Universal deformation rings of strings modules over a certain symmetric special biserial algebra

Abstract

Let $\mathbb {k}$ be an algebraically closed field, let $\Lambda$ be a finite dimensional $\mathbb {k}$-algebra and let $V$ be a $\Lambda$-module whose stable endomorphism ring is isomorphic to $\mathbb {k}$. If $\Lambda$ is self-injective then $V$ has a universal deformation ring $R(\Lambda ,V)$, which is a complete local commutative Noetherian $\mathbb {k}$-algebra with residue field $\mathbb {k}$. Moreover, if $\Lambda$ is also a Frobenius $\mathbb {k}$-algebra then $R(\Lambda ,V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $\Lambda _{\bar{r}}$-modules whose stable endomorphism rings are isomorphic to $\mathbb {k}$ that belong to a component ${\mathfrak {C}}$ of the stable Auslander–Reiten quiver of $\Lambda _{\bar{r}}$, where $\Lambda _{\bar{r}}$ is a symmetric special biserial $\mathbb {k}$-algebra that has quiver with relations depending on the four parameters $\bar{r}=(r_0,r_1,r_2,k)$ with $r_0,r_1,r_2\ge 2$ and $k\ge 1$, and where ${\mathfrak {C}}$ is either of type ${\mathbb {ZA}}_\infty ^\infty$ containing a module with endomorphism ring isomorphic to $\mathbb {k}$ or a $3$-tube.