CONTENTS Abstract iv 0. Introduction 1 1. Statements of results 4 2. Preliminaries 10 3. The trace ring 17 4. Affine fixed rings 23 5. Borho theory 33 6. The correspondence $:Speci2 —o Spec RG 45 7. Lying over 58 8. Characterizations of linearly reductive groups through actions on affine Pi-algebras 67 9. Actions by inner automorphisms 80 9.1. Spec-inner automorphisms 81 9.2. A Bergman-Isaacs theorem, chain conditions, and integrality 86 9.3. Prime ideals in the ring extension RG C R 91 9.4. Connections with actions of finite groups 98 References 103 iii
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