Countably generated flat modules are quite flat

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F22%3A00557869" target="_blank" >RIV/67985840:_____/22:00557869 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/00216208:11320/22:10454755

Výsledek na webu

<a href="http://https:dx.doi.org/10.1216/jca.2022.14.37" target="_blank" >http://https:dx.doi.org/10.1216/jca.2022.14.37</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1216/jca.2022.14.37" target="_blank" >10.1216/jca.2022.14.37</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Countably generated flat modules are quite flat

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Countably generated flat modules are quite flat

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA17-23112S" target="_blank" >GA17-23112S: Strukturní teorie reprezentací algeber (lokalizace a vychylující teorie)</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Commutative Algebra

ISSN

1939-0807

e-ISSN

1939-2346

Svazek periodika

14

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

18

Strana od-do

37-54

Kód UT WoS článku

000808049400004

EID výsledku v databázi Scopus

2-s2.0-85131455805