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 $\mathrm{\Lambda }$ be a finite dimensional $\mathbf{k}$-algebra. In this short note we prove that if $V$ is a finitely generated strongly Gorenstein-projective left $\mathrm{\Lambda }$-module whose stable endomorphism ring ${\underset{‾}{\mathrm{E}\mathrm{n}\mathrm{d}}}_{\mathrm{\Lambda }}(V)$ is isomorphic to $\mathbf{k}$, then $V$ has an universal deformation ring $R(\mathrm{\Lambda },V)$ 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 $\mathrm{\Lambda }=\mathbf{k}Q[\epsilon ]=\mathbf{k}Q{\otimes }_{\mathbf{k}}\mathbf{k}[\epsilon ]$. If $V$ is a finitely generated Gorenstein-projective left $\mathrm{\Lambda }$-module with ${\underset{‾}{\mathrm{E}\mathrm{n}\mathrm{d}}}_{\mathrm{\Lambda }}(V)=\mathbf{k}$, then $V$ has an universal deformation ring $R(\mathrm{\Lambda },V)$ isomorphic to $$\mathbf{k}$[ε]