Proposition 12

Let M be a projective A-module. For elements $x_1, x_2, \ldots, x_n$ of M to be linearly dependent, it is necessary and sufficient that there exist $\lambda \neq 0$ in A such that

The condition is necessary (with no hypothesis on M), for if for example $\lambda x_1$ (with $\lambda \neq 0$) is a linear combination of $x_2, \ldots, x_n$, relation (22) holds (no. 3, Proposition 5). We show that the condition is sufficient by induction on $n$; for $n = 1$, it is a trivial consequence of the definition. Suppose therefore that $n > 1$ and that condition (22) holds for some $\lambda \neq 0$. If

then the induction hypothesis implies that $x_2, \ldots, x_n$ are linearly dependent and hence a fortiori $x_1, x_2, \ldots, x_n$ are. If $\lambda x_2 \wedge x_3 \wedge \cdots \wedge x_n \neq 0$, it follows from no. 8, Corollary 3 to Theorem 1 that there exists an alternating $(n - 1)$-linear form $f$ such that $f(\lambda x_2 \wedge x_3 \wedge \cdots \wedge x_n) = \mu \neq 0$. Since it follows from no. 8, Corollary 3 to Theorem 1 that $\mu x_1$ is a linear combination of $\lambda x_2, x_3, \ldots, x_n$; hence $x_1, x_2, \ldots, x_n$ are linearly dependent.