Peter Scholze and Jared Weinstein
- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0010
- Subject:
- Mathematics, Geometry / Topology
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 ...
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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.Less
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.
