= Cocharacter
{title2=$\lambda:\mathbb G_m\to T$}
A $k$-defined cocharacter of an <algebraic torus> $T$ is a homomorphism of algebraic groups
$$
\lambda:\mathbb G_m\longrightarrow T
$$
over $k$. A geometric cocharacter is defined after passing to a <separable closure>; the <cocharacter lattice of an algebraic torus> uses these geometric maps with their Galois action. The $k$-defined maps are the Galois-fixed elements.
For $T=\mathbb G_m^2$, an integer pair $(a,b)$ gives $\lambda(t)=(t^a,t^b)$. Composing with the character $\chi(x,y)=x^my^n$ yields $\chi(\lambda(t))=t^{ma+nb}$, defining the integer character-cocharacter pairing. This makes the two geometric lattices dual.
Cocharacters describe algebraic one-parameter subgroups and the integral directions underlying <toric cones>. For instance, $t\mapsto(t^2,t^3)$ has primitive exponent vector $(2,3)$; Bézout recovers $t$ from the coordinates by $t^3/t^2$. The distinction between geometric and ground-field-defined cocharacters matters for nonsplit tori.
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