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

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