Contents IX Chapter 7. Sheaves and Sheaf Cohomology 145 §7.1. Sheaves 145 §7.2. Morphisms of Sheaves 150 §7.3. Operations on Sheaves 152 §7.4. Sheaf Cohomology 157 §7.5. Classes of Acyclic Sheaves 163 §7.6. Ringed Spaces 168 §7.7. De Rham Cohomology 172 §7.8. Cech Cohomology 174 §7.9. Line Bundles and Cech Cohomology 180 Exercises 182 Chapter 8. Coherent Algebraic Sheaves 185 §8.1. Abstract Varieties 186 §8.2. Localization 189 §8.3. Coherent and Quasi-coherent Algebraic Sheaves 194 §8.4. Theorems of Artin-Rees and Krull 197 §8.5. The Vanishing Theorem for Quasi-coherent Sheaves 199 §8.6. Cohomological Characterization of Affine Varieties 200 §8.7. Morphisms - Direct and Inverse Image 204 §8.8. An Open Mapping Theorem 207 Exercises 212 Chapter 9. Coherent Analytic Sheaves 215 §9.1. Coherence in the Analytic Case 215 §9.2. Oka's Theorem 217 §9.3. Ideal Sheaves 221 §9.4. Coherent Sheaves on Varieties 225 §9.5. Morphisms between Coherent Sheaves 226 §9.6. Direct and Inverse Image 229 Exercises 234 Chapter 10. Stein Spaces 237 §10.1. Dolbeault Cohomology 237 §10.2. Chains of Syzygies 243 §10.3. Functional Analysis Preliminaries 245
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