  We apply geometric techniques from representation theory to the study of
homologically finite differential graded (DG) modules $M$ over a finite
dimensional, positively graded, commutative DG algebra $U$. In particular, in
this setting we prove a version of a theorem of Voigt by exhibiting an
isomorphism between the Yoneda Ext group $\operatorname{YExt}^1_U(M,M)$ and a
quotient of tangent spaces coming from an algebraic group action on an
algebraic variety. As an application, we answer a question of Vasconcelos from
1974 by showing that a local ring has only finitely many semidualizing
complexes up to shift-isomorphism in the derived category $\mathcal{D}(R)$.
