Adjoint groups over ℚ_{𝕡}(𝕏) and R-equivalence - revisited

Preeti, R.; Soman, A.

doi:10.1090/proc/13304

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2019

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“…In [PS15,Thm. 4.1], this result is stated with the extra assumptions that F has have characteristic 0 and cohomological dimension ≤ 3.…”

Section: Introductionmentioning

confidence: 99%

On the Rost divisibility of henselian discrete valuation fields of cohomological dimension 3

Hu,

2019

Preprint

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Let F be a field, ℓ a prime and D a central division F -algebra of ℓ-power degree. By the Rost kernel of D we mean the subgroup of F * consisting of elements λ such that the cohomology class (D) ∪ (λ) ∈ H 3 (F, Q ℓ /Z ℓ (2)) vanishes. In 1985, Suslin conjectured that the Rost kernel is generated by i-th powers of reduced norms from D ⊗i , ∀i ≥ 1. Despite of known counterexamples, we prove some new cases of Suslin's conjecture. We assume F is a henselian discrete valuation field with residue field k of characteristic different from ℓ. When D has period ℓ, we show that Suslin's conjecture holds if either k is a 2-local field or the cohomological ℓ-dimension cd ℓ (k) of k is ≤ 2. When the period is arbitrary, we prove the same result when k itself is a henselian discrete valuation field with cd ℓ (k) ≤ 2. In the case ℓ = char(k) an analog is obtained for tamely ramified algebras. We conjecture that Suslin's conjecture holds for all fields of cohomological dimension 3.

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“…In [PS15,Thm. 4.1], this result is stated with the extra assumptions that F has have characteristic 0 and cohomological dimension ≤ 3.…”

Section: Introductionmentioning

confidence: 99%