Generically trivial torsors under constant groups
Alexis Bouthier, Kestutis Cesnavicius, Federico Scavia
TL;DR
This work resolves the Grothendieck–Serre conjecture for generically trivial torsors over arbitrary base fields by developing a comprehensive framework that handles imperfect fields and nonreductive group schemes. Central to the approach are purity theorems for torsors under pseudo-finite, pseudo-proper, and pseudo-complete k-groups, and a refined structure theory for k-group schemes locally of finite type, including the fundamental filtration and comparison maps i_G. The authors derive an Auslander–Buchsbaum extension theorem for torsors under quasi-reductive groups, classify G-torsors over P^1_k, and establish Birkhoff, Cartan, and Iwasawa decompositions for G(k((t))). A key strategic theme is reducing to well-understood reductive or abelian cases via the kernels and cokernels of i_G and the pseudo-parabolic subgroups, then bootstrapping local extendability to global triviality through purity and deformation arguments. The results have broad implications for torsor theory, unramified Whitehead groups, and relative P^1-torsor theory, and they illuminate subtle arithmetic phenomena that arise over imperfect base fields.
Abstract
We resolve the Grothendieck-Serre question over an arbitrary base field $k$: for a smooth $k$-group scheme $G$ and a smooth $k$-variety $X$, we show that every generically trivial $G$-torsor over $X$ trivializes Zariski semilocally on $X$. This was known when $G$ is reductive or when $k$ is perfect, and to settle it in general we uncover a wealth of new arithmetic phenomena over imperfect $k$. We build our arguments on new purity theorems for torsors under pseudo-complete, pseudo-proper, and pseudo-finite $k$-groups, for instance, respectively, under wound unipotent $k$-groups, under pseudo-abelian varieties, and under the kernels $\mathrm{Ker}(i_G)$ of comparison maps $i_G$ that relate pseudo-reductive groups to restrictions of scalars of reductive groups. We then deduce an Auslander-Buchsbaum extension theorem for torsors under quasi-reductive $k$-groups; for instance, we show that torsors over $\mathbb{A}^2_k \setminus \{(0,0)\}$ under wound unipotent $k$-groups extend to torsors over $\mathbb{A}^2_k$. For a quasi-reductive $k$-group $G$, this extension theorem allows us to quickly classify $G$-torsors over $\mathbb{P}^1_k$ by an argument that already simplifies the reductive case and to establish Birkhoff, Cartan, and Iwasawa decompositions for $G(k((t)))$. We combine these new results with deep inputs from recent work on the structure of pseudo-reductive and quasi-reductive $k$-groups to show an unramifiedness statement for the Whitehead group (the unstable $K_1$-group) of a quasi-reductive $k$-group, and then use it to argue that, for a smooth $k$-group $G$ and a semilocal $k$-algebra $A$, every $G$-torsor over $\mathbb{P}^1_A$ trivial at $\{t = \infty\}$ is also trivial at $\{t = 0\}$, which is known to imply the Grothendieck--Serre conclusion via geometric arguments. To achieve all this, we develop and heavily use the structure theory of $k$-group schemes locally of finite type.