Quasi-complete Semilocal Rings and Modules

Abstract

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 $\left \{A_{n}\right \}_{n=1}^{\infty }$ with $\bigcap \nolimits _{n=1}^{\infty }A_{n} = 0$, for each positive integer k there exists an s _k with $A_{s_{k}} \subseteq J^{k}$. A finitely generated R-module M is said to be (weakly) quasi-complete if for any descending chain $\left \{A_{n}\right \}_{n=1}^{\infty }$ of R-submodules of M (with $\bigcap \nolimits _{n=1}^{\infty }A_{n} = 0$) and k ≥ 1, there exists an s _k with $A_{s_{k}} \subseteq (\bigcap \nolimits _{n=1}^{\infty }A_{n}) + J^{k}M$. 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.