*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/9780691202082.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter reviews the theory of adic spaces as developed by Huber. There are two familiar categories of geometric objects which arise in nonarchimedean geometry: formal schemes and rigid-analytic ...
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This chapter reviews the theory of adic spaces as developed by Huber. There are two familiar categories of geometric objects which arise in nonarchimedean geometry: formal schemes and rigid-analytic varieties. The goal is to construct a category of adic spaces which contains both formal schemes and rigid-analytic spaces as full subcategories. Just as formal schemes are built out of affine formal schemes associated to adic rings, and rigid-analytic spaces are built out of affinoid spaces associated to affinoid algebras, adic spaces are built out of affinoid adic spaces, which are associated to pairs of topological rings. The affinoid adic space associated to such a pair is the adic spectrum. The chapter then looks at Huber rings and defines the set of continuous valuations on a Huber ring, which constitute the points of an adic space.Less

This chapter reviews the theory of adic spaces as developed by Huber. There are two familiar categories of geometric objects which arise in nonarchimedean geometry: formal schemes and rigid-analytic varieties. The goal is to construct a category of adic spaces which contains both formal schemes and rigid-analytic spaces as full subcategories. Just as formal schemes are built out of affine formal schemes associated to adic rings, and rigid-analytic spaces are built out of affinoid spaces associated to affinoid algebras, adic spaces are built out of affinoid adic spaces, which are associated to pairs of topological rings. The affinoid adic space associated to such a pair is the adic spectrum. The chapter then looks at Huber rings and defines the set of continuous valuations on a Huber ring, which constitute the points of an adic space.

*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/9780691202082.003.0007
- Subject:
- Mathematics, Geometry / Topology

This chapter offers a second lecture on perfectoid spaces. A perfectoid Tate ring R is a complete, uniform Tate ring containing a pseudo-uniformizer. A perfectoid space is an adic space covered by ...
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This chapter offers a second lecture on perfectoid spaces. A perfectoid Tate ring R is a complete, uniform Tate ring containing a pseudo-uniformizer. A perfectoid space is an adic space covered by affinoid adic spaces with R perfectoid. The term “affinoid perfectoid space” is ambiguous. The chapter then looks at the tilting process and the tilting equivalence. The tilting equivalence extends to the étale site of a perfectoid space. Why is it important to study perfectoid spaces? The chapter puts forward a certain philosophy which indicates that perfectoid spaces may arise even when one is only interested in classical objects.Less

This chapter offers a second lecture on perfectoid spaces. A perfectoid Tate ring *R* is a complete, uniform Tate ring containing a pseudo-uniformizer. A perfectoid space is an adic space covered by affinoid adic spaces with *R* perfectoid. The term “affinoid perfectoid space” is ambiguous. The chapter then looks at the tilting process and the tilting equivalence. The tilting equivalence extends to the étale site of a perfectoid space. Why is it important to study perfectoid spaces? The chapter puts forward a certain philosophy which indicates that perfectoid spaces may arise even when one is only interested in classical objects.