  In this paper, we introduce the theory of local cohomology and local duality
to Notherian connected cochain DG algebras. We show that the notion of local
cohomology functor can be used to detect the Gorensteinness of a homologically
smooth DG algebra. For any Gorenstein homologically smooth locally finite DG
algebra $\mathcal{A}$, we define a group homomorphism $\mathrm{Hdet}:
\mathrm{Aut}_{dg}(\mathcal{A})\to k^{\times},$ called the homological
determinant. As applications, we present a sufficient condition for the
invariant DG subalgebra $\mathcal{A}^G$ to be Gorensten, where $\mathcal{A}$ is
a homologically smooth DG algebra such that $H(\mathcal{A})$ is a Noetherian
AS-Gorenstein graded algebra and $G$ is a finite subgroup of
$\mathrm{Aut}_{dg}(\mathcal{A})$. Especially, we can apply this result to DG
down-up algebras and non-trivial DG free algebras generated in two degree-one
elements.
