Hermitian K-theory for stable ∞-categories III: Grothendieck-Witt groups of rings

• 成果类型: Article
• 署名作者: Calmes, Baptiste; Dotto, Emanuele; Harpaz, Yonatan; Hebestreit, Fabian; Land, Markus; Moi, Kristian; Nardin, Denis; Nikolaus, Thomas; Steimle, Wolfgang
• 署名单位: Universite d'Artois; University of Warwick; Sorbonne Universite; Universite Paris Cite; University of Bielefeld; Johannes Gutenberg University of Mainz; Royal Institute of Technology; University of Regensburg; University of Munster; University of Augsburg
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X; 1939-8980
• DOI: 10.4007/annals.2026.204.1.2
• 发表日期: 2026-07
• 页码: 119-220
• 关键词: Grothendieck-Witt groups; L-groups; algebraic surgery; homotopy limit problem; devissage; GROUP-COMPLETION; periodicity; forms; localization; HOMOLOGY; functors

摘要

We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring R to the homotopy C-2-orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that 2 is a unit in R from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of Z, show that the Grothendieck-Witt groups of rings of integers in number fields are finitely generated and that the comparison map from quadratic to symmetric Grothendieck-Witt theory of coherent rings of global dimension d is an equivalence in degrees >= d + 3. As an important tool, we establish the hermitian analogue of Quillen's localisation-devissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi.