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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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  • $\begingroup$ If you want Hopf algebras + combinatorics, I think it is a good idea to look at Marcelo Aguiar's works $\endgroup$
    – Marco Farinati
    Commented Jun 25, 2019 at 11:21

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

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About your question (2). When removing commutativity it is an open problem/conjecture that every noetherian Hopf algebra is Auslander-Gorenstein, see question 3.5 in https://www.cambridge.org/core/journals/glasgow-mathematical-journal/article/noetherian-hopf-algebras/DAC39E641D0715E94C7459AD31EF0315

Note that Auslander-Gorenstein (Auslander regular) rings are the non-commutative generalisation due to Auslander (motivated by results of Bass) of the commutative Gorenstein (regular) rings.

Here a ring $R$ is Auslander-Gorenstein when it has finite injective dimension on both sides and there exists an injective coresolution $0 \rightarrow R \rightarrow I^0 \rightarrow I^1 \rightarrow ...$ such that the flat dimension of $I^i$ is at most $i$. It is called Auslander regular if additionally it has finite global dimension.

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