Groups of multiplicative type and tori

A group scheme of finite type over a field $k$ is of multiplicative type if it is fpqc locally diagonalizable: there is a faithfully flat quasi-compact covering $S^{\mathrm{\prime }}\to \mathrm{Spec}k$ on which it becomes a diagonalizable group. A torus over $k$ is a finite-type group scheme fpqc locally isomorphic to ${\mathbf{G}}_m^r$, for a finite integer $r\ge 0$. A group or torus is split if the relevant isomorphism already exists over $k$.

Diagonalizable means the group-algebra construction in Diagonalizable groups and their character modules. These definitions include the trivial torus of rank zero and nonsmooth multiplicative-type groups. No affineness or separable splitting condition is imposed by definition. Affineness and field splitting are proved locally on this page, followed by finite separable splitting.