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

TL;DR

This paper demonstrates limitations in extending the Bloch-Suslin complex to strongly e2
a1e2
a1e2
a1e2
a1-invariant sheaves on smooth schemes over a field of characteristic zero, particularly regarding canonical symbol maps.

Contribution

It proves the non-existence of certain extensions of the Bloch-Suslin complex to strongly e2
a1e2
a1e2
a1e2
a1-invariant sheaves that preserve canonical symbol maps.

Findings

No extension of the abelian groups in the Bloch-Suslin complex to strongly e2
a1e2
a1e2
a1e2
a1-invariant sheaves exists.

Extensions cannot preserve the canonical symbol maps from g_m^{\u0000a0n}.

Results hold over fields of characteristic zero.

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