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

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web

Let $k$ be a field with prime characteristic $p$.

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

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

The same is true for $k$-formal groups.

Let $G$ be a commutative affine $k$-group. Then for the Cartier dual $D(-)$ we have

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

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

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

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

$$V_G\circ F_G=p{\mathrm{id}}_G$$V_G\circ F_G=p id_G

and

$$F_G\circ V_G=p{\mathrm{id}}_{G^{(p)}}$$F_G\circ V_G=p id_{G^{(p)}}

Examples

$V_{{\mu }_k}$ is the identity and $V_{{\alpha }_k}$ is zero.

This follows since $F$ is an epimorphism and $p{\mathrm{id}}_{{\mu }_k}=F_{{\mu }_k}$ and $p{\mathrm{id}}_{{\alpha }_k}=0$