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 $k$.Comment: 25 pages, 2 figures. Some typos have been fixed, the outline of the paper has been changed to improve readabilit

Let k be a field and let Λ be a finite dimensional k-algebra. We prove that every bounded complex V ‚ of finitely generated Λ-modules has a well-defined versal deformation ring RpΛ, V ‚ q which is a complete local commutative Noetherian k-algebra with residue field k. We also prove that nice two-sided tilting complexes between Λ and another finite dimensional k-algebra Γ preserve these versal deformation rings. Additionally, we investigate stable equivalences of Morita type between self-injective algebras in this context. We apply these results to the derived equivalence classes of the members of a particular family of algebras of dihedral type that were introduced by Erdmann and shown by Holm to be not derived equivalent to any block of a group algebra.2010 Mathematics Subject Classification. Primary 16G10; Secondary 16G20, 20C20.

Let k be an algebraically closed field, 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 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 Λr-modules whose stable endomorphism ring isomorphic to k, where Λr is a symmetric special biserial k-algebra that has quiver with relations depending on the four parametersr = (r 0 , r 1 , r 2 , k) with r 0 , r 1 , r 2 ≥ 2 and k ≥ 1. Universal deformation rings and Frobenius algebras and Stable endomorphism rings and Special biserial algebras [2000]16G10 and 16G20 and 20C20Key words and phrases. Universal deformation rings and Frobenius algebras and Stable endomorphism rings.

Let k be a field of arbitrary characteristic, let Λ be a finite dimensional k-algebra, and let V be a finitely generated Λ-module. F. M. Bleher and the third author previously proved that V has a well-defined versal deformation ring R(Λ, V ). If the stable endomorphism ring of V is isomorphic to k, they also proved under the additional assumption that Λ is self-injective that R(Λ, V ) is universal. In this paper, we prove instead that if Λ is arbitrary but V is Gorenstein-projective then R(Λ, V ) is also universal when the stable endomorphism ring of V is isomorphic to k. Moreover, we show that singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over Gorenstein algebras. We also provide examples. In particular, if Λ is a monomial algebra in which there is no overlap (as introduced by X. W. Chen, D. Shen and G. Zhou) we prove that every finitely generated indecomposable Gorenstein-projective Λ-module has a universal deformation ring that is isomorphic to either k or to k[[t]]/(t 2 ).2010 Mathematics Subject Classification. 16G10 and 16G20 and 20C20.

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.