Reflecting Perfection for Finite Dimensional Differential Graded Algebras

We generalise two facts about finite dimensional algebras to finite dimensional differential graded algebras. The first is the Nakayama Lemma and the second is that the simples can detect finite projective dimension. We prove two dual versions which relate to Gorenstein differential graded algebras and Koszul duality respectively. As an application, we prove a corepresentability result for finite dimensional differential graded algebras.