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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 $G$ 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. The talk is based on joint work with Roman Fedorov.

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