The Stacks project

Lemma 33.25.9. Let $k$ be a field. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $k$. Let $x \in X$ be a point and set $y = f(x)$. If $X \to \mathop{\mathrm{Spec}}(k)$ is smooth at $x$ and $f$ is flat at $x$ then $Y \to \mathop{\mathrm{Spec}}(k)$ is smooth at $y$. In particular, if $X$ is smooth over $k$ and $f$ is flat and surjective, then $Y$ is smooth over $k$.

Proof. It suffices to show that $Y$ is geometrically regular at $y$, see Lemma 33.12.6. This follows from Lemma 33.12.5 (and Lemma 33.12.6 applied to $(X, x)$). $\square$

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