Tori correspond exactly to torsion-free character lattices

A finitely generated abelian group admits a finite presentation: for a surjection ${\mathbb{Z}}^d\to M$, its kernel is finitely generated by induction on $d$, projecting to the last coordinate, whose image is cyclic; lift a generator of that image and use the induction hypothesis for the kernel of the projection. For its integer relation matrix, integer row and column operations give diagonal form. Specifically move a nonzero entry of smallest positive absolute value to the first position and divide entries in its row and column by it with remainder. A nonzero remainder lowers that pivot, so this process terminates. If the pivot does not divide an entry of the remaining block, add the row containing that entry to the pivot row and repeat the division; again the pivot decreases. Thus it eventually divides the whole block, its row and column can be cleared, and induction diagonalizes the smaller block. These invertible operations yield $M\cong {\mathbb{Z}}^r\oplus {⨁}_i\mathbb{Z}\mathrm{/}{n}_i\mathbb{Z}$, with $n_i>1$ after removing unit entries. Consequently $M$ is torsion-free exactly when it is free of finite rank.

If $M$ is free of rank $r$, F1 and F2 give $G_{k_s}\cong D_{k_s}(M)\cong {\mathbf{G}}_m^r$, so $G$ is a torus. Conversely, if $G_K\cong {\mathbf{G}}_m^r$ for some extension $K\mathrm{/}k$, choose a finite Galois splitting field $L\mathrm{/}k$ from F1. The finite-dimensional nonzero $K$-algebra $L{\otimes }_{k}K$ has a maximal ideal: select a proper ideal of largest vector-space dimension. Its residue field $E$ contains both $L$ and $K$, since the maps from these fields are unital. Base change of the splitting isomorphism $G_L\cong D_L(M)$ to $E$ identifies $G_E$ with $D_E(M)$, so F2 computes $X^{\ast }(G_E)=M$, while base change of $G_K\cong {\mathbf{G}}_m^r$ computes that same module as ${\mathbb{Z}}^r$. Hence $M\cong {\mathbb{Z}}^r$ and is torsion-free. The natural bijections of F1 restrict to these objects, proving the asserted equivalence and the exclusion of every nonzero torsion module.