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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 from "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?

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?

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 from "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?

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 from "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?

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

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

I wonder if there's a natural reason to call these groups "unramified" in certain sense. Is there any relation to terminology of "unramifiedness" in context of (local) algebraic number theory?

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?

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 from "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?