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308.
4734
v5 [
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2017
Foundations of Rigid Geometry I
ArXiv version
Kazuhiro Fujiwara
Graduate School of MathematicsNagoya UniversityNagoya
464-8502
Japan
[email protected]
Fumiharu Kato
Department of MathematicsTokyo Institute of Technology
Tokyo 152-8551Japan
[email protected]
http://arxiv.org/abs/1308.4734v5
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To the memory of Professor Masayoshi Nagata
and Professor Masaki Maruyama
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Contents
Introduction 1
0 Preliminaries 211 Basic Languages . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 22
1.1 Sets and ordered sets . . . . . . . . . . . . . . . . . . .
. . . 221.1. (a) Sets . . . . . . . . . . . . . . . . . . . . . . .
. . . 221.1. (b) Ordered sets and order types . . . . . . . . . . .
. 22
1.2 Categories . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 231.2. (a) Conventions . . . . . . . . . . . . . . . . . . .
. . 231.2. (b) Frequently used categories . . . . . . . . . . . . .
231.2. (c) Functor category . . . . . . . . . . . . . . . . . . .
241.2. (d) Groupoids and discrete categories . . . . . . . . .
241.2. (e) Category associated to an ordered set . . . . . . .
24
1.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 241.3. (a) Definition and universal property . . . . . . . .
. 241.3. (b) Limits over ordered sets . . . . . . . . . . . . . . .
251.3. (c) Final and cofinal functors . . . . . . . . . . . . . .
25
1.4 Several stabilities for properties of arrows . . . . . . . .
. . 261.4. (a) Base-change stability . . . . . . . . . . . . . . .
. 261.4. (b) Topology associated to base-change stable sub-
category . . . . . . . . . . . . . . . . . . . . . . . 281.4.
(c) Stability and effective descent . . . . . . . . . . . 291.4.
(d) Categorical equivalence relations . . . . . . . . . . 30
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 312 General topology . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 33
2.1 Some general prerequisites . . . . . . . . . . . . . . . . .
. . 342.1. (a) Generization and specialization . . . . . . . . . .
. 342.1. (b) Sober spaces . . . . . . . . . . . . . . . . . . . . .
342.1. (c) Completely regular spaces . . . . . . . . . . . . . .
352.1. (d) Quasi-compact spaces and quasi-separated spaces 36
2.2 Coherent spaces . . . . . . . . . . . . . . . . . . . . . .
. . 362.2. (a) Definition and first properties . . . . . . . . . .
. 362.2. (b) Stone’s representation theorem . . . . . . . . . . .
37
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4 Contents
2.2. (c) Projective limit of coherent sober spaces . . . . .
392.2. (d) Locally coherent spaces . . . . . . . . . . . . . . .
45
2.3 Valuative spaces . . . . . . . . . . . . . . . . . . . . . .
. . 472.3. (a) Valuative spaces . . . . . . . . . . . . . . . . . .
. 472.3. (b) Closures and tubes . . . . . . . . . . . . . . . . . .
482.3. (c) Separated quotients and separation maps . . . . . 492.3.
(d) Overconvergent sets . . . . . . . . . . . . . . . . . 502.3.
(e) Valuative maps . . . . . . . . . . . . . . . . . . . . 522.3.
(f) Structure of separated quotients . . . . . . . . . . 532.3. (g)
Overconvergent interior . . . . . . . . . . . . . . . 54
2.4 Reflexive valuative spaces . . . . . . . . . . . . . . . . .
. . 552.4. (a) Reflexive valuative spaces . . . . . . . . . . . . .
. 552.4. (b) Reflexivization . . . . . . . . . . . . . . . . . . .
. 562.4. (c) Coherent case . . . . . . . . . . . . . . . . . . . .
562.4. (d) General case . . . . . . . . . . . . . . . . . . . . .
58
2.5 Locally strongly compact valuative spaces . . . . . . . . .
. 592.5. (a) Locally strongly compact valuative spaces . . . . .
592.5. (b) Characteristic properties . . . . . . . . . . . . . .
612.5. (c) Paracompact spaces . . . . . . . . . . . . . . . . .
62
2.6 Valuations of locally Hausdorff spaces . . . . . . . . . . .
. 642.6. (a) Nets and coverings . . . . . . . . . . . . . . . . . .
642.6. (b) Valuations of compact spaces . . . . . . . . . . . .
652.6. (c) Valuations of locally Hausdorff spaces . . . . . . .
672.6. (d) Saturation and associated valuations . . . . . . . .
682.6. (e) Reflexive locally strongly compact valuative spaces
70
2.7 Some generalities on topoi . . . . . . . . . . . . . . . . .
. . 712.7. (a) Spacial topoi . . . . . . . . . . . . . . . . . . .
. . 712.7. (b) Points . . . . . . . . . . . . . . . . . . . . . . .
. . 722.7. (c) Localic topoi . . . . . . . . . . . . . . . . . . .
. . 722.7. (d) Coherent topoi . . . . . . . . . . . . . . . . . . .
. 732.7. (e) Fibered topoi and projective limits . . . . . . . . .
742.7. (f) Projective limit of spacial topoi . . . . . . . . . .
762.7. (g) Quasi-compact topoi and projective limits . . . . 78
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 783 Homological algebra . . . . . . . . . . . . . . . . .
. . . . . . . . . 81
3.1 Inductive limits . . . . . . . . . . . . . . . . . . . . . .
. . . 813.1. (a) Preliminaries . . . . . . . . . . . . . . . . . .
. . . 813.1. (b) Inductive limits and Noetherness . . . . . . . . .
. 833.1. (c) Inductive limit of sheaves . . . . . . . . . . . . . .
843.1. (d) Sheaves on limit spaces . . . . . . . . . . . . . . .
863.1. (e) Canonical flasque resolution . . . . . . . . . . . .
893.1. (f) Inductive limit and cohomology . . . . . . . . . .
903.1. (g) Cohomology of sheaves on limit spaces . . . . . . 91
3.2 Projective limits . . . . . . . . . . . . . . . . . . . . .
. . . 92
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Contents 5
3.2. (a) The Mittag-Leffler condition . . . . . . . . . . . .
923.2. (b) Canonical strict resolution . . . . . . . . . . . . .
943.2. (c) Projective limit of sheaves . . . . . . . . . . . . .
953.2. (d) Canonical s-flasque resolution . . . . . . . . . . . .
963.2. (e) Projective limit and cohomology . . . . . . . . . .
98
3.3 Coherent rings and modules . . . . . . . . . . . . . . . . .
. 102Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 104
4 Ringed spaces . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 1054.1 Generalities . . . . . . . . . . . . . . . . . .
. . . . . . . . . 105
4.1. (a) Ringed spaces and locally ringed spaces . . . . . .
1054.1. (b) Generization map . . . . . . . . . . . . . . . . . .
1064.1. (c) Sheaves of modules . . . . . . . . . . . . . . . . .
1064.1. (d) Cohesive ringed spaces . . . . . . . . . . . . . . .
1084.1. (e) Filtered projective limit of ringed spaces . . . . .
108
4.2 Sheaves on limit spaces . . . . . . . . . . . . . . . . . .
. . 1094.2. (a) Finitely presented sheaves on limit spaces . . . .
. 1094.2. (b) Limits and direct images . . . . . . . . . . . . . .
112
4.3 Cohomologies of sheaves on ringed spaces . . . . . . . . . .
1144.3. (a) Derived category formalism . . . . . . . . . . . . .
1144.3. (b) Calculation of cohomologies . . . . . . . . . . . . .
1144.3. (c) Module structures on cohomologies . . . . . . . .
115
4.4 Cohomologies of module sheaves on limit spaces . . . . . .
116Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 118
5 Schemes and algebraic spaces . . . . . . . . . . . . . . . . .
. . . . 1185.1 Schemes . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 119
5.1. (a) Schemes . . . . . . . . . . . . . . . . . . . . . . . .
1195.1. (b) Universally cohesive schemes . . . . . . . . . . . .
119
5.2 Algebraic spaces . . . . . . . . . . . . . . . . . . . . . .
. . 1205.2. (a) Conventions . . . . . . . . . . . . . . . . . . . .
. 1205.2. (b) Basic notions . . . . . . . . . . . . . . . . . . . .
. 1215.2. (c) Universally cohesive algebraic spaces . . . . . . . .
121
5.3 Derived category calculus . . . . . . . . . . . . . . . . .
. . 1225.3. (a) Quasi-coherent sheaves on affine schemes . . . . .
1225.3. (b) Permanence of coherency . . . . . . . . . . . . . .
123
5.4 Cohomology of quasi-coherent sheaves . . . . . . . . . . . .
1245.4. (a) Cohomologies on affine schemes . . . . . . . . . .
1245.4. (b) Some finiteness results . . . . . . . . . . . . . . . .
1265.4. (c) Cohomologies on projective spaces . . . . . . . . .
1265.4. (d) Ample and very ample sheaves . . . . . . . . . . .
127
5.5 More basics on algebraic spaces . . . . . . . . . . . . . .
. . 1285.5. (a) The stratification by subschemes . . . . . . . . .
. 1285.5. (b) Affineness criterion . . . . . . . . . . . . . . . .
. 1285.5. (c) Limit theorem . . . . . . . . . . . . . . . . . . . .
129
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 130
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6 Contents
6 Valuation rings . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 1306.1 Prerequisites . . . . . . . . . . . . . . . . .
. . . . . . . . . 131
6.1. (a) Totally ordered commutative group . . . . . . . .
1316.1. (b) Invertible ideals . . . . . . . . . . . . . . . . . . .
133
6.2 Valuation rings and valuations . . . . . . . . . . . . . . .
. 1346.2. (a) Valuation rings . . . . . . . . . . . . . . . . . . .
. 1346.2. (b) Valuations . . . . . . . . . . . . . . . . . . . . .
. 1356.2. (c) Height and rational rank of valuation rings . . . .
136
6.3 Spectrum of valuation rings . . . . . . . . . . . . . . . .
. . 1376.3. (a) General description . . . . . . . . . . . . . . . .
. 1376.3. (b) Valuation rings of finite height . . . . . . . . . .
. 1386.3. (c) Non-archimedean norms . . . . . . . . . . . . . . .
138
6.4 Composition and decomposition of valuation rings . . . . .
1396.5 Center of a valuation and height estimates for
Noetherian
domains . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 1416.6 Examples of valuation rings . . . . . . . . . . . . . . .
. . . 142
6.6. (a) Divisorial valuations . . . . . . . . . . . . . . . . .
1426.6. (b) The case dim(R) = 1 . . . . . . . . . . . . . . . .
1426.6. (c) The case dim(R) = 2 . . . . . . . . . . . . . . . .
142
6.7 a-adically separated valuation rings . . . . . . . . . . . .
. . 143Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 145
7 Topological rings and modules . . . . . . . . . . . . . . . .
. . . . 1457.1 Topology defined by a filtration . . . . . . . . . .
. . . . . . 146
7.1. (a) Filtrations . . . . . . . . . . . . . . . . . . . . . .
1467.1. (b) Topology defined by a filtration . . . . . . . . . .
1467.1. (c) Hausdorff completion . . . . . . . . . . . . . . . .
1487.1. (d) Hausdorff completion and exact sequence . . . . .
