Continuous Galois character modules

Fix a separable closure $k_s\mathrm{/}k$ and ${\mathrm{\Gamma }}_k=\mathrm{Gal}(k_s\mathrm{/}k)$ with its Krull topology. A continuous Galois character module is a finitely generated abelian group $M$ with a ${\mathrm{\Gamma }}_k$-action by group automorphisms, continuous for the discrete topology on $M$. Equivalently every $m$ has an open stabilizer. Module morphisms are equivariant group homomorphisms.

For a multiplicative-type group $G$, set $X^{\ast }(G)={\mathrm{Hom}}_{k_s\text{-groups}}(G_{k_s},{\mathbf{G}}_{m,k_s})$. The action is transport of structure: on a group-like coordinate function $a$, it is the canonical semilinear action on $\mathcal{O}(G)\otimes k_s$. On points this reads $(\sigma \chi )(g)=\sigma (\chi ({\sigma }^{-1}g))$. Assuming the Axiom of Choice, the separable splitting in Multiplicative type groups split over a finite Galois extension permits using the full character group over $k_s$ even when $G$ is nonsmooth.