viii CONTENTS §3. Group representations and the McKay-Gabriel quiver 71 §4. Exercises 77 Chapter 6. Kleinian Singularities and Finite CM Type 81 §1. Invariant rings in dimension two 81 §2. Kleinian singularities 83 §3. McKay-Gabriel quivers of the Kleinian singularities 91 §4. Geometric McKay correspondence 97 §5. Exercises 106 Chapter 7. Isolated Singularities and Dimension Two 109 §1. Miyata’s theorem 109 §2. Isolated singularities 113 §3. Two-dimensional CM rings of finite CM type 117 §4. Exercises 120 Chapter 8. The Double Branched Cover 123 §1. Matrix factorizations 123 §2. The double branched cover 128 §3. Kn¨ orrer’s periodicity 133 §4. Exercises 139 Chapter 9. Hypersurfaces with Finite CM Type 141 §1. Simple singularities 141 §2. Hypersurfaces in good characteristics 144 §3. Gorenstein rings of finite CM type 150 §4. Matrix factorizations for the Kleinian singularities 151 §5. Bad characteristics 159 §6. Exercises 160 Chapter 10. Ascent and Descent 163 §1. Descent 163 §2. Ascent to the completion 164 §3. Ascent along separable field extensions 170 §4. Equicharacteristic Gorenstein singularities 172 §5. Exercises 173 Chapter 11. Auslander-Buchweitz Theory 175 §1. Canonical modules 175 §2. MCM approximations and FID hulls 179 §3. Numerical invariants 189
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