Upper bounds, cofiniteness, and artinianness of local cohomology modules defined by a pair of ideals

• Published in: arXiv.org
• Main Authors: Aghapournahr, M, Ahmadi-amoli, KH, Sadeghi, M Y
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 18.11.2012
• Subjects: Homology; Minimax technique; Modules; Upper bounds

Let \(R\) be a commutative noetherian ring, \(I,J\) be two ideals of \(R\), \(M\) be an \(R\)-module, and \(\mathcal{S}\) be a Serre class of \(R\)-modules. A positive answer to the Huneke\(^,\)s conjecture is given for a noetherian ring \(R\) and minimax \(R\)-module \(M\) of krull dimension less than 3, with respect to \(\mathcal{S}\). There are some results on cofiniteness and artinianness of local cohomology modules with respect to a pair of ideals. For a ZD-module \(M\) of finite krull dimension and an integer \(n\in\mathbb{N}\), if \(\lc^{i}_{I,J}(M)\in\mathcal{S}\) for all \(i>n\), then \(\lc^{i}_{I,J}(M)/\fa^{j}\lc^{i}_{I,J}(M)\in\mathcal{S}\) for any \(\fa\in\tilde{W}(I,J)\), all \(i\geq n\), and all \(j\geq0\). By introducing the concept of Seree cohomological dimension of \(M\) with respect to \((I,J)\), for an integer \(r\in\mathbb{N}_0\), \(\lc^{j}_{I,J}(R)\in\mathcal{S}\) for all \(j>r\) iff \(\lc^{j}_{I,J}(M)\in\mathcal{S}\) for all \(j>r\) and any finite \(R\)-module \(M\).