A note on deformations of finite dimensional modules over $\mathbb{k}$-algebras

Abstract

Let $\mathbb{k}$ be a field, and let $Λ$ be a (not necessarily finite dimensional) $\mathbb{k}$-algebra. Let $V$ be a left $Λ$-module such that is finite dimensional over $\mathbb{k}$. Assume further that $V$ has a weak universal deformation ring $R^w(Λ,V)$, which is a complete Noetherian commutative local $\mathbb{k}$-algebra with residue field $\mathbb{k}$. We prove in this note that under certain conditions on the $Λ$-module $V$, that if $R^w(Λ,V)$ is a quotient of $\mathbb{k}[\![t]\!]$, then $R^w(Λ,V)$ is either isomorphic to $\mathbb{k}$, or $\mathbb{k}[\![t]\!]$, or to $\mathbb{k}[\![t]\!]/(t^N)$ for some integer $N\geq 2$.