Smoothness of equivariant derived categories

We introduce the notion of (homological) G-smoothness for a complex G-variety X, where G is a connected affine algebraic group. This is based on the notion of smoothness for dg algebras and uses a suitable enhancement of the G-equivariant derived category of X. If there are only finitely many G-orbits and all stabilizers are connected, we show that X is G-smooth if and only if all orbits O satisfy H*(O; R) = R. On the way, we prove several results concerning smoothness of dg categories over a graded commutative dg ring.