Remark 42.6.4 (0EAY): Milnor K-theory

Remark 42.6.4 (Milnor K-theory). For a field $k$ let us denote $K^ M_*(k)$ the quotient of the tensor algebra on $k^*$ divided by the two-sided ideal generated by the elements $x \otimes 1 - x$ for $x \in k \setminus \{ 0, 1\} $. Thus $K^ M_0(k) = \mathbf{Z}$, $K_1^ M(k) = k^*$, and

\[ K^ M_2(k) = k^* \otimes _\mathbf {Z} k^* / \langle x \otimes 1 - x \rangle \]

If $A$ is a discrete valuation ring with fraction field $F = \text{Frac}(A)$ and residue field $\kappa $, there is a tame symbol

\[ \partial _ A : K_{i + 1}^ M(F) \to K_ i^ M(\kappa ) \]

defined as in Section 42.5; see [Kato-Milnor-K]. More generally, this map can be extended to the case where $A$ is an excellent local domain of dimension $1$ using normalization and norm maps on $K_ i^ M$, see [Kato-Milnor-K]; presumably the method in Section 42.5 can be used to extend the construction of the tame symbol $\partial _ A$ to arbitrary Noetherian local domains $A$ of dimension $1$. Next, let $X$ be a Noetherian scheme with a dimension function $\delta $. Then we can use these tame symbols to get the arrows in the following:

\[ \bigoplus \nolimits _{\delta (x) = j + 1} K^ M_{i + 1}(\kappa (x)) \longrightarrow \bigoplus \nolimits _{\delta (x) = j} K^ M_ i(\kappa (x)) \longrightarrow \bigoplus \nolimits _{\delta (x) = j - 1} K^ M_{i - 1}(\kappa (x)) \]

However, it is not clear, that the composition is zero, i.e., that we obtain a complex of abelian groups. For excellent $X$ this is shown in [Kato-Milnor-K]. When $i = 1$ and $j$ arbitrary, this follows from Lemma 42.6.3.

The Stacks project

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