A Zariski-local notion of F-total acyclicity for complexes of sheaves

TL;DR

This paper introduces a Zariski-local concept of F-total acyclicity for complexes of flat sheaves on schemes, aligning with existing notions and extending results on sheaf covers and precovers.

Contribution

It defines a Zariski-local notion of total acyclicity for complexes of flat sheaves and extends known theorems on sheaf covers and precovers.

Findings

The notion is Zariski-local and verifiable on affine covers.

It aligns with Murfet and Salarian's notion for noetherian semi-separated schemes.

Establishes the existence of an adjoint functor related to totally acyclic complexes.

Abstract

We study a notion of total acyclicity for complexes of flat sheaves over a scheme. It is Zariski-local - i.e. it can be verified on any open affine covering of the scheme - and it agrees, in their setting, with the notion studied by Murfet and Salarian for sheaves over a noetherian semi-separated scheme. As part of the study we recover, and in several cases extend the validity of, recent theorems on existence of covers and precovers in categories of sheaves. One consequence is the existence of an adjoint to the inclusion of these totally acyclic complexes into the homotopy category of complexes of flat sheaves.