Continuity of the norm map on Milnor \(K\)-theory
For a finite extension \(L/F\) of complete, discrete valuation fields, the norm map on the Milnor \(K\)-groups \(K_m(L)\) is continuous w.r.t. the unit group filtrations, more precisely, for all \(i \geq 1,\) \(N_{L/F} (U^{ie} K_m(L)) \subseteq U^i K_m (F),\) where \(e\) is the ramification degree of \(L/F.\) This allows to define a norm map from the completion \(\widehat K_m (L) := \displaystyle\mathop{\lim_{\leftarrow\atop j}}\;K_m (L)/U^j \;K_m (L)\) to the analogous completion \(\widehat K(F),\) which is at the heart of Kato's higher local class field theory. Kato's original proof appealed to two-dimensional local rings and Weierstrass preparation. In the present article, the author gives an ``elementary'' proof relying on explicit calculations of symbols of polynomial expressions, using in particular the so called ``reciprocity property'' of the norm : the map \(\displaystyle\sum_v N_{F(v)/F} \partial_v\;:\;K_m (F(X)) \to K_{m-1} (F)\) is zero, where \(v\) ranges over all discrete valuations on \(F(X)\) which are trivial on \(F\) and \(F(v)\) denotes the residue field of \(F(X)\) at \(v\).
scientific article; zbMATH DE number 6072573