A diagonalizable group scheme over a ring is a group scheme satisfying the following equivalent conditions:
Let be a constant group scheme. Let be its Cartier dual. By definition we have and hence . Here is the group algebra of and the last isomorphism is given by the adjunction called the universal property of group rings and each is a character of . Note that -as is any group algebra- is a Hopf algebra.
Conversely, if is affine and generated by characters, let H be the group of all characters of G; then the canonical map is surjective. But it is always injective (by Dedekind's lemma on linear independence of characters?), hence .