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

TL;DR

This paper investigates deformation rings of Gorenstein-projective modules over Nakayama and triangular matrix algebras, proving universality and isomorphism properties that extend previous results in algebra representation theory.

Contribution

It establishes conditions under which deformation rings are universal and shows isomorphisms between deformation rings over specific classes of algebras, extending prior work.

Findings

Universal deformation rings are proven for Gorenstein-projective modules over certain Nakayama algebras.

Deformation rings of a module and its syzygy are isomorphic in this setting.

Deformation rings over triangular matrix algebras are isomorphic to those over component algebras.

Abstract

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.