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

TL;DR

This paper investigates the structure of endomorphisms of the projective plane in a motivic homotopy context and proves Suslin's conjecture on the Suslin-Hurewicz homomorphism in degree four.

Contribution

It characterizes the endomorphism ring in the motivic setting and confirms Suslin's conjecture using recent advances in algebraic K-theory.

Findings

Endomorphism ring is non-commutative if a certain sixth root condition is met.

Proof of Suslin's conjecture in degree four.

Comparison of endomorphism structures in different motivic categories.

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 the 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, Fasel, and Williams.