Universal deformation rings of modules for algebras of dihedral type of polynomial growth

TL;DR

This paper classifies the universal deformation rings of certain modules over dihedral type algebras of polynomial growth, showing only three possible ring types occur.

Contribution

It explicitly describes all modules with stable endomorphism ring k and determines their universal deformation rings for dihedral type algebras.

Findings

Universal deformation rings are isomorphic to k, k[[t]]/(t^2), or k[[t]]

Only three isomorphism types of deformation rings occur

Provides a complete classification for modules with stable endomorphism ring k

Abstract

Let k be an algebraically closed field, and let \Lambda\ be an algebra of dihedral type of polynomial growth as classified by Erdmann and Skowro\'{n}ski. We describe all finitely generated \Lambda-modules V whose stable endomorphism rings are isomorphic to k and determine their universal deformation rings R(\Lambda,V). We prove that only three isomorphism types occur for R(\Lambda,V): k, k[[t]]/(t^2) and k[[t]].