Proposition 4

Let $C$ and $C'$ be two $p$-rings such that $l(C) \geqslant l(C')$, and let $\pi$ (resp. $\pi'$) be the canonical homomorphism of $C$ (resp. $C'$) onto $\kappa_C$ (resp. $\kappa_{C'}$). Let $(x_\lambda)_{\lambda \in \Lambda}$ (resp. $(x'_\lambda)_{\lambda \in \Lambda}$) be a family of elements of $C$ (resp. $C'$) whose image under $\pi$ (resp. $\pi'$) is a $p$-basis of $\kappa_C$ (resp. $\kappa_{C'}$). Let $v$ be an isomorphism of $\kappa_C$ onto $\kappa_{C'}$ such that $v(\pi(x_\lambda)) = \pi'(x'_\lambda)$ for every $\lambda \in \Lambda$. Then there exists a unique homomorphism $u$ of $C$ into $C'$, such that $v \circ \pi = \pi' \circ u$ and $u(x_\lambda) = x'_\lambda$ for every $\lambda \in \Lambda$. It is surjective. If $l(C) = l(C')$, it is an isomorphism.

Let us prove the existence of $u$. Let $A$ be the subring of $C \times C'$ formed by the pairs $(x, x')$ such that $v(\pi(x)) = \pi'(x')$. The mapping $(x, x') \mapsto \pi(x)$ is a surjective homomorphism of rings from A onto $\kappa_C$. Its kernel $m$, equal to $pC \times pC'$, is therefore a maximal ideal of A. The topological subspace A of $C \times C'$ is closed in $C \times C'$, hence complete, and the topology induced on A by that of $C \times C'$ is the $m$-adic topology. Hence A is a separated complete local ring with maximal ideal $m$ (III, § 2, No. 13, Prop. 19). For every $\lambda \in \Lambda$, we have $(x_\lambda, x'_\lambda) \in A$ by hypothesis; if $\xi_\lambda$ is the class of $(x_\lambda, x'_\lambda)$ modulo $m$, the family $(\xi_\lambda)_{\lambda \in \Lambda}$ is a $p$-basis of the field $A/m$. By theorem 1 of No. 2, there exists a Cohen subring $C''$ of A, and only one, containing $(x_\lambda, x'_\lambda)$ for every $\lambda \in \Lambda$. We have $l(C'') = l(C) \geq l(C')$. The restriction to $C''$ of the projection of $C \times C'$ onto C is a homomorphism $h : C'' \to C$ which induces an isomorphism of $\kappa_{C''}$ onto $\kappa_C$. By Prop. 2, c) of No. 2, $h$ is an isomorphism of $C''$ onto C. One sees analogously that the restriction $h'$ to $C''$ of the projection of $C \times C'$ onto $C'$ is a surjective homomorphism of $C''$ onto $C'$. Hence, $C''$ is the graph of a surjective homomorphism $u = h' \circ h^{-1}$ of C onto $C'$, and one obviously has $v \circ \pi = \pi' \circ u$, $u(x_\lambda) = x'_\lambda$ for every $\lambda \in \Lambda$. Moreover, if $l(C) = l(C')$, $u$ is an isomorphism.

Let us prove the uniqueness of $u$. Let $u_1$ be a homomorphism of C into $C'$ such that $v \circ \pi = \pi' \circ u_1$ and $u_1(x_\lambda) = x'_\lambda$ for every $\lambda \in \Lambda$, and let $C_1$ be the graph of $u_1$. It is immediate that $C_1$ is a Cohen subring of A, containing $(x_\lambda, x'_\lambda)$ for every $\lambda \in \Lambda$, whence $C_1 = C''$ (theorem 1 of No. 2) and finally $u_1 = u$.