On ideals preserving generalized local cohomology modules (Q2788764)

Let \(R\) be a Noetherian commutative ring with nonzero identity. Let \(\mathfrak a\) be an ideal of \(R\) and \(M, N\) two finitely generated \(R\)-modules. As a generalization of the notion of local cohomology, \textit{J. Herzog} [Komplexe, Auflösungen und Dualität in der lokalen Algebra. Habilitationsschrift, Univ. Regensburg, Regensburg (1970)]. has introduced the notion of generalized local cohomology. For any nonegative integer \(i,\) the \(i\)th generalized local cohomology module of \(M\) and \(N\) with respect to \(\mathfrak a\) is defined by NEWLINE\[CARRIAGE_RETURNNEWLINEH_{\mathfrak a}^i(M,N):={\varinjlim}_n \mathrm{Ext}^i_R(M/{\mathfrak a}^nM,N).CARRIAGE_RETURNNEWLINE\]NEWLINENEWLINENEWLINELet \(t\in \mathbb{N}\cup \{+\infty\}\). The author shows that the set NEWLINE\[CARRIAGE_RETURNNEWLINE\Omega_t:=\{{\mathfrak c}\mid {\mathfrak c} \text{ is an ideal of } R \text{ and } H_{\mathfrak c}^i(M,N)\cong H_{\mathfrak a}^i(M,N) \;\text{for \;all} \;i<t\}CARRIAGE_RETURNNEWLINE\]NEWLINEhas a largest member \({\mathfrak b}_t\). The author also proves that the ideal \({\mathfrak b}_t\) has the following two nice properties: NEWLINE\begin{itemize} NEWLINE\item[i)] \(\dim R/\mathfrak b_t=\sup \{\dim (\mathrm{Supp}_R(H_{\mathfrak a}^i(M,N)))| i<t\}\). NEWLINE\item [ii)] \(\;H_{\mathfrak c}^i(M,N)\cong H_{\mathfrak a}^i(M,N)\) for every ideal \(\mathfrak c\) such that \({\mathfrak a}\subseteq \mathfrak c\subseteq {\mathfrak b}_t\) and every \(i<t\).NEWLINE\end{itemize}