Successive extensions of minimal \(p\)-divisible groups
type: Item
instance of: scholarly article
title: Successive extensions of minimal \(p\)-divisible groups
zbMATH Open document ID: 1170.14034
author: Shushi Harashita
published in: Journal of Pure and Applied Algebra
publication date: 2009-06-04T00:00:00Z
review text: Let \(K\) be a field, let \(q\in {\mathbb Z}\) be prime, let \(T\) be an indeterminate, and let \(K[\overline T]=K[T]/(T^q-1)\). Then \(K[\overline T]\) is a \(K\)-Hopf algebra of dimension \(q\) with comultiplication \(\Delta: K[\overline T]\rightarrow K[\overline T]\otimes K[\overline T]\) defined by \(\overline T\mapsto\overline T\otimes \overline T\). Thus \(\mu_q=\text{Spec}\;K[\overline T]=\text{Hom}_{K-alg}(K[\overline T],-)\) is a representable group functor from the category of commutative \(K\)-algebras to the category of abelian groups. For a commutative \(K\)-algebra \(A\), \(\mu_q(A)\) consists of functions \(f_a: K[\overline T]\rightarrow A\) with \(\overline T\mapsto a\), \(a^q=1\). The group product in \(\mu_q(A)\) is defined as \((f_a*f_b)(T)=(f_a,f_b)\Delta(\overline T)=f_a(T)f_b(T)=ab\). We assume that \(K\) is a perfect field of characteristic \(p\) (with \(p\) not necessarily \(= q\)). Then the map \(\phi: K\rightarrow K\), \(r\mapsto r^p\), is an automorphism. Viewing \(K\) as a \(K\)-module via \(r\cdot s=\phi(r)s=r^ps\), one defines the group scheme \(\mu_q^{(p)}= \text{Spec}\;(K[\overline T]\otimes_K K)\), called the twist of \(\mu_q\). Since \(K\) has characteristic \(p\), \({\mathcal F}: K[\overline T]\otimes_K K\rightarrow K[\overline T]\) defined as \({\overline T\otimes 1}\rightarrow {\overline T}^p\) is an algebra map (in fact, it is a Hopf algebra map), and so by Yoneda's Lemma, \({\mathcal F}\) corresponds to a endomorphism of group schemes \(F_{\mu_q}: \mu_q\rightarrow \mu_q^{(p)}\). This is the Frobenius homomorphism on \(\mu_q\). One also has a shift homomorphism \(V_{\mu_q}: \mu_q^{(p)}\rightarrow \mu_q\) which satisfies \(V_{\mu_q}\circ F_{\mu_q}=[p]_{\mu_q}\) and \(F_{\mu_q}\circ V_{\mu_q}=[p]_{\mu_q^{(p)}}\) where \([p]_{\mu_q}\), \([p]_{\mu_q^{(p)}}\) are the \(p\)th power maps on \(\mu_q\), \(\mu_q^{(p)}\). In \(\mu_q(A)\), one has \([p](f_a)({\overline T})=F_{\mu_q}(f_a)({\overline T})\). Thus \(F_{\mu_q}=[p]_{\mu_q}\) and …
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 14L15
zbMATH DE Number: 5561734
zbMATH Keywords: Barsotti-Tate group of level one
zbMATH Keywords: \(p\)-divisible group
zbMATH Keywords: Dieudonné module
reviewed by: Robert G. Underwood
MaRDI profile type: Publication
full work available at URL: j.jpaa.2009.01.007
OpenAlex ID: W2067472658
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DOI: 10.1016/J.JPAA.2009.01.007
description: scientific article; zbMATH DE number 5561734