Remark 33.36.11. Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. For $n \geq 1$

viewed as a scheme over $S$. Observe that $X \mapsto X^{(p^ n)}$ is a functor. Applying Lemma 33.36.2 we see $F_{X/S, n} = (F_ X^ n, \text{id}_ S) : X \longrightarrow X^{(p^ n)}$ is a morphism over $S$ fitting into the commutative diagram

where the right square is cartesian. The morphism $F_{X/S, n}$ is sometimes called the *$n$-fold relative Frobenius morphism of $X/S$*. This makes sense because we have the formula

which shows that $F_{X/S, n}$ is the composition of $n$ relative Frobenii. Since we have

(details omitted) we get also that

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