Abstract
Let \((R,\mathfrak {m})\) be a commutative Noetherian local ring, M be a finitely generated R-module and \(\mathfrak {a}\), I and J are ideals of R. We investigate the structure of formal local cohomology modules of \(\mathfrak {F}^i_{\mathfrak {a},I,J}(M)\) and \(\check{\mathfrak {F}}^i_{\mathfrak {a},I,J}(M)\) with respect to a pair of ideals, for all \(i\ge 0\). The main subject of the paper is to study the finiteness properties and artinianness of \(\mathfrak {F}^i_{\mathfrak {a},I,J}(M)\) and \(\check{\mathfrak {F}}^i_{\mathfrak {a},\mathfrak {m},J}(M)\). We study the maximum and minimum integer \(i\in \mathbb {N}\) such that \(\mathfrak {F}^i_{\mathfrak {a},\mathfrak {m},J}(M)\) and \(\check{\mathfrak {F}}^i_{\mathfrak {a},\mathfrak {m},J}(M)\) are not Artinian and we obtain some results involving cosupport, coassociated and attached primes for formal local cohomology modules with respect to a pair of ideals. Also, we give an criterion involving the concepts of finiteness and vanishing of formal local cohomology modules and Čech-formal local cohomology modules with respect to a pair of ideals.
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Acknowledgements
The authors would like to thank the referee for his/her useful suggestions. This research was carried out during visits by the authors to the Department of Mathematics-Purdue University and we would like to thanks that department for its hospitality. Especially we are deeply grateful to Professors B. Ulrich and G. Caviglia for some conversations.
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T.H. Freitas’s work was partially supported by FAPESP-Brazil-Grant 2012/01084-0 and 2013/20723-7. V. H. J. Pérez’s work was partially supported by CNPq-Brazil-Grant 245872/2012-4 and FAPESP-Brazil-Grant 2012/20304-1.
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Freitas, T.H., Jorge Pérez, V.H. Artinianness and finiteness of formal local cohomology modules with respect to a pair of ideals. Beitr Algebra Geom 58, 319–340 (2017). https://doi.org/10.1007/s13366-016-0322-6
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DOI: https://doi.org/10.1007/s13366-016-0322-6