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 ƒ! ........................................ 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 -  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 ........................ 348
7.5 Cohomological Dimension of Rƒ* ........................ 366
7.6 Local Acyclicity ...................................... 375
7.7 Smooth Base Change Theorem ............................ 389
7.8 Finiteness of Rƒ! ..................................... 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 Rƒ! ....................................... 441
8.5 Poincarй 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 Rψη ..................................... 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 Lefschetz Trace Formula ............................... 591
10.5 Grothendieck's Formula of L-functions ................. 602
Bibliography .................................................. 607
Index ......................................................... 609
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