TITLE: Is every monomorphism of commutative Hopf algebras (over a field) injective?
QUESTION [5 upvotes]: Is it true that any monomorphism of commutative Hopf algebras over a field is injective? Moreover, is it true that any epimorphism of commutative Hopf algebras over a field is surjective?

REPLY [2 votes]: I haven't read through yet but the following paper by Chirvasitu seems to answer your second question and discusses closely related problems. See discussion in page 7 after Proposition 2.5.
http://arxiv.org/abs/0907.2881
Namely an epimorphism of Hopf algebras over a field
$$f : H \longrightarrow K$$
is necessarily surjective if $K$ is commutative.