Let $k$ be an algebraically closed field.
By a finitely generated subfield of $k$ I mean a subfield $k\_0\subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $\mathbb Q$ or $\mathbb F\_p$).


I need a reference or a proof for the following (well-known? evident?) proposition:



> 
> **Proposition.**
> *Let
>  $$f\colon X\to V$$
>  be a morphism of $k$-varieties.
>  Then the triple $(X,Y,f)$ can be defined over a finitely generated subfield of $k$.
>  In other words, there exists a finitely generated subfield $k\_0\subset k$
>  and a morphism of $k\_0$-varieties
>  $$f\_0\colon X\_0\to Y\_0$$
>  such that
>  $(X\_0,Y\_0,f\_0)\times\_{k\_0} k$ is isomorphic to $(X,Y,f)$.*
> 
> 
> 


A referee asked me to prove this assertion. I know how to prove this for affine and projective varieties, but not for *abstract* varieties.



Defining abstract varieties and their morphisms over a finitely generated subfield of the base field
This is treated (in much greater generality) in EGA IV$\_3$, **Théorème 8.8.2**. The existence of $X\_0$ and $Y\_0$ follows from part (ii), whereas the existence of $f\_0$ is part (i).


**References.**


[EGA IV$\_3$] A. Grothendieck, [*Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Troisième partie)*](http://www.numdam.org/articles/PMIHES_1966__28__5_0). Publ. Math., Inst. Hautes Étud. Sci. **28**, 1-255 (1966). [ZBL0144.19904](https://zbmath.org/?q=an:0144.19904). DOI: [10.1007/BF02684343](http://dx.doi.org/10.1007/BF02684343) 


