Geometric models for higher Grothendieck-Witt groups in A1-homotopy theory

TL;DR

This paper demonstrates that higher Grothendieck-Witt groups can be represented by an infinite orthogonal Grassmannian within the A1-homotopy category, providing geometric models for related hermitian K-theory spaces.

Contribution

It introduces geometric models for higher Grothendieck-Witt groups and related loop spaces in A1-homotopy theory, advancing the understanding of hermitian K-theory in algebraic geometry.

Findings

Higher Grothendieck-Witt groups are represented by an infinite orthogonal Grassmannian.

Provides geometric models for P1- and S1-loop spaces of hermitian K-theory.

Applicable over regular bases where 2 is invertible.

Abstract

We show that the higher Grothendieck-Witt groups, a.k.a. algebraic hermitian K-groups, are represented by an infinite orthogonal Grassmannian in the A1-homotopy category of smooth schemes over a regular base for which 2 is a unit in the ring of regular functions. We also give geometric models for various P1- and S1-loop spaces of hermitian K-theory.