Noetherian Dimension and Co-localization of Artinian Modules over Local Rings

Abstract

Let (R, 𝔪) be a Noetherian local ring. Denote by N-dim RA the Noetherian dimension of an Artinian R-module A. In this paper, we give some characterizations for the ring R to satisfy N-dim RA = dim (R/ Ann RA) for certain Artinian R-modules A. Then the existence of a co-localization compatible with Artinian R-modules is studied and it is shown that if it is compatible with local cohomologies of finitely generated modules, then the base ring is universally catenary and all of its formal fibers are Cohen-Macaulay.