Artinian Cofinite Modules and Going-up for R ⊆ R ̂ $R\subseteq \widehat {R}$

Abstract

Let \((R,\operatorname {\frak m})\) be a commutative Noetherian local ring. In this paper, it is shown that the going-up theorem holds for \(R\subseteq \widehat {R}\) if and only if \(\operatorname {Rad}(I+\operatorname {Ann}_{R} A)=\operatorname {\frak m}\) for any proper ideal I of R and any non-zero Artinian I-cofinite module A. Furthermore, using the main result of Zoschinger, Arch. Math. 95, 225–231 (2010), it is shown that these equivalent conditions are equivalent to R being formal catenary with α(R) = 0 and to \(\operatorname {Att}_{R} H^{\dim M}_{I}(M)=\{\operatorname {\frak p} \in \operatorname {Assh}_{R}(M)\,:\,\operatorname {Rad}(\operatorname {\frak p}+I)=\operatorname {\frak m}\}\) for any ideal I of R and any non-zero finitely generated R-module M.