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Let $X$ be an integral locally noetherian smooth scheme over base field $k$. Then for every $x \in X^{(1)}$ point 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 have 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 $F(X)$ 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

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

where

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

Question: Is there a natural reason to call these groups "unramified"? I hoped that this terminology may have somehow arised and therefore be directly related to notation of "unramifiedness" in local algebraic number theory but I not see til now a direct connection which may justify the choice of this name.

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 etale cohomology, which involve so called unramified etale 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 to what is the origin of the the name "unramified" for unramified etale cohomology groups. Or are these groups are called unramified from different reasons?

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  • $\begingroup$ You probably want to lower the index $n$ to $n -1$. As I understand, for the étale case, each $\partial$ comes from the Gysin morphism (and purity), the closed immersion in this case being simply the respective point. Now, that coincides with the definition of unramified Galois cohomology. As for the name "unramified", I'm not sure, but I suppose that is because one is restricting an étale sheaf over the spectrum of a field to the unramified extensions only. $\endgroup$
    – user40276
    Nov 29, 2022 at 1:27
  • $\begingroup$ @user40276: when you say that an étale sheaf over the spectrum of a field $k$ is restricting to the unramified extensions only, you mean "unramified extensions" in sense of [unramified morphisms ](en.wikipedia.org/wiki/Unramified_morphism), ie in that case to finite coproducts of specs of separable field extensions of $k$. Is that what you mean in your last sentence? $\endgroup$
    – user267839
    Dec 6, 2022 at 22:43
  • $\begingroup$ It seems that "unramified cohomology" has also a closed relation to the unramified extensions in sense of number theory for local fields; the connection is explained in Peyre's paper "Unramified cohomology and rationality problems" $\endgroup$
    – user267839
    Dec 6, 2022 at 22:49
  • $\begingroup$ The field at the end of my comment is the function field of the respective irreducible scheme. By unramified extension, I mean an extension that lifts to an étale covering of the irreducible scheme. So unramified here coincides with the notion of unramified as in "unramified extension of a local field". For a fixed closed point, the residue morphism is just the one coming from Gysin sequence applied to the $X = Spec (\mathcal{O}_{X, x})$ and $Z = Spec (\kappa (x))$. Now, what's $H_Z (X)$? What's the relation between that and the Galois cohomology of the inertia subgroup? $\endgroup$
    – user40276
    Dec 13, 2022 at 22:13
  • $\begingroup$ I don't think that Galois cohomology can yield any Milnor-Witt groups. $\endgroup$ Mar 27, 2023 at 6:46

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