# Is there a clean definition of the residue map in Milnor K-theory?

If K is a field, v a discrete valuation, and k the residue field, there is a residue map $\partial: K^M_n(K) \to K^M_{n - 1}(k)$. All the definitions I have seen for this map involve two pages of combinatorics of different cases (e.g. the book of Fesenko Vostokov).

Does anyone know of a clean, concise, compact definition of this map, for example via some universal property that it might satisfy?

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Look at Lemma 2.1 in Milnor's original paper, $\partial$ is characterized by the following properties: given $\pi, u_i \in K$ with $v(\pi)=1$ and $v(u_i)=0$,
$$\partial(u_1 \cdots u_n)=0,$$
$$\partial(\pi\cdot u_2 \cdots u_n)=\bar{u}_2\cdots \bar{u}_n.$$