Corollary

(i) When $F$ is a projective (resp. finitely generated projective) module, the canonical homomorphism (11) is injective (resp. bijective). (ii) When $E$ is a finitely generated projective module, the canonical homomorphism (11) is bijective.

Even when $E$ and $F$ are both finitely generated, $\theta$ is not necessarily surjective, as is shown by the example $A = \mathbf{Z}, E = F = \mathbf{Z}/2\mathbf{Z}$; the right hand side of (11) is non-zero but $E^* = 0$. On the other hand, examples can be given where $E$ is free, but (11) is neither injective nor surjective (Exercise 3(b)).

When E admits a finite basis $(e_i)$, the inverse isomorphism $\theta^{-1}$ of $\theta$ can be found explicitly as follows. Let $(e_i^*)$ be the dual basis of $(e_i)$ (§ 2, no. 6); for all $u \in \mathrm{Hom}(E, F)$ and all $x = \sum_i \xi_i e_i$ with $\xi_i \in A$,

and therefore $u = \sum_i \theta(e_i^* \otimes u(e_i))$, in other words

In particular, if further $F = E$, it is seen that the image under $\theta_E^{-1}$ of the identity mapping $1_E$ is the element $\sum_i e_i^* \otimes e_i$, which is therefore independent of the basis $(e_i)$ considered in E.

Note on the other hand that when E is a finitely generated projective module the ring structure on $\mathrm{End}_A(E)$ can be transported by $\theta_E^{-1}$ to $E^* \otimes_A E$; it is immediately verified that, for $x, y$ in E, $x^*, y^*$ in $E^*$, in the ring $\mathrm{End}_A(E)$,

Remark (2). Let E be a right A-module; replacing E by $E^*$ in (11), we obtain a canonical $\mathbf{Z}$-homomorphism

On the other hand, there is a canonical A-homomorphism $c_E : E \to E^{**}$, whence there is a $\mathbf{Z}$-homomorphism $c_E \otimes 1_F : E \otimes_A F \to E^{**} \otimes_A F$; composing the latter with the homomorphism (14), we hence obtain a canonical $\mathbf{Z}$-homomorphism

such that $\theta'(x \otimes y)$ is the linear mapping

If E and F are projective modules, the mapping (15) is injective. For $c_E$ is then injective (§ 2, no. 7, Corollary 4 to Proposition 13) and as F is projective, the $\mathbf{Z}$-homomorphism $c_E \otimes 1_F : E \otimes_A F \to E^{**} \otimes_A F$ is also injective (§ 3, no. 7, Corollary 6 to Proposition 7); finally, it has been seen (Proposition 2) that the homomorphism (14) is injective, whence the conclusion.

If E is projective and finitely generated, the mapping (15) is bijective for the two mappings of which it is composed are then bijective (§ 2, no. 7, Corollary 4 to Proposition 13 and Proposition 2 above).