  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.
