Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems

Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems / Igor Burban, Yuriy Drozd.

In this article the authors develop a new method to deal with maximal Cohen-Macaulay modules over non-isolated surface singularities. In particular, they give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen-Macaulay m...

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Bibliographic Details
Main Authors: Burban, Igor, 1977- (Author), Drozd, Yurij A. (Author)
Format: eBook
Language:English
Published: Providence, Rhode Island : American Mathematical Society, 2017.
Series:Memoirs of the American Mathematical Society ; no. 1178.
Subjects:
Cohen-Macaulay modules.
Modules (Algebra)
Singularities (Mathematics)
Matrices.
Cohen-Macaulay modules
Matrices
Online Access:Click for online access
Table of Contents:
  • Cover; Title page; Introduction, motivation and historical remarks; Chapter 1. Generalities on maximal Cohen-Macaulay modules; 1.1. Maximal Cohen-Macaulay modules over surface singularities; 1.2. On the category \CM^{ }(\rA); Chapter 2. Category of triples in dimension one; Chapter 3. Main construction; Chapter 4. Serre quotients and proof of Main Theorem; Chapter 5. Singularities obtained by gluing cyclic quotient singularities; 5.1. Non-isolated surface singularities obtained by gluing normal rings; 5.2. Generalities about cyclic quotient singularities.
  • 5.3. Degenerate cusps and their basic properties5.4. Irreducible degenerate cusps; 5.5. Other cases of degenerate cusps which are complete intersections; Chapter 6. Maximal Cohen-Macaulay modules over \kk\llbracket, \rrbracket/(²+ ³- ); Chapter 7. Representations of decorated bunches of chains-I; 7.1. Notation; 7.2. Bimodule problems; 7.3. Definition of a decorated bunch of chains; 7.4. Matrix description of the category \Rep(\dX); 7.5. Strings and Bands; 7.6. Idea of the proof; 7.7. Decorated Kronecker problem; Chapter 8. Maximal Cohen-Macaulay modules over degenerate cusps-I.
  • 8.1. Maximal Cohen-Macaulay modules on cyclic quotient surface singularities8.2. Matrix problem for degenerate cusps; 8.3. Reconstruction procedure; 8.4. Cohen-Macaulay representation type and tameness of degenerate cusps; Chapter 9. Maximal Cohen-Macaulay modules over degenerate cusps-II; 9.1. Maximal Cohen-Macaulay modules over \kk\llbracket, \rrbracket/(); 9.2. Maximal Cohen-Macaulay modules over \kk\llbracket, \rrbracket/(,); 9.3. Degenerate cusp \kk\llbracket, \rrbracket/(,); Chapter 10. Schreyer's question.
  • Chapter 11. Remarks on rings of discrete and tame CM-representation type11.1. Non-reduced curve singularities; 11.2. Maximal Cohen-Macaulay modules over the ring ̃ ((1,0)); 11.3. Other surface singularities of discrete and tame CM-representation type; 11.4. On deformations of certain non-isolated surface singularities; Chapter 12. Representations of decorated bunches of chains-II; 12.1. Decorated conjugation problem; 12.2. Some preparatory results from linear algebra; 12.3. Reduction to the decorated chessboard problem; 12.4. Reduction procedure for the decorated chessboard problem.
  • 12.5. Indecomposable representations of a decorated chessboard12.6. Proof of the Classification Theorem; References; Back Cover.

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