Flat morphisms of finite presentation are very flat

0524626 - MÚ 2021 RIV DE eng J - Journal Article
Positselski, Leonid - Slávik, A.
Flat morphisms of finite presentation are very flat.
Annali di Matematica Pura ed Applicata. Roč. 199, č. 3 (2020), s. 875-924. ISSN 0373-3114. E-ISSN 1618-1891
Institutional support: RVO:67985840
Keywords : flat morphisms * very flat modules * contramodules
OECD category: Pure mathematics
Impact factor: 0.969, year: 2020
Method of publishing: Open access
https://doi.org/10.1007/s10231-019-00905-1

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.
Permanent Link: http://hdl.handle.net/11104/0308963