*Let A be a commutative ring and M a free A-module with a basis ( e j ) j ∈ J (e_j)_{j \in J} ( e j ) j ∈ J p ≤ C a r d ( J ) p \leq \mathrm{Card}(J) p ≤ Card ( J ) f : M p → N f : M^p \to N f : M p → N x i = ∑ j ∈ J ξ j i e j x_i = \sum_{j \in J} \xi_{ji} e_j x i = ∑ j ∈ J ξ j i e j ( 1 ≤ i ≤ p ) (1 \leq i \leq p) ( 1 ≤ i ≤ p )
f ( x 1 , x 2 , … , x p ) = ∑ j 1 < j 2 < ⋯ < j p ( ∑ σ ∈ S p ε σ ⋅ ξ j σ ( 1 ) , 1 ξ j σ ( 2 ) , 2 ⋯ ξ j σ ( p ) , p ) f ( e j 1 , … , e j p )
f(x_1, x_2, \ldots, x_p)
= \sum_{j_1 < j_2 < \cdots < j_p} \left( \sum_{\sigma \in S_p} \varepsilon_\sigma \cdot \xi_{j_{\sigma(1)}, 1} \xi_{j_{\sigma(2)}, 2} \cdots \xi_{j_{\sigma(p)}, p} \right) f(e_{j_1}, \ldots, e_{j_p})
f ( x 1 , x 2 , … , x p ) = j 1 < j 2 < ⋯ < j p ∑ σ ∈ S p ∑ ε σ ⋅ ξ j σ ( 1 ) , 1 ξ j σ ( 2 ) , 2 ⋯ ξ j σ ( p ) , p f ( e j 1 , … , e j p ) where ( j k ) 1 ≤ k ≤ p (j_k)_{1 \leq k \leq p} ( j k ) 1 ≤ k ≤ p runs through the set of strictly increasing sequences of p elements of J.
Now
f ( x 1 , … , x p ) = ∑ ( j k ) ξ j 1 , 1 ξ j 2 , 2 ⋯ ξ j p , p f ( e j 1 , e j 2 , … , e j p )
f(x_1, \ldots, x_p) = \sum_{(j_k)} \xi_{j_1, 1} \xi_{j_2, 2} \cdots \xi_{j_p, p} f(e_{j_1}, e_{j_2}, \ldots, e_{j_p})
f ( x 1 , … , x p ) = ( j k ) ∑ ξ j 1 , 1 ξ j 2 , 2 ⋯ ξ j p , p f ( e j 1 , e j 2 , … , e j p ) where ( j k ) 1 ≤ k ≤ p (j_k)_{1 \leq k \leq p} ( j k ) 1 ≤ k ≤ p all the sequences of p elements of J; it then suffices to apply Corollary 1 to Proposition 7 of § 7, no. 4 to f.
In particular, if J is finite and has n elements and x i = ∑ j ∈ J ξ j i e j x_i = \sum_{j \in J} \xi_{ji} e_j x i = ∑ j ∈ J ξ j i e j ( 1 ≤ i ≤ n ) (1 \leq i \leq n) ( 1 ≤ i ≤ n )
(11)
x 1 ∧ x 2 ∧ ⋯ ∧ x n = ( ∑ σ ∈ S n ε σ ξ j σ ( 1 ) , 1 ξ j σ ( 2 ) , 2 ⋯ ξ j σ ( n ) , n ) e j 1 ∧ e j 2 ∧ ⋯ ∧ e j n
x_1 \wedge x_2 \wedge \cdots \wedge x_n
= \left( \sum_{\sigma \in S_n} \varepsilon_\sigma \xi_{j_{\sigma(1)}, 1} \xi_{j_{\sigma(2)}, 2} \cdots \xi_{j_{\sigma(n)}, n} \right) e_{j_1} \wedge e_{j_2} \wedge \cdots \wedge e_{j_n}
x 1 ∧ x 2 ∧ ⋯ ∧ x n = ( σ ∈ S n ∑ ε σ ξ j σ ( 1 ) , 1 ξ j σ ( 2 ) , 2 ⋯ ξ j σ ( n ) , n ) e j 1 ∧ e j 2 ∧ ⋯ ∧ e j n where ( j k ) 1 ≤ k ≤ n (j_k)_{1 \leq k \leq n} ( j k ) 1 ≤ k ≤ n
(12)
det ( x 1 , x 2 , … , x n ) = ∑ σ ∈ S ε σ ⋅ ξ j σ ( 1 ) , 1 ξ j σ ( 2 ) , 2 ⋯ ξ j σ ( n ) , n .
\det(x_1, x_2, \ldots, x_n) = \sum_{\sigma \in S} \varepsilon_\sigma \cdot \xi_{j_{\sigma(1)}, 1} \xi_{j_{\sigma(2)}, 2} \cdots \xi_{j_{\sigma(n)}, n}.
det ( x 1 , x 2 , … , x n ) = σ ∈ S ∑ ε σ ⋅ ξ j σ ( 1 ) , 1 ξ j σ ( 2 ) , 2 ⋯ ξ j σ ( n ) , n . With the notation of Lemma 1, comparing formulae (10) and (12) gives
x 1 ∧ x 2 ∧ ⋯ ∧ x p = ∑ H ∈ F p ( J ) det ( x H , 1 , x H , 2 , … , x H , p ) e H
x_1 \wedge x_2 \wedge \cdots \wedge x_p = \sum_{H \in \mathscr{F}_p(J)} \det(x_{H,1}, x_{H,2}, \ldots, x_{H,p}) e_H
x 1 ∧ x 2 ∧ ⋯ ∧ x p = H ∈ F p ( J ) ∑ det ( x H , 1 , x H , 2 , … , x H , p ) e H where F p ( J ) \mathscr{F}_p(J) F p ( J ) J J J p p p H ∈ F p ( J ) H \in \mathscr{F}_p(J) H ∈ F p ( J ) x H , i = ∑ j ∈ H ξ j i e j x_{H,i} = \sum_{j \in H} \xi_{ji} e_j x H , i = ∑ j ∈ H ξ j i e j e H = e j 1 ∧ e j 2 ∧ ⋯ ∧ e j p e_H = e_{j_1} \wedge e_{j_2} \wedge \cdots \wedge e_{j_p} e H = e j 1 ∧ e j 2 ∧ ⋯ ∧ e j p ( j k ) 1 ≤ k ≤ p (j_k)_{1 \leq k \leq p} ( j k ) 1 ≤ k ≤ p H H H det ( x H , 1 , … , x H , p ) \det(x_{H,1}, \ldots, x_{H,p}) det ( x H , 1 , … , x H , p ) ( e j k ) 1 ≤ k ≤ p (e_{jk})_{1 \leq k \leq p} ( e j k ) 1 ≤ k ≤ p
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
Nothing.