Andrei Druzhinin (St. Petersburg): On the homomorphism K^MW_* -> pi^*,*_s

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