Definition 33.36.4. Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. We define

viewed as a scheme over $S$. Applying Lemma 33.36.2 we see there is a unique morphism $F_{X/S} : X \longrightarrow X^{(p)}$ over $S$ fitting into the commutative diagram

where the right square is cartesian. The morphism $F_{X/S}$ is called the *relative Frobenius morphism of $X/S$*.

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