# Defining abstract varieties and their morphisms over a finitely generated subfield of the base field

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.

• In the case you need it, you may also be interested by this question and its answers mathoverflow.net/questions/211867/… Commented Oct 5, 2018 at 17:06
• This is the notion of "spreading out", discussed here. Commented Nov 7, 2018 at 9:37

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). Publ. Math., Inst. Hautes Étud. Sci. 28, 1-255 (1966). ZBL0144.19904. DOI: 10.1007/BF02684343

• If one prefers English, it is also in [Görtz-Wedhorn], Chapter 10, "Schemes over inductive limits of rings".
– user19475
Commented Oct 7, 2018 at 14:52

A variety $$V$$ is a finite union of open affine varieties $$V_i$$, and because $$V$$ is separated (usually part of the definition of variety) the intersections $$V_i\cap V_j$$ are also affine. Now $$V$$ can be reconstructed from the affine varieties $$V_i,V_i\cap V_j$$ and the maps of affine varieties $$V_i\cap V_j\to V_i$$. Obviously, this system is defined over a finitely generated field. Hence $$X$$ and $$V$$ are defined over a finitely generated field. The graph of $$f$$ is a closed subvariety of $$X\times V$$, and so is defined by a coherent sheaf of ideals in the structure sheaf of $$X\times V$$, which is obviously defined over a finitely generated field.

• Thank you, @anon! I have included your proof in my paper with a reference to "anonymous Mathoverflow user". Commented Oct 6, 2018 at 13:57
• Ex ungue leonem! Commented Oct 6, 2018 at 13:59
• Dear Mikhail: and who might this modern Newton be? Commented Oct 29, 2018 at 17:51