An etale cover of a semiperfect ring

Question

Assume that $R$ is a semiperfect ring in characteristic $p$,i.e the frobenius is surjective on $R$. I think one can prove that an etale cover of $R$ should again be semiperfect by considering the basic etale maps simmilar to the fact that an algebraic extension of a perfect field is again perfect. Is this true and has anyone a better proof or a reference for this fact?

Answer 1

This is true. A non-ideal reference is in the proof of Lemma 1.5 of my joint paper, but the proof is simple enough that one should just write the proof themselves if they need it.

Proposition: Let $R$ be a semi-perfect $\mathbb{F}_p$-algebra, and $R\to S$ and etale map. Then, $S$ is semi-perfect.

Proof: By definition $\mathrm{Frob}_R\colon R\to R$ is surjective, and thus the same holds true for $S\to S\otimes_{R,\mathrm{Frob}_R}R$. But, as $R\to S$ is etale, this map may be identified with $\mathrm{Frob}_S$ (e.g., see Tag 0EBS). $\blacksquare$