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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