Theorem F-Tori Are Classified by Galois Modules
theorem admitted

F-Tori Are Classified by Galois Modules

Let \(F_s/F\) be a separable closure with Galois group \(\Gamma=\operatorname{Gal}(F_s/F)\). The functor \[ T\longmapsto X^*(T_{F_s}) \] gives an anti-equivalence between \(F\)-tori and finite-rank free abelian groups with continuous \(\Gamma\)-action.

Under this correspondence:

  • \(T\) is \(F\)-split if and only if \(\Gamma\) acts trivially on \(X^*(T_{F_s})\), equivalently on \(X_*(T_{F_s})\);
  • \(T\) is anisotropic if and only if \(X^*(T_{F_s})^\Gamma=0\), equivalently \(X_*(T_{F_s})^\Gamma=0\);
  • every \(F\)-torus is split over \(F_s\);
  • every \(F\)-subtorus has an almost complementary \(F\)-subtorus: if \(T_1\subseteq T\), then there is \(T_2\subseteq T\) with \(T=T_1T_2\) and \(T_1\cap T_2\) finite.
Proof

The statement asserts an anti-equivalence of categories; per the working reduction to a finite level (every torus is split by a finite Galois extension, and the continuous \(\Gamma\)-action factors through its group — see the discussion in Character and Cocharacter Lattices), it suffices to prove, for each finite Galois \(E/k\) with group \(\Gamma_E = \operatorname{Gal}(E/k)\), an anti-equivalence between tori split by \(E\) and \(\Gamma_E\)-lattices, compatibly with enlarging \(E\). Three steps, each with its own node:

  1. Well-definedness. For \(T\) split by \(E\), the character lattice \(X^*(T_E)\) is a finite free \(\mathbb{Z}\)-module and the Galois action on \(E[T_E]\) preserves the character basis, so \(X^*(T)\) is a \(\Gamma_E\)-lattice, functorially in \(T\) (Character and Cocharacter Lattices, with the characters realized as group-likes of the coordinate algebra by Characters Are Group-Like Elements).

  2. Full faithfulness. The functor is bijective on homomorphism sets: \(\operatorname{Hom}_k(T, T') \cong \operatorname{Hom}_{\Gamma_E}(X^*(T'), X^*(T))\) — this is Homomorphisms of Tori Correspond to Lattice Maps, whose proof combines the hom part of Galois descent for Hopf algebras (Galois Descent for Hopf Algebras) with Cartier duality over the splitting field (Diagonalizable Groups Are Dual to Character Groups).

  3. Essential surjectivity. Every \(\Gamma_E\)-lattice \((M, \sigma)\) is the character lattice of a torus: the twisted form \(T(M,\sigma) = \operatorname{Spec} E[M]^{\Gamma_E}\) constructed in The Twisted Form of a Galois Lattice is a torus split by \(E\) with \(X^*(T(M,\sigma)) \cong (M,\sigma)\).

The four bullets under the correspondence:

  • Split criterion. A split torus has a basis of \(k\)-defined characters, so the action is trivial; conversely trivial action gives \(T = D(M)\) split (The Twisted Form of a Galois Lattice, property (4)). The \(X_*\) formulation follows by the perfect pairing (Character and Cocharacter Lattices).
  • Anisotropy criterion. This is the working definition and the averaging argument recorded in Anisotropic Torus.
  • Splitting over \(F_s\). Continuity of the action on the discrete lattice \(X^*\) forces a finite-index kernel, whose fixed field is a finite Galois splitting extension.
  • Almost-complement. Over \(\mathbb{Q}\), Maschke's theorem for the finite quotient through which \(\Gamma\) acts splits off \(X^*(T_1)_{\mathbb{Q}}\) as a direct summand; scaling the projection by a suitable integer produces a sublattice complementary up to finite index, i.e. a subtorus \(T_2\) with \(T = T_1 T_2\) and \(T_1 \cap T_2\) finite.

\(\square\)

Remarks

Downstream nodes that develop the bullets: the minimal splitting field and its Galois group are identified in The Minimal Splitting Field of a Torus; the split–anisotropic decomposition theory built on the almost-complement statement is Split-Anisotropic Decomposition of a Torus. Both consume this classification, so they are cited here only as forward pointers, not as inputs.

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Also in