Cohen-Macaulay modules over some non-reduced curve singularities

TL;DR

This paper investigates Cohen-Macaulay modules over specific non-reduced curve singularities, establishing their tame representation type and explicitly classifying modules for certain cases, with applications to matrix factorizations.

Contribution

It proves tameness for certain singularities and provides explicit classifications of Cohen-Macaulay modules, including matrix factorizations, over these non-reduced curves.

Findings

Rings $k[[x,y,z]]/(xy, y^q -z^2)$ have tame Cohen-Macaulay representation type.

Explicit classification of indecomposable Cohen-Macaulay modules for $k[[x,y,z]]/(xy, z^2)$.

Constructed explicit families of indecomposable matrix factorizations.

Abstract

In this article, we study Cohen-Macaulay modules over non-reduced curve singularities. We prove that the rings $k[[x,y,z]]/(xy, y^q -z^2)$ have tame Cohen-Macaulay representation type. For the singularity $k[[x,y,z]]/(xy, z^2)$ we give an explicit description of all indecomposable Cohen--Macaulay modules and apply the obtained classification to construct explicit families of indecomposable matrix factorizations of $(xy)^2 \in k[[x,y]]$.