Framed correspondences and the Milnor-Witt K-theory

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.