Does a $K_{\upsilon}$-point of a variety $V$ give a point of $V$ in $K_{(\upsilon)}=K^{sep}\cap K_{\upsilon}$ for a global field $K$?

Question

Let $K$ be a global field and $\upsilon$ a place of $K$. Let $K_{\upsilon}$ denote the completion of $K$ at $\upsilon$ and $K_{(\upsilon)}:=K^{sep}\cap K_{\upsilon}$ the henselization (which is the quotient field of the henselization of the discrete valuation ring in case of a non-archimedean valuation).

Let $V$ be a $K_{(\upsilon)}$ variety with a point $\operatorname{Spec}(K_{\upsilon}) \to V$. I have seen used in a paper that in this case $V$ has to have a $K_{(\upsilon)}$-point. Why is this true?

I am thankful for any thoughts and ideas as I am not so familiar with algebraic geometry and maybe missed a result that would be helpful here.

Thoughts so far:

• If $\upsilon$ is complex, then $K_{(\upsilon)}$ is separably closed and the statement is true.
• If $\upsilon$ is non-archimedian, let $R^h$ denote the henselization of the corresponding discrete valuation ring and let $t$ be a generator of the maximal ideal. By a result of Greenberg a $R^h$-variety has a $R^h$-point iff it has a $R^h/t^n$-point for all $n \geq 1$. I fail to see how a point $\operatorname{Spec}(K_{\upsilon})=\operatorname{Spec}(\operatorname{Quot}(\varprojlim R^h/t^n)) \to V$ gives a point $\operatorname{Spec}(\varprojlim R^h/t^n) \to V$ (in the case that $V$ is not proper), such that the result can be used.
• If $\upsilon$ is real, I have no idea where to start.

Answer 1

Let $A$ be a henselian discrete valuation ring and let $A \subset B$ be an extension of discrete valuation rings which has ramification index $1$, induces a trivial extension of residue fields, and a separable extension $L/K$ of fraction fields. Let $X$ be a scheme of finite type over $K$. We claim that if $X$ has a $L$-valued point $y$, then $X$ has a $K$-point.

To prove this we may assume $X$ is affine, say $X = \text{Spec}(R)$. Then $y$ determines a $K$-algebra map $R \to L$ which we may assume is injective (this replaces $X$ by a closed subscheme). We can find a finite type $A$-algebra $R' \subset R$ such that $A \subset R' \subset B$ and $R = R' \otimes_A K$. By Neron desingularization (Tag 0BJ7), we may assume that $A \to R'$ is a smooth ring map (this step replaces $X$ by some finite type $K$-scheme lying over $X$). Then we finish because $A$ is assumed henselian and $\text{Spec}(R') \to \text{Spec}(A)$ has a section over the closed point, namely given by $R' \to B \to B/\mathfrak m_B = A/\mathfrak m_A$.

This argument is one of the reasons why Neron desingularization is useful. You can apply this in your setup (use one of your previous questions here on mathoverflow to check assumptions).

Answer 2

If one is prepared to invoke some big theorems, these three situations can all be understood simultaneously in the language of model theory.

We say a field extension $E/F$ is elementary just when $E$ and $F$ satisfy the same first-order formulas (in the language of fields, with parameters in $F$). The importance of this notion in the context of this question is that if $E/F$ is an elementary extension, then a scheme $X/F$ of finite type has an $F$-point if and only if it has an $E$-point. Thus the original question is actually a special case of the following result (at least in characteristic $0$):

Proposition: Let $v$ be a place of a number field $K$. Then the extension $K_v/K_{(v)}$ is elementary.

This theorem is the culmination of a huge amount of work on determination of fields by their Galois groups, the essential idea being that in certain circumstances, an extension $E/F$ being elementary is the same as it inducing an isomorphism $G_E\simeq G_F$ on absolute Galois groups.

Theorem: Let $E/F$ be an extension of characteristic $0$ fields inducing an isomorphism on absolute Galois groups. Suppose moreover that this common Galois group is trivial, is $C_2$, or is the absolute Galois group of a finite extension of $\mathbb Q_p$ for some $p$. Then the extension $E/F$ is elementary.

To very roughly indicate where these results come from, the three cases (Galois group trivial, $C_2$, open in $G_{\mathbb Q_p}$) correspond to three well-studied types of fields: algebraically closed fields; real closed fields; and $p$-adically closed fields (see Theorem 4.1 of [1]). The above theorem is thus an amalgam of several results:

• Extensions of algebraically closed fields are elementary by Chevalley's Theorem on the images of constructible sets.
• Extensions of real closed fields are elementary by the Tarksi--Seidenberg Theorem on the image of semialgebraic sets.
• Extensions of $p$-adically closed fields inducing isomorphisms on absolute Galois groups are elementary by Theorem 5.1 of [2].

[1] Koenigsmann, Jochen, From $p$-rigid elements to valuations (with a Galois-characterization of $p$-adic fields), J. Reine Angew. Math. 465, 165-182 (1995). ZBL0824.12006.

[2] Prestel, Alexander; Roquette, Peter, Formally $p$-adic fields, Lecture Notes in Mathematics. 1050. Berlin etc.: Springer-Verlag. V, 167 p. DM 24.00; $ 9.00 (1984). ZBL0523.12016.