For an algebraically closed field $k$, there exists a fully faithful functor from the category of $k$-varieties to $k$-schemes.

(A $k$-variety is an affine, quasi-affine, projective, or quasi-projective variety. And an affine (resp. projective) variety is an irreducible closed subset of $\mathbb{A}_k$ (resp. $\mathbb{P}_k$), and a quasi-affine (resp. quasi-projective) variety is an open subset of an affine (resp. projective) variety, with respect to the Zariski topology, where $\mathbb{A}^n_k = \{ (a_i)_{i=1, \cdots n} | a_i \in k \}, \mathbb{P}^n_k = \{ (a_0 : \cdots : a_n) | a_i \in k, $ for some $i, a_i \neq 0 \}.$)

I think this is the very motivation to study schemes in order to understand varieties.

Now, but, it is natural desires to understand varieties over non-algebraically closed field, and also I want to, because I want to study diophantine geometry.

But the functor only works over algebraically closed fields.

Is there such a functor over non-algebraically closed fields?

And if not, is there a reasonable reason to study schemes for people who want to study diophantine geometry?


1 Answer 1


If the base field is not algebraically closed, the usual formalism of varieties doesn't work, the main problem being the absence of Nullstellensatz: if you take the equation $x^2+1=0$ in $\mathbb{R}$, it has no solution, and hence defines the empty variety in $\mathbb{A}^1_{\mathbb{R}}$. The functor you want cannot exist: which should be the image of the empty set, the empty set again or the scheme defined by $x^2+1$? You can construct a functor in the other direction, from quasi-projective schemes over $k$ to $k$-varieties by simply taking rational points, but it will not be faithful. The point is that your notion of variety simply doesn't work well over non algebraically closed fields. You would like it to define some sort of nonempty "variety" associated to $x^2+1$, because when you switch to complex numbers two solutions of $x^2+1$ pop up from nowhere.

There have been attempts of constructing a theory of varieties over arbitrary fields before Grothendieck, the main one being Weil's fundations (see here). The advent of the theory schemes relegated these attempts to the history of mathematics (only the fundations, not the ideas nor the results, since the seeds planted by Weil proved to be among the most fruitful of the last century), and nowadays scheme theory is considered the standard fundation for a lot of theories, diophantine geometry being one of them. I suppose that you can do diophantine geometry also without schemes, and in the beginning it will be easier, but I think that studying them pays back in the long run.

There are still a lot of mathematicians working in the fields "conquered" by scheme theory without actually using scheme theory, and if you want you can do it too, it really depends a lot on which particular problems you are interested in. From my point of view, scheme theory provides both powerful machinery and an enlightening perspective useful to study a lot of things, diophantine geometry being just one of them.

  • $\begingroup$ Thank you very much! So, in diophantine geometry do mathematicians study the $k$-rational points of $k$-scheme? In the Silverman's book "The Arithmetic of Elliptic Curves" he studies the set of solutions in $k$ of equations in $k$, for arbitary field. $\endgroup$
    – k.j.
    Nov 27, 2017 at 12:58
  • $\begingroup$ Yes, that's right. Mainly, they do it when $k$ is a number field. $\endgroup$ Nov 27, 2017 at 13:06
  • $\begingroup$ Un' ottima risposta, caro Giulio! $\endgroup$ Feb 24, 2018 at 22:57
  • $\begingroup$ Do you have a reference for "you can construct a functor in the other direction, from quasi-projective schemes over $k$ to $k$-varieties by simply taking rational points"? I tried to construct the functor myself, but failed. $\endgroup$ Jul 8, 2023 at 8:34

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