A short note on deformations of (strongly) Gorenstein-projective modules over the dual numbers

Let $\mathbf{k}$ be a field of arbitrary characteristic, and let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra. In this short note we prove that if $V$ is a finitely generated strongly Gorenstein-projective left $\Lambda$-module whose stable endomorphism ring $\underline{\mathrm{End}}_{\Lambda}(V)$ is isomorphic to $\mathbf{k}$, then $V$ has an universal deformation ring $R(\Lambda,V)$ isomorphic to the ring of dual numbers $\mathbf{k}[\epsilon]$ with $\epsilon^2=0$. As a consequence, we obtain the following result. Assume that $Q$ is a finite connected acyclic quiver, let $\mathbf{k} Q$ be the corresponding path algebra and let $\Lambda = \mathbf{k} Q[\epsilon] = \mathbf{k} Q\otimes_{\mathbf{k}} \mathbf{k}[\epsilon]$. If $V$ is a finitely generated Gorenstein-projective left $\Lambda$-module with $\underline{\mathrm{End}}_{\Lambda}(V)=\mathbf{k}$, then $V$ has an universal deformation ring $R(\Lambda,V)$ isomorphic to $\mathbf{k}[\epsilon]