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Countably generated flat modules are quite flat
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SYSNO ASEP 0557869 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Countably generated flat modules are quite flat Author(s) Hrbek, Michal (MU-W) SAI, ORCID, RID
Positselski, Leonid (MU-W) SAI, ORCID, RID
Slávik, A. (CZ)Source Title Journal of Commutative Algebra. - : Rocky Mountain Mathematics Consortium - ISSN 1939-0807
Roč. 14, č. 1 (2022), s. 37-54Number of pages 18 s. Language eng - English Country US - United States Keywords countably presented modules ; quite flat modules ; strongly discrete valuation domains Subject RIV BA - General Mathematics OECD category Pure mathematics Method of publishing Limited access Institutional support MU-W - RVO:67985840 UT WOS 000808049400004 EID SCOPUS 85131455805 DOI 10.1216/jca.2022.14.37 Annotation 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. Workplace Mathematical Institute Contact Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Year of Publishing 2023 Electronic address https://dx.doi.org/10.1216/jca.2022.14.37
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