(untitled) (European Journal of Pure and Applied Mathematics, 2021)
The aim of this paper is to study the localization of hopfian and cohopfian objects in the categories A-Mod of left A-modules, AGr(A-Mod) of graded left A-modules and COMP(AGr(A-Mod)) of complexÂ sequences associated to graded left A-modules. We have among othersÂ the main following results : 1. Let M be a noetherian graded left A-module, S a saturated multiplicative part formed by the non-zero homogeneous elements of A verifying the left Ore conditions, N a submodule of M, M_* is a noetherian quasi-injective complex sequence associated with M and N_* is an essential and completely invariant complex sub-sequence of M_*. Then, S^-1(N_*) the complex sequence of morphisms of left S^-1A-modules is cohopfian if, and only, if S^-1(M_*) is cohopfian ; 2. let M be a graded left A-module and S a saturated multiplicative part formed by the non-zero homogeneous elements of A verifying the left Ore conditions. If M_* is a hopfian, noetherian and quasi-injective complex sequence associated with M, then the complex sequence of morphisms of left S^-1(A)-modules S^-1(M_*) has the following property : any epimorphism of sub-complex S^-1(N_*) of S^-1(M_*) is an isomorphism ; 3. let M be a graded left A-module, N a graded submodule of M, S a saturated multiplicative part formed by the non-zero homogeneous elements of A verifying the left Ore conditions. M_* the quasi-projective complex sequence associated with M and N_* a superfluous and completely invariant complex sub-sequence of M_*. Then the complex morphism sequence of left S^-1(A)-modules S^-1(N_*) is hopfian if, and only if, S^-1(M_*/N_*) the complex sequence associated with S^-1(M/N) is hopfian. Â