# Duality in Hopf algebras and Milnor-Moore paper

I am going through Milnor and Moore - On the structure of Hopf algebras (MSN) (I have already posted one question on that, another one is coming). My question is about Proposition 4.9, more specifically about the proof. It is stated that it is in parts already proved, but I am struggling with seeing what is going on in the proof that a) implies b), namely that:

Let $$A$$, $$B$$, $$C$$ be connected Hopf algebras over $$K$$ and $$i:A\to B$$, $$\pi :B\to C$$ be morphisms of Hopf algebras. Then a) implies b), where:

a) $$i$$ is a left normal monomorphism of algebras, $$C=K\otimes_A B$$ and the sequences $$0\to A\to B$$, $$B\to C\to 0$$ are split exact as sequences of graded $$K$$-modules,

b) $$\pi$$ is a right normal epimorphism of coalgebras, $$A\Box_C K$$, and the sequences as above are split exact as sequences of graded $$K$$-modules.

I will be very grateful for any help in seeing how the implication is proven.

• Is the other question you mention mathoverflow.net/questions/324407/free-hopf-algebra ? – LSpice Mar 5 at 22:19
• Yes, it is not directly about Milnor-Moore, but strictly connected with it. – Igor Sikora Mar 6 at 0:16
• The related questions sidebar brings up this Q&A for me, the accepted answer for which includes a (still valid) link to a book which May says "reworks" Milnor and Moore in chapters 20 to 23. I'm not familiar enough with this book and your particular question to know if it can definitely help you, but I wanted to point it out as something you might want to look into. – zibadawa timmy Mar 6 at 6:57