Let $X$ be an integral locally noetherian smooth scheme over base field $k$. Then for every point $x \in X^{(1)}$ of codimension $1$, the stalk $\mathcal{O}_{X,x}$ is a discrete valuation ring. Choose some uniformizer $\pi_x \in \mathcal{O}_{X,x}$. Then we get a residue homomorphism

$$ \partial_{MW}^{\pi_x}: \text{K}_n^{MW}(F(X)) \to \text{K}_{n-1}^{MW}(F(x)), \ (\{\pi_x, u_2, ... , u_n \}) \mapsto (\{ \overline{u_2}, ... , \overline{u_n} \}) $$

where here $F(X)$ is the function field of $X$, $k(x)= \mathcal{O}_{X,x}/m_x$ the residue fieeld of $x$, $u_i \in \mathcal{O}_{X,x}^{\times} $ units and $\overline{u_i}$ their residues with respect residue map $v_x: \mathcal{O}_{X,x} \to k(x)$. (the terminology can be looked up eg in F. Morel's $\mathbb{A}^1$-Algebraic Topology over a Field).

The $n$-th unramified Milnor-Witt K-group of $X$ is defined as

$$ \text{K}^{MW}_{n, unr}(X) := \text{Ker}(\sum_{x \in X^{(1)}} \partial_{MW}^{\pi_x}) $$


$$\sum_{x \in X^{(1)}} \partial_{MW}^{\pi_x}: \text{K}_n^{MW}(F(X)) \to \bigoplus_{x \in X^{(1)}} \text{K}_{n-1}^{MW}(F(x)) $$

Question: Is there a natural reason to call these groups "unramified" and is there any relation known to the concept of "unramifiedness" in context of splitting behavoir of prime ideals in number fields in algebraic number theory, justifying this name? Or are these two notions completely unrelated, but then the question is where this notion else come from; surely it not falls from heaven.

A naive guess of mine where the notion might come from: There exist so called Gersten complex for Milnor K-theory which is structurally similar to Gersten complex for étale cohomology, which involve so called unramified étale cohomology groups $H^i_{ur}(X, \mu_2^{\otimes i})$. If that's the origin of the name of "unramified" for unramified Milnor-Witt K-groups, the question turn's into what is the origin of the the name "unramified" for unramified étale cohomology groups.

Rmk.: The identical question was already posted in MO, but I think that maybe that's a more appropriate plattform for it.



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