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

Abstract

Let $\mathbb{k}$ be a field, and let $\Lambda$ be a (not necessarily finite dimensional) $\mathbb{k}$-algebra. Let $V$ be a left $\Lambda$-module such that is finite dimensional over $\mathbb{k}$. Assume further that $V$ has a weak universal deformation ring $R^w(\Lambda,V)$, 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 $\Lambda$-module $V$, that if $R^w(\Lambda,V)$ is a quotient of $\mathbb{k}[\![t]\!]$, then $R^w(\Lambda,V)$ is either isomorphic to $\mathbb{k}$, or $\mathbb{k}[\![t]\!]$, or to $\mathbb{k}[\![t]\!]/(t^N)$ for some integer $N\geq 2$.