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

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 $Sigma=begin{pmatrix} Lambda & B0& Gammaend{pmatrix}$ 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 $begin{pmatrix} VWend{pmatrix}_f$ is a finitely generated Gorenstein-projective left $Sigma$-module, then the versal deformation rings $Rleft(Sigma,begin{pmatrix} VWend{pmatrix}_fright)$ and $R(Lambda,V)$ are isomorphic.