Grothendieck–Serre in the quasi-split unramified case

Abstract

AbstractThe Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group schemeGover a regular local ringRis trivial. We settle it in the case whenGis quasi-split andRis unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.