Vol. 11, No. 3, 2017

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Geometry on totally separably closed schemes

Stefan Schröer

Vol. 11 (2017), No. 3, 537–582
Abstract

We prove, for quasicompact separated schemes over ground fields, that Čech cohomology coincides with sheaf cohomology with respect to the Nisnevich topology. This is a partial generalization of Artin’s result that for noetherian schemes such an equality holds with respect to the étale topology, which holds under the assumption that every finite subset admits an affine open neighborhood (AF-property). Our key result is that on the absolute integral closure of separated algebraic schemes, the intersection of any two irreducible closed subsets remains irreducible. We prove this by establishing general modification and contraction results adapted to inverse limits of schemes. Along the way, we characterize schemes that are acyclic with respect to various Grothendieck topologies, study schemes all local rings of which are strictly henselian, and analyze fiber products of strict localizations.

Keywords
Absolute algebraic closure, acyclic schemes, étale and Nisnevich topology, henselian rings, Čech and sheaf cohomology, contractions
Mathematical Subject Classification 2010
Primary: 14F20
Secondary: 14E05, 13B22, 13J15
Milestones
Received: 24 March 2015
Revised: 18 October 2016
Accepted: 18 November 2016
Published: 6 May 2017
Authors
Stefan Schröer
Mathematisches Institut
Heinrich-Heine-Universität, 40204
Düsseldorf, Germany