Vol. 5, No. 4, 2020

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

Yong Hu and Zhengyao Wu

Vol. 5 (2020), No. 4, 677–707
Abstract

Let F be a field, a prime and D a central division F-algebra of -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) H3(F, (2)) vanishes. In  1985, Suslin conjectured that the Rost kernel is generated by i-th powers of reduced norms from Di 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 . When D has period , we show that Suslin’s conjecture holds if either k is a 2-local field or the cohomological -dimension cd(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(k) 2. In the case = char(k), an analog is obtained for tamely ramified algebras. We conjecture that Suslin’s conjecture holds for all fields of cohomological dimension 3.

Keywords
reduced norms, division algebras over henselian fields, Rost invariant, biquaternion algebras
Mathematical Subject Classification
Primary: 11S25
Secondary: 11R52, 16K50, 17A35
Milestones
Received: 31 May 2019
Revised: 19 October 2019
Accepted: 2 July 2020
Published: 26 December 2020
Authors
Yong Hu
Department of Mathematics
Southern University of Science and Technology
Shenzhen, Guangdong
China
Zhengyao Wu
Department of Mathematics
Shantou University
Shantou, Guangdong
China