Corollary 1

Tag 0635, permanent. Cite it as [Bourbaki, Tag 0635].

For every alternating n-linear mapping $g : M^n \to N$ , for every permutation $\sigma \in \mathfrak{S}_n$ ,

g(x_{\sigma(1)}, x_{\sigma(2)}, \ldots, x_{\sigma(n)}) = \varepsilon_\sigma \cdot g(x_1, x_2, \ldots, x_n)

for all

x_i \in M

; moreover if

x_i = x_j

for two distinct indices

i, j

, then

g(x_1, x_2, \ldots, x_n) = 0.

This is an obvious consequence of Proposition 7 and no. 3, Proposition 5.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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