Proposition 53.13.7. Let $k$ be a field of characteristic $p > 0$. Let $f : X \to Y$ be a nonconstant morphism of proper smooth curves over $k$. Then we can factor $f$ as

$X \longrightarrow X^{(p^ n)} \longrightarrow Y$

where $X^{(p^ n)} \to Y$ is a nonconstant morphism of proper smooth curves inducing a separable field extension $k(X^{(p^ n)})/k(Y)$, we have

$X^{(p^ n)} = X \times _{\mathop{\mathrm{Spec}}(k), F_{\mathop{\mathrm{Spec}}(k)}^ n} \mathop{\mathrm{Spec}}(k),$

and $X \to X^{(p^ n)}$ is the $n$-fold relative frobenius of $X$.

Proof. By Fields, Lemma 9.14.6 there is a subextension $k(X)/E/k(Y)$ such that $k(X)/E$ is purely inseparable and $E/k(Y)$ is separable. By Theorem 53.2.6 this corresponds to a factorization $X \to Z \to Y$ of $f$ with $Z$ a nonsingular proper curve. Apply Lemma 53.13.4 to the morphism $X \to Z$ to conclude. $\square$

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