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

DOI10.7146/MATH.SCAND.A-129704zbMATH Open1487.14095arXiv2012.10196OpenAlexW4214617011MaRDI QIDQ5080428

Tilman Bauer

Publication date: 31 May 2022

Published in: MATHEMATICA SCANDINAVICA (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/2012.10196

Mathematics Subject Classification ID

Group schemes (14L15) Witt vectors and related rings (13F35)

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