The Stacks project

Lemma 41.14.3. Let $U \to X$ be an 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 étale as a morphism between schemes étale over $X$ (Morphisms, Lemma 29.36.18). Hence $F_{U/X}$ is an isomorphism by Theorem 41.14.1. $\square$

Comments (2)

Comment #5554 by Peng DU on

Wiki https://en.wikipedia.org/wiki/Frobenius_endomorphism#Relative_Frobenius says the converse is also true, is it? I can't find a proof.

Comment #5738 by on

Don't think the converse is true as stated in this lemma (for example flat closed immersions should give counter examples). If you want to add it to the Stacks project (under suitable hypotheses), then please go and write a latex thing and send it to the email address of the Stacks project.

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