Quasi-complete Semilocal Rings and Modules

Daniel D Anderson

doi:10.1007/978-1-4939-0925-4_2

Let R be a (commutative Noetherian) semilocal ring with Jacobon radical J. Chevalley has shown that if R is complete, then R satisfies the following condition: given any descending chain of ideals Ann=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left \{A_{n}\right \}_{n=1}^{\infty }$$ \end{document} with ⋂n=1∞An=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bigcap \nolimits _{n=1}^{\infty }A_{n} = 0$$ \end{document}, for each positive integer k there exists an sk with Ask⊆Jk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$A_{s_{k}} \subseteq J^{k}$$ \end{document}. A finitely generated R-module M is said to be (weakly) quasi-complete if for any descending chain Ann=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left \{A_{n}\right \}_{n=1}^{\infty }$$ \end{document} of R-submodules of M (with ⋂n=1∞An=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bigcap \nolimits _{n=1}^{\infty }A_{n} = 0$$ \end{document}) and k ≥ 1, there exists an sk with Ask⊆(⋂n=1∞An)+JkM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$A_{s_{k}} \subseteq (\bigcap \nolimits _{n=1}^{\infty }A_{n}) + J^{k}M$$ \end{document}. An easy modification of Chevalley’s proof shows that a finitely generated R-module over a complete semilocal ring is quasi-complete. However, the converse is false as any DVR is quasi-complete. In this paper we survey known results about (weakly) quasi-complete rings and modules and prove some new results.

Quasi-complete Semilocal Rings and Modules

View Online

Abstract

Details

Metrics