Surjective étale morphisms étale locally split

The actual question is slightly more general than that in the title:

Let $$p: U\to Y$$ be a surjective étale morphism and $$Y\to X$$ be a finite morphism of schemes. Is there an étale cover $$V\to X$$ (surjective) such that the base change $$p': V\times_X U\to V\times_X Y$$ admits a section?

If necessary, one can also assume that $$Y\to X$$ is surjective.

I think this is implicitly used in Lecture Notes on Motivic Cohomology by Carlo Mazza, Vladimir Voevodsky, Charles Weibel, Lemma 6.16. I don't know how to go through. I can prove this when $$p$$ is also a finite morphism and $$Y=X$$ and $$Y\to X$$ is the identity but not in general.

It would be very helpful to me to have a reference or an answer to this question.

• Consider the case $X=Y$. If $p:U\to Y$ is not surjective, then neither is $p'$, so it cannot have a section. Conversely, if $p$ is surjective, then you can take $V=U$ and then the diagonal morphism gives a section. Apr 9 '20 at 16:06
• Sorry, I forgot to say $p$ is surjective...edited just now. But $V=U$ may be not étale over $X$... Apr 9 '20 at 16:07

We can work locally on $$X$$ and even (by standard limit arguments) assume that $$X=\mathrm{Spec}(R)$$ where $$R$$ is local and strictly henselian. Then $$Y=\coprod_{i=1}^{n}Y_i$$ where each $$Y_i$$ is local and finite over $$X$$, in particular strictly henselian too. So $$U\times_Y Y_i\to Y_i$$ has a section since it is étale and surjective.
• I was meant to have an étale cover $V\to X$ (surjective) of $X$ such that the base change $p': V\times_X U\to V\times_X Y$ admits a section. I think your argument gives an étale cover of $Y$. Apr 9 '20 at 16:53
• @Lao-tzu No, the argument gives an etale cover of $X$, which by the limit argument is equivalent to assuming $X$ the spectrum of a strictly henselian local ring. Apr 9 '20 at 17:29
• @Lao-tzu It suffices to show that for each point there is an etale neighborhood of that point where the base change admits a section, as then the disjoint union of these over all points (or, if $X$ is quasicompact, a finite disjoint union) will be a cover with the desired properties. Now Laurent's argument gives a section over the inverse limit of all etale neighborhoods of that point. We want to show the section is defined over one of them. Apr 9 '20 at 18:51
• @Lao-tzu This follows because $p$, being etale, is of finite presentation, so the finitely many generators of the ring of functions on $p$ must go to functions defined on one of these etale neighborhoods (because, by definition, all functions on the limit are on one of the neighborhoods). Apr 9 '20 at 18:52