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.



Reference request for generic smoothness
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.


