Totally acyclic complexes over noetherian schemes

TL;DR

This paper extends the concept of totally acyclic complexes from rings to noetherian schemes, establishing new characterizations of Gorenstein schemes and simplifying existing results without requiring a dualising complex.

Contribution

It introduces a notion of total acyclicity for complexes of flat sheaves over schemes, generalizing ring-based concepts and removing the need for dualising complexes in key results.

Findings

A scheme is Gorenstein iff every acyclic flat sheaf complex is totally acyclic.

Extended properties of totally acyclic complexes from rings to schemes.

Removed the necessity of a dualising complex in several Gorenstein-related results.

Abstract

We define a notion of total acyclicity for complexes of flat quasi-coherent sheaves over a semi-separated noetherian scheme, generalising complete flat resolutions over a ring. By studying these complexes as objects of the pure derived category of flat sheaves we extend several results about totally acyclic complexes of projective modules to schemes; for example, we prove that a scheme is Gorenstein if and only if every acyclic complex of flat sheaves is totally acyclic. Our formalism also removes the need for a dualising complex in several known results for rings, including Jorgensen's proof of the existence of Gorenstein projective precovers.