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

• Published in: Beiträge zur Algebra und Geometrie 2015-03, Vol.56 (1), p.129-146
• Main Author: Vélez-Marulanda, José A.
• Format: Article
• Language: English
• Subjects: Algebra; Algebraic Geometry; Convex and Discrete Geometry; Geometry; Mathematics; Mathematics and Statistics; Original Paper
• ISSN: 0138-4821; 2191-0383
• DOI: 10.1007/s13366-014-0201-y

Let k be an algebraically closed field, let Λ be a finite dimensional k -algebra and let V be a Λ -module whose stable endomorphism ring is isomorphic to k . If Λ is self-injective then V has a universal deformation ring R ( Λ , V ) , which is a complete local commutative Noetherian k -algebra with residue field k . Moreover, if Λ is also a Frobenius k -algebra then R ( Λ , V ) is stable under syzygies. We use these facts to determine the universal deformation rings of string Λ r ¯ -modules whose stable endomorphism rings are isomorphic to k that belong to a component C of the stable Auslander–Reiten quiver of Λ r ¯ , where Λ r ¯ is a symmetric special biserial k -algebra that has quiver with relations depending on the four parameters r ¯ = ( r 0 , r 1 , r 2 , k ) with r 0 , r 1 , r 2 ≥ 2 and k ≥ 1 , and where C is either of type ZA ∞ ∞ containing a module with endomorphism ring isomorphic to k or a 3 -tube.