On the top local cohomology modules
Let \((R,\mathfrak{m})\) denote a local Noetherian ring. Let \(N\) be a finitely generated \(R\)-module. Then it is a well known fact that \(\text{Ann}_R (N/\mathfrak{p}N) = \mathfrak{p}\) for all \(\mathfrak{p} \in V(\text{Ann}_RN)\). Let \(A\) denote an Artinian \(R\)-module. Then the dual property (*) ``\(\text{Ann}_R (0:_A \mathfrak{p}) = \mathfrak{p}\) for all \(\mathfrak{p} \in V(\text{Ann}_RA)\)'' is true whenever \(R\) is complete but does not hold in general (see [\textit{Nguyen Tu Cung} and \textit{Le Thanh Nhan}, ``On the Noetherian dimension of Artinian modules'', Vietnam J. Math. 30, No. 2, 121--130 (2002; Zbl 1096.13523)]). See also \textit{H. Zöschinger} [Arch. Math. 95, No. 3, 225--231 (2010; Zbl 1206.13011)], where it is shown that (*) is satisfied for all Artinian \(R\)-moduls if and anly if \(R \to \hat{R}\) satisfies the Going Up Theorem.NEWLINENEWLINELet \(I \subset R\) denote and ideal and let \(M\) be a finitely generated \(R\)-module \(M\) and \(d = \dim M\). The authors investigate the property (*) for the case of the (Artinian) local cohomology module \(H^d_I(M)\). They use it in order to study the catenarity of the ring \(R/\text{Ann}_R H^d_I(M)\), the set of attached prime ideals, the co-support and the multiplicity of \(H^d_I(M)\).
scientific article; zbMATH DE number 6037581
scientific article