PREFACE xiii In addition to telling the basic story of finite CM type, our choice of material is guided by a number of themes. (i) For a homomorphism of local rings R −→ S, which maximal Cohen-Macaulay modules over S “come from” R is a basic question. It is especially important when S = R, the completion of R, for then the Krull-Remak-Schmidt uniqueness theorem holds for direct-sum de- compositions of R-modules. (ii) The failure of the Krull-Remak-Schmidt theorem is often more interesting than its success. We can often quantify exactly how badly it fails. (iii) A certain amount of non-commutativity can be useful even in pure commutative algebra. In particular, the endomorphism ring of a module, while technically a non-commutative ring, should be a standard object of consideration in commutative algebra. (iv) An abstract, categorical point of view is not always a good thing in and of itself. We tend to be stubbornly concrete, emphasizing explicit constructions over universal properties. The main material of the book is divided into 17 chapters. The first chapter contains some vital background information on the Krull- Remak-Schmidt Theorem, which we view as a version of the Fundamen- tal Theorem of Arithmetic for modules, and on the relationship between modules over a local ring R and over its completion R. Chapter 2 is devoted to an analysis of exactly how badly the Krull-Remak-Schmidt Theorem can fail. Nothing here is specifically about Cohen-Macaulay rings or maximal Cohen-Macaulay modules. Chapters 3 and 4 contain the classification theorems for Cohen- Macaulay local rings of finite CM type in dimensions zero and one. Here essentially everything is known. In particular Chapter 3 introduces an auxiliary representation-theoretic problem, the Artinian pair, which is then used in Chapter 4 to solve the problem of finite CM type over one-dimensional rings via the conductor-square construction. The two-dimensional Cohen-Macaulay local rings of finite CM type are at a focal point in our telling of the theory, with connections to alge- braic geometry, invariant theory, group representations, solid geometry, representations of quivers, and other areas, by way of the McKay cor- respondence. Chapter 5 sets the stage for this material, introducing (in

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