Contents
Foreword
1 Introduction
1.1 Motivation: Drinfeld, L. Lafforgue, and V. Lafforgue
1.2 The possibility of shtukas in mixed characteristic
2 Adic spaces
2.1 Motivation: Formal schemes and their generic fibers
2.2 Huber rings
2.3 Continuous valuations
3 Adic spaces II
3.1 Rational Subsets
3.2 Adic spaces
3.3 The role of A+
3.4 Pre-adic spaces
Appendix: Pre-adic spaces
4 Examples of adic spaces
4.1 Basic examples
4.2 Example: The adic open unit disc over Zp
4.3 Analytic points
5 Complements on adic spaces
5.1 Adic morphisms
5.2 Analytic adic spaces
5.3 Cartier divisors
6 Perfectoid rings
6.1 Perfectoid Rings
6.2 Tilting
6.3 Sousperfectoid rings
7 Perfectoid spaces
7.1 Perfectoid spaces: Definition and tilting equivalence
7.2 Why do we study perfectoid spaces?
7.3 The equivalence of รฉtale sites
7.4 Almost mathematics, after Faltings
7.5 The รฉtale site
8 Diamonds
8.1 Diamonds: Motivation
8.2 Pro-รฉtale morphisms
8.3 Definition of diamonds
8.4 The example of 3942"613A``4547"603ASpdQp
9 Diamonds II
9.1 Complements on the pro-รฉtale topology
9.2 Quasi-pro-รฉtale morphisms
9.3 G-torsors
9.4 The diamond3942"613A45`47`"603ASpdQp
10 Diamonds associated with adic spaces
10.1 The functor XX
10.2 Example: Rigid spaces
10.3 The underlying topological space of diamonds
10.4 The รฉtale site of diamonds
Appendix: Cohomology of local systems
11 Mixed-characteristic shtukas
11.1 The equal characteristic story: Drinfeld's shtukas and local shtukas
11.2 The adic spaceS3942"613A``4547"603ASpaZp''
11.3 Sections of (S3942"613A45`47`"603ASpaZp)S
11.4 Definition of mixed-characteristic shtukas
12 Shtukas with one leg
12.1 p-divisible groups over OC
12.2 Shtukas with one leg and p-divisible groups: An overview
12.3 Shtukas with no legs, and -modules over the integral Robba ring
12.4 Shtukas with one leg, and BdR-modules
13 Shtukas with one leg II
13.1 Y is an adic space
13.2 The extension of shtukas over xL
13.3 Full faithfulness
13.4 Essential surjectivity
13.5 The Fargues-Fontaine curve
14 Shtukas with one leg III
14.1 Fargues' theorem
14.2 Extending vector bundles over the closed point of `39`42`"613A4547"603ASpecAinf
14.3 Proof of Theorem 14.2.1
14.4 Description of the functor ?''
Appendix: Integral p-adic Hodge theory
14.6 Cohomology of rigid-analytic spaces
14.7 Cohomology of formal schemes
14.8 p-divisible groups
14.9 The results of BMS
15 Examples of diamonds
15.1 The self-product `39`42`"613A4547"603ASpdQp3942"613A``4547"603ASpdQp
15.2 Banach-Colmez spaces
16 Drinfeld's lemma for diamonds
16.1 The failure of 1(XY)=1(X)1(Y)
16.2 Drinfeld's lemma for schemes
16.3 Drinfeld's lemma for diamonds
17 The v-topology
17.1 The v-topology on3942"613A45`47`"603APerfd
17.2 Small v-sheaves
17.3 Spatial v-sheaves
17.4 Morphisms of v-sheaves
Appendix: Dieudonnรฉ theory over perfectoid rings
18 v-sheaves associated with perfect and formal schemes
18.1 Definition
18.2 Topological spaces
18.3 Perfect schemes
18.4 Formal schemes
19 The BdR+-affine Grassmannian
19.1 Definition of the BdR+-affine Grassmannian
19.2 Schubert varieties
19.3 The Demazure resolution
19.4 Minuscule Schubert varieties
Appendix: G-torsors
20 Families of affine Grassmannians
20.1 The convolution affine Grassmannian
20.2 Over `39`42`"613A4547"603ASpdQp
20.3 Over 3942"613A``4547"603ASpdZp
20.4 Over3942"613A45`47`"603ASpdQpโฆ`39`42`"613A4547"603ASpdQp
20.5 Over 3942"613A``4547"603ASpdZpโฆ3942"613A`4547`"603ASpdZp
21 Affine flag varieties
21.1 Over Fp
21.2 Over Zp
21.3 Affine flag varieties for tori
21.4 Local models
21.5 Dรฉvissage
Appendix: Examples
21.7 An EL case
21.8 A PEL case
22 Vector bundles and G-torsors
22.1 Vector bundles
22.2 Semicontinuity of the Newton polygon
22.3 The รฉtale locus
22.4 Classification of G-torsors
22.5 Semicontinuity of the Newton point
22.6 Extending G-torsors
23 Moduli spaces of shtukas
23.1 Definition of mixed-characteristic local shtukas
23.2 The case of no legs
23.3 The case of one leg
23.4 The case of two legs
23.5 The general case
24 Local Shimura varieties
24.1 Definition of local Shimura varieties
24.2 Relation to Rapoport-Zink spaces
24.3 General EL and PEL data
25 Integral models of local Shimura varieties
25.1 Definition of the integral models
25.2 The case of tori
25.3 Non-parahoric groups
25.4 The EL case
25.5 The PEL case
Bibliography
Index