On Deformations of Gorenstein-projective modules over Nakayama and triangular matrix algebras

Abstract

Let $\mathbf{k}$ be a fixed field of arbitrary characteristic, and let $Λ$ be a finite dimensional $\mathbf{k}$-algebra. Assume that $V$ is a left $Λ$-module of finite dimension over $\mathbf{k}$. F. M. Bleher and the author previously proved that $V$ has a well-defined versal deformation ring $R(Λ,V)$ which is a local complete commutative Noetherian ring with residue field isomorphic to $\mathbf{k}$. Moreover, $R(Λ,V)$ is universal if the endomorphism ring of $V$ is isomorphic to $\mathbf{k}$. In this article we prove that if $Λ$ is a basic connected cycle Nakayama algebra without simple modules and $V$ is a Gorenstein-projective left $Λ$-module, then $R(Λ,V)$ is universal. Moreover, we also prove that the universal deformation rings $R(Λ,V)$ and $R(Λ, ΩV)$ are isomorphic, where $ΩV$ 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 $Σ=\begin{pmatrix} Λ& B\\0& Γ\end{pmatrix}$ be a triangular matrix finite dimensional Gorenstein $\mathbf{k}$-algebra with $Γ$ of finite global dimension and $B$ projective as a left $Λ$-module. If $\begin{pmatrix} V\\W\end{pmatrix}_f$ is a finitely generated Gorenstein-projective left $Σ$-module, then the versal deformation rings $R\left(Σ,\begin{pmatrix} V\\W\end{pmatrix}_f\right)$ and $R(Λ,V)$ are isomorphic.