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 $x \in X$?

5$\begingroup$ Smooth morphisms factor into an étale morphism into an affine nspace 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 LiouNov 15, 2013 at 17:00
1 Answer
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.