Grothendieck–Witt groups of henselian valuation rings

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.The computation of K-groups of various fields has been a classic question in algebraic K-theory.Without delving into the history, let us only mention the work [1] of Daniel Quillen, where he completely computes the K-theory of finite fields.From an algebro-geometric point of view, K-groups of fields carry on some information about geometric points of an algebraic variety.The Henselization of a local ring of a point is an infinitesimally small ètale neighborhood, which is, in most of the cases, non-contractible.Thus, besides the K-functors of fields, it is also important to study K-groups of local rings.We are trying to answer a natural question whether K-functors can "distinguish" closed points and their infinitesimal neighborhoods.For algebraic K-theory, the answer to this question was obtained by Andrei Suslin [2], who has computed, in particular, the algebraic K-groups with finite coefficients of a local Henselian ring.As it turned out, these groups coincide with K-groups of corresponding residue fields.So that, taken with finite coefficients, K-functors are locally constant.In other words, they are "non-sensitive" to the passage from a point to its infinitesimal neighborhood.Let us write down this property in a more rigorous mathematical form.We reproduce here Definition C below.Definition.We say that the functor F satisfies the local constancy condition (LCC) for a local pair pR, Iq if the natural map R Ñ R{I induces an isomorphism FpRq -Ñ FpR{Iq.In this paper, we are going to check the LCC for several functors (cohomology theories).Among them are unitary, orthogonal, and symplectic K-functors.In these cases, the corresponding computation of K-groups for finite fields was performed by Eric Friedlander [3].