Universal deformation rings of modules over Frobenius algebras

TL;DR

This paper studies universal deformation rings of modules over Frobenius algebras, proving their existence and properties for self-injective algebras, and explicitly determining these rings for a specific dihedral type algebra.

Contribution

It establishes the existence and stability of universal deformation rings for modules over Frobenius algebras and computes these rings for a particular algebra of dihedral type.

Findings

Universal deformation rings exist for modules over self-injective algebras with stable endomorphism ring k.

These rings are stable under syzygies in Frobenius algebras.

Explicit determination of deformation rings for modules over a dihedral type algebra.

Abstract

Let $k$ be a field, and let $\Lambda$ be a finite dimensional $k$-algebra. We prove that if $\Lambda$ is a self-injective algebra, then every finitely generated $\Lambda$-module $V$ whose stable endomorphism ring is isomorphic to $k$ has a universal deformation ring $R(\Lambda,V)$ which is a complete local commutative Noetherian $k$-algebra with residue field $k$. If $\Lambda$ is also a Frobenius algebra, we show that $R(\Lambda,V)$ is stable under taking syzygies. We investigate a particular Frobenius algebra $\Lambda_0$ of dihedral type, as introduced by Erdmann, and we determine $R(\Lambda_0,V)$ for every finitely generated $\Lambda_0$-module $V$ whose stable endomorphism ring is isomorphic to $k$.