The Grothendieck–serre Conjecture Over Semilocal Dedekind Rings

Guo, Ning

doi:10.1007/s00031-020-09619-8

Cited by 26 publications

(31 citation statements)

References 18 publications

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“…In the GL n case the conjecture was proved by Dorin Popescu [Pop2,Thm. 1] (see also [BR]), assuming that R contains a field. Ning Guo proved a version of this conjecture for the rings of formal power series over valuation rings [Guo,Cor. 7.5].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the purity conjecture of Nisnevich for torsors under reductive group schemes

Fedorov¹

2021

Preprint

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Let R be a regular semilocal integral domain containing a field k. Assume that k is infinite or there is a homomorphism R → k left inverse to the inclusion. Let f ∈ R be an element such that for all maximal ideals m of R we have f / ∈ m 2 . Let G be a reductive group scheme over R. Under an isotropy assumption on G we show that a G-torsor over R f is trivial, provided it is rationally trivial. We show that it is not true without the isotropy assumption.The first statement is derived from its abstract version concerning Nisnevich sheaves satisfying some properties. The counterexample is constructed by providing a torsor over a local family of affine lines that cannot be extended to the projective lines. The latter is accomplished using the technique of affine Grassmannians.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the purity conjecture of Nisnevich for torsors under reductive group schemes

Fedorov¹

2021

Preprint

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“…Since ξ is trivial over the completion F w of F at the discrete valuation w associated to the special fiber of C, we claim that the specialisation ζ κ ∈ H 1 (C κ , G) of ζ is generically trivial. This follows from the fact that H 1 (O w , G) injects in H 1 (F w , G) due to Bruhat-Tits (see [Gu,Th. 5.1]).…”

Section: Extension To Reductive Groupsmentioning

confidence: 96%

Local triviality for G-torsors

2021

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“…2, Theorem 7.1] and [48,Theorem 4.2] in the case where the residue field of A is perfect. The general case is treated in [29,Theorem 1]. This result has the following important consequence.…”

Section: Groups With Good Reductionmentioning

confidence: 99%

Simple algebraic groups with the same maximal tori, weakly commensurable Zariski-dense subgroups, and good reduction

Chernousov¹,

Rapinchuk²,

Rapinchuk³

2021

Preprint

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We provide a new characterization of the existence of good reduction for an absolutely almost simple algebraic group with respect to a discrete valuation of the base field that is formulated in terms of the existence of maximal tori with special properties. This characterization, in particular, shows that the Finiteness Conjecture for forms of an absolutely almost simple algebraic group over a finitely generated field that have good reduction at a divisorial set of places of the field (cf. [62]) would imply the finiteness of the genus of the group at hand. It also leads to a new phenomenon that we refer to as "killing of the genus by a purely transcendental extension." Yet another application deals with the investigation of "eigenvalue rigidity" of Zariski-dense subgroups (cf. [60]), which in turn is related to the analysis of length-commensurable Riemann surfaces and general locally symmetric spaces. Finally, we analyze the genus problem for simple algebraic groups of type F4.

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The Grothendieck–serre Conjecture Over Semilocal Dedekind Rings

Cited by 26 publications

References 18 publications

On the purity conjecture of Nisnevich for torsors under reductive group schemes

On the purity conjecture of Nisnevich for torsors under reductive group schemes

Local triviality for G-torsors

Simple algebraic groups with the same maximal tori, weakly commensurable Zariski-dense subgroups, and good reduction

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