Proposition 8

If one of the A-modules E, $\rho_* (B_s)$ is projective and finitely generated, the homomorphism $v_E$ is bijective.

This follows from the above and § 4, no. 2, Proposition 2.

Suppose in particular that E is a finitely generated free A-module and let $(e_i)_{1 \leq i \leq n}$ be a basis of E and $(e_i^*)$ the dual basis; then the canonical isomorphism (19) maps the basis $(e_i^* \otimes 1)$ of $(E^*)_{(B)}$ to the dual basis of the basis $(1 \otimes e_i)$ of $E_{(B)}$.