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

TL;DR

This paper introduces a new method for studying maximal Cohen-Macaulay modules over non-isolated surface singularities, resolving longstanding questions and classifying certain singularities as tame in their module representation type.

Contribution

It develops a novel approach to analyze Cohen-Macaulay modules over complex surface singularities and introduces a new class of linear algebra problems called representations of decorated bunches of chains.

Findings

Negative answer to Schreyer's old question about countably many indecomposable modules

Degenerate cusp singularities are of tame Cohen-Macaulay representation type

Matrix problems of decorated bunches of chains have tame representation type

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 $k\llbracket x,y,z\rrbracket/(xyz)$ 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.