Deformations of complexes for finite dimensional algebras

Abstract

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 R(Λ,V•) 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.