The Stacks project

Lemma 37.61.15. Let $U \to X$ be a weakly étale morphism of schemes where $X$ is a scheme in characteristic $p$. Then the relative Frobenius $F_{U/X} : U \to U \times _{X, F_ X} X$ is an isomorphism.

Proof. The morphism $F_{U/X}$ is a universal homeomorphism by Varieties, Lemma 33.36.6. The morphism $F_{U/X}$ is weakly étale as a morphism between schemes weakly étale over $X$ by Lemma 37.61.13. Hence $F_{U/X}$ is an isomorphism by Lemma 37.61.14. $\square$

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