The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes

TL;DR

This paper establishes localization and Mayer-Vietoris sequences for higher Grothendieck-Witt groups of schemes, including singular schemes, without characteristic restrictions, advancing hermitian K-theory and its foundational properties.

Contribution

It proves localization and Mayer-Vietoris principles for hermitian K-theory of schemes with ample line bundles, extending to singular schemes and removing characteristic constraints.

Findings

Proved localization and Mayer-Vietoris sequences for hermitian K-groups.

Established additivity, fibration, and approximation theorems for hermitian K-theory.

Extended results to singular schemes without characteristic restrictions.

Abstract

We prove localization and Zariski-Mayer-Vietoris for higher Grothendieck-Witt groups, alias hermitian $K$-groups, of schemes admitting an ample family of line-bundles. No assumption on the characteristic is needed, and our schemes can be singular. Along the way, we prove additivity, fibration and approximation theorems for the hermitian $K$-theory of exact categories with weak equivalences and duality.