A note on deformations of Gorenstein-projective modules over finite dimensional algebras

Abstract

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 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 = Λ B 0 Γ be a triangular matrix finite dimensional Gorenstein k-algebra with Γ of finite global dimension and B projective as a left Λ-module. If V W f is a finitely generated Gorenstein-projective left Σ-module, then the versal deformation rings R Σ, V W f and R(Λ, V ) are isomorphic.2010 Mathematics Subject Classification. 16G10 and 16G20 and 20C20. Key words and phrases. (Uni)versal deformation rings and finitely generated Gorenstein-projective modules and Nakayama algebras and triangular matrix algebras.