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Berkeley Lectures on p-adic Geometry(AMS-207)$
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Peter Scholze and Jared Weinstein

Print publication date: 2020

Print ISBN-13: 9780691202082

Published to Princeton Scholarship Online: January 2021

DOI: 10.23943/princeton/9780691202082.001.0001

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Diamonds associated with adic spaces

Diamonds associated with adic spaces

(p.74) Lecture 10 Diamonds associated with adic spaces
Berkeley Lectures on p-adic Geometry

Peter Scholze

Jared Weinstein

Princeton University Press

This chapter focuses on diamonds associated with adic spaces. The goal is to construct a functor which forgets the structure morphism to Spa Zp, but retains topological information. The chapter studies how much information is lost when applying this construction. The intuition is that only topological information is kept. A morphism of adic spaces is a universal homeomorphism if all pullbacks are homeomorphisms. As in the case of schemes, in characteristic 0 the map f is a universal homeomorphism if and only if it is a homeomorphism and induces isomorphisms on completed residue fields. In keeping with the intuition, universal homeomorphisms induce isomorphisms of diamonds. The chapter then considers the underlying topological space of diamonds, as well as the étale site of diamonds.

Keywords:   diamonds, adic spaces, topological information, universal homeomorphism, isomorphisms, topological space, étale site

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