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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

  • Czech description

Classification

  • Type

    J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database

  • CEP classification

  • OECD FORD branch

    10101 - Pure mathematics

Result continuities

  • Project

  • 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

  • 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