Let $L|F$ be a extension of perfect fields of characteristic $p$, $\phi_F:F \to F_{\phi}$, $\phi_L:L \to L_{\phi}$ the Frobenius isomorphisms ($F_{\phi}=F$ but considered as $F$-algebra via $\phi_F$). The induced homomorphisms on the differential modules $\Omega_{L|K}\otimes_LL_{\phi} \to \Omega_{L_{\phi}|K}$ and $\Omega_{L_{\phi}|K} \to \Omega_{L_{\phi}|K_{\phi}}$ are then isomorphisms, but their composition $\Omega_{L|K}\otimes_LL_{\phi} \to \Omega_{L_{\phi}|K_{\phi}}$ takes $dX$ to $dX^p=pX^{p-1}dX = 0$ and then $\Omega_{L|K}=0$.

I find it a little strange (when $L|F$ is transcendental). Am I wrong?