Maximal Cohen-Macaulay modules over non-isolated surface singularities

Details Maximal Cohen-Macaulay modules over non-isolated surface singularities Maximal Cohen-Macaulay modules over non-isolated surface singularities

2010-02-16

arxiv.org/abs/1002.3042v1

Mathematics

Algebraic Geometry

60 pages

Scientific paper

In this paper we develop a new method to deal with 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 Cohen-Macaulay modules. Next, we show that the degenerate cusp singularities have tame Cohen-Macaulay representation type. Our approach is illustrated on the case of the rings k[[x,y,z]]/(x^3 + y^2 - xyz), k[[x,y,z]]/(xyz) and k[[x,y,u,v]]/(xy, uv). This study of 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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