Lemma 33.35.5. Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $f : X \to Y$ be a morphism of schemes over $S$ . Then the diagram

$\xymatrix{ X \ar[d]_ f \ar[r]_{F_{X/S}} & X^{(p)} \ar[d]^{f^{(p)}} \\ Y \ar[r]^{F_{Y/S}} & Y^{(p)} }$

commutes.

Proof. This follows from Lemma 33.35.2 and the definitions. $\square$

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