Exercise 4, § 2

Let C be a Cohen $p$-ring of infinite length, and $k$ its residue field. Let A be a ring, and $(J_n)_{n \in \mathbf{N}}$ a decreasing sequence of ideals of A satisfying $pJ_n + J_n^p \subset J_{n+1}$ for all $n \in \mathbf{N}$. Finally, let $\overline{\varphi}: k \to A/J_0$ be a ring homomorphism, $(\xi_\lambda)_{\lambda \in \Lambda}$ a $p$-basis of $k$, $(x_\lambda)_{\lambda \in \Lambda}$ a family of elements of C lifting this $p$-basis, $(a_\lambda)_{\lambda \in \Lambda}$ a family of elements of A, such that the image of $a_\lambda$ in $A/J_0$ is $\overline{\varphi}(\xi_\lambda)$. a) For every $n \in \mathbf{N}$, there exists a unique homomorphism $\varphi_n$ of $C/p^{n+1}C$ into $A/J_n$ such that $\varphi_n \circ \rho_{n,i}^C = \rho_{n,i}^A \circ \overline{\varphi}$ and that the image under $\varphi_n$ of the class of $x_\lambda$ is the class of $a_\lambda$. (Use Exercises 2 and 3.) b) Suppose that A is separated and complete for the topology defined by the filtration $(J_n)_{n \in \mathbf{N}}$. Then there exists a unique ring homomorphism $\varphi$ of C into A, such that $\varphi(x_\lambda) = a_\lambda$ for $\lambda \in \Lambda$, and such that $\varphi$ induces $\overline{\varphi}$ by passing to quotients. This homomorphism is continuous. c) Suppose that the ring A is local, separated and complete, that $J_n = m_A^{n+1}$ for all $n \in \mathbf{N}$, that $A/J_0 = k$ and that $\overline{\varphi}$ is the identity mapping. Prove that the image of C in A under $\varphi$ is the unique Cohen subring of $A'$ containing each of the elements $a_\lambda$, and thus give another proof of part a) of Theorem 1 of § 2, No. 2; if $k$ is perfect and C is the ring W(k), one obtains another proof of Theorem 2 of § 2, No. 4.