The category of groupoid graded modules

Tom 100 / 2004

Patrik Lundström Colloquium Mathematicum 100 (2004), 195-211 MSC: 16D10, 16D40, 16D50, 16D90. DOI: 10.4064/cm100-2-4


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.


  • Patrik LundströmDepartment of Informatics and Mathematics
    University of Trollhättan/Uddevalla
    Gärdhemsvägen 4
    Box 957
    461 29 Trollhättan, Sweden

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