Proposition 9

Let $A$ be a commutative field endowed with an automorphism $J$, $E_i, F_i$ finite-dimensional vector spaces over $A$, and $\Phi_i$ a sesquilinear form for $J$ on $E_i \times F_i$ ($1 \leqslant i \leqslant m$). If the forms $\Phi_i$ are non-degenerate, the same is true of their tensor product $\Phi$. In this case the inverse form $\widehat{\Phi}$ of $\Phi$ is the tensor product of the inverse forms $\widehat{\Phi}_i$.

Indeed, since $A$ is a field, it follows from props. 6 and 7 of chap. III, § 1, no. 3 that a tensor product of injective (resp. surjective) linear mappings of $A$-modules is an injective (resp. surjective) linear mapping. Since the $s_{\Phi_i}$ are bijective by assumption (prop. 6, no. 6), the same is therefore true of their tensor product. On the other hand the canonical mapping $j_s$ of $\bigotimes_i F_i^*$ into $(\bigotimes_i F_i)^*$ is bijective (chap. III, § 1, no. 5, prop. 11). Therefore, by virtue of (36), $s_\Phi$ is bijective, and this establishes our first assertion (prop. 6, no. 6). Analogously $d_\Phi$ is bijective.

In the second assertion we have implicitly identified $\bigotimes_i F_i^*$ with $(\bigotimes_i F_i)^*$ and $\bigotimes_i E_i^*$ with $(\bigotimes_i E_i)^*$ by means of the mappings $j_s$ and $j_d$, which are here isomorphisms. The inverse forms in the statement exist since the $s_{\Phi_i}$, the $d_{\Phi_i}$, $s_\Phi$ and $d_\Phi$ are bijective (no. 7). Let us then put $x' = x'_1 \otimes \cdots \otimes x'_m$, $y' = y'_1 \otimes \cdots \otimes y'_m$ ($x'_i \in E_i^*$, $y'_i \in F_i^*$, $i = 1, \ldots, m$). By definition of the inverse forms, and viewed (36), we have

whence our second assertion.

C. Q. F. D.

Let E and F be two modules over the commutative ring A, and $\Phi$ a sesquilinear form for $J$ on $E \times F$. The mapping

of $E^m \times (F^j)^m$ into A is A-multilinear. It therefore defines a bilinear form $\Phi'$ on $(\bigotimes^m E) \times (\bigotimes^m F^j)$ characterized by Since the first member is zero when $x_i = x_k$ or when $y_i = y_k$ ($i \neq k$), $\Phi'$ defines, by passing to the quotients, a bilinear form on $(\wedge^m E) \times (\wedge^m F^j)$, or again, since $\wedge^m F^j$ is identified with $(\wedge^m F)^j$, a sesquilinear form $\Phi_{(m)}$ for J on $(\wedge^m E) \times (\wedge^m F)$. This one is characterized by (37)