  We investigate under which assumptions a subclass of flat quasi-coherent
shea\-ves on a quasi-compact and semi-separated scheme allows to "mock" the
homotopy category of projective modules. Our methods are based on module
theoretic properties of the subclass of flat modules involved as well as their
behaviour with respect to Zariski localizations. As a consequence we get that,
for such schemes, the derived category of flats is equivalent to the derived
category of very flats. If, in addition, the scheme satisfies the resolution
property then both derived categories are equivalent to the derived category of
infinite-dimensional vector bundles. The equivalences are inferred from a
Quillen equivalence between the corresponding models.
