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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.18580 |
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Table of Contents:
- Let $\mathbf{k}$ be a field of arbitrary characteristic, and let $Λ$ be a finite dimensional $\mathbf{k}$-algebra. In this short note we prove that if $V$ is a finitely generated strongly Gorenstein-projective left $Λ$-module whose stable endomorphism ring $\underline{\mathrm{End}}_Λ(V)$ is isomorphic to $\mathbf{k}$, then $V$ has an universal deformation ring $R(Λ,V)$ isomorphic to the ring of dual numbers $\mathbf{k}[ε]$ with $ε^2=0$. 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 $Λ= \mathbf{k} Q[ε] = \mathbf{k} Q\otimes_{\mathbf{k}} \mathbf{k}[ε]$. If $V$ is a finitely generated Gorenstein-projective left $Λ$-module with $\underline{\mathrm{End}}_Λ(V)=\mathbf{k}$, then $V$ has an universal deformation ring $R(Λ,V)$ isomorphic to $\mathbf{k}[ε]