Grothendieck--Witt Groups of Henselian Valuation Rings

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.