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 $\left(D\right)\cup \left(\lambda \right)\in H^3\left(F,{\mathbb{Q}}_{\ell }∕{\mathbb{Z}}_{\ell }\left(2\right)\right)$ vanishes. In 1985, Suslin conjectured that the Rost kernel is generated by $i$-th powers of reduced norms from $D^{\otimes i}$ for all $i\ge 1$. 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 $\ell$. 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 ${cd}_{\ell }\left(k\right)$ 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 ${cd}_{\ell }\left(k\right)\le 2$. In the case $\ell =char\left(k\right)$, an analog is obtained for tamely ramified algebras. We conjecture that Suslin’s conjecture holds for all fields of cohomological dimension 3.

Let $p$ be a prime, and suppose that $F$ is a field of characteristic zero which is $p$-special (that is, every finite field extension of $F$ has dimension a power of $p$). Let $\alpha \in {\mathcal{𝒦}}_n^M\left(F\right)∕p$ be a nonzero symbol and $X∕F$ a norm variety for $\alpha$. We show that $X$ has a ${\mathcal{𝒦}}_m^M$-norm principle for any $m$, extending the known ${\mathcal{𝒦}}_1^M$-norm principle. As a corollary we get an improved description of the kernel of multiplication by a symbol. We also give a new proof for the norm principle for division algebras over $p$-special fields by proving a decomposition theorem for polynomials over $F$-central division algebras. Finally, for $p=n=m=2$ we show that the known ${\mathcal{𝒦}}_1^M$-multiplication principle cannot be extended to a ${\mathcal{𝒦}}_2^M$-multiplication principle for $X$.

This paper provides a generalization of excision theorems in controlled algebra in the context of equivariant $G$-theory with fibred control and families of bounded actions. It also states and proves several characteristic features of this theory such as existence of the fibred assembly and the fibrewise trivialization.

Let $D$ be an effective Cartier divisor on a regular quasiprojective scheme $X$ of dimension $d\ge 1$ over a field. For an integer $n\ge 0$, we construct a cycle class map from the higher Chow groups with modulus ${\left\{{CH}^{n+d}\left(X|mD,n\right)\right\}}_{m\ge 1}$ to the relative $K$-groups ${\left\{K_n\left(X,mD\right)\right\}}_{m\ge 1}$ in the category of pro-abelian groups. We show that this induces a proisomorphism between the additive higher Chow groups of relative $0$-cycles and the reduced algebraic $K$-groups of truncated polynomial rings over a regular semilocal ring which is essentially of finite type over a characteristic zero field.