On the \'etale site of a marked scheme
Alexander Schmidt
TL;DR
This paper studies the properties of the étale site of marked schemes, focusing on coverings that split over marked points, and proves that Nisnevich coverings of quasi-compact schemes have finite subcoverings.
Contribution
It introduces the étale site of marked schemes with specific coverings and proves a key finiteness property for Nisnevich coverings, filling a gap in the literature.
Findings
Verified basic properties of the étale site of marked schemes.
Proved that any Nisnevich covering of a quasi-compact scheme has a finite subcovering.
Connected the concepts of small étale, Nisnevich, and marked schemes sites.
Abstract
We verify basic properties of the \'etale site of a `scheme with marking'. The coverings are \'etale coverings that split over all marked points. Familiar cases are the small \'etale site (no marking) and the Nisnevich site (all points are marked). This version 2 closes a small gap in the literature by showing that any Nisnevich covering of a quasi-compact scheme has a finite subcovering.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology · Polynomial and algebraic computation