1507.1. (e) Completeness of sub and quotient modules . . . .
152
7.2 Adic topology . . . . . . . . . . . . . . . . . . . . . . .
. . . 1527.2. (a) Adic filtration and adic topology . . . . . . . .
. . 1527.2. (b) I-adic completion . . . . . . . . . . . . . . . . .
. 1557.2. (c) Criterion for adicness . . . . . . . . . . . . . . .
. 1577.2. (d) Existence of I-adic completions . . . . . . . . . . .
160
7.3 Henselian rings and Zariskian rings . . . . . . . . . . . .
. . 1617.3. (a) Henselian rings . . . . . . . . . . . . . . . . . .
. . 1617.3. (b) Zariskian rings . . . . . . . . . . . . . . . . . .
. . 1627.3. (c) Interrelation of the conditions . . . . . . . . . .
. 163
7.4 Preservation of adicness . . . . . . . . . . . . . . . . . .
. . 1647.4. (a) General observation . . . . . . . . . . . . . . . .
. 1647.4. (b) I-adicness of quotient topologies . . . . . . . . . .
1657.4. (c) I-adicness of subspace topologies . . . . . . . . . .
1667.4. (d) Useful consequences of the conditions . . . . . . .
169
7.5 Rees algebra and I-goodness . . . . . . . . . . . . . . . .
. 171Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 172
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Contents 7
8 Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 1748.1 Pairs . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 175
8.1. (a) Generalities . . . . . . . . . . . . . . . . . . . . .
. 1758.1. (b) Pairs of finite ideal type . . . . . . . . . . . . .
. . 1758.1. (c) Torsions and saturation . . . . . . . . . . . . . .
. 176
8.2 Bounded torsion condition and preservation of adicness . .
1788.2. (a) Bounded torsion condition . . . . . . . . . . . . .
1788.2. (b) Preservation of adicness . . . . . . . . . . . . . . .
1798.2. (c) The properties (BT) and (AP) . . . . . . . . . .
1818.2. (d) Bounded torsion condition for complete pairs . . .
183
8.3 Pairs and flatness . . . . . . . . . . . . . . . . . . . . .
. . . 1838.3. (a) Gluing of flatness . . . . . . . . . . . . . . .
. . . 1838.3. (b) Local criterion of flatness . . . . . . . . . . .
. . . 1858.3. (c) Formal fpqc descent of ‘Noetherian outside I’ . .
. 187
8.4 Restricted formal power series ring . . . . . . . . . . . .
. . 1888.5 Adhesive pairs . . . . . . . . . . . . . . . . . . . . .
. . . . 191
8.5. (a) Adhesive pairs and universally adhesive pairs . . .
1918.5. (b) Some examples . . . . . . . . . . . . . . . . . . . .
1958.5. (c) Preservation of adicness . . . . . . . . . . . . . . .
1968.5. (d) Topologically universally adhesive pairs . . . . . .
1978.5. (e) Adhesiveness and coherence . . . . . . . . . . . . .
198
8.6 Scheme-theoretic pairs . . . . . . . . . . . . . . . . . . .
. . 1998.7 I-valuative rings . . . . . . . . . . . . . . . . . . .
. . . . . 201
8.7. (a) I-valuative rings . . . . . . . . . . . . . . . . . . .
2018.7. (b) Structure theorem . . . . . . . . . . . . . . . . . .
2038.7. (c) Patching method . . . . . . . . . . . . . . . . . . .
205
8.8 Pairs and complexes . . . . . . . . . . . . . . . . . . . .
. . 2108.8. (a) Set-up . . . . . . . . . . . . . . . . . . . . . .
. . . 2108.8. (b) Results in case I is finitely generated . . . . .
. . 2108.8. (c) Results in case I is principal . . . . . . . . . .
. . 213
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 2159 Topological algebras of type (V) . . . . . . . . . .
. . . . . . . . . 216
9.1 a-adic completion of valuation rings . . . . . . . . . . . .
. 2179.1. (a) Fundamental structure theorem . . . . . . . . . .
2179.1. (b) Proof of Theorem 9.1.1 . . . . . . . . . . . . . . .
2189.1. (c) Corollaries . . . . . . . . . . . . . . . . . . . . . .
220
9.2 Topologically finitely generated V -algebras . . . . . . . .
. 2219.2. (a) Adhesiveness . . . . . . . . . . . . . . . . . . . .
. 2219.2. (b) Noether normalization . . . . . . . . . . . . . . . .
224
9.3 Classical affinoid algebras . . . . . . . . . . . . . . . .
. . . 2269.3. (a) Tate algebra and classical affinoid algebras . .
. . 2269.3. (b) Ring-theoretic properties . . . . . . . . . . . . .
. 228
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 230A Appendix: Further techniques for topologically of
finite type algebras230
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8 Contents
A.1 Nagata’s idealization trick . . . . . . . . . . . . . . . .
. . . 230A.2 Standard basis and division algorithm . . . . . . . .
. . . . 231
A.2. (a) Setting . . . . . . . . . . . . . . . . . . . . . . . .
231A.2. (b) Division algorithm . . . . . . . . . . . . . . . . . .
232A.2. (c) Standard bases . . . . . . . . . . . . . . . . . . . .
233
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 234B Appendix: f-adic rings . . . . . . . . . . . . . . .
. . . . . . . . . . 235
B.1 f-adic rings . . . . . . . . . . . . . . . . . . . . . . . .
. . . 235B.1. (a) Extension of adic topologies . . . . . . . . . .
. . 235B.1. (b) f-adic rings . . . . . . . . . . . . . . . . . . .
. . . 235B.1. (c) Extremal f-adic rings . . . . . . . . . . . . . .
. . 237B.1. (d) Complete f-adic rings . . . . . . . . . . . . . . .
. 238B.1. (e) Banach f-adic rings and classical affinoid algebras
239
B.2 Modules over f-adic rings . . . . . . . . . . . . . . . . .
. . 240B.2. (a) Topological modules . . . . . . . . . . . . . . . .
. 240B.2. (b) Open mapping theorem . . . . . . . . . . . . . . .
241
C Appendix: Addendum on derived category . . . . . . . . . . . .
. . 242C.1 Prerequisites on triangulated categories . . . . . . . .
. . . 242C.2 The category of complexes . . . . . . . . . . . . . .
. . . . . 243
C.2. (a) Definitions . . . . . . . . . . . . . . . . . . . . . .
243C.2. (b) Shifts . . . . . . . . . . . . . . . . . . . . . . . .
. 244C.2. (c) Cohomology functor . . . . . . . . . . . . . . . . .
244C.2. (d) Truncations . . . . . . . . . . . . . . . . . . . . . .
245
C.3 The triangulated category K(A ) . . . . . . . . . . . . . .
. 245C.3. (a) Homotopies . . . . . . . . . . . . . . . . . . . . .
. 245C.3. (b) Mapping cones . . . . . . . . . . . . . . . . . . . .
246
C.4 The derived category D(A ) . . . . . . . . . . . . . . . . .
. 247C.4. (a) Definition and first properties . . . . . . . . . . .
247C.4. (b) Canonical cohomology functor and canonical t-
structure . . . . . . . . . . . . . . . . . . . . . . . 248C.4.
(c) Representation by complexes and amplitude . . . 249
C.5 Subcategories of D(A ) . . . . . . . . . . . . . . . . . . .
. 250
I Formal geometry 2521 Formal schemes . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 253
1.1 Formal schemes and ideals of definition . . . . . . . . . .
. 2541.1. (a) Admissible rings . . . . . . . . . . . . . . . . . .
. 2541.1. (b) Formal spectrum . . . . . . . . . . . . . . . . . . .
2571.1. (c) Formal schemes . . . . . . . . . . . . . . . . . . .
2581.1. (d) Ideals of definition . . . . . . . . . . . . . . . . .
. 2591.1. (e) Noetherian formal schemes . . . . . . . . . . . . .
261
1.2 Fiber products . . . . . . . . . . . . . . . . . . . . . . .
. . 2611.2. (a) Complete tensor product of admissible rings . . .
2611.2. (b) Fiber products of formal schemes . . . . . . . . .
263
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1.2. (c) Fiber products and open immersions . . . . . . . .
2631.3 Adic morphisms . . . . . . . . . . . . . . . . . . . . . . .
. 264
1.3. (a) Adic morphisms . . . . . . . . . . . . . . . . . . .
2641.3. (b) Adicness of diagonal maps . . . . . . . . . . . . .
266
1.4 Formal completion . . . . . . . . . . . . . . . . . . . . .
. . 2661.4. (a) Formal schemes as inductive limits of schemes . .
2661.4. (b) Formal completion of schemes . . . . . . . . . . .
2671.4. (c) Formal completion of quasi-coherent sheaves . . .
268
1.5 Categories of formal schemes . . . . . . . . . . . . . . . .
. 2691.5. (a) Notation . . . . . . . . . . . . . . . . . . . . . .
. 2691.5. (b) Properties of morphisms in Fs . . . . . . . . . . .
2701.5. (c) Properties of morphisms in AcFs . . . . . . . . .
2711.5. (d) Adicalization . . . . . . . . . . . . . . . . . . . . .
273
1.6 Quasi-compact and quasi-separated morphisms . . . . . . .
2741.6. (a) Quasi-compact morphisms and some preliminary
facts on diagonal morphisms . . . . . . . . . . . . 2741.6. (b)
Quasi-separatedmorphisms and coherent morphisms2751.6. (c) Notation
. . . . . . . . . . . . . . . . . . . . . . . 278
1.7 Morphisms of finite type . . . . . . . . . . . . . . . . . .
. . 278Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 280
2 Universally rigid-Noetherian formal schemes . . . . . . . . .
. . . . 2812.1 Universally rigid-Noetherian and universally
adhesive for-
mal schemes . . . . . . . . . . . . . . . . . . . . . . . . . .
. 2822.1. (a) T.u. rigid-Noetherian rings and t.u. adhesive rings
2822.1. (b) Universally adhesive and universally
rigid-Noetherian
formal schemes . . . . . . . . . . . . . . . . . . . . 2842.1.
(c) Categories of universally rigid-Noetherian formal
schemes . . . . . . . . . . . . . . . . . . . . . . . . 2852.2
Morphisms of finite presentation . . . . . . . . . . . . . . .
2852.3 Relation with other notions . . . . . . . . . . . . . . . .
. . 287
2.3. (a) Admissible formal schemes . . . . . . . . . . . . .
2872.3. (b) Interrelations between the classes . . . . . . . . .
288
3 Adically quasi-coherent sheaves . . . . . . . . . . . . . . .
. . . . . 2893.1 Complete sheaves and adically quasi-coherent
sheaves . . . 289
3.1. (a) Hausdorff completion of OX -modules . . . . . . .
2893.1. (b) Adically quasi-coherent (a.q.c.) sheaves . . . . . .
290
3.2 A.q.c. sheaves on affine formal schemes . . . . . . . . . .
. . 2923.2. (a) The ∆-sheaves . . . . . . . . . . . . . . . . . . .
. 2923.2. (b) Adically quasi-coherent ∆-sheaves . . . . . . . . .
