A Zariski-local notion of F-Total acyclicity for complexes of sheaves1

Abstract

We study a notion of total acyclicity for complexes of  at sheaves over a scheme. It is Zariski-local|i.e. it can be veried on any open affine covering of the scheme|and for sheaves over a quasi-compact semi-separated scheme it agrees with the categorical notion. In particular, 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 results 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  at sheaves.

Key words: F-total acyclicity, ascent-descent property, Gorenstein at precover.