Abstract
Let R be a commutative Noetherian ring with non-zero identity, \(\mathcal {F}\) a filtration of \({{\mathrm{Spec}}}(R)\) which admits an R-module X, and \({{\mathrm{C}}}_R(\mathcal {F},X)\) the Cousin complex for X with respect to \(\mathcal {F}\). In this paper, we first introduce the Cousin functor and the Cousin spectral sequences. Then for non-negative integers s, t and a finite R-module N, we study the membership of R-modules \({{\mathrm{H}}}^{s-1}({{\mathrm{Ext}}}^t_R(N,{{\mathrm{C}}}_R(\mathcal {F},X)))\) and \({{\mathrm{Ext}}}^{s}_R(N,{{\mathrm{H}}}^{t-1}({{\mathrm{C}}}_R(\mathcal {F},X)))\) in Serre subcategories of the category of R-modules and find some conditions for validity of an isomorphism between them. Finally, we use these results to present some facts about the vanishing and finiteness of Cousin cohomology modules.
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Acknowledgements
The authors would like to thank the referee for the invaluable comments on the manuscript. This research of Alireza Vahidi was in part supported by a grant from Payame Noor University.
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Communicated by Mohammad-Taghi Dibaei.
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Bamdad, H., Vahidi, A. Extension Functors of Cousin Cohomology Modules. Bull. Iran. Math. Soc. 44, 253–267 (2018). https://doi.org/10.1007/s41980-018-0009-x
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DOI: https://doi.org/10.1007/s41980-018-0009-x