293
3.3 A.q.c. algebras of finite type . . . . . . . . . . . . . . .
. . . 2973.4 A.q.c. sheaves as projective limits . . . . . . . . .
. . . . . . 2973.5 A.q.c. sheaves on locally universally
rigid-Noetherian for-
mal schemes . . . . . . . . . . . . . . . . . . . . . . . . . .
. 298
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3.5. (a) ∆-sheaves on affine universally rigid-Noetherianformal
schemes . . . . . . . . . . . . . . . . . . . . 298
3.5. (b) Adically quasi-coherent sheaves of finite presen-tation
. . . . . . . . . . . . . . . . . . . . . . . . . 299
3.5. (c) Adically quasi-coherent algebras of finite
presen-tation . . . . . . . . . . . . . . . . . . . . . . . . .
301
3.6 Complete pull-back of a.q.c. sheaves . . . . . . . . . . . .
. 3013.7 Admissible ideals . . . . . . . . . . . . . . . . . . . .
. . . . 303
3.7. (a) Pull-back of quasi-coherent sheaves on closed
sub-schemes . . . . . . . . . . . . . . . . . . . . . . . . 303
3.7. (b) Admissible ideals . . . . . . . . . . . . . . . . . . .
3053.7. (c) Extension of admissible ideals . . . . . . . . . . .
306
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 3084 Several properties of morphisms . . . . . . . . . .
. . . . . . . . . . 309
4.1 Affine morphisms . . . . . . . . . . . . . . . . . . . . . .
. . 3094.1. (a) Definition of affine morphisms . . . . . . . . . .
. 3094.1. (b) Affine adic morphisms and adically quasi-coherent
sheaves . . . . . . . . . . . . . . . . . . . . . . . . 3094.1.
(c) Formal spectra of a.q.c. algebras . . . . . . . . . . 3104.1.
(d) Basic properties of affine adic morphisms . . . . . 311
4.2 Finite morphisms . . . . . . . . . . . . . . . . . . . . . .
. . 3124.3 Closed immersions . . . . . . . . . . . . . . . . . . .
. . . . 314
4.3. (a) A preliminary result . . . . . . . . . . . . . . . . .
3144.3. (b) Definitions and first properties . . . . . . . . . . .
3154.3. (c) Universally rigid-Noetherian case . . . . . . . . . .
3184.3. (d) Closed immersions and admissible ideals . . . . .
318
4.4 Immersions . . . . . . . . . . . . . . . . . . . . . . . . .
. . 3194.5 Surjective, closed and universally closed morphisms . .
. . . 321
4.5. (a) Surjective morphisms . . . . . . . . . . . . . . . .
3214.5. (b) Closed and universally closed morphisms . . . . .
321
4.6 Separated morphisms . . . . . . . . . . . . . . . . . . . .
. 3234.6. (a) Definition and fundamental properties . . . . . . .
3234.6. (b) Separatedness and properties of morphisms . . . .
326
4.7 Proper morphisms . . . . . . . . . . . . . . . . . . . . . .
. 3274.8 Flat and faithfully flat morphisms . . . . . . . . . . . .
. . 328
4.8. (a) First properties of flatness . . . . . . . . . . . . .
3284.8. (b) Faithfully flat morphisms . . . . . . . . . . . . . .
3304.8. (c) Adically flat morphisms . . . . . . . . . . . . . . .
332
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 3345 Differential calculus on formal schemes . . . . . .
. . . . . . . . . . 335
5.1 Differential calculi for topological rings . . . . . . . . .
. . . 3355.1. (a) Continuous derivations . . . . . . . . . . . . .
. . 3355.1. (b) Differentials and canonical topology . . . . . . .
. 3355.1. (c) Completion and differentials . . . . . . . . . . . .
336
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Contents 11
5.1. (d) Differentials and finiteness conditions . . . . . . .
3395.2 Differential invariants on formal schemes . . . . . . . . .
. . 340
5.2. (a) The sheaf of differentials . . . . . . . . . . . . . .
3405.2. (b) Differentials on universally rigid-Noetherian for-
mal schemes . . . . . . . . . . . . . . . . . . . . . 3415.3
Étale and smooth morphisms . . . . . . . . . . . . . . . . .
342
5.3. (a) Neat morphisms . . . . . . . . . . . . . . . . . . .
3425.3. (b) Étale morphisms . . . . . . . . . . . . . . . . . . .
3445.3. (c) Smooth morphisms . . . . . . . . . . . . . . . . .
346
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 3486 Formal algebraic spaces . . . . . . . . . . . . . .
. . . . . . . . . . 348
6.1 Adically flat descent . . . . . . . . . . . . . . . . . . .
. . . 3496.1. (a) Basic assertions . . . . . . . . . . . . . . . .
. . . 3496.1. (b) Descent of morphisms . . . . . . . . . . . . . .
. . 3516.1. (c) Descent of properties of morphisms . . . . . . . .
3526.1. (d) Effective descent . . . . . . . . . . . . . . . . . . .
3536.1. (e) Adically flat descent and finiteness conditions . . .
355
6.2 Étale topology on adic formal schemes . . . . . . . . . . .
. 3556.2. (a) Étale sites . . . . . . . . . . . . . . . . . . . .
. . 3556.2. (b) Adically quasi-coherent sheaves on the étale site
. 358
6.3 Formal algebraic spaces . . . . . . . . . . . . . . . . . .
. . 3606.3. (a) Formal algebraic spaces . . . . . . . . . . . . . .
. 3606.3. (b) Formal algebraic spaces by quotients . . . . . . . .
3626.3. (c) Fiber products . . . . . . . . . . . . . . . . . . . .
3666.3. (d) Étale topology on formal algebraic spaces . . . . .
3666.3. (e) Ideal of definition and adic morphisms . . . . . . .
3676.3. (f) Formal completion of algebraic spaces . . . . . . .
3696.3. (g) Adically quasi-coherent sheaves on formal alge-
braic spaces . . . . . . . . . . . . . . . . . . . . . 3706.4
Several properties of morphisms . . . . . . . . . . . . . . .
3716.5 Universally adhesive and universally rigid-Noetherian
for-
mal algebraic spaces . . . . . . . . . . . . . . . . . . . . . .
374Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 375
7 Cohomology theory . . . . . . . . . . . . . . . . . . . . . .
. . . . . 3767.1 Cohomologies of adically quasi-coherent sheaves .
. . . . . . 3777.2 Coherent sheaves . . . . . . . . . . . . . . . .
. . . . . . . . 3787.3 Calculi in derived categories . . . . . . .
. . . . . . . . . . . 379Exercises . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 380
8 Finiteness theorem for proper algebraic spaces . . . . . . . .
. . . . 3808.1 Finiteness theorem: Announcement . . . . . . . . . .
. . . 3818.2 Generalized Serre’s theorem . . . . . . . . . . . . .
. . . . . 382
8.2. (a) Announcement . . . . . . . . . . . . . . . . . . . .
3828.2. (b) Reduction process . . . . . . . . . . . . . . . . . .
3828.2. (c) Proof of Proposition 8.2.2 . . . . . . . . . . . . . .
383
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8.3 The carving method . . . . . . . . . . . . . . . . . . . . .
. 3848.3. (a) The main assertion . . . . . . . . . . . . . . . . .
3848.3. (b) Preparation for the proof and carving lemma . . .
3858.3. (c) Proof of Proposition 8.3.1 . . . . . . . . . . . . . .
387
8.4 Proof of Theorem 8.1.3 . . . . . . . . . . . . . . . . . . .
. 3888.4. (a) Reduction process . . . . . . . . . . . . . . . . . .
3888.4. (b) End of the proof . . . . . . . . . . . . . . . . . . .
389
8.5 Application to I-goodness . . . . . . . . . . . . . . . . .
. . 389Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 390
9 GFGA comparison theorem . . . . . . . . . . . . . . . . . . .
. . . 3909.1 Announcement of the theorem . . . . . . . . . . . . .
. . . 391
9.1. (a) Formal completion functor . . . . . . . . . . . . .
3919.1. (b) The statement . . . . . . . . . . . . . . . . . . . .
392
9.2 The classical comparison theorem . . . . . . . . . . . . . .
. 3939.3 Proof of Theorem 9.1.3 . . . . . . . . . . . . . . . . . .
. . 396
9.3. (a) Reduction process . . . . . . . . . . . . . . . . . .
3969.3. (b) Projective case . . . . . . . . . . . . . . . . . . . .
397
9.4 Comparison of Ext modules . . . . . . . . . . . . . . . . .
. 39710 GFGA existence theorem . . . . . . . . . . . . . . . . . .
. . . . . 398
10.1 Announcement of the theorem. . . . . . . . . . . . . . . .
. 39810.2 Proof of Theorem 10.1.2 . . . . . . . . . . . . . . . . .
. . . 399
10.2. (a) Modification of the carving method . . . . . . . .
39910.2. (b) Reduction process . . . . . . . . . . . . . . . . . .
40010.2. (c) Projective case . . . . . . . . . . . . . . . . . . .
. 401
10.3 Applications . . . . . . . . . . . . . . . . . . . . . . .
. . . 404Exercises . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 405
11 Finiteness theorem and Stein factorization . . . . . . . . .
. . . . . 40511.1 Finiteness theorem for proper morphisms . . . . .
. . . . . 40611.2 Proof of Theorem 11.1.1 . . . . . . . . . . . . .
. . . . . . . 406
11.2. (a) Invertible ideal case . . . . . . . . . . . . . . . .
. 40611.2. (b) Blow-up case . . . . . . . . . . . . . . . . . . . .
. 40811.2. (c) General case . . . . . . . . . . . . . . . . . . . .
. 408
11.3 Stein factorization . . . . . . . . . . . . . . . . . . . .
. . . 40911.3. (a) Announcement of the theorem . . . . . . . . . .
. 40911.3. (b) Proof of Proposition 11.3.2 . . . . . . . . . . . .
. 41111.3. (c) Proof of Theorem 11.3.1 . . . . . . . . . . . . . .
. 412
A Appendix: Stein factorization for schemes . . . . . . . . . .
. . . . 413A.1 Pseudo-affine morphisms of schemes . . . . . . . . .
. . . . 413
A.1. (a) Definition and the first properties . . . . . . . . .
413A.1. (b) Pseudo-affineness and compactifications . . . . . .
414
A.2 Cohomological criterion . . . . . . . . . . . . . . . . . .
. . 416B Appendix: Zariskian schemes . . . . . . . . . . . . . . .
. . . . . . 417
B.1 Zariskian schemes . . . . . . . . . . . . . . . . . . . . .
. . 417B.1. (a) Zariskian rings and Zariskian spectra . . . . . . .
417
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Contents 13
B.1. (b) Zariskian schemes . . . . . . . . . . . . . . . . . .
419B.2 Fiber products . . . . . . . . . . . . . . . . . . . . . . .
. . 420B.3 Ideals of definition and adic morphisms . . . . . . . .
. . . 420B.4 Morphism of finite type and morphism of finite
presentation421
C Appendix: FP-approximated sheaves and GFGA theorems . . . . .
422C.1 Finiteness up to bounded torsion . . . . . . . . . . . . . .
. 422
C.1. (a) Weak isomorphisms . . . . . . . . . . . . . . . . .
422C.1. (b) Weakly finitely presented modules . . . . . . . . .
