On the Attached Primes of Top Local Cohomology Modules

Let 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} be an ideal of a Noetherian ring R and M be a finitely generated R-module with cd(a,M)=c >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{cd}(\mathfrak {a},M)=c\ge 1$$\end{document}. In this paper, we prove that mAttRHac(M)subset of mAssRM boolean OR{p is an element of SuppM:AnnRHac-1(R/p)=p=AnnRHac(R/p)}.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textrm{mAtt}_R\,H<^>c_{\mathfrak {a}}(M) \subseteq \textrm{mAss}_R\,M \cup \{\mathfrak {p}\in \textrm{Supp}\,M: \textrm{Ann}_R\,H<^>{c-1}_{\mathfrak {a}}(R/\mathfrak {p})=\mathfrak {p}=\textrm{Ann}_R\,H<^>{c}_{\mathfrak {a}}(R/\mathfrak {p})\}. $$\end{document}Moreover, we show that AttRHac(M)subset of mAssRM boolean OR{p is an element of SuppM:AnnRHac-1(R/p)=p=AnnRHac(R/p)},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textrm{Att}_R\,H<^>c_{\mathfrak {a}}(M)\subseteq \textrm{mAss}_R\,M \cup \{\mathfrak {p}\in \textrm{Supp}\,M: \textrm{Ann}_R\,H<^>{c-1}_{\mathfrak {a}}(R/\mathfrak {p})=\mathfrak {p}=\textrm{Ann}_R\,H<^>{c}_{\mathfrak {a}}(R/\mathfrak {p})\}, $$\end{document}whenever the R-module Hac(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>c_{\mathfrak {a}}(M)$$\end{document} is Artinian.