Countably generated flat modules are quite flat

0557869 - MÚ 2023 RIV US eng J - Journal Article
Hrbek, Michal - Positselski, Leonid - Slávik, A.
Countably generated flat modules are quite flat.
Journal of Commutative Algebra. Roč. 14, č. 1 (2022), s. 37-54. ISSN 1939-0807. E-ISSN 1939-2346
Institutional support: RVO:67985840
Keywords : countably presented modules * quite flat modules * strongly discrete valuation domains
OECD category: Pure mathematics
Impact factor: 0.6, year: 2022
Method of publishing: Limited access
https://dx.doi.org/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 kappa, where kappa is an uncountable regular cardinal, then every flat R-module is a transfinite extension of flat modules with less than kappa 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.
Permanent Link: http://hdl.handle.net/11104/0331714