Library / Algebra / Chapter III / § 7
Corollary 1
Tag 0630, permanent. Cite it as [Bourbaki, Tag 0630].
Algebra, chapter III, § 7. Exterior algebras
Let H , K H, K H , K [ 1 , n ] [1, n] [ 1 , n ] N \mathbf{N} N ( i h ) 1 ≤ h ≤ p , ( j k ) 1 ≤ k ≤ n − p (i_h)_{1 \leq h \leq p}, (j_k)_{1 \leq k \leq n-p} ( i h ) 1 ≤ h ≤ p , ( j k ) 1 ≤ k ≤ n − p H H H K K K
x H = x i 1 ∧ x i 2 ∧ ⋯ ∧ x i p , x K = x j 1 ∧ x j 2 ∧ ⋯ ∧ x j n − p ;
x_H = x_{i_1} \wedge x_{i_2} \wedge \cdots \wedge x_{i_p}, \quad x_K = x_{j_1} \wedge x_{j_2} \wedge \cdots \wedge x_{j_{n-p}};
x H = x i 1 ∧ x i 2 ∧ ⋯ ∧ x i p , x K = x j 1 ∧ x j 2 ∧ ⋯ ∧ x j n − p ; then
x H ∧ x K = ( − 1 ) ν x 1 ∧ x 2 ∧ ⋯ ∧ x n
x_H \wedge x_K = (-1)^{\nu} x_1 \wedge x_2 \wedge \cdots \wedge x_n
x H ∧ x K = ( − 1 ) ν x 1 ∧ x 2 ∧ ⋯ ∧ x n where ν \nu ν ( i , j ) ∈ H × K (i, j) \in H \times K ( i , j ) ∈ H × K i > j i > j i > j
By Proposition 5 this reduces to proving
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Transcribed from the printed edition, 1998, Springer. Not translated.