This is a widely revised version of lectures I gave at Tokyo Metropolitan University in 1987, originally written in Japanese.

Throughout the book our attention is directed to the point – how we can classify CM modules over a given CM ring, or classify CM rings which have essentially a finite number of CM modules? Being analogous to lattices over orders, this question seems to have arised very naturally. The first approach to CM modules in this direction was, perhaps, done by Herzog, and it turned out that there are, in themselves, two basic aspects of this problem. The first is an algebraic or represent at ion-theoretic side, in which Auslander and Reiten made remarkable progress by the powerful use of AR sequences. The second is a geometric side, more precisely, the spectra of CM local rings having only a finite number of CM modules should be well-behaved singularities. Artin, Verdier, Knorrer, Buchweitz, Eisenbud, Greuel and many others are concerned with this direction.

In this book I have tried to give a systematic treatment of the subject, as self-contained as possible, but there is no intention to make this book an encyclopedia of CM modules. Therefore, even in the case that more general treatments for definitions or proofs exist, I have prefered to give direct expositions, which, I am afraid, the experienced reader might feel unwise of me.

I conclude this preface with acknowledgements and thanks to all who supported the preparation of the book.