nLab
Demazure, lectures on p-divisible groups, II.4, k-formal groups, Cartier duality

This entry is about a section of the text

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

A $k$-formal group is a $k$-group whose underlying $k$-functor is a $k$-formal functor.

The previous constructions in chapter II carry over to $k$-formal groups.

Definition

Let $G$ be a commutative $k$-group functor. Then the Cartier dual $D(G)$ of $G$ is defined by

$$D(G)(R)={\mathrm{Gr}}_R(G{\otimes }_{k}R,{\mu }_R)$$D(G)(R)=Gr_R(G\otimes_k R,\mu_R)

Moreover we have

$$\mathrm{hom}(G,D(H))=\mathrm{hom}(H,D(G))=\mathrm{hom}(G\times H,{\mu }_k)$$hom(G,D(H))=hom(H,D(G))=hom(G\times H,\mu_k)