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 $Sm_{k}$ (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 $\mathbb{G}_{m}^{\wedge2}$).