Gabriel Dimension for Graded Rings

Abstract

Using techniques of localization for Grothendieck categories with a family of projective generators, we show that for a graded ring R = ⊕ σ∈G R σ with finite support if R e has Gabriel dimension then $R\hbox{-}{\rm gr}$ has Gabriel dimension. Moreover, adding some lattice results, we prove that if $R\hbox{-}{\rm mod}$ has Gabriel dimension then $R\hbox{-}{\rm gr}$ also has Gabriel dimension.