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

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 $F^{\ast }$ 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.

30 pages

References in corpus (3)

• Applications of patching to quadratic forms and central simple algebras
• Period-index and u-invariant questions for function fields over complete discretely valued fields
• Local-global principle for reduced norms over function fields of p-adic curves