Attached primes of local cohomology modules under localization and completion

Abstract

Let $(R,\m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module. Following I. G. Macdonald \cite{Mac}, the set of all attached primes of the Artinian local cohomology module $H^i_{\m}(M)$ is denoted by $\Att_R(H^i_{\m}(M))$. In \cite[Theorem 3.7]{Sh}, R. Y. Sharp proved that if $R$ is a quotient of a Gorenstein local ring then the shifted localization principle always holds true, i.e. $$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Att_{R_{\p}}\big(H^{i-\dim (R/\p)}_{\p R_{\p}}(M_{\p})\big)=\big\{\q R_{\p}\mid \q\in\Att_RH^i_{\m}(M), \q\subseteq \p\big\} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$ for any local cohomology modules $H^i_{\m}(M)$ and any $\p\in\Spec (R).$ In this paper, we improve Sharp's result as follows: the shifted localization principle always holds true if and only if $R$ is universally catenary and all its formal fibers are Cohen-Macaulay, if and only if $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \displaystyle \Att_{\R}(H^i_{\m}(M))=\bigcup_{\p\in\Att_R(H^i_{\m}(M))}\Ass_{\R}(\R/\p\R)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$$ holds true for any finitely generated $R$-module $M$ and any integer $i\geq 0.$ This also improves the main result of the paper \cite{CN}.