Etale Cohomology Theory

Contents......Page 8
Preface......Page 6
1.1 Flat Modules......Page 12
1.2 Faithfully Flat Modules......Page 14
1.3 Local Criteria for Flatness......Page 21
1.4 Constructible Sets......Page 26
1.5 Flat Morphisms......Page 29
1.6 Descent of Quasi-coherent Sheaves......Page 32
1.7 Descent of Properties of Morphisms......Page 39
1.8 Descent of Schemes......Page 46
1.9 Quasi-finite Morphisms......Page 52
1.10 Passage to Limit......Page 56
2.1 The Sheaf of Relative Differentials......Page 70
2.2 Unramified Morphisms......Page 75
2.3 Etale Morphisms......Page 77
2.4 Smooth Morphisms......Page 84
2.5 Jacobian Criterion......Page 86
2.6 Infinitesimal Liftings of Morphisms......Page 94
2.7 Direct Limits and Inverse Limits......Page 98
2.8 Henselization......Page 101
2.9 Etale Morphisms between Normal Schemes......Page 124
3.1 Finite Group Actions on Schemes......Page 128
3.2 Etale Covering Spaces and Fundamental Groups......Page 132
3.3 Functorial Properties of Fundamental Groups......Page 142
4.1 Group Cohomology......Page 150
4.2 Profinite Groups......Page 157
4.3 Cohomology of Profinite Groups......Page 163
4.4 Cohomological Dimensions......Page 172
4.5 Galois Cohomology......Page 175
5.1 Presheaves and ech Cohomology......Page 182
5.2 Etale Sheaves......Page 187
5.3 Stalks of Sheaves......Page 204
5.4 Recollement of Sheaves......Page 212
5.5 The Functor f!......Page 216
5.6 Etale Cohomology......Page 221
5.7 Calculation of Etale Cohomology......Page 233
5.8 Constructible Sheaves......Page 256
5.9 Passage to Limit......Page 268
6.1 Triangulated Categories......Page 278
6.2 Derived Categories......Page 283
6.3 Derived Functors......Page 290
6.4 RHom(-,-) and -  L A -......Page 298
6.5 Way-out Functors......Page 314
7.1 Divisors......Page 322
7.2 Cohomology of Curves......Page 328
7.3 Proper Base Change Theorem......Page 342
7.4 Cohomology with Proper Support......Page 361
7.5 Cohomological Dimension of Rf......Page 379
7.6 Local Acyclicity......Page 388
7.7 Smooth Base Change Theorem......Page 402
7.8 Finiteness of Rf......Page 417
8.1 Extensions of Henselian Discrete Valuation Rings......Page 422
8.2 Trace Morphisms......Page 430
8.3 Duality for Curves......Page 443
8.4 The Functor Rf!......Page 454
8.5 Poincare Duality......Page 473
8.6 Cohomology Classes of Algebraic Cycles......Page 489
9.1 Sheaves with Group Actions......Page 508
9.2 Nearby Cycle and Vanishing Cycle......Page 513
9.3 Generic Base Change Theorem and Generic Local Acyclicity......Page 520
9.4 Finiteness of R	......Page 528
9.5 Finiteness Theorems......Page 532
9.6 Biduality......Page 537
10.1 Adic Formalism......Page 542
10.2 Grothendieck–Ogg–Shafarevich Formula......Page 577
10.3 Frobenius Correspondences......Page 596
10.4 Lefschetz Trace Formula......Page 603
10.5 Grothendieck’s Formula of L-functions......Page 614
Bibliography......Page 618
Index......Page 620