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?



Is there a clean definition of the residue map in Milnor K-theory?
Look at Lemma 2.1 in [Milnor's original paper](http://0-www.springerlink.com.fama.us.es/content/t025u1152j330325/), $\partial$ is characterized by the following properties: given $\pi, u\_i \in K$ with $v(\pi)=1$ and $v(u\_i)=0$,


$$\partial(u\_1 \cdots u\_n)=0,$$


$$\partial(\pi\cdot u\_2 \cdots u\_n)=\bar{u}\_2\cdots \bar{u}\_n.$$


