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

TL;DR

This paper introduces the concept of p-polar k-algebras, showing that certain functors for p-typical co-Witt vectors and p-adic group schemes depend only on this weaker structure, expanding the understanding of algebraic structures in characteristic p.

Contribution

It defines p-polar k-algebras and demonstrates that key functors and Hopf algebra constructions depend solely on this structure, generalizing previous algebraic frameworks.

Findings

Functor of p-typical co-Witt vectors depends only on p-polar structures.

Cofree cocommutative Hopf algebra can be constructed on p-polar k-algebras.

Dual results for free commutative Hopf algebras on finite k-coalgebras.

Abstract

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.