Universal deformation rings and self-injective Nakayama algebras

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