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Let $R$ be a DVR (possibly mixed characteristic) with fraction field $K$. Let $V \to \operatorname{Spec} R$ be a smooth affine scheme with a transitive action of $GL_{n,R}$ so that each geometric fiber is a homogeneous space.

Suppose that $V$ has a $K$-point. Does $V$ admit an $R$ point, possibly after making a totally ramified base change by adjoining a root of the uniformizer of $R$?

Edit: I'm not assuming that the action is free so $V \to \operatorname{Spec} R$ is not a torsor.

The geometric fibers of $V$ are isomorphic to $GL_{n,\overline{K}}/G_\overline{K}$ (resp. $GL_{n,\overline{k}}/G_\overline{k}$) for some reductive group $G_\overline{K}$ (resp. $G_\overline{k}$) and the assumption on a $K$-point guarantees that the (non-geometric) generic fiber is isomorphic to $GL_{n,K}/G_K$ for some form of $G_\overline{K}$. My real question is whether $G_K$ has a reductive model $G_R$ over $\operatorname{Spec} R$ so that $V \cong GL_{n,R}/G_R$ and this is equivalent to the existence of an $R$-point by taking $G$ to be the stabilizer of the $R$-point.

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    $\begingroup$ For $B_{n,R}\hookrightarrow \text{GL}_{n,R}$ the standard Borel, the quotient scheme $V/B_{n,R}$ is proper (and even projective) over $\text{Spec}\ R$. By the valuative criterion, your $K$-point of $V$ extends to an $R$-point of $V/B_{n,R}$. The inverse image in $V$ is a $B_{n,R}$-torsor over $\text{Spec}\ R$. The induced torsor for the maximal multiplicative quotient $B_{n,R}/U_{n,R}$ is trivial since $R$ is a local ring. Every $\mathbb{A}^1$-torsor over $\text{Spec}\ R$ is trivial since $\text{Spec}\ R$ is affine. $\endgroup$
    – Jason Starr
    Commented Jul 17 at 23:07
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    $\begingroup$ Just to summarize, the quotient of sets of $R$-points, $V(R)/B_{n,R}(R)$, equals the set of $R$-points of the geometric quotient $V/B_{n,R}$. $\endgroup$
    – Jason Starr
    Commented Jul 18 at 1:17
  • $\begingroup$ The assumption that $V$ has a $K$-point is also unneeded since by Hilbert 90 $H^1(K, GL_n)$ is trivial and thus $V$ always has a $K$-point. $\endgroup$
    – Will Sawin
    Commented Jul 18 at 15:12
  • $\begingroup$ @JasonStarr Thanks for the argument and answer below! I believe that works when $V$ is a torsor (and this is Grothendieck-Serre as you pointed out) but I am not assuming that the action of $GL_n$ is free, only transitive. I've edited the question to make this clear. $\endgroup$
    – Dori Bejleri
    Commented Jul 18 at 15:22
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    $\begingroup$ Ok. You probably already know this, but I just wanted to mention that there is an etale base change $R \to R'$ such that you have a section. $\endgroup$
    – David Benjamin Lim
    Commented Jul 21 at 0:56

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I am posting an answer to call attention to a much more general theorem, proved by Nisnevich (I believe the origin of his Nisnevich topology).

Nisnevich, Yevsey.
Espaces homogènes principaux rationellement triviaux et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind.
C. R. Acad. Sc. Paris, t. 299, Série I, no 1, 1984.
https://cims.nyu.edu/~nisnevic/articles/yncr11.pdf

In this article, Nisnevich proves that for every discrete valuation ring $R$ with fraction field $K$, for every reductive group scheme $G_R$ over $\text{Spec}\ R$, for every (étale locally trivial) $G_R$-torsor $V_R$ over $\text{Spec}\ R$, if the base change $G_K$-torsor $V_K$ has a $K$-point, then also $V_R$ has an $R$-point (thus confirming the Grothendieck-Serre Conjecture for reductive group schemes, hence for all linear algebraic group schemes over a Dedekind domain).

