Cohen-Macaulay representations
"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 indecomp…
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- Full title :Cohen-Macaulay representations / Graham J. Leuschke, Roger Wiegand.
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" --Back cover.
Read More ↓Edition Statements
- Publication Date:©2012.
Publication Statements
- Publisher(s):American mathematical Society
- Place of Publication:Providence, R.I
- Language:eng
- Publication Date:©2012.
The Physical Object
- Description:xvii, 367 pages : : illustrations ; : 26 cm.
Edition Identifiers
- ISBN:9780821875810, 0821875817
- oclc:ocn778074251
- LC Classification:QA251.3
- Dewey Decimal:512/.44
Table of Contents
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.
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