Character and Cocharacter Lattices
To every algebraic torus \(T\) over a field \(k\) one associates two free abelian groups: the character lattice \(X^*(T)\) and the cocharacter lattice \(X_*(T)\). Together with the action of \(\Theta = \operatorname{Gal}(k_s/k)\), they form the complete invariant of \(T\) (see F-Tori Are Classified by Galois Modules) and, downstream, the foundation of root data.
Definitions
For a torus \(T\) over \(k\) of rank \(r\), with \(k_s\) a separable closure:
- Character lattice \[ X^*(T) \;:=\; \operatorname{Hom}_{k_s\text{-grp}}(T_{k_s}, \mathbb{G}_m), \] a free \(\mathbb{Z}\)-module of rank \(r\).
- Cocharacter lattice \[ X_*(T) \;:=\; \operatorname{Hom}_{k_s\text{-grp}}(\mathbb{G}_m, T_{k_s}), \] a free \(\mathbb{Z}\)-module of rank \(r\); elements are one-parameter subgroups.
- Perfect pairing \[ \langle -, - \rangle : X^*(T) \times X_*(T) \;\to\; \mathbb{Z}, \] defined by \(\chi \circ \lambda = (z \mapsto z^{\langle \chi, \lambda \rangle})\) in \(\operatorname{End}(\mathbb{G}_m) = \mathbb{Z}\).
Both Homs are taken over \(k_s\); the Galois action is separate data, not built into the Hom. A separable closure suffices because tori split over finite separable extensions (The Minimal Splitting Field of a Torus); no algebraic closure is used.
Galois action
\(\Theta\) acts on \(X^*(T)\) by \(({}^{\gamma}\chi)(t) = \gamma\bigl(\chi(\gamma^{-1} t)\bigr)\) and compatibly on \(X_*(T)\); the pairing is \(\Theta\)-invariant. The action is continuous for the discrete topology on the lattices: each character is defined over a finite subextension of \(k_s/k\). This makes both lattices finitely generated continuous \(\Theta\)-modules, free as \(\mathbb{Z}\)-modules.
Two natural \(\Theta\)-equivariant identifications describe the points of \(T\) over \(k_s\): \[ X_*(T) \otimes_{\mathbb{Z}} k_s^\times \xrightarrow{\;\sim\;} T(k_s), \quad \lambda \otimes x \mapsto \lambda(x); \qquad T(k_s) \xrightarrow{\;\sim\;} \operatorname{Hom}(X^*(T), k_s^\times), \quad t \mapsto (\chi \mapsto \chi(t)). \]
Functoriality
For a morphism of tori \(f : T \to T'\) over \(k\):
- \(X^*(f) : X^*(T') \to X^*(T)\) (contravariant);
- \(X_*(f) : X_*(T) \to X_*(T')\) (covariant);
both \(\Theta\)-equivariant. Under the anti-equivalence of F-Tori Are Classified by Galois Modules, closed immersions of tori correspond to surjections of character lattices and surjections of tori to injections; the full dictionary for subtori and quotients is Subtori and Quotients via the Character Lattice.
Examples
| \(T\) | \(X^*(T)\) | \(\Theta\)-action |
|---|---|---|
| \(\mathbb{G}_m\) | \(\mathbb{Z}\) | trivial |
| \(\mathbb{G}_m^n\) | \(\mathbb{Z}^n\) | trivial |
| \(R_{E/k}\mathbb{G}_m\) | \(\mathbb{Z}[\Theta/\Theta_E]\) | permutation |
| \(R^{1}_{E/k}\mathbb{G}_m\) | \(\mathbb{Z}[\Theta/\Theta_E]/\mathbb{Z}\!\cdot\!N\) | quotient of permutation |
| anisotropic \(T\) | lattice with \(X_*(T)^\Theta = 0\) | no non-zero fixed vector |
Downstream role
For a connected reductive group with a maximal torus \(T\), the lattice \(X^*(T)\) carries the roots and \(X_*(T)\) the coroots; the quadruple \((X^*(T), \Phi, X_*(T), \Phi^\vee)\) is the root datum (root data). Valuation-theoretic invariants of \(T(k)\) over discretely valued fields are expressed through \(X_*(T)\), which is why the cocharacter lattice is the preferred working lattice of this topic.