  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.
