  Contraherent cosheaves are module objects over algebraic varieties defined by
gluing using the colocalization functors. Contraherent cosheaves are designed
to be used for globalizing contramodules and contraderived categories for the
purposes of Koszul duality and semi-infinite algebraic geometry. One major
technical problem associated with contraherent cosheaves is that the
colocalization functors, unlike the localizations, are not exact. The reason is
that, given a commutative ring homomorphism $R\rightarrow S$ arising in
connection with a typical covering in algebraic geometry, the ring $S$ is
usually a flat, but not a projective $R$-module. We argue that the relevant
difference between projective and flat modules, from the standpoint of
homological algebra, is not that big, as manifested by the flat/projective and
cotorsion periodicity theorems. The difference becomes even smaller if one
observes that the ring $S$ is often a very flat $R$-module.
