nLab
Cartier duality

Idea

Cartier duality is a refinement of Pontryagin duality form topological groups to group schemes.

Definition

Definition

Let $G$ be a commutative $k$-group functor (in cases of interest this is a finite flat commutative group scheme). 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)