Unramified Grothendieck–Serre for isotropic groups

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 $\mathcal{E}$ on ${\mathbb{P}}_A^1$ satisfies $\mathcal{E}{|}_{\{t=0\}}\simeq \mathcal{E}{|}_{\{t=\infty \}}$. 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 $\ge 2$.