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

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)U(Λ)εH(F,Qℓ/Zℓ(2)) vanishes. In 1985, Suslin conjectured that the Rost kernel is generated by i-th powers of reduced norms from D for all i≥1. Despite known counterexamples, we prove some new special 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 cdℓ(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 cdℓ(k) ≤ 2. In the case ℓ D char(k), an analog is obtained for tamely ramified algebras. We conjecture that Suslin’s conjecture holds for all fields of cohomological dimension 3.