Demazure, lectures on p-divisible groups, II.5, the Frobenius and the Verschiebung morphism

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Let kk be a field with prime characteristic pp.

The Frobenius morphism F G:GG (p)F_G: G\to G^{(p)} commutes with finite products and hence if GG is a k-group-functor G (p)G^{(p)} is a kk-group functor, too, and F GF_G is a kk-group morphism.

We abbreviate F G n:GG (p n)F_G^n:G\to G^{(p^n)}.

The same is true for kk-formal groups.

Let GG be a commutative affine kk-group. Then for the Cartier dual D()D(-) we have

D(G (p))=D(G) (p)D(G^{(p)})=D(G)^{(p)}

By Cartier duality we obtain the Verschiebung morphism V G:G (p)GV_G:G^{(p)}\to G for which holds D^(V G)=F D^(G)\hat D(V_G)=F_{\hat D(G)}. We abbreviate V G n:G (p n)GV_G^n:G^{(p^n)}\to G.

Let f:GHf:G\to H be a morphism of commutative affine kk-groups. The the following diagram is commutative

G (p) V G G F G G (p) f (p) f f (p) F (p) V H H F H H (p)\array{ G^{(p)} &\stackrel{V_G}{\to}& G &\stackrel{F_G}{\to}& G^{(p)} \\ \downarrow^{f^{(p)}}&&\downarrow^f&&\downarrow^{f^{(p)}} \\ F^{(p)} &\stackrel{V_H}{\to}& H &\stackrel{F_H}{\to}& H^{(p)} }

Moreover we have

V GF G=pid GV_G\circ F_G=p id_G


F GV G=pid G (p)F_G\circ V_G=p id_{G^{(p)}}


V μ kV_{\mu_k} is the identity and V α kV_{\alpha_k} is zero.

This follows since FF is an epimorphism and pid μ k=F μ kp id_{\mu_k}=F_{\mu_k} and pid α k=0p id_{\alpha_k}=0

Revised on June 25, 2015 03:56:42 by David Corfield (