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