xiv PREFACE arbitrary dimension) the necessary invariant theory and results of Aus- lander relating a ring of invariants to the associated skew group ring. These results are applied in Chapter 6 to show that two-dimensional rings of invariants have finite CM type. In particular this applies to the Kleinian singularities, also known as Du Val singularities, rational double points, or ADE hypersurface singularities. We also describe some aspects of the McKay correspondence, including the geometric results due to M. Artin and J.-L. Verdier. Finally Chapter 7 gives the full classification of complete local two-dimensional C-algebras of finite CM type. This chapter also includes Auslander’s theorem mentioned earlier that finite CM type implies isolated singularity. In dimensions higher than two, our understanding of finite CM type is imperfect. We do, however, understand the Gorenstein case more or less completely. By a result of J. Herzog, a complete Gorenstein lo- cal ring of finite CM type is a hypersurface ring these are completely classified in the equicharacteristic case. This classification is detailed in Chapter 9, including the theorem of R.-O. Buchweitz, G.-M. Greuel, and Schreyer which states that if a complete equicharacteristic hyper- surface singularity over an algebraically closed field has finite CM type, then it is a simple singularity in the sense of V. I. Arnol d. We also write down the matrix factorizations for the indecomposable MCM modules over the Kleinian singularities, from which the matrix fac- torizations in arbitrary dimension can be obtained. Our proof of the Buchweitz-Greuel-Schreyer result is by reduction to dimension two via the double branched cover construction and H. Kn¨ orrer’s periodicity theorem. Chapter 8 contains these background results, after a brief presentation of the theory of matrix factorizations. Chapter 10 addresses the critical questions of ascent and descent of finite CM type along ring extensions, particularly between a Cohen- Macaulay local ring and its completion, as well as passage to a local ring with a larger residue field. This allows us to extend the clas- sification theorem for hypersurface singularities of finite CM type to non-algebraically closed fields. Chapters 11 and 13 describe two powerful tools in the study of maximal Cohen-Macaulay modules over Cohen-Macaulay rings: MCM approximations and Auslander-Reiten sequences. We are not aware of another complete, concise and explicit treatment of Auslander and Buchweitz’s theory of MCM approximations and hulls of finite injective
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