Exercise 5, § 2

Let A be a ring and let $(J_n)_{n \in \mathbf{N}}$ be a decreasing sequence of ideals of A, satisfying $pJ_n + J_n^p \subset J_{n+1}$ for every integer $n \geqslant 0$. Let R be a ring of characteristic $p$ and let $\overline{\varphi}$ be a ring homomorphism of R into $A/J_0$. A lifting of $\overline{\varphi}$ to $A/J_n$ is by definition a mapping $\varphi_n : R \to A/J_n$ such that $\varphi_n(x^p) = \varphi_n(x)^p$ for $x \in R$, and which yields $\overline{\varphi}$ again by passing to the quotient. a) Let $\varphi_n, \tilde{\varphi}_n$ be two liftings of $\overline{\varphi}$ to $A/J_n$. Then $\varphi_n$ and $\tilde{\varphi}_n$ coincide on $R^{p^n}$. b) If R is perfect, there exists a unique lifting $\varphi_n$ of $\overline{\varphi}$ to $A/J_n$. One has $\varphi_n(1) = 1$ and $\varphi_n(xy) = \varphi_n(x) \varphi_n(y)$ for $x, y \in R$.