by SH , then: (i) The relation CH(−) : Gr(S −1 H A − Mod) −→ COMP (Gr(S −1 H A − Mod)) which that for all graded left S −1 H A−module S −1 H M of Gr(S −1 H A−Mod) we correspond the associate complex sequence (S −1 H M)∗ to a graded S −1 H A−module S −1 H M and for all graded morphism of graded left S −1 H A−modules S −1 H f : S −1 H M −→ S −1 H N of degree k we correspond the associate complex chain (S −1 H f)k∗ to a morphism of graded left S −1 H A−module S −1 H f : S −1 H M −→ S −1 H N is additively exact covariant functor. (ii) The relation (CH ◦ S−1 H )(−) : Gr(A −Mod) −→ COMP (Gr(S −1 H A −Mod)) which that for all graded left A−module M of Gr(A−Mod) we correspond the associate complex sequence (CH ◦ S −1 H )(M) = (S −1 H M)∗ to a graded A−module M and for all graded morphism of graded left A−modules f : M −→ N of degree k we correspond the associate complex chain (CH ◦S−1 H )(f) = (S −1 H f)k∗ to a morphism of graded left A−module f : M −→ N is additively exact covariant functor. (iii) For all n ∈ Z fixed and for all M ∈ Gr(A−Mod) we have: S −1 H ((Hn ◦ C)(M)) ∼= Hn(CH ◦ S−1 H )(M)). 2020 Mathematics Subject Classifications: 13A02, 16W50, 18C40, 18G35, 13D45 Key Words and Phrases: Duo-ring, graded ring, graded module, multiplicatively closed subset of duo-ring generated by regular homogeneous elements, Category, sequence complex,Complex chain and homology functor ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4753 Email addresses: achbih@gmail.com (A. O. Chbih), mohamed-ben.maaouia@ugb.edu.sn (M. B. Maaouia), mamadou.sanghare@ucad.edu.sn ( M. Sanghare) https://www.ejpam.com 1913 © 2023 EJPAM All rights reserved. A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1914 1. Introduction In this article A is supposed unitary graded ring and all left A−module is a unitary. In this article, we study the localization in the category COMP (Gr(A −Mod)) of com- plexes of graded left A−modules so for this gaol we used the localization in the cat- egory Gr(A − Mod), of graded left A−modules, the functor S−1 H : Gr(A − Mod) −→ Gr(S −1 H A − Mod) with SH is a multiplicatively closed subset satisfying the left con- ditions of Ore formed of homogeneous elements of a graded ring A and the functor Hn : COMP (Gr(A−Mod)) −→ Gr(A−Mod). This work finds its roots in particular as regards the functor S−1 : A−Mod −→ S−1A− Mod in [8], and as regards the graduation of graded module of fractions in [2] and [1]. This article is presented as follows: In the second section we present a reminder containing the definitions and background results of graded rings and modules and homological algebra extracted in [10], [11],[4] and [9]. In section 3, the following results have been shown, among others: If A = ⊕ n∈Z An is a graded duo-ring, SH is a part formed of regular homogeneous elements of A, SH is the homogeneous multiplicatively closed subset of A generated by SH , then we have: (i) S −1 H (f) : S −1 H M −→ S −1 H N m s 7−→ S −1 H (f)( m s ) = f(m) s is a graded morphism of degree k ∈ Z of graded left S −1 H A−module; (ii) The relation S −1 H (−) : Gr(A−Mod) −→ Gr(S −1 H A−Mod) which that for any graded left A−module M we made to correspond S −1 H (M) and for all graded morphism of degree k ∈ Z of graded left A−modules f : M −→ N we correspond S −1 H (f) of degree k ∈ Z is a exact additively covariant functor; (iii) Furthermore let P a prime ideal of A and SH is a part formed of regular homogeneous elements of A\P and SPH is the homogeneous multiplicatively closed subset of A generated by SH , then the relation S −1 PH (−) : Gr(A−Mod) −→ S −1 PH A−Mod which that for any graded left A−module M we correspond S −1 PH (M) and for all graded morphism of degree k ∈ Z of graded left A−modules f : M −→ N we correspond S −1 PH (f) of degree k ∈ Z is a exact additively covariant functor. In the last section the following results among others have been shown: if A = ⊕ n∈Z An is a graded duo-ring, SH is a part formed of regular homogeneous elements of A, SH is the homogeneous multiplicatively closed subset of A generated by SH , then we have the A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1915 following results: (i) The following complex sequence : S −1 H (M∗) : · · · −→ S −1 H (M(n+1)) S −1 H (dn+1)−→ S −1 H (M(n)) S −1 H (dn)−→ S −1 H (M(n−1)) −→ · · · with dn : M(n) −→ M(n− 1) x = y + z 7−→ y with (y, z) ∈ Mn ×M(n+ 1); (ii) The following complex chain : S −1 H (M∗) : · · · // S −1 H (fk ∗ ) �� // S −1 H (M(n+ 1)) S −1 H (dn+1)// S −1 H (fk(n+1)) �� S −1 H (M(n)) S −1 H (dn)// S −1 H (fk(n)) �� S −1 H (M(n− 1)) // S −1 H (fk(n−1)) �� · · · S −1 H (N∗) : · · · // S −1 H (N(n+ 1)) S −1 H (d′n+1+k)// S −1 H (N(n)) S −1 H (d′n+k)// S −1 H (N(n− 1)) // · · · (iii) The relation CH(−) : Gr(S −1 H A−Mod) −→ COMP (Gr(S −1 H A−Mod)) which that for all graded left S −1 H A−module S −1 H M of Gr(S −1 H A − Mod) we correspond the associate complex sequence (S −1 H M)∗ to a graded S −1 H A−module S −1 H M and for all graded morphism of graded left S −1 H A−modules S −1 H f : S −1 H M −→ S −1 H N of degree k we correspond the associate complex chain (S −1 H f)k∗ to a morphism of graded left S −1 H A−module S −1 H f : S −1 H M −→ S −1 H N is additively exact covariant functor. (iv) The relation (CH ◦ S−1 H )(−) : Gr(A −Mod) −→ COMP (Gr(S −1 H A −Mod)) which that for all graded left A−module M of Gr(A−Mod) we correspond the associate complex sequence (CH ◦S−1 H )(M) = (S −1 H M)∗ to a graded A−module M and for all graded morphism of graded left A−modules f : M −→ N of degree k we correspond the associate complex chain (CH ◦ S−1 H )(f) = (S −1 H f)k∗ to a morphism