Abstract
Let be a field, a prime and a central division -algebra of -power degree. By the Rost kernel of we mean the subgroup of consisting of elements such that the cohomology class vanishes. In 1985, Suslin conjectured that the Rost kernel is generated by -th powers of reduced norms from for all . Despite known counterexamples, we prove some new special cases of Suslin’s conjecture. We assume is a henselian discrete valuation field with residue field of characteristic different from . When has period , we show that Suslin’s conjecture holds if either is a -local field or the cohomological -dimension of is . When the period is arbitrary, we prove the same result when itself is a henselian discrete valuation field with . In the case , an analog is obtained for tamely ramified algebras. We conjecture that Suslin’s conjecture holds for all fields of cohomological dimension 3.
Citation
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