We introduce the abelian category R-gr of groupoid graded modules and give an answer to the following general question: If U: R-gr → R-mod denotes the functor which associates to any graded left R-module M the underlying ungraded structure U(M), when does either of the following two implications hold: (I) M has property X ⇒ U(M) has property X; (II) U(M) has property X ⇒ M has property X? We treat the cases when X is one of the properties: direct summand, free, finitely generated, finitely presented, projective, injective, essential, small, and flat. We also investigate when exact sequences are pure in R-gr. Some relevant counterexamples are indicated.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm100-2-4,
author = {Patrik Lundstr\"om},
title = {The category of groupoid graded modules},
journal = {Colloquium Mathematicae},
volume = {100},
year = {2004},
pages = {195-211},
zbl = {1073.16041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm100-2-4}
}
Patrik Lundström. The category of groupoid graded modules. Colloquium Mathematicae, Tome 100 (2004) pp. 195-211. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm100-2-4/