of graded left A−module f : M −→ N is additively exact covariant functor. (v) We have the composed functor Hn = Hn ◦ C, Hn : Gr(A − Mod) −→ Gr(A − Mod). With C() : Gr(A − Mod) −→ COMP (Gr(A − Mod)) and Hn : COMP (Gr(A−Mod)) −→ Gr(A−Mod). (vi) For all n ∈ Z fixed and for all M ∈ Gr(A−Mod) we have: S −1 H ((Hn ◦ C)(M)) ∼= Hn(CH ◦ S−1 H )(M)). A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1916 2. Reminder and preliminary results Definition 1. Let A be a ring, then we say that A is a graded ring if there exists a suite (An)n∈Z of additive subgroups of A such that (i) A = ⊕ n∈Z An; (ii) An ·Am ⊂ An+m, ∀ n, m ∈ Z. Definition 2. Let A be a graded ring, and x be a non-zero element of A. then we say that x is homogeneous of degree n, if there exist n such that x ∈ An and we note deg(x) = n. In all that follows, A and M are supposed unitary. Definition 3. Let A = ⊕ n∈Z An be a graded ring and M be a left A−module, we say that M is a graded left A−module if there exists a suite (Mn)n∈Z of sub-groups of M such that: (i) M = ⊕ n∈Z Mn; (ii) An ·Md ⊂ Mn+d, ∀ n, d ∈ Z. Definition 4. Let A = ⊕ n∈Z An be a graded ring, M = ⊕ n∈Z Mn be a graded left A−module and N is a sub-module of M , then we say that N is a graded sub-module of M , if ∀x ∈ N such that x = ∑ n∈Z xn, then xn ∈ N , ∀n ∈ Z. Proposition 1. Let A = ⊕ n∈Z An be a graded ring and M = ⊕ n∈Z Mn is graded left A−module, then for all n ∈ Z fixed, we have M(n) = ⊕ k≥n Mk is a graded sub-module of M and we have the descendant sequence: · · ·M(n+ 2) ⊂ M(n+ 1) ⊂ M(n) ⊂ · · ·. Proof. For all n ∈ Z fixed, M(n) = ⊕ k≥n Mk is a sub-group of M and As ·M(n)k = As ·Mn+k ⊂ Mn+k+s = Mn+(k+s) = M(n)k+s. In the other hand, it suffices to remark that M(n) = ⊕ k≥n Mk = Mn ⊕ M(n+ 1). Hence M(n+ 1) ⊂ M(n). Thus · · ·M(n+ 2) ⊂ M(n+ 1) ⊂ M(n) ⊂ · · ·. A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1917 Definition 5. Let A = ⊕ n∈ZAn be a graded ring, M = ⊕ n∈Z Mn and N = ⊕ n∈Z Nn two graded left A−modules and f : M −→ N is a morphism of left A−modules, then we say that f is a graded morphism of degree k ∈ Z if for any m ∈ Ms then f(m) ∈ Ns+k. Theorem 1. Let A be a graded ring, then the following information: (i) The class of objects are the graded left A−modules; (ii) The class of morphisms are the graded morphisms of degree k ∈ Z. constitute a category called the category of graded left A−module and it is denoted by Gr(A−Mod). Proof. See [3] Definition 6. A complex sequence (C, d) : . . . → Cn+1 dn+1→ Cn dn→ Cn−1 dn−1→ . . . is a sequence of morphisms of A− modules satisfying dn ◦ dn+1 = 0, for all n ∈ Z. Definition 7. A complex chain f : (C, d) → (C ′, d′) is a sequence of homomorphisms (fn : Cn −→ C ′ n)n∈Z of A− modules making the following diagram commute: (C, d) : · · · // f �� Cn+1 dn+1 // fn+1 �� Cn dn // fn �� Cn−1 // fn−1 �� · · · (C ′, d′) : · · · // C ′ n+1 d′n+1 // C ′ n d′n // C ′ n−1 // · · · i.e d′n+1 ◦ fn+1 = fn ◦ dn+1, for all n ∈ Z. Proposition 2. We called the category of complexes of A−modules and we denote COMP , the category whose: (i) The objects are the sequences complex; (ii) The morphisms are the complex chains. Proof. See [3] Proposition 3. We called functor homology Hn the functor Hn : COMP −→ Ab defined by: (i) For all objet (C, d) of COMP , Hn((C, d)) = ker dn/Imdn+1 (ii) For all chain f : (C, d) → (C ′, d′) of COMP Hn(f) : Hn((C, d)) −→ Hn((C ′, d′)) zn 7−→ fn(zn) A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1918 Proof. See [3] Theorem 2. Let (0) −→ ((M,d)) f−→ ((N, d ′ )) g−→ ((L, d ′′ )) −→ (0) be a short exact complex sequence, then for all n ∈ Z there exist a morphism of left A−module δn : Hn((L, d ′ ) −→ Hn−1((M,d)) called connecting morphism such that the following long exact sequence is exact · · · −→ Hn((M,d)) Hn(f)−→ Hn((N, d ′ )) Hn(g)−→ Hn((L, d ′′ )) δn−→ Hn−1(M,d) Hn−1(f)−→ Hn−1(N, d ′ ) −→ · · · Proof. See [3] Definition 8. Let A be a ring, we say that A is duo ring if every left ideal of A is two-sided, and any right ideal of A is two-sided. Proposition 4. Let A be a ring, then A is a duo-ring if, and only if, ∀a ∈ A, aA = Aa. Proof. See [6]. Proposition 5. Let A be a duo-ring then, the set of all regular elements of A is a multi- plicatively closed subset of A verifies the conditions Ore. Proof. See [6]. Proposition 6. Let A be a duo-ring and S be a nonempty subset formed of regular el- ements of A, then there exists a multiplicatively closed subset of A satisfying the left conditions of Ore containing S. Proof. It suffices to note that the set of all regular elements of A is a multiplicatively closed subset satisfying the conditions Ore and containing S. Definition 9. Let A be a duo-ring and S be a nonempty subset formed of regular elements of A, then the smaller multiplicatively closed subset of A satisfying the conditions of Ore containing S is called the multiplicatively closed subset of A satisfying the left conditions of Ore generated by S and denoted by S. Proposition 7. Let A = ⊕ n∈Z An be a graded duo-ring and SH be a nonempty subset formed of regular homogeneous elements of A, then there exists a homogeneous multiplicatively closed subset of A satisfying the left conditions of Ore containing S, and denoted by SH . A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1919 Proof. Put SH the the smaller multiplicatively closed subset of A satisfying the condi- tions Ore containing S, SH exist because the set of regular elements of A is a multiplica- tively closed subset of A satisfying the conditions Ore containing S. Then it is enough to proof that SH is homogeneous. We have the elements of SH are of the form ∏ i si, si ∈ S which ∏ i si is homogeneous. Corollary 1. Let A = ⊕ n∈Z An be a graded duo-ring then the set of all regular homogeneous of A is multiplicatively closed subset satisfying the left conditions of Ore. Proof. Put S the set of all regular homogeneous of A then SH = S. Proposition 8. Let A = ⊕ n∈Z An be a graded duo-ring, P is a prime ideal of A and SPH is the set formed of homogeneous regular elements of A\P , then SPH ⊂ (A\P ). Proof. The set of regular elements of A\P is a multiplicatively closed subset satisfying the conditions of Ore, (see [5] and [7]) and containing SPH , then SPH ⊂ (A\P ). Corollary 2. Let A = ⊕ n∈Z An be a graded duo-ring and P is a prime ideal of A, then the set of regular homogeneous of A\P is a multiplicatively closed subset satisfying the conditions of Ore. Proof. Put S the set of all regular homogeneous of A\P , then SPH = S. 3. Functor Graduation S −1 H and Functorization of graded modules Theorem 3. Let A = ⊕ n∈Z An be a graded ring, M = ⊕ n∈Z Mn and N = ⊕ n∈Z Nn be a two graded left A−modules and S is a multiplicatively closed subset satisfying the left conditions of Ore formed of homogeneous elements of a graded ring A. Let f : M −→ N be graded morphism of degree k ∈ Z of graded left A−modules, then: S−1(f) : S−1M −→ S−1N m s 7−→ S−1(f)( m s ) = f(m) s is a graded morphism of degree k ∈ Z of graded left S−1A−module. Proof. Since [1], S−1(f) is a morphism of left S−1A−module. Show that S−1(f) is graded morphism of degree k ∈ Z, let m ∈ M homogeneous such that m s ∈ S−1M is of degree d, then d = deg( m s ) = deg(m)− deg(s), on the other hand deg(S−1(f)( m s )) = deg( f(m) s ) A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1920 = deg(f(m))− deg(s) = deg(f(m))− deg(s) = (deg(m) + k)− deg(s) = d+ k because f is graded of degree k ∈ Z, thus S−1(f) has degree k, hence S−1(f) is graded morphism of degree k of graded left S−1A−module. Proposition 9. Let A = ⊕ n∈Z An be a graded duo-ring, M = ⊕ n∈Z Mn and N = ⊕ n∈Z Nn be a two graded left A−modules and SH be a part formed of regulars homogeneous elements of A. Let f : M −→ N be graded morphism of degree k ∈ Z of graded left A−modules, then: S −1 H (f) : S −1 H M −→ S −1 H N m s 7−→ S −1 H (f)( m s ) = f(m) s is a graded morphism of degree k ∈ Z of graded left S −1 H A−module. Proof. Since the proposition 6, SH is a multiplicatively closed subset satisfying the left conditions of Ore formed of homogeneous elements of A and from 9, S −1 H (f) is graded morphism of degree k of graded left S −1 H A−module. Proposition 10. Let A = ⊕ n∈Z An be a graded ring, M = ⊕ n∈Z Mn, N = ⊕ n∈Z Nn and L =⊕ n∈Z Ln be three graded left A−modules and S be a multiplicatively closed subset satisfying the left conditions of Ore formed of homogeneous elements of A, then for every short exact sequences of a graded morphisms of degree k ∈ Z of a graded left A−module 0 −→ M φ−→ N ϕ−→ L −→ 0, we have the following short exact sequences of a graded morphisms of degree k ∈ Z of a graded left S−1A−modules: 0 −→ S−1M S−1(φ)−→ S−1N S−1(ϕ)−→ S−1L −→ 0. Proof. Since the theorem 3.4 of [8], if 0 −→ M φ−→ N ϕ−→ L −→ 0 is a short exact sequences of a morphisms of degree k ∈ Z of a left A−modules, then 0 −→ S−1M S−1(φ)−→ S−1N S−1(ϕ)−→ S−1L −→ 0 A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1921 is a short exact sequences of a morphisms of degree k ∈ Z of a left S−1A−modules, and as S is a set formed of no null homogeneous elements of A and S−1(−) preserve degree, then we have 0 −→ S−1M S−1(φ)−→ S−1N S−1(ϕ)−→ S−1L −→ 0 is a short exact sequences of a graded morphisms of degree k ∈ Z of a graded left S−1A−modules. Corollary 3. Let A = ⊕ n∈Z An be a graded duo-ring, M = ⊕ n∈Z Mn, N = ⊕ n∈Z Nn and L = ⊕ n∈Z Ln be three graded left A−modules and SH be part formed of regulars homogeneous elements of A, then for every short exact sequences of a graded morphisms of degree k ∈ Z of a graded left A−module 0 −→ M φ−→ N ϕ−→ L −→ 0 we have the following short exact sequences of a graded morphisms of degree k ∈ Z of a graded left S −1 H A−modules: 0 −→ S −1 H M S −1 H (φ)−→ S −1 H N S −1 H (ϕ)−→ S −1 H L −→ 0 Proof. Since the proposition 6, SH is a multiplicatively closed subset satisfying the left conditions of Ore formed of homogeneous elements of A. Corollary 4. Let A = ⊕ n∈Z An be a graded duo-ring, M = ⊕ n∈Z Mn, N = ⊕ n∈Z Nn and L = ⊕ n∈Z Ln be three graded left A−modules and SH the set of all regular homogeneous of A, then for every short exact sequences of a graded morphisms of degree k ∈ Z of a graded left A−module 0 −→ M φ−→ N ϕ−→ L −→ 0 we have the following short exact sequences of a graded morphisms of degree k ∈ Z of a graded left S−1 H A−modules: 0 −→ S−1 H M S−1 H (φ) −→ S−1 H N S−1 H (ϕ) −→ S−1 H L −→ 0 Proof. Similarly to the proof of the corollary precedent 3 with SH = SH A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1922 Theorem 4. Let A = ⊕ n∈Z An be a graded ring and S be a multiplicatively closed subset satisfying the left conditions of Ore formed of homogeneous elements of A, then the relation S−1(−) : Gr(A − Mod) −→ Gr(S −1A − Mod) which that for any graded left A−module M we correspond S−1(M) and for all graded morphism of degree k ∈ Z of graded left A−modules f : M −→ N we correspond S−1(f) of degree k ∈ Z is a exact additively covariant functor. Proof. Let f : M −→ N be a graded morphism of degree k ∈ Z of a graded left A−modules, then S−1(f) : S−1M −→ S−1N m s 7−→ f(m) s is a morphism of degree k ∈ Z of left S−1A−modules. So (i) Let M ∈ Gr(A −Mod), then S−1M is a graded left S−1A−module, thus S−1M ∈ Gr(S −1A−Mod). (ii) Let f : M −→ N be a graded morphism of the graded left