Universal deformation rings of modules over Frobenius algebras

Let $k$ be a field, and let $Λ$ be a finite dimensional $k$-algebra. We prove that if $Λ$ is a self-injective algebra, then every finitely generated $Λ$-module $V$ whose stable endomorphism ring is isomorphic to $k$ has a universal deformation ring $R(Λ,V)$ which is a complete local commutative Noetherian $k$-algebra with residue field $k$. If $Λ$ is also a Frobenius algebra, we show that $R(Λ,V)$ is stable under taking syzygies. We investigate a particular Frobenius algebra $Λ_0$ of dihedral type, as introduced by Erdmann, and we determine $R(Λ_0,V)$ for every finitely generated $Λ_0$-module $V$ whose stable endomorphism ring is isomorphic to $k$.

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