2025/05/01 by Bouthier, Alexis, Cesnavicius, Kestutis, Scavia, Federico · 1 citation
#14G17 #14M17 #20G10 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary 14L10 #Secondary 14L30
paper · doi:10.48550/arxiv.2505.00505
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 Ker(iG) of comparison maps iG 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 \mathbbA2k ∖ \(0,0)\ under wound unipotent k-groups extend to torsors over \mathbbA2k. For a quasi-reductive k-group G, this extension theorem allows us to quickly classify G-torsors over ℙ1k 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 K1-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 ℙ1A trivial at \t = ∞\ 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.