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

$X^{(p)} = X^{(p/S)} = X \times _{S, F_ S} S$

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

$\xymatrix{ X \ar[rr]_{F_{X/S}} \ar[rrd] \ar@/^1em/[rrrr]^{F_ X} & & X^{(p)} \ar[rr] \ar[d] & & X \ar[d] \\ & & S \ar[rr]^{F_ S} & & S }$

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

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