Local cohomology for Gorenstein homologically smooth DG algebras

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.