Local cohomology for Gorenstein homologically smooth DG algebras

TL;DR

This paper develops local cohomology and duality theories for Gorenstein homologically smooth DG algebras, providing tools to detect Gorensteinness and analyze invariants under automorphisms.

Contribution

It introduces local cohomology for DG algebras, defines the homological determinant, and offers criteria for Gorensteinness of invariant subalgebras, extending the theory to DG down-up and free algebras.

Findings

Local cohomology detects Gorensteinness of DG algebras.

Defined the homological determinant for automorphisms of DG algebras.

Provided conditions for invariant DG subalgebras to be Gorenstein.

Abstract

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.