Exercise 9, § 2

Let $k$ be a field of characteristic $p$, $C_k$ a $p$-ring of infinite length, with residue field $k$. Let $A$ be a separated complete local ring and $u : C_k \to A$ a local homomorphism such that the homomorphism deduced from $u$ by passing to quotients makes the residue field $K$ of $A$ into a separable extension of $k$. Let $C_K$ be a $p$-ring of infinite length, with residue field $K$. Prove that there exist local homomorphisms $i : C_k \to C_K$ and $v : C_K \to A$ such that $u = v \circ i$ and such that $v$ induces the identity on $K$ by passing to quotients.