Geodesic
Grothendieck–Serre in the quasi-split unramified case
Forum of Mathematics, Pi, Tome 10 (2022)

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The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is 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. 
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Kęstutis Česnavičius. Grothendieck–Serre in the quasi-split unramified case. Forum of Mathematics, Pi, Tome 10 (2022). doi: 10.1017/fmp.2022.5

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