The Steinberg relation in the stable motivic homotopy theory over a base

A. Druzhinin

Published 2018-08-31Version 1

We prove the Steinberg relation in the sections of the sheaf $\underline{\pi}^{2,2}$ in $\mathbf{SH}(S)$ over an arbitrary base scheme $S$. Namely we prove that the class of the morphism $$(1-x,x)\colon \mathbf (A^1-\{0,1\})\times S\to \mathbf G_m\times\mathbf G_m/(1\times\mathbf G_m\cup \mathbf G_m\times 1)$$ is trivial in the stable motivic homotopy category $$[(1-x,x)]=0\in [A^1_S-\{0_S,1_S\},\mathbf G_m^{\wedge 2}]_{\mathbf{SH}(S)}.$$