Unramified Grothendieck-Serre for isotropic groups
Kestutis Cesnavicius, Roman Fedorov
TL;DR
The paper proves the unramified Grothendieck--Serre conjecture for reductive groups whose adjoint quotient $G^{\mathrm{ad}}$ is totally isotropic over a regular semilocal ring $R$, establishing that generically trivial $G$-torsors are trivial. Central to the approach is a Bun$_G$-driven geometric framework and a weak $\mathbb{P}^1$-invariance principle for torsors, together with a lifting construction to families over $\mathbb{P}^1_R$ away from codimension-$2$ loci and a Quillen/Gabber-type patching technology in mixed characteristic. The authors develop reembedding and excision techniques for relative curves with quasi-finiteness, enabling a reduction to simply connected groups via Brauer-purity arguments and a sequence of structural reductions (to semisimple, simply connected cases, and finally to isotropic instances). As a result, the unramified case is settled for totally isotropic adjoint groups, with the reduction strategy providing a pathway toward broader anisotropic situations in mixed characteristic. This advances the nonabelian to abelian cohomology strategy for Grothendieck--Serre-type questions and sharpens the geometric toolkit (notably Bun$_G$) for torsor problems over $\mathbb{P}^1$ and related spaces.
Abstract
The Grothendieck-Serre conjecture predicts that every generically trivial torsor under a reductive group $G$ over a regular semilocal ring $R$ is trivial. We establish this for unramified $R$ granted that $G^{\mathrm{ad}}$ is totally isotropic, that is, has a "maximally transversal" parabolic $R$-subgroup. We also use purity for the Brauer group to reduce the conjecture for unramified $R$ to simply connected $G$--a much less direct such reduction of Panin had been a step in solving the equal characteristic case of Grothendieck-Serre. We base the group-theoretic aspects of our arguments on the geometry of the stack $\mathrm{Bun}_G$, instead of the affine Grassmannian used previously, and we quickly reprove the crucial weak $\mathbb{P}^1$-invariance input: for any reductive group $H$ over a semilocal ring $A$, every $H$-torsor $\mathscr{E}$ on $\mathbb{P}^1_A$ satisfies $\mathscr{E}|_{\{t = 0\}} \simeq \mathscr{E}|_{\{t = \infty\}}$. For the geometric aspects, we develop reembedding and excision techniques for relative curves with finiteness weakened to quasi-finiteness, thus overcoming a known obstacle in mixed characteristic, and show that every generically trivial torsor over $R$ under a totally isotropic $G$ trivializes over every affine open of $\mathrm{Spec}(R) \setminus Z$ for some closed $Z$ of codimension $\ge 2$.