On annihilators and associated primes of local cohomology modules. (English) Zbl 0968.13010

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 \({\mathfrak a}\). 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.
Reviewer: T.Porter (Bangor)


13D45 Local cohomology and commutative rings
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13E05 Commutative Noetherian rings and modules
14B15 Local cohomology and algebraic geometry


Zbl 0451.13008
Full Text: DOI


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