The Stacks project

Proof. Let $P_\bullet $ be the standard resolution of $B$ over $A$. By Lemma 92.10.1 the map $P_\bullet \to P_\bullet $ induced by $F_ A$ and $F_ B$ is homotopic to the map $F_{P_\bullet } : P_\bullet \to P_\bullet $ given by Frobenius on each term. Hence we obtain what we want as clearly $F_{P_\bullet }$ induces the zero map $\Omega _{P_ n/A} \to \Omega _{P_ n/A}$ (since the derivative of a $p$th power is zero). $\square$