Let $S$ be a scheme and $X\to Y$ be a morphism over $S$. Then we have an induced homomorphism of sheaves $h_X=\operatorname{Hom}_S({-}, X)\to h_Y=\operatorname{Hom}_S({-}, Y)$ over the small étale site $S_\text{étale}$.
Question: When is $h_X\to h_Y$ a surjective homomorphism between sheaves of sets over $S_\text{étale}$?
I find this result in the comment of Ehsan M. Kermani under Étale Local Sections of a Smooth Surjective Morphism. Is there any proof or reference of this claim?
Best Answer
Proposition. Let $f\colon X\to Y$ be a surjective morphism such that $\forall y\in Y$ there exists a point $x\in f^{-1}(y)$ where $f$ is smooth (i.e. every fiber has a non-reduced point). Then $h_X\to h_Y$ is surjective on the (small or big) étale site.
Proof: Take $f\in h_Y(S)$. The property of having a smooth point in every fiber is stable under base change so we may assume $S=Y$. Consider the open subset $X^\circ$ of $X$ where $f$ is smooth. The restriction $f^\circ$ of $f$ to $X^\circ$ is smooth and surjective. Apply EGA IV 4 Cor. 17.16.3 (ii) to $f^\circ$. We find an étale covering $Y'\to Y$ such that $Y'\times_Y X$ admits a section $s\colon Y'\to Y'\times_Y X$ which is what we need.
Remark 1. By EGA IV 4 Cor. 17.16.2 if $f\colon X\to Y$ is locally of finite presentation and faithfully flat then it admits a section after a faithfully flat base change. Therefore, in this case $h_X$ surjects over $h_Y$ on the (small or big) fppf site.
Remark 2. The property in the proposition is equivalent to the surjectivity of $h_X\to h_Y$. Indeed, the surjectivity implies that $f$ admits a section étale locally on $Y$.