**Mathematical Surveys and Monographs**

Volume: 181;
2012;
367 pp;
Hardcover

MSC: Primary 13; 16;

Print ISBN: 978-0-8218-7581-0

Product Code: SURV/181

List Price: $96.00

AMS Member Price: $76.80

MAA member Price: $86.40

**Electronic ISBN: 978-0-8218-8790-5
Product Code: SURV/181.E**

List Price: $96.00

AMS Member Price: $76.80

MAA member Price: $86.40

#### Supplemental Materials

# Cohen-Macaulay Representations

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*Graham J. Leuschke; Roger Wiegand*

This book is a comprehensive treatment of the representation theory of
maximal Cohen-Macaulay (MCM) modules over local rings. This topic is
at the intersection of commutative algebra, singularity theory, and
representations of groups and algebras.

Two introductory chapters treat the Krull-Remak-Schmidt Theorem on
uniqueness of direct-sum decompositions and its failure for modules
over local rings. Chapters 3–10 study the central problem of
classifying the rings with only finitely many indecomposable MCM
modules up to isomorphism, i.e., rings of finite CM type. The
fundamental material—ADE/simple singularities, the double branched
cover, Auslander-Reiten theory, and the Brauer-Thrall conjectures—is
covered clearly and completely. Much of the content has never before
appeared in book form. Examples include the representation theory of
Artinian pairs and Burban-Drozd's related construction in dimension
two, an introduction to the McKay correspondence from the point of
view of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM
approximation theory, and a careful treatment of nonzero
characteristic. The remaining seven chapters present results on
bounded and countable CM type and on the representation theory of
totally reflexive modules.

#### Readership

Research mathematicians interested in algebra, in particular, theory of rings and modules.

#### Table of Contents

# Table of Contents

## Cohen-Macaulay Representations

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Introduction xi12 free
- The Krull-Remak-Schmidt theorem 120 free
- Semigroups of modules 1332
- Dimension zero 2948
- Dimension one 4160
- Invariant theory 6180
- Kleinian singularities and finite CM type 81100
- Isoalted singularities and classification in dimension two 109128
- The double branched cover 123142
- Hypersurfaces with finite CM type 141160
- Ascent and descent 163182
- Auslander-Buchweitz theory 175194
- Totally reflexive modules 203222
- Auslander-Reiten theory 217236
- Countable Cohen-Macaulay type 241260
- The Brauer-Thrall conjectures 267286
- Finite CM type in higher dimensions 287306
- Bounded CM type 297316
- Basics and background 309328
- Ramification theory 321340
- Bibliography 341360
- Index 355374 free
- Back Cover Back Cover1390