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

TL;DR

This paper determines the universal deformation rings of certain string modules over a symmetric special biserial algebra, expanding understanding of module deformations in this algebraic context.

Contribution

It explicitly computes universal deformation rings for string modules over a specific class of symmetric special biserial algebras, a novel contribution in this area.

Findings

Universal deformation rings are determined for string modules with stable endomorphism ring isomorphic to .

Results apply to algebras with quivers depending on four parameters , , , and .

The deformation rings are complete local Noetherian -algebras.

Abstract

Let $\k$ be an algebraically closed field, let $\A$ be a finite dimensional $\k$-algebra and let $V$ be a $\A$-module with stable endomorphism ring isomorphic to $\k$. If $\A$ is self-injective then $V$ has a universal deformation ring $R(\A,V)$, which is a complete local commutative Noetherian $\k$-algebra with residue field $\k$. Moreover, if $\Lambda$ is also a Frobenius $\k$-algebra then $R(\A,V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $\Ar$-modules whose stable endomorphism ring isomorphic to $\k$, where $\Ar$ is a symmetric special biserial $\k$-algebra that has quiver with relations depending on the four parameters $ \bar{r}=(r_0,r_1,r_2,k)$ with $r_0,r_1,r_2\geq 2$ and $k\geq 1$.