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Weekly seminar of the Laboratory of Algebraic Geometry. Speaker: Guo Ning (The Euler International Mathematical Institute, St. Petersburg)

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The Grothendieck--Serre conjecture on smooth projective schemes over mixed characteristic DVR

Abstract: The Grothendieck—Serre conjecture predicts that over a regular local ring, no nontrivial reductive group torsors become trivial over the fraction field. The conjecture is settled in equicharacteristic case but still open in the mixed characteristic case. In this talk, I will explain a progress for this conjecture in a mixed characteristic case: for a discrete valuation ring R, an irreducible R-smooth, projective scheme X, and a reductive R-group scheme G, if a G-torsor P on X becomes trivial over the function field K(X), then P is Zariski-locally trivial. First, we recollect useful preliminary knowledge about reductive groups and torsors.
Then, we transfer the 
aforementioned problem on relative curves and even affine line. Finally, we discuss the behavior of torsors on affine line.