Universal deformation rings of modules over Frobenius algebras

Abstract

Let k be a field, and let Λ be a finite dimensional k-algebra. We prove that if Λ is a self-injective algebra, then every finitely generated Λ-module V whose stable endomorphism ring is isomorphic to k has a universal deformation ring R(Λ,V) which is a complete local commutative Noetherian k-algebra with residue field k. If Λ is also a Frobenius algebra, we show that R(Λ,V) is stable under taking syzygies. We investigate a particular Frobenius algebra Λ0 of dihedral type, as introduced by Erdmann, and we determine R(Λ0,V) for every finitely generated Λ0-module V whose stable endomorphism ring is isomorphic to k.