Perfectoid spaces : lectures from the 2017 Arizona Winter School

313 Pages·2019·2.563 MB·English

by  Bhatt, Bhargav, Cais, Bryden

Mathematical
Surveys
and
Monographs
Volume 242
Perfectoid Spaces
Lectures from the 2017
Arizona Winter School
Bhargav Bhatt
Ana Caraiani
Kiran S. Kedlaya
Jared Weinstein
With an introduction by Peter Scholze
Bryden Cais, Editor
Perfectoid Spaces
Lectures from the 2017
Arizona Winter School

Mathematical
Surveys
and
Monographs
Volume 242
Perfectoid Spaces
Lectures from the 2017
Arizona Winter School
Bhargav Bhatt
Ana Caraiani
Kiran S. Kedlaya
Jared Weinstein
With an introduction by Peter Scholze
Bryden Cais, Editor
EDITORIAL COMMITTEE
Robert Guralnick, Chair Benjamin Sudakov
Natasa Sesum ConstantinTeleman
2010 Mathematics Subject Classification. Primary 11F80, 11G25,14C30, 14F30, 14F40,
14G20, 14G22,14G35, 14G40, 14L05.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-242
Library of Congress Cataloging-in-Publication Data
Names: Bhatt, Bhargav, 1983- | Cais, Bryden, editor. | Arizona Winter School on Perfectoid
Spaces(20th: 2017: Tucson,Ariz.)
Title: Perfectoid spaces : lectures from the 2017 Arizona Winter School / Bhargav Bhatt [and
threeothers];withanIntroductionbyPeterScholze;BrydenCais,editor.
Description: Providence, Rhode Island : AmericanMathematicalSociety, [2019]| Series: Math-
ematical surveys and monographs ; volume 242 | Prepared for the twentieth Arizona Winter
SchoolonPerfectoidSpaces,heldMarch11-17,2017,attheUniversityofArizonainTucson.
|Includesbibliographicalreferences.
Identifiers: LCCN2019016582|ISBN9781470450151(alk. paper)
Subjects: LCSH:Topologicalfields. |AMS:Numbertheory–Discontinuousgroupsandauto-
morphicforms–Galoisrepresentations. msc|Numbertheory–Arithmeticalgebraicgeometry
(Diophantinegeometry)–Varietiesoverfiniteandlocalfields. msc|Algebraicgeometry–Cy-
cles and subschemes – Transcendental methods, Hodge theory. msc | Algebraic geometry –
(Co)homology theory – p-adic cohomology, crystalline cohomology. msc | Algebraic geome-
try – (Co)homology theory – de Rham cohomology. msc | Algebraic geometry – Arithmetic
problems. Diophantine geometry – Local ground fields. msc | Algebraic geometry – Arith-
metic problems. Diophantine geometry– Rigid analytic geometry. msc | Algebraic geometry
– Arithmetic problems. Diophantine geometry – Modular and Shimura varieties. msc | Al-
gebraic geometry – Arithmetic problems. Diophantine geometry – Arithmetic varieties and
schemes; Arakelov theory; heights. msc | Algebraic geometry – Algebraic groups – Formal
groups,p-divisiblegroups. msc
Classification: LCCQA247.P44852019|DDC516.3/5–dc23
LCrecordavailableathttps://lccn.loc.gov/2019016582
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Contents
Preface
by Bryden Cais vii
Introduction
by Peter Scholze ix
Adic spaces
by Jared Weinstein 1
1. An introduction to adic spaces 1
2. Perfectoid fields 11
3. Perfectoid spaces and diamonds 22
4. Banach-Colmez spaces 32
5. Projects 39
References 42
Sheaves, stacks, and shtukas
by Kiran S. Kedlaya 45
1. Sheaves on analytic adic spaces 46
2. Perfectoid rings and spaces 93
3. Sheaves on Fargues–Fontaine curves 121
4. Shtukas 148
Appendix A. Project descriptions 179
References 184
TheHodge-Tatedecompositionviaperfectoidspaces
by Bhargav Bhatt 193
1. Lecture 1: Introduction 193
2. Lecture 2: The Hodge-Tate decomposition for abelian schemes 200
3. Lecture 3: The Hodge-Tate decomposition in general 203
4. Lecture 4: Integral aspects 218
5. Exercises 230
6. Projects 237
References 241
PerfectoidShimuravarieties
by Ana Caraiani 245
1. Introduction 245
2. Locally symmetric spaces and Shimura varieties 250
v
vi CONTENTS
3. Background from p-adic Hodge theory 265
4. The canonical subgroup and the anticanonical tower 270
5. Perfectoid Shimura varieties and the Hodge–Tate period morphism 279
6. The cohomology of locally symmetric spaces: conjectures and results 287
References 295
Preface
The Southwest Center for Arithmetic Geometry (SWC) was founded in 1997 by a
group of seven mathematicians working in the southwest United States, and has
been continuously supported by the National Science Foundation since that time.
In the beginning, the SWC served as a true center for Arithmetic Geometry in the
Southwest,butsurvivestodayinnameonly,havingbeensubsumedbyitsprincipal
activity, the Arizona Winter School (AWS).
The AWS was started with the ambitious goal of creating an intense and im-
mersive workshop in which graduate students—especially those who may not be
studying at traditional centers of number theory—would work under the guidance
of leading expertstosolve research-level problems at the forefrontof the field. The
very first school was held in the Spring of 1998 under the title “Workshop on Dio-
phantine GeometryRelatedtotheABCConjecture.” Inthetwenty-one yearsthat
have followed, the AWS has been held annually each March on a different topic
in arithmetic geometry and related areas, and has become a pillar of graduate
education and training in these subjects throughout the country and abroad.
Over the years, the Arizona Winter School model has been adjusted and re-
