Remark 4

The homomorphisms of $\mathbf{Z}$-modules $\delta : X(S)\rightarrow$ Hom$_{\mathbf{C}}(L(S)_{(\mathbf{C})},\mathbf{C})$ and $\iota :\Gamma (S)\rightarrow L(S)_{(\mathbf{C})}(\iota$ is induced by the canonical injection of $\Gamma (S)$ into L(S)) extend to isomorphisms of $\mathbf{C}$-vector spaces

$u:\mathbf{C}\otimes X(S)\rightarrow$ Hom$_{\mathbf{C}}(L(S)_{(\mathbf{C})},\mathbf{C})$

which we shall call canonical in the sequel. Note that, if we extend the pairing between X(S) and $\Gamma (S)$ by $\mathbf{C}$-linearity to a bilinear form $\ll ,\gg$ on $(\mathbf{C}\otimes X(S))\times (\mathbf{C}\otimes \Gamma$(S)), then