A−modules, then S−1(g ◦ f) : S−1M −→ S−1N S−1(g ◦ f)(m s ) = (g ◦ f)(m) s = g(f(m)) s = g ( f(m) s ) = S−1(g)( f(m) s ) = S−1(g) ◦ S−1(f)( m s ) Thus ∀ m s ∈ S−1M , S−1(g ◦ f) = S−1(g) ◦ S−1(f). S−1(1M ) : S−1M −→ S−1M m s 7−→ 1M (m) s = m s = 1S−1M ( m s ) so ∀m s ∈ S−1M we have S−1(1M ) = 1S−1M , so S−1(−) : Gr(A−Mod) −→ Gr(S −1A−Mod) is a covariant functor. A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1923 Furthermore deg( m s ) = deg(m)−deg(s) or f is graded of degree k ∈ Z, then deg(m)+k = deg(f(m)) so deg(S−1(f)( m s )) = deg( f(m) s ) = deg(f(m))− deg(s) = (deg(m) + k)− deg(s) = deg( m s ) + k. Thus S−1(−) is additively exact covariant functor. Or S−1(−) is exact then additively exact covariant functor. Proposition 11. Let A = ⊕ n∈Z An be a graded duo-ring and SH be the part formed of all regulars homogeneous elements of A, then the relation S −1 H (−) : Gr(A − Mod) −→ Gr(S −1 H A − Mod) which that for any graded left A−module M we correspond S −1 H (M) and for all graded morphism of degree k ∈ Z of graded left A−modules f : M −→ N we correspond S −1 H (f) of degree k ∈ Z is a exact additively covariant functor. Proof. Similarly to the proof of the theorem precedent 4. Corollary 5. Let A = ⊕ n∈Z An be a graded duo-ring and SH the set of all regular homoge- neous of A, then the relation S−1 H (−) : Gr(A−Mod) −→ S−1 H A−Mod which that for any graded left A−module M we correspond S−1 H (M) and for all graded morphism of degree k ∈ Z of graded left A−modules f : M −→ N we correspond S−1 H (f) is additively exact covariant functor. Proof. SH is the set of regular homogeneous of A then SH is homogeneous multiplica- tively closed subset so SH = SH then according to proposition precedent 11. Proposition 12. Let A = ⊕ n∈Z An be a graded duo-ring , P be a prime ideal of A and SPH be a set formed of homogeneous regular elements of A\P , then the relation S −1 PH (−) : Gr(A −Mod) −→ Gr(S −1 PH A −Mod) which that for any graded left A−module M we correspond S −1 PH (M) and for all graded morphism of degree k ∈ Z of graded left A−modules f : M −→ N we correspond S −1 PH (f) of degree k ∈ Z is additively exact covariant functor. Proof. Since the proposition 8 SPH homogeneous multiplicatively closed subset, so S −1 PH (−) : Gr(A − Mod) −→ Gr(S −1 PH A − Mod) is a covariant functor indeed. Let M,N two graded left A−modules and f : M −→ N is a graded morphism of degree k ∈ Z, then S −1 PH (−)(f) : S −1 PH M −→ S −1 PH N A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1924 m s 7−→ f(m) s is a graded morphism of degree k ∈ Z of S −1 PH A−modules and for any graded left A−module M , S−1 PH (−)(M) = S −1 PH M is a graded left S −1 PH A−module, so S−1 PH (−) is a functor covariant for the category Gr(A−Mod) to the category Gr(S −1 PH A−Mod). Furthermore S −1 PH (−) is of degree k ∈ Z, indeed let (s,m) ∈ S ×M such that deg( m s ) = deg(m)− deg(s) = d1 so S −1 PH (−)(f)( m s ) = f(m) s , and deg(S −1 PH (f)( m s )) = deg( f(m) s ) = deg(f(m))− deg(s) = (deg(m) + k)− deg(s) = deg( m s ) + k = d1 + k. Thus S −1 PH (−) is additively exact covariant functor, since S −1 PH (−) preserve the exactness. Definition 10. Let A = ⊕ n∈Z An is a graded duo-ring, M = ⊕ n∈Z Mn be a left graded A−module, P is a prime ideal of A and SH be the set of homogeneous regular elements of A\P then: (i) S−1 PH A is called homogeneous localized to A in P . and denoted by APH ; (ii) S−1 PH M is called homogeneous localized to M in P . and denoted by MPH . Corollary 6. Let A = ⊕ n∈Z An be a graded duo-ring , P be a prime ideal of A and SPH be the set of all homogeneous regular elements of A\P , then the relation S−1 PH (−) : Gr(A−Mod) −→ APH −Mod which that for any graded left A−module M we correspond MPH and for all graded morphism of degree k ∈ Z of graded left A−modules f : M −→ N we correspond S−1 PH (f) of degree k ∈ Z is additively exact covariant functor. Proof. It is enough to note that SPH = SPH since the corollary 2. 4. Localization of complex in COMP (Gr(A−Mod)) over a duo-ring Proposition 13. Let A = ⊕ n∈Z An be a graded ring, M = ⊕ n∈Z Mn and N = ⊕ n∈Z Nn are two graded left A−module f : M −→ N is a graded morphism of degree k ∈ Z of a graded left A−modules, then for all n ∈ Z fk(n) : M(n) −→ N(n) A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1925 m 7−→ fk(n)(m) = f(m) is graded morphism of degree k ∈ Z of graded left A−modules. Proof. We have f : M −→ N is graded morphism of degree k ∈ Z of graded left A−modules, and M(n) is a sub-module of graded left A−module M then let m ∈ M(n), so m = ∑ i∈Z mi+n =⇒ fk(n)(m) = f(m) = f( ∑ i∈Z mi+n) = ∑ i∈Z f(mi+n) or f(mi+n) ∈ Ni+n+k = (N(n))i+k thus f is graded morphism of degree k ∈ Z of a graded left A−modules. Corollary 7. Let A = ⊕ n∈Z An be a graded ring, M = ⊕ n∈Z Mn and N = ⊕ n∈Z Nn are two graded left A−module f : M −→ N is a graded morphism of degree k ∈ Z of a graded left A−modules, then f : M −→ N(k) is graded morphism of graded left A−modules. Proof. We have f : M −→ N is graded morphism of degree k ∈ Z of graded left A−modules, and N(k) is a sub-module of graded left A−module N then let m ∈ M , so m = ∑ i∈Z mi =⇒ f(m) = f( ∑ i∈Z mi) = ∑ i∈Z f(mi). Or f(mi) ∈ Ni+k = (N(k))i thus f is graded morphism of a graded left A−modules. Theorem 5. Let A be a graded ring and Gr(A − Mod) the category of a graded left A−modules, then for all n ∈ Z the relation (−)(n) : Gr(A − Mod) −→ Gr(A − Mod) which that for any M ∈ Gr(A − Mod) we made to correspond M(n) and for all graded morphism of degree k ∈ Z of a graded left A−modules f : M −→ N we correspond fk(n) is a additively exact covariant functor. Proof. Let