Hermitian K-theory, derived equivalences and Karoubi's Fundamental Theorem

TL;DR

This paper proves invariance of higher Grothendieck-Witt groups under derived equivalences, establishes long exact sequences, and generalizes Karoubi's Fundamental Theorem within the framework of dg categories with duality.

Contribution

It introduces new invariance results for hermitian K-theory under derived equivalences and extends classical theorems to broader algebraic and geometric contexts.

Findings

Higher Grothendieck-Witt groups are invariant under derived equivalences.

Derived Morita sequences induce long exact sequences of Grothendieck-Witt groups.

Generalization of Karoubi's Fundamental Theorem and new localization results.

Abstract

Within the framework of dg categories with weak equivalences and duality that have uniquely 2-divisible mapping complexes, we show that higher Grothendieck-Witt groups (aka. hermitian K-groups) are invariant under derived equivalences and that Morita exact sequences induce long exact sequences of Grothendieck-Witt groups. This implies an algebraic Bott sequence and a new proof and generalization of Karoubi's Fundamental Theorem. For the higher Grothendieck-Witt groups of vector bundles of (possibly singular) schemes with an ample family of line-bundles such that 2 is invertible in the ring of regular functions, we obtain Mayer-Vietoris long exact sequences for Nisnevich coverings and blow-ups along regularly embedded centers, projective bundle formulas, and a Bass fundamental theorem. For coherent Grothendieck-Witt groups, we obtain a localization theorem analogous to Quillen's K'-localization theorem.