Étale Local Sections of a Smooth Surjective Morphism

Question

Why does a smooth surjective morphism of schemes admit a section étale-locally?

Answer 1

Let $f : X\to Y$ be a morphism of schemes. A section of $f$ for some topology is an open covering $T \to Y$ for this topology which factorizes into $T\to X$ and $f : X\to Y$. Note that this induces $T\to X\times_Y T$ which is a section of $X\times_Y T\to T$ in the usual terminology.

Let $f : X\to Y$ be a smooth morphism. The local structure of $f$ can be described as follows. For any $x\in X$ and for $y=f(x)$, there exists an open neighborhood $V=V_y$ of $y$ and an open neigbhorhood $U$ of $x$, contained in $f^{-1}(V)$, such that $f|_U$ factorizes as an étale morphism $g : U\to \mathbb A^d_V$ followed by the canonical projection $s : \mathbb A^d_V \to V$.

Consider a section $\sigma : V\to \mathbb A^d_V$ of $s$. The base change $$ \sigma' : W_y:=U\times_{\mathbb A^d_V} V\to V$$ is étale, and the canonical maps $U\times_{\mathbb A^d_V} V\to U\to X$ makes $\sigma'$ a local étale section of $U\to V$.

Now suppose $Y$ is quasi-compact for simplicity. Suppose moreover $f$ is surjective. Cover $Y$ by finitely many $V_y$ and consider the disjoint union of the corresponding $W_y$. It produces naturally an étale section of $f$.

Answer 2

This is EGA IV, 17.16.3 (ii).