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

TL;DR

The paper proves that the abelian groups underlying the Bloch–Suslin complex cannot be extended to strongly $\mathbb{A}^{1}$-invariant Nisnevich sheaves on $Sm_{k}$ in a way that preserves the canonical symbol maps from $\mathbb{G}_{m}$ and $\mathbb{G}_{m}^{\wedge2}$. By leveraging the universal property of Milnor–Witt $K$-theory via Theorem 2.3, the author shows that although $K_{2}^{M}$ admits such an extension, the intermediate groups $\mathcal{P}$ and $\mathcal{A}$ cannot be promoted to strongly $\mathbb{A}^{1}$-invariant sheaves without violating the symbol-relations when evaluated on certain fields. A concrete obstruction is derived using a field $K$ with chosen square roots, where relations force $\{3\}=0$ but map to a nonzero element in $\wedge^{2}_{\mathbb{Z}}(K^{*})$, yielding a contradiction. Consequently, no such extensions exist, highlighting incompatibilities between five-term/dilogarithmic structures and Milnor–Witt framework in the strongly $\mathbb{A}^{1}$-invariant setting.

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

Theorems & Definitions (17)

• Theorem 1.1
• Definition 2.1: Milnor-Witt K-groups
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.5: Scissor congruence group
• Remark 2.6
• Proposition 2.7
• proof
• Proposition 2.8