Proposition 3

The bilinear form $(a, X) \rightarrow \langle a, X\rangle$ on $X(S)\times \Gamma (S)$ is invertible.

Recall (Algebra, Chap. IX) that, by definition, this means that the linear maps $X(S)\rightarrow$ Hom$_{\mathbf{Z}}(\Gamma (S),\mathbf{Z})$ and $\Gamma (S)\rightarrow$ Hom$_{\mathbf{Z}}(X(S),\mathbf{Z})$ associated to this bilinear form are bijective.

It is immediate that if the conclusion of the proposition is true for two tori, it is also true for their product. Thus, since every torus of dimension $n$ is isomorphic to $\mathbf{U}^n$, we are reduced to the case in which $S =\mathbf{U}$. In this particular case, the assertion is immediate.

Let $f: S\rightarrow S'$ be a morphism of tori. Then, each of the linear maps $X(f) : X(S')\rightarrow X(S)$ and $\Gamma (f) :\Gamma (S)\rightarrow \Gamma (S')$ is the transpose of the other: for all $a'\in X(S')$ and all $X\in \Gamma$(S),