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. 1. Introduction Our aim is to give a survey on recent progress on submersions and new results that commutative algebraists may find useful. We also recall results that are needed. The paper is written in the language of schemes because it is sometimes necessary to enlarge the category of commutative rings to get proofs, but the results can be easily translated. Submersive morphisms of schemes (or submersions) are surjective morphisms inducing the quotient topology on ; that is, is an open (closed) subset if and only if is open (closed). They are also called topological epimorphisms by some authors like Voevodsky who defines and uses the and -(Grothendieck) topologies [1]. They appear naturally in many situations such as when studying quotients, homology, descent, and the fundamental group of schemes. A morphism of schemes is called universally submersive if is submersive for each morphism . The first proper treatment of submersive morphisms was settled by Grothendieck, with applications to the fundamental group of a scheme. We singled out a subclass of submersive morphisms in [2] and dubbed them subtrusive morphisms (or subtrusions). Submersive morphisms used in practice are subtrusive. Our study was established in the affine schemes context. But as Rydh showed, the theory can be extended to the arbitrary schemes context
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