Endomorphisms of the projective plane and the image of the Suslin–Hurewicz map

Abstract The endomorphism ring of the projective plane over a field F characteristic neither two nor three is slightly more complicated in Morel–Voevodsky motivic stable homotopy category than Voevodsky’s derived motives. In particular, it not commutative precisely if there exists square which does admit sixth root. A byproduct these computations proof Suslin’s conjecture on Suslin–Hurewicz homomorphism from Quillen to Milnor K -theory degree four, based work Asok et al. (Invent Math 219:39-73, 2020).