A perfect $\mathbb{F}_p$-algebra $A$ is formally étale

Question

Suppose $A$ is a perfect $\mathbb{F}_p$-algebra, i.e., the Frobenius morphism $x\mapsto x^p$ of $A$ is bijective. I want to prove $\operatorname{Spec}A\to\operatorname{Spec}\mathbb{F}_p$ is formally étale. This means that given a $\mathbb{F}_p$-algebra morphism $g:A\to R/I$, where $I^2=0$, there is a unqiue lift $A\to R$. How to use the condition that $A$ is perfect?

Answer 1

For each $a\in A$, choose a lift $b\in R$ of $g(a^{1/p})\in R/I$, and let $\tilde g(a):=b^p$. Then $\tilde g(a)$ does not depend on the choice of the lift $b$, and $\tilde g\colon A\to R$ is the unique lift of $g$.