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

Abstract

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}[ε]

arXiv admin note: substantial text overlap with arXiv:2109.08015