Lemma 33.36.2. Let $p > 0$ be a prime number. Let $f : X \to Y$ be a morphism of schemes in characteristic $p$. Then the diagram

$\xymatrix{ X \ar[d]_ f \ar[r]_{F_ X} & X \ar[d]^ f \\ Y \ar[r]^{F_ Y} & Y }$

commutes.

Proof. This follows from the following trivial algebraic fact: if $\varphi : A \to B$ is a homomorphism of rings of characteristic $p$, then $\varphi (a^ p) = \varphi (a)^ p$. $\square$

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