Flat morphisms of finite presentation are very flat

TL;DR

This paper proves that flat, finitely presented algebra extensions are very flat modules, establishing their ubiquity in algebraic geometry and demonstrating descent properties for very flatness.

Contribution

It proves finitely presented flat algebras are very flat modules, confirming a prior conjecture and exploring descent of very flatness under certain ring homomorphisms.

Findings

Finitely presented flat algebras are very flat modules.

Very flat modules and sheaves are widespread in algebraic geometry.

Very flatness satisfies descent for certain ring homomorphisms.

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.