Flat morphisms of finite presentation are very flat
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00524626" target="_blank" >RIV/67985840:_____/20:00524626 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1007/s10231-019-00905-1" target="_blank" >https://doi.org/10.1007/s10231-019-00905-1</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s10231-019-00905-1" target="_blank" >10.1007/s10231-019-00905-1</a>
Alternative languages
Result language
angličtina
Original language name
Flat morphisms of finite presentation are very flat
Original language description
Principal affine open subsets in affine schemes are an important tool in the foundations of algebraic geometry. Given a commutative ring R, R-modules built from the rings of functions on principal affine open subschemes in SpecR using ordinal-indexed filtrations and direct summands are called very flat. The related class of very flat quasi-coherent sheaves over a scheme is intermediate between the classes of locally free and flat sheaves, and has serious technical advantages over both. In this paper, we show that very flat modules and sheaves are ubiquitous in algebraic geometry: if S is a finitely presented commutative R-algebra which is flat as an R-module, then S is a very flat R-module. This proves a conjecture formulated in the February 2014 version of the first author’s long preprint on contraherent cosheaves (Positselski in Contraherent cosheaves, arXiv:1209.2995 [math.CT]). We also show that the (finite) very flatness property of a flat module satisfies descent with respect to commutative ring homomorphisms of finite presentation inducing surjective maps of the spectra.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Annali di Matematica Pura ed Applicata
ISSN
0373-3114
e-ISSN
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Volume of the periodical
199
Issue of the periodical within the volume
3
Country of publishing house
DE - GERMANY
Number of pages
50
Pages from-to
875-924
UT code for WoS article
000535508400003
EID of the result in the Scopus database
2-s2.0-85074024917