Smoothness of commutative Hopf algebras
Kensuke Egami, Akira Masuoka, Kenta Suzuki
TL;DR
This work develops a comprehensive framework for smoothness of commutative Hopf algebras beyond finite type, establishing several equivalent conditions that connect ordinary algebraic smoothness with equivariant and coflatness concepts. It proves, in characteristic zero, that every Hopf algebra satisfies these equivalences, while in positive characteristic weaker properties hold and are tied to finite type substructures and André-Quillen cohomology vanishing. The authors extend the theory to Hopf algebras in general symmetric monoidal categories and identify stronger smoothness phenomena in categories such as sVec and Ver_p^ind, including explicit structural decompositions and cohomological classifications via H^2_s(H, -). They also provide concrete computations of H^2_s(H,k) for explicit examples using augmented H cleft extensions, illustrating the practical use of the developed cohomological and categorical methods. The results have implications for understanding torsors, affine group schemes, and obstructions to lifting morphisms in equivariant settings, with connections to Hochschild and André-Quillen cohomology.
Abstract
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.