**Mathematical Surveys and Monographs**

Volume: 242;
2019;
297 pp;
Softcover

MSC: Primary 11; 14;

**Print ISBN: 978-1-4704-6510-0
Product Code: SURV/242.S**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

**Electronic ISBN: 978-1-4704-5411-1
Product Code: SURV/242.E**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

#### Supplemental Materials

# Perfectoid Spaces: Lectures from the 2017 Arizona Winter School

Share this page *Edited by *
*Bryden Cais*

Introduced by Peter Scholze in 2011, perfectoid
spaces are a bridge between geometry in characteristic 0 and
characteristic \(p\), and have been used to solve many
important problems, including cases of the weight-monodromy conjecture
and the association of Galois representations to torsion classes in
cohomology. In recognition of the transformative impact perfectoid
spaces have had on the field of arithmetic geometry, Scholze was
awarded a Fields Medal in 2018.

This book, originating from a series of lectures given at the 2017
Arizona Winter School on perfectoid spaces, provides a broad
introduction to the subject. After an introduction with insight into
the history and future of the subject by Peter Scholze, Jared
Weinstein gives a user-friendly and utilitarian account of the theory
of adic spaces. Kiran Kedlaya further develops the foundational
material, studies vector bundles on Fargues–Fontaine curves, and
introduces diamonds and shtukas over them with a view toward the local
Langlands correspondence. Bhargav Bhatt explains the application of
perfectoid spaces to comparison isomorphisms in \(p\)-adic Hodge theory.
Finally, Ana Caraiani explains the application of perfectoid spaces to
the construction of Galois representations associated to torsion
classes in the cohomology of locally symmetric spaces for the general
linear group.

This book will be an invaluable asset for any graduate student or
researcher interested in the theory of perfectoid spaces and their
applications.

#### Readership

Graduate students and researchers interested in new developments in algebraic geometry and algebraic number theory.

#### Table of Contents

# Table of Contents

## Perfectoid Spaces: Lectures from the 2017 Arizona Winter School

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Introduction ix10
- Adic spaces 114
- Sheaves, stacks, and shtukas 4558
- The Hodge-Tate decomposition via perfectoid spaces 193206
- Perfectoid Shimura varieties 245258
- 1. Introduction 245258
- 2. Locally symmetric spaces and Shimura varieties 250263
- 3. Background from 𝑝-adic Hodge theory 265278
- 4. The canonical subgroup and the anticanonical tower 270283
- 5. Perfectoid Shimura varieties and the Hodge–Tate period morphism 279292
- 6. The cohomology of locally symmetric spaces: conjectures and results 287300
- References 295308

- Back Cover Back Cover1313