Recent Progress on Submersions: A Survey and New Properties

Abstract

This paper is a survey about recent progress on submersive morphisms of schemes combined with new results that we prove. They concern the class of quasicompact universally subtrusive morphisms that we introduced about 30 years ago. They are revisited in a recent paper by Rydh, with substantial complements and key results. We use them to show Artin-Tate-like results about the 14th problem of Hilbert, for a base scheme either Noetherian or the spectrum of a valuation domain. We look at faithfully flat morphisms and get “almost” Artin-Tate-like results by considering the Goldman (finite type) points of a scheme. Bjorn Poonen recently proved that universally closed morphisms are quasicompact. By introducing incomparable morphisms of schemes, we are able to characterize universally closed surjective morphisms that are either integral or finite. Next we consider pure morphisms of schemes introduced by Mesablishvili. In the quasicompact case, they are universally schematically dominant morphisms. This leads us to a characterization of universally subtrusive morphisms by purity. Some results on the schematically dominant property are given. The paper ends with properties of monomorphisms and topological immersions, a dual notion of submersions.

On the canonical, fpqc, and finite topologies on affine schemes. The state of the art

“…Both admit interpretations in terms of classical topologies: the v-topology coincides with the "universally substrusive" topology, which was already investigated by Picavet in the 80's, cf. [51] [52,8] (revisited by Rydh [54] in the non Noetherian situation); the (finer) arc-topology coincides with the "universally spectrally submersive" topology, cf. [11, 2.19].…”

Quasi-Prüfer Extensions of Rings

“…We denote by Ass(R) the set of all (Bourbaki) prime ideals P associated to the R-module R; that is, P ∈ Min(V(0 : r)) for some r ∈ R. Recall that a ring morphism f : R → S is called schematically dominant if for each open subset U of Spec(R), the map Γ(U, R) → Γ( a f −1 (U), S) is injective [23,Proposition I.5.4.1]. The first author proved that a flat ring morphism f : R → S is schematically dominant if and only if Ass(R) ⊆ X(S) [40,Proposition 52]. Clearly if Min(R) = Ass(R) (for example, if R is an integral domain) and f is injective and flat, then f is schematically dominant.…”