Tori Are Classified by Cocycles in GL_n(Z)
Let \(k\) be a field, \(k_s\) a separable closure, \(\Theta = \operatorname{Gal}(k_s/k)\), and \(n \geq 0\). Since the automorphism group of the split torus \(\mathbb{G}_m^{\,n}\) is \(\operatorname{GL}_n(\mathbb{Z})\) — acting on the character lattice — with trivial \(\Theta\)-action, the general form-cocycle machinery (Nonabelian H1) simplifies:
- Isomorphism classes of rank-\(n\) tori over \(k\) are in bijection with \[ H^1\bigl(\Theta,\, \operatorname{GL}_n(\mathbb{Z})\bigr) \;=\; \operatorname{Hom}_{\mathrm{cont}}\bigl(\Theta,\, \operatorname{GL}_n(\mathbb{Z})\bigr)\big/\text{conj}, \] continuous homomorphisms up to conjugacy (cocycles for a trivial action are homomorphisms, and cohomologous means conjugate).
- For a fixed finite Galois extension \(E/k\) with group \(\Gamma\), tori split by \(E\) correspond to \(\operatorname{Hom}(\Gamma, \operatorname{GL}_n(\mathbb{Z}))/\text{conj}\).
Proof
By F-Tori Are Classified by Galois Modules, rank-\(n\) tori up to isomorphism correspond to \(\Theta\)-lattices of rank \(n\) up to isomorphism. Choosing a \(\mathbb{Z}\)-basis of the lattice turns the action into a continuous homomorphism \(\Theta \to \operatorname{GL}_n(\mathbb{Z})\) (continuity = open kernel, by The Minimal Splitting Field of a Torus the kernel cuts out the minimal splitting field); changing the basis conjugates the homomorphism, and two lattices are isomorphic iff the homomorphisms are conjugate. A torus is split by \(E\) iff its homomorphism factors through \(\Gamma = \operatorname{Gal}(E/k)\), giving (2). That this bijection agrees with the general nonabelian-\(H^1\) classification of forms is the observation that both sides classify descent data on \(\mathbb{G}_{m,k_s}^{\,n}\), with trivial action on \(\operatorname{Aut} = \operatorname{GL}_n(\mathbb{Z})\) because every automorphism of the split torus is defined over the prime field. \(\square\)