A short note on deformations of (strongly) Gorenstein-projective modules over the dual numbers
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Velez-Marulanda, Jose A. Suarez, Hector |
| author_facet | Velez-Marulanda, Jose A. Suarez, Hector |
| 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] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18580 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A short note on deformations of (strongly) Gorenstein-projective modules over the dual numbers Velez-Marulanda, Jose A. Suarez, Hector Representation Theory 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] |
| title | A short note on deformations of (strongly) Gorenstein-projective modules over the dual numbers |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2402.18580 |