Let $\mathbf{k}$ be a fixed field of arbitrary characteristic, and let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra. Assume that $V$ is a left $\Lambda$-module of finite dimension over $\mathbf{k}$. F. M. Bleher and the author previously proved that $V$ has a well-defined versal deformation ring $R(\Lambda,V)$ which is a local complete commutative Noetherian ring with residue field isomorphic to $\mathbf{k}$. Moreover, $R(\Lambda,V)$ is universal if the endomorphism ring of $V$ is isomorphic to $\mathbf{k}$. In this article we prove that if $\Lambda$ is a basic connected cycle Nakayama algebra without simple modules and $V$ is a Gorenstein-projective left $\Lambda$-module, then $R(\Lambda,V)$ is universal. Moreover, we also prove that the universal deformation rings $R(\Lambda,V)$ and $R(\Lambda, \Omega V)$ are isomorphic, where $\Omega V$ denotes the first syzygy of $V$. This result extends the one obtained by F. M. Bleher and D. J. Wackwitz concerning universal deformation rings of finitely generated modules over self-injective Nakayama algebras. In addition, we also prove the following result concerning versal deformation rings of finitely generated modules over triangular matrix finite dimensional algebras. Let $\Sigma=\begin{pmatrix} \Lambda & B\\0& \Gamma\end{pmatrix}$ be a triangular matrix finite dimensional Gorenstein $\mathbf{k}$-algebra with $\Gamma$ of finite global dimension and $B$ projective as a left $\Lambda$-module. If $\begin{pmatrix} V\\W\end{pmatrix}_f$ is a finitely generated Gorenstein-projective left $\Sigma$-module, then the versal deformation rings $R\left(\Sigma,\begin{pmatrix} V\\W\end{pmatrix}_f\right)$ and $R(\Lambda,V)$ are isomorphic.

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关于 Gorenstein-projective 模在 Nakayama 和三角矩阵代数上的变形

令 $\mathbf{k}$ 为任意特征的固定域，令 $\Lambda$ 为有限维的 $\mathbf{k}$-代数。假设 $V$ 是 $\mathbf{k}$ 上有限维的左 $\Lambda$-模。FM Bleher 和作者之前证明了 $V$ 有一个定义明确的通用变形环 $R(\Lambda,V)$，它是一个局部完全可交换的 Noetherian 环，剩余场与 $\mathbf{k}$ 同构。此外，如果 $V$ 的自同态环与 $\mathbf{k}$ 同构，则 $R(\Lambda,V)$ 是通用的。在本文中，我们证明如果 $\Lambda$ 是没有简单模的基本连通环中山代数，并且 $V$ 是 Gorenstein 射影左 $\Lambda$-模，则 $R(\Lambda,V)$ 是通用的. 此外，我们还证明了通用变形环 $R(\Lambda,V)$ 和 $R(\Lambda, \Omega V)$ 是同构的，其中 $\Omega V$ 表示 $V$ 的第一个 syzygy。这一结果扩展了 FM Bleher 和 DJ Wackwitz 关于有限生成模块在自注入 Nakayama 代数上的通用变形环的结果。此外，我们还证明了以下关于有限生成模块在三角矩阵有限维代数上的变形环的结果。令 $\Sigma=\begin{pmatrix} \Lambda & B\\0& \Gamma\end{pmatrix}$ 是一个三角矩阵有限维 Gorenstein $\mathbf{k}$-代数与 $\Gamma$ 的有限全局维度和 $B$ 投影作为左 $\Lambda$ 模块。如果 $\begin{pmatrix} V\\W\end{pmatrix}_f$ 是一个有限生成的 Gorenstein-projective left $\Sigma$-module，那么通用变形环 $R\left(\Sigma,\begin{pmatrix } V\\W\end{pmatrix}_f\right)$ 和 $R(\Lambda,