The Stacks project

Lemma 53.2.3. Let $k$ be a field. Let $f : X \to Y$ be a nonconstant morphism of curves over $k$. If $Y$ is normal, then $f$ is flat.

Proof. Pick $x \in X$ mapping to $y \in Y$. Then $\mathcal{O}_{Y, y}$ is either a field or a discrete valuation ring (Varieties, Lemma 33.43.8). Since $f$ is nonconstant it is dominant (as it must map the generic point of $X$ to the generic point of $Y$). This implies that $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is injective (Morphisms, Lemma 29.8.7). Hence $\mathcal{O}_{X, x}$ is torsion free as a $\mathcal{O}_{Y, y}$-module and therefore $\mathcal{O}_{X, x}$ is flat as a $\mathcal{O}_{Y, y}$-module by More on Algebra, Lemma 15.22.10. $\square$

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