Proposition 7

Let M be a finitely generated projective A-module. Then the canonical homomorphisms $\theta_T : T(M^*) \to T(M)^{*gr}$ and $\theta_\Lambda : \bigwedge(M^*) \to (\bigwedge(M)^{*gr})^0$ are bijective. Also the graded dual $\bigwedge(M)^{*gr}$ is then equal to the dual $\bigwedge(M)^*$ of the A-module $\bigwedge(M)$.

Suppose first that M has a finite basis $(e_i)_{1 \leq i \leq m}$ and let $(e_i^*)_{1 \leq i \leq m}$ be the dual basis of $M^*$ (II, § 10, no. 4). Formula (28 bis) shows that, for every finite sequence $s = (j_k)_{1 \leq k \leq n}$ of n elements of the interval $[1, m]$ of $\mathbf{N}$,

is the element of index s in the basis of $(T^n(M))^*$, dual to the basis of $T^n(M)$ consisting of the $e_s = e_{j_1} \otimes \cdots \otimes e_{j_n}$ (§ 5, no. 5, Theorem 1). Hence $\theta_T$ is bijective.

Similarly, formula (30 bis) shows that, for every finite subset H of $[1, m]$ with n elements, $(-1)^{n(n-1)/2} \theta_\Lambda(e_H^*)$ (notation of § 7, no. 8, Theorem 1) is the element of index H in the basis of $(\bigwedge^n(M))^*$, dual to the basis of $\bigwedge^n(M)$ consisting of the $e_H$. Hence $\theta_\Lambda$ is bijective.

Suppose now only that M is finitely generated and projective; then M is a direct factor of a finitely generated free A-module L, so that there exist two

A-linear mappings $M \xrightarrow{j} L \xrightarrow{p} M$ whose composition is the identity $l_M$. We deduce a commutative diagram

and an analogous commutative diagram where $T$ is replaced by $\Lambda$. The proposition then follows from the following lemma: