PREFACE xv dimension, which we believe deserves to be better known. The the- ory of Auslander-Reiten sequences and quivers, of course, is essential. Chapter 12 establishes some homological tools and introduces totally reflexive modules, whose homological behavior over general local rings mimics that of MCM modules over Gorenstein rings. The last four chapters consider other representation types, namely countable and bounded CM type, and finite CM type in higher dimen- sions. Chapter 14 uses recent results of I. Burban and Drozd, based on a modification of the conductor-square construction, to prove Buchweitz- Greuel-Schreyer’s classification of the hypersurface singularities with countable CM type. It also proves certain structural results for rings of countable CM type, due to Huneke and Leuschke. Chapter 15 con- tains a proof of the first Brauer-Thrall conjecture, that an excellent isolated singularity with bounded CM type necessarily has finite CM type. Our presentation follows the original proofs of E. Dieterich and Yoshino. The Brauer-Thrall theorem is then used, in Chapter 16, to prove that two three-dimensional examples have finite CM type. We also quote the theorem of D. Eisenbud and Herzog which classifies the homogeneous rings of finite CM type in particular, their result says that there are no examples in dimension 3 other than the ones we have described in the text. Finally, in Chapter 17, we consider the rings of bounded but infinite CM type. It happens that for hypersurface rings they are precisely the same as the rings of countable but infinite CM type. We also classify the one-dimensional rings of bounded CM type. We include two Appendices. In Appendix A, we gather for ease of reference some basic definitions and results of commutative algebra that are prerequisites for the book. Appendix B, on the other hand, contains material that we require from ramification theory that is not generally covered in a general commutative algebra course. It includes the basics on unramified and ´ etale homomorphisms, Henselian rings, ramification of prime ideals, and purity of the branch locus. We make essential use of these concepts, but they are peripheral to the main material of the book. The knowledgeable reader will have noticed significant overlap be- tween the topics mentioned above and those covered by Yoshino in his book [Yos90]. To a certain extent this is unavoidable the basics of the area are what they are, and any book on Cohen-Macaulay represen- tation types will mention them. However, the reader should be aware
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