Proposition 12

*Let M be a free A-module of dimension n; let $e \in \wedge^n(M)$ be an element forming a basis of $\wedge^n(M)$ and let $e^*$ be the element of $\wedge^n(M^*)$ such that $\{(-1)^{n(n-1)/2} \theta_{\wedge}(e^*)\}$ is the dual basis of $\{e\}$ in $(\wedge^n(M))^*$. Let $\phi : \wedge(M^*) \to \wedge(M)$ be the mapping $z \mapsto z \wedge e^*$ and $\phi' : \wedge(M^*) \to \wedge(M)$ the mapping $z^* \mapsto z^* \wedge e$. Let $\phi_p$ (resp. $\phi'_p$) be the restriction of $\phi$ (resp. $\phi'$) to $\wedge^p(M)$ (resp. $\wedge^p(M^*)$). Then:

(i) The mapping $\phi$ is a left $\wedge(M)$-module isomorphism and the mapping $\phi'$ is a left $\wedge(M^*)$-module isomorphism; moreover the mappings $\phi$ and $\phi'$ are inverses of one another.

(ii) *The mapping $\phi_p$ is an isomorphism of the A-module $\wedge^p(M)$ onto the A-module $\wedge^{n-p}(M^*)$ and the mapping $\phi'_p$ is an isomorphism of the A-module $\wedge^p(M^*)$ onto the A-module $\wedge^{n-p}(M)$.

(iii) *If we write $B(u, v^*) = \langle u, \theta_{\wedge}(v^*) \rangle$ for $u \in \wedge(M)$ and $v^* \in \wedge(M^*)$ then, for $u^* \in \wedge^p(M^*)$ and $v^* \in \wedge^{n-p}(M^*)$,

The fact that $\phi$ is $\wedge(M)$-linear and $\phi'$ is $\wedge(M^*)$-linear follows from the formulae $(u \wedge v) \wedge e^* = u \wedge (v \wedge e^*)$ and $(u^* \wedge v^*) \wedge e = u^* \wedge (v^* \wedge e)$ (no. 6, formula (37)), using the fact that $\theta_{\wedge}$ is an isomorphism of $\wedge(M^*)$ onto the opposite algebra to $\wedge(M)^*$. On the other hand there exists a basis $(e_i)_{1 \leq i \leq n}$ of M such that

where $(e_i^*)$ is the basis dual to $(e_i)$. We write $I = \{1, n\}$; it follows from (76) that, for every subset J of I with p elements,

This proves that $\phi$ and $\phi'$ are bijective; moreover $\rho_{J, I-J} \rho_{I-J, J} = (-1)^{p(n-p)}$ (§ 7, no. 8, formula (21)); as the number

is even, it follows that $\phi$ and $\phi'$ are inverses of one another. Finally, to prove (77), it suffices to take $u^* = e_J^*$ and $v^* = e_{I-J}^*$; the verification also follows from the definition of $\theta_{\wedge}$, formulae (78) and the relation $\rho_{J, I-J} \rho_{I-J, J} = (-1)^{p(n-p)}$ (§ 7, no. 8, formula (21)). Note that, for $u^* \in \wedge^p(M^*)$ and $v^* \in \wedge^{n-p}(M^*)$,

B(\phi_p'(u^*), v^*) is, to within a sign, the coefficient of $u^* \wedge v^*$ with respect to the basis $\{e^*\}$ of $\Lambda^n(M^*)$.