Table of Contents
Preface v
1 Descent Theory 1
1.1 Flat Modules 1
1.2 Faithfully Flat Modules 3
1.3 Local Criteria for Flatness 10
1.4 Constructible Sets 15
1.5 Flat Morphisms 18
1.6 Descent of Quasi-coherent Sheaves 21
1.7 Descent of Properties of Morphisms 28
1.8 Descent of Schemes 35
1.9 Quasi-finite Morphisms 41
1.10 Passage to Limit 45
2 Etale Morphisms and Smooth Morphisms 59
2.1 The Sheaf of Relative Differentials 59
2.2 Unramified Morphisms 64
2.3 Etale Morphisms 66
2.4 Smooth Morphisms 73
2.5 Jacubian Criterion 75
2.6 Infinitesimal Liftings of Morphisms 83
2.7 Direct Limits and Inverse Limits 87
2.8 Henselizalion 90
2.9 Etale Morphisms between Normal Schemes 113
3 Etale Fundamental Groups 117
3.1 Finite Group Actions on Schemes 117
3.2 Etale Covering Spaces and Fundamental Groups 121
3.3 Functorial Properties of Fundamental Groups 131
4 Group Cohomology and Galois Cohomology 139
4.1 Group Cohomology 139
4.2 Profinite Groups 146
4.3 Cohomology of Profhrite Groups 152
4.4 Cohomological Dimensions 161
4.5 Galois Cohomology 164
5 Etale Cohomology 171
5.1 Presheaves and Cech Cohomology 171
5.2 Etale Sheaves 176
5.3 Stalks of Sheaves 193
5.4 Recohement of Sheaves 201
5.5 The Functor f! 205
5.6 Etale Cohomology 210
5.7 Calculation of Etale Cohomology 223
5.8 Constructible Sheaves 246
5.9 Passage to Limit 257
6 Derived Categories and Derived Functors 267
6.1 Triangulated Categories 267
6.2 Derived Categories 272
6.3 Derived Functors 279
6.4 RHom{-,-) and - ⊗$$$ - 287
6.5 Way-out Functors 303
7 Base Change Theorems 311
7.1 Divisors 311
7.2 Cohomology of Curves 317
7.3 Proper Base Change Theorem 331
7.4 Cohomology with Proper Support 348
7.5 Cohomological Dimension of Rf* 366
7.6 Local Acyclicity 375
7.7 Smooth Base Change Theorem 389
7.8 Finiteness of Rf! 404
8 Duality 409
8.1 Extensions of Henselian Discrete Valuation Rings 409
8.2 Trace Morphisms 417
8.3 Duality for Curves 430
8.4 The Functor Rf! 441
8.5 Poincaré Duality 460
8.6 Cohomology Classes of Algebraic Cycles 476
9 Finiteness Theorems 495
9.1 Sheaves with Group Actions 495
9.2 Nearby Cycle and Vanishing Cycle 500
9.3 Generic Base Change Theorem and Generic Local Acyclicity 507
9.4 Finiteness of Rn 515
9.5 Finiteness Theorems 519
9.6 Biduality 524
10 l-adic Cohomology 529
10.1 Adic Formalism 529
10.2 Grothendieck-Ogg-Shafarevich Formula 565
10.3 Frobenius Correspondences 584
10.4 Lcfschetz Trace Formula 591
10.5 Grothendieck's Formula of L-functions 602
Bibliography 607
Index 609
