Preface

This book is about the representation theory of commutative local

rings, specifically the study of maximal Cohen-Macaulay modules over

Cohen-Macaulay local rings.

The guiding principle of representation theory, broadly speaking,

is that we can understand an algebraic structure by studying the sets

upon which it acts. Classically, this meant understanding finite groups

by studying the vector spaces they act upon; the powerful tools of lin-

ear algebra can then be brought to bear, revealing information about

the group that was otherwise hidden. In other branches of represen-

tation theory, such as the study of finite-dimensional associative alge-

bras, sophisticated technical machinery has been built to investigate

the properties of modules, and how restrictions on modules over a ring

restrict the structure of the ring.

The representation theory of maximal Cohen-Macaulay modules

began in the late 1970s and grew quickly, inspired by three other ar-

eas of algebra. Spectacular successes in the representation theory of

finite-dimensional algebras during the 1960s and 70s set the standard

for what one might hope for from a representation theory. In partic-

ular, this period saw: P. Gabriel’s introduction of the representations

of quivers and his theorem that a quiver has finite representation type

if and only if it is a disjoint union of ADE Coxeter-Dynkin diagrams;

M. Auslander’s influential Queen Mary notes applying his work on

functor categories to representation theory; Auslander and I. Reiten’s

foundational work on AR sequences; and key insights from the Kiev

school, particularly Y. Drozd, L. A. Nazarova, and A. V. Ro˘ ıter. All

these advances continued the work on finite representation type be-

gun in the 1940s and 50s by T. Nakayama, R. Brauer, R. Thrall, and

J. P. Jans. Secondly, the study of lattices over orders, a part of integral

representation theory, blossomed in the late 1960s. Restricting atten-

tion to lattices rather than arbitrary modules allowed a rich theory to

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