423
C.2 Global approximation by finitely presented sheaves . . . . .
423C.2. (a) FP-approximation of sheaves on schemes . . . . .
423C.2. (b) FP-approximation of sheaves on formal schemes . 426
C.3 Finiteness theorem and GFGA theorems . . . . . . . . . . .
427C.3. (a) Finiteness theorem for FP-approximated sheaves .
427C.3. (b) GFGA comparison theorem in rigid-Noetherian
situation . . . . . . . . . . . . . . . . . . . . . . . 428C.3.
(c) GFGA existence theorem in rigid-Noetherian sit-
uation . . . . . . . . . . . . . . . . . . . . . . . . .
429Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 429
II Rigid spaces 4301 Admissible blow-ups . . . . . . . . . . . .
. . . . . . . . . . . . . . 432
1.1 Admissible blow-ups . . . . . . . . . . . . . . . . . . . .
. . 4321.1. (a) Admissible blow-ups . . . . . . . . . . . . . . . .
. 4321.1. (b) Explicit local description . . . . . . . . . . . . .
. 4331.1. (c) Universal mapping property . . . . . . . . . . . .
4351.1. (d) Some basic properties . . . . . . . . . . . . . . . .
436
1.2 Strict transform . . . . . . . . . . . . . . . . . . . . . .
. . 4401.3 The cofiltered category of admissible blow-ups . . . . .
. . 443Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 444
2 Rigid spaces . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 4452.1 Coherent rigid spaces and their formal models .
. . . . . . 446
2.1. (a) Coherent rigid spaces . . . . . . . . . . . . . . . .
4462.1. (b) Formal models . . . . . . . . . . . . . . . . . . . .
4482.1. (c) Comma category CRfS . . . . . . . . . . . . . . 4492.1.
(d) Coherent universally Noetherian and universally
adhesive rigid spaces . . . . . . . . . . . . . . . . . 4502.2
Admissible topology and general rigid spaces . . . . . . . .
451
2.2. (a) Coherent admissible sites . . . . . . . . . . . . . .
4512.2. (b) Properties of coherent admissible sites . . . . . . .
4532.2. (c) General rigid space . . . . . . . . . . . . . . . . .
4552.2. (d) Universally Noetherian and universally adhesive
rigid spaces . . . . . . . . . . . . . . . . . . . . . . 4572.2.
(e) Admissible sites . . . . . . . . . . . . . . . . . . . 457
2.3 Morphism of finite type . . . . . . . . . . . . . . . . . .
. . 458
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14 Contents
2.4 Fiber products of rigid spaces . . . . . . . . . . . . . . .
. . 4592.5 Examples of rigid spaces . . . . . . . . . . . . . . . .
. . . . 460
2.5. (a) Rigid spaces of type (V) . . . . . . . . . . . . . . .
4602.5. (b) Rigid spaces of type (N) . . . . . . . . . . . . . . .
4602.5. (c) Unit disk over a rigid space . . . . . . . . . . . . .
4612.5. (d) Projective space over a rigid space . . . . . . . . .
461
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 4613 Visualization . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 461
3.1 Zariski-Riemann spaces . . . . . . . . . . . . . . . . . . .
. 4633.1. (a) Construction in coherent case . . . . . . . . . . . .
4633.1. (b) Functoriality . . . . . . . . . . . . . . . . . . . . .
4643.1. (c) Topological feature . . . . . . . . . . . . . . . . .
4643.1. (d) Non-emptiness . . . . . . . . . . . . . . . . . . . .
4653.1. (e) General construction . . . . . . . . . . . . . . . .
4663.1. (f) Connectedness . . . . . . . . . . . . . . . . . . . .
4673.1. (g) Notation . . . . . . . . . . . . . . . . . . . . . . .
467
3.2 Structure sheaves and local rings . . . . . . . . . . . . .
. . 4683.2. (a) Integral structure sheaf . . . . . . . . . . . . .
. . 4683.2. (b) Rigid structure sheaf . . . . . . . . . . . . . . .
. 4693.2. (c) Zariski-Riemann triple . . . . . . . . . . . . . . .
. 4713.2. (d) Reducedness . . . . . . . . . . . . . . . . . . . . .
4713.2. (e) Description of the local rings . . . . . . . . . . . .
4713.2. (f) Generization maps . . . . . . . . . . . . . . . . . .
473
3.3 Points on Zariski-Riemann spaces . . . . . . . . . . . . . .
. 4753.3. (a) Rigid points . . . . . . . . . . . . . . . . . . . .
. 4753.3. (b) Seminorms associated to points . . . . . . . . . . .
4783.3. (c) Spectral seminorms . . . . . . . . . . . . . . . . .
479
3.4 Comparison of topologies . . . . . . . . . . . . . . . . . .
. 4793.5 Finiteness conditions and consistency of terminologies . .
. 482
3.5. (a) Finiteness conditions . . . . . . . . . . . . . . . .
4823.5. (b) Consistency of open immersions . . . . . . . . . .
4833.5. (c) Rigid space as quotient . . . . . . . . . . . . . . .
4833.5. (d) Consistency of finiteness conditions . . . . . . . .
4833.5. (e) Rigid spaces associated to adic formal schemes . .
484
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 4844 Topological properties . . . . . . . . . . . . . . .
. . . . . . . . . . 485
4.1 Generization and specialization . . . . . . . . . . . . . .
. . 4864.2 Tubes . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 488
4.2. (a) Tubes . . . . . . . . . . . . . . . . . . . . . . . . .
4884.2. (b) Explicit description . . . . . . . . . . . . . . . . .
489
4.3 Separation map and overconvergent sets . . . . . . . . . . .
4914.3. (a) Separation map . . . . . . . . . . . . . . . . . . .
4914.3. (b) Overconvergent sets and tube subsets . . . . . . .
4924.3. (c) Overconvergent interior . . . . . . . . . . . . . . .
493
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Contents 15
4.4 Locally quasi-compact and paracompact rigid spaces . . . .
4934.4. (a) Locally quasi-compact rigid spaces . . . . . . . . .
4934.4. (b) Paracompact rigid spaces . . . . . . . . . . . . . .
494
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 4945 Coherent sheaves . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 495
5.1 Formal models of sheaves . . . . . . . . . . . . . . . . . .
. 4955.1. (a) The ‘rig’ functor for OX -modules . . . . . . . . . .
4955.1. (b) Formal models and lattice models . . . . . . . . .
4985.1. (c) Weak isomorphisms . . . . . . . . . . . . . . . . .
500
5.2 Existence of finitely presented formal models (weak version)
5015.3 Existence of finitely presented formal models (strong
version)5045.4 Integral models . . . . . . . . . . . . . . . . . .
. . . . . . . 506Exercises . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 508
6 Affinoids . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 5086.1 Affinoids and affinoid coverings . . . . . . .
. . . . . . . . . 509
6.1. (a) Definition and basic properties . . . . . . . . . . .
5096.1. (b) Affinoid subdomains . . . . . . . . . . . . . . . . .
511
6.2 Morphisms between affinoids . . . . . . . . . . . . . . . .
. 5126.3 Coherent sheaves on affinoids . . . . . . . . . . . . . .
. . . 5156.4 Comparison theorem for affinoids . . . . . . . . . . .
. . . . 5166.5 Stein affinoids . . . . . . . . . . . . . . . . . .
. . . . . . . . 518
6.5. (a) Stein affinoids and Stein affinoid coverings . . . .
5186.5. (b) Theorem A and Theorem B . . . . . . . . . . . . .
520
6.6 Associated schemes . . . . . . . . . . . . . . . . . . . . .
. . 5206.6. (a) Definition and functoriality . . . . . . . . . . .
. . 5206.6. (b) The comparison map . . . . . . . . . . . . . . . .
522
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 5237 Basic properties of morphisms of rigid spaces . . .
. . . . . . . . . 524
7.1 Quasi-compact and quasi-separated morphisms . . . . . . .
5247.2 Finite morphism . . . . . . . . . . . . . . . . . . . . . .
. . 5257.3 Closed immersions . . . . . . . . . . . . . . . . . . .
. . . . 528
7.3. (a) Definition and first properties . . . . . . . . . . .
5287.3. (b) Irreducible rigid spaces . . . . . . . . . . . . . . .
5327.3. (c) Open complement . . . . . . . . . . . . . . . . . .
5327.3. (d) Closed subspaces of an affinoid . . . . . . . . . . .
533
7.4 Immersions . . . . . . . . . . . . . . . . . . . . . . . . .
. . 5337.4. (a) Immersions and rigid subspaces . . . . . . . . . .
533
7.5 Separated morphisms and proper morphisms . . . . . . . .
5357.5. (a) Closed morphisms . . . . . . . . . . . . . . . . . .
5357.5. (b) Separated morphisms and proper morphisms . . . 5367.5.
(c) Valuative criterion . . . . . . . . . . . . . . . . . . 5407.5.
(d) Finiteness theorem . . . . . . . . . . . . . . . . . . 543
7.6 Projective morphisms . . . . . . . . . . . . . . . . . . . .
. 544Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 545
-
16 Contents
8 Classical points . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 5468.1 Spectral functor . . . . . . . . . . . . . . . .
. . . . . . . . 546
8.1. (a) Definitions . . . . . . . . . . . . . . . . . . . . . .
5468.1. (b) Continuity . . . . . . . . . . . . . . . . . . . . . .
5478.1. (c) Regularity . . . . . . . . . . . . . . . . . . . . . .
5498.1. (d) Density argument . . . . . . . . . . . . . . . . . .
549
8.2 Classical points . . . . . . . . . . . . . . . . . . . . . .
. . . 5508.2. (a) Point-like rigid spaces . . . . . . . . . . . . .
. . . 5508.2. (b) Structure of point-like rigid spaces . . . . . .
. . . 5508.2. (c) Classical points . . . . . . . . . . . . . . . .
. . . 5538.2. (d) Functoriality . . . . . . . . . . . . . . . . . .
. . . 5558.2. (e) Spectrality . . . . . . . . . . . . . . . . . . .
. . . 556
8.3 Noetherness theorem . . . . . . . . . . . . . . . . . . . .
. . 5598.3. (a) Comparison of complete local rings . . . . . . . .
5598.3. (b) Reducedness and irreducibility . . . . . . . . . . .
5608.3. (c) Noetherness theorem . . . . . . . . . . . . . . . .
561
9 GAGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 5639.1 Construction of GAGA functor . . . . . . . . . . .
. . . . . 563
9.1. (a) The category EmbX|S . . . . . . . . . . . . . . .
5639.1. (b) Construction of Xan . . . . . . . . . . . . . . . . .
5659.1. (c) Some basic properties of the GAGA functor . . . 5679.1.
(d) Some examples . . . . . . . . . . . . . . . . . . . . 5699.1.
(e) GAGA functor for non-separated schemes . . . . . 569
9.2 Affinoid valued points . . . . . . . . . . . . . . . . . . .
. . 5699.3 Comparison map and comparison functor . . . . . . . . .
. 572
9.3. (a) Comparison map . . . . . . . . . . . . . . . . . . .
5729.3. (b) Comparison functor . . . . . . . . . . . . . . . . .
