Contents Introduction vii Chapter 1. The Intersection Pairing for One-Dimensional Schemes 1 1. Preliminaries 2 2. The Determinant of Cohomology Functor 4 3. The Norm Functor for Zero-Dimensional Schemes 16 4. The Definition of the Intersection Pairing 20 5. The Intersection Pairing and Norms 28 6. The Norm Functor for Divisors 32 7. Other Properties of the Intersection Pairing 38 8. Extensions of the Base Field 44 Chapter 2. The Intersection Pairing for Families of One-dimensional Schemes 47 1. Introduction 47 2. Horizontal Divisors 48 3. The Norm Functor 51 4. The Intersection Pairing 56 5. The Determinant of Cohomology Line Bundle 62 6. Flat Base Change 78 Chapter 3. The Riemann-Roch Isomorphism 80 1. The Relative Dualizing Sheaf 80 2. The Adjunction Formula 86 3. The Duality Isomorphism on Determinants 94 4. The Riemann-Roch Isomorphism 99 Chapter 4. Intersection Functions on Complex Curves 101 1. Motivation: The Non-Archimedean Situation 101 2. The Archimedean Case: Basic Definitions 109 3. Intersection Functions on Nonsingular Curves 115 4. Intersection Functions on Singular Curves: The Existence Theorem 123 5. Classification of Intersection Functions: Preliminaries 128 6. Classification of Intersection Functions 137 7. Chern Forms of Intersection Functions 144 8. Chern Forms of Intersection Functions: Proofs 147 9. Chern Forms of Intersection Functions: Another Interpretation 153 Chapter 5. The Arithmetic Riemann-Roch Isomorphism 161 1. Norms for Determinants of Cohomology 161 2. The Riemann-Roch Isomorphism for Arithmetic Surfaces 171 Bibliography 174 v

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