Universal deformation rings of modules over Frobenius algebras

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 $k$.

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

References in corpus (3)

• Universal deformation rings and dihedral defect groups
• Universal deformation rings and generalized quaternion defect groups
• Universal deformation rings for the symmetric group S_5 and one of its double covers

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• On Deformations of Gorenstein-Projective Modules over Monomial Algebras with no Overlaps
• Universal deformation rings for a class of self-injective special biserial algebras
• Universal deformation rings of string modules over a certain symmetric special biserial algebra