573
9.4 GAGA comparison theorem . . . . . . . . . . . . . . . . . .
5739.5 GAGA existence theorem . . . . . . . . . . . . . . . . . . .
5769.6 Adic part for non-adic morphisms . . . . . . . . . . . . . .
577
9.6. (a) Adic part . . . . . . . . . . . . . . . . . . . . . . .
5789.6. (b) Functoriality . . . . . . . . . . . . . . . . . . . . .
5799.6. (c) Adic part for formally locally of finite type mor-
phisms . . . . . . . . . . . . . . . . . . . . . . . . 5809.6.
(d) Examples . . . . . . . . . . . . . . . . . . . . . . . 580
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 58110 Dimension of rigid spaces . . . . . . . . . . . . .
. . . . . . . . . . 582
10.1 Dimension of rigid spaces . . . . . . . . . . . . . . . . .
. . 58210.1. (a) Dimension . . . . . . . . . . . . . . . . . . . .
. . 58210.1. (b) Germs of rigid subspaces . . . . . . . . . . . . .
. 58310.1. (c) Dimension of rigid spaces of type (V) or of type
(N) . . . . . . . . . . . . . . . . . . . . . . . . . . 58510.1.
(d) Calculation of the dimension . . . . . . . . . . . . 58710.1.
(e) GAGA comparison of the dimensions . . . . . . . 588
-
Contents 17
10.1. (f) Dimension function . . . . . . . . . . . . . . . . .
59010.2 Codimension . . . . . . . . . . . . . . . . . . . . . . . .
. . 59110.3 Relative dimension . . . . . . . . . . . . . . . . . .
. . . . . 591
11 Maximum modulus principle . . . . . . . . . . . . . . . . . .
. . . 59211.1 Classification of points . . . . . . . . . . . . . .
. . . . . . . 592
11.1. (a) Basic inequality . . . . . . . . . . . . . . . . . . .
59211.1. (b) Divisorial points . . . . . . . . . . . . . . . . . .
. 59411.1. (c) Example: Unit disk . . . . . . . . . . . . . . . . .
594
11.2 Maximum modulus principle . . . . . . . . . . . . . . . . .
59711.2. (a) Spectral seminorm formula . . . . . . . . . . . . .
59711.2. (b) Maximum modulus principle . . . . . . . . . . . .
60011.2. (c) Reduction scheme . . . . . . . . . . . . . . . . . .
601
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 603A Appendix: Adic spaces . . . . . . . . . . . . . . .
. . . . . . . . . . 605
A.1 Triples . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 605A.2 Rigid f-adic rings . . . . . . . . . . . . . . . . .
. . . . . . . 607
A.2. (a) T.u. rigid-Noetherian f-adic rings . . . . . . . . . .
607A.2. (b) Finite type extensions . . . . . . . . . . . . . . . .
608A.2. (c) Rigidification of f-adic rings . . . . . . . . . . . .
. 608
A.3 Adic spaces . . . . . . . . . . . . . . . . . . . . . . . .
. . . 609A.3. (a) Affinoid rings . . . . . . . . . . . . . . . . .
. . . . 609A.3. (b) Adic spectrum . . . . . . . . . . . . . . . . .
. . . 610A.3. (c) Adic spaces . . . . . . . . . . . . . . . . . . .
. . . 611A.3. (d) Analytic adic spaces . . . . . . . . . . . . . .
. . . 612
A.4 Rigid geometry and affinoid rings . . . . . . . . . . . . .
. . 613A.4. (a) Affinoid rings associated to f-r-pairs . . . . . .
. . 613A.4. (b) Stein affinoids and analytic affinoid pairs . . . .
. 614A.4. (c) Visualization and adic spectrum . . . . . . . . . .
615A.4. (d) Description of power-bounded elements . . . . . .
617A.4. (e) Rigidification and finite type extensions . . . . . .
618A.4. (f) Analytic rings of type (N) . . . . . . . . . . . . . .
619A.4. (g) Canonical rigidifications of classical affinoid
alge-
bras . . . . . . . . . . . . . . . . . . . . . . . . . . 620A.5
Rigid geometry and adic spaces . . . . . . . . . . . . . . . .
622
B Appendix: Tate’s rigid analytic geometry . . . . . . . . . . .
. . . 623B.1 Admissibility . . . . . . . . . . . . . . . . . . . .
. . . . . . 623
B.1. (a) Admissibility with respect to a spectral functor . .
623B.1. (b) G-topology on a topological space . . . . . . . . .
625B.1. (c) G-topology associated to a spectral functor . . . .
625
B.2 Rigid analytic geometry . . . . . . . . . . . . . . . . . .
. . 627B.2. (a) Classical affinoids . . . . . . . . . . . . . . . .
. . 627B.2. (b) Affinoid subdomains . . . . . . . . . . . . . . . .
. 627B.2. (c) Rigid analytic spaces . . . . . . . . . . . . . . . .
629B.2. (d) Comparison with rigid spaces . . . . . . . . . . . .
630
-
18 Contents
B.2. (e) Coherent sheaves . . . . . . . . . . . . . . . . . . .
632C Appendix: Non-archimedean analytic space of Banach type . . .
. 633
C.1 Seminorms and norms . . . . . . . . . . . . . . . . . . . .
. 633C.2 Graded valuations . . . . . . . . . . . . . . . . . . . .
. . . 635
C.2. (a) Graded rings and modules . . . . . . . . . . . . .
635C.2. (b) Graded valuation rings . . . . . . . . . . . . . . .
636C.2. (c) Graded valuations . . . . . . . . . . . . . . . . . .
638C.2. (d) Generization and specialization of graded
valuations639C.2. (e) Unit-element part . . . . . . . . . . . . . .
. . . . 639C.2. (f) The space of graded valuations . . . . . . . .
. . . 642
C.3 Filtered valuations . . . . . . . . . . . . . . . . . . . .
. . . 643C.3. (a) Filtered rings . . . . . . . . . . . . . . . . .
. . . . 643C.3. (b) Filtrations and seminorms . . . . . . . . . . .
. . 646C.3. (c) Filtered polynomial and power series algebras . .
647C.3. (d) Filtered valuation fields . . . . . . . . . . . . . . .
648C.3. (e) Filtered valuation via valuation . . . . . . . . . .
650C.3. (f) Non-degenerate filtered valuations . . . . . . . . .
651C.3. (g) Examples of filtered valuations . . . . . . . . . . .
653
C.4 Valuative spectrum of non-archimedean Banach rings . . .
655C.4. (a) Gelfand-Berkovich spectrum . . . . . . . . . . . .
655C.4. (b) R+-finite type algebras . . . . . . . . . . . . . . .
658C.4. (c) Integrally closed filtrations . . . . . . . . . . . . .
659C.4. (d) Power bounded filtration . . . . . . . . . . . . . .
662C.4. (e) R+-affinoid rings . . . . . . . . . . . . . . . . . . .
664C.4. (f) Valuative spectrum . . . . . . . . . . . . . . . . .
666C.4. (g) Basic properties of valuative spectrum . . . . . . .
668C.4. (h) Proof of Theorem C.4.29 . . . . . . . . . . . . . .
670C.4. (i) Relation with adic spectrum . . . . . . . . . . . .
672C.4. (j) R+-affinoid algebras of R+-finite type over K . . .
676C.4. (k) Reflexivity of valuative spectrum . . . . . . . . . .
678
C.5 Non-archimedean analytic space of Banach type . . . . . .
680C.5. (a) Admissible site of R+-affinoid rings . . . . . . . .
680C.5. (b) Sheaf condition of Banach type . . . . . . . . . . .
683C.5. (c) Metrized Banach ringed spaces . . . . . . . . . . .
684C.5. (d) Relation with adic spaces . . . . . . . . . . . . . .
689
C.6 Berkovich’s analytic geometry . . . . . . . . . . . . . . .
. . 691C.6. (a) Gerritzen-Grauert theorem . . . . . . . . . . . . .
691C.6. (b) Berkovich analytic spaces . . . . . . . . . . . . . .
693C.6. (c) G-topology on Berkovich analytic spaces . . . . .
694C.6. (d) Berkovich analytic spaces and R+-metrized ana-
lytic spaces . . . . . . . . . . . . . . . . . . . . . . 695C.6.
(e) Comparison with rigid spaces . . . . . . . . . . . . 698
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 700D Appendix: Rigid Zariskian spaces . . . . . . . . . .
. . . . . . . . . 701
-
Introduction 1
D.1 Admissible blow-ups . . . . . . . . . . . . . . . . . . . .
. . 701D.2 Coherent rigid Zariskian spaces . . . . . . . . . . . .
. . . . 702
D.2. (a) The category of coherent rigid Zariskian spaces . .
702D.2. (b) Visualization . . . . . . . . . . . . . . . . . . . . .
704
E Appendix: Classical Zariski-Riemann spaces . . . . . . . . . .
. . . 705E.1 Birational geometry . . . . . . . . . . . . . . . . .
. . . . . 705
E.1. (a) Basic terminologies . . . . . . . . . . . . . . . . .
705E.1. (b) U -admissible blow-ups . . . . . . . . . . . . . . . .
706E.1. (c) The correspondence diagram . . . . . . . . . . . .
708E.1. (d) Birational category . . . . . . . . . . . . . . . . .
710
E.2 Classical Zariski-Riemann spaces . . . . . . . . . . . . . .
. 711E.2. (a) The cofiltered category of modifications . . . . . .
711E.2. (b) The classical Zariski-Riemann spaces . . . . . . .
712E.2. (c) Comparison maps . . . . . . . . . . . . . . . . . .
714E.2. (d) Relation with rigid Zariskian spaces . . . . . . . .
714E.2. (e) Points of the Zariski-Riemann space . . . . . . . .
715
F Appendix: Nagata’s embedding theorem . . . . . . . . . . . . .
. . 717F.1 Announcement of the theorem . . . . . . . . . . . . . .
. . 717F.2 Preparation for the proof . . . . . . . . . . . . . . .
. . . . 717
F.2. (a) Canonical compactification . . . . . . . . . . . . .
717F.2. (b) General construction . . . . . . . . . . . . . . . .
719F.2. (c) Properties of canonical compactification . . . . . .
722
F.3 Proof of Theorem F.1.1 . . . . . . . . . . . . . . . . . . .
. 723F.3. (a) Lemmas . . . . . . . . . . . . . . . . . . . . . . .
. 723F.3. (b) Proof of the theorem . . . . . . . . . . . . . . . .
725
F.4 Application: Removing Noetherian hypothesis . . . . . . .
725F.5 Nagata embedding for algebraic spaces . . . . . . . . . . .
. 726Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 727
Solutions and Hints for Exercises 728
List of Notations 752
Index 757
-
Introduction
In the early stage of its history, rigid geometry has been first
envisaged in anattempt to construct a non-archimedean analytic
geometry, an analogue overnon-archimedean valued fields, such as
p-adic fields, of complex analytic geome-try. Later, in the course
of its development, rigid geometry has acquired severalrich
structures, much richer than being merely as a ‘copy’ of complex
analyticgeometry, which endowed the theory with a great potential
of applications. Thistheory is nowadays recognized by many
mathematicians of various research fieldsto be an important and
independent discipline in arithmetic and algebraic ge-ometry. This
book is the first volume of our prospective book project, whichaims
to discuss the rich overall structures of rigid geometry, and to
explore itsapplications.
