Generically trivial torsors under algebraic group

I am attending a number theory seminar in the Ramanujan guest house at TIFR. The speaker is Federico Scavia. This is joint work with A. Bouthier and K. Česnavičius.

Let $k$ be a field.

1. The Grothendieck-Serre conjecture (constant case)

(BCS 2025)

Let $X$ be a smooth $k$-scheme , let $G$ be a smooth $k$-group. Then every generically trivial $G$-torsors over $X$ is Zariski (semi-)locally trivial.

This was asked Serre (1958) for $k = \overline{k}$.

When $G= \mathrm{PGL}_{n}$, this was solved by Serre himself in 1958.
When $k$ is perfect infinite for every $G$ over $k$, or for all $k$ when $G$ over $k$ is reductive topologically isotropic (Collot-Thélène, Ojongureu 1992)
When $k$ is infinite and $G$ over $k$ is reductive, this was solved by Ragunathan 1994-1995
When $k$ is finite, for every $G$ over $k$ (Gobber 1990s, unpublished)
When $G$ over $k$ is reductive, $k$ infinite (Fedorov-Penin 2015) and for $k$ finite (Penin 2020)
When $X = \mathrm{Spec}R[[t]]$, and $G/k$ is affine (Florence-Gille 2021)

What is remaining is the case of $k$ being imperfect and $G/k$ is non-reductive.

It is enough to assume the following:

$G/k$ l.f.t (what is that?) and every $\overline{k}$-torus of $G_{\overline{k}}$ lies in $(G^{\mathrm{grad}})_{\overline{k}}$, where $G^{\mathrm{grad}} \leq G$ maximal and smooth $k$-subgroup of $G$.
- For example, when $G$ is unipotent, $G$ commutative (ex $G = \mathrm{Pic}X$ for some proper $X/k$).
- This condition cannot be dropped
Can replace $X$ by $\mathrm{Spec}R$ where $R/k$ is a geometry regular (semi-)local ring.

2.The difficulty: dèvissage

Assume $G/k$ is smooth, $X = \mathrm{Spec}R$, $(R/k)$ is geometric regular semi-local. If $k$ is perfect, the main theorem reduces to the reductive case.

We have the short exact sequence \[\begin{equation} 1 \xrightarrow[]{} N \rightarrow G \rightarrow H \rightarrow 1 \end{equation}\]

Then we have \[\begin{equation} \begin{cases} \text{ Thm for $N$ } \\ E(\mathrm{Frac}R) = E(R) \\ \forall H-\text{torsion } E/R \end{cases} \implies \text{Thm for }G \end{equation}\]

\[\begin{equation} 1 \rightarrow \underbrace{G^{\circ}}_{\text{ smooth comm }} \rightarrow G \rightarrow G / G^{\circ} \rightarrow 1 \end{equation}\] Hence this reduces the problem to $G^{\circ}$.

\[\begin{equation} 1 \rightarrow \underbrace{H}_{\text{ smooth affine connected}} \rightarrow G^{\infty} \rightarrow \underbrace{A}_{\text{ ab var }} \rightarrow 1 \end{equation}\] By (Chevalley), we can reduce to $H$.

\[\begin{equation} 1 \rightarrow \underbrace{R_{u,k}(H)}_{\substack{\text{ split unipotent } \\ H^{1}(\text{ affine }, R_{u,k}(H)) = \{*\} }} \rightarrow H \rightarrow \underbrace{H/R_{u,t}(H)}_{\text{ reductive }} \rightarrow 1 \end{equation}\] This will finish the proof.

For arbitrary (imperfect) $k$,

$A$ is just a a pseudo Abelian variety (totorus) when $R_{k'/k}$ is an Abelian variety. ($k'/k$ inseperable)
$R_{u,k}(H)$ not necessarily split, $H'$ not necessarily trivial.
$H/R_{u,k}(H)$ just psuedo-reductive $R_{k'/k}$ (red.) ($(k'/k)$ is inseparable).
If $A$ is pseudo-Abelian $\rightarrow$ $R_{k'/k}(\underbrace{A_{k'}/R_{u}(A_{k'})}_{\text{ab}}) \implies$ can still reduce to $H$.
We can assume: $G$ smooth affine connected and no $\mathbb{G}_{k} \subseteq G$ ($G$ quasi-reductive)

Proof of main theorem

Geometric method of Federov-Penin (Gobber-Quillen presentation lemmas) gives: \[\begin{align} & \forall R/k \text{ geom reg semilocal } \forall E \rightarrow \mathrm{Spec}R G-\text{torsor}, \\ & \exists \mathcal{E} \rightarrow \mathbb{P}^{1}_{R} \text{ and } Z \text{ does not embed in } \mathbb{A}^{1}_{R}, R-\text{finite}\\ & \text{such that } \mathcal{E}|_{t=0} \simeq E \text{ and } \mathcal{E}_{\mathbb{P}^{1}_{R} \setminus \mathbb{Z} } \text{ trivial } \end{align}\]

So main theorem reduces to showing that for all $\mathcal{E} \rightarrow \mathbb{P}^{1}_{R} G$-torsor, \[\begin{equation} \mathcal{E}_{\{t = \infty\}} \text{ trivial } \implies \mathcal{E}|_{\{t=0\}} \text{ trivial}. \end{equation}\]

(BCS 2025)

For $G/k$ smooth, $E \rightarrow \mathbb{P}_{k}^{1}$ $G$-torsion, the following are equivalent:

$E|_{t=\infty}$ trivial
$E|_{\mathbb{A}^{1}_{k}}$ trival
$E$ Zariski locally trivial

If $G$ is quasi reductive, the above three are equivalent to

$\exists \lambda: \mathbb{G}_{m} \rightarrow G$, $E \simeq \lambda_{*} O(1)$ and moreover \[\begin{align} H^{1}(\mathbb{P}^{1}_{k}, G ) & \simeq \mathrm{Hom}(\mathbb{G}_{m},G) / G(k) \\ \lambda_{k} O(1) & \leftarrow \lambda \end{align}\]

For $G=\mathrm{GL}_{n}$, this is known due to Grothendieck 1957
$G$ reductive is Aerloer 1968 ($k=\overline{k}$) and generalized by Bizwas-Nagaraj (for all $k$)
False if $G= \mathbb{G}_{m} \times \mathbb{G}_{m}$

Proof idea: $\mathbb{P}^{1}_{k} = \mathbb{A}^{2}_{k} \setminus \{0\}/ \mathbb{G}_{m} \rightarrow [\mathbb{A}^{2}/ \mathbb{G}_{m}]$. \[\begin{equation} H^{1}(\mathbb{P}^{1},G) \leftarrow H^{1}([\mathbb{A}^{2}/\mathbb{G}_{m}, G ]) \xrightarrow[]{} H^{1}(B \mathbb{G}_{m}, G). \end{equation}\]

At this point the talk got extremely technical and I gave up on taking notes.

This page was updated on February 2, 2026.
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