On universal deformation rings and stable homogeneous tubes

TL;DR

This paper investigates the deformation rings of modules in stable homogeneous tubes over finite dimensional algebras, showing that under certain conditions these rings are isomorphic to formal power series rings, thus providing explicit descriptions.

Contribution

It proves that modules in stable homogeneous tubes have versal deformation rings isomorphic to  t , and establishes universality in specific algebraic settings.

Findings

Deformation rings are isomorphic to  t  for modules in stable homogeneous tubes.

Under certain conditions, these deformation rings are universal.

Results apply to symmetric special biserial algebras with band modules.

Abstract

Let $\mathbf{k}$ be a field of any characteristic and let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra. We prove that if $V$ is a finite dimensional right $\Lambda$-module that lies in the mouth of a stable homogeneous tube $\mathfrak{T}$ of the Auslander-Reiten quiver $\Lambda$ with $\underline{\mathrm{End}}_\Lambda(V)$ a division ring, then $V$ has a versal deformation ring $R(\Lambda,V)$ isomorphic to $\mathbf{k}[\![t]\!]$. As consequence we obtain that if $\mathbf{k}$ is algebraically closed, $\Lambda$ is a symmetric special biserial $\mathbf{k}$-algebra and $V$ is a band $\Lambda$-module with $\underline{\mathrm{End}}_\Lambda(V) \cong \mathbf{k}$ that lies in the mouth of its homogeneous tube, then $R(\Lambda,V)$ is universal and isomorphic to $\mathbf{k}[\![t]\!]$.