Total separable closure and contractions

TL;DR

This paper extends Artin's result from rings to schemes, showing local intersections of local schemes on certain integral normal schemes and applying this to relate cech and sheaf cohomology in the Nisnevich topology.

Contribution

It generalizes the characterization of contractible curves to higher dimensions and establishes new results on local intersections and cohomology equivalences.

Findings

Local intersections of schemes are local under specified conditions.

cech cohomology coincides with sheaf cohomology for the Nisnevich topology.

Generalization of contractible curves characterization to higher dimensions.

Abstract

We show that on integral normal separated schemes whose function field is separably closed, for each pair of points the intersection of the resulting local schemes is local. This extends a result of Artin from rings to schemes. The argument relies on the existence of certain modifications in inverse limits. As an application, we show that \v{C}ech cohomology coincides with sheaf cohomology for the Nisnevich topology. Along the way, we generalize the characterization of contractible curves on surfaces by negative-definiteness of the intersection matrix to higher dimensions, using bigness of invertible sheaves on non-reduced schemes.