Corollary 3

Let M be a projective A-module and $(x_i)_{1 \leq i \leq n}$ a finite sequence of elements of M. For there to exist on M an alternating n-linear form f such that

it is necessary and sufficient that $x_1 \wedge x_2 \wedge \cdots \wedge x_n \neq 0$.

We know already (with no hypothesis on M) that the condition is necessary (no. 4, Proposition 7). Suppose now that M is projective and that

Then $\bigwedge^n(M)$ is projective (Corollary 2) and hence the canonical mapping is injective (II, § 2, no. 7, Corollary 4 to Proposition 13); we conclude that there exists a linear form $g : \bigwedge^n(M) \to A$ such that $g(x_1 \wedge x_2 \wedge \cdots \wedge x_n) \neq 0$. If $f$ is the alternating n-linear form corresponding to $g$ (no. 4, Proposition 7), then $f(x_1, \ldots, x_n) \neq 0$.