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

About this Title

Igor Burban, Universität zu Köln, Mathematisches Institut, Weyertal 86-90, D-50931 Köln, Germany and Yuriy Drozd, Institute of Mathematics National Academy of Sciences, Tereschenkivska str. 3, 01004 Kyiv, Ukraine

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)
Published electronically: March 16, 2017
Keywords: Maximal Cohen–Macaulay modules, matrix factorizations, non–isolated surface singularities, degenerate cusps, tame matrix problems
MSC: Primary 16G50, 16G60, 13C14, 13H10

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Table of Contents


  • Introduction, motivation and historical remarks
  • 1. Generalities on maximal Cohen–Macaulay modules
  • 2. Category of triples in dimension one
  • 3. Main construction
  • 4. Serre quotients and proof of Main Theorem
  • 5. Singularities obtained by gluing cyclic quotient singularities
  • 6. Maximal Cohen–Macaulay modules over $\mathbb {k}\lBrack x, y, z\rBrack /(x^2 + y^3 - xyz)$
  • 7. Representations of decorated bunches of chains–I
  • 8. Maximal Cohen–Macaulay modules over degenerate cusps–I
  • 9. Maximal Cohen–Macaulay modules over degenerate cusps–II
  • 10. Schreyer’s question
  • 11. Remarks on rings of discrete and tame CM–representation type
  • 12. Representations of decorated bunches of chains–II


In this article we develop a new method to deal with maximal Cohen–Macaulay modules over non–isolated surface singularities. In particular, we give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen–Macaulay modules. Next, we prove that the degenerate cusp singularities have tame Cohen–Macaulay representation type. Our approach is illustrated on the case of $\mathbb {k}\lBrack x,y,z\rBrack /(xyz)$ as well as several other rings. This study of maximal Cohen–Macaulay modules over non–isolated 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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