Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems
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Bibliographic Information
Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems
（Memoirs of the American Mathematical Society, no. 1178）
American Mathematical Society, 2017
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Note
"Volume 248, number 1178 (fourth of 5 numbers), July 2017"
Bibliography: p. 111-114
Description and Table of Contents
Description
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 $\mathbb{k}[[ x,y,z]]/(xyz)$ 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 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.
Table of Contents
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 $\mathbb{k}[[ x, y, z]]/(x^2 + y^3 - xyz)$
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.
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