fined to meet the needs of an ever-growing and increasingly diverse audience: the
five-daymeeting, organizedaroundadifferentcentraltopiceachyear, nowfeatures
a set of four lecture series by leading and emerging experts. A month before the
school begins, each speaker proposes one or more research projects related to their
lectures, and is assigned 10–15 graduate students who will work on these projects.
At that time, speakers also provide detailed lecture notes for their courses. During
theschool, students attendlecturesdailyfrom9amto5pm, andwork eachevening
(oftenintotheearlyhoursofthemorning!) withspeakersanddesignatedassistants
on these research projects. Students not assigned to these research project groups
have the option to join one of two problem sessions devoted to solving advanced
exercises related to the lecture series, or one of four study groups which focus on
understanding the course lecture notes in detail; these additional activities are su-
pervisedbyyoungresearchersandallowstudentsnotassignedtooneoftheresearch
projects to meaningfully engage with the workshop material on many levels. On
the last day, the students from each research project group present their work to
the whole school. The result is an extremely focused and immersive five days of
mathematical activity for all.
This volume is comprised of the lecture notes which were prepared for the
twentieth Arizona Winter School on “Perfectoid Spaces,” held March 11–17, 2017
at the University of Arizona in Tucson, and attended by over 367 participants,
makingitthelargestWinterSchooltodate. ThespeakerswereBhargavBhatt,Ana
Caraiani, Kiran Kedlaya, and Jared Weinstein. We are greatly indebted to these
vii
viii PREFACE
authorsfortheirhardworkinmakingboththetwentiethAWSandthisproceedings
volume a reality. Peter Scholze gave opening and closing lectures at the Winter
School, whichprovidedhistorical background aswell asinsights intohispioneering
effortsandvisionsforthefutureofthesubject,andwethankhimmostheartilyfor
furnishing the introduction to this volume. We extend our sincere gratitude to the
anonymous reviewers for their expert and careful readings of the articles, and for
the valuable feedback they provided. We thank the National Science Foundation
for their longstanding and continued support of the Arizona Winter School, the
Clay Mathematics Institute for their partnership in organizing the 2017 AWS, and
the University of Arizona Department of Mathematics for their support. Finally,
we owe a great deal to the other members of the Southwest Center, both past and
present, for their effort, perseverance, and vision in running the Arizona Winter
School for more than twenty years, and for helping it to become the one-of-a-kind
workshop that it is.
Bryden Cais
Introduction
Peter Scholze
The Arizona Winter School 2017 was around the subject of perfectoid spaces.
These spaces belong to the world of p-adic geometry, and despite their rather com-
plicated definition and appearance, they have found applications in many different
areas of mathematics, including arithmetic geometry, number theory, representa-
tion theory, commutative algebra and algebraic topology.
The problem that motivated the author to introduce perfectoidspaces was the
weight-monodromy conjecture. This is a conjecture on the (cid:2)-adic cohomology of
projective smooth varieties X over a p-adic field K, whose analogue over an equal-
characteristic field F ((t)) was proved by Deligne, [Del80], as a consequence of
q
his work on the Weil conjectures. Let us recall the statement. For any integer i,
considerthe´etalecohomologygroupV =Hi (X ,Q )forsomeprime(cid:2)(cid:3)=p,which
´et K (cid:2)
is a representation of the absolute Galois group G = Gal(K/K). For simplicity,
K
let us fix an element Φ∈G projecting to the geometric Frobenius on the residue
K
field. Similarly to the case over a finite field, the element Φ acts via Weil numbers
on Hi (X ,Q ), but now different weights can appear:
´et K (cid:2)
(cid:2)2i
V = V ,
j
j=0
where Φ acts through Weil numbers of weight j on V , i.e. all eigenvalues α ∈ Q
j (cid:2)
of Φ on V are algebraic numbers whose complex absolute value is qj/2 under any
j
embedding Q (cid:4)→ C. This follows from de Jong’s alterations, [dJ96], and the
Rapoport-Zink spectral sequence, [RZ82].
On the other hand, there is a monodromy operator N : V → V given as the
logarithmoftheactionofinertia(usingGrothendieck’squasi-unipotencetheorem).
This satisfies NΦ=qΦN, and soN maps Vj into Vj−2.
Conjecture 1 (Deligne, [Del71]). For all j =0,...,i, the map Nj :V →
i+j
Vi−j is an isomorphism.
This is similar to the hard Lefschetz theorem, and is sometimes said to be
“Mirror dual” to it. Let us quickly explain what happens for an elliptic curve E.
If it has (potentially) good reduction, then N = 0, and V = V , as follows from
1
proper smooth base change and the Weil conjectures for the special fibre. If it has
(potentially)semistablereduction,thenN isnonzero,andsoasV is2-dimensional,
one must have V = V ⊕V with both V and V being 1-dimensional; and this is
2 0 0 2
indeed what happens, as can be deduced from an analysis of the N´eron model,
ix