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Consider the following: Suppose that $K$ is a perfect field, $V$ and $W$ are integral $K$-varieties, $V \to W$ is a dominant morphism, and the function field of $V$ is a separable extension of the function field of $W$. Then there is a dense open subvariety $U$ of $V$ such that $U \to W$ is smooth.

I would like a reference for this. I need this for something, and I have a proof, but I think this should be very well known so it's probably silly to write out a proof.

Vakil's notes give a proof in characteristic zero, but he doesn't give a citation. I have seen a sketch of a proof on mathoverflow, but no citation.

I apologize if this is a silly question. I'm doing some algebraic geometry without having any real background in the subject. I think this might be so well-known that you wouldn't need a citation for an algebraic geometry paper, but it's going to be used in a model theory paper, so I want to give references for basically everything from algebraic geometry that we use.

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    $\begingroup$ this fact is true in more general context.extending a property from generic point(function field) of an integral variety is called "spreading out". and you can do this for smoothness. a reference could be EGA IV_4 theorem 8.10.5 $\endgroup$
    – ali
    Commented Oct 3, 2020 at 19:02
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    $\begingroup$ Certainly you didn't mean to write $U\to W$ smooth, but $V\times_W U\to U$ smooth. $\endgroup$
    – Ariyan Javanpeykar
    Commented Oct 3, 2020 at 20:15
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    $\begingroup$ A "user-friendly" reference is Poonen's "Rational points on varieties", a large part of which is something like a "reader's guide" to EGA. It contains in particular a helpful table collecting which properties of morphisms of schemes satisfy base change/descent/spreading out and where in EGA this is proven. EGA IV_4 Proposition 17.7.8(ii) is indeed the right reference for this result. $\endgroup$
    – Dan Petersen
    Commented Oct 4, 2020 at 3:17
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    $\begingroup$ I think spreading out is actually not relevant here. First, the openness of the smooth locus on the base is something you can cite from Liu's book or the stacks project. However, this open locus can of course be empty. What you are trying to show is that it is non-empty by showing that the generic point is contained in it. Thus, all you really have to argue is that the generic fibre of your morphism $V\to W$ is smooth (which has nothing to do with spreading out) and cite the relevant openness result. $\endgroup$
    – Ariyan Javanpeykar
    Commented Oct 4, 2020 at 20:20
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    $\begingroup$ @ErikWalsberg I am still puzzled by your formulation. Are you not asking for the existence of a dense open $U\subset W$ such that $V\to W$ is smooth over $U$? If so, let $U$ be the locus of points in $W$ over which $V\to W$ is smooth. This is open. Let $V_U$ be the inverse image of $U$ along $V\to W$. If $U$ is non-empty, then it contains the generic point of $U$ (hence $V$). Base-changing along the inclusion of the generic point you end up again with a smooth morphism, because smoothness is stable under base-change. Conversely, if $U$ contains the generic point, then it is non-empty...tbc $\endgroup$
    – Ariyan Javanpeykar
    Commented Oct 5, 2020 at 5:07

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Well, thanks everyone, but in the end I found a good reference. I will post it here as this might be useful for someone else.

The following is (a special case of) Corollary 5.4.3 in Mumford and Oda's Algebraic Geometry II: Suppose that $V$ and $W$ are integral $K$-varieties, $W$ is regular, and $f : V \to W$ is a dominant morphism. Then $f$ is smooth on a dense open subset of $V$ if and only if the function field of $V$ is a separable extension of the function field of $W$. The result in the book is written for schemes.

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