Let $k$ be a field and let $Lambda$ be a finite dimensional $k$-algebra. We prove that every bounded complex $V^bullet$ of finitely generated $Lambda$-modules has a well-defined versal deformation ring $R(Lambda,V^bullet)$ which is a complete local commutative Noetherian $k$-algebra with residue field $k$. We also prove that nice two-sided tilting complexes between $Lambda$ and another finite dimensional $k$-algebra $Gamma$ 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.
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