A1-invariants in Galois cohomology and a claim of Morel

TL;DR

This paper develops a new splitting principle for homotopy invariant sheaf invariants and proves a folklore result relating motivic localization of finite étale schemes to unramified Grothendieck-Witt groups.

Contribution

It introduces a variant of the splitting principle for invariants in homotopy invariant sheaves and confirms a longstanding conjecture about the structure of motivic localizations.

Findings

Established a new splitting principle for homotopy invariant sheaves.

Proved the sheaf of unramified Grothendieck-Witt groups describes pi_0 of the motivic localization.

Confirmed Morel's folklore result on the structure of finite étale schemes.

Abstract

We establish a variant of the splitting principle of Garibaldi-Merkurjev-Serre for invariants taking values in a strictly homotopy invariant sheaf. As an application, we prove the folklore result of Morel that pi_0 of the motivic localization of the group completion of the stack of finite \'etale schemes is given by the sheaf of unramified Grothendieck-Witt groups.