# Andrei Druzhinin(St. Petersburg): On the homomorphism K^MW_* to \pi^*_s

We construct the homomorphism of presheaves $\mathrm{K}^\mathrm{MW}_{*}\to \pi^{*,*}_s$, where $\mathrm{K}^\mathrm{MW}_{*}$ is the naive Milnor-Witt K-theory presheaf, and $\pi^{*,*}_s$ are stable motivic homotopy groups over a base $S$. The Garkusha-Panin’s theory of framed motives and the Neshitov’s computation of $\pi^{*,*}_s(k)$ for $char k=0$, gives the alternative proof of the stable version of Morel’s theorem on zero motivic homotopy groups, namely the isomorphism $\mathrm{K}^\mathrm{MW}_{*}(k)\to \pi^{*,*}_s(k)$, for the case of fields $k$, $char k=0$. We extend this proof to the case of perfect fields of odd characteristic, and deduce that the above homomorphism induces isomorphism of unramified Milnor-Witt K-theory sheaf $\mathbf{K}^\mathrm{MW}_*$ and the associated (Nisnevich and Zariski) sheaf $\underline{\pi}^{*,*}_s$ over such fields. The talk is based on the joint work with Jonas Irgens Kylling.