f : M −→ N be a graded morphism of degree k ∈ Z of a graded left A−modules, we denote by (−)(n)(f) = fk(n) the morphism of left A−modules of M(n) to N(n) thus (−)(n)(M) = M(n) is in A −Mod, furthermore M(n) and N(n) are both graded left A−module then M(n), N(n) ∈ Gr(A−Mod). Thus (−)(n) : M(n) −→ N(n) has a sense. (i) Let f : M −→ N is graded morphism of degree k ∈ Z of graded left A−modules, then (−)(n)(f) : M(n) −→ N(n) fk(n) : M(n) −→ N(n) m 7−→ fk(n)(m) = f(m) is a graded morphism of a graded left A−modules. Furthermore (−)(n)(g ◦ f)(m) = (g ◦ f)k(n)(m) = (g ◦ f)(m) = g[f(m)] = g[fk(n)(m)] A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1926 = gk(n)[fk(n)(m)] = (gk(n) ◦ fk(n))(m) = (−)(n)(g) ◦ (−)(n)(f)(m). So (−)(n)(g ◦ f)(m) = (−)(n)(g) ◦ (−)(n)(f)(m) ∀ m ∈ M(n). Thus (−)(n)(g ◦ f) = (−)(n)(g) ◦ (−)(n)(f). On the other hand (−)(n)(1M(n)) : M(n) −→ M(n) 1M (n) : M(n) −→ M(n) m 7→ 1M(n)(n)(m) = 1M(n)(m) = m = 1(−)(n)(M)(m), so (−)(n)(1M(n)) = 1(−)(n)(M(n)), ∀m ∈ M(n), so (−)(n) is a functor of Gr(A −Mod) to Gr(A−Mod). Thus (−)(n) : Gr(A−Mod) −→ Gr(A−Mod) is a functor covariant. Let m ∈ M be homogeneous of degree d, then (−)(n)(f)(m) = fk(n)(m) = f(m) is of degree k + n thus (−)(n) is a additively exact covariant functor of degree k ∈ Z. Proposition 14. Let A = ⊕ n∈Z An be a graded ring and M = ⊕ n∈Z Mn be a graded left A−module, then we have the following associate complex sequence M∗ of a graded A−module M = ⊕ n∈Z Mn : M∗ : · · · → M(n+ 1) dn+1→ M(n) dn→ M(n− 1) → · · · with M(n) = ⊕ k∈Z Mn+k and dn : M(n) −→ M(n− 1) x = y + z 7−→ y with (y, z) ∈ Mn ×M(n+ 1). Proof. We have M(n) = ⊕ k∈Z Mn+k = ⊕ k≥n Mk = Mn ⊕ Mn+1 and M(n− 1) = Mn−1 ⊕ M(n) = Mn−1 ⊕ Mn ⊕ M(n+ 1). Let x ∈ M(n), then it is exist a unique (y, z) ∈ Mn ×M(n+ 1) such that x = y + z. Put dn : M(n) −→ M(n− 1) x = y + z 7−→ y, A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1927 so Im(dn) = Mn; On the other hand dn−1 : M(n− 1) −→ M(n− 2) w = u+ v 7−→ v with (u, v) ∈ Mn−1 ×M(n), so ker(dn−1) = M(n) so Im(dn) ⊂ ker(dn−1), so dn−1 ◦ dn = 0 ,thus M∗ : · · · → M(n+ 1) dn+1→ M(n) dn→ M(n− 1) → · · · is a complex sequence. Proposition 15. Let A = ⊕ n∈Z An be a graded ring, M = ⊕ n∈Z Mn, N = ⊕ n∈Z Nn are two graded left A−modules and f : M = ⊕ n∈Z Mn −→ N = ⊕ n∈Z Nn is a graded morphism of degree k ∈ Z of a graded A−modules, then we have the following associate complex fk ∗ of graded morphism f : M = ⊕ n∈Z Mn −→ N = ⊕ n∈Z Nn of a graded A−modules : M∗ : · · · // fk ∗ �� M(n+ 1) dn+1 // fk(n+1) �� M(n) dn // fk(n) �� M(n− 1) // fk(n−1) �� . . . N∗ : · · · // N(n+ 1) d′n+1+k // N(n) d′n+k// N(n− 1) // · · · . Proof. Prove that for all n ∈ Z, fk(n) ◦ dn+1 = d ′ n+1+k ◦ fk(n+ 1). Let x ∈ M(n+1), then there exist the unique couple (y, z) ∈ Mn+1 ×M(n+2) such that x = y + z, so (fk(n) ◦ dn+1)(x) = fk(n)[dn+1(x)] = f [dn+1(x)] = f [y] = f(y), and (d ′ n+1+k ◦ fk(n + 1))(x) = d ′ n+1+k[f k(n + 1)(x)] = d ′ n+1+k[f(x)] = d ′ n+1+k[f(y + z)] = d ′ n+1+k[f(y) + f(z)] = f(y), because f(y) ∈ Nn+1+k and f(z) ∈ N(n+ 2 + k), =⇒ (fk(n) ◦ dn+1)(x) = (d ′ n+1+k ◦ fk(n+ 1))(x), ∀ x ∈ M(n+ 1), so fk(n) ◦ dn+1 = d ′ n+1+k ◦ fk(n+ 1), thus fk ∗ is a complex chain. A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1928 Theorem 6. Let A = ⊕ n∈Z An be a graded ring, then the following information: (i) The objets are the associate complex sequences of a graded left A−modules; (ii) The morphisms are the associate complex chains of a graded morphism of a graded left A−modules. formed a category called the category of associate complex of a graded left A−modules and denoted by COMP (Gr(A−Mod)). Proof. Let M∗ and N∗ two objets of COMP (Gr(A−Mod)),then: (i) HomCOMP (Gr(A−Mod))(M∗, N∗) = { the set of associate complex chains fk ∗ , of M∗ to N∗}; (ii) The morphisms are the associate complex chains of a graded morphism of degrees k of a graded left A−modules. then we have : (a) ∀ fk ∗ ∈ HomCOMP (Gr(A−Mod))(M∗, N∗); ∀ gr∗ ∈ HomCOMP (Gr(A−Mod))(N∗, P∗); ∀ hs∗ ∈ HomCOMP (Gr(A−Mod))(P∗, Q∗) on a : M∗ : · · · // fk ∗ �� //M(n+ 1) fk(n+1) �� dn+1 //M(n) fk(n) �� dn // . . . N∗ : · · · gr∗ �� // N(n+ 1) gr(n+1) �� d ′ n+1+k // N(n) gr(n) �� d ′ n+k // . . . P∗ : · · · hs ∗ �� // P (n+ 1) hs(n+1) �� d ′′ n+1+k+r // P (n) hs(n) �� d ′′ n+k+r // . . . Q∗ : · · · // Q(n+ 1) d ′′′ n+1+k+r+s// Q(n) d ′′′ n+k+r+s// . . . So (hs∗ ◦ gr∗) ◦ fk ∗ = hs∗ ◦ (gr∗ ◦ fk ∗ ); (b) Let M∗ the object of COMP (Gr(A−Mod)), we have: 1M∗ : M∗ −→ M∗ M∗ : · · · // 1M∗ �� //M(n+ 1) 1(n+1) �� dn+1 //M(n) 1(n) �� dn // . . . M∗ : · · · //M(n+ 1) dn+1 //M(n) dn // . . . 1M∗ verified f∗ ◦ 1M∗ = f∗ ∀ f∗ ∈ HomCOMP (Gr(A−Mod))(M∗, N∗). Furthermore 1M∗ ◦ g∗ = g∗ ∀ g∗ ∈ HomCOMP (Gr(A−Mod))(N∗,M∗). A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1929 Thus COMP (Gr(A−Mod)) is a category. Proposition 16. Let A = ⊕ n∈Z An be a graded ring, M = ⊕ n∈Z Mn and N = ⊕ n∈Z Nn are two graded left A−modules, f : M −→ N is a graded morphism of degree k and S be a multiplicatively closed subset satisfying the left conditions of Ore formed of homogeneous elements of A, then we have: (i) The following complex sequence: S−1(M∗) : · · · −→ S−1(M(n+ 1)) S−1(dn+1)−→ S−1(M(n)) S−1(dn)−→ S−1(M(n− 1)) −→ · · · (ii) The following complex chain: S−1(M∗) : · · · // S−1(fk ∗ ) �� // S−1(M(n+ 1)) S−1(dn+1)// S−1(fk(n+1)) �� S−1(M(n)) S−1(dn)// S−1(fk(n)) �� S−1(M(n− 1)) // S−1(fk(n−1)) �� · · · S−1(N∗) : · · · // S−1(N(n+ 1)) S−1(d′n+1+k)// S−1(N(n)) S−1(d′n+k)// S−1(N(n− 1)) // · · · Proof. As for all n ∈ Z, M∗ and N∗ are two complex sequences of graded left A−module, then S−1(M∗) and S−1(N∗) are two complex sequences of a graded left S−1A−module. Prove that for