Spring 2022 Countably generated flat modules are quite flat
Michal Hrbek, Leonid Positselski, Alexander Slávik
J. Commut. Algebra 14(1): 37-54 (Spring 2022). DOI: 10.1216/jca.2022.14.37


We prove that if R is a commutative Noetherian ring, then every countably generated flat R-module is quite flat, i.e., a direct summand of a transfinite extension of localizations of R in countable multiplicative subsets. We also show that if the spectrum of R is of cardinality less than κ, where κ is an uncountable regular cardinal, then every flat R-module is a transfinite extension of flat modules with less than κ generators. This provides an alternative proof of the fact that over a commutative Noetherian ring with countable spectrum, all flat modules are quite flat. More generally, we say that a commutative ring is CFQ if every countably presented flat R-module is quite flat. We show that all von Neumann regular rings and all S-almost perfect rings are CFQ. A zero-dimensional local ring is CFQ if and only if it is perfect. A domain is CFQ if and only if all its proper quotient rings are CFQ. A valuation domain is CFQ if and only if it is strongly discrete.


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Michal Hrbek. Leonid Positselski. Alexander Slávik. "Countably generated flat modules are quite flat." J. Commut. Algebra 14 (1) 37 - 54, Spring 2022. https://doi.org/10.1216/jca.2022.14.37


Received: 8 July 2019; Revised: 18 November 2019; Accepted: 19 November 2019; Published: Spring 2022
First available in Project Euclid: 31 May 2022

MathSciNet: MR4430700
zbMATH: 1495.13018
Digital Object Identifier: 10.1216/jca.2022.14.37

Primary: 13C11 , 13E05
Secondary: 16D40

Keywords: almost perfect domains , commutative rings , countably presented modules , flat modules , Noetherian rings , perfect rings , quite flat modules , strongly discrete valuation domains , Strongly flat modules

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium


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Vol.14 • No. 1 • Spring 2022
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