Berkeley lectures on p-adic geometry:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Princeton and Oxford
Princeton University Press
2020
|
| Schriftenreihe: | Annals of mathematics studies
number 207 |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xx, 264 Seiten Illustrationen, Diagramme |
| ISBN: | 9780691202082 9780691202099 |
Internformat
MARC
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| 100 | 1 | |a Scholze, Peter |d 1987- |0 (DE-588)1143603133 |4 aut | |
| 245 | 1 | 0 | |a Berkeley lectures on p-adic geometry |c Peter Scholze and Jared Weinstein |
| 264 | 1 | |a Princeton and Oxford |b Princeton University Press |c 2020 | |
| 300 | |a xx, 264 Seiten |b Illustrationen, Diagramme | ||
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| 490 | 1 | |a Annals of mathematics studies |v number 207 | |
| 700 | 1 | |a Weinstein, Jared |0 (DE-588)1198156260 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-0-691-20215-0 |
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Datensatz im Suchindex
| _version_ | 1804181495877730304 |
|---|---|
| adam_text | Berkeley Lectures on p-adic
Geometry
Peter Scholze and Jared Weinstein
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2020
Universitäts- und
LandesbibSiothek
Darmstadt
Contents
Foreword ix
Lecture 1: Introduction 1
1 1 Motivation: Drinfeld, L Lafforgue, and V Lafforgue 1
1 2 The possibility of shtukas in mixed characteristic 4
Lecture 2: Adic spaces 7
2 1 Motivation: Formal schemes and their generic fibers 7
2 2 Huber rings 9
2 3 Continuous valuations 13
Lecture 3: Adic spaces II 17
3 1 Rational Subsets 17
3 2 Adic spaces 20
3 3 The role of A+ 20
3 4 Pre-adic spaces 21
Appendix: Pre-adic spaces 23
Lecture 4: Examples of adic spaces 27
4 1 Basic examples 27
4 2 Example: The adic open unit disc over Zp 29
4 3 Analytic points 32
Lecture 5: Complements on adic spaces 35
5 1 Adic morphisms 35
5 2 Analytic adic spaces 36
5 3 Cartier divisors 38
Lecture 6: Perfectoid rings 41
6 1 Perfectoid Rings 41
6 2 Tilting 43
6 3 Sousperfectoid rings 47
Lecture 7: Perfectoid spaces 49
7 1 Perfectoid spaces: Definition and tilting equivalence 49
7 2 Why do we study perfectoid spaces? 50
VI
CONTENTS
7 3 The equivalence of etale sites 50
7 4 Almost mathematics, after Faltings 52
7 5 The etale site 55
Lecture 8: Diamonds 56
8 1 Diamonds: Motivation 56
8 2 Pro-etale morphisms 57
8 3 Definition of diamonds 00
8 4 The example of Spd Qp 62
Lecture 9: Diamonds II 64
9 1 Complements on the pro-etale topology 64
9 2 Quasi-pro-etale morphisms 67
9 3 G-torsors 68
9 4 The diamond Spd Qp 69
Lecture 10: Diamonds associated with adic spaces 74
10 1 The functor X - X 74
10 2 Example: Rigid spaces 77
10 3 The underlying topological space of diamonds 79
10 4 The etale site of diamonds 80
Appendix: Cohomology of local systems 82
Lecture 11: Mixed-characteristic shtukas 90
11 1 The equal characteristic story: Drinfeld’s shtukas and local shtukas 90
11 2 The adic space “S x SpaZp:’ 91
11 3 Sections of (Sx SpaZp)^ — S 94
11 4 Definition of mixed-characteristic shtukas 95
Lecture 12: Shtukas with one leg 98
12 1 p-divisible groups over Oc 98
12 2 Shtukas with one leg and p-divisible groups: An overview 100
12 3 Shtukas with no legs, and ^-modules over the integral Robba ring 103
12 4 Shtukas with one leg, and BdR-m°dules 105
Lecture 13: Shtukas with one leg II 108
13 1 y is an adic space 108
13 2 The extension of shtukas over Xl 109
13 3 Full faithfulness 109
13 4 Essential surjectivity 111
13 5 The Fargues-Fontaine curve 112
Lecture 14: Shtukas with one leg III 115
14 1 Fargues’ theorem 115
14 2 Extending vector bundles over the closed point of Spec Alnf 116
14 3 Proof of Theorem 14 2 1 119
CONTENTS vii
14 4 Description of the functor “? 121
Appendix: Integral p-adic Hodge theory 123
14 6 Cohomology of rigid-analytic spaces 124
14 7 Cohomology of formal schemes 124
