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diagonalizable group scheme

A diagonalizable group scheme $G$ over a ring $k$ is a group scheme satisfying the following equivalent conditions:

1. $G$ is the Cartier dual of a constant group scheme.

2. $O(G)=\mathrm{hom}(G,O_k)$ is a character group i.e. $O(G)$ is generated by morphisms $G\to {\mu }_k$ into the multiplicative group scheme.

Proof

Let $H$ be a constant group scheme. Let $G=D(H)$ be its Cartier dual. By definition we have $D(H)(R)={\mathrm{hom}}_{\mathrm{Grp}}(H{\otimes }_{k}R,{\mu }_R)\simeq {\mathrm{hom}}_{\mathrm{Grp}}(H,R^{\times })\simeq {\mathrm{hom}}_{\mathrm{Alg}}(k[H],R)$ and hence $G=\mathrm{Spec}k[H]$. Here $k[H]$ is the group algebra of $H$ and the last isomorphism is given by the adjunction called the universal property of group rings and each $\zeta \in H\subset k[H]$ is a character of $G$. Note that -as is any group algebra- $k[H]$ is a Hopf algebra.

Conversely, if $G$ is affine and $O(G)$ generated by characters, let H be the group of all characters of G; then the canonical map $k[H]\to O(G)$ is surjective. But it is always injective (by Dedekind's lemma on linear independence of characters?), hence $k[H]\simeq O(G)$.