Finiteness conditions for graded modules (Gr-\(\Sigma\) (\(\Delta\) )- injective modules)
This paper is mainly concerned with providing ungraded finiteness properties from graded finiteness properties for \({\mathbb{Z}}\)-graded rings. It is shown in Section 2 of the paper that a graded R-module M which is \(\kappa^ g\)-Noetherian with respect to a graded localization \(\kappa^ g\) is in fact \(\kappa\)-Noetherian with respect to the ungraded localization \(\kappa\) generated by \(\kappa^ g\). The section also contains a strengthening of Hopkins' theorem for Grothendieck categories and this allows to prove a graded version of the relative Hopkins- Levitzki theorem. In Section 3 the authors study gr-\(\Sigma\) (\(\Delta)\)- injective modules and establish similar results i.e. one can forget about the grading and obtain ungraded \(\Sigma\) (\(\Delta)\)-injectivity properties. As an application it is mentioned that \(E_{R[X]}(Q[X])\) is a \(\Sigma\) (\(\Delta)\)-injective R[X]-module whenever Q is a \(\Sigma\) (\(\Delta)\)-injective R-module. In a Section 4 the authors study some dual results concerning \(\Sigma^*(\Delta^*)\)-projective modules.
scientific article; zbMATH DE number 3941723
scientific article