- Graham J. Leuschke, Roger Wiegand.
- Providence, R.I. : American Mathematical Society, c2012.
- Physical description
- xvii, 367 p. : ill ; 26 cm.
- Mathematical surveys and monographs ; no. 181.
Math & Statistics Library
QA3 .A4 V.181
- Unknown QA3 .A4 V.181
- Includes bibliographical references and index.
- Chapter 1. The Krull-Remak-Schmidt theorem
- Chapter 2. Semigroups of modules
- Chapter 3. Dimension zero
- Chapter 5. Invariant theory
- Chapter 6. Kleinian singularities and finite CM type
- Chapter 7. Isolated singularities and dimension two
- Chapter 8. The double branched cover
- Chapter 9. Hypersurfaces with finite CM type
- Chapter. 10. Ascent and descent
- Chapter 11. Auslander-Buchweitz theory
- Chapter 12. Totally reflexive modules
- Chapter 13. Auslander-Reiten theory
- Chapter 14. Countable Cohen-Macaulay type
- Chapter 15. The Brauer-Thrall conjectures
- Chapter 16. Finite CM type in higher dimensions
- Chapter 17. Bounded CM type
- Appendix A.
- Basics and background
- Appendix B.
- Ramification theory.
- Publisher's Summary
- 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.
(source: Nielsen Book Data)
- Cohen-Macaulay modules.
- Representations of rings (Algebra)
- Commutative algebra -- Theory of modules and ideals -- Cohen-Macaulay modules.
- Associative rings and algebras -- Representation theory of rings and algebras -- Cohen-Macaulay modules.
- Commutative algebra -- Ring extensions and related topics -- Étale and flat extensions; Henselization; Artin approximation.
- Commutative algebra -- Theory of modules and ideals -- Structure, classification theorems.
- Commutative algebra -- Theory of modules and ideals -- Module categories.
- Commutative algebra -- Local rings and semilocal rings -- Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
- Associative rings and algebras -- Representation theory of rings and algebras -- Representations of Artinian rings.
- Associative rings and algebras -- Representation theory of rings and algebras -- Representation type (finite, tame, wild, etc.)
- Associative rings and algebras -- Representation theory of rings and algebras -- Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers.
- Publication date
- Mathematical surveys and monographs ; v. 181
- 9780821875810 (alk. paper)
- 0821875817 (alk. paper)