The category of groupoid graded modules

Patrik Lundström¹

¹ Department of Informatics and Mathematics University of Trollhättan/Uddevalla Gärdhemsvägen 4 Box 957 461 29 Trollhättan, Sweden

Colloquium Mathematicum, Tome 100 (2004) no. 2, pp. 195-211

Résumé

We introduce the abelian category $R$-gr of groupoid graded modules and give an answer to the following general question: If $U:R \hbox {-gr}\rightarrow R\hbox{-mod}$ 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 $X$ $\Rightarrow $ $U(M)$ has property $X$; (II) $U(M)$ has property $X$ $\Rightarrow $ $M$ 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.

Détail

DOI : 10.4064/cm100-2-4

Keywords: introduce abelian category r gr groupoid graded modules answer following general question hbox gr rightarrow hbox mod denotes functor which associates graded r module underlying ungraded structure does either following implications has property rightarrow has property has property rightarrow has property treat cases properties direct summand finitely generated finitely presented projective injective essential small flat investigate exact sequences pure r gr relevant counterexamples indicated