On annihilators and associated primes of local cohomology modules (Q1587980)

Let \(R\) be a commutative Noetherian ring, \(\mathfrak{a}\) and ideal of \(R\) and \(M\) a finitely generated \(R\)-module, then \(H_{\mathfrak a }^i(M)\) is the local cohomology module of \(M\) with respect to \({\mathfrak a}\). \textit{G. Faltings} proved [Math. Ann. 255, 45-56 (1981; Zbl 0451.13008)] a `local-global' principle: If \(r\) is a positive integer, then the \(R_{\mathfrak p}\)-module \(H^i_{\mathfrak a} {R}_{\mathfrak p}(M_{\mathfrak p})\) is finitely generated for all \(i \leq r\) and all \({\mathfrak p}\in \text{Spec}(R)\) if and only if \(H_{\mathfrak a}^i(M)\) is finitely generated for all \(i \leq r\). This leads to a notion of finiteness dimension \(f_{\mathfrak a}(M)\), being the infimum of those \(i\) for which \(H_{\mathfrak a}^i(M)\) is not finitely generated. If \(\mathfrak{b}\) is a second ideal, a variant of this finiteness dimension is the \(\mathfrak{b}\)-finiteness dimension, \(f^{\mathfrak b}_{\mathfrak a}(M)\), defined to be \(\inf\{i\in\mathbb{N}_0 : {\mathfrak b}^{nHi}_{\mathfrak a}(M) \neq 0\) for all \(n \in \mathbb{N} \}\). This paper continues the study of analogues of Faltings' local-global principle for this \(\mathfrak{b}\)-finiteness dimension, establishing it for arbitrary \(R\) `at level 2', and for \(R\) with \(\dim R \leq 4\) at all levels.