2012/09/30 by Marco Schlichting · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Cohomology #Discrete mathematics #Exact sequence #Grothendieck group #Hermitian matrix #Homotopy and Cohomology in Algebraic Topology #Invariant (physics) #Invertible matrix #Mathematics #Morita equivalence #Pure mathematics #Spectral sequence #Witt vector #math.KT
paper · pdf · doi:10.1016/j.jpaa.2016.12.026
to appear in J. Pure Appl. Algebra
arxiv created 2016/09/07 · openalex publication_date 2016/12/21 · arxiv updated 2017/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.