Contents -- Preface -- 1. Descent Theory -- 1.1 Flat Modules -- 1.2 Faithfully Flat Modules -- 1.3 Local Criteria for Flatness -- 1.4 Constructible Sets -- 1.5 Flat Morphisms -- 1.6 Descent of Quasi-coherent Sheaves -- 1.7 Descent of Properties of Morphisms -- 1.8 Descent of Schemes -- 1.9 Quasi-finite Morphisms -- 1.10 Passage to Limit -- 2. Etale Morphisms and Smooth Morphisms -- 2.1 The Sheaf of Relative Differentials -- 2.2 Unramified Morphisms -- 2.3 Etale Morphisms -- 2.4 Smooth Morphisms -- 2.5 Jacobian Criterion -- 2.6 Infinitesimal Liftings of Morphisms -- 2.7 Direct Limits and Inverse Limits -- 2.8 Henselization -- 2.9 Etale Morphisms between Normal Schemes -- 3. Etale Fundamental Groups -- 3.1 Finite Group Actions on Schemes -- 3.2 Etale Covering Spaces and Fundamental Groups -- 3.3 Functorial Properties of Fundamental Groups -- 4. Group Cohomology and Galois Cohomology -- 4.1 Group Cohomology -- 4.2 Profinite Groups -- 4.3 Cohomology of Profinite Groups -- 4.4 Cohomological Dimensions -- 4.5 Galois Cohomology -- 5. Etale Cohomology -- 5.1 Presheaves and ech Cohomology -- 5.2 Etale Sheaves -- 5.3 Stalks of Sheaves -- 5.4 Recollement of Sheaves -- 5.5 The Functor f! -- 5.6 Etale Cohomology -- 5.7 Calculation of Etale Cohomology -- 5.8 Constructible Sheaves -- 5.9 Passage to Limit -- 6. Derived Categories and Derived Functors -- 6.1 Triangulated Categories -- 6.2 Derived Categories -- 6.3 Derived Functors -- 6.4 RHom(-,-) and - L A - -- 6.5 Way-out Functors -- 7. Base Change Theorems -- 7.1 Divisors -- 7.2 Cohomology of Curves -- 7.3 Proper Base Change Theorem -- 7.4 Cohomology with Proper Support -- 7.5 Cohomological Dimension of Rf -- 7.6 Local Acyclicity -- 7.7 Smooth Base Change Theorem -- 7.8 Finiteness of Rf -- 8. Duality -- 8.1 Extensions of Henselian Discrete Valuation Rings -- 8.2 Trace Morphisms -- 8.3 Duality for Curves.

8.4 The Functor Rf! -- 8.5 Poincare Duality -- 8.6 Cohomology Classes of Algebraic Cycles -- 9. Finiteness Theorems -- 9.1 Sheaves with Group Actions -- 9.2 Nearby Cycle and Vanishing Cycle -- 9.3 Generic Base Change Theorem and Generic Local Acyclicity -- 9.4 Finiteness of R -- 9.5 Finiteness Theorems -- 9.6 Biduality -- 10. `-adic Cohomology -- 10.1 Adic Formalism -- 10.2 Grothendieck-Ogg-Shafarevich Formula -- 10.3 Frobenius Correspondences -- 10.4 Lefschetz Trace Formula -- 10.5 Grothendieck's Formula of L-functions -- Bibliography -- Index.