A new proof of Faltings’ local-global principle for the finiteness of local cohomology modules

Let R be a commutative Noetherian ring, and let b⊆a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{b} \subseteq \mathfrak{a}}$$\end{document} be ideals of R. The goal of this paper is to show that, for a finitely generated R-module M, if the set AssR(Hafab(M)(M))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Ass}_R (H_{\mathfrak{a}}^{f_{\mathfrak{a}}^{\mathfrak{b}}(M)}(M))}$$\end{document} is finite or fa(M)≠cab(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f_{\mathfrak{a}}(M) \neq c_{\mathfrak{a}}^{\mathfrak{b}}(M)}$$\end{document}, then fab(M)=inf{faRpbRp(Mp)|p∈Spec(R)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f_{\mathfrak{a}}^{\mathfrak{b}}(M) = {\rm inf} \{f_{\mathfrak{a} R_{\mathfrak{p}}}^{\mathfrak{b} R_{\mathfrak{p}}}(M_{\mathfrak{p}})|\,\,\,\mathfrak{p} \in {\rm Spec}(R)\}}$$\end{document}, where cab(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_{\mathfrak{a}}^{\mathfrak{b}}(M)}$$\end{document} denotes the first non b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{b}}$$\end{document}-cofiniteness of the local cohomology module Hai(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H^i_{\mathfrak{a}}(M)}$$\end{document}. As a consequence of this, we provide a new and short proof of the Faltings’ local-global principle for finiteness dimensions. Also, several new results concerning the finiteness dimensions are given.