I was talking this morning to a colleague who thinks about combinatorial Hopf algebras. He mentioned several rings, which are of interest in combinatorics, for which he didn't know whether a Hopf structure existed. I was able to rule out several by the following result:


If A is a finitely generated commutative algebra over a field of characteristic 0, and A has a Hopf structure, then A is a regular ring.


So, two questions:


(1) The only reference I know for this is Tate's article on group schemes in "Modular Forms and Fermat's Last Theorem." Does anyone know a version which targeted towards a reader who likes algebra better than geometry? (So, for example, "Hopf algebra" is a friendlier term than "group scheme".)


(2) Are there any useful generalizations that take out "commutative" or "finitely generated"?



Hopf algebra reference
Oort has an elementary proof that group schemes in char. 0 are reduced -- see MR0206005.


