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Maximal Cohen–Macaulay Modules Over Non–Isolated Surface Singularities and Matrix Problems


About this Title

Igor Burban and Yuriy Drozd

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 248, Number 1178
ISBNs: 978-1-4704-2537-1 (print); 978-1-4704-4058-9 (online)
DOI: https://doi.org/10.1090/memo/1178
Published electronically: March 16, 2017
Keywords:Maximal Cohen–Macaulay modules, matrix factorizations, non–isolated surface singularities, degenerate cusps, tame matrix problems

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Table of Contents


Chapters

  • Introduction, motivation and historical remarks
  • Chapter 1. Generalities on maximal Cohen–Macaulay modules
  • 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
  • Chapter 6. Maximal Cohen–Macaulay modules over $\kk \llbracket x, y, z\rrbracket /(x^2 + y^3 - xyz)$
  • Chapter 7. Representations of decorated bunches of chains–I
  • Chapter 8. Maximal Cohen–Macaulay modules over degenerate cusps–I
  • Chapter 9. Maximal Cohen–Macaulay modules over degenerate cusps–II
  • Chapter 10. Schreyer’s question
  • Chapter 11. Remarks on rings of discrete and tame CM–representation type
  • Chapter 12. Representations of decorated bunches of chains–II

Abstract


In this article we develop a new method to deal with maximal CohenâĂŞMacaulay modules over nonâĂŞisolated surface singularities. In particular, we give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal CohenâĂŞMacaulay modules. Next, we prove that the degenerate cusp singularities have tame CohenâĂŞMacaulay representation type. Our approach is illustrated on the case of as well as several other rings. This study of maximal CohenâĂŞMacaulay modules over nonâĂŞisolated singularities leads to a new class of problems of linear algebra, which we call representations of decorated bunches of chains. We prove that these matrix problems have tame representation type and describe the underlying canonical forms.

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