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## Perfectoid spaces

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- 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

## Diamonds

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0008
- Subject:
- Mathematics, Geometry / Topology

This chapter investigates the notion of a diamond. The idea is that there should be a functor which “forgets the structure morphism to Zp.” The desired quotient in the example provided in the chapter ... More

## Diamonds II

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0009
- Subject:
- Mathematics, Geometry / Topology

This chapter evaluates the complements on the pro-étale topology. It addresses two issues raised in the previous lecture on the pro-étale topology. The first issue concerned descent, or more ... More

## Perfectoid rings

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- 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

## Berkeley Lectures on p-adic Geometry: (AMS-207)

*Peter Scholze and Jared Weinstein*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.001.0001
- Subject:
- Mathematics, Geometry / Topology

This book presents an important breakthrough in arithmetic geometry. In 2014, this book's author delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory ... More

## Examples of diamonds

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0015
- Subject:
- Mathematics, Geometry / Topology

This chapter assesses some interesting examples of diamonds. So far, the only example encountered is the self-product of copies of Spd Qp. The chapter first studies this self-product. It is useful to ... More

## Introduction

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0001
- Subject:
- Mathematics, Geometry / Topology

This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case ... More

## Mixed-characteristic shtukas

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0011
- Subject:
- Mathematics, Geometry / Topology

This chapter looks at mixed-characteristic shtukas. Much of the theory of mixed-characteristic shtukas is motivated by the structures appearing in (integral) p-adic Hodge theory. The chapter assesses ... More

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