Definition 6

Given an isomorphism u of an A-module E onto an A-module F, the transpose of the inverse isomorphism of u (equal to the inverse isomorphism of the transpose of u) is called the contragredient isomorphism of u and denoted by $\tilde{u}$.

The isomorphism $\tilde{u}$ is thus characterized by the relation (19)

If $v : F \to G$ is an isomorphism, the contragredient isomorphism of $v \circ u$ is $\tilde{v} \circ \tilde{u}$.

In particular, the mapping $u \mapsto \tilde{u}$ is an isomorphism of the linear group $\mathbf{GL}(E)$ onto a subgroup of the linear group $\mathbf{GL}(E^*)$.

Let $\sigma : A \to B$ be an isomorphism of a ring $A$ onto a ring $B$, $E$ a left $A$-module, $F$ a left $B$-module and $u : E \to F$ a semi-linear mapping (§ 1, no. 13) relative to $\sigma$. Let $\sigma^{-1}$ be the inverse isomorphism of $\sigma$; for all $y^* \in F^*$, the mapping $x \mapsto \langle u(x), y^* \rangle^{\sigma^{-1}}$ of $E$ into $A$ is a linear form; if it is also denoted by ${}^t u(y^*)$, a mapping ${}^t u : F^* \to E^*$ is defined which is also called the transpose of the semi-linear mapping $u$; it is thus characterized by the identity

for $x \in E, y^* \in F^*$. It is immediately verified that ${}^t u$ is a semi-linear mapping relative to $\sigma^{-1}$. If $v$ denotes the mapping $u$ considered as an A-linear mapping of $E$ into $\sigma_*(F)$ (§ 1, no. 13), we may write $u = \phi \circ v$, where $\phi$ is the identity mapping $\sigma_*(F) \to F$, considered as a semi-linear mapping relative to $\sigma$. It is immediate that ${}^t u = {}^t v \circ {}^t \phi$ and $({}^t \phi, \sigma^{-1})$ is a di-isomorphism of $F^*$ onto $(\sigma_*(F))^*$ relative to the isomorphism $\sigma^{-1}$; this relation allows us immediately to extend the properties of transposes of linear mappings to transposes of semi-linear mappings.