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We apply geometric techniques from representation theory to the study of homologically finite differential graded (DG) modules over a finite dimensional, positively graded, commutative DG algebra . In particular, in this setting we prove a version of a theorem of Voigt by exhibiting an isomorphism between the Yoneda Ext group 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 .