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The geometric cocharacter lattice of an algebraic torus $T$ over $k$ isusing a separable closure $k_s$. Composition with an algebraic torus character gives a homomorphism ${\mathbb{G}}_m\to {\mathbb{G}}_m$, necessarily $z\mapsto z^a$ for an integer $a$. This defines the perfect pairingFor a split algebraic torus, the cocharacter with exponent vector $(n_1,\dots ,n_r)$ is $z\mapsto (z^{n_1},\dots ,z^{n_r})$ and the pairing is ${\sum }_i{m}_i{n}_i$. The two geometric toric lattices carry compatible Galois group actions; $k$-defined cocharacters are the fixed elements. This is the duality used for the toric lattice $N$ and its dual $M$.^[ref]

A primitive lattice vector is a nonzero element $u$ of a finite-rank free abelian group $N$ that cannot be written $u=kv$ with $v\in N$ and an integer $|k|>1$. In an integral basis, this is equivalent to the greatest common divisor of its coordinates being one. The property is independent of the chosen lattice basis.

For example, $(2,3)$ is primitive although neither coordinate is one: the integer functional $\ell (x,y)=-x+y$ satisfies $\ell (2,3)=1$. Hence $N=\mathbb{Z}u\oplus \ker \ell$, showing that $u$ extends to a lattice basis. In contrast, $(2,4)=2(1,2)$ is not primitive.

Every rational ray of a fan contains a unique primitive generator pointing along it. This generator gives the integral normalization used in toric divisors and cocharacters; replacing it by a larger positive multiple leaves the geometric ray unchanged but changes its lattice data.

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]