My comments above show that this is elementary in the case that $G_R$ is split, i.e., if there exists a Borel subgroup scheme $B_R$ of $G_R$ whose associated maximal multiplicative quotient $B_R/U_R$ is a product of copies of the multiplicative group scheme $\mathbb{G}_{m,R}$. As explained in the comments, the quotient scheme $V_R/B_R$ is proper, hence every $K$-point extends to an $R$-point by the valuative criterion of properness. Thus, every $G_R$-torsor is induced from a $B_R$-torsor. Since the quotient $B_R/U_R$ is split, also every $B_R$-torsor is induced from a $U_R$-torsor. Every $U_R$-torsor over an affine scheme is trivial.

Edit. The original poster clairified that $V$ is a homogeneous space for $\text{GL}_{n,R}$, but not necessarily a principal homogeneous space for $\text{GL}_{n,R}$. I believe we can complete the argument in this case using the Cartan-Iwahori-Matsumoto Theorem (for $\text{GL}_{n,R}$, this is the Birkhoff factorization, apparently proved earlier by Dedekind-Weber and equivalent to a geometric theorem proved by del Pezzo and Bertini). The idea is to find a $K$-point $g_K$ of $\text{GL}_{n,R}$ such that translation of the $K$-point $x_K$ of $V$ by $g_K$ then extends to an $R$-point. Of course we can translate both the $K$-point $x_K$ by an $R$-point and we can multiply $g_K$ on the left by an $R$-point without changing the problem, so the problem "factors" through the $\text{GL}_{n,R}(R)$-double coset of the element $g_K$ of $\text{GL}_{n,R}(K)$. The Cartan-Iwahori-Matsumoto Theorem classifies these double cosets in a purely "root theoretic" way as coming from morphisms $\rho$ of $R$-group schemes $\mathbb{G}_{m,R}\to T_{n,R}$, where $T_{n,R}$ is the set of diagonal matrices. Since this set is "rigid", we can now consider the same problem after an étale base change of $R$ to a local ring $\widetilde{R}$. Since $V$ is $R$-smooth, there exists such a base change that has $\widetilde{R}$-points.

In other words, there is a morphism $\rho$ of $R$-group schemes $\mathbb{G}_{m,R} \to T_{n,R}$ such that the associated double $\text{GL}_{n,R}(\widetilde{R})$-coset of $\rho$ does contain an element translating $x_K$ to the $\widetilde{K}$-point of a $\widetilde{R}$-point of $V$. I suspect that the corresponding double $\text{GL}_{n,R}(R)$-coset of $\rho$ also contains an element translating $x_K$ to the $K$-point of an $R$-point of $V$, but I am having a little trouble making this precise.

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    $\begingroup$ Can you clarify the claim “… hence for all linear algebraic group schemes, over a Dedekind domain…”? This seems not obvious to me. For instance, what if $R=\mathbb{Z}_p$ and $G$ is a parahoric group scheme, what is the argument? $\endgroup$
    – Alex Youcis
    Commented Jul 18 at 13:10
  • $\begingroup$ I guess $\mathbb{Z}_p$ is a bad example of because of Lang’s theorem ,but hopefully you get my point. $\endgroup$
    – Alex Youcis
    Commented Jul 18 at 13:41
  • $\begingroup$ @AlexYoucis Let me clarify: for every group scheme $G_R$ that is étake locally an extension of an $R$-flat reductive group scheme by an $R$-flat unipotent group scheme, Nisnevich’s theorem applies. $\endgroup$
    – Jason Starr
    Commented Jul 18 at 14:55
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    $\begingroup$ Thanks! I wasn't familiar with the Cartan-Iwahori-Matsumoto Theorem. This seems very promising. If we take $g \in GL_n(K)$ to be an element in the double coset which translates $x_K$ to an $\tilde{R}$-point, then we consider the closure of $gx_K$ in $V$. I want to say something like the only thing that can go wrong is that this closure goes off to infinity and misses the central fiber, but when we pass to $\tilde{R}$, this point extends to a section over $\tilde{R}$ by the way we picked $g$ and so the closure can't go off to infinity. $\endgroup$
    – Dori Bejleri
    Commented Jul 19 at 15:58

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