Grothendieck--Witt Groups of Henselian Valuation Rings

TL;DR

This paper proves that algebraic K-theory and Grothendieck--Witt groups exhibit local constancy over Henselian valuation rings, maintaining isomorphisms with residue fields when using finite coefficients, in both equal and mixed characteristic cases.

Contribution

It establishes the local constancy property for higher Grothendieck--Witt groups and related functors over Henselian valuation rings, extending known results to mixed characteristic.

Findings

Functorial local constancy for Grothendieck--Witt groups

Isomorphisms induced by residue maps with finite coefficients

Applicable to both equal and mixed characteristic cases

Abstract

We show that functors like algebraic $K$-theory (such as unitary or symplectic $K$-functors), as well as the higher Grothendieck--Witt groups, possess the local constancy condition for Henselian valuation rings. Namely, taken with finite coefficients, these functors send canonical residue maps into isomorphisms. This statement holds in cases of both equal and mixed characteristics. The proof is based on a slight modification of Suslin's methods. In particular, we use his notion of universal homotopy.