Infinite Comatrix Corings

We characterize the corings whose category of comodules has a generating set of small projective comodules in terms of the (non commutative) descent theory. In order to extricate the structure of these corings, we give a generalization of the notions of comatrix coring and Galois comodule which avoid finiteness conditions. A sufficient condition for a coring to be isomorphic to an infinite comatrix coring is found. We deduce in particular that any coalgebra over a field and the coring associated to a group-graded ring are isomorphic to adequate infinite comatrix corings. We also characterize when the free module canonically associated to a (not necessarily finite) set of group like elements is Galois.