The Stacks project

Lemma 53.13.5. Let $k$ be a field of characteristic $p > 0$. Let $f : X \to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. Assume

  1. $X$ is smooth,

  2. $H^0(X, \mathcal{O}_ X) = k$,

  3. $k(X)/k(Y)$ is purely inseparable.

Then $Y$ is smooth, $H^0(Y, \mathcal{O}_ Y) = k$, and the genus of $Y$ is equal to the genus of $X$.

Proof. By Lemma 53.13.4 we see that $Y = X^{(p^ n)}$ is the base change of $X$ by $F_{\mathop{\mathrm{Spec}}(k)}^ n$. Thus $Y$ is smooth and the result on the cohomology and genus follows from Lemma 53.8.2. $\square$

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