Universal deformation rings of modules over Frobenius algebras

Details Universal deformation rings of modules over Frobenius algebras Universal deformation rings of modules over Frobenius algebras

2009-11-05

arxiv.org/abs/0911.1100v2

Mathematics

Representation Theory

25 pages, 2 figures. Some typos have been fixed, the outline of the paper has been changed to improve readability

Scientific paper

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 $k$.

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