Generically trivial torsors under constant groups

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.