Framed correspondences and the Milnor-Witt K-theory

TL;DR

This paper constructs an isomorphism linking framed correspondences to Milnor-Witt K-theory, advancing the understanding of motivic cohomotopy groups in algebraic geometry.

Contribution

It establishes a graded ring isomorphism between framed correspondences and Milnor-Witt K-theory for fields of characteristic zero, providing new insights into motivic homotopy theory.

Findings

Constructed a graded ring isomorphism between $H_0(ZF(ullet))$ and $K^{MW}_*$.

Partially recovers the computation of motivic cohomotopy groups.

Links algebraic K-theory with framed correspondences.

Abstract

The article is to construct a graded ring isomorphism between $H_0(ZF(\Delta^{\bullet}_k,\mathbb{G}_m^{\wedge *}))$ and the Milnor-Witt K-theory ring $K^{MW}_{*\geqslant 0}(k)$, where $k$ is a field of characteristic zero and $ZF_*(k)$ is the category of linear framed correspondences of algebraic k-varieties, introduced by Garkusha and Panin. As it was shown by Garkusha and Panin, this partially recovers the computation of the motivic cohomotopy groups ${\pi}^{n,n}(\Sigma^{\infty}_{S^1}\Sigma^{\infty}_{\mathbb{G}_m}S^0)(k)$, which is originally due to Morel.