Torsors on the complement of a smooth divisor

2022/04/18 by Kęstutis Česnavičius, Cesnavicius, Kestutis · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Algebraic structures and combinatorial models

paper · doi:10.48550/arxiv.2204.08233

Abstract

We complete the proof of the Nisnevich conjecture in equal characteristic: for a smooth algebraic variety X over a field k, a k-smooth divisor D ⊂ X, and a reductive X-group G whose base change G_D is totally isotropic, we show that each generically trivial G-torsor on X∖ D trivializes Zariski semilocally on X. In mixed characteristic, we show the same when k is a replaced by a discrete valuation ring O, the divisor D is the closed O-fiber of X, and either G is quasi-split or G is only defined over X ∖ D but descends to a quasi-split group over Frac(O) (a Kisin-Pappas type variant). Our arguments combine Gabber-Quillen style presentation lemmas with excision and reembedding dévissages to reduce to analyzing generically trivial torsors over a relative affine line. As a byproduct of this analysis, we give a new proof for the Bass-Quillen conjecture for reductive group torsors over \mathbbA^d_R in equal characteristic.

Cited by

• On the generalized Bass--Quillen conjecture in dimension 2

Related

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• On the purity conjecture of Nisnevich for torsors under reductive group schemes
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• Generically trivial torsors under constant groups
• Grothendieck-Serre in the quasi-split unramified case