Diagonalizable groups and their character modules

For an abelian group $M$, its group algebra $k[M]$ has $k$-basis $e_m$ for $m\in M$, product $e_m{e}_n=e_{m+n}$, unit $e_0$, and Hopf maps $\mathrm{\Delta }(e_m)=e_m\otimes e_m$, $ϵ(e_m)=1$, $S(e_m)=e_{-m}$. Write $D_k(M)=\mathrm{Spec}k[M]$. These formulas define an affine group by The affine Hopf dictionary used for multiplicative type. For a general base scheme $S$, write $D_S(M)$ for the group obtained by gluing $\mathrm{Spec}R[M]$ on affine opens $\mathrm{Spec}R\subset S$ with the same formulas; the group-algebra construction commutes with localization, so these affine charts agree on overlaps and glue by Gluing affine schemes along compatible open isomorphisms. A diagonalizable group over $S$ is one isomorphic to $D_S(M)$ for an abelian group $M$. For a group scheme $G$ over $S$, its character group is $X(G)={\mathrm{Hom}}_{S\text{-groups}}(G,{\mathbf{G}}_{m,S})$, with pointwise multiplication of characters as its addition; for $S=\mathrm{Spec}k$ this is ${\mathrm{Hom}}_{k\text{-groups}}(G,{\mathbf{G}}_m)$. For every $k$-algebra $R$ the group-algebra universal property gives $D_k(M)(R)=\mathrm{Hom}(M,R^{\times })$: an algebra map $k[M]\to R$ is determined by the units it assigns to the basis elements $e_m$, and conversely any group homomorphism $M\to R^{\times }$ extends linearly.