The Nakayama functor and its completion for Gorenstein algebras

TL;DR

This paper explores duality properties of Gorenstein algebras over their centers, establishing a local duality theorem using the Nakayama functor and homotopy categories, extending classical duality results.

Contribution

It introduces a local duality theorem for Gorenstein algebras via the Nakayama functor, extending duality concepts to the homotopy category of injective modules.

Findings

Established a local duality theorem for acyclic complexes of Gorenstein algebras.

Extended the Nakayama functor to the full homotopy category of injective modules.

Connected duality properties of Gorenstein algebras with classical geometric dualities.

Abstract

Duality properties are studied for a Gorenstein algebra that is finite and projective over its center. Using the homotopy category of injective modules, it is proved that there is a local duality theorem for the subcategory of acyclic complexes of such an algebra, akin to the local duality theorems of Grothendieck and Serre in the context of commutative algebra and algebraic geometry. A key ingredient is the Nakayama functor on the bounded derived category of a Gorenstein algebra, and its extension to the full homotopy category of injective modules.