The Stacks project

Lemma 33.36.3. Let $p > 0$ be a prime number. Let $X$ be a scheme in characteristic $p$. Then the absolute frobenius $F_ X : X \to X$ is a universal homeomorphism, is integral, and induces purely inseparable residue field extensions.

Proof. This follows from the corresponding results for the frobenius endomorphism $F_ A : A \to A$ of a ring $A$ of characteristic $p > 0$. See the discussion in Algebra, Section 10.46, for example Lemma 10.46.7. $\square$

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