# Universal deformation rings and self-injective Nakayama algebras

## TL;DR

This paper proves that indecomposable modules over certain self-injective algebras, including Brauer tree algebras, have explicitly describable universal deformation rings, extending deformation theory to a broad class of algebras.

## Contribution

It establishes the existence and explicit description of universal deformation rings for modules over algebras stably equivalent to Nakayama algebras, including Brauer tree algebras.

## Key findings

- Universal deformation rings exist for all indecomposable modules in the studied class.
- Explicit descriptions of these rings as quotients of power series rings.
- Application to p-modular blocks with cyclic defect groups.

## Abstract

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.