Let $f:X \to Y$ be a smooth surjective morphism of irreducible Noetherian schemes over $\mathbb{C}$. Assume $X, Y$ are not necessarily smooth over $\mathbb{C}$. Is it true that the induced morphism of tangent spaces from $T_xX$ to $T_{f(x)} Y$ is surjective for all $x \in X$?

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    $\begingroup$ Smooth morphisms factor into an étale morphism into an affine n-space over Y, followed by the projection onto Y. The indeuced morphisms of tangent spaces of both maps are clearly surjective (and bijective for the étale one.) $\endgroup$
    – Wille Liou
    Nov 15, 2013 at 17:00

1 Answer 1


Willy Liu answered this in a comment. By EGA 4v4 Corollary 17.11.4, a necessary and sufficient condition that $X \to Y$ be smooth at $x \in X$ is that there be an open neighborhood $U \subset X$ of $x$, and an étale morphism $g: U \to \mathbb{A}^n_Y$ for some $n$. Étale maps are unramified, so the map $T_x U \to T_{g(x)} \mathbb{A}^n_Y$ is an isomorphism. The projection $\mathbb{A}^n_Y \to Y$ induces a surjection $T_{g(x)}\mathbb{A}^n_Y \to T_{f(x)} Y$. The composition is the surjection you want.


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