*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/9780691202082.003.0006
- Subject:
- Mathematics, Geometry / Topology

This chapter examines perfectoid spaces. A Huber ring R is Tate if it contains a topologically nilpotent unit; such elements are called pseudo-uniformizers. One can more generally define when an ...
More

This chapter examines perfectoid spaces. A Huber ring R is Tate if it contains a topologically nilpotent unit; such elements are called pseudo-uniformizers. One can more generally define when an analytic Huber ring is perfectoid. There are also notions of integral perfectoid rings which are not analytic. In this course, the perfectoid rings are all Tate. It would have been possible to proceed with the more general definition of perfectoid ring as a kind of analytic Huber ring. However, being analytic is critical for the purposes of the course. The chapter then looks at tilting and sousperfectoid rings. The class of sousperfectoid rings has good stability properties.Less

This chapter examines perfectoid spaces. A Huber ring *R* is Tate if it contains a topologically nilpotent unit; such elements are called pseudo-uniformizers. One can more generally define when an analytic Huber ring is perfectoid. There are also notions of integral perfectoid rings which are not analytic. In this course, the perfectoid rings are all Tate. It would have been possible to proceed with the more general definition of perfectoid ring as a kind of analytic Huber ring. However, being analytic is critical for the purposes of the course. The chapter then looks at tilting and sousperfectoid rings. The class of sousperfectoid rings has good stability properties.