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2020 On the Rost divisibility of henselian discrete valuation fields of cohomological dimension 3
Yong Hu, Zhengyao Wu
Ann. K-Theory 5(4): 677-707 (2020). DOI: 10.2140/akt.2020.5.677

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 i1. 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.

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Yong Hu. Zhengyao Wu. "On the Rost divisibility of henselian discrete valuation fields of cohomological dimension 3." Ann. K-Theory 5 (4) 677 - 707, 2020. https://doi.org/10.2140/akt.2020.5.677

Information

Received: 31 May 2019; Revised: 19 October 2019; Accepted: 2 July 2020; Published: 2020
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.2140/akt.2020.5.677

Subjects:
Primary: 11S25
Secondary: 11R52 , 16K50 , 17A35

Keywords: biquaternion algebras , division algebras over henselian fields , reduced norms , Rost invariant

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.5 • No. 4 • 2020
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