  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.
