Philosophy of contraherent cosheaves

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.