Let k be an algebraically closed field of arbitrary characteristic, let Λ be a finite dimensional k-algebra and let V be a Λ-module with stable endomorphism ring 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 further 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 N -modules whose corresponding stable endomorphism ring is isomorphic to k, and which lie either in a connected component of the stable Auslander-Reiten quiver of Λ m,N containing a module with endomorphism ring isomorphic to k or in a periodic component containing only string Λ m,N -modules, where m ≥ 3 and N ≥ 1 are integers, and Λ m,N is a self-injective special biserial k-algebra.2010 Mathematics Subject Classification. 16G10, 16G20, 20C20. Key words and phrases. Universal deformation rings and self-injective algebras and self-injective special biserial algebras and stable endomorphism rings.

Let k be an algebraically closed field, let Λ be a finite dimensional k-algebra, and let Λ be the repetitive algebra of Λ. For the stable category of finitely generated left Λ-modules Λ-mod, we show that the irreducible morphisms fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. We next use this fact in order to describe the shape of the Auslander-Reiten triangles in Λ-mod. We use the fact (and prove) that every Auslander-Reiten triangle in Λ-mod is induced from an Auslander-Reiten sequence of finitely generated left Λ-modules.2010 Mathematics Subject Classification. 16G10 and 16G20 and 20C20.

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