Universal deformation rings and self-injective Nakayama algebras

Abstract

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring R(Λ, V ) and we describe R(Λ, V ) explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.2000 Mathematics Subject Classification. Primary 16G10; Secondary 16G20, 20C20.

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

“…Observe that Theorem 1.2 extends the results of [8,Thm. 1.3] to indecomposable non-projective Gorensteinprojective modules over arbitrary basic connected Nakayama algebras without simple projective modules.…”

“…2.1]) together with Proposition 3.3 imply that the versal deformation rings R(Λ, V ) and R(Λ , H X (Λ) (V )) are isomorphic in Ĉ. Moreover, since R(Λ , H X (Λ) (V )) is universal by [8,Thm. 1.3], we also have that R(Λ, V ) is universal.…”

“…2.6 (iv)]. Moreover, they also proved in [8,Thm. 1.3] that if Λ is an indecomposable k-algebra that is stably Morita equivalent (in the sense of [9]) to a self-injective split basic Nakayama algebra and V is further indecomposable, then R(Λ, V ) is universal and its isomorphism class is either k or k[[t]]/(t 2 ), or a quotient of a ring of power series in finitely many variables and which is also determined by the closets distance of the isomorphism class of V to the boundary of the stable Auslander-Reiten quiver of Λ.…”

“…2.5] that if V is a left Λ-module, then V has a well-defined versal deformation ring R(Λ, V ), which is an object in Ĉ, and which also an invariant under Morita equivalence. More recently, F. M. Bleher and D. J. Wackwitz proved in [8,Prop. 2.4] that if Λ is a Frobenius k-algebra (i.e., Λ Λ and Hom k (Λ Λ , k) are isomorphic as left Λ-modules) and V is non-projective, then the versal deformation rings R(Λ, V ) and R(Λ, ΩV ) are isomorphic in Ĉ, which generalizes [6,Thm.…”

Universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras

“…3.2.6] that the isomorphism class of versal deformation rings of modules is preserved by stable equivalences of Morita type (as introduced by M. Broué in [11]) between self-injective k-algebras. Moreover, in [10], F. M. Bleher and D. J. Wackwitz studied universal deformation rings of modules over self-injective Nakayama k-algebras.…”