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Artinian Cofinite Modules and Going-up for $R\subseteq \widehat {R}$
Gholamreza Pirmohammadi
,
Khadijeh Ahmadi Amoli
,
Kamal Bahmanpour
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 Zöschinger, Arch. Math. 95, 225–231 (2010), it is shown that these equivalent conditions are equivalent to $R$ being formal catenary with $\alpha(R) = 0$ 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$.