Gersten complexes in Quillen's and Milnor's K-theories

Question

Consider a good enough scheme $X$ (e.g. an algebraic variety over a field). Let $X_i$ be the set of points of dimension $i$ in $X$. Then we have the Gersten complex in Quillen's K-theory:

$$ \oplus_{x\in X_{i+1}}K_{n+1}(\kappa(x))\to \oplus_{x\in X_i}K_n(\kappa(x))\to \oplus_{x\in X_{i-1}}K_{n-1}(\kappa(x)) $$

Kato proved that there is a Gersten complex of the same type using Milnor's K-groups instead. As there are natural maps from Milnor K-groups to Quillen K-groups, I would like to know whether there is a compatibility between (the differential maps in) these two complexes.

For the differentials from $K_2$ to $K_1$ or from $K_1$ to $K_0$, it is well-known that the two complexes give the same thing (up to a sign). But I'm not sure if someone has checked this for higher $K$-groups. Could anyone give some hints or references?

Answer 1

Yes, the natural multiplication morphisms induce a morphism of Gersten complexes from Milnor to Quillen K-theory. The basic points are made in the paper

• M. Rost. "Chow groups with coefficients", Doc. Math. 1 (1996), pp. 319-393, link to DocMath page.

That paper contains a definition of "cycle modules" and constructions of Gersten complexes for these. Milnor K-theory is almost by definition a cycle module, and Quillen K-theory is a cycle module by Remark 1.12. Remark 5.4 in Rost's paper states that the multiplication morphisms yield a morphism of cycle modules, and by the results and constructions in the paper, a morphism of cycle modules induces a morphism of corresponding Gersten complexes. (Note that Rost references the 1987 paper of Merkurjev and Suslin for an identification of the differentials for $K_3$.)

Since Remark 5.4 doesn't contain a proof, we trace back a bit to Remark 1.12 in Rost's paper which discusses the structure of Quillen K-theory as cycle module. The notion of cycle module incorporates an action of Milnor K-theory which, by Remark 1.12, is given exactly by the multiplication morphism. In particular, the compatibility of the multiplication morphism with the differentials of the Gersten complexes is already present in the claim that Quillen K-theory yields a cycle module -- it is the compatibility of the action of Milnor K-theory (part D3 of the definition of cycle premodule with the differentials (part D4 of the definition of cycle premodule) via relations R3e resp. R3f.

Now the claim that Quillen K-theory is a cycle module (and hence that all the compatibilities required above are satisfied) is also not done in Remark 1.12, but that remark contains references to the literature from which the verification of the rules "is a lengthy but straightforward exercise". The key point in the end is a K-linearity of the boundary map in the localization sequence for Quillen K-theory, which is discussed e.g. in Section V.6 of Weibel's K-book.