Algebra & Number Theory

Geometry on totally separably closed schemes

Stefan Schröer

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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.

Article information

Algebra Number Theory, Volume 11, Number 3 (2017), 537-582.

Received: 24 March 2015
Revised: 18 October 2016
Accepted: 18 November 2016
First available in Project Euclid: 19 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F20: Étale and other Grothendieck topologies and (co)homologies
Secondary: 14E05: Rational and birational maps 13B22: Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.) 13J15: Henselian rings [See also 13B40]

Absolute algebraic closure acyclic schemes étale and Nisnevich topology henselian rings Čech and sheaf cohomology contractions


Schröer, Stefan. Geometry on totally separably closed schemes. Algebra Number Theory 11 (2017), no. 3, 537--582. doi:10.2140/ant.2017.11.537.

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