*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.

*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/9780691202082.003.0004
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses various examples of adic spaces. These examples include the adic closed unit disc; the adic affine line; the closure of the adic closed unit disc in the adic affine line; the ...
More

This chapter discusses various examples of adic spaces. These examples include the adic closed unit disc; the adic affine line; the closure of the adic closed unit disc in the adic affine line; the open unit disc; the punctured open unit disc; and the constant adic space associated to a profinite set. The chapter focuses on one example: the adic open unit disc over Zp. The adic spectrum Spa Zp consists of two points, a special point and a generic point. The chapter then studies the structure of analytic points. It also clarifies the relations between analytic rings and Tate rings.Less

This chapter discusses various examples of adic spaces. These examples include the adic closed unit disc; the adic affine line; the closure of the adic closed unit disc in the adic affine line; the open unit disc; the punctured open unit disc; and the constant adic space associated to a profinite set. The chapter focuses on one example: the adic open unit disc over **Z**p. The adic spectrum Spa **Z**p consists of two points, a special point and a generic point. The chapter then studies the structure of analytic points. It also clarifies the relations between analytic rings and Tate rings.

*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 ...
More

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.