Cofiniteness of top local cohomology modules

Let R be a commutative Noetherian ring with non-zero identity, a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {a}$$\end{document} an ideal of R, M a finitely generated R-module with finite Krull dimension d, and n a non-negative integer. In this paper, we prove that the top local cohomology module Had-n(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {H}}<^>{d-n}_{\mathfrak {a}}(M)$$\end{document} is an (FD<n,a)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\text {FD}}_{<n},\mathfrak {a})$$\end{document}-cofinite R-module and {p is an element of AssR(Had-n(M)):dim(R/p)>= n}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\mathfrak {p}\in {{\text {Ass}}_R({\text {H}}<^>{d-n}_{\mathfrak {a}}(M))}:\dim (R/\mathfrak {p})\ge {n}\}$$\end{document} is a finite set. As a consequence, we observe that SuppR(Had-1(M))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Supp}}_R({\text {H}}<^>{d-1}_{\mathfrak {a}}(M))$$\end{document} is a finite set when R is a semi-local ring.