The category of groupoid graded modules

Patrik Lundström. "The category of groupoid graded modules." Colloquium Mathematicae 100.2 (2004): 195-211. <http://eudml.org/doc/283784>.

@article{PatrikLundström2004,
abstract = {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.},
author = {Patrik Lundström},
journal = {Colloquium Mathematicae},
keywords = {groupoid graded modules; groupoid graded rings; categories of graded modules; projective modules; injective modules; flat modules; weak Hopf algebras},
language = {eng},
number = {2},
pages = {195-211},
title = {The category of groupoid graded modules},
url = {http://eudml.org/doc/283784},
volume = {100},
year = {2004},
}

TY - JOUR
AU - Patrik Lundström
TI - The category of groupoid graded modules
JO - Colloquium Mathematicae
PY - 2004
VL - 100
IS - 2
SP - 195
EP - 211
AB - 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.
LA - eng
KW - groupoid graded modules; groupoid graded rings; categories of graded modules; projective modules; injective modules; flat modules; weak Hopf algebras
UR - http://eudml.org/doc/283784
ER -