Framed correspondences and the Milnor-Witt K-theory

Abstract

The article is to construct a graded ring isomorphism between $H_0(ZF({\Delta }_k^{\bullet },{\mathbb{G}}_m^{\wedge \ast }))$ and the Milnor-Witt K-theory ring $K_{\ast ⩾0}^{MW}(k)$, where $k$ is a field of characteristic zero and $Z{F}_{\ast }(k)$ 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 ${\pi }^{n,n}({\Sigma }_{S^1}^{\infty }{\Sigma }_{{\mathbb{G}}_m}^{\infty }{S}^0)(k)$, which is originally due to Morel.

26 pages, improved introduction, typos corrected

Cited by in corpus (3)

• The unit map of the algebraic special linear cobordism spectrum
• Cohomological correspondence categories
• Zariski-local framed ${\mathbb{A}}^1$-homotopy theory