On the attached primes and shifted localization principle for local cohomology modules (Q2862504)

Let \((R,\mathfrak{m})\) be a commutative Noetherian ring, \(M\) a finitely generated \(R\)-module and \(H^i_{\mathfrak{m}}(M)\) the \(i\)th local cohomology module of \(M\). \textit{R. Y. Sharp} [Proc. Lond. Math. Soc. (3) 30, 177--195 (1975; Zbl 0298.13011)] shows that in general NEWLINE\[NEWLINE\text{Att}_{R_{\mathfrak{P}}}(H^{i-\dim(R/\mathfrak{P})}_{\mathfrak{P} R_{\mathfrak{P}}}(M_{\mathfrak{P}}))\subseteq \{\mathfrak{q} R_{\mathfrak{P}}\mid \mathfrak{q} \in \text{Att}_{R}(H^i_{\mathfrak{m}}(M))\},NEWLINE\]NEWLINE and that the equality holds if \(R\) is a homomorphic image of a Gorenstein local ring. \(H^i_{\mathfrak{m}}(M)\) is said to satisfy the \textit{shifted localization principle} if the inclusion above is in fact an equality. Here \(\text{Att}\) stands for \textit{attached primes}, a concept that was introduced by \textit{I. G. Macdonald} [in: Sympos. math. 11, Algebra commut., Geometria, Convegni 1971/72, 23--43 (1973; Zbl 0271.13001)] and is considered as a dual to associated primes, for Artinian modules.NEWLINENEWLINEIn the paper under review, the author studies shifted localization principle in general and relates it to a property defined by \textit{Nguyen Tu Cung} and \textit{Le Thanh Nhan} [Vietnam J. Math. 30, No. 2, 121--130 (2002; Zbl 1096.13523)] as follows: An Artinian \(R-\)module \(A\) is said to satisfy property \((*)\) if NEWLINE\[NEWLINE\text{Ann}_R(0:_A\mathfrak{P})=\mathfrak{P} \text{~~for all~~} \mathfrak{P} \supseteq\text{Ann}_R(A).NEWLINE\]NEWLINE In the main Theorem of the paper it is shown that \(H^{\dim(M)}_{\mathfrak{m}}(M)\) satisfies the shifted localization principle if and only if it satisfies the property \((*)\) and both of these facts are equivalent to \(R/\text{Ann}_R(H^{\dim(M)}_{\mathfrak{m}}(M))\) being catenary.NEWLINENEWLINEIn addition, some relations between the property \((*)\) and the pseudo support of \(M\) are stated.