all n ∈ Z, S−1(fk(n)) ◦ S−1(dn+1) = S−1(d ′ n+1+k) ◦ S−1(fk(n+ 1)). Let x s ∈ S−1(M(n+1)), then it is exist a unique couple ( y t , z r ) ∈ S−1Mn+1×S−1M(n+2) such that x s = y t + z r , so (S−1fk(n) ◦ S−1dn+1)( x s ) = S−1fk(n)[S−1dn+1( x s )] = S−1fk(n)[ y t ] = S−1f [ y t ] = f(y) t , and (S−1d ′ n+1+k ◦ S−1fk(n+ 1))( x s ) = S−1d ′ n+1+k[S −1fk(n+ 1)( y t + z r )] = S−1d ′ n+1+k[S −1f( y t + z r )] = S−1d ′ n+1+k[S −1f( y t ) + S−1f( z r )] = S−1f( y t ) = f(y) t A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1930 because S−1f(yt ) ∈ S−1Nn+1+k and S−1f( zr ) ∈ S−1N(n+ 2 + k) =⇒ (S−1d ′ n+1+k◦S−1fk(n+1))( x s ) = (S−1fk(n)◦S−1dn+1)( x s ) ∀ x s ∈ S−1M(n+1) so (S−1d ′ n+1+k ◦ S−1fk(n+ 1))) = (S−1fk(n) ◦ S−1dn+1) thus S−1(fk ∗ ) is a complex chain. Corollary 8. Let A = ⊕ n∈Z An be a graded duo-ring, M = ⊕ n∈Z Mn and N = ⊕ n∈Z Nn are two graded left A−modules, f : M −→ N is a graded morphism of degree k and SH be a part formed of regulars homogeneous elements of A, then we have: (i) The following complex sequence: S −1 H (M∗) : · · · −→ S −1 H (M(n+1)) S −1 H (dn+1)−→ S −1 H (M(n)) S −1 H (dn)−→ S −1 H (M(n−1)) −→ · · · (ii) The following complex chain: S −1 H (M∗) : · · · // S −1 H (fk ∗ ) �� // S −1 H (M(n+ 1)) S −1 H (dn+1)// S −1 H (fk(n+1)) �� S −1 H (M(n)) S −1 H (dn)// S −1 H (fk(n)) �� S −1 H (M(n− 1)) // S −1 H (fk(n−1)) �� · · · S −1 H (N∗) : · · · // S −1 H (N(n+ 1)) S −1 H (d′n+1+k)// S −1 H (N(n)) S −1 H (d′n+k)// S −1 H (N(n− 1)) // · · · Proof. Since the proposition 6 SH is multiplicatively closed subset satisfying the left conditions of Ore formed of homogeneous elements of A, and the rest is similarly to the proof of the proposition 16. Proposition 17. Let A = ⊕ n∈Z An be a graded ring, M = ⊕ n∈Z Mn and N = ⊕ n∈Z Nn two graded left A−modules, f : M −→ N is graded morphism of degree k ∈ Z and S be a multiplicatively closed subset satisfying the left conditions of Ore formed of homogeneous elements of A, then we have: (i) The following complex sequence: B∗ : · · · −→ S−1A⊗ (M(n+ 1)) S−1A⊗(dn+1)−→ S−1A⊗ (M(n)) S−1A⊗(dn)−→ S−1A⊗ (M(n− 1)) −→ · · · (ii) The following complex chain: B∗ : · · · // S−1A⊗fk ∗ �� // S−1A⊗ (M(n+ 1)) S−1A⊗(dn+1)// S−1A⊗(fk(n+1)) �� S−1A⊗ (M(n)) S−1A⊗(dn)// S−1A⊗(fk(n)) �� S−1A⊗ (M(n− 1)) // S−1A⊗(fk(n−1)) �� · · · D∗ : · · · // S−1A⊗ (N(n+ 1)) S−1A⊗(d′ n+1+k)// S−1A⊗N(n)) S−1A⊗(d′ n+k)// S−1A⊗ (N(n− 1)) // · · · A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1931 With B∗ = S−1A⊗ (M∗) and D∗ = S−1A ⊗ A(N∗). Proof. We have the functor S−1() and the functor S−1A ⊗ A() are isomorphs. On the other hand it suffices to prove that the following diagram is commutative B∗ : · · · // γ �� // S−1A ⊗ M(n+ 1) γn+1 �� S−1A ⊗ dn+1// S−1A ⊗ M(n) γn �� S−1A ⊗ dn // . . . S−1(M∗) : · · · S−1fk ∗ �� // S−1M(n+ 1) S−1fk(n+1) �� S−1dn+1 // S−1M(n) S−1fk(n) �� S−1dn // . . . (S−1(N∗) : · · · λ �� // S−1N(n+ 1) λn+1 �� S−1d′n+1+k // S−1N(n) λn �� S−1d′n+k // . . . D∗ : · · · // S−1A ⊗ N(n+ 1) S−1A ⊗ d′n+1+k// S−1A ⊗ N(n) S−1A ⊗ d′n+k// . . . i.e. prove that for all n ∈ Z we have λn ◦ S−1fk(n) ◦ γn ◦ S−1A ⊗ dn+1 = S−1A ⊗ d′n+1+k ◦ λn+1 ◦ S−1fk(n+ 1) ◦ γn+1 or for all n ∈ Z, we have λn ◦ S−1fk(n) ◦ γn = 1S−1A ⊗ fk(n). Let 1 s ⊗m ∈ S−1A ⊗ A M(n+1), then it is exist an unique couple (x, y) ∈ Mn+1×M(n+2) such that m = x+ y so λn◦S−1fk(n)◦γn◦S−1A ⊗ dn+1[ 1 s ⊗m] = λn◦S−1fk(n)◦γn◦S−1A ⊗ dn+1[ 1 s ⊗(x+y)] = λn ◦ S−1fk(n) ◦ γn[ 1 s ⊗ x] = 1S−1A ⊗ fk(n)[ 1 s ⊗ x] = 1 s ⊗ f(x). On the other hand we have S−1A ⊗ d′n+1+k ◦ λn+1 ◦ S−1fk(n+ 1) ◦ γn+1[ 1 s ⊗m] = S−1A ⊗ d′n+1+k ◦ λn+1 ◦ S−1fk(n) ◦ γn+1[ 1 s ⊗ (x+ y)] = S−1A ⊗ d′n+1+k[ 1 s ⊗ f(x+ y)] = 1 s ⊗ f(x) thus S−1A⊗ fk ∗ is a complex the chain. Corollary 9. Let A = ⊕ n∈Z An be a graded duo-ring, M = ⊕ n∈Z Mn and N = ⊕ n∈Z Nn two graded left A−modules, f : M −→ N is graded morphism of degree k ∈ Z and SH be a part formed of regulars homogeneous elements of A, then we have : A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1932 (i) The following complex sequence: B∗ : · · · −→ S −1 H A⊗ (M(n+ 1)) S −1 H A⊗(dn+1)−→ S −1 H A⊗ (M(n)) S −1 H A⊗(dn)−→ S −1 H A⊗ (M(n− 1)) −→ · · · (ii) The following complex chain: B∗ : · · · // S −1 H A⊗fk ∗ �� // S −1 H A⊗ (M(n+ 1)) S −1 H A⊗(dn+1)// S −1 H A⊗(fk(n+1)) �� S −1 H A⊗ (M(n)) S −1 H A⊗(dn)// S −1 H A⊗(fk(n)) �� S −1 H A⊗ (M(n− 1)) // S −1 H A⊗(fk(n−1)) �� · · · D∗ : · · · // S −1 H A⊗ (N(n+ 1)) S −1 H A⊗(d′ n+1+k)// S −1 H A⊗N(n)) S −1 H A⊗(d′ n+k)// S −1 H A⊗ (N(n− 1)) // · · · With B∗ = S −1 H A⊗ (M∗) and D∗ = S −1 H A ⊗ A(N∗). Proof. it is sufficient to note that SH is a multiplicatively closed subset satisfying the left conditions of Ore formed of homogeneous elements of A. Corollary 10. Let A = ⊕ n∈Z An be a graded duo-ring, M = ⊕ n∈Z Mn and N = ⊕ n∈Z Nn two graded left A−modules, f : M −→ N is graded morphism of degree k and SH be the set of all regulars homogeneous elements of A, then we have: (i) The following complex sequence: B∗ : · · · −→ S−1A⊗(M(n+1)) S−1A⊗(dn+1)−→ S−1A⊗(M(n)) S−1A⊗(dn)−→ S−1A⊗(M(n−1)) −→ · · · (ii) The following complex chain: B∗ : · · · // S−1 H A⊗fk ∗ �� // S−1 H A⊗ (M(n+ 1)) S−1 H A⊗(dn+1)// S−1 H A⊗(fk(n+1)) �� S−1 H A⊗ (M(n)) S−1 H A⊗(dn)// S−1 H A⊗(fk(n)) �� S−1 H A⊗ (M(n− 1)) // S−1 H A⊗(fk(n−1)) �� · · · D∗ : · · · // S−1 H A⊗ (N(n+ 1)) S−1 H A⊗(d′n+1+k)// S−1 H A⊗N(n)) S−1 H A⊗(d′n+k)// S−1 H A⊗ (N(n− 1)) // · · · With B∗ = S−1 H A⊗ (M∗) and D∗ = S−1 H A ⊗ A(N∗). Proof. it is sufficient to note that SH = SH . Theorem 7. Let A = ⊕ n∈Z An be a graded ring and Gr(A−Mod) the category of graded left A−modules, then the relation C(−) : Gr(A−Mod) −→ COMP (Gr(A−Mod)) which that for all