Table of Contents
Algebra
Volume 2013 (2013), Article ID 128064, 14 pages
http://dx.doi.org/10.1155/2013/128064
Research Article

Recent Progress on Submersions: A Survey and New Properties

Université Blaise Pascal, Laboratoire de Mathématiques UMR 6620 CNRS, Les Cézeaux, 24 Avenue des Landais BP 80026, 63177 Aubière Cedex, France

Received 27 November 2012; Accepted 15 March 2013

Academic Editor: Prasanna Kumar Sahoo

Copyright © 2013 Gabriel Picavet. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.