Affine and formal abelian group schemes on p-polar rings - Bauer, Tilman (MATHEMATICA SCANDINAVICA, 2022)
We show that the functor of p-typical co-Witt vectors on commutative algebras over a perfect field k of characteristic p is defined on, and in fact only depends on, a weaker structure than that of a k-algebra. We call this structure a p-polar k-algebra. By extension, the functors of points for any p-adic affine commutative group scheme and for any formal group are defined on, and only depend on, p-polar structures. In terms of abelian Hopf algebras, we show that a cofree cocommutative Hopf algebra can be defined on any p-polar k-algebra P, and it agrees with the cofree commutative Hopf algebra on a commutative k-algebra A if P is the p-polar algebra underlying A; a dual result holds for free commutative Hopf algebras on finite k-coalgebras.