The Stacks project

Lemma 33.35.6. Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. Then the relative frobenius $F_{X/S} : X \to X^{(p)}$ is a universal homeomorphism, is integral, and induces purely inseparable residue field extensions.

Proof. By Lemma 33.35.3 the morphisms $F_ X : X \to X$ and the base change $h : X^{(p)} \to X$ of $F_ S$ are universal homeomorphisms. Since $h \circ F_{X/S} = F_ X$ we conclude that $F_{X/S}$ is a universal homeomorphism. By Morphisms, Lemmas 29.45.5 and 29.10.2 we conclude that $F_{X/S}$ has the other properties as well. $\square$


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