  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.
