Definition 12

Tag 0DJL, permanent. Cite it as [Bourbaki, Tag 0DJL].

Algebra, chapter IX, § 1. Formes sesquilinéaires

Given two A-modules E, F and a form $\Phi$ sesquilinear for J on $E \times F$ , the form $\Phi_{(m)}$ sesquilinear for J on $(\wedge^m E) \times (\wedge^m F)$ characterized by (37) is called the extension of $\Phi$ to the m-th exterior powers.

The notations being those of Def. 12, let us study the mappings associated with $\Phi_{(m)}$ . From formula (24) (No. 6) and (37) we obtain the relations

\Phi_{(m)}(x_1 \wedge \ldots \wedge x_m, y_1 \wedge \ldots \wedge y_m) = \det(\langle x_i, d_\Phi(y_k) \rangle) = \det(\langle y_i, s_\Phi(x_k) \rangle^j).

We therefore have

s_{\Phi(m)} = k_s \circ (\wedge^m s_\Phi), \quad d_{\Phi m} = k_d \circ (\wedge^m d_\Phi),

where $k_s$ (resp. $k_d$ ) denotes the canonical mapping of $\wedge^m F^*$ into $(\wedge^m F)^*$ (resp. of $\wedge^m E^*$ into $(\wedge^m E)^*$ ) (cf. Chap. III, § 8, No. 2).

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

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