Gorenstein acyclic complexes and finitistic dimensions

TL;DR

This paper establishes a connection between the finiteness of the big finitistic dimension of a noetherian ring with a dualizing complex and the contractibility of certain Gorenstein-projective-acyclic complexes, extending to Artin algebras.

Contribution

It provides a new characterization of finitistic dimension finiteness via Gorenstein-projective complexes and extends Rickard's theorem to Gorenstein contexts for Artin algebras.

Findings

Finitistic dimension is finite iff Gorenstein-projective-acyclic complexes are contractible.

Gorenstein variant of Rickard's theorem for Artin algebras.

Finiteness of Gorenstein-injective derived category generated by Gorenstein-injective modules.

Abstract

Given a two-sided noetherian ring $A$ with a dualizing complex, we show that the big finitistic dimension of $A$ is finite if and only if every bounded below Gorenstein-projective-acyclic cochain complex of Gorenstein-projective $A$-modules is contractible. If $A$ is further assumed to be an Artin algebra, we also prove a Gorenstein variant of a theorem of Rickard, showing its finitistic dimension is finite in case its Gorenstein-injective derived category is generated by the Gorenstein-injective modules.