Serre condition for category of gradual modules

Abstract

In this paper, we first describe the gradual modules and the homomorphisms between two gradual modules. Next, we introduce the category G-Mod, whose objects are all gradual modules and morphisms are all homomorphisms between two gradual modules. Also, according to the definition of the gradual module’s structure induced on each homomorphism [Formula: see text], we consider the gradual module’s structure on [Formula: see text] and [Formula: see text]. We show that being a monomorphism (respectively, epimorphism) in the category G-Mod is equivalent to a monomorphism (respectively, epimorphism) in the category [Formula: see text]. As the main result, we prove that in any short exact sequence [Formula: see text], in [Formula: see text], if the [Formula: see text]-module [Formula: see text] has a gradual module’s structure, then there are gradual module’s structure on the modules [Formula: see text] and [Formula: see text] where the short exact sequence created is a short exact sequence in the category G-Mod.