Über die Bedingung Going up für ${R \subset \hat{R}}$

Abstract

Let ${(R,\mathfrak m)}$ be a noetherian, local ring with completion ${\hat{R}}$ . We show that ${R \subset \hat{R}}$ satisfies the condition Going up if and only if there exists to every artinian R-module M with ${{\rm Ann}_R(M) \subset \mathfrak{p}}$ a submodule ${U \subset M}$ with ${{\rm {Ann}}_R(U)=\mathfrak{p}.}$ This is further equivalent to R being formal catenary, to α(R) = 0 and to ${H^d_{\mathfrak{q}/\mathfrak{p}}(R/\mathfrak{p})=0}$ for all prime ideals ${\mathfrak{p} \subset \mathfrak{q} \subsetneq \mathfrak{m}}$ where ${d = {\rm {dim}}(R/\mathfrak{p})}$.