Proposition 5

(i) Let $(x_i)_{1 \leq i \leq n}$ be a finite sequence of elements of the module $M$; for every permutation $\sigma$ in the symmetric group $S_n$,

where $\varepsilon_\sigma$ denotes the signature of the permutation $\sigma$.

(ii) If there exist two distinct indices $i, j$ such that $x_i = x_j$, the product

is zero.

(i) First of all, since $x \wedge x = 0$ for all $x \in M$ by definition of the ideal $\mathfrak{g}'$, also, for $x, y$ in $M$,

This establishes (6) in the case $n = 2$. The general case then follows from § 4, no. 6, Lemma 3.

(ii) Under the hypothesis of (ii), there exists a permutation $\sigma \in S_n$ such that $\sigma(1) = i$ and $\sigma(2) = j$; then the left hand side of (6) is zero for this permutation and hence so is the right hand side.