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I am interested in the topology on schemes where surjective morphisms of finite presentation are coverings. In particular, I am interested in the topology on Noetherian schemes where surjective morphisms of finite type are coverings. So, the morphism $U\sqcup Z\to X$, where $Z\subset X$ is a closed subscheme of a Noetherian scheme $X$ and $U = X\setminus Z$ is the open complement, would be a typical example of covering in this topology (in addition to faithfully flat morphisms of finite presentation, etc.)

Is there a name for this topology? How does it compare to other known topologies on schemes, such as fpqc?

The motivation is that I am studying a property of quasi-coherent sheaves which I call very flatness. I believe I can prove that very flatness of flat sheaves is local in this topology for Noetherian schemes, i.e., given a Noetherian ring $R$ and a finitely generated $R$-algebra $S$ such that the induced map of spectra $\operatorname{Spec}S\to\operatorname{Spec}R$ is surjective (as a map of sets), a flat $R$-module $F$ is very flat whenever the $S$-module $S\otimes_RF$ is very flat.

Moreover, there is a stronger version of very flatness called "finite very flatness", which I think I can prove is local for the topology on arbitrary schemes in which surjective morphisms of finite presentation are coverings (assuming that the sheaf is known to be flat). In other words, given a commutative ring $R$ and a finitely presented $R$-algebra $S$ such that the induced map of spectra $\operatorname{Spec}S\to\operatorname{Spec}R$ is surjective, a flat $R$-module $F$ is finitely very flat whenever the $S$-module $S\otimes_R F$ is finitely very flat.

So I want to understand what this topology on schemes is and what does locality in it entail.

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  • $\begingroup$ I should add that the very flatness is a nontrivial property quite different from (stronger than) flatness. E.g., the projective dimension of any very flat module over a commutative ring does not exceed 1. Rational numbers are not a very flat module over the integers. $\endgroup$ – Leonid Positselski Jun 27 '17 at 18:54

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