2017/09/15 by Velez-Marulanda, Jose A.
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1709.05391
Let k be a fixed field of arbitrary characteristic, and let Λ be a finite dimensional k-algebra. Assume that V is a left Λ-module of finite dimension over 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 k. Moreover, R(Λ,V) is universal if the endomorphism ring of V is isomorphic to 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=\beginpmatrix Λ& B 0& Γ\endpmatrix be a triangular matrix finite dimensional Gorenstein k-algebra with Γ of finite global dimension and B projective as a left Λ-module. If \beginpmatrix V W\endpmatrixf is a finitely generated Gorenstein-projective left Σ-module, then the versal deformation rings R(Σ,\beginpmatrix V W\endpmatrixf) and R(Λ,V) are isomorphic.