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

## TL;DR

This paper investigates the Rost divisibility of henselian discrete valuation fields of cohomological dimension 3, proving new cases of Suslin's conjecture related to the generation of the Rost kernel by reduced norms.

## Contribution

The paper proves new instances where Suslin's conjecture holds for henselian discrete valuation fields with specific residue field conditions.

## Key findings

- Suslin's conjecture holds when the residue field is a 2-local field.
- Suslin's conjecture holds when the residue field has cohomological $oldsymbol{ ext{cd}_ ext{ell}}$ $oldsymbol{ extleq 2}$.
- Analogous results are obtained for tamely ramified algebras when $oldsymbol{ ext{char}(k)=oldsymbol{ ext{ell}}}$.

## Abstract

Let $F$ be a field, $\ell$ a prime and $D$ a central division $F$-algebra of $\ell$-power degree. By the Rost kernel of $D$ we mean the subgroup of $F^*$ consisting of elements $\lambda$ such that the cohomology class $(D)\cup (\lambda)\in H^3(F,\,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}(2))$ vanishes. In 1985, Suslin conjectured that the Rost kernel is generated by $i$-th powers of reduced norms from $D^{\otimes i},\,\forall i\ge 1$. Despite of known counterexamples, we prove some new cases of Suslin's conjecture. We assume $F$ is a henselian discrete valuation field with residue field $k$ of characteristic different from $\ell$. When $D$ has period $\ell$, we show that Suslin's conjecture holds if either $k$ is a $2$-local field or the cohomological $\ell$-dimension $\mathrm{cd}_{\ell}(k)$ of $k$ is $\le 2$. When the period is arbitrary, we prove the same result when $k$ itself is a henselian discrete valuation field with $\mathrm{cd}_{\ell}(k)\le 2$. In the case $\ell=\text{char}(k)$ an analog is obtained for tamely ramified algebras. We conjecture that Suslin's conjecture holds for all fields of cohomological dimension 3.