Examples of degenerations of Cohen-Macaulay modules

TL;DR

This paper investigates how maximal Cohen-Macaulay modules degenerate, providing examples and characterizing degenerations over certain singularities as extensions, with implications for modules over finite representation type algebras.

Contribution

It offers new examples of degenerations and characterizes them as extensions over specific singularities and algebras, advancing understanding of Cohen-Macaulay module degenerations.

Findings

Degenerations over type (A_n) hypersurface singularities are given by extensions.

All extended degenerations over finite representation type algebras are generated by Auslander-Reiten sequences.

Provided explicit examples illustrating degeneration phenomena.

Abstract

We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of such degenerations. It is proved that such degenerations over an even-dimensional simple hypersurface singularity of type $(A_n)$ are given by extensions. We also prove that all extended degenerations of maximal Cohen-Macaulay modules over a Cohen-Macaulay complete local algebra of finite representation type are obtained by iteration of extended degenerations of Auslander-Reiten sequences.