xii PREFACE develop. In particular, the work of Drozd-Ro˘ ıter and H. Jacobinski around this time introduced the conditions we call “the Drozd-Ro˘ıter conditions” classifying commutative orders with only a finite number of non-isomorphic indecomposable lattices. Finally, M. Hochster’s study of the homological conjectures emphasized the importance of the max- imal Cohen-Macaulay condition (even for non-finitely generated mod- ules). The equality of the geometric invariant of dimension with the arithmetic one of depth makes this class of modules easy to work with, simultaneously ensuring that they faithfully reflect the structure of the ring. The main focus of this book is on the problem of classifying Cohen- Macaulay local rings having only a finite number of indecomposable maximal Cohen-Macaulay modules, that is, having finite CM type. No- tice that we wrote “the problem,” rather than “the solution.” Indeed, there is no complete classification to date. There are many partial results, however, including complete classifications in dimensions zero and one, a characterization in dimension two under some mild assump- tions, and a complete understanding of the hypersurface singularities with this property. The tools developed to obtain these classifications have many applications to other problems as well, in addition to their inherent beauty. In particular there are applications to the study of other representation types, including countable type and bounded type. This is not the first book about the representation theory of Cohen- Macaulay modules over Cohen-Macaulay local rings. The text [Yos90] by Y. Yoshino is a fantastic book and an invaluable resource, and has inspired us both on countless occasions. It has been the canonical ref- erence for the subject for twenty years. In those years, however, there have been many advances. To give just two examples, we mention C. Huneke and Leuschke’s elementary proof in 2002 of Auslander’s the- orem that finite CM type implies isolated singularity, and R. Wiegand’s 2000 verification of F.-O. Schreyer’s conjecture that finite CM type as- cends to and descends from the completion. These developments alone might justify a new exposition. Furthermore, there are many facets of the subject not covered in Yoshino’s book, some of which we are quali- fied to describe. Thus this book might be considered simultaneously an updated edition of [Yos90], a companion volume, and an alternative.
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