14 8 p-divisible groups 126
14 9 The results of [BMS18] 127
Lecture 15: Examples of diamonds 131
15 1 The self-product SpdQp x SpdQp 131
15 2 Banach-Colmez spaces 133
Lecture 16:Drinfeld’s lemma for diamonds 140
16 1 The failure of ni(X xY) = rri(A) x 7Ti(F) 140
16 2 Drinfeld’s lemma for schemes 141
16 3 Drinfeld’s lemma for diamonds 143
Lecture 17: The v-topology 149
17 1 The v-topology on Perfd 149
17 2 Small v-sheaves 152
17 3 Spatial v-sheaves 152
17 4 Morphisms of v-sheaves 155
Appendix: Dieudonne theory over perfectoid rings 158
Lecture 18: v-sheaves associated with perfect and formal schemes 161
18 1 Definition 161
18 2 Topological spaces 162
18 3 Perfect schemes 163
18 4 Formal schemes 167
Lecture 19: The B^-affine Grassmannian 169
19 1 Definition of the B^-affine Grassmannian 169
19 2 Schubert varieties 172
19 3 The Demazure resolution 173
19 4 Minuscule Schubert varieties 176
Appendix: fj-torsors 178
Lecture 20: Families of affine Grassmannians 182
20 1 The convolution affine Grassmannian 183
20 2 Over Spd Qp 184
20 3 Over Spd Zp 185
20 4 Over Spd Qp x x Spd Qp 186
20 5 Over Spd Zp x x Spd Zp 189
Lecture 21: Affine flag varieties 191
21 1 Over Fp 191
21 2 Over Zp 192
viii
21 3 Affine flag varieties for tori
21 4 Local models
21 5 Devissage
Appendix: Examples
21 7 An EL case
21 8 A PEL case
Lecture 22: Vector bundles and G-torsors
22 1 Vector bundles
22 2 Semicontinuity of the Newton polygon
22 3 The etale locus
22 4 Classification of G-torsors
22 5 Semicontinuity of the Newton point
22 6 Extending G-torsors
Lecture 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
Lecture 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
Lecture 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
CONTENTS
194
194
196
198
203
204
207
207
208
209
210
212
213
215
216
217
218
220
223
225
225
226
229
232
232
235
236
237
238
241
Index
|
| adam_txt |
Berkeley Lectures on p-adic
Geometry
Peter Scholze and Jared Weinstein
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2020
Universitäts- und
LandesbibSiothek
Darmstadt
Contents
Foreword ix
Lecture 1: Introduction 1
1 1 Motivation: Drinfeld, L Lafforgue, and V Lafforgue 1
1 2 The possibility of shtukas in mixed characteristic 4
Lecture 2: Adic spaces 7
2 1 Motivation: Formal schemes and their generic fibers 7
2 2 Huber rings 9
2 3 Continuous valuations 13
Lecture 3: Adic spaces II 17
3 1 Rational Subsets 17
3 2 Adic spaces 20
3 3 The role of A+ 20
3 4 Pre-adic spaces 21
Appendix: Pre-adic spaces 23
Lecture 4: Examples of adic spaces 27
4 1 Basic examples 27
4 2 Example: The adic open unit disc over Zp 29
4 3 Analytic points 32
Lecture 5: Complements on adic spaces 35
5 1 Adic morphisms 35
5 2 Analytic adic spaces 36
5 3 Cartier divisors 38
Lecture 6: Perfectoid rings 41
6 1 Perfectoid Rings 41
6 2 Tilting 43
6 3 Sousperfectoid rings 47
Lecture 7: Perfectoid spaces 49
7 1 Perfectoid spaces: Definition and tilting equivalence 49
7 2 Why do we study perfectoid spaces? 50
VI
CONTENTS
7 3 The equivalence of etale sites 50
7 4 Almost mathematics, after Faltings 52
7 5 The etale site 55
Lecture 8: Diamonds 56
8 1 Diamonds: Motivation 56
8 2 Pro-etale morphisms 57
8 3 Definition of diamonds 00
8 4 The example of Spd Qp 62
Lecture 9: Diamonds II 64
9 1 Complements on the pro-etale topology 64
9 2 Quasi-pro-etale morphisms 67
9 3 G-torsors 68
9 4 The diamond Spd Qp 69
Lecture 10: Diamonds associated with adic spaces 74
10 1 The functor X - X 74
10 2 Example: Rigid spaces 77
10 3 The underlying topological space of diamonds 79
10 4 The etale site of diamonds 80
Appendix: Cohomology of local systems 82
Lecture 11: Mixed-characteristic shtukas 90
11 1 The equal characteristic story: Drinfeld’s shtukas and local shtukas 90
11 2 The adic space “S x SpaZp:’ 91
11 3 Sections of (Sx SpaZp)^ — S 94
11 4 Definition of mixed-characteristic shtukas 95
Lecture 12: Shtukas with one leg 98