Before explaining our general perspective of this book project,
we first beginwith an overview of the past developments of the
theory.
0. Background. After K. Hensel introduced p-adic numbers by the
end of the19th century, there had emerged the idea of constructing
p-adic analogues ofalready existing mathematical theories that were
formerly considered only overreal or complex number field. One of
such theories was the theory of complexanalytic functions, which
had by then already matured to be one of the mostsuccessful and
rich branches of mathematics. Complex analysis saw further
de-velopments and innovations later on. Most importantly, from
extensive works oncomplex analytic spaces and analytic sheaves by
H. Cartan and J.P. Serre in themid-20th century, after the
pioneering work by K. Oka, arose a new idea thatthe theory of
complex analytic functions should be regarded as part of
complexanalytic geometry. According to this view, it was only
natural to expect thenotion of p-adic analytic geometry, or more
generally, non-archimedean analyticgeometry.
However, all first attempts had to encounter with essential
difficulties, espe-cially in establishing reasonable
local-to-global linkage of the notion of analyticfunctions. Such a
naive approach is, generally speaking, characterized by its
in-clination to produce a faithful imitation of complex analytic
geometry, which canbe already seen at the level of point-sets and
topology of the putative analyticspaces. For example, for the
‘complex plane’ over Cp (= the completion of thealgebraic closure
Qp of Qp), one takes the naive point-set, that is, Cp itself,
and
-
Introduction 3
the topology just induced from the p-adic metric. Starting from
the situation likethis, it goes on to construct a locally ringed
space X = (X,OX) by introducingthe sheaf OX of ‘holomorphic
functions’, of which a conventional definition issomething like as
follows: OX(U) for any open subset U is the set of all functionson
U that admit the convergent power series expansion at every point.
But thisleads to an extremely cumbersome situation. Indeed, since
the topology of Xis totally disconnected, there are too many open
subsets, and this causes thepatching of the functions to be
extremely ‘wobbly’, so much so that one failsto have good control
of global behavior of the analytic functions. For example,if X is
the ‘p-adic Riemann sphere’ Cp ∪ {∞}, one would expect that
OX(X)consists only of constant functions, which is, however, far
from being true in thissituation.
Let us call the problem of this kind the Globalization Problem.1
Although theproblem in its essence may be seen, inasmuch as being
concerned with patching ofanalytic functions, as a topological one,
as it will turn out, it deeply links with theissue of notion of
points. In the prehistory of rigid geometry, this
GlobalizationProblem has been one, and perhaps the most crucial
one, of the obstacles in thequest for a good non-archimedean
analytic geometry.2
1. Tate’s rigid analytic geometry. The Globalization Problem
found itsfundamental solution when J. Tate introduced his rigid
analytic geometry [86]at a seminar at Harvard University in 1961.
Tate’s motivation was to justifyhis construction of the so-called
Tate curves, a non-archimedean analogue of 1-dimensional complex
tori, constructed by means of an infinite quotient [87].3
Tate’s solution to the problem consists of the following
items:
• ‘reasonable’ and ‘sufficiently large’ class of analytic
functions,• ‘correct’ notion of analytic coverings.
Here, one can find behind this idea an influence of A.
Grothendieck in at leasttwo ways: First, Tate introduced spaces via
local characterization by means oftheir function rings, as typified
in scheme theory; second, he used the machineryof Grothendieck
topology to define analytic coverings.
Now, let us briefly review Tate’s theory. First of all, Tate
introduced thecategory AffK of so-called affinoid algebras over a
complete non-archimedeanvaluation field K. Each affinoid algebra A,
which is a K-Banach algebra, is con-sidered to be the ring of
‘reasonable’ analytic functions over the ‘space’ SpA,called the
affinoid, which is the corresponding object in the dual category
AffoppK
1This problem is, in classical literature, usually referred to
as the problem in analytic con-tinuation.
2In his pioneering works [61][62], M. Krasner conducted deep
research into the problem andgave a first general recipe to manage
a meaningful analytic continuation of non-archimedeananalytic
functions.
3Elliptic curves and elliptic functions over p-adic fields have
already been studied by É. Lutzunder the suggestion of A. Weil,
who is inspired by classical works of Eisenstein (cf. [94,
p.538]).
-
4 Introduction
of AffK . Moreover, based on the notion of admissible coverings,
he introduced anew ‘topology’, in fact, a Grothendieck topology, on
SpA, which we call the ad-missible topology. The admissibility
imposes, most importantly, a strong finite-ness condition on
analytic coverings, which establishes the closer ties betweenlocal
and global behaviors of analytic functions, as well-described by
the famousTate’s acyclicity theorem (II.B.2.3). An important
consequence of this nice local-to-global linkage is the good notion
of ‘patching’ affinoids, by which Tate was ableto solve the
Globalization Problem, and thus to construct global analytic
spaces.
In summary, Tate overcame the difficulty by ‘rigidifying’ the
topology itself byimposing the admissibility condition, which puts
strong restriction on patchingof local analytic functions. It is
for this reason that this theory is nowadays calledrigid analytic
geometry.
Aside from the fact that it gave a beautiful solution to the
GlobalizationProblem, one should also find it remarkable that
Tate’s rigid anaytic geometryproved it possible to apply
Grothendieck’s way of constructing geometric objectsto the
situation of non-archimedean analytic geometry. Thus, rather than
com-plex analytic geometry, Tate’s rigid analytic geometry
resembles scheme theory.There seemed to be, however, one technical
difference between scheme theoryand rigid analytic geometry, which
was considered to be quite essential at thetime when rigid analytic
geometry appeared: Rigid analytic geometry had to useGrothendieck
topology, not classical point-set topology.
There is yet another aspect of rigid analytic geometry
reminiscent of algebraicgeometry. In order to have a better grasp
of the abstractly defined analyticspaces, Tate introduced a notion
of points. He defined points of an affinoid SpAto be maximal ideals
of the affinoid algebra A; viz., his affinoids are visualized bythe
maximal spectra, that is, the set of all maximal ideals of affinoid
algebras,just like affine varieties in the classical algebraic
geometry are visualized by themaximal spectra of finite type
algebras over a field. Notice that this choice ofpoints is
essentially the same as the naive one that we have mentioned
before.This notion of points was, despite its naivety, considered
to be natural, especiallyin view of his construction of Tate
curves, and practically good enough as faras being concerned with
rigid analytic geometry over a fixed non-archimedeanvalued
field.4
2. Functoriality and topological visualization. Tate’s rigid
analytic geom-etry has, since its first appearance, proven itself
to be useful for many purposes,and been further developed by
several authors. For example, Grauert-Remmert[44] laid foundations
of topological and ring theoretic aspects of affinoid alge-bras,
and R. Kiehl [63][64] promoted the theory of coherent sheaves and
theircohomologies on rigid analytic spaces.
4One might be apt to think that Tate’s choice of points is an
‘easygoing’ analogue of thespectra of complex commutative Banach
algebras, for which the justification, Gelfand-Mazurtheorem, is,
however, only valid in complex analytic situation, and actually
fails in p-adicsituation (see below).
-
Introduction 5
However, it was widely perceived that rigid analytic geometry
still has someessential difficulties. For example:
• Functoriality of points does not hold: If K ′/K is an
extension of completenon-archimedean valuation fields, then one
expects to have, for any rigid analyticspace X over K, a mapping
from the points of the base change XK′ to the pointsof X , which,
however, does not exist in general in Tate’s framework.
Let us call this problem the Functoriality Problem. The problem
is linkedwith the following more fundamental one:
• The analogue of the Gelfand-Mazur theorem does not hold: The
Gelfand-Mazur theorem states that there exist no Banach field
extension of C other thanitself. In non-archimedean situation, in
contrast, there exist many Banach K-fields other than finite
extensions of K. This would imply that there should beplenty of
‘valued points’ of an affinoid algebra not factoring through the
residuefield of a maximal ideal; in other words, there should be
much more points thanthose that Tate has introduced.
It is clear that, in order to overcome the difficulties of this
kind, one has tochange the notion of points. More precisely, the
problem lies in what to chooseas the spectrum of an affinoid
algebra. To this, there are at least two solutions:
(I) Gromov-Berkovich style spectrum;
(II) Stone-Zariski style spectrum.
The spectrum of the first style, which turns out to be the
‘smallest’ spectrumto solve the Functoriality Problem in the
category of Banach algebras, consistsof height one valuations, that
is, seminorms (of a certain type) on affinoid al-gebras. The
resulting point-sets carry a natural topology, the so-called
Gelfandtopology. This kind of spectra is adopted by V.G. Berkovich
in his approach tonon-archimedean analytic geometry, so-called
Berkovich analytic geometry [9]. Anice point of this approach is
that it can deal with, in principle, a wide class ofBanach
K-algebras, including affinoid algebras, and thus solve the
FunctorialityProblem (in the category of Banach algebras).
Moreover, the spectra of affi-noid algebras in this approach are
Hausdorff, hence providing intuitively familiarspaces as the
underlying topological spaces of the analytic spaces.
However, the Gelfand topology differs from the admissible
topology; it is evenweaker, in the sense that, as we will see
later, the former topology is a quotientof the latter. Therefore,
this topology does not solve the Globalization Problemfor affinoid
algebras compatibly with Tate’s solution, and, in order to do
analyticgeometry, one still has to use the Grothendieck topology
just imported fromTate’s theory.
It is thus necessary, in order to solve the Globalization
Problem (for affinoids)and Functoriality Problem at the same time,
to further improve the notion ofpoints and the topology. In the
second style, the Stone-Zariski style, whichwe will take up in this
book, each spectrum has more points by valuations,
-
6 Introduction
not only of height one, but of higher height.5 It turns out that
the topologyon the point-set thus obtained coincides with the
admissible topology on thecorresponding affinoid, thus solving the
Globalization Problem without usingGrothendieck topology. Moreover,
the spectra have plenty enough points to solvethe Functoriality
Problem as well.
As we have seen, to sum up, both the Globalization Problem and
the Func-toriality Problem are closely linked with the more
fundamental issue concernedwith the notion of points and topology,
that is, the problem for the choice of spec-tra. What lies behind
all this is the philosophical tenet that every notion of spacein
commutative geometry should be accompanied with ‘visualization’ by
meansof topological spaces, which we call the topological
visualization (Figure 1). It
Commutative
geometry+3 Topological
spaces
Figure 1. Topological visualization
can be said, therefore, that the original difficulties in the
early non-archimedeananalytic geometry in general, Globalization
and Functoriality, are rooted in thelack of adequate topological
visualizations. We will discuss more on this topiclater.
3. Raynaud’s approach to rigid analytic geometry. To adopt the
spectraas in the Stone-Zariski style, in which points are described
in terms of valuationrings of arbitrary height, it is more or less
inevitable to deal with finer structures,somewhat related to
integral structures, of affinoid algebras.6 The approach is,then,
further divided into the following two branches:
(II-a) R. Huber’s adic spaces7 [51][52][53];
(II-b) M. Raynaud’s viewpoint via formal geometry8 as a model
geometry [81].
