Quasi-Gorenstein morphisms of commutative local dg-algebras

We introduce quasi-Gorenstein morphisms of commutative local dg-algebras and use a Gorenstein version of the virtually small property to characterize them, a result which is new even for homomorphisms of local rings. In a different direction, we characterize exact sequences in a noetherian local ring, in the sense of Avramov, Henriques, and Şega, in terms of quasi-Gorenstein morphisms involving Koszul complexes.