The Stacks project

Lemma 53.13.2. Let $k$ be a field of characteristic $p > 0$. Let $f : X \to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. If the extension $k(X)/k(Y)$ of function fields is purely inseparable, then there exists a factorization

\[ X = X_0 \to X_1 \to \ldots \to X_ n = Y \]

such that each $X_ i$ is a proper nonsingular curve and $X_ i \to X_{i + 1}$ is a degree $p$ morphism with $k(X_{i + 1}) \subset k(X_ i)$ inseparable.

Proof. This follows from Theorem 53.2.6 and the fact that a finite purely inseparable extension of fields can always be gotten as a sequence of (inseparable) extensions of degree $p$, see Fields, Lemma 9.14.5. $\square$

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