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

ISBN-13: 9781470425371 / Angielski

Igor Burban;Yuriy Drozd

Opis

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 modules. Next, the authors prove that the degenerate cusp singularities have tame Cohen-Macaulay representation type. The authors' approach is illustrated on the case of $mathbb[[ x,y,z]]/(xyz)$ 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 the authors call representations of decorated bunches of chains. They prove that these matrix problems have tame representation type and describe the underlying canonical forms.

Spis treści

• Generalities on maximal Cohen-Macaulay modules
• Category of triples in dimension one
• Main construction
• Serre quotients and proof of Main Theorem
• Singularities obtained by gluing cyclic quotient singularities
• Maximal Cohen-Macaulay modules over $\mathbb[[ x, y, z]]/(x^2 + y^3 - xyz)$
• Representations of decorated bunches of chains-I
• Maximal Cohen-Macaulay modules over degenerate cusps-I
• Maximal Cohen-Macaulay modules over degenerate cusps-II
• Schreyer's question
• Remarks on rings of discrete and tame CM-representation type
• Representations of decorated bunches of chains-II
• References.