graded left A−module M = ⊕ n∈Z Mn of Gr(A − Mod) we correspond the associate A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1933 complex sequence M∗ to a graded A−module M = ⊕ n∈Z Mn and for all graded morphism of graded left A−modules f : M = ⊕ n∈Z Mn −→ N = ⊕ n∈Z Nn of degree k we correspond the associate complex chain fk ∗ to a morphism of graded left A−module f : M = ⊕ n∈Z Mn −→ N = ⊕ n∈Z Nn is exact additively covariant functor. Proof. Let M , N two graded left A−modules and f : M −→ N graded morphism of graded A−modules, we note that C(M) = M∗ (respectively C(N) = N∗) the associate complex sequence M∗ ( respectively N∗) to a graded A−module M = ⊕ n∈Z Mn ( respectively to a graded A−module N = ⊕ n∈Z Nn) so M∗, N∗ ∈ COMP (Gr(A−Mod)). So C(f) : M∗ −→ N∗ has a sense. (i) Let M ∈ Gr(A − Mod) then C(M) = M∗ is the associate complex sequence to a graded A−module M = ⊕ n∈Z Mn then M∗ ∈ COMP (Gr(A−Mod)). (ii) Let f : M −→ N graded morphism of degree k of graded A−modules then : C(f) = fk ∗ : M∗ −→ N∗ the associate complex chain to a graded morphism of degree k of graded left A−module. Furthermore C(g ◦ f) = (g ◦ f)k∗ = g[f ]k∗ = g[fk ∗ ] k ∗ = gk∗ ◦ fk ∗ = C(g) ◦ C(f). On other hand C(1M(n)) : M(n)∗ −→ M(n)∗ 1M(n)∗ = 1C(M) Thus C() is a covariant functor of Gr(A−Mod) to COMP (Gr(A−Mod)). Let 0 −→ M f−→ N g−→ L −→ 0 be the short exact sequence of graded left A−modules then we make the functor C() then we have A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1934 0 : · · · // �� // 0 �� // 0 �� // . . . M∗ : · · · // fk ∗ �� //M(n+ 1) fk(n+1) �� dn+1 //M(n) fk(n) �� dn // . . . N∗ : · · · gr∗ �� // N(n+ 1) gr(n+1) �� d ′ n+1+k // N(n) gr(n) �� d ′ n+k // . . . L∗ : · · · �� // L(n+ 1) �� d ′′ n+1+k+r // L(n) �� d ′′ n+k+r // . . . 0 : · · · // 0 // 0 // . . . is a short exact complex chain associate to short exact sequence of graded left A−modules then 0 −→ M∗ fk ∗−→ N∗ gk∗−→ L∗ −→ 0 is exact complex chain. Thus C() is exact additively covariant functor of Gr(A−Mod) to COMP (Gr(A−Mod)). Theorem 8. Let A = ⊕ n∈Z An be a graded ring, S is a multiplicatively closed subset sat- isfying the left conditions of Ore formed of homogeneous elements of A and Gr(S −1A − Mod) the category of graded left S−1A−modules, then the relation CH(−) : Gr(S −1A − Mod) −→ COMP (Gr(S −1A−Mod)) which that for all graded left S−1A−module S−1M of Gr(S −1A −Mod) we correspond the associate complex sequence (S−1M)∗ to a graded S−1A−module S−1M and for all graded morphism of graded left S−1A−modules S−1f : S−1M −→ S−1N of degree k we correspond the associate complex chain (S−1f)k∗ to a mor- phism of graded left S−1A−module S−1f : S−1M −→ S−1N is additively exact covariant functor. Proof. Similarly to the proof of theorem precedent 7 Theorem 9. Let A = ⊕ n∈Z An be a graded duo-ring, SH is a part formed of regulars homoge- neous elements of A and Gr(S −1 H A−Mod) the category of graded left S −1 H A−modules, then the relation CH(−) : Gr(S −1 H A−Mod) −→ COMP (Gr(S −1 H A−Mod)) which that for all graded left S −1 H A−module S −1 H M of Gr(S −1 H A−Mod) we correspond the associate complex sequence (S −1 H M)∗ to a graded S −1 H A−module S −1 H M and for all graded morphism of graded left S −1 H A−modules S −1 H f : S −1 H M −→ S −1 H N of degree k we correspond the associate com- plex chain (S −1 H f)k∗ to a morphism of graded left S −1 H A−module S −1 H f : S −1 H M −→ S −1 H N is additively exact covariant functor. A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1935 Proof. Similarly to the proof of theorem precedent 7 Theorem 10. Let A = ⊕ n∈Z An be a graded ring, S is a multiplicatively closed subset satisfying the left conditions of Ore formed of homogeneous elements of A and Gr(S −1A− Mod) the category of graded left S−1A−modules, then the relation (CH ◦S−1)(−) : Gr(A− Mod) −→ COMP (Gr(S −1A − Mod)) which that for all graded left A−module M of Gr(A−Mod) we correspond the associate complex sequence (CH◦S−1)(M) = (S−1M)∗ to a graded A−module M and for all graded morphism of graded left A−modules f : M −→ N of degree k we correspond the associate complex chain (CH ◦ S−1)(f) = (S−1f)k∗ to a morphism of graded left A−module f : M −→ N is additively exact covariant functor. Proof. Similarly to the proof of theorem precedent 7 Theorem 11. Let A = ⊕ n∈Z An be a graded duo-ring, SH is a part formed of regulars ho- mogeneous elements of A and Gr(S −1 H A−Mod) the category of graded left S −1 H A−modules, then the relation (CH ◦S−1 H )(−) : Gr(A−Mod) −→ COMP (Gr(S −1 H A−Mod)) which that for all graded left A−module M of Gr(A−Mod) we correspond the associate complex se- quence (CH ◦S−1 H )(M) = (S −1 H M)∗ to a graded A−module M and for all graded morphism of graded left A−modules f : M −→ N of degree k we correspond the associate complex chain (CH ◦ S−1 H )(f) = (S −1 H f)k∗ to a morphism of graded left A−module f : M −→ N is additively exact covariant functor. Proof. Similarly to the proof of theorem precedent7 Lemma 1. Let A = ⊕ n∈Z An be a graded ring, M = ⊕ n∈Z Mn and a graded left A−module then for all n ∈ Z Hn(M∗) ∼= M(n+ 2). Proof. Let M∗ : · · · → M(n+1) dn+1→ M(n) dn→ M(n− 1) → · · · the complex sequence, then ker(dn) = M(n+ 1) and Im(dn) = Mn so Hn(M∗) = ker(dn)/Im(dn+1) = M(n+1)/Mn+1 = (Mn+1⊕M(n+2))/Mn+1 ∼= M(n+2). Theorem 12. Let A = ⊕ n∈Z An be a graded ring, we have the induced functor of Hn : COMP (Gr(A − Mod)) −→ Gr(A − Mod) which that for all associate complex sequence M∗ to a graded A−module M = ⊕ n∈Z Mn we correspond Hn(M∗) = M(n + 2) and for all associate complex chain fk ∗ to a morphism of graded left A−module f : M =⊕ n∈Z Mn −→ N = ⊕ n∈Z