Universal deformation rings and self-injective Nakayama algebras

Let \(k\) be a field and let \(\Lambda\) be an indecomposable finite dimensional \(k\)-algebra such that there is a stable equivalence of Morita type between \(\Lambda\) and a self-injective split basic Nakayama algebra over \(k\). We show that every indecomposable finitely generated \(\Lambda\)-module \(V\) has a universal deformation ring \(R(\Lambda,V)\) and we describe \(R(\Lambda,V)\) 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.