Universal deformation rings for a class of self-injective special biserial algebras

Hernan Giraldo, Jose A. Velez-Marulanda

Published 2016-05-31Version 1

Let $\mathbf{k}$ be an algebraically closed field, let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra and let $V$ be a $\Lambda$-module with stable endomorphism ring isomorphic to $\mathbf{k}$. If $\Lambda$ is self-injective then $V$ has a universal deformation ring $R(\Lambda,V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$. Moreover, if $\Lambda$ is also a Frobenius $\mathbf{k}$-algebra then $R(\Lambda,V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $\Lambda_N$-modules with stable endomorphism ring isomorphic to $\mathbf{k}$, where $N\geq 1$ and $\Lambda_N$ is a self-injective special biserial $\mathbf{k}$-algebra whose Hochschild cohomology ring is a finitely generated $\mathbf{k}$-algebra as proved by N. Snashall and R. Taillefer.