Gabriel dimension for graded rings. (Q863167)

Let \(G\) be a group with identity element \(e\), and let \(R=\bigoplus_{\sigma\in G}R_\sigma\) be a ring graded by \(G\), such that the grading has finite support. Using colocalization for Grothendieck categories with a family of projective generators, it is proved that if the category \(R_e\)-mod has Gabriel dimension, then so does the category \(R\)-gr of \(G\)-graded \(R\)-modules. Then using graded Clifford theory, it is shown that if a graded module \(M\) has Gabriel dimension \(\alpha\) in \(R\)-gr, then \(M\) has Gabriel dimension \(\leq\alpha\) in \(R\)-mod. Finally, modular lattices are used to show that if \(R\)-mod has Gabriel dimension, then so does \(R\)-gr.