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Flat morphisms of finite presentation are very flat

  • Leonid PositselskiEmail author
  • Alexander Slávik
Article
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Abstract

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 \({{\,\mathrm{Spec}\,}}R\) 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.

Keywords

Commutative rings Finitely presented commutative algebras Flat modules Flat morphisms of schemes Very flat modules Very flat morphisms of schemes Finitely very flat modules Contraadjusted modules Contramodules Cotorsion theories Approximation sequences Surjective descent 

Mathematics Subject Classification

14B25 13B30 13J10 13C60 13D99 
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Notes

Acknowledgements

This paper grew out of the first author’s visits to Prague in 2015–2017, and he wishes to thank to Jan Trlifaj for inviting him. Our discussions with Jan Trlifaj also played a particularly important role in the development of this project. The first author is grateful to Silvana Bazzoni, Jan Št’ovíček, and Amnon Yekutieli for helpful conversations. The first author’s research is supported by research plan RVO: 67985840, by the Israel Science Foundation Grant # 446/15, and by the Grant Agency of the Czech Republic under the Grant P201/12/G028. The second author’s research is supported by the Grant Agency of the Czech Republic under the Grant 17-23112S and by the SVV project under the Grant SVV-2017-260456.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Mathematics of the Czech Academy of SciencesPrague 1Czech Republic
  2. 2.Laboratory of Algebraic GeometryNational Research University Higher School of EconomicsMoscowRussia
  3. 3.Laboratory of Algebra and Number TheoryInstitute for Information Transmission ProblemsMoscowRussia
  4. 4.Department of Mathematics, Faculty of Natural SciencesUniversity of HaifaMount Carmel, HaifaIsrael
  5. 5.Department of Algebra, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

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