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

Authors

  • Tilman Bauer

DOI:

https://doi.org/10.7146/math.scand.a-129704

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.

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Published

2022-02-24

How to Cite

Bauer, T. (2022). Affine and formal abelian group schemes on $p$-polar rings. MATHEMATICA SCANDINAVICA, 128(1). https://doi.org/10.7146/math.scand.a-129704