Extension Functors of Cousin Cohomology Modules

Let R be a commutative Noetherian ring with non-zero identity, F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document} a filtration of Spec(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{Spec}}}(R)$$\end{document} which admits an R-module X, and CR(F,X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{C}}}_R(\mathcal {F},X)$$\end{document} the Cousin complex for X with respect to F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document}. 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 Hs-1(ExtRt(N,CR(F,X)))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{H}}}^{s-1}({{\mathrm{Ext}}}^t_R(N,{{\mathrm{C}}}_R(\mathcal {F},X)))$$\end{document} and ExtRs(N,Ht-1(CR(F,X)))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{Ext}}}^{s}_R(N,{{\mathrm{H}}}^{t-1}({{\mathrm{C}}}_R(\mathcal {F},X)))$$\end{document} 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.