  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.
