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

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Date
2015-05-13MFO Scientific Program
Research in Pairs 2013Series
Oberwolfach Preprints;2015,06Author
Burban, Igor
Drozd, Yuriy
Metadata
Show full item recordOWP-2015-06
Abstract
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 $\mathbb{k}[[x, y, z]]/(xyz)$ 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 we call representations of decorated bunches of chains. We prove that these matrix problems have tame representation type and describe the underlying canonical forms.