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The main results of this paper are : If A=_nZ A_n is a graded duo-ring, S_H is a part formed of regulars homogeneous elements of A, S_H is the homogeneous multiplicatively closed subset of A generated by S_H, then: enumerate The relation C_H(-) :G_r(S_H^-1A-Mod)COMP(G_r(S_H^-1A-Mod)) which that for all graded left S_H^-1A-module S_H^-1M of G_r(S_H^-1A-Mod) we correspond the associate complex sequence (S_H^-1M)_* to a graded S_H^-1A-module S_H^-1M and for all graded morphism of graded left S_H^-1A-modules S_H^-1f : S_H^-1MS_H^-1N of degree k we correspond the associated complex chain (S_H^-1f)_*^k to a morphism of graded left S_H^-1A-module S_H^-1f : S_H^-1MS_H^-1N is an additively exact covariant functor. The relation (C_H°S_H^-1)(-) :G_r(A-Mod)COMP(G_r(S_H^-1A-Mod)) which that for all graded left A-module M of G_r(A-Mod) we correspond the associate complex sequence (C_H°S_H^-1)(M)=(S_H^-1M)_* to a graded A-module M and for all graded morphism of graded left A-modules f : MN of degree k we correspond the associated complex chain (C_H°S_H^-1)(f)=(S_H^-1f)_*^k to a morphism of graded left A-module f : MN is an additively exact covariant functor. oindent For all nZ fixed and for all M G_r(A-Mod) we have: S^-1_H((H_n° C)(M))H_n(C_H° S^-1_H)(M)). enumerate , number=3, journal=European Journal of Pure and Applied Mathematics, year=2023, month=Jul., pages=1913–1939