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 Jacobian Criterion 75
2.6 Infinitesimal Liftings of Morphisms 83
2.7 Direct Limits and Inverse Limits 87
2.8 Henselization 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 Profinite 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 Recollement of Sheaves 201
5.5 The Functor f! 205
5.6 Etale Cohomology 210
5.7 Calculation of Etale Cohomology 222
5.8 Constructible Sheaves 245
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 - ⊗LA - 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 350
7.5 Cohomological Dimension of Rf* 368
7.6 Local Acyclicity 377
7.7 Smooth Base Change Theorem 391
7.8 Finiteness of Rf! 406
8 Duality 411
8.1 Extensions of Henselian Discrete Valuation Rings 411
8.2 Trace Morphisms 419
8.3 Duality for Curves 432
8.4 The Functor Rf! 443
8.5 Poincaré Duality 462
8.6 Cohomology Classes of Algebraic Cycles 478
9 Finiteness Theorems 497
9.1 Sheaves with Group Actions 497
9.2 Nearby Cycle and Vanishing Cycle 502
9.3 Generic Base Change Theorem and Generic Local Acyclicity 509
9.4 Finiteness of RΨη 517
9.5 Finiteness Theorems 521
9.6 Biduality 526
10 l-adic Cohomology 531
10.1 Adic Formalism 531
10.2 Grothendieck-Ogg-Shafarevich Formula 566
10.3 Frobenius Correspondences 585
10.4 Lefschetz Trace Formula 592
10.5 Grothendieck's Formula of L-functions 603
Bibliography 607
Index 609
