Hopfian and Cohopfian Objects in the Categories of Gr(A - Mod) and COMP(Gr(A - Mod))

Abstract

We study in this work the notions of hopficity and cohopficity in the categories AGr(A - Mod) and COMP(AGr(A - Mod)) of associate complex to a graded left A-module and we show that:
 1. Let M a graded left A-module, N a graded submodule of M, M_* be a complex associate to M. Suppose that M_* be a quasi-projective and N be a completely invariant and essential sub-complex of M_* associate to N. Then N_* is cohopfian if, and only, if M_* is cohopfian.
 2. Let M a graded left A-module, N a graded submodule of M, M_* quasi-injective and M_* a completely invariant and superfluous sub-complex of M_*. Then M_* is cohopfian if, and only, if M_*=M_* is cohopfian.

Localization of Hopfian and Cohopfian Objects in the Categories of A − Mod, AGr(A − Mod) and COMP(AGr(A − Mod))

“…In this article, we study the localization of hopfian and cohopfian objects in the categories A−M od of left A-modules, AGr(A−M od) of graded left A-modules and COM P (AGr(A-− M od)) of complex sequences associated to graded left A-modules. We rely on the articles, Graduation of Module of Fraction on a Graded Domain Ring not Necessarily Commutative [2], Factorization of Graded Modules of Fractions [3], Module de Fractions, Sous-modules S−saturée et Foncteur S −1 [18] and Hopfian and Cohopfian Objects in the Categories of Gr(A − M od) and COM P (Gr(A − M od)) [23], which are used as a basis for studying the notions of localization, hopficity and cohopficity. The transition to localization and the study of hopficity and cohopficity from the category of left A − M od whose objects are the left A-modules and the morphisms are the left A-module morphisms to the category AGr(A − M od) whose objects are the graded left A-module and the morphisms are the graded left A-module graded morphisms and AGr(A − M od) to the category of COM P (AGr(A − M od)) whose the objects are the complex sequences of graded left A-modules and the morphisms are the chain complexes associated to the graded morphims of graded left A-modules is not easy.…”