Multiplicative Group Scheme \(\mathbb{G}_m\)
Let \(S\) be a scheme. The multiplicative group scheme over \(S\), denoted \(\mathbb{G}_{m,S}\), is defined by base change:
\[ \mathbb{G}_{m,S} \;=\; S \,\times_{\operatorname{Spec} \mathbb{Z}}\, \operatorname{Spec} \mathbb{Z}[T, T^{-1}], \]
where \(\mathbb{Z}[T, T^{-1}]\) is the Laurent polynomial ring (the localization of \(\mathbb{Z}[T]\) at the multiplicative set generated by \(T\)). Its coordinate Hopf algebra is \(\mathbb{Z}[T, T^{-1}]\) with the multiplicative Hopf structure:
\[ \Delta(T) = T \otimes T, \qquad \varepsilon(T) = 1, \qquad S(T) = T^{-1}. \]
The structure morphism \(\mathbb{G}_{m,S} \to S\) is affine and locally of finite type. Hence \(\mathbb{G}_{m,S}\) is an algebraic group over \(S\); it is smooth of relative dimension one and is commutative. Its functor of points is identified by the multiplicative points formula.
The group scheme \(\mathbb{G}_m\) is the rank-one case of the family of split tori \(T = \mathbb{G}_m^n\); its character lattice is \(X^*(\mathbb{G}_m) \cong \mathbb{Z}\), generated by the identity character \(T\).