12 1 p-divisible groups over Oc 98
12 2 Shtukas with one leg and p-divisible groups: An overview 100
12 3 Shtukas with no legs, and ^-modules over the integral Robba ring 103
12 4 Shtukas with one leg, and BdR-m°dules 105
Lecture 13: Shtukas with one leg II 108
13 1 y is an adic space 108
13 2 The extension of shtukas over Xl 109
13 3 Full faithfulness 109
13 4 Essential surjectivity 111
13 5 The Fargues-Fontaine curve 112
Lecture 14: Shtukas with one leg III 115
14 1 Fargues’ theorem 115
14 2 Extending vector bundles over the closed point of Spec Alnf 116
14 3 Proof of Theorem 14 2 1 119
CONTENTS vii
14 4 Description of the functor “? 121
Appendix: Integral p-adic Hodge theory 123
14 6 Cohomology of rigid-analytic spaces 124
14 7 Cohomology of formal schemes 124
14 8 p-divisible groups 126
14 9 The results of [BMS18] 127
Lecture 15: Examples of diamonds 131
15 1 The self-product SpdQp x SpdQp 131
15 2 Banach-Colmez spaces 133
Lecture 16:Drinfeld’s lemma for diamonds 140
16 1 The failure of ni(X xY) = rri(A) x 7Ti(F) 140
16 2 Drinfeld’s lemma for schemes 141
16 3 Drinfeld’s lemma for diamonds 143
Lecture 17: The v-topology 149
17 1 The v-topology on Perfd 149
17 2 Small v-sheaves 152
17 3 Spatial v-sheaves 152
17 4 Morphisms of v-sheaves 155
Appendix: Dieudonne theory over perfectoid rings 158
Lecture 18: v-sheaves associated with perfect and formal schemes 161
18 1 Definition 161
18 2 Topological spaces 162
18 3 Perfect schemes 163
18 4 Formal schemes 167
Lecture 19: The B^-affine Grassmannian 169
19 1 Definition of the B^-affine Grassmannian 169
19 2 Schubert varieties 172
19 3 The Demazure resolution 173
19 4 Minuscule Schubert varieties 176
Appendix: fj-torsors 178
Lecture 20: Families of affine Grassmannians 182
20 1 The convolution affine Grassmannian 183
20 2 Over Spd Qp 184
20 3 Over Spd Zp 185
20 4 Over Spd Qp x x Spd Qp 186
20 5 Over Spd Zp x x Spd Zp 189
Lecture 21: Affine flag varieties 191
21 1 Over Fp 191
21 2 Over Zp 192
viii
21 3 Affine flag varieties for tori
21 4 Local models
21 5 Devissage
Appendix: Examples
21 7 An EL case
21 8 A PEL case
Lecture 22: Vector bundles and G-torsors
22 1 Vector bundles
22 2 Semicontinuity of the Newton polygon
22 3 The etale locus
22 4 Classification of G-torsors
22 5 Semicontinuity of the Newton point
22 6 Extending G-torsors
Lecture 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
Lecture 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
Lecture 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
CONTENTS
194
194
196
198
203
204
207
207
208
209
210
212
213
215
216
217
218
220
223
225
225
226
229
232
232
235
236
237
238
241
Index |
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| spelling | Scholze, Peter 1987- (DE-588)1143603133 aut Berkeley lectures on p-adic geometry Peter Scholze and Jared Weinstein Princeton and Oxford Princeton University Press 2020 xx, 264 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Annals of mathematics studies number 207 Weinstein, Jared (DE-588)1198156260 aut Erscheint auch als Online-Ausgabe 978-0-691-20215-0 Annals of mathematics studies number 207 (DE-604)BV000000991 207 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032152337&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Scholze, Peter 1987- Weinstein, Jared Berkeley lectures on p-adic geometry Annals of mathematics studies |
| title | Berkeley lectures on p-adic geometry |
| title_auth | Berkeley lectures on p-adic geometry |
| title_exact_search | Berkeley lectures on p-adic geometry |
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| title_full | Berkeley lectures on p-adic geometry Peter Scholze and Jared Weinstein |
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| title_short | Berkeley lectures on p-adic geometry |
| title_sort | berkeley lectures on p adic geometry |
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