PERFECTOID SPACES: LECTURES FROM THE 2017 ARIZONA WINTER SCHOOL

PERFECTOID SPACES: LECTURES FROM THE 2017 ARIZONA WINTER SCHOOL

Editorial:
AMS (AMERICAN MATHEMATICAL SOCIETY)
Año de edición:
Materia
Matematicas
ISBN:
978-1-4704-5015-1
Páginas:
297
N. de edición:
1
Idioma:
Inglés
Disponibilidad:
Disponibilidad inmediata

Descuento:

-5%

Antes:

152,00 €

Despues:

144,40 €

Cover1
Title page4
Contents6
Preface8
Introduction10

References12
Adic spaces14

1. An introduction to adic spaces14
2. Perfectoid fields24
3. Perfectoid spaces and diamonds35
4. Banach-Colmez spaces45
5. Projects52
References55
Sheaves, stacks, and shtukas58

1. Sheaves on analytic adic spaces59
2. Perfectoid rings and spaces106
3. Sheaves on Fargues–Fontaine curves134
4. Shtukas161
Appendix A. Project descriptions192
References197
The Hodge-Tate decomposition via perfectoid spaces206

1. Lecture 1: Introduction206
2. Lecture 2: The Hodge-Tate decomposition for abelian schemes213
3. Lecture 3: The Hodge-Tate decomposition in general216
4. Lecture 4: Integral aspects231
5. Exercises243
6. Projects250
References254
Perfectoid Shimura varieties258

1. Introduction258
2. Locally symmetric spaces and Shimura varieties263
3. Background from ??-adic Hodge theory278
4. The canonical subgroup and the anticanonical tower283
5. Perfectoid Shimura varieties and the Hodge–Tate period morphism292
6. The cohomology of locally symmetric spaces: conjectures and results300
References

Introduced by Peter Scholze in 2011, perfectoid spaces are a bridge between geometry in characteristic 0 and characteristic pp, 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 pp-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.

Authors
Bryden Cais: University of Arizona, Tucson, AZ,
Bhargav Bhatt: University of Michigan, Ann Arbor, MI,
Ana Caraiani: Imperial College, London, United Kingdom,
Kiran S. Kedlaya: University of California, San Diego, La Jolla, CA,
Peter Scholze: University of Bonn, Bonn, Germany,
Jared Weinstein: Boston University, Boston, MA