The homomorphism of presheaves ${\mathrm{K}}^\mathrm{MW}_*\to {\pi}^{*,*}_s$ over a base

TL;DR

This paper constructs a homomorphism between Milnor-Witt K-theory presheaves and stable motivic homotopy sheaves over any base scheme, and discusses proofs of their isomorphism over fields.

Contribution

It introduces a new construction of the homomorphism over arbitrary bases and explores alternative proofs of the sheaf isomorphism over fields.

Findings

Homomorphism of presheaves constructed over arbitrary base schemes.

Discussion of alternative proofs for the sheaf isomorphism.

Confirmation of the isomorphism over fields as previously established.

Abstract

We construct the homomorphism of presheaves ${\mathrm{K}}^\mathrm{MW}_* \to {\pi}^{*,*}$ over an arbitrary base scheme $S$, where $\mathrm{K}^\mathrm{MW}$ is the (naive) Milnor-Witt K-theory presheave. Also we discuss some partly alternative proof (or proofs) of the isomorphism of sheaves $\unKMW_n\simeq \underline{\pi}^{n,n}_s$, $n\in \mathbb Z$, over a filed $k$ originally proved in \cite{M02} and \cite{M-A1Top}.