Smoothness of commutative Hopf algebras
Hopf algebras, most generally in a semisimple abelian symmetric monoidal category, are here supposed to be commutative but not to be of finite-type, and their (equivariant) smoothness are discussed. Given a Hopf algebra $H$ in a category such as above, it is proved that the following are equivalent: (i) $H$ is smooth as an algebra; (ii) $H$ is smooth as an $H$-comodule algebra; (iii) the product morphism $S_H^2(H^+) \to H^+$ defined on the 2nd symmetric power is monic. Working over a field $k$ of characteristic zero, we prove: (1) every ordinary Hopf algebra, i.e., such in the category $\mathsf{Vec}$ of vector spaces, satisfies the equivalent conditions (i)--(iii) and some others; (2) every Hopf algebra in the category $\mathsf{sVec}$ of super-vector spaces has a certain property that is stronger than (i). In the case where $\operatorname{char}k=p>0$, there are shown weaker properties of ordinary Hopf algebras and of Hopf algebras in $\mathsf{sVec}$ or in the ind-completion $\mathsf{Ver}_p^{\mathrm{ind}}$ of the Verlinde category.