Flat morphisms of finite presentation are very flat

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 $\operatorname{Spec}R$ 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 long preprint arXiv:1209.2995. 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.