Split algebraic torus ($T{\cong }_k{\mathbb{G}}_m^r$)

A split algebraic torus over a field $k$ is an algebraic group isomorphic over $k$ to a product ${\mathbb{G}}_m^r$ of multiplicative group schemes. Its coordinate ring is the Laurent polynomial ringand its $k$-points are $(k^{\times }{)}^r$. Choosing such an isomorphism supplies torus coordinates; the torus need not come with preferred coordinates.

Every algebraic torus character is a monomial $x_1^{m_1}\cdots x_r^{m_r}$ with integer exponents. The character group is therefore ${\mathbb{Z}}^r$, and cocharacters form the dual lattice. This integral structure makes split tori the dense groups in ordinary fan-based toric geometry.

The adjective split concerns the field of definition, not connectedness of the real points. For example, ${\mathbb{G}}_m$ is split over $\mathbb{R}$ although ${\mathbb{R}}^{\times }$ has two connected components. The real norm-one algebraic torus from $\mathbb{C}/\mathbb{R}$ is connected as a real circle but is nonsplit. ^[ref]