Algebraic Torus

Let \(k\) be a field and \(k_s\) a separable closure. An algebraic torus over \(k\) is a linear algebraic group \(T\) over \(k\) such that \[ T_{k_s} \;\cong\; \mathbb{G}_{m,k_s}^{\,n} \] for some \(n \geq 0\). The integer \(n\) is called the rank of \(T\).

Equivalently (see The Minimal Splitting Field of a Torus) there exists a finite separable extension \(E/k\) with \(T_E \cong \mathbb{G}_{m,E}^{\,n}\); the separable closure in the definition is not a genuine infinite limit.

Working form (hierarchy). A torus is a special diagonalizable group: \(T\) is a torus split by \(E\) if and only if \(T_E\) is diagonalizable and its character group \(X^*(T_E)\) is a lattice — free of finite rank (Diagonalizable Groups Are Dual to Character Groups: groups with free abelian character group of finite rank are exactly the split tori). The formalization takes this form as the definition, with \(T_E \cong \mathbb{G}_m^n\) recovered by choosing a basis of the lattice.

The character lattice \(X^*(T)\) and cocharacter lattice \(X_*(T)\) of a torus of rank \(n\) are free \(\mathbb{Z}\)-modules of rank \(n\) in perfect duality, carrying continuous actions of \(\Theta = \operatorname{Gal}(k_s/k)\); see Character and Cocharacter Lattices. They form the complete invariant of \(T\) by F-Tori Are Classified by Galois Modules.

A torus is smooth, connected, and commutative. Smoothness and connectedness hold because both can be checked after base change to \(k_s\), where \(\mathbb{G}_m^n\) is a smooth connected variety; commutativity holds because \(T(k_s) \subseteq T(k_s)\) is the group \((k_s^\times)^n\) under the defining isomorphism and \(T(k_s)\) is schematically dense in the smooth \(k_s\)-group \(T_{k_s}\).