Proposition 9

Tag 064K, permanent. Cite it as [Bourbaki, Tag 064K].

Let $M$ be an $\mathbf{A}$ -module with a basis $(e_i)_{i \in J}$ (finite or otherwise) whose indexing set $J$ is totally ordered. For every integer $p > 0$ , let $(e_H)_{H \in \mathfrak{S}_p(J)}$ be the corresponding basis of $\wedge^p(M)$ (§ 7, no. 8). Let $(x_i)_{1 \leq i \leq p}$ be a sequence of $p$ elements of $M$ ; let

x_i = \sum_{j \in J} \xi_{ji} e_j \quad \text{for } i \in I = \{1, p\}

and let $X$ denote the matrix $(\xi_{ji})$ of type $(J, I)$ . Then

x_1 \wedge x_2 \wedge \cdots \wedge x_p = \sum_{H \in \mathfrak{F}_p(J)} (\det X_{H,I}) e_H.

where $H$ runs through the set $\mathfrak{F}_p(J)$ of subsets of $J$ with $p$ elements.

This follows immediately from formula (12) of no. 4 and formula (6) of no. 3.

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

Cited by

Proposition 10, § 8
Proposition 4, § 4

Cites

§ 7, no. 8