  We introduce a new filtration on Hopf algebras, the standard filtration,
generalizing the coradical filtration. Its zeroth term, called the Hopf
coradical, is the subalgebra generated by the coradical. We give a structure
theorem: any Hopf algebra with injective antipode is a deformation of the
bosonization of the Hopf coradical by its diagram, a connected graded Hopf
algebra in the category of Yetter-Drinfeld modules over the latter. We discuss
the steps needed to classify Hopf algebras in suitable classes accordingly. For
the class of co-Frobenius Hopf algebras, we prove that a Hopf algebra is
co-Frobenius if and only if its Hopf coradical is so and the diagram is finite
dimensional. We also prove that the standard filtration of such Hopf algebras
is finite. Finally, we show that extensions of co-Frobenius (resp.
cosemisimple) Hopf algebras are co-Frobenius (resp. cosemisimple).