Nn we correspond Hn(f∗) = fk(n+ 2), is a covariant functor. A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1936 Theorem 13. Let A = ⊕ n∈Z An be a graded ring, For all n ∈ Z and for all short exact sequence 0 −→ M f−→ N g−→ L −→ 0 of a graded left A−modules of graded morphism of degree k ∈ Z we have the following long exact sequence · · · −→ M(n+2) fk(n+2)−→ N(n+2) gk(n+2)−→ L(n+2) δn−→ M(n+1) fk(n+1)−→ N(n+1) −→ · · · of a graded left A−modules of graded morphism of degree k ∈ Z. Furthermore, if S is a multiplicatively closed subset satisfying the left conditions of Ore formed of homogeneous elements of A, we have the following longs exacts sequences of a graded left S−1A−modules · · ·S−1M(n+ 2) S−1fk(n+2)−→ S−1N(n+ 2) S−1gk(n+2)−→ S−1L(n+ 2) S−1δn−→ S−1M(n+ 1) · · · · · ·S−1⊗M(n+2) S−1⊗fk(n+2)−→ S−1⊗N(n+2) S−1⊗gk(n+2)−→ S−1⊗L(n+2) S−1⊗δn−→ S−1⊗M(n+1) · · · Proof. Let 0 −→ M f−→ N g−→ L −→ 0 be the short exact sequence of graded left A−modules then we make the functor C() to the short exact sequence of graded A−modules then we have 0 −→ M∗ fk ∗−→ N∗ gk∗−→ L∗ −→ 0 the associate short exact complex to a of graded A−modules or since precedent theorem 12, it exist a morphism of left A−module Hn(L∗) δn−→ Hn−1(M∗) such that we have the following long exact sequence of graded left A−modules · · · −→ Hn(M∗) Hn(fk ∗ )−→ Hn(N∗) Hn(gk∗ )−→ Hn(L∗) δn−→ Hn−1(M∗) Hn−1(fk ∗ )−→ Hn−1(N∗) −→ · · · i.e · · · −→ M(n+2) fk(n+2)−→ N(n+2) gk(n+2)−→ L(n+2) δn−→ M(n+1) fk(n+1)−→ Nk(n+1) −→ · · · We have the functor S−1() is exact then we have · · ·S−1M(n+ 2) S−1fk(n+2)−→ S−1N(n+ 2) S−1gk(n+2)−→ S−1L(n+ 2) S−1δn−→ S−1M(n+ 1) · · · Or the functor S−1() and the functor S−1A ⊗ A are isomorph then we have also · · ·S−1⊗M(n+2) S−1⊗fk(n+2)−→ S−1⊗N(n+2) S−1⊗gk(n+2)−→ S−1⊗L(n+2) S−1⊗δn−→ S−1⊗M(n+1) · · · A. O. Chbih, M. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 16 (3) (2023), 1913-1939 1937 Proposition 18. Let A = ⊕ n∈Z An be a graded ring, For all n ∈ Z and for all short exact sequence 0 −→ M f−→ N g−→ L −→ 0 of a graded left A−modules of graded morphism of degree k ∈ Z we have the following long exact sequence · · · −→ M(n+2) fk(n+2)−→ N(n+2) gk(n+2)−→ L(n+2) δn−→ M(n+1) fk(n+1)−→ N(n+1) −→ · · · of a graded left A−modules of graded morphism of degree k ∈ Z. Furthermore, if S is a multiplicatively closed subset satisfying the left conditions of Ore formed of homogeneous elements of A, we have the following longs exacts sequences of a graded left S−1A−modules · · ·S−1M(n+ 2) S−1fk(n+2)−→ S−1N(n+ 2) S−1gk(n+2)−→ S−1L(n+ 2) S−1δn−→ S−1M(n+ 1) · · · · · ·S−1⊗M(n+2) S−1⊗fk(n+2)−→ S−1⊗N(n+2) S−1⊗gk(n+2)−→ S−1⊗L(n+2) S−1⊗δn−→ S−1⊗M(n+1) · · · Proof. Similarly to the proof of theorem precedent13 Corollary 11. Let A = ⊕ n∈Z An be a graded duo-ring, For all n ∈ Z and for all short exact sequence 0 −→ M f−→ N g−→ L −→ 0 of a graded left A−modules we have the following long exact sequence · · · −→ M(n+2) fk(n+2)−→ N(n+2) gk(n+2)−→ L(n+2) δn−→ M(n+1) fk(n+1)−→ N(n+1) −→ · · · of a graded left A−modules. Furthermore, if SH is the part of regulars homogeneous elements of A, we have the fol- lowing longs exacts sequences of a graded left S −1 H A−modules · · ·S−1 H M(n+ 2) S −1 H fk(n+2)−→ S −1 H N(n+ 2) S −1 H gk(n+2)−→ S −1 H L(n+ 2) S −1 H δn−→ S −1 H M(n+ 1) · · · · · ·S−1 H ⊗M(n+2) S −1 H ⊗fk(n+2)−→ S −1 H ⊗N(n+2) S −1 H ⊗gk(n+2)−→ S −1 H ⊗L(n+2) S −1 H ⊗δn−→ S −1 H ⊗M(n+1) · · · Proof. it is sufficient to note that SH is a multiplicatively closed subset satisfying the left conditions of Ore formed of homogeneous elements of A and according to proposition 18 REFERENCES 1938 Proposition 19. Let A = ⊕ n∈Z An be a graded duo-ring, M = ⊕ n∈Z Mn a graded left A−module and SH be the part formed of regulars homogeneous elements of A, then for all n ∈ Z S −1 H (Hn(M∗)) ∼= S −1 H (A) ⊗ M(n+ 2). Moreover S −1 H (Hn(M∗)) ∼= Hn((S −1 H (M)∗). Proof. We have Hn(M∗) ∼= M(n+2) and as SH is a multiplicatively closed subset satis- fying the left conditions of Ore formed of homogeneous elements of A then S −1 H (Hn(M∗)) ∼= S −1 H (M(n+ 2)) or S −1 H A ⊗ M(n) ∼= S −1 H M(n) so S −1 H (Hn(M∗)) ∼= S −1 H A ⊗ (M(n+ 2)). On other hand Hn((S −1 H (M))∗) ∼= (S −1 H (M))(n + 2) ∼= S −1 H (M(n + 2)) ∼= S −1 H (Hn(M∗)) Thus S −1 H (Hn(M∗)) ∼= Hn((S −1 H (M))∗). Corollary 12. Let A = ⊕ n∈Z An be a graded duo-ring, M = ⊕ n∈Z Mn and be a graded left A−modules and SH be the set of all regular homogeneous of A, then for all n ∈ Z S−1 H (Hn(M∗)) ∼= S−1 H (A) ⊗ M(n+ 2). Moreover S−1 H (Hn(M∗)) ∼= Hn((S −1 H (M)∗). Proof. it is sufficient to note that SH = SH . 5. Conclusion In this article, we study the localization in the category COMP (Gr(A −Mod)) and we used the localization in the category Gr(A − Mod), and we proof that for all n ∈ Z fixed and for all M ∈ Gr(A−Mod) we have: S −1 H ((Hn ◦ C)(M)) ∼= Hn(CH ◦ S−1 H )(M)). References [1] O C Ahmed. Graduation et filtration des modules de fractions sur des anneaux non nécessairement commutatifs. PhD thesis, 2016. [2] O C Ahmed, M F Maaouia, and M Sanghare. Graduation of module of fraction on a graded domain ring not necessarily commutative. International Journal of Algebra, 10:457–474, 2015. REFERENCES 1939 [3] E C Dade. Group graded rings and modules. Math. Z., pages 241–262, 1980. [4] D Faye, M F Maaouia, and M Sanghare. Localization in a duo-ring and polynomi- als algebra. Non-Associative Algebra and Operator Theory, Springer Proceedings in Mathematics and Statistics, Switzerland, 160:183–191, 2016. [5] M F Maaouia. Localisation et enveloppe plate dans un duo-anneau. PhD thesis, 2003. 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