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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"?

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up vote 6 down vote accepted

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

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