Units in Grothendieck-Witt rings and A^1-spherical fibrations

TL;DR

This paper demonstrates that units in Grothendieck-Witt rings form an unramified sheaf, leading to the construction of an A^1-local classifying space for spherical fibrations, advancing the understanding of algebraic topology in algebraic geometry.

Contribution

It introduces an unramified strictly A^1-invariant sheaf structure for units in Grothendieck-Witt rings and constructs an associated classifying space for spherical fibrations.

Findings

Units in Grothendieck-Witt rings extend to an A^1-invariant sheaf

Existence of an A^1-local classifying space for spherical fibrations

Advancement in algebraic topology within algebraic geometry

Abstract

In this note, we show that the units in Grothendieck-Witt rings extend to an unramified strictly A^1-invariant sheaf of abelian groups on the category of smooth schemes. This implies that there is an A^1-local classifying space of spherical fibrations.