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Let $K\subset L$ be a field extension and let $K\subset F_1,F_2,...,F_n\subset L$ be proper intermediate fields. Consider the "mixed" affine space $\mathbb{A}_{(F_i)}:=\prod_{i=1}^n F_i$ instead of $\mathbb{A}_K^n$ (or $\mathbb{A}_L^n$ for that matter) and consider zero sets of polynomials $f\in K[X_1,X_2,...,X_n]$ in $\mathbb{A}_{(F_i)}$. From the perspective of classic algebraic geometry, it only seems natural to look for such a generalization.

Has this specific type of problems been studied/discussed in more detail anywhere and, if yes, under what wording? A reference will be most appreciated. I am mostly interested in the case of algebraic number fields, but other examples are most welcome as well.

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    $\begingroup$ What is the natural definition of a regular function coming out of that mixed affine space? $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jun 24, 2013 at 22:04
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    $\begingroup$ As far as I can tell, this is just affine space over $K$, but you are looking at $L$-points whose coordinate maps are defined over the various subfields. $\endgroup$
    – S. Carnahan ♦
    Commented Jun 24, 2013 at 23:39
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    $\begingroup$ You can think of this in terms of the Weil restriction functor. For each $i=1,\dots,n$, there is a natural map of $K$-algebras, $K[X_i] \to \text{Res}_{F_i/K}(F_i[X_i])$. The tensor product of these natural maps is $K[X_1,\dots,X_n] \to \bigotimes_{i=1}^n_K \text{Res}_{F_i/K}(F_i[X_i])$. $\endgroup$
    – Jason Starr
    Commented Jun 25, 2013 at 0:38
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    $\begingroup$ We used such a construction in papers with Tschinkel. It serves us to describe an étale neighborhood of a point in a variety over a number field endowed with a divisor which is geometrically SNC. See Igusa integrals and volume asymptotics in analytic and adelic geometry, Confluentes Math., 2010. $\endgroup$
    – ACL
    Commented Jun 25, 2013 at 6:07

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The product of Weil restrictions $P = \prod_{i=1}^n {\rm{R}}_{F_i/K}(\mathbf{A}^1_{F_i})$ is naturally a closed subscheme of the direct product $\prod_{i=1}^n {\rm{R}}_{L/K}(\mathbf{A}^1_L) = {\rm{R}}_{L/K}(\mathbf{A}^n_L)$. For any closed subscheme $Z \subset \mathbf{A}^n_K$, we also get a closed subscheme ${\rm{R}}_{L/K}(Z_L) \subset {\rm{R}}_{L/K}(\mathbf{A}^n_L)$ and can see where it meets $P$. Those intersections are the $K$-schemes you're wondering about (e.g., the $K$-points are precisely the loci inside $\prod F_i$ defined by the vanishing of several polynomials in $n$ variables over $K$).

From the viewpoint of Weil restriction of affine schemes of finite type, this looks like a rather special kind of construction. It isn't clear why it should be useful or a natural thing to consider from the viewpoint of classical algebraic geometry (though Weil restriction is very useful). Is there a situation you have in mind for which this is interesting to consider?

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