Separable Functors and Formal Smoothness

We prove that given a bialgebra surjection π : E → H with nilpotent kernel such that H is a Hopf algebra which is formally smooth as a K-algebra, then π has a section which is a right H-colinear algebra homomorphism. Moreover, if H is also endowed with an ad-invariant integral, then the section can be chosen to be H-bicolinear. We also deal with the dual case. As an application, we prove that every connected Hopf algebra E over a field K with char (K) = 0 has a weak projection π : E → K [x], for every x ∈ P (E)\{0}. To these aims we investigate the relation between formal smoothness and separability of certain functors. We also characterize the existence of ad-(co)invariant integrals by means of the separability of certain functors.