Local cohomology for Gorenstein homologically smooth DG algebras

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 DG algebra A, we define a group homomorphism Hdet : Aut dg (A) → k × , called the homological determinant. As applications, we present a sufficient condition for the invariant DG subalgebra A G to be Gorensten, where A is a homologically smooth DG algebra such that H(A) is AS-Gorenstein and G is a finite subgroup of Aut dg (A). Especially, we can apply this result to nontrivial DG free algebras generated in two degree one elements, DG polynomial algebras and DG down-up algebras. introductionLocal cohomology is an indispensable tool in almost all branches of analytic and algebraic geometry as well as in commutative and combinatorial algebra. Since the flourishing of non-commutative algebraic geometry in 1990s, people have done a lot to develop a theory of local cohomology in the context of non-commutative cases. In [Yek], Yekutieli introduced balanced dualizing complexes for non-commutative graded algebras by using local cohomology functors. After his work, Jørgensen [Jor1] generalized the theory of local cohomology and local duality theorem to non-commutative positively graded Noetherian algebras, which are quotients of Artin-Schelter Gorenstein algebras. Later, Van den Bergh gave a more general local duality formula in [VDB]. In [JoZ], local cohomology functor is used to define homological determinants for AS-Gorenstein algebras. In the invariant theory of Hopf algebra action (and formerly group actions) on non-commutative graded algebras, the notion of homological determinant usually serves as an effective tool in precise analysis (cf. [CWZ, JoZ, JZ, KKZ1, LMZ1]).In this paper, we are concerned with connected cochain DG k-algebras, where k is an algebraically closed field with zero characteristic. Any connected graded algebra can be seen as a connected cochain DG algebra with zero differential. The motivation of this paper is to introduce an analogous local cohomology theory under the background of DG homological algebra. Let A be a connected cochain DG algebra with maximal DG ideal m = A ≥1 . For any DG A-module M , its subset Γ m (M ) = {m ∈ M | A ≥n m = 0, for some n ≥ 1} is a DG A-submodule of M . We actually define a left exact covariant A-linear functor Γ m (−) = lim − →