The last approach, which we will adopt in this book, fits in the
general frame-work in which a geometry as a whole is a package
derived from a so-called modelgeometry. Here is a toy model that
exemplifies the framework: Consider, for
5Notice that this height tolerance is necessary even for rigid
spaces defined over completevaluation fields of height one.
6Such a structure, which we call a rigidification, will be
discussed in detail in II, §A.2. (c).In the original Tate’s rigid
analytic geometry, the rigidifications are canonically determined
byclassical affinoid algebras themselves, and this fact should come
as the reason why Tate’s rigidanalytic geometry, unlike more
general Huber’s adic geometry, could work without reference
tointegral models of affinoid algebras.
7Notice that Huber’s theory is based on the choice of integral
structures of topological rings.We will give, mainly in II, §A, a
reasonably detailed account of Huber’s theory.
8By formal geometry, we mean in this book the geometry of formal
schemes, developed byA. Grothendieck.
-
Introduction 7
example, the category of finite dimensional Qp-vector spaces. We
observe thatthis category is equivalent to the quotient category of
the category of finitelygenerated Zp-modules mod out by the Serre
subcategory consisting of p-torsionZp-modules, since any finite
dimensional Qp-vector space has a Zp-lattice, thatis, a ‘model’
over Zp. This suggests that the overall theory of finite
dimensionalQp-vector spaces is derived from the theory of models,
in this case, the theory offinitely generated Zp-modules.
In our context, what Raynaud discovered on rigid analytic
geometry consistsof the following:
• Formal geometry, which has already been established by
Grothendieckprior to Tate’s work, can be adopted as a model
geometry for Tate’s rigidanalytic geometry.
• Consequently, the overall theory of rigid analytic geometry
arises fromGrothendiek’s formal geometry (Figure 2), from which one
obtains anextremely useful idea that, between formal geometry and
Tate’s rigid an-alytic geometry, one can use theorems in one side
to prove theorems inthe other.
Formal
geometry+3 Rigid analytic
geometry
Figure 2. Raynaud’s approach to rigid geometry
To make more precise what it means to say formal geometry can be
a modelgeometry for rigid analytic geometry, consider, just as in
the toy model as above,the category of rigid analytic spaces over
K. Raynaud showed that the categoryof Tate’s rigid analytic spaces
(with some finiteness conditions) is equivalentto the quotient
category of the category of finite type formal schemes over
thevaluation ring V of K. Here the ‘quotient’ means inverting all
‘modifications’(especially, blow-ups) that are ‘isomorphisms over
K’, the so-called admissiblemodifications (blow-ups).
There are several impacts of Raynaud’s discovery; let us mention
a few ofthem. First, guided by the principle that rigid analytic
geometry is derivedfrom formal geometry, one can build the theory
afresh, starting from defining thecategory of rigid analytic spaces
as the quotient category of the category of formalschemes mod out
by all admissible modifications.9 Second, Raynaud’s theoremsays
that rigid analytic geometry can be seen as birational geometry of
formalschemes, a novel viewpoint, which attracts one to explore the
link with traditional
9The rigid spaces obtained in this way are, more precisely, what
we call coherent (= quasi-compact and quasi-separated) rigid
spaces, from which general rigid spaces are constructed
bypatching.
-
8 Introduction
birational geometry. Third, as already mentioned above, the
bridge betweenformal schemes and rigid analytic spaces, established
by Raynaud’s viewpoint,gives rise to fruitful interactions between
these theories. Especially useful is thefact that theorems in the
rigid analytic side can be deduced, at least in thesituation over
complete discrete valuation rings, from theorems in the
formalgeometry side, available in EGA and SGA works by Grothendieck
et al, at leastin the Noetherian situation.
4. Rigid geometry of formal schemes. We can now describe, along
the lineof Raynaud’s discovery, the basic framework of our rigid
geometry that we areto promote in this book project. Here is what
rigid geometry is for us: Rigidgeometry is a geometry obtained from
a birational geometry of model geometries.This being so, the main
purpose of this book project is to develop such a theory forformal
geometry, thus generalizing Tate’s rigid analytic geometry and
providingmore general analytic geometry. Thus to each formal scheme
X is associatedan object of a resulting category, denoted as usual
by Xrig, which itself shouldalready be regarded as a rigid space.
Then we define general rigid spaces bypatching these objects.
Notice that, here, the rigid spaces are introduced as an‘absolute’
object without reference to a base space.
Among several classes of formal schemes we start with, one of
the most im-portant is the class of what we call locally
universally rigid-Noetherian formalschemes (I.2.1.7). The rigid
spaces obtained from this class of formal schemesare called locally
universally Noetherian rigid spaces (II.2.2.23), which cover mostof
the analytic spaces that appear in contemporary arithmetic
geometry. Noticethat the formal schemes of the above kind are not
themselves locally Noetherian.A technical point imposed from the
demand of removing Noetherian hypothe-sis is that one has to treat
non-Noetherian adic rings of fairly general kind, forwhich
classical theories including EGA do not give us enough tools; for
example,valuation rings of arbitrary height are necessary for
describing points on rigidspaces, and we accordingly need to treat
fairly wide class of adic rings over themfor describing fibers of
finite type morphisms.
Besides, we would like to propose another viewpoint, which
classical theorydoes not offer. Among what Raynaud’s theory
suggests, the most inspiring is,we think, the suggestion that rigid
geometry should be a birational geometry offormal schemes. We would
like to put this perspective to be one of the core ideasof our
theory. In fact, as we will see soon below, it tells us what should
be themost natural notion of points of the rigid spaces, and thus
leads to an extremelyrich structures concerned with visualizations
(that is, spectra), whereby to obtaina satisfactory solution to the
above-mentioned Globalization and Functorialityproblems. Let us see
this next in the sequel.
5. Revival of Zariski’s idea. The birational geometric aspect of
our rigidgeometry is best explained by means of O. Zariski’s
classical approach to bira-tional geometry as a model example.
Around 1940’s, in his attempt to attackthe desingularization
problem for algebraic varieties, Zariski introduced abstract
-
Introduction 9
Riemann spaces for function fields, which we call
Zariski-Riemann spaces, gen-eralizing the classical
valuation-theoretic construction of Riemann surfaces
byDedekind-Weber. This idea has been applied to several other
problems in al-gebraic geometry, including, for example, Nagata’s
compactification theorem foralgebraic varieties.
Let us briefly overview Zariski’s idea. Let Y →֒ X be a closed
immersionof schemes (with some finiteness conditions), and set U =
X \ Y . We considerU -admissible modifications of X , which are by
definition proper birational mapsX ′ → X that is an isomorphism
over U . This class of morphisms containsthe subclass consisting of
U -admissible blow-ups, that is, blow-ups along closedsubschemes
contained in Y . In fact, U -admissible blow-ups are cofinal in the
setof all U -admissible modifications (due to flattening theorem;
cf. II, §E.1. (b)).The Zariski-Riemann space, denoted by 〈X〉U , is
the topological space defined asthe projective limit taken along
the ordered set of all U -admissible modifications,or equivalently,
U -admissible blow-ups, of X . Especially important is the factthat
the Zariski-Riemann space 〈X〉U is quasi-compact (essentially due to
Zariski[98]; cf. II.E.2.5), a fact that is crucial in proving many
theorems, for example,the above-mentioned Nagata’s theorem.10
As is classically known, points of the Zariski-Riemann space
〈X〉U are de-scribed in terms of valuation rings. More precisely,
these points are in one-to-onecorrespondence with the set of all
morphisms, up to equivalence by ‘domination’,of the form SpecV → X
where V is a valuation ring (possibly of height 0) thatmap the
generic point to points in U (see II, §E.2. (e) for details). Since
thespectra of valuation rings are viewed as ‘long paths’ (cf.
Figure 1 in 0, §6),one can say intuitively that the space 〈X〉U is
like a ‘path space’ in algebraicgeometry (Figure 3).
X
Y
Figure 3. Set-theoretical description of 〈X〉U
Now, what we have meant by putting birational geometry into one
of the coreideas in our theory is that we apply Zariski’s approach
to birational geometry tothe main body of our rigid geometry. Our
basic dictionary for doing this, e.g.,for rigid geometry over the
p-adic field, is as follows:
10Zariski-Riemann spaces are also used in O. Gabber’s
unpublished works in 1980’s on al-gebraic geometry problems. Its
first appearance in literature in the context of rigid
geometryseems to be in [33].
-
10 Introduction
• X ↔ formal scheme of finite type over Spf Zp;• Y ↔ the closed
fiber, that is, the closed subscheme defined by ‘p = 0’.
In this, the notion of U -admissible blow-ups corresponds
precisely to the admis-sible blow-ups of the formal schemes.
6. Birational approach to rigid geometry. As we have already
mentionedabove, our approach to rigid geometry, called the
birational approach to rigidgeometry, is, so to speak, the
combination of Raynaud’s algebro-geometric inter-pretation of rigid
analytic geometry, which regards rigid geometry as a
birationalgeometry of formal schemes, and Zariski’s classical
birational geometry (Figure4). Most notably, it will turn out that
this approach naturally gives rise to the
Raynaud’s viewpoint ofrigid geometry
+Zariski’s viewpoint ofbirational geometry
Figure 4. Birational approach to rigid geometry
Stone-Zariski style spectrum, which we have already mentioned
before.A nice point in combining Raynaud’s viewpoint and Zariski’s
viewpoint is
that, while the former gives the fundamental recipe for defining
rigid spaces, thelatter provides them with a ‘visualization’. Let
us see this more precisely, andalongside, explain what kind of
visualization we mean here to attach to rigidspaces.
As already described earlier, from an adic formal scheme X (of
finite idealtype; cf. I.1.1.16), we obtain the associated rigid
space X = Xrig. Then, sug-gested by what we have seen in the
previous section, we define the associatedZariski-Riemann space 〈X
〉 as the projective limit
〈X 〉 = lim←−X′,
taken in the category of topological spaces, of all admissible
blow-ups X ′ → X(Definition II.3.2.11). We adopt this space 〈X 〉 as
the topological visualizationof the rigid space X . In fact, this
space is exactly what we have expected as thetopological
visualization in the case of Tate’s theory, since it can be shown
thatthe canonical topology (the projective limit topology) of 〈X 〉
actually coincideswith the admissible topology.
To explain more about the visualization of rigid spaces, we
would like tointroduce three kinds of visualizations in general
context. One is the topologicalvisualization, which we have already
discussed. The second one, which we namestandard visualization, is
the one that appears in ordinary geometries, as typifiedby scheme
theory; that is, visualization by locally ringed spaces. Recall
that anaffine scheme, first defined abstractly as an object of the
dual category of thecategory of all commutative rings, can be
visualized by a locally ringed space
-
Introduction 11
supported on the prime spectrum of the corresponding commutative
ring. Thethird visualization, which we call the enriched
visualization, or just visualizationin this book, is given by what
we call triples11: this is an object of the form(X,O+X ,OX)
consisting of a topological space X and two sheaves of
topologicalrings together with an injective ring homomorphism O+X
→֒ OX that identifies O+Xwith an open subsheaf of OX such that the
pairsX = (X,OX) andX+ = (X,O
+X)
are locally ringed spaces; normally speaking, OX is regarded as
the structure sheafof X , while O+X represents the enriched
structure, such as an integral structure(whenever it makes sense)
of OX .
The enriched visualization is typified by rigid spaces. Indeed,
the Zariski-Riemann space 〈X 〉 has two natural structure sheaves,
the integral structuresheaf O intX , defined as the inductive limit
of the structure sheaves of all admissibleblow-ups ofX , and the
rigid structure sheaf OX , obtained from O intX by ‘invertingthe
ideal of definition’. What is intended here is that, while the
rigid structuresheaf OX should, as in Tate’s rigid analytic
geometry, normally come as the‘genuine’ structure sheaf of the
rigid space X , the integral structure sheaf O intXrepresents its
integral structure. These data comprise the triple
ZR(X ) = (〈X 〉,O intX ,OX ),
called the associated Zariski-Riemann triple, which gives the
enriched visualiza-tion of the rigid space X . That the rigid
structure sheaf should be the structuresheaf of X means that the
locally ringed space (〈X 〉,OX ) visualizes the rigidspace in an
ordinary sense, that is, in the sense of standard
visualization.
Notice that the Zariski-Riemann triple ZR(X ) for a rigid space
X coincideswith Huber’s adic space associated to X ; in fact, the
notion of Zariski-Riemanntriple not only gives an interpretation of
adic spaces, but it also gives a foundationfor them via formal
geometry, which we establish in this book; see II, §A.5 formore
details.
Figure 5 illustrates the basic design of our birational approach
to rigid ge-ometry, summarizing all what we have discussed so far.
The figure shows a‘commutative’ diagram, in which the arrow (∗1) is
Raynaud’s approach to rigidgeometry (Figure 2), and the arrow (∗2)
is the enriched visualization by Zariski-Riemann triples, coming
from Zariski’s viewpoint. The other visualizations arealso
indicated in the diagram, the standard visualization by (∗3), and
the topolog-ical visualization by (∗4); the right-hand vertical
arrows represent the respectiveforgetful functors.
All these are the outline of what we will discuss in this
volume. Here, beforefinishing this overview, let us add a few words
on the outgrowth of our theory.Our approach to rigid geometry, in
fact, gives rise to a new perspective of rigidgeometry itself:
Rigid geometry in general is an analysis along a closed subspacein
a ringed topos. This idea, which tells us what rigid-geometrical
idea in math-ematics should ultimately be, is linked with the idea
of tubular neighborhoods in
11See II, §A.1 for the generalities of triples.
-
12 Introduction
Formal
Geometry
Rigid
GeometryTriples
Locally ringed
spaces
Topological
spaces
>
>
(∗1) +3 (∗2) +3
��
��
(∗3)
(∗4)
Figure 5. Birational approach to rigid geometry
algebraic geometry, already discussed in [33]. From this
viewpoint, Raynaud’schoice, for example, of formal schemes as
models of rigid spaces can be interpretedas capturing the ‘tubular
neighborhoods’ along a closed subspace by means of theformal
completion. Now that there are several other ways to capture such
struc-tures, e.g., henselian schemes etc., there are several other
choices for the modelgeometry of rigid geometry.12 This yields
several variants, e.g. rigid henseliangeometry, rigid Zariskian
geometry, etc., all of which are encompassed within ourbirational
approach.13
7. Relation with other theories. In the first three sections II,
§A, II, §B,and II, §C of the appendices to Chapter II, we give the
comparisons of ourtheory with other theories related to rigid
geometry. Here we give a digest of thecontents of these sections
for the reader’s convenience.14
• Relation with Tate’s rigid analytic geometry. Let V be an
a-adicallycomplete valuation ring of height one, and set K = Frac(V
) (the fractional field),which is a complete non-archimedean valued
field with a non-trivial valuation‖ · ‖ : K → R≥0. In II, §8.2. (c)
we will define the notion of classical points (inthe sense of Tate)
for rigid spaces of a certain kind including locally of finite
typerigid spaces over S = (Spf V )rig. If X is a rigid space of the
latter kind, itwill turn out that the classical points of X are
reduced zero dimensional closedsubvarieties in X (cf.
II.8.2.6).
12There is, in addition to formal geometry and henselian
geometry, the third possibility forthe model geometry, by Zariskian
schemes. We put a general account of the theory of Zariskianschemes
and the associated rigid spaces, so-called, rigid Zariskian spaces,
in the appendices I,§B and II, §D.
13The reader might notice that this idea is also related to the
cdh-topology in the theory ofmotivic cohomology.
14A. Abbes has recently published another foundational book [1]
on rigid geometry, in which,similarly to ours, he developed and
generalized Raynaud’s approach to rigid geometry.
-
Introduction 13
We set X0 to be the set of all classical points of X . The
assignment X 7→X0has several nice properties, some of which are put
together into the notion of(continuous) spectral functor (cf. II,
§8.1). Among them is an important propertythat classical points
detect quasi-compact open subspaces: for quasi-compactopen
subspaces U ,V ⊆ X , U0 = V0 implies U = V . In view of all this,
onecan introduce on X0 a Grothendieck topology τ0 and sheaf of
rings OX0 , whichare naturally constructed from the topology and
the structure sheaf of X ; forexample, for a quasi-compact open
subspace U ⊆ X , U0 is an admissible opensubset of X0, and we have
OX0(U0) = OX (〈U 〉). It will turn out that theresulting triple X0 =
(X0, τ0,OX0) is a Tate’s rigid analytic variety over K, andthus one
has the canonical functor
X 7−→X0
from the category of locally of finite type rigid spaces over S
to the category ofrigid analytic varieties over K.
Theorem (Theorem II.B.2.5, Corollary II.B.2.6). The functor X 7→
X0 is acategorical equivalence from the category of quasi-separated
locally of finite typerigid spaces over S = (Spf V )rig to the
category of quasi-separated Tate analyticvarieties over K.
Moreover, under this functor, affinoids (resp. coherent
spaces)correspond to affinoid spaces (resp. coherent analytic
spaces).
Notice that the Raynaud’s theorem (the existence of formal
models) gives thecanonical quasi-inverse functor to the above
functor.15
• Relation with Huber’s adic geometry. As we have already
remarkedabove, the Zariski-Riemann triple ZR(X ), at least in the
situation as before,is an adic space. This is true in much more
general situation, for example, incase X is locally universally
Noetherian (II.2.2.23). In fact, by the enrichedvisualization, we
have the functor
ZR : X 7−→ ZR(X )
from the category of locally universally Noetherian rigid spaces
to the categoryof adic spaces (Theorem II.A.5.1), which gives rise
to a categorical equivalencein most important cases. In particular,
we have:
Theorem (Theorem II.A.5.2). Let S be a locally universally
Noetherian rigidspace. Then ZR gives a categorical equivalence from
the category of locally offinite type rigid spaces over S to the
category of adic spaces locally of finite typeover ZR(S ).
15To show the theorem, we need Gerritzen-Grauert theorem [40],
which we assume wheneverdiscussing Tate’s rigid analytic geometry.
Notice that, when it comes to the rigid geometryover valuation
rings, this volume is self-contained only with this exception. We
will proveGerritzen-Grauert theorem without vicious circle in the
next volume.
-
14 Introduction
• Relation with Berkovich analytic geometry. Let V and K be
asbefore. We will construct a natural functor
X 7−→XB
from the category of locally quasi-compact16 (II.4.4.1) and
locally of finite typerigid spaces over S = (Spf V )rig to the
category of strictly K-analytic spaces (inthe sense of
Berkovich).
Theorem (Theorem II.C.6.12). The functor X 7→ XB gives a
categorical equiv-alence from the category of all locally
quasi-compact locallly of finite type rigidspaces over (Spf V )rig
to the category of all strictly K-analytic spaces. More-over, XB is
Hausdorff (resp. paracompact Hausdorff, resp. compact Hausdorff)if
and only if X is quasi-separated (resp. paracompact and
quasi-separated, resp.coherent).
The underlying topological space of XB is what we call the
separated quotient(II, §4.3. (a)) of 〈X 〉, denoted by [X ], which
comes with the quotient mapsepX : 〈X 〉 → [X ] (separation map). In
particular, the topology of XB is thequotient topology of the
topology of 〈X 〉.
Figure 6 illustrates the interrelations among those theories we
have discussedso far. In the diagram,
• the functors (∗1) (∗2) are fully faithful; the functor (∗3),
defined on locallyquasi-compact rigid analytic spaces, is fully
faithful to the category ofstrictly K-analytic spaces;
• the functor (∗4): X → X0, defined on locally of finite type
rigid spacesover (Spf V )rig, is quasi-inverse to (∗1) restricted
on quasi-separated spaces;
• the functor (∗5) is given by the enriched visualization,
defined on locallyuniversally Noetherian rigid spaces; it is fully
faithful in practical situ-ations including those of locally of
finite type rigid spaces over a fixedlocally universally Noetherian
rigid space, and of rigid spaces of type (N)(II.A.5.3);
• the functor (∗6): X 7→ XB, defined on locally quasi-compact
locally offinite type rigid spaces over (Spf V )rig, gives a
categorical equivalence tothe category of strictly K-analytic
spaces.
Finally, we would like to mention that it has recently become
known to theexperts that some of the non-archimedean spaces that
come naturally in contem-porary arithmetic geometry may not
possibly handled in Berkovich’s analyticgeometry (e.g. [50, 4.4]).
This state of affair makes it important to investigate indetail the
relationship between Berkovich’s analytic geometry and rigid
geometry(or adic geometry). In II, §C.5, we will study a spectral
theory of filtered rings
16Note that, if X is quasi-separated, then X is locally
quasi-compact if and only if 〈X 〉 istaut in the sen of Huber [53,
5.1.2] (cf. 0.2.5.6).
-
Introduction 15
Tate’s rigid
analytic varieties
Rigid spaces
(in our sense)
Adic spaces
Berkovich
spaces
(∗1)33
(∗4)||
(∗2)//
(∗3) ,,
(∗5)��
(∗6)
��
Figure 6. Relation with other theories
and introduce a new category of spaces, the so-called metrized
analytic spaces.This new notion of spaces generalizes Berkovich’s
K-analytic spaces, and givesa clear picture of the comparison; see
II, §C.6. (d). Also, the newly introducedspaces turn out to be
equivalent to Kedlaya’s reified adic spaces [59], to whichour
filtered ring approach in this book offers a new perspective.
8. Applications. We expect that our rigid geometry will have
rich applications,not only in arithmetic geometry, but also in
various other fields. A few of themhave already been sketched in
[37], which include
• arithmetic moduli spaces (e.g. Shimura varieties) and their
compactifica-tions,
• trace formula in characteristic p > 0 (Deligne’s
conjecture).In addition to these, since our theory has set out from
Zariski’s birational
geometry, applications to problems in birational geometry,
modern or classical,are also expected. For example, this volume
already contains Nagata’s compact-ification theorem for schemes and
a proof of it (II, §F), as an application of thegeneral idea of our
rigid geometry to algebraic geometry.
Some other prospective applications may be to p-adic Hodge
theory (cf.[83][84]) and to rigid cohomology theory for algebraic
varieties in positive char-acteristic. Here the visualization in
our sense of rigid spaces will give concretepictures for tubes and
the dagger construction. One of such applications in th