Hermitian K-theory, derived equivalences and Karoubi's fundamental theorem

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 generalisation of Karoubi's Fundamental Theorem. For the higher Grothendieck–Witt groups of vector bundles of (possibly singular) schemes X with an ample family of line-bundles such that 12∈Γ(X,OX), we obtain Mayer–Vietoris long exact sequences for Nisnevich coverings and blow-ups along regularly embedded centres, 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.