Quasi-complete Semilocal Rings and Modules

Daniel D. Anderson ()
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Daniel D. Anderson: University of Iowa, Department of Mathematics

A chapter in Commutative Algebra, 2014, pp 25-37 from Springer

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

Keywords: Quasi-complete rings; Quasi-complete modules; Noether lattices; 13E05; 13H10; 13A15; 06F10 (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4939-0925-4_2

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

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