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Grothendieck–Witt groups of henselian valuation rings

Serge Yagunov

Vol. 9 (2024), No. 1, 23–39
Abstract

We compute higher Grothendieck–Witt groups of henselian discrete valuation rings. More specifically, we demonstrate that, when taken with finite coefficients, the corresponding Grothendieck–Witt functors send the canonical residue maps to isomorphisms. In the first part of the manuscript this property is verified for symplectic K-theory. For this aim modified methods of Suslin are employed. Subsequently, by applying Karoubi’s induction, the result is extended to all higher Grothendieck–Witt groups. The utilization of the universal homotopy method enables us to address the case of mixed characteristics.

Keywords
Grothendieck–Witt groups, symplectic $K$-theory, rigidity, universal homotopy
Mathematical Subject Classification
Primary: 19D50, 19E20, 19G38
Secondary: 14F35, 14F43
Milestones
Received: 16 November 2022
Revised: 28 March 2024
Accepted: 16 April 2024
Published: 25 May 2024
Authors
Serge Yagunov
St. Petersburg Department of Steklov Mathematical Institute
St. Petersburg
Russia