Acyclicity versus total acyclicity for complexes over noetherian rings

TL;DR

This paper establishes a deep connection between acyclic and totally acyclic complexes over noetherian rings with dualizing complexes, revealing new equivalences and characterizations in homotopy categories.

Contribution

It proves an equivalence between homotopy categories of projective and injective modules and characterizes complexes in Auslander and Bass categories over such rings.

Findings

Homotopy category of projectives is equivalent to that of injectives.

Quotients of acyclic complexes relate to categories of injectives.

New characterizations for complexes in Auslander and Bass categories.

Abstract

It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is equivalent, as a triangulated category, to the homotopy category of injective modules. Restricted to compact objects, this statement is a reinterpretation of Grothendieck's duality theorem. Using this equivalence it is proved that the (Verdier) quotient of the category of acyclic complexes of projectives by its subcategory of totally acyclic complexes and the corresponding category consisting of injective modules are equivalent. A new characterization is provided for complexes in Auslander categories and in Bass categories of such rings.