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Abstract
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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.
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Keywords
reduced norms, division algebras over henselian fields,
Rost invariant, biquaternion algebras
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Mathematical Subject Classification
Primary: 11S25
Secondary: 11R52, 16K50, 17A35
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Milestones
Received: 31 May 2019
Revised: 19 October 2019
Accepted: 2 July 2020
Published: 26 December 2020
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