Bloch-Suslin Complex and Strong ${\mathbb{A}}^1-$invariance

Abstract

We prove that there is no extension of the abelian groups appearing in the Bloch-Suslin complex to strongly ${\mathbb{A}}^1-$invariant sheaves on $S{m}_k$ (char($k$)=0) that also extend the canonical symbol maps from the respective ${\mathbb{G}}_m^{\wedge n}$ (i.e., from ${\mathbb{G}}_m$ and ${\mathbb{G}}_m^{\wedge 2}$).