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Foundations of Rigid Geometry I

ArXiv version

Kazuhiro Fujiwara

Graduate School of MathematicsNagoya UniversityNagoya 464-8502

Japan

fujiwara@math.nagoya-u.ac.jp

Fumiharu Kato

Department of MathematicsTokyo Institute of Technology

Tokyo 152-8551Japan

bungen@math.titech.ac.jp

http://arxiv.org/abs/1308.4734v5

To the memory of Professor Masayoshi Nagata

and Professor Masaki Maruyama

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

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

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

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

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

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

Contents 9

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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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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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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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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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

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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