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Contramodules over pro-perfect topological rings

  • Leonid Positselski ORCID logo EMAIL logo
From the journal Forum Mathematicum

Abstract

For four wide classes of topological rings R, we show that all flat left R-contramodules have projective covers if and only if all flat left R-contramodules are projective if and only if all left R-contramodules have projective covers if and only if all descending chains of cyclic discrete right R-modules terminate if and only if all the discrete quotient rings of R are left perfect. Three classes of topological rings for which this holds are the complete, separated topological associative rings with a base of neighborhoods of zero formed by open two-sided ideals such that either the ring is commutative, or it has a countable base of neighborhoods of zero, or it has only a finite number of semisimple discrete quotient rings. The fourth class consists of all the topological rings with a base of neighborhoods of zero formed by open right ideals which have a closed two-sided ideal with certain properties such that the quotient ring is a topological product of rings from the previous three classes. The key technique on which the proofs are based is the contramodule Nakayama lemma for topologically T-nilpotent ideals.

MSC 2010: 16W80; 16S90; 16D40; 16N40; 16L30; 13J10

Award Identifier / Grant number: 446/15

Award Identifier / Grant number: 20-13778S

Funding statement: The author was supported by the Israel Science Foundation grant # 446/15, research plan RVO: 67985840, and the GAČR project 20-13778S.

Acknowledgements

I owe a debt of gratitude to Silvana Bazzoni for her invaluable participation in the early stages of my work on this paper, which was started when I was visiting her in Padova in January–February 2018. I would like to thank Jan Trlifaj and Jan Št’ovíček for helpful discussions. Finally, I wish to thank an anonymous referee for reading the manuscript carefully and making several helpful suggestions.

  1. Communicated by: Manfred Droste

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Received: 2021-01-11
Revised: 2021-09-18
Published Online: 2021-11-30
Published in Print: 2022-01-01

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