The Twisted Form of a Galois Lattice

Let \(E/k\) be a finite Galois extension, \(\Gamma = \operatorname{Gal}(E/k)\), and let \((M, \sigma)\) be a \(\Gamma\)-lattice: a finite free \(\mathbb{Z}\)-module \(M\) with an action \(\sigma : \Gamma \to \operatorname{Aut}(M)\).

On the group algebra \(E[M]\) (the coordinate Hopf algebra of the split torus \(D(M)\), see The Diagonalizable Group Scheme D(M)), define the semilinear \(\Gamma\)-action \[ \rho_\gamma\bigl(a\,e^m\bigr) \;=\; \gamma(a)\, e^{\sigma_\gamma m}, \qquad a \in E,\; m \in M . \] Each \(\rho_\gamma\) permutes the group-like basis \(\{e^m\}\) and is therefore compatible with the Hopf structure (\(\Delta e^m = e^m \otimes e^m\), \(\varepsilon(e^m) = 1\), \(S(e^m) = e^{-m}\)), so Galois Descent for Hopf Algebras applies. The twisted form of \((M,\sigma)\) is the affine group scheme \[ T(M, \sigma) \;:=\; \operatorname{Spec}\Bigl( E[M]^{\Gamma} \Bigr) \] over \(k\), with coordinate \(k\)-Hopf algebra the fixed algebra \(E[M]^\Gamma\).

Properties

1. Descent. \(E[M]^\Gamma\) is a \(k\)-Hopf algebra and the natural map \(E \otimes_k E[M]^\Gamma \to E[M]\) is an isomorphism of \(E\)-Hopf algebras — this is Galois Descent for Hopf Algebras (1)–(2), applicable because \(\rho\) permutes the group-like basis.
2. \(T(M,\sigma)\) is a torus split by \(E\). By (1) the base change \(T(M,\sigma)_E\) is \(D(M) \cong \mathbb{G}_m^{\,n}\) (\(n = \operatorname{rk} M\), after a choice of basis), which is the defining property in Algebraic Torus.
3. Character lattice. \(X^*(T(M,\sigma)) \cong (M, \sigma)\) as \(\Gamma\)-lattices: the characters of \(T(M,\sigma)_E = D(M)\) are the group-likes \(\{e^m\}\) of \(E[M]\) (Characters Are Group-Like Elements), and the Galois action transported through the isomorphism of (1) sends \(e^m \mapsto e^{\sigma_\gamma m}\), i.e. is \(\sigma\) on \(M\).
4. Trivial action. If \(\sigma\) is trivial then \(E[M]^\Gamma = k[M]\) (coefficientwise Galois fixed points), so \(T(M, \mathrm{triv}) = D(M)\) is the split torus (Split Torus).

This construction is the essential-surjectivity half of the classification F-Tori Are Classified by Galois Modules: every \(\Gamma\)-lattice arises as the character lattice of a torus, namely of its twisted form.