Affine and formal abelian group